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Fundamentals of Electrical Engineering I

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Fundamentals of Electrical Engineering I
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Johnson, Don

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Engineering, Electrical Engineering, Communication Systems, Complex Numbers, Elemental Signals, Signal Decomposition, Simple Systems, Analog Signal Processing, Voltage, Current, Generic Circuit Elements, Ideal Circuit Elements, Power Dissipation, Resistor Circuits, Series and Parallel Circuits, Capacitors, Inductors, Transfer Functions, Node Method, Power …
Electromechanical Technology, Electronics, Engineering
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The text focuses on the creation, manipulation, transmission, and reception of information by electronic means. Contents: 1) Introduction. 2) Signals and Systems. 3) Analog Signal Processing. 4) Frequency Domain. 5) Digital Signal Processing. 6) Information Communication. 7) Appendices: Decibels; Permutations and Combinations, Frequency Allocations.
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Community College, Higher Education
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http://www.ogtp-cart.com/product.aspx?ISBN=9781616100377
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Don Johnson, Rice University, Houston, Texas
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http://florida.theorangegrove.org/og/file/c7d013ec-2b4d-6971-65f9-b1a6af2d9092/1/ElectricalEngineering.pdf

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FundamentalsofElectricalEngineeringIBy:DonJohnson

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FundamentalsofElectricalEngineeringIBy:DonJohnsonOnline:CONNEXIONSRiceUniversity,Houston,Texas

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2008DonJohnsonThisselectionandarrangementofcontentislicensedundertheCreativeCommonsAttributionLicense:http://creativecommons.org/licenses/by/1.0

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TableofContents1Introduction1.1Themes......................................................................................11.2SignalsRepresentInformation...............................................................21.3StructureofCommunicationSystems........................................................61.4TheFundamentalSignal.....................................................................71.5IntroductionProblems.......................................................................8Solutions........................................................................................112SignalsandSystems2.1ComplexNumbers..........................................................................132.2ElementalSignals...........................................................................172.3SignalDecomposition.......................................................................202.4Discrete-TimeSignals.......................................................................212.5IntroductiontoSystems....................................................................242.6SimpleSystems.............................................................................262.7SignalsandSystemsProblems..............................................................29Solutions........................................................................................343AnalogSignalProcessing3.1Voltage,Current,andGenericCircuitElements.............................................353.2IdealCircuitElements......................................................................363.3IdealandReal-WorldCircuitElements.....................................................393.4ElectricCircuitsandInterconnectionLaws..................................................393.5PowerDissipationinResistorCircuits......................................................433.6SeriesandParallelCircuits.................................................................443.7EquivalentCircuits:ResistorsandSources..................................................493.8CircuitswithCapacitorsandInductors.....................................................543.9TheImpedanceConcept....................................................................543.10TimeandFrequencyDomains.............................................................563.11PowerintheFrequencyDomain...........................................................583.12EquivalentCircuits:ImpedancesandSources..............................................593.13TransferFunctions........................................................................603.14DesigningTransferFunctions..............................................................653.15FormalCircuitMethods:NodeMethod....................................................663.16PowerConservationinCircuits............................................................713.17Electronics................................................................................733.18DependentSources........................................................................733.19OperationalAmpliers....................................................................763.20TheDiode................................................................................813.21AnalogSignalProcessingProblems........................................................84Solutions.......................................................................................1064FrequencyDomain4.1IntroductiontotheFrequencyDomain.....................................................1094.2ComplexFourierSeries....................................................................1094.3ClassicFourierSeries......................................................................1144.4ASignal'sSpectrum.......................................................................1164.5FourierSeriesApproximationofSignals...................................................1174.6EncodingInformationintheFrequencyDomain...........................................1224.7FilteringPeriodicSignals..................................................................124

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iv4.8DerivationoftheFourierTransform.......................................................1264.9LinearTimeInvariantSystems............................................................1314.10ModelingtheSpeechSignal..............................................................1344.11FrequencyDomainProblems.............................................................140Solutions.......................................................................................1525DigitalSignalProcessing5.1IntroductiontoDigitalSignalProcessing..................................................1555.2IntroductiontoComputerOrganization...................................................1555.3TheSamplingTheorem....................................................................1595.4AmplitudeQuantization...................................................................1625.5Discrete-TimeSignalsandSystems........................................................1655.6Discrete-TimeFourierTransformDTFT.................................................1675.7DiscreteFourierTransformsDFT........................................................1725.8DFT:ComputationalComplexity..........................................................1745.9FastFourierTransformFFT.............................................................1745.10Spectrograms............................................................................1775.11Discrete-TimeSystems...................................................................1805.12Discrete-TimeSystemsintheTime-Domain..............................................1815.13Discrete-TimeSystemsintheFrequencyDomain.........................................1855.14FilteringintheFrequencyDomain.......................................................1865.15EciencyofFrequency-DomainFiltering.................................................1895.16Discrete-TimeFilteringofAnalogSignals................................................1925.17DigitalSignalProcessingProblems.......................................................193Solutions.......................................................................................2046InformationCommunication6.1InformationCommunication...............................................................2096.2TypesofCommunicationChannels........................................................2106.3WirelineChannels.........................................................................2106.4WirelessChannels.........................................................................2156.5Line-of-SightTransmission.................................................................2166.6TheIonosphereandCommunications......................................................2176.7CommunicationwithSatellites............................................................2176.8NoiseandInterference.....................................................................2186.9ChannelModels...........................................................................2196.10BasebandCommunication................................................................2206.11ModulatedCommunication...............................................................2216.12Signal-to-NoiseRatioofanAmplitude-ModulatedSignal.................................2226.13DigitalCommunication...................................................................2246.14BinaryPhaseShiftKeying...............................................................2246.15FrequencyShiftKeying..................................................................2276.16DigitalCommunicationReceivers........................................................2286.17DigitalCommunicationinthePresenceofNoise..........................................2306.18DigitalCommunicationSystemProperties................................................2316.19DigitalChannels.........................................................................2336.20Entropy..................................................................................2346.21SourceCodingTheorem..................................................................2356.22CompressionandtheHumanCode......................................................2366.23SubtliesofCoding.......................................................................2376.24ChannelCoding..........................................................................2386.25RepetitionCodes.........................................................................240

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v6.26BlockChannelCoding...................................................................2416.27Error-CorrectingCodes:HammingDistance..............................................2426.28Error-CorrectingCodes:ChannelDecoding...............................................2456.29Error-CorrectingCodes:HammingCodes................................................2466.30NoisyChannelCodingTheorem..........................................................2486.31CapacityofaChannel...................................................................2496.32ComparisonofAnalogandDigitalCommunication.......................................2506.33CommunicationNetworks................................................................2516.34MessageRouting.........................................................................2526.35Networkarchitecturesandinterconnection................................................2536.36Ethernet.................................................................................2546.37CommunicationProtocols................................................................2566.38InformationCommunicationProblems....................................................258Solutions.......................................................................................2727Appendix7.1Decibels...................................................................................2797.2PermutationsandCombinations...........................................................2807.3FrequencyAllocations.....................................................................281Solutions.......................................................................................283Index...............................................................................................284Attributions........................................................................................290

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Chapter1Introduction1.1Themes1Fromitsbeginningsinthelatenineteenthcentury,electricalengineeringhasblossomedfromfocusingonelectricalcircuitsforpower,telegraphyandtelephonytofocusingonamuchbroaderrangeofdisciplines.However,theunderlyingthemesarerelevanttoday:Powercreationandtransmissionandinformationhavebeentheunderlyingthemesofelectricalengineeringforacenturyandahalf.Thiscourseconcentratesonthelattertheme:therepresentation,manipulation,transmission,andreceptionofinformationbyelectricalmeans.Thiscoursedescribeswhatinformationis,howengineersquantifyinformation,andhowelectricalsignalsrepresentinformation.Informationcantakeavarietyofforms.Whenyouspeaktoafriend,yourthoughtsaretranslatedbyyourbrainintomotorcommandsthatcausevariousvocaltractcomponentsthejaw,thetongue,thelipstomoveinacoordinatedfashion.Informationarisesinyourthoughtsandisrepresentedbyspeech,whichmusthaveawelldened,broadlyknownstructuresothatsomeoneelsecanunderstandwhatyousay.Utterancesconveyinformationinsoundpressurewaves,whichpropagatetoyourfriend'sear.There,soundenergyisconvertedbacktoneuralactivity,and,ifwhatyousaymakessense,sheunderstandswhatyousay.YourwordscouldhavebeenrecordedonacompactdiscCD,mailedtoyourfriendandlistenedtobyheronherstereo.Informationcantaketheformofatextleyoutypeintoyourwordprocessor.Youmightsendtheleviae-mailtoafriend,whoreadsitandunderstandsit.Fromaninformationtheoreticviewpoint,allofthesescenariosareequivalent,althoughtheformsoftheinformationrepresentationsoundwaves,plasticandcomputerlesareverydierent.Engineers,whodon'tcareaboutinformationcontent,categorizeinformationintotwodierentforms:analoganddigital.Analoginformationiscontinuousvalued;examplesareaudioandvideo.Digitalinformationisdiscretevalued;examplesaretextlikewhatyouarereadingnowandDNAsequences.Theconversionofinformation-bearingsignalsfromoneenergyformintoanotherisknownasenergyconversionortransduction.Allconversionsystemsareinecientsincesomeinputenergyislostasheat,butthislossdoesnotnecessarilymeanthattheconveyedinformationislost.Conceptuallywecoulduseanyformofenergytorepresentinformation,butelectricsignalsareuniquelywell-suitedforinformationrepresentation,transmissionsignalscanbebroadcastfromantennasorsentthroughwires,andmanipulationcircuitscanbebuilttoreducenoiseandcomputerscanbeusedtomodifyinformation.Thus,wewillbeconcernedwithhowtorepresentallformsofinformationwithelectricalsignals,encodeinformationasvoltages,currents,andelectromagneticwaves,manipulateinformation-bearingelectricsignalswithcircuitsandcomputers,andreceiveelectricsignalsandconverttheinformationexpressedbyelectricsignalsbackintoausefulform. 1Thiscontentisavailableonlineat.1

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2CHAPTER1.INTRODUCTIONTelegraphyrepresentstheearliestelectricalinformationsystem,anditdatesfrom1837.Atthattime,electricalsciencewaslargelyempirical,andonlythosewithexperienceandintuitioncoulddeveloptelegraphsystems.ElectricalsciencecameofagewhenJamesClerkMaxwell2proclaimedin1864asetofequationsthatheclaimedgovernedallelectricalphenomena.Theseequationspredictedthatlightwasanelectro-magneticwave,andthatenergycouldpropagate.BecauseofthecomplexityofMaxwell'spresentation,thedevelopmentofthetelephonein1876wasduelargelytoempiricalwork.OnceHeinrichHertzconrmedMaxwell'spredictionofwhatwenowcallradiowavesinabout1882,Maxwell'sequationsweresimpliedbyOliverHeavisideandothers,andwerewidelyread.Thisunderstandingoffundamentalsledtoaquicksuccessionofinventionsthewirelesstelegraph1899,thevacuumtube,andradiobroadcastingthatmarkedthetrueemergenceofthecommunicationsage.Duringtherstpartofthetwentiethcentury,circuittheoryandelectromagnetictheorywereallanelectricalengineerneededtoknowtobequaliedandproducerst-ratedesigns.Consequently,circuittheoryservedasthefoundationandtheframeworkofallofelectricalengineeringeducation.Atmid-century,three"inventions"changedthegroundrules.Theseweretherstpublicdemonstrationoftherstelectroniccomputer1946,theinventionofthetransistor,andthepublicationofAMathematicalTheoryofCommunicationbyClaudeShannon3.Althoughconceivedseparately,thesecreationsgavebirthtotheinformationage,inwhichdigitalandanalogcommunicationsystemsinteractandcompetefordesignpreferences.Abouttwentyyearslater,thelaserwasinvented,whichopenedevenmoredesignpossibilities.Thus,theprimaryfocusshiftedfromhowtobuildcommunicationsystemsthecircuittheoryeratowhatcommunicationssystemswereintendedtoaccomplish.Onlyoncetheintendedsystemisspeciedcananimplementationbeselected.Today'selectricalengineermustbemindfulofthesystem'sultimategoal,andunderstandthetradeosbetweendigitalandanalogalternatives,andbetweenhardwareandsoftwarecongurationsindesigninginformationsystems.VisionImpairedAccess:ThankstothetranslationeortsofRiceUniversity'sDisabilitySup-portServices4,thiscollectionisnowavailableinaBraille-printableversion.Pleaseclickhere5todownloada.ziplecontainingallthenecessary.dxbandimageles.1.2SignalsRepresentInformation6Whetheranalogordigital,informationisrepresentedbythefundamentalquantityinelectricalengineering:thesignal.Statedinmathematicalterms,asignalismerelyafunction.Analogsignalsarecontinuous-valued;digitalsignalsarediscrete-valued.Theindependentvariableofthesignalcouldbetimespeech,forexample,spaceimages,ortheintegersdenotingthesequencingoflettersandnumbersinthefootballscore.1.2.1AnalogSignalsAnalogsignalsareusuallysignalsdenedovercontinuousindependentvariables.SpeechSection4.10isproducedbyyourvocalcordsexcitingacousticresonancesinyourvocaltract.Theresultispressurewavespropagatingintheair,andthespeechsignalthuscorrespondstoafunctionhavingindependentvariablesofspaceandtimeandavaluecorrespondingtoairpressure:sx;tHereweusevectornotationxtodenotespatialcoordinates.Whenyourecordsomeonetalking,youareevaluatingthespeechsignalataparticularspatiallocation,x0say.Anexampleoftheresultingwaveformsx0;tisshowninthisgureFigure1.1:SpeechExample. 2http://www-groups.dcs.st-andrews.ac.uk/history/Mathematicians/Maxwell.html3http://www.lucent.com/minds/infotheory/4http://www.dss.rice.edu/5http://cnx.org/content/m0000/latest/FundElecEngBraille.zip6Thiscontentisavailableonlineat.

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3 SpeechExample Figure1.1:Aspeechsignal'samplituderelatestotinyairpressurevariations.Shownisarecordingofthevowel"e"asin"speech". Photographsarestatic,andarecontinuous-valuedsignalsdenedoverspace.Black-and-whiteimageshaveonlyonevalueateachpointinspace,whichamountstoitsopticalreectionproperties.InFig-ure1.2Lena,animageisshown,demonstratingthatitandallotherimagesaswellarefunctionsoftwoindependentspatialvariables.

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4CHAPTER1.INTRODUCTION Lena a bFigure1.2:OntheleftistheclassicLenaimage,whichisusedubiquitouslyasatestimage.Itcontainsstraightandcurvedlines,complicatedtexture,andaface.OntherightisaperspectivedisplayoftheLenaimageasasignal:afunctionoftwospatialvariables.Thecolorsmerelyhelpshowwhatsignalvaluesareaboutthesamesize.Inthisimage,signalvaluesrangebetween0and255;whyisthat? Colorimageshavevaluesthatexpresshowreectivitydependsontheopticalspectrum.Painterslongagofoundthatmixingtogethercombinationsoftheso-calledprimarycolorsred,yellowandbluecanproduceveryrealisticcolorimages.Thus,imagestodayareusuallythoughtofashavingthreevaluesateverypointinspace,butadierentsetofcolorsisused:Howmuchofred,greenandblueispresent.Mathematically,colorpicturesaremultivaluedvector-valuedsignals:sx=rx;gx;bxT.Interestingcasesaboundwheretheanalogsignaldependsnotonacontinuousvariable,suchastime,butonadiscretevariable.Forexample,temperaturereadingstakeneveryhourhavecontinuousanalogvalues,butthesignal'sindependentvariableisessentiallytheintegers.1.2.2DigitalSignalsTheword"digital"meansdiscrete-valuedandimpliesthesignalhasaninteger-valuedindependentvariable.Digitalinformationincludesnumbersandsymbolscharacterstypedonthekeyboard,forexample.Com-putersrelyonthedigitalrepresentationofinformationtomanipulateandtransforminformation.Symbolsdonothaveanumericvalue,andeachisrepresentedbyauniquenumber.TheASCIIcharactercodehastheupper-andlowercasecharacters,thenumbers,punctuationmarks,andvariousothersymbolsrepresentedbyaseven-bitinteger.Forexample,theASCIIcoderepresentstheletteraasthenumber97andtheletterAas65.Figure1.3showstheinternationalconventiononassociatingcharacterswithintegers.

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5 ASCIITable 00 nul 01 soh 02 stx 03 etx 04 eot 05 enq 06 ack 07 bel 08 bs 09 ht 0A nl 0B vt 0C np 0D cr 0E so 0F si 10 dle 11 dc1 12 dc2 13 dc3 14 dc4 15 nak 16 syn 17 etb 18 car 19 em 1A sub 1B esc 1C fs 1D gs 1E rs 1F us 20 sp 21 22 23 # 24 $ 25 % 26 & 27 28 29 2A 2B + 2C 2D 2E 2F / 30 0 31 1 32 2 33 3 34 4 35 5 36 6 37 7 38 8 39 9 3A : 3B ; 3C < 3D = 3E > 3F ? 40 @ 41 A 42 B 43 C 44 D 45 E 46 F 47 G 48 H 49 I 4A J 4B K 4C L 4D M 4E N 4F 0 50 P 51 Q 52 R 53 S 54 T 55 U 56 V 57 W 58 X 59 Y 5A Z 5B [ 5C n 5D ] 5E 5F 60 61 a 62 b 63 c 64 d 65 e 66 f 67 g 68 h 69 i 6A j 6B k 6C l 6D m 6E n 6F o 70 p 71 q 72 r 73 s 74 t 75 u 76 v 77 w 78 x 79 y 7A z 7B { 7C | 7D } 7E 7F del Figure1.3:TheASCIItranslationtableshowshowstandardkeyboardcharactersarerepresentedbyintegers.Inpairsofcolumns,thistabledisplaysrsttheso-called7-bitcodehowmanycharactersinaseven-bitcode?,thenthecharacterthenumberrepresents.Thenumericcodesarerepresentedinhexadecimalbase-16notation.Mnemoniccharacterscorrespondtocontrolcharacters,someofwhichmaybefamiliarlikecrforcarriagereturnandsomenotbelmeansa"bell".

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6CHAPTER1.INTRODUCTION1.3StructureofCommunicationSystems7 Fundamentalmodelofcommunication Figure1.4:TheFundamentalModelofCommunication. Denitionofasystem Figure1.5:Asystemoperatesonitsinputsignalxttoproduceanoutputyt. ThefundamentalmodelofcommunicationsisportrayedinFigure1.4Fundamentalmodelofcommuni-cation.Inthisfundamentalmodel,eachmessage-bearingsignal,exempliedbyst,isanalogandisafunctionoftime.Asystemoperatesonzero,one,orseveralsignalstoproducemoresignalsortosimplyabsorbthemFigure1.5Denitionofasystem.Inelectricalengineering,werepresentasystemasabox,receivinginputsignalsusuallycomingfromtheleftandproducingfromthemnewoutputsignals.Thisgraphicalrepresentationisknownasablockdiagram.Wedenoteinputsignalsbylineshavingarrowspointingintothebox,outputsignalsbyarrowspointingaway.Astypiedbythecommunicationsmodel,howinformationows,howitiscorruptedandmanipulated,andhowitisultimatelyreceivedissummarizedbyinterconnectingblockdiagrams:Theoutputsofoneormoresystemsserveastheinputstoothers.Inthecommunicationsmodel,thesourceproducesasignalthatwillbeabsorbedbythesink.Examplesoftime-domainsignalsproducedbyasourcearemusic,speech,andcharacterstypedonakeyboard.Signalscanalsobefunctionsoftwovariablesanimageisasignalthatdependsontwospatialvariablesormoretelevisionpicturesvideosignalsarefunctionsoftwospatialvariablesandtime.Thus,informationsourcesproducesignals.Inphysicalsystems,eachsignalcorrespondstoanelectricalvoltageorcurrent.Tobeabletodesignsystems,wemustunderstandelectricalscienceandtechnology.However,werstneedtounderstandthebigpicturetoappreciatethecontextinwhichtheelectricalengineerworks.Incommunicationsystems,messagessignalsproducedbysourcesmustberecastfortransmission.Theblockdiagramhasthemessagestpassingthroughablocklabeledtransmitterthatproducesthesignalxt.Inthecaseofaradiotransmitter,itacceptsaninputaudiosignalandproducesasignalthat 7Thiscontentisavailableonlineat.

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7physicallyisanelectromagneticwaveradiatedbyanantennaandpropagatingasMaxwell'sequationspredict.Inthecaseofacomputernetwork,typedcharactersareencapsulatedinpackets,attachedwithadestinationaddress,andlaunchedintotheInternet.Fromthecommunicationsystemsbigpictureperspective,thesameblockdiagramappliesalthoughthesystemscanbeverydierent.Inanycase,thetransmittershouldnotoperateinsuchawaythatthemessagestcannotberecoveredfromxt.Inthemathematicalsense,theinversesystemmustexist,elsethecommunicationsystemcannotbeconsideredreliable.Itisridiculoustotransmitasignalinsuchawaythatnoonecanrecovertheoriginal.However,cleversystemsexistthattransmitsignalssothatonlytheincrowdcanrecoverthem.Suchcrytographicsystemsunderliesecretcommunications.Transmittedsignalsnextpassthroughthenextstage,theevilchannel.Nothinggoodhappenstoasignalinachannel:Itcanbecomecorruptedbynoise,distorted,andattenuatedamongmanypossibilities.Thechannelcannotbeescapedtherealworldiscruel,andtransmitterdesignandreceiverdesignfocusonhowbesttojointlyfendothechannel'seectsonsignals.Thechannelisanothersysteminourblockdiagram,andproducesrt,thesignalreceivedbythereceiver.Ifthechannelwerebenigngoodluckndingsuchachannelintherealworld,thereceiverwouldserveastheinversesystemtothetransmitter,andyieldthemessagewithnodistortion.However,becauseofthechannel,thereceivermustdoitsbesttoproduceareceivedmessage^stthatresemblesstasmuchaspossible.Shannon8showedinhis1948paperthatreliableforthemoment,takethiswordtomeanerror-freedigitalcommunicationwaspossibleoverarbitrarilynoisychannels.Itisthisresultthatmoderncommunicationssystemsexploit,andwhymanycommunicationssystemsaregoingdigital.ThemoduleonInformationCommunicationSection6.1detailsShannon'stheoryofinformation,andtherewelearnofShannon'sresultandhowtouseit.Finally,thereceivedmessageispassedtotheinformationsinkthatsomehowmakesuseofthemessage.Inthecommunicationsmodel,thesourceisasystemhavingnoinputbutproducinganoutput;asinkhasaninputandnooutput.Understandingsignalgenerationandhowsystemsworkamountstounderstandingsignals,thenatureoftheinformationtheyrepresent,howinformationistransformedbetweenanaloganddigitalforms,andhowinformationcanbeprocessedbysystemsoperatingoninformation-bearingsignals.Thisunderstandingdemandstwodierenteldsofknowledge.Oneiselectricalscience:Howaresignalsrepresentedandma-nipulatedelectrically?Thesecondissignalscience:Whatisthestructureofsignals,nomatterwhattheirsource,whatistheirinformationcontent,andwhatcapabilitiesdoesthisstructureforceuponcommunicationsystems?1.4TheFundamentalSignal91.4.1TheSinusoidThemostubiquitousandimportantsignalinelectricalengineeringisthesinusoid.SineDenitionst=Acos2ft+orAcos!t+.1Aisknownasthesinusoid'samplitude,anddeterminesthesinusoid'ssize.Theamplitudeconveysthesinusoid'sphysicalunitsvolts,lumens,etc.ThefrequencyfhasunitsofHzHertzors)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1,anddetermineshowrapidlythesinusoidoscillatesperunittime.Thetemporalvariabletalwayshasunitsofseconds,andthusthefrequencydetermineshowmanyoscillations/secondthesinusoidhas.AMradiostationshavecarrierfrequenciesofabout1MHzonemega-hertzor106Hz,whileFMstationshavecarrierfrequenciesofabout100MHz.Frequencycanalsobeexpressedbythesymbol!,whichhasunitsofradians/second.Clearly,!=2f.Incommunications,wemostoftenexpressfrequencyinHertz.Finally,isthephase,anddeterminesthesinewave'sbehaviorattheorigint=0.Ithasunitsofradians,butwecanexpressitin 8http://www.lucent.com/minds/infotheory/9Thiscontentisavailableonlineat.

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8CHAPTER1.INTRODUCTIONdegrees,realizingthatincomputationswemustconvertfromdegreestoradians.Notethatif=)]TJ/F1 9.9626 Tf 9.4091 8.0698 Td[()]TJ/F10 6.9738 Tf 5.7618 -4.1472 Td[( 2,thesinusoidcorrespondstoasinefunction,havingazerovalueattheorigin.Asinft+=Acos2ft+)]TJ/F11 9.9626 Tf 11.1581 6.7398 Td[( 2.2Thus,theonlydierencebetweenasineandcosinesignalisthephase;wetermeitherasinusoid.Wecanalsodeneadiscrete-timevariantofthesinusoid:Acos2fn+.Here,theindependentvariableisnandrepresentstheintegers.Frequencynowhasnodimensions,andtakesonvaluesbetween0and1.Exercise1.1Solutiononp.11.Showthatcos2fn=cosf+1n,whichmeansthatasinusoidhavingafrequencylargerthanonecorrespondstoasinusoidhavingafrequencylessthanone.Analogordigital?:Noticethatweshallcalleithersinusoidananalogsignal.Onlywhenthediscrete-timesignaltakesonanitesetofvaluescanitbeconsideredadigitalsignal.Exercise1.2Solutiononp.11.Canyouthinkofasimplesignalthathasanitenumberofvaluesbutisdenedincontinuoustime?Suchasignalisalsoananalogsignal.1.4.2CommunicatingInformationwithSignalsThebasicideaofcommunicationengineeringistouseasignal'sparameterstorepresenteitherrealnumbersorothersignals.Thetechnicaltermistomodulatethecarriersignal'sparameterstotransmitinformationfromoneplacetoanother.Toexplorethenotionofmodulation,wecansendarealnumbertoday'stemperature,forexamplebychangingasinusoid'samplitudeaccordingly.Ifwewantedtosendthedailytemperature,wewouldkeepthefrequencyconstantsothereceiverwouldknowwhattoexpectandchangetheamplitudeatmidnight.WecouldrelatetemperaturetoamplitudebytheformulaA=A0+kT,whereA0andkareconstantsthatthetransmitterandreceivermustbothknow.Ifwehadtwonumberswewantedtosendatthesametime,wecouldmodulatethesinusoid'sfrequencyaswellasitsamplitude.Thismodulationschemeassumeswecanestimatethesinusoid'samplitudeandfrequency;weshalllearnthatthisisindeedpossible.Nowsupposewehaveasequenceofparameterstosend.Wehaveexploitedallofthesinusoid'stwoparameters.WhatwecandoismodulatethemforalimitedtimesayTseconds,andsendtwoparameterseveryT.Thissimplenotioncorrespondstohowamodemworks.Here,typedcharactersareencodedintoeightbits,andtheindividualbitsareencodedintoasinusoid'samplitudeandfrequency.We'lllearnhowthisisdoneinsubsequentmodules,andmoreimportantly,we'lllearnwhatthelimitsareonsuchdigitalcommunicationschemes.1.5IntroductionProblems10Problem1.1:RMSValuesThermsroot-mean-squarevalueofaperiodicsignalisdenedtobes=s 1 TZT0s2tdtwhereTisdenedtobethesignal'speriod:thesmallestpositivenumbersuchthatst=st+T. 10Thiscontentisavailableonlineat.

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9aWhatistheperiodofst=Asinf0t+?bWhatisthermsvalueofthissignal?Howisitrelatedtothepeakvalue?cWhatistheperiodandrmsvalueofthedepictedFigure1.6squarewave,genericallydenotedbysqt?dByinspectinganydeviceyouplugintoawallsocket,you'llseethatitislabeled"110voltsAC".Whatistheexpressionforthevoltageprovidedbyawallsocket?Whatisitsrmsvalue? Figure1.6 Problem1.2:ModemsTheword"modem"isshortfor"modulator-demodulator."Modemsareusednotonlyforconnectingcom-puterstotelephonelines,butalsoforconnectingdigitaldiscrete-valuedsourcestogenericchannels.Inthisproblem,weexploreasimplekindofmodem,inwhichbinaryinformationisrepresentedbythepres-enceorabsenceofasinusoidpresencerepresentinga"1"andabsencea"0".Consequently,themodem'stransmittedsignalthatrepresentsasinglebithastheformxt=Asinf0t,0tTWithineachbitintervalT,theamplitudeiseitherAorzero.aWhatisthesmallesttransmissionintervalthatmakessensewiththefrequencyf0?bAssumingthattencyclesofthesinusoidcompriseasinglebit'stransmissioninterval,whatisthedatarateofthistransmissionscheme?cNowsupposeinsteadofusing"on-o"signaling,weallowoneofseveraldierentvaluesfortheampli-tudeduringanytransmissioninterval.IfNamplitudevaluesareused,whatistheresultingdatarate?dTheclassiccommunicationsblockdiagramappliestothemodem.Discusshowthetransmittermustinterfacewiththemessagesourcesincethesourceisproducinglettersofthealphabet,notbits.Problem1.3:AdvancedModemsTotransmitsymbols,suchaslettersofthealphabet,RUcomputermodemsusetwofrequenciesand1800Hzandseveralamplitudelevels.AtransmissionissentforaperiodoftimeTknownasthetransmissionorbaudintervalandequalsthesumoftwoamplitude-weightedcarriers.xt=A1sinf1t+A2sinf2t,0tTWesendsuccessivesymbolsbychoosinganappropriatefrequencyandamplitudecombination,andsendingthemoneafteranother.aWhatisthesmallesttransmissionintervalthatmakessensetousewiththefrequenciesgivenabove?Inotherwords,whatshouldTbesothatanintegernumberofcyclesofthecarrieroccurs?

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10CHAPTER1.INTRODUCTIONbSketchusingMatlabthesignalthatmodemproducesoverseveraltransmissionintervals.Makesureyouaxesarelabeled.cUsingyoursignaltransmissioninterval,howmanyamplitudelevelsareneededtotransmitASCIIcharactersatadatarateof3,200bits/s?Assumeuseoftheextended-bitASCIIcode.Note:WeuseadiscretesetofvaluesforA1andA2.IfwehaveN1valuesforamplitudeA1,andN2valuesforA2,wehaveN1N2possiblesymbolsthatcanbesentduringeachTsecondinterval.Toconvertthisnumberintobitsthefundamentalunitofinformationengineersusetoqualifythings,computelog2N1N2.

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11SolutionstoExercisesinChapter1SolutiontoExercise1.1p.8Ascos+=coscos)]TJ/F8 9.9626 Tf 13.8458 0 Td[(sinsin,cos2f+1n=cosfncosn)]TJ/F8 9.9626 Tf -460.2513 -11.9552 Td[(sinfnsinn=cosfn.SolutiontoExercise1.2p.8Asquarewavetakesonthevalues1and)]TJ/F8 9.9626 Tf 7.7488 0 Td[(1alternately.SeetheplotinthemoduleElementalSignalsSection2.2.6:SquareWave.

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12CHAPTER1.INTRODUCTION

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Chapter2SignalsandSystems2.1ComplexNumbers1Whilethefundamentalsignalusdinelectricalengineeringisthesinusoid,itcanbeexpressedmathematicallyintermsofanevenmorefundamentalsignal:thecomplexexponential.Representingsinusoidsintermsofcomplexexponentialsisnotamathematicaloddity.Fluencywithcomplexnumbersandrationalfunctionsofcomplexvariablesisacriticalskillallengineersmaster.Understandinginformationandpowersystemdesignsanddevelopingnewsystemsallhingeonusingcomplexnumbers.Inshort,theyarecriticaltomodernelectricalengineering,arealizationmadeoveracenturyago.2.1.1DenitionsThenotionofthesquarerootof)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1originatedwiththequadraticformula:thesolutionofcertainquadraticequationsmathematicallyexistsonlyiftheso-calledimaginaryquantityp )]TJ/F8 9.9626 Tf 7.7487 0 Td[(1couldbedened.Euler2rstusedifortheimaginaryunitbutthatnotationdidnottakeholduntilroughlyAmpre'stime.Ampre3usedthesymbolitodenotecurrentintensitdecurrent.Itwasn'tuntilthetwentiethcenturythattheimportanceofcomplexnumberstocircuittheorybecameevident.Bythen,usingiforcurrentwasentrenchedandelectricalengineerschosejforwritingcomplexnumbers.Animaginarynumberhastheformjb=p )]TJ/F8 9.9626 Tf 9.4091 0 Td[(b2.Acomplexnumber,z,consistsoftheorderedpaira,b,aistherealcomponentandbistheimaginarycomponentthejissuppressedbecausetheimaginarycomponentofthepairisalwaysinthesecondposition.Theimaginarynumberjbequals0,b.Notethataandbarereal-valuednumbers.Figure2.1TheComplexPlaneshowsthatwecanlocateacomplexnumberinwhatwecallthecomplexplane.Here,a,therealpart,isthex-coordinateandb,theimaginarypart,isthey-coordinate.Fromanalyticgeometry,weknowthatlocationsintheplanecanbeexpressedasthesumofvectors,withthevectorscorrespondingtothexandydirections.Consequently,acomplexnumberzcanbeexpressedasthevectorsumz=a+jbwherejindicatesthey-coordinate.ThisrepresentationisknownastheCartesianformofz.Animaginarynumbercan'tbenumericallyaddedtoarealnumber;rather,thisnotationforacomplexnumberrepresentsvectoraddition,butitprovidesaconvenientnotationwhenweperformarithmeticmanipulations.Someobviousterminology.Therealpartofthecomplexnumberz=a+jb,writtenasRez,equalsa.Weconsidertherealpartasafunctionthatworksbyselectingthatcomponentofacomplexnumbernotmultipliedbyj.Theimaginarypartofz,Imz,equalsb:thatpartofacomplexnumberthatismultipliedbyj.Again,boththerealandimaginarypartsofacomplexnumberarereal-valued. 1Thiscontentisavailableonlineat.2http://www-groups.dcs.st-and.ac.uk/history/Mathematicians/Euler.html3http://www-groups.dcs.st-and.ac.uk/history/Mathematicians/Ampere.html13

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14CHAPTER2.SIGNALSANDSYSTEMSTheComplexPlane Figure2.1:Acomplexnumberisanorderedpaira,bthatcanberegardedascoordinatesintheplane.Complexnumberscanalsobeexpressedinpolarcoordinatesasr.Thecomplexconjugateofz,writtenasz,hasthesamerealpartaszbutanimaginarypartoftheoppositesign.z=Rez+jImzz=Rez)]TJ/F11 9.9626 Tf 9.9626 0 Td[(jImz.1UsingCartesiannotation,thefollowingpropertieseasilyfollow.Ifweaddtwocomplexnumbers,therealpartoftheresultequalsthesumoftherealpartsandtheimaginarypartequalsthesumoftheimaginaryparts.Thispropertyfollowsfromthelawsofvectoraddition.a1+jb1+a2+jb2=a1+a2+jb1+b2Inthisway,therealandimaginarypartsremainseparate.Theproductofjandarealnumberisanimaginarynumber:ja.Theproductofjandanimaginarynumberisarealnumber:jjb=)]TJ/F11 9.9626 Tf 7.7487 0 Td[(bbecausej2=)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1.Consequently,multiplyingacomplexnumberbyjrotatesthenumber'spositionby90degrees.Exercise2.1Solutiononp.34.Usethedenitionofadditiontoshowthattherealandimaginarypartscanbeexpressedasasum/dierenceofacomplexnumberanditsconjugate.Rez=z+z 2andImz=z)]TJ/F10 6.9738 Tf 6.2267 0 Td[(z 2j.Complexnumberscanalsobeexpressedinanalternateform,polarform,whichwewillndquiteuseful.Polarformarisesarisesfromthegeometricinterpretationofcomplexnumbers.TheCartesianformofacomplexnumbercanbere-writtenasa+jb=p a2+b2a p a2+b2+jb p a2+b2Byformingarighttrianglehavingsidesaandb,weseethattherealandimaginarypartscorrespondtothecosineandsineofthetriangle'sbaseangle.Wethusobtainthepolarformforcomplexnumbers.z=a+jb=rr=jzj=p a2+b2a=rcosb=rsin=arctan)]TJ/F10 6.9738 Tf 6.1707 -4.1472 Td[(b a

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15Thequantityrisknownasthemagnitudeofthecomplexnumberz,andisfrequentlywrittenasjzj.Thequantityisthecomplexnumber'sangle.Inusingthearc-tangentformulatondtheangle,wemusttakeintoaccountthequadrantinwhichthecomplexnumberlies.Exercise2.2Solutiononp.34.Convert3)]TJ/F8 9.9626 Tf 9.9626 0 Td[(2jtopolarform.2.1.2Euler'sFormulaSurprisingly,thepolarformofacomplexnumberzcanbeexpressedmathematicallyasz=rej.2Toshowthisresult,weuseEuler'srelationsthatexpressexponentialswithimaginaryargumentsintermsoftrigonometricfunctions.ej=cos+jsin.3cos=ej+e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j 2.4sin=ej)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j 2jTherstoftheseiseasilyderivedfromtheTaylor'sseriesfortheexponential.ex=1+x 1!+x2 2!+x3 3!+:::Substitutingjforx,wendthatej=1+j 1!)]TJ/F11 9.9626 Tf 11.1581 6.7398 Td[(2 2!)]TJ/F11 9.9626 Tf 9.9626 0 Td[(j3 3!+:::becausej2=)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1,j3=)]TJ/F11 9.9626 Tf 7.7487 0 Td[(j,andj4=1.Groupingseparatelythereal-valuedtermsandtheimaginary-valuedones,ej=1)]TJ/F11 9.9626 Tf 11.1581 6.7398 Td[(2 2!++j 1!)]TJ/F11 9.9626 Tf 11.1581 6.7398 Td[(3 3!+:::Thereal-valuedtermscorrespondtotheTaylor'sseriesforcos,theimaginaryonestosin,andEuler'srstrelationresults.Theremainingrelationsareeasilyderivedfromtherst.Becauseof,weseethatmultiplyingtheexponentialin.3byarealconstantcorrespondstosettingtheradiusofthecomplexnumberbytheconstant.2.1.3CalculatingwithComplexNumbersAddingandsubtractingcomplexnumbersexpressedinCartesianformisquiteeasy:Youaddsubtracttherealpartsandimaginarypartsseparately.z1z2=a1a2+jb1b2.5TomultiplytwocomplexnumbersinCartesianformisnotquiteaseasy,butfollowsdirectlyfromfollowingtheusualrulesofarithmetic.z1z2=a1+jb1a2+jb2=a1a2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(b1b2+ja1b2+a2b1.6

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16CHAPTER2.SIGNALSANDSYSTEMSNotethatweare,inasense,multiplyingtwovectorstoobtainanothervector.Complexarithmeticprovidesauniquewayofdeningvectormultiplication.Exercise2.3Solutiononp.34.Whatistheproductofacomplexnumberanditsconjugate?Divisionrequiresmathematicalmanipulation.Weconvertthedivisionproblemintoamultiplicationproblembymultiplyingboththenumeratoranddenominatorbytheconjugateofthedenominator.z1 z2=a1+jb1 a2+jb2=a1+jb1 a2+jb2a2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(jb2 a2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(jb2=a1+jb1a2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(jb2 a22+b22=a1a2+b1b2+ja2b1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(a1b2 a22+b22.7Becausethenalresultissocomplicated,it'sbesttorememberhowtoperformdivisionmultiplyingnumeratoranddenominatorbythecomplexconjugateofthedenominatorthantryingtorememberthenalresult.Thepropertiesoftheexponentialmakecalculatingtheproductandratiooftwocomplexnumbersmuchsimplerwhenthenumbersareexpressedinpolarform.z1z2=r1ej1r2ej2=r1r2ej1+2.8z1 z2=r1ej1 r2ej2=r1 r2ej1)]TJ/F10 6.9738 Tf 6.2267 0 Td[(2Tomultiply,theradiusequalstheproductoftheradiiandtheanglethesumoftheangles.Todivide,theradiusequalstheratiooftheradiiandtheanglethedierenceoftheangles.WhentheoriginalcomplexnumbersareinCartesianform,it'susuallyworthtranslatingintopolarform,thenperformingthemultiplicationordivisionespeciallyinthecaseofthelatter.AdditionandsubtractionofpolarformsamountstoconvertingtoCartesianform,performingthearithmeticoperation,andconvertingbacktopolarform.Example2.1Whenwesolvecircuitproblems,thecrucialquantity,knownasatransferfunction,willalwaysbeexpressedastheratioofpolynomialsinthevariables=j2f.Whatwe'llneedtounderstandthecircuit'seectisthetransferfunctioninpolarform.Forinstance,supposethetransferfunctionequalss+2 s2+s+1.9s=j2f.10Performingtherequireddivisionismosteasilyaccomplishedbyrstexpressingthenumeratorand

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17denominatoreachinpolarform,thencalculatingtheratio.Thus,s+2 s2+s+1=j2f+2 )]TJ/F8 9.9626 Tf 7.7487 0 Td[(42f2+j2f+1.11=p 4+42f2ejarctanf q )]TJ/F8 9.9626 Tf 9.9626 0 Td[(42f22+42f2ejarctan2f 1)]TJ/F6 4.9813 Tf 5.3965 0 Td[(42f2.12=s 4+42f2 1)]TJ/F8 9.9626 Tf 9.9626 0 Td[(42f2+164f4ejarctanf)]TJ/F7 6.9738 Tf 6.2266 0 Td[(arctan2f 1)]TJ/F6 4.9813 Tf 5.3965 0 Td[(42f2.132.2ElementalSignals4Elementalsignalsarethebuildingblockswithwhichwebuildcomplicatedsignals.Bydenition,elementalsignalshaveasimplestructure.Exactlywhatwemeanbythe"structureofasignal"willunfoldinthissectionofthecourse.Signalsarenothingmorethanfunctionsdenedwithrespecttosomeindependentvariable,whichwetaketobetimeforthemostpart.Veryinterestingsignalsarenotfunctionssolelyoftime;onegreatexampleofwhichisanimage.Forit,theindependentvariablesarexandytwo-dimensionalspace.Videosignalsarefunctionsofthreevariables:twospatialdimensionsandtime.Fortunately,mostoftheideasunderlyingmodernsignaltheorycanbeexempliedwithone-dimensionalsignals.2.2.1SinusoidsPerhapsthemostcommonreal-valuedsignalisthesinusoid.st=Acos2f0t+.14Forthissignal,Aisitsamplitude,f0itsfrequency,anditsphase.2.2.2ComplexExponentialsThemostimportantsignaliscomplex-valued,thecomplexexponential.st=Aejf0t+=Aejej2f0t.15Here,jdenotesp )]TJ/F8 9.9626 Tf 7.7487 0 Td[(1.Aejisknownasthesignal'scomplexamplitude.Consideringthecomplexampli-tudeasacomplexnumberinpolarform,itsmagnitudeistheamplitudeAanditsanglethesignalphase.Thecomplexamplitudeisalsoknownasaphasor.Thecomplexexponentialcannotbefurtherdecomposedintomoreelementalsignals,andisthemostimportantsignalinelectricalengineering!Mathematicalmanip-ulationsatrstappeartobemoredicultbecausecomplex-valuednumbersareintroduced.Infact,earlyinthetwentiethcentury,mathematiciansthoughtengineerswouldnotbesucientlysophisticatedtohandlecomplexexponentialseventhoughtheygreatlysimpliedsolvingcircuitproblems.Steinmetz5introducedcomplexexponentialstoelectricalengineering,anddemonstratedthat"mere"engineerscouldusethemtogoodeectandevenobtainrightanswers!SeeComplexNumbersSection2.1forareviewofcomplexnumbersandcomplexarithmetic.Thecomplexexponentialdenesthenotionoffrequency:itistheonlysignalthatcontainsonlyonefrequencycomponent.Thesinusoidconsistsoftwofrequencycomponents:oneatthefrequency+f0andtheotherat)]TJ/F11 9.9626 Tf 7.7487 0 Td[(f0. 4Thiscontentisavailableonlineat.5http://www.invent.org/hall_of_fame/139.html

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18CHAPTER2.SIGNALSANDSYSTEMSEulerrelation:ThisdecompositionofthesinusoidcanbetracedtoEuler'srelation.cos2ft=ej2ft+e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2ft 2.16sinft=ej2ft)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2ft 2j.17ej2ft=cosft+jsinft.18Decomposition:ThecomplexexponentialsignalcanthusbewrittenintermsofitsrealandimaginarypartsusingEuler'srelation.Thus,sinusoidalsignalscanbeexpressedaseithertherealortheimaginarypartofacomplexexponentialsignal,thechoicedependingonwhethercosineorsinephaseisneeded,orasthesumoftwocomplexexponentials.Thesetwodecompositionsaremathematicallyequivalenttoeachother.Acos2ft+=Re)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(Aejej2ft.19Asinft+=Im)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(Aejej2ft.20Usingthecomplexplane,wecanenvisionthecomplexexponential'stemporalvariationsasseenintheabovegureFigure2.2.ThemagnitudeofthecomplexexponentialisA,andtheinitialvalueofthecomplexexponentialatt=0hasanangleof.Astimeincreases,thelocusofpointstracedbythecomplexexponentialisacircleithasconstantmagnitudeofA.Thenumberoftimespersecondwegoaroundthecircleequalsthefrequencyf.ThetimetakenforthecomplexexponentialtogoaroundthecircleonceisknownasitsperiodT,andequals1 f.TheprojectionsontotherealandimaginaryaxesoftherotatingvectorrepresentingthecomplexexponentialsignalarethecosineandsinesignalofEuler'srelation.16.2.2.3RealExponentialsAsopposedtocomplexexponentialswhichoscillate,realexponentialsFigure2.3decay.st=e)]TJ/F8 9.9626 Tf 6.2266 -0.7472 Td[(t .21Thequantityisknownastheexponential'stimeconstant,andcorrespondstothetimerequiredfortheexponentialtodecreasebyafactorof1 e,whichapproximatelyequals0:368.Adecayingcomplexexponentialistheproductofarealandacomplexexponential.st=Aeje)]TJ/F8 9.9626 Tf 6.2266 -0.7472 Td[(t ej2ft=Aeje)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(1 +j2ft.22Inthecomplexplane,thissignalcorrespondstoanexponentialspiral.Forsuchsignals,wecandenecomplexfrequencyasthequantitymultiplyingt.2.2.4UnitStepTheunitstepfunctionFigure2.4isdenotedbyut,andisdenedtobeut=8<:0ift<01ift>0.23

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19 Figure2.2:Graphically,thecomplexexponentialscribesacircleinthecomplexplaneastimeevolves.Itsrealandimaginarypartsaresinusoids.TherateatwhichthesignalgoesaroundthecircleisthefrequencyfandthetimetakentogoaroundistheperiodT.AfundamentalrelationshipisT=1 f. Figure2.3:Therealexponential.

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20CHAPTER2.SIGNALSANDSYSTEMS Figure2.4:Theunitstep. originwarning:Thissignalisdiscontinuousattheorigin.Itsvalueattheoriginneednotbedened,anddoesn'tmatterinsignaltheory.Thiskindofsignalisusedtodescribesignalsthat"turnon"suddenly.Forexample,tomathematicallyrepre-sentturningonanoscillator,wecanwriteitastheproductofasinusoidandastep:st=Asinftut.2.2.5PulseTheunitpulseFigure2.5describesturningaunit-amplitudesignalonforadurationofseconds,thenturningito.pt=8>><>>:0ift<01if0.24 Figure2.5:Thepulse.Wewillndthatthisisthesecondmostimportantsignalincommunications.2.2.6SquareWaveThesquarewaveFigure2.6sqtisaperiodicsignallikethesinusoid.Ittoohasanamplitudeandaperiod,whichmustbespeciedtocharacterizethesignal.Wendsubsequentlythatthesinewaveisasimplersignalthanthesquarewave.2.3SignalDecomposition6Asignal'scomplexityisnotrelatedtohowwigglyitis.Rather,asignalexpertlooksforwaysofdecomposingagivensignalintoasumofsimplersignals,whichwetermthesignaldecomposition.Thoughwewill 6Thiscontentisavailableonlineat.

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21 Figure2.6:Thesquarewave. nevercomputeasignal'scomplexity,itessentiallyequalsthenumberoftermsinitsdecomposition.Inwritingasignalasasumofcomponentsignals,wecanchangethecomponentsignal'sgainbymultiplyingitbyaconstantandbydelayingit.Morecomplicateddecompositionscouldcontainderivativesorintegralsofsimplesignals.Example2.2Asanexampleofsignalcomplexity,wecanexpressthepulseptasasumofdelayedunitsteps.pt=ut)]TJ/F11 9.9626 Tf 9.9626 0 Td[(ut)]TJ/F8 9.9626 Tf 9.9626 0 Td[(.25Thus,thepulseisamorecomplexsignalthanthestep.Bethatasitmay,thepulseisveryusefultous.Exercise2.4Solutiononp.34.ExpressasquarewavehavingperiodTandamplitudeAasasuperpositionofdelayedandamplitude-scaledpulses.Becausethesinusoidisasuperpositionoftwocomplexexponentials,thesinusoidismorecomplex.Wecouldnotpreventourselvesfromthepuninthisstatement.Clearly,theword"complex"isusedintwodierentwayshere.ThecomplexexponentialcanalsobewrittenusingEuler'srelation.16asasumofasineandacosine.Wewilldiscoverthatvirtuallyeverysignalcanbedecomposedintoasumofcomplexexponentials,andthatthisdecompositionisveryuseful.Thus,thecomplexexponentialismorefundamental,andEuler'srelationdoesnotadequatelyrevealitscomplexity.2.4Discrete-TimeSignals7Sofar,wehavetreatedwhatareknownasanalogsignalsandsystems.Mathematically,analogsignalsarefunctionshavingcontinuousquantitiesastheirindependentvariables,suchasspaceandtime.Discrete-timesignalsSection5.5arefunctionsdenedontheintegers;theyaresequences.OneofthefundamentalresultsofsignaltheorySection5.3willdetailconditionsunderwhichananalogsignalcanbeconvertedintoadiscrete-timeoneandretrievedwithouterror.Thisresultisimportantbecausediscrete-timesignalscanbemanipulatedbysystemsinstantiatedascomputerprograms.Subsequentmodulesdescribehowvirtuallyallanalogsignalprocessingcanbeperformedwithsoftware.Asimportantassuchresultsare,discrete-timesignalsaremoregeneral,encompassingsignalsderivedfromanalogonesandsignalsthataren't.Forexample,thecharactersformingatextleformasequence,whichisalsoadiscrete-timesignal.Wemustdealwithsuchsymbolicvaluedp.167signalsandsystemsaswell. 7Thiscontentisavailableonlineat.

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22CHAPTER2.SIGNALSANDSYSTEMSAswithanalogsignals,weseekwaysofdecomposingreal-valueddiscrete-timesignalsintosimplercom-ponents.Withthisapproachleadingtoabetterunderstandingofsignalstructure,wecanexploitthatstructuretorepresentinformationcreatewaysofrepresentinginformationwithsignalsandtoextractin-formationretrievetheinformationthusrepresented.Forsymbolic-valuedsignals,theapproachisdierent:Wedevelopacommonrepresentationofallsymbolic-valuedsignalssothatwecanembodytheinformationtheycontaininauniedway.Fromaninformationrepresentationperspective,themostimportantissuebecomes,forbothreal-valuedandsymbolic-valuedsignals,eciency;Whatisthemostparsimoniousandcompactwaytorepresentinformationsothatitcanbeextractedlater.2.4.1Real-andComplex-valuedSignalsAdiscrete-timesignalisrepresentedsymbolicallyassn,wheren=f:::;)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1;0;1;:::g.Weusuallydrawdiscrete-timesignalsasstemplotstoemphasizethefacttheyarefunctionsdenedonlyontheintegers.Wecandelayadiscrete-timesignalbyanintegerjustaswithanalogones.Adelayedunitsamplehastheexpressionn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(m,andequalsonewhenn=m. Discrete-TimeCosineSignal Figure2.7:Thediscrete-timecosinesignalisplottedasastemplot.Canyoundtheformulaforthissignal? 2.4.2ComplexExponentialsThemostimportantsignalis,ofcourse,thecomplexexponentialsequence.sn=ej2fn.262.4.3SinusoidsDiscrete-timesinusoidshavetheobviousformsn=Acos2fn+.Asopposedtoanalogcomplexexponentialsandsinusoidsthatcanhavetheirfrequenciesbeanyrealvalue,frequenciesoftheirdiscrete-timecounterpartsyielduniquewaveformsonlywhenfliesintheinterval)]TJ/F14 9.9626 Tf 4.5662 -8.0699 Td[()]TJ/F1 9.9626 Tf 9.4092 8.0699 Td[()]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(1 2;1 2.Thispropertycanbeeasilyunderstoodbynotingthataddinganintegertothefrequencyofthediscrete-timecomplexexponentialhasnoeectonthesignal'svalue.ej2f+mn=ej2fnej2mn=ej2fn.27Thisderivationfollowsbecausethecomplexexponentialevaluatedatanintegermultipleof2equalsone.

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232.4.4UnitSampleThesecond-mostimportantdiscrete-timesignalistheunitsample,whichisdenedtoben=8<:1ifn=00otherwise.28 UnitSample Figure2.8:Theunitsample. Examinationofadiscrete-timesignal'splot,likethatofthecosinesignalshowninFigure2.7Discrete-TimeCosineSignal,revealsthatallsignalsconsistofasequenceofdelayedandscaledunitsamples.Becausethevalueofasequenceateachintegermisdenotedbysmandtheunitsampledelayedtooccuratmiswrittenn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(m,wecandecomposeanysignalasasumofunitsamplesdelayedtotheappropriatelocationandscaledbythesignalvalue.sn=1Xm=smn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(m.29Thiskindofdecompositionisuniquetodiscrete-timesignals,andwillproveusefulsubsequently.Discrete-timesystemscanactondiscrete-timesignalsinwayssimilartothosefoundinanalogsignalsandsystems.Becauseoftheroleofsoftwareindiscrete-timesystems,manymoredierentsystemscanbeenvisionedandconstructedwithprogramsthancanbewithanalogsignals.Infact,aspecialclassofanalogsignalscanbeconvertedintodiscrete-timesignals,processedwithsoftware,andconvertedbackintoananalogsignal,allwithouttheincursionoferror.Forsuchsignals,systemscanbeeasilyproducedinsoftware,withequivalentanalogrealizationsdicult,ifnotimpossible,todesign.2.4.5Symbolic-valuedSignalsAnotherinterestingaspectofdiscrete-timesignalsisthattheirvaluesdonotneedtoberealnumbers.Wedohavereal-valueddiscrete-timesignalslikethesinusoid,butwealsohavesignalsthatdenotethesequenceofcharacterstypedonthekeyboard.Suchcharacterscertainlyaren'trealnumbers,andasacollectionofpossiblesignalvalues,theyhavelittlemathematicalstructureotherthanthattheyaremembersofaset.Moreformally,eachelementofthesymbolic-valuedsignalsntakesononeofthevaluesfa1;:::;aKgwhichcomprisethealphabetA.Thistechnicalterminologydoesnotmeanwerestrictsymbolstobeingmem-bersoftheEnglishorGreekalphabet.Theycouldrepresentkeyboardcharacters,bytes8-bitquantities,integersthatconveydailytemperature.Whethercontrolledbysoftwareornot,discrete-timesystemsareultimatelyconstructedfromdigitalcircuits,whichconsistentirelyofanalogcircuitelements.Furthermore,thetransmissionandreceptionofdiscrete-timesignals,likee-mail,isaccomplishedwithanalogsignalsandsystems.Understandinghowdiscrete-timeandanalogsignalsandsystemsintertwineisperhapsthemaingoalofthiscourse.

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24CHAPTER2.SIGNALSANDSYSTEMS2.5IntroductiontoSystems8Signalsaremanipulatedbysystems.Mathematically,werepresentwhatasystemdoesbythenotationyt=Sxt,withxrepresentingtheinputsignalandytheoutputsignal. Denitionofasystem Figure2.9:Thesystemdepictedhasinputxtandoutputyt.Mathematically,systemsoperateonfunctionstoproduceotherfunctions.Inmanyways,systemsarelikefunctions,rulesthatyieldavalueforthedependentvariableouroutputsignalforeachvalueofitsindependentvariableitsinputsignal.Thenotationyt=Sxtcorrespondstothisblockdiagram.WetermStheinput-outputrelationforthesystem. Thisnotationmimicsthemathematicalsymbologyofafunction:Asystem'sinputisanalogoustoanindependentvariableanditsoutputthedependentvariable.Forthemathematicallyinclined,asystemisafunctional:afunctionofafunctionsignalsarefunctions.Simplesystemscanbeconnectedtogetheronesystem'soutputbecomesanother'sinputtoaccomplishsomeoveralldesign.Interconnectiontopologiescanbequitecomplicated,butusuallyconsistofweavesofthreebasicinterconnectionforms.2.5.1CascadeInterconnection cascade Figure2.10:Themostrudimentarywaysofinterconnectingsystemsareshownintheguresinthissection.Thisisthecascadeconguration. Thesimplestformiswhenonesystem'soutputisconnectedonlytoanother'sinput.Mathematically,wt=S1xt,andyt=S2wt,withtheinformationcontainedinxtprocessedbytherst,thenthesecondsystem.Insomecases,theorderingofthesystemsmatter,inothersitdoesnot.Forexample,inthefundamentalmodelofcommunicationFigure1.4:Fundamentalmodelofcommunicationtheorderingmostcertainlymatters. 8Thiscontentisavailableonlineat.

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252.5.2ParallelInterconnection parallel Figure2.11:Theparallelconguration. Asignalxtisroutedtotwoormoresystems,withthissignalappearingastheinputtoallsystemssimultaneouslyandwithequalstrength.Blockdiagramshavetheconventionthatsignalsgoingtomorethanonesystemarenotsplitintopiecesalongtheway.Twoormoresystemsoperateonxtandtheiroutputsareaddedtogethertocreatetheoutputyt.Thus,yt=S1xt+S2xt,andtheinformationinxtisprocessedseparatelybybothsystems.2.5.3FeedbackInterconnection feedback Figure2.12:Thefeedbackconguration. Thesubtlestinterconnectioncongurationhasasystem'soutputalsocontributingtoitsinput.Engineerswouldsaytheoutputis"fedback"totheinputthroughsystem2,hencetheterminology.ThemathematicalstatementofthefeedbackinterconnectionFigure2.12:feedbackisthatthefeed-forwardsystemproducestheoutput:yt=S1et.Theinputetequalstheinputsignalminustheoutputofsomeothersystem'soutputtoyt:et=xt)]TJ/F11 9.9626 Tf 10.3061 0 Td[(S2yt.Feedbacksystemsareomnipresentincontrolproblems,withtheerrorsignalusedtoadjusttheoutputtoachievesomeconditiondenedbytheinputcontrollingsignal.Forexample,inacar'scruisecontrolsystem,xtisaconstantrepresentingwhatspeedyouwant,andyt

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26CHAPTER2.SIGNALSANDSYSTEMSisthecar'sspeedasmeasuredbyaspeedometer.Inthisapplication,system2istheidentitysystemoutputequalsinput.2.6SimpleSystems9Systemsmanipulatesignals,creatingoutputsignalsderivedfromtheirinputs.Whythefollowingarecate-gorizedas"simple"willonlybecomeevidenttowardstheendofthecourse.2.6.1SourcesSourcesproducesignalswithouthavinginput.Weliketothinkoftheseashavingcontrollableparameters,likeamplitudeandfrequency.Exampleswouldbeoscillatorsthatproduceperiodicsignalslikesinusoidsandsquarewavesandnoisegeneratorsthatyieldsignalswitherraticwaveformsmoreaboutnoisesubsequently.Simplywritinganexpressionforthesignalstheyproducespeciessources.Asinewavegeneratormightbespeciedbyyt=Asinf0tut,whichsaysthatthesourcewasturnedonatt=0toproduceasinusoidofamplitudeAandfrequencyf0.2.6.2AmpliersAnamplierFigure2.13:ampliermultipliesitsinputbyaconstantknownastheampliergain.yt=Gxt.30 amplier Figure2.13:Anamplier. Thegaincanbepositiveornegativeifnegative,wewouldsaythattheamplierinvertsitsinputandcanbegreaterthanoneorlessthanone.Iflessthanone,theamplieractuallyattenuates.Areal-worldexampleofanamplierisyourhomestereo.Youcontrolthegainbyturningthevolumecontrol.2.6.3DelayAsystemservesasatimedelayFigure2.14:delaywhentheoutputsignalequalstheinputsignalatanearliertime.yt=xt)]TJ/F11 9.9626 Tf 9.9626 0 Td[(.31 9Thiscontentisavailableonlineat.

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27 delay Figure2.14:Adelay. Here,isthedelay.Thewaytounderstandthissystemistofocusonthetimeorigin:Theoutputattimet=equalstheinputattimet=0.Thus,ifthedelayispositive,theoutputemergeslaterthantheinput,andplottingtheoutputamountstoshiftingtheinputplottotheright.Thedelaycanbenegative,inwhichcasewesaythesystemadvancesitsinput.Suchsystemsarediculttobuildtheywouldhavetoproducesignalvaluesderivedfromwhattheinputwillbe,butwewillhaveoccasiontoadvancesignalsintime.2.6.4TimeReversalHere,theoutputsignalequalstheinputsignalippedaboutthetimeorigin.yt=x)]TJ/F11 9.9626 Tf 7.7487 0 Td[(t.32 timereversal Figure2.15:Atimereversalsystem. Again,suchsystemsarediculttobuild,butthenotionoftimereversaloccursfrequentlyincommuni-cationssystems.Exercise2.5Solutiononp.34.Mentionedearlierwastheissueofwhethertheorderingofsystemsmattered.Inotherwords,ifwehavetwosystemsincascade,doestheoutputdependonwhichcomesrst?Determineiftheorderingmattersforthecascadeofanamplierandadelayandforthecascadeofatime-reversalsystemandadelay.2.6.5DerivativeSystemsandIntegratorsSystemsthatperformcalculus-likeoperationsontheirinputscanproducewaveformssignicantlydierentthanpresentintheinput.Derivativesystemsoperateinastraightforwardway:Arst-derivativesystem

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28CHAPTER2.SIGNALSANDSYSTEMSwouldhavetheinput-outputrelationshipyt=d dtxt.Integralsystemshavethecomplicationthattheintegral'slimitsmustbedened.Itisasignaltheoryconventionthattheelementaryintegraloperationhavealowerlimitof,andthatthevalueofallsignalsatt=equalszero.Asimpleintegratorwouldhaveinput-outputrelationyt=Ztxd.332.6.6LinearSystemsLinearsystemsareaclassofsystemsratherthanhavingaspecicinput-outputrelation.Linearsystemsformthefoundationofsystemtheory,andarethemostimportantclassofsystemsincommunications.Theyhavethepropertythatwhentheinputisexpressedasaweightedsumofcomponentsignals,theoutputequalsthesameweightedsumoftheoutputsproducedbyeachcomponent.WhenSislinear,SG1x1t+G2x2t=G1Sx1t+G2Sx2t.34forallchoicesofsignalsandgains.Thisgeneralinput-outputrelationpropertycanbemanipulatedtoindicatespecicpropertiessharedbyalllinearsystems.SGxt=GSxtThecolloquialismsummarizingthispropertyis"Doubletheinput,youdoubletheoutput."Notethatthispropertyisconsistentwithalternatewaysofexpressinggainchanges:Since2xtalsoequalsxt+xt,thelinearsystemdenitionprovidesthesameoutputnomatterwhichoftheseisusedtoexpressagivensignal.S=0Iftheinputisidenticallyzeroforalltime,theoutputofalinearsystemmustbezero.ThispropertyfollowsfromthesimplederivationS=Sxt)]TJ/F11 9.9626 Tf 9.9626 0 Td[(xt=Sxt)]TJ/F11 9.9626 Tf 9.9626 0 Td[(Sxt=0.Justwhylinearsystemsaresoimportantisrelatednotonlytotheirproperties,whicharedivulgedthroughoutthiscourse,butalsobecausetheylendthemselvestorelativelysimplemathematicalanalysis.Saidanotherway,"They'retheonlysystemswethoroughlyunderstand!"Wecanndtheoutputofanylinearsystemtoacomplicatedinputbydecomposingtheinputintosimplesignals.Theequationabove.34saysthatwhenasystemislinear,itsoutputtoadecomposedinputisthesumofoutputstoeachinput.Forexample,ifxt=e)]TJ/F10 6.9738 Tf 6.2266 0 Td[(t+sinf0ttheoutputSxtofanylinearsystemequalsyt=S)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(e)]TJ/F10 6.9738 Tf 6.2267 0 Td[(t+Ssinf0t2.6.7Time-InvariantSystemsSystemsthatdon'tchangetheirinput-outputrelationwithtimearesaidtobetime-invariant.Themathemat-icalwayofstatingthispropertyistousethesignaldelayconceptdescribedinSimpleSystemsSection2.6.3:Delay.yt=Sxtyt)]TJ/F11 9.9626 Tf 9.9626 0 Td[(=Sxt)]TJ/F11 9.9626 Tf 9.9626 0 Td[(.35Ifyoudelayoradvancetheinput,theoutputissimilarlydelayedadvanced.Thus,atime-invariantsystemrespondstoaninputyoumaysupplytomorrowthesamewayitrespondstothesameinputappliedtoday;today'soutputismerelydelayedtooccurtomorrow.Thecollectionoflinear,time-invariantsystemsarethemostthoroughlyunderstoodsystems.Muchofthesignalprocessingandsystemtheorydiscussedhereconcentratesonsuchsystems.Forexample,electric

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29circuitsare,forthemostpart,linearandtime-invariant.Nonlinearonesabound,butcharacterizingthemsothatyoucanpredicttheirbehaviorforanyinputremainsanunsolvedproblem. Linear,Time-InvariantTable Input-OutputRelation Linear Time-Invariant yt=d dtx yes yes yt=d2 dt2x yes yes yt=)]TJ/F10 6.9738 Tf 7.2665 -4.1472 Td[(d dtx2 no yes yt=d dtx+x yes yes yt=x1+x2 yes yes yt=xt)]TJ/F11 9.9626 Tf 9.9626 0 Td[( yes yes yt=cosftxt yes no yt=x)]TJ/F11 9.9626 Tf 7.7487 0 Td[(t yes no yt=x2t no yes yt=jxtj no yes yt=mxt+b no yes Figure2.16 2.7SignalsandSystemsProblems10Problem2.1:ComplexNumberArithmeticFindtherealpart,imaginarypart,themagnitudeandangleofthecomplexnumbersgivenbythefollowingexpressions.a)]TJ/F8 9.9626 Tf 7.7488 0 Td[(1b1+p 3j 2c1+j+ej 2dej 3+ej+e)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(j 3Problem2.2:DiscoveringRootsComplexnumbersexposealltherootsofrealandcomplexnumbers.Forexample,thereshouldbetwosquare-roots,threecube-roots,etc.ofanynumber.Findthefollowingroots.aWhatarethecube-rootsof27?Inotherwords,whatis271 3?bWhatarethefthrootsof331 5?cWhatarethefourthrootsofone? 10Thiscontentisavailableonlineat.

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30CHAPTER2.SIGNALSANDSYSTEMSProblem2.3:CoolExponentialsSimplifythefollowingcoolexpressions.ajjbj2jcjjjProblem2.4:Complex-valuedSignalsComplexnumbersandphasorsplayaveryimportantroleinelectricalengineering.Solvingsystemsforcomplexexponentialsismucheasierthanforsinusoids,andlinearsystemsanalysisisparticularlyeasy.aFindthephasorrepresentationforeach,andre-expresseachastherealandimaginarypartsofacomplexexponential.WhatisthefrequencyinHzofeach?Ingeneral,areyouranswersunique?Ifso,proveit;ifnot,ndanalternativeanswerforthecomplexexponentialrepresentation.i3sintiip 2cos)]TJ/F8 9.9626 Tf 4.5662 -8.0698 Td[(260t+ 4iii2cos)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(t+ 6+4sin)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(t)]TJ/F10 6.9738 Tf 11.1581 3.9226 Td[( 3bShowthatforlinearsystemshavingreal-valuedoutputsforrealinputs,thatwhentheinputistherealpartofacomplexexponential,theoutputistherealpartofthesystem'soutputtothecomplexexponentialseeFigure2.17.S)]TJ/F8 9.9626 Tf 4.5662 -8.0698 Td[(Re)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(Aej2ft=Re)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(S)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(Aej2ft Figure2.17 Problem2.5:Foreachoftheindicatedvoltages,writeitastherealpartofacomplexexponentialvt=ReVest.ExplicitlyindicatethevalueofthecomplexamplitudeVandthecomplexfrequencys.RepresenteachcomplexamplitudeasavectorintheV-plane,andindicatethelocationofthefrequenciesinthecomplexs-plane.avt=costbvt=sin)]TJ/F8 9.9626 Tf 4.5663 -8.0699 Td[(8t+ 4cvt=e)]TJ/F10 6.9738 Tf 6.2266 0 Td[(tdvt=e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(tsin)]TJ/F8 9.9626 Tf 4.5662 -8.0698 Td[(4t+3 4evt=5etsint+2fvt=)]TJ/F8 9.9626 Tf 7.7487 0 Td[(2gvt=4sint+3costhvt=2cos)]TJ/F8 9.9626 Tf 4.5662 -8.0698 Td[(100t+ 6)]TJ 9.9626 8.2413 Td[(p 3sin)]TJ/F8 9.9626 Tf 4.5663 -8.0698 Td[(100t+ 2

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31Problem2.6:ExpresseachofthefollowingsignalsFigure2.18asalinearcombinationofdelayedandweightedstepfunctionsandrampstheintegralofastep. a b c d eFigure2.18 Problem2.7:Linear,Time-InvariantSystemsWhentheinputtoalinear,time-invariantsystemisthesignalxt,theoutputisthesignalytFig-ure2.19.aFindandsketchthissystem'soutputwhentheinputisthedepictedsignalFigure2.20.bFindandsketchthissystem'soutputwhentheinputisaunitstep.

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32CHAPTER2.SIGNALSANDSYSTEMS Figure2.19 Figure2.20 Problem2.8:LinearSystemsThedepictedinputFigure2.21xttoalinear,time-invariantsystemyieldstheoutputyt. Figure2.21 aWhatisthesystem'soutputtoaunitstepinputut?bWhatwilltheoutputbewhentheinputisthedepictedsquarewaveFigure2.22?

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33 Figure2.22 Problem2.9:CommunicationChannelAparticularlyinterestingcommunicationchannelcanbemodeledasalinear,time-invariantsystem.Whenthetransmittedsignalxtisapulse,thereceivedsignalrtisasshownFigure2.23. Figure2.23 aWhatwillbethereceivedsignalwhenthetransmittersendsthepulsesequenceFigure2.24x1t?bWhatwillbethereceivedsignalwhenthetransmittersendsthepulsesignalFigure2.24x2tthathashalfthedurationastheoriginal? Figure2.24

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34CHAPTER2.SIGNALSANDSYSTEMSSolutionstoExercisesinChapter2SolutiontoExercise2.1p.14z+z=a+jb+a)]TJ/F11 9.9626 Tf 9.9626 0 Td[(jb=2a=2Rez.Similarly,z)]TJ/F11 9.9626 Tf 9.9626 0 Td[(z=a+jb)]TJ/F8 9.9626 Tf 9.9626 0 Td[(a)]TJ/F11 9.9626 Tf 9.9626 0 Td[(jb=2jb=2jImzSolutiontoExercise2.2p.15Toconvert3)]TJ/F8 9.9626 Tf 8.7358 0 Td[(2jtopolarform,werstlocatethenumberinthecomplexplaneinthefourthquadrant.Thedistancefromtheorigintothecomplexnumberisthemagnituder,whichequalsp 13=q 32+)]TJ/F8 9.9626 Tf 7.7488 0 Td[(22.Theangleequals)]TJ/F1 9.9626 Tf 9.4091 8.0698 Td[()]TJ/F8 9.9626 Tf 4.5663 -8.0698 Td[(arctan)]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(2 3or)]TJ/F8 9.9626 Tf 7.7487 0 Td[(0:588radians)]TJ/F8 9.9626 Tf 7.7488 0 Td[(33:7degrees.Thenalanswerisp 13)]TJ/F8 9.9626 Tf 7.7487 0 Td[(33:7degrees.SolutiontoExercise2.3p.16zz=a+jba)]TJ/F11 9.9626 Tf 9.9626 0 Td[(jb=a2+b2.Thus,zz=r2=jzj2.SolutiontoExercise2.4p.21sqt=P1n=)]TJ/F8 9.9626 Tf 4.5663 -8.0698 Td[()]TJ/F8 9.9626 Tf 7.7488 0 Td[(1nApT=2)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(t)]TJ/F11 9.9626 Tf 9.9626 0 Td[(nT 2SolutiontoExercise2.5p.27Intherstcase,orderdoesnotmatter;intheseconditdoes."Delay"meanst!t)]TJ/F11 9.9626 Tf 10.2083 0 Td[(."Time-reverse"meanst!)]TJ/F11 9.9626 Tf 20.4787 0 Td[(tCase1yt=Gxt)]TJ/F11 9.9626 Tf 9.9626 0 Td[(,andthewayweapplythegainanddelaythesignalgivesthesameresult.Case2Time-reversethendelay:yt=x)]TJ/F8 9.9626 Tf 9.4092 0 Td[(t)]TJ/F11 9.9626 Tf 9.9626 0 Td[(=x)]TJ/F11 9.9626 Tf 7.7487 0 Td[(t+.Delaythentime-reverse:yt=x)]TJ/F11 9.9626 Tf 7.7488 0 Td[(t)]TJ/F11 9.9626 Tf 9.9626 0 Td[(.

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Chapter3AnalogSignalProcessing3.1Voltage,Current,andGenericCircuitElements1Weknowthatinformationcanberepresentedbysignals;nowweneedtounderstandhowsignalsarephysicallyrealized.Overtheyears,electricsignalshavebeenfoundtobetheeasiesttouse.Voltageandcurrentscomprisetheelectricinstantiationsofsignals.Thus,weneedtodelveintotheworldofelectricityandelectromagnetism.Thesystemsusedtomanipulateelectricsignalsdirectlyarecalledcircuits,andtheyrenetheinformationrepresentationorextractinformationfromthevoltageorcurrent.Inmanycases,theymakeniceexamplesoflinearsystems.Agenericcircuitelementplacesaconstraintbetweentheclassicvariablesofacircuit:voltageandcurrent.Voltageiselectricpotentialandrepresentsthe"push"thatdriveselectricchargefromoneplacetoanother.Whatcauseschargetomoveisaphysicalseparationbetweenpositiveandnegativecharge.Abatterygenerates,throughelectrochemicalmeans,excesspositivechargeatoneterminalandnegativechargeattheother,creatinganelectriceld.Voltageisdenedacrossacircuitelement,withthepositivesigndenotingapositivevoltagedropacrosstheelement.Whenaconductorconnectsthepositiveandnegativepotentials,currentows,withpositivecurrentindicatingthatpositivechargeowsfromthepositiveterminaltothenegative.Electronscomprisecurrentowinmanycases.Becauseelectronshaveanegativecharge,electronsmoveintheoppositedirectionofpositivecurrentow:Negativechargeowingtotherightisequivalenttopositivechargemovingtotheleft.Itisimportanttounderstandthephysicsofcurrentowinconductorstoappreciatetheinnovationofnewelectronicdevices.Electricchargecanarisefrommanysources,thesimplestbeingtheelectron.Whenwesaythat"electronsowthroughaconductor,"whatwemeanisthattheconductor'sconstituentatomsfreelygiveupelectronsfromtheiroutershells."Flow"thusmeansthatelectronshopfromatomtoatomdrivenalongbytheappliedelectricpotential.Amissingelectron,however,isavirtualpositivecharge.Electricalengineerscalltheseholes,andinsomematerials,particularlycertainsemiconductors,currentowisactuallyduetoholes.Currentowalsooccursinnervecellsfoundinyourbrain.Here,neurons"communicate"usingpropagatingvoltagepulsesthatrelyontheowofpositiveionspotassiumandsodiumprimarily,andtosomedegreecalciumacrosstheneuron'souterwall.Thus,currentcancomefrommanysources,andcircuittheorycanbeusedtounderstandhowcurrentowsinreactiontoelectricelds. 1Thiscontentisavailableonlineat.35

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36CHAPTER3.ANALOGSIGNALPROCESSING GenericCircuitElement Figure3.1:Thegenericcircuitelement. Currentowsthroughcircuitelements,suchasthatdepictedinFigure3.1GenericCircuitElement,andthroughconductors,whichweindicatebylinesincircuitdiagrams.Foreverycircuitelementwedeneavoltageandacurrent.Theelementhasav-irelationdenedbytheelement'sphysicalproperties.Indeningthev-irelation,wehavetheconventionthatpositivecurrentowsfrompositivetonegativevoltagedrop.Voltagehasunitsofvolts,andboththeunitandthequantityarenamedforVolta2.Currenthasunitsofamperes,andisnamedfortheFrenchphysicistAmpre3.Voltagesandcurrentsalsocarrypower.AgainusingtheconventionshowninFigure3.1GenericCircuitElementforcircuitelements,theinstantaneouspowerateachmomentoftimeconsumedbytheelementisgivenbytheproductofthevoltageandcurrent.pt=vtitApositivevalueforpowerindicatesthatattimetthecircuitelementisconsumingpower;anegativevaluemeansitisproducingpower.Withvoltageexpressedinvoltsandcurrentinamperes,powerdenedthiswayhasunitsofwatts.Justasinallareasofphysicsandchemistry,poweristherateatwhichenergyisconsumedorproduced.Consequently,energyistheintegralofpower.Et=ZtpdAgain,positiveenergycorrespondstoconsumedenergyandnegativeenergycorrespondstoenergyproduc-tion.Notethatacircuitelementhavingapowerprolethatisbothpositiveandnegativeoversometimeintervalcouldconsumeorproduceenergyaccordingtothesignoftheintegralofpower.Theunitsofenergyarejoulessinceawattequalsjoules/second.Exercise3.1Solutiononp.106.Residentialenergybillstypicallystateahome'senergyusageinkilowatt-hours.Isthisreallyaunitofenergy?Ifso,howmanyjoulesequalsonekilowatt-hour?3.2IdealCircuitElements4Theelementarycircuitelementstheresistor,capacitor,andinductorimposelinearrelationshipsbetweenvoltageandcurrent. 2http://www.bioanalytical.com/info/calendar/97/volta.htm3http://www-groups.dcs.st-and.ac.uk/history/Mathematicians/Ampere.html4Thiscontentisavailableonlineat.

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373.2.1Resistor Resistor Figure3.2:Resistor.v=Ri Theresistorisfarandawaythesimplestcircuitelement.Inaresistor,thevoltageisproportionaltothecurrent,withtheconstantofproportionalityR,knownastheresistance.vt=RitResistancehasunitsofohms,denotedby,namedfortheGermanelectricalscientistGeorgOhm5.Sometimes,thev-irelationfortheresistoriswritteni=Gv,withG,theconductance,equalto1 R.ConductancehasunitsofSiemensS,andisnamedfortheGermanelectronicsindustrialistWernervonSiemens6.Whenresistanceispositive,asitisinmostcases,aresistorconsumespower.Aresistor'sinstantaneouspowerconsumptioncanbewrittenoneoftwoways.pt=Ri2t=1 Rv2tAstheresistanceapproachesinnity,wehavewhatisknownasanopencircuit:Nocurrentowsbutanon-zerovoltagecanappearacrosstheopencircuit.Astheresistancebecomeszero,thevoltagegoestozeroforanon-zerocurrentow.Thissituationcorrespondstoashortcircuit.Asuperconductorphysicallyrealizesashortcircuit.3.2.2Capacitor Capacitor Figure3.3:Capacitor.i=Cd dtvt 5http://www-groups.dcs.st-and.ac.uk/history/Mathematicians/Ohm.html6http://w4.siemens.de/archiv/en/persoenlichkeiten/werner_von_siemens.html

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38CHAPTER3.ANALOGSIGNALPROCESSINGThecapacitorstoreschargeandtherelationshipbetweenthechargestoredandtheresultantvoltageisq=Cv.Theconstantofproportionality,thecapacitance,hasunitsoffaradsF,andisnamedfortheEnglishexperimentalphysicistMichaelFaraday7.Ascurrentistherateofchangeofcharge,thev-irelationcanbeexpressedindierentialorintegralform.it=Cd dtvtorvt=1 CZtid.1Ifthevoltageacrossacapacitorisconstant,thenthecurrentowingintoitequalszero.Inthissituation,thecapacitorisequivalenttoanopencircuit.Thepowerconsumed/producedbyavoltageappliedtoacapacitordependsontheproductofthevoltageanditsderivative.pt=Cvtd dtvtThisresultmeansthatacapacitor'stotalenergyexpenditureuptotimetisconciselygivenbyEt=1 2Cv2tThisexpressionpresumesthefundamentalassumptionofcircuittheory:allvoltagesandcurrentsinanycircuitwerezerointhefardistantpastt=.3.2.3Inductor Inductor Figure3.4:Inductor.v=Ld dtit Theinductorstoresmagneticux,withlargervaluedinductorscapableofstoringmoreux.InductancehasunitsofhenriesH,andisnamedfortheAmericanphysicistJosephHenry8.Thedierentialandintegralformsoftheinductor'sv-irelationarevt=Ld dtitorit=1 LZtvd.2Thepowerconsumed/producedbyaninductordependsontheproductoftheinductorcurrentanditsderivativept=Litd dtitanditstotalenergyexpenditureuptotimetisgivenbyEt=1 2Li2t 7http://www.iee.org.uk/publish/faraday/faraday1.html8http://www.si.edu/archives//ihd/jhp/

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393.2.4Sources Sources a bFigure3.5:Thevoltagesourceontheleftandcurrentsourceontherightarelikeallcircuitelementsinthattheyhaveaparticularrelationshipbetweenthevoltageandcurrentdenedforthem.Forthevoltagesource,v=vsforanycurrenti;forthecurrentsource,i=)]TJ/F58 8.9664 Tf 7.1675 0 Td[(isforanyvoltagev. Sourcesofvoltageandcurrentarealsocircuitelements,buttheyarenotlinearinthestrictsenseoflinearsystems.Forexample,thevoltagesource'sv-irelationisv=vsregardlessofwhatthecurrentmightbe.Asforthecurrentsource,i=)]TJ/F11 9.9626 Tf 7.7487 0 Td[(isregardlessofthevoltage.Anothernameforaconstant-valuedvoltagesourceisabattery,andcanbepurchasedinanysupermarket.Currentsources,ontheotherhand,aremuchhardertoacquire;we'lllearnwhylater.3.3IdealandReal-WorldCircuitElements9Sourceandlinearcircuitelementsareidealcircuitelements.Onecentralnotionofcircuittheoryiscombiningtheidealelementstodescribehowphysicalelementsoperateintherealworld.Forexample,the1kresistoryoucanholdinyourhandisnotexactlyanideal1kresistor.Firstofall,physicaldevicesaremanufacturedtoclosetolerancesthetighterthetolerance,themoremoneyyoupay,butneverhaveexactlytheiradvertisedvalues.Thefourthbandonresistorsspeciestheirtolerance;10%iscommon.Morepertinenttothecurrentdiscussionisanotherdeviationfromtheideal:Ifasinusoidalvoltageisplacedacrossaphysicalresistor,thecurrentwillnotbeexactlyproportionaltoitasfrequencybecomeshigh,sayabove1MHz.Atveryhighfrequencies,thewaytheresistorisconstructedintroducesinductanceandcapacitanceeects.Thus,thesmartengineermustbeawareofthefrequencyrangesoverwhichhisidealmodelsmatchrealitywell.Ontheotherhand,physicalcircuitelementscanbereadilyfoundthatwellapproximatetheideal,buttheywillalwaysdeviatefromtheidealinsomeway.Forexample,aashlightbattery,likeaC-cell,roughlycorrespondstoa1.5Vvoltagesource.However,itceasestobemodeledbyavoltagesourcecapableofsupplyinganycurrentthat'swhatidealonescando!whentheresistanceofthelightbulbistoosmall.3.4ElectricCircuitsandInterconnectionLaws10Acircuitconnectscircuitelementstogetherinaspeciccongurationdesignedtotransformthesourcesignaloriginatingfromavoltageorcurrentsourceintoanothersignaltheoutputthatcorrespondstothecurrentorvoltagedenedforaparticularcircuitelement.AsimpleresistivecircuitisshowninFigure3.6. 9Thiscontentisavailableonlineat.10Thiscontentisavailableonlineat.

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40CHAPTER3.ANALOGSIGNALPROCESSINGThiscircuitistheelectricalembodimentofasystemhavingitsinputprovidedbyasourcesystemproducingvint. a b cFigure3.6:Thecircuitshowninthetoptwoguresisperhapsthesimplestcircuitthatperformsasignalprocessingfunction.Onthebottomistheblockdiagramthatcorrespondstothecircuit.TheinputisprovidedbythevoltagesourcevinandtheoutputisthevoltagevoutacrosstheresistorlabelR2.Asshowninthemiddle,weanalyzethecircuitunderstandwhatitaccomplishesbydeningcurrentsandvoltagesforallcircuitelements,andthensolvingthecircuitandelementequations. Tounderstandwhatthiscircuitaccomplishes,wewanttodeterminethevoltageacrosstheresistorlabeledbyitsvalueR2.Recastingthisproblemmathematically,weneedtosolvesomesetofequationssothatwerelatetheoutputvoltagevouttothesourcevoltage.Itwouldbesimplealittletoosimpleatthispointifwecouldinstantlywritedowntheoneequationthatrelatesthesetwovoltages.Untilwehavemoreknowledgeabouthowcircuitswork,wemustwriteasetofequationsthatallowustondallthevoltagesandcurrentsthatcanbedenedforeverycircuitelement.Becausewehaveathree-elementcircuit,wehaveatotalofsixvoltagesandcurrentsthatmustbeeitherspeciedordetermined.Youcandenethedirectionsforcurrentowandpositivevoltagedropanywayyoulike.Whentwopeoplesolveacircuittheirownways,thesignsoftheirvariablesmaynotagree,butcurrentowandvoltagedropvaluesforeachelementwillagree.DorecallindeningyourvoltageandcurrentvariablesSection3.2thatthev-irelationsfortheelementspresumethatpositivecurrentowisinthesamedirectionaspositivevoltagedrop.Onceyoudenevoltagesandcurrents,weneedsixnonredundantequationstosolveforthesixunknownvoltagesandcurrents.Byspecifyingthesource,wehaveone;thisamountstoprovidingthesource'sv-irelation.Thev-irelationsfortheresistorsgiveustwomore.Weareonlyhalfwaythere;wheredowegettheotherthree

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41equationsweneed?Whatweneedtosolveeverycircuitproblemaremathematicalstatementsthatexpresshowthecircuitelementsareinterconnected.Saidanotherway,weneedthelawsthatgoverntheelectricalconnectionofcircuitelements.Firstofall,theplaceswherecircuitelementsattachtoeachotherarecallednodes.TwonodesareexplicitlyindicatedinFigure3.6;athirdisatthebottomwherethevoltagesourceandresistorR2areconnected.Electricalengineerstendtodrawcircuitdiagramsschematicsinarectilinearfashion.Thusthelonglineconnectingthebottomofthevoltagesourcewiththebottomoftheresistorisintendedtomakethediagramlookpretty.Thislinesimplymeansthatthetwoelementsareconnectedtogether.Kircho'sLaws,oneforvoltageSection3.4.2:Kircho'sVoltageLawKVLandoneforcurrentSection3.4.1:Kircho'sCurrentLaw,determinewhataconnectionamongcircuitelementsmeans.Theselawscanhelpusanalyzethiscircuit.3.4.1Kircho'sCurrentLawAteverynode,thesumofallcurrentsenteringanodemustequalzero.Whatthislawmeansphysicallyisthatchargecannotaccumulateinanode;whatgoesinmustcomeout.Intheexample,Figure3.6,belowwehaveathree-nodecircuitandthushavethreeKCLequations.)]TJ/F11 9.9626 Tf 7.7487 0 Td[(i)]TJ/F11 9.9626 Tf 9.9626 0 Td[(i1=0i1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(i2=0i+i2=0Notethatthecurrententeringanodeisthenegativeofthecurrentleavingthenode.GivenanytwooftheseKCLequations,wecanndtheotherbyaddingorsubtractingthem.Thus,oneofthemisredundantand,inmathematicalterms,wecandiscardanyoneofthem.Theconventionistodiscardtheequationfortheunlabelednodeatthebottomofthecircuit. a bFigure3.7:Thecircuitshownisperhapsthesimplestcircuitthatperformsasignalprocessingfunction.TheinputisprovidedbythevoltagesourcevinandtheoutputisthevoltagevoutacrosstheresistorlabelledR2. Exercise3.2Solutiononp.106.InwritingKCLequations,youwillndthatinann-nodecircuit,exactlyoneofthemisalwaysredundant.Canyousketchaproofofwhythismightbetrue?Hint:Ithastodowiththefactthatchargewon'taccumulateinoneplaceonitsown.

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42CHAPTER3.ANALOGSIGNALPROCESSING3.4.2Kircho'sVoltageLawKVLThevoltagelawsaysthatthesumofvoltagesaroundeveryclosedloopinthecircuitmustequalzero.Aclosedloophastheobviousdenition:Startingatanode,traceapaththroughthecircuitthatreturnsyoutotheoriginnode.KVLexpressesthefactthatelectriceldsareconservative:Thetotalworkperformedinmovingatestchargearoundaclosedpathiszero.TheKVLequationforourcircuitisv1+v2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(v=0InwritingKVLequations,wefollowtheconventionthatanelement'svoltageenterswithaplussignwhentraversingtheclosedpath,wegofromthepositivetothenegativeofthevoltage'sdenition. a bFigure3.8:Thecircuitshownisperhapsthesimplestcircuitthatperformsasignalprocessingfunction.TheinputisprovidedbythevoltagesourcevinandtheoutputisthevoltagevoutacrosstheresistorlabelledR2. FortheexamplecircuitFigure3.8,wehavethreev-irelations,twoKCLequations,andoneKVLequationforsolvingforthecircuit'ssixvoltagesandcurrents.v-i:v=vinv1=R1i1vout=R2ioutKCL:)]TJ/F11 9.9626 Tf 7.7487 0 Td[(i)]TJ/F11 9.9626 Tf 9.9626 0 Td[(i1=0i1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(iout=0KVL:)]TJ/F11 9.9626 Tf 7.7487 0 Td[(v+v1+vout=0Wehaveexactlytherightnumberofequations!Eventually,wewilldiscovershortcutsforsolvingcircuitproblems;fornow,wewanttoeliminateallthevariablesbutvoutanddeterminehowitdependsonvinandonresistorvalues.TheKVLequationcanberewrittenasvin=v1+vout.Substitutingintoittheresistor'sv-irelation,wehavevin=R1i1+R2iout.Yes,wetemporarilyeliminatethequantityweseek.Thoughnotobvious,itisthesimplestwaytosolvetheequations.OneoftheKCLequationssaysi1=iout,whichmeansthatvin=R1iout+R2iout=R1+R2iout.Solvingforthecurrentintheoutputresistor,wehaveiout=vin R1+R2.Wehavenowsolvedthecircuit:Wehaveexpressedonevoltageorcurrentintermsofsourcesandcircuit-elementvalues.Tondanyothercircuitquantities,wecanbacksubstitutethisanswerintoouroriginalequationsoroneswedevelopedalongtheway.Usingthev-irelationfortheoutputresistor,weobtainthequantityweseek.vout=R2 R1+R2vin

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43Exercise3.3Solutiononp.106.ReferringbacktoFigure3.6,acircuitshouldservesomeusefulpurpose.Whatkindofsystemdoesourcircuitrealizeand,intermsofelementvalues,whatarethesystem'sparameters?3.5PowerDissipationinResistorCircuits11Wecanndvoltagesandcurrentsinsimplecircuitscontainingresistorsandvoltageorcurrentsources.WeshouldexaminewhetherthesecircuitsvariablesobeytheConservationofPowerprinciple:sinceacircuitisaclosedsystem,itshouldnotdissipateorcreateenergy.Forthemoment,ourapproachistoinvestigaterstaresistorcircuit'spowerconsumption/creation.Later,wewillprovethatbecauseofKVLandKCLallcircuitsconservepower.Asdenedonp.36,theinstantaneouspowerconsumed/createdbyeverycircuitelementequalstheproductofitsvoltageandcurrent.Thetotalpowerconsumed/createdbyacircuitequalsthesumofeachelement'spower.P=XkvkikRecallthateachelement'scurrentandvoltagemustobeytheconventionthatpositivecurrentisdenedtoenterthepositive-voltageterminal.Withthisconvention,apositivevalueofvkikcorrespondstoconsumedpower,anegativevaluetocreatedpower.BecausethetotalpowerinacircuitmustbezeroP=0,somecircuitelementsmustcreatepowerwhileothersconsumeit.ConsiderthesimpleseriescircuitshouldinSection3.4.Inperformingourcalculations,wedenedthecurrentiouttoowthroughthepositive-voltageterminalsofbothresistorsandfoundittoequaliout=vin R1+R2.ThevoltageacrosstheresistorR2istheoutputvoltageandwefoundittoequalvout=R2 R1+R2vin.Consequently,calculatingthepowerforthisresistoryieldsP2=R2 R1+R22vin2Consequently,thisresistordissipatespowerbecauseP2ispositive.Thisresultshouldnotbesurprisingsinceweshowedp.37thatthepowerconsumedbyanyresistorequalseitherofthefollowing.v2 Rori2R.3Sinceresistorsarepositive-valued,resistorsalwaysdissipatepower.Butwheredoesaresistor'spowergo?ByConversationofPower,thedissipatedpowermustbeabsorbedsomewhere.Theanswerisnotdirectlypredictedbycircuittheory,butisbyphysics.Currentowingthrougharesistormakesithot;itspowerisdissipatedbyheat.Note:Aphysicalwirehasaresistanceandhencedissipatespoweritgetswarmjustlikearesistorinacircuit.Infact,theresistanceofawireoflengthLandcross-sectionalareaAisgivenbyR=L AThequantityisknownastheresistivityandpresentstheresistanceofaunit-lengthofmaterialconstitutingthewire.Mostmaterialshaveapositivevaluefor,whichmeansthelongerthewire,thegreatertheresistanceandthusthepowerdissipated.Thethickerthewire,thesmallertheresistance.Superconductorshavenoresistanceandhencedonotdissipatepower.Ifaroom-temperaturesuperconductorcouldbefound,electricpowercouldbesentthroughpowerlineswithoutloss! 11Thiscontentisavailableonlineat.

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44CHAPTER3.ANALOGSIGNALPROCESSINGExercise3.4Solutiononp.106.Calculatethepowerconsumed/createdbytheresistorR1inoursimplecircuitexample.Weconcludethatbothresistorsinourexamplecircuitconsumepower,whichpointstothevoltagesourceastheproducerofpower.Thecurrentowingintothesource'spositiveterminalis)]TJ/F11 9.9626 Tf 7.7487 0 Td[(iout.Consequently,thepowercalculationforthesourceyields)]TJ/F8 9.9626 Tf 9.4092 0 Td[(viniout=)]TJ/F1 9.9626 Tf 9.4091 14.0474 Td[(1 R1+R2vin2Weconcludethatthesourceprovidesthepowerconsumedbytheresistors,nomore,noless.Exercise3.5Solutiononp.106.Conrmthatthesourceproducesexactlythetotalpowerconsumedbybothresistors.Thisresultisquitegeneral:sourcesproducepowerandthecircuitelements,especiallyresistors,consumeit.Butwheredosourcesgettheirpower?Again,circuittheorydoesnotmodelhowsourcesareconstructed,butthetheorydecreesthatallsourcesmustbeprovidedenergytowork.3.6SeriesandParallelCircuits12 a bFigure3.9:Thecircuitshownisperhapsthesimplestcircuitthatperformsasignalprocessingfunction.TheinputisprovidedbythevoltagesourcevinandtheoutputisthevoltagevoutacrosstheresistorlabelledR2. TheresultsshowninothermodulescircuitelementsSection3.4,KVLandKCLSection3.4,intercon-nectionlawsSection3.4withregardtothiscircuitFigure3.9,andthevaluesofothercurrentsandvoltagesinthiscircuitaswell,haveprofoundimplications.Resistorsconnectedinsuchawaythatcurrentfromonemustowonlyintoanothercurrentsinallresistorsconnectedthiswayhavethesamemagnitudearesaidtobeconnectedinseries.Forthetwoseries-connectedresistorsintheexample,thevoltageacrossoneresistorequalstheratioofthatresistor'svalueandthesumofresistancestimesthevoltageacrosstheseriescombination.Thisconceptissopervasiveithasaname:voltagedivider.Theinput-outputrelationshipforthissystem,foundinthisparticularcasebyvoltagedivider,takestheformofaratiooftheoutputvoltagetotheinputvoltage.vout vin=R2 R1+R2 12Thiscontentisavailableonlineat.

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45Inthisway,weexpresshowthecomponentsusedtobuildthesystemaecttheinput-outputrelationship.Becausethisanalysiswasmadewithidealcircuitelements,wemightexpectthisrelationtobreakdowniftheinputamplitudeistoohighWillthecircuitsurviveiftheinputchangesfrom1volttoonemillionvolts?orifthesource'sfrequencybecomestoohigh.Inanycase,thisimportantwayofexpressinginput-outputrelationshipsasaratioofoutputtoinputpervadescircuitandsystemtheory.Thecurrenti1isthecurrentowingoutofthevoltagesource.Becauseitequalsi2,wehavethatvin i1=R1+R2:Resistorsinseries:Theseriescombinationoftworesistorsacts,asfarasthevoltagesourceisconcerned,asasingleresistorhavingavalueequaltothesumofthetworesistances.Thisresultistherstofseveralequivalentcircuitideas:Inmanycases,acomplicatedcircuitwhenviewedfromitsterminalsthetwoplacestowhichyoumightattachasourceappearstobeasinglecircuitelementatbestorasimplecombinationofelementsatworst.Thus,theequivalentcircuitforaseriescombinationofresistorsisasingleresistorhavingaresistanceequaltothesumofitscomponentresistances. Figure3.10:Theresistorontherightisequivalenttothetworesistorsontheleftandhasaresistanceequaltothesumoftheresistancesoftheothertworesistors. Thus,thecircuitthevoltagesource"feels"throughthecurrentdrawnfromitisasingleresistorhavingresistanceR1+R2.Notethatinmakingthisequivalentcircuit,theoutputvoltagecannolongerbedened:TheoutputresistorlabeledR2nolongerappears.Thus,thisequivalenceismadestrictlyfromthevoltagesource'sviewpoint. Figure3.11:Asimpleparallelcircuit. OneinterestingsimplecircuitFigure3.11hastworesistorsconnectedside-by-side,whatwewilltermaparallelconnection,ratherthaninseries.Here,applyingKVLrevealsthatallthevoltagesareidentical:

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46CHAPTER3.ANALOGSIGNALPROCESSINGv1=vandv2=v.Thisresulttypiesparallelconnections.TowritetheKCLequation,notethatthetopnodeconsistsoftheentireupperinterconnectionsection.TheKCLequationisiin)]TJ/F11 9.9626 Tf 9.9794 0 Td[(i1)]TJ/F11 9.9626 Tf 9.9794 0 Td[(i2=0.Usingthev-irelations,wendthatiout=R1 R1+R2iinExercise3.6Solutiononp.106.SupposethatyoureplacedthecurrentsourceinFigure3.11byavoltagesource.Howwouldioutberelatedtothesourcevoltage?Basedonthisresult,whatpurposedoesthisrevisedcircuithave?Thiscircuithighlightssomeimportantpropertiesofparallelcircuits.Youcaneasilyshowthatthepar-allelcombinationofR1andR2hasthev-irelationofaresistorhavingresistance1 R1+1 R2)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1=R1R2 R1+R2.AshorthandnotationforthisquantityisR1kR2.AsthereciprocalofresistanceisconductanceSec-tion3.2.1:Resistor,wecansaythatforaparallelcombinationofresistors,theequivalentconductanceisthesumoftheconductances. Figure3.12 Similartovoltagedividerp.44forseriesresistances,wehavecurrentdividerforparallelresistances.Thecurrentthrougharesistorinparallelwithanotheristheratiooftheconductanceofthersttothesumoftheconductances.Thus,forthedepictedcircuit,i2=G2 G1+G2i.Expressedintermsofresistances,currentdividertakestheformoftheresistanceoftheotherresistordividedbythesumofresistances:i2=R1 R1+R2i. Figure3.13

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47 Figure3.14:ThesimpleattenuatorcircuitFigure3.9isattachedtoanoscilloscope'sinput.Theinput-outputrelationfortheabovecircuitwithoutaloadis:vout=R2 R1+R2vin. Supposewewanttopasstheoutputsignalintoavoltagemeasurementdevice,suchasanoscilloscopeoravoltmeter.Insystem-theoryterms,wewanttopassourcircuit'soutputtoasink.Formostapplications,wecanrepresentthesemeasurementdevicesasaresistor,withthecurrentpassingthroughitdrivingthemeasurementdevicethroughsometypeofdisplay.Incircuits,asinkiscalledaload;thus,wedescribeasystem-theoreticsinkasaloadresistanceRL.Thus,wehaveacompletesystembuiltfromacascadeofthreesystems:asource,asignalprocessingsystemsimpleasitis,andasink.Wemustanalyzeafreshhowthisrevisedcircuit,showninFigure3.14,works.Ratherthandeningeightvariablesandsolvingforthecurrentintheloadresistor,let'stakeahintfromotheranalysisseriesrulesp.44,parallelrulesp.46.ResistorsR2andRLareinaparallelconguration:Thevoltagesacrosseachresistorarethesamewhilethecurrentsarenot.Becausethevoltagesarethesame,wecanndthecurrentthrougheachfromtheirv-irelations:i2=vout R2andiL=vout RL.Consideringthenodewhereallthreeresistorsjoin,KCLsaysthatthesumofthethreecurrentsmustequalzero.Saidanotherway,thecurrententeringthenodethroughR1mustequalthesumoftheothertwocurrentsleavingthenode.Therefore,i1=i2+iL,whichmeansthati1=vout1 R2+1 R1.LetReqdenotetheequivalentresistanceoftheparallelcombinationofR2andRL.UsingR1'sv-irelation,thevoltageacrossitisv1=R1vout Req.TheKVLequationwrittenaroundtheleftmostloophasvin=v1+vout;substitutingforv1,wendvin=voutR1 Req+1orvout vin=Req R1+ReqThus,wehavetheinput-outputrelationshipforourentiresystemhavingtheformofvoltagedivider,butitdoesnotequaltheinput-outputrelationofthecircuitwithoutthevoltagemeasurementdevice.Wecannotmeasurevoltagesreliablyunlessthemeasurementdevicehaslittleeectonwhatwearetryingtomeasure.Weshouldlookmorecarefullytodetermineifanyvaluesfortheloadresistancewouldlessenitsimpactonthecircuit.Comparingtheinput-outputrelationsbeforeandafter,whatweneedisReqR2.AsReq=1 R2+1 RL)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1,theapproximationwouldapplyif1 R21 RLorR2RL.Thisistheconditionweseek:Voltagemeasurement:Voltagemeasurementdevicesmusthavelargeresistancescomparedwiththatoftheresistoracrosswhichthevoltageistobemeasured.Exercise3.7Solutiononp.106.Let'sbemoreprecise:Howmuchlargerwouldaloadresistanceneedtobetoaecttheinput-outputrelationbylessthan10%?bylessthan1%?

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48CHAPTER3.ANALOGSIGNALPROCESSINGExample3.1 Figure3.15Wewanttondthetotalresistanceoftheexamplecircuit.Toapplytheseriesandparallelcombinationrules,itisbesttorstdeterminethecircuit'sstructure:Whatisinserieswithwhatandwhatisinparallelwithwhatatbothsmall-andlarge-scaleviews.WehaveR2inparallelwithR3;thiscombinationisinserieswithR4.ThisseriescombinationisinparallelwithR1.Notethatindeterminingthisstructure,westartedawayfromtheterminals,andworkedtowardthem.Inmostcases,thisapproachworkswell;tryitrst.Thetotalresistanceexpressionmimicsthestructure:RT=R1kR2kR3+R4RT=R1R2R3+R1R2R4+R1R3R4 R1R2+R1R3+R2R3+R2R4+R3R4Suchcomplicatedexpressionstypifycircuit"simplications."Asimplecheckforaccuracyistheunits:Eachcomponentofthenumeratorshouldhavethesameunitshere3aswellasinthedenominator2.Theentireexpressionistohaveunitsofresistance;thus,theratioofthenumerator'sanddenominator'sunitsshouldbeohms.Checkingunitsdoesnotguaranteeaccuracy,butcancatchmanyerrors.Anothervaluablelessonemergesfromthisexampleconcerningthedierencebetweencascadingsystemsandcascadingcircuits.Insystemtheory,systemscanbecascadedwithoutchangingtheinput-outputrelationofintermediatesystems.Incascadingcircuits,thisidealisrarelytrueunlessthecircuitsaresodesigned.Designisinthehandsoftheengineer;heorshemustrecognizewhathavecometobeknownasloadingeects.Inoursimplecircuit,youmightthinkthatmakingtheresistanceRLlargeenoughwoulddothetrick.BecausetheresistorsR1andR2canhavevirtuallyanyvalue,youcannevermaketheresistanceofyourvoltagemeasurementdevicebigenough.Saidanotherway,acircuitcannotbedesignedinisolationthatwillworkincascadewithallothercircuits.Electricalengineersdealwiththissituationthroughthenotionofspecications:Underwhatconditionswillthecircuitperformasdesigned?Thus,youwillndthatoscilloscopesandvoltmetershavetheirinternalresistancesclearlystated,enablingyoutodeterminewhetherthevoltageyoumeasurecloselyequalswhatwaspresentbeforetheywereattachedtoyourcircuit.Furthermore,sinceourresistorcircuitfunctionsasanattenuator,withtheattenuationafancywordforgainslessthanonedependingonlyontheratioofthetworesistorvaluesR2 R1+R2=1+R1 R2)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1,wecanselectanyvaluesforthetworesistanceswewanttoachievethedesiredattenuation.Thedesignerofthiscircuitmustthusspecifynotonlywhattheattenuationis,butalsotheresistancevaluesemployedsothatintegratorspeoplewhoputsystemstogetherfromcomponentsystemscancombinesystemstogetherandhaveachanceofthecombinationworking.Figure3.16seriesandparallelcombinationrulessummarizestheseriesandparallelcombinationresults.Theseresultsareeasytorememberandveryuseful.Keepinmindthatforseriescombinations,voltageand

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49resistancearethekeyquantities,whileforparallelcombinationscurrentandconductancearemoreimportant.Inseriescombinations,thecurrentsthrougheachelementarethesame;inparallelones,thevoltagesarethesame. seriesandparallelcombinationrules aseriescombinationrule bparallelcombinationruleFigure3.16:Seriesandparallelcombinationrules.aRT=PNn=1Rnvn=Rn RTvbGT=PNn=1Gnin=Gn GTi Exercise3.8Solutiononp.106.Contrastaseriescombinationofresistorswithaparallelone.Whichvariablevoltageorcurrentisthesameforeachandwhichdiers?Whataretheequivalentresistances?Whenresistorsareplacedinseries,istheequivalentresistancebigger,inbetween,orsmallerthanthecomponentresistances?Whatisthisrelationshipforaparallelcombination?3.7EquivalentCircuits:ResistorsandSources13Wehavefoundthatthewaytothinkaboutcircuitsistolocateandgroupparallelandseriesresistorcombinations.Thoseresistorsnotinvolvedwithvariablesofinterestcanbecollapsedintoasingleresistance.Thisresultisknownasanequivalentcircuit:fromtheviewpointofapairofterminals,agroupofresistorsfunctionsasasingleresistor,theresistanceofwhichcanusuallybefoundbyapplyingtheparallelandseriesrules.Thisresultgeneralizestoincludesourcesinaveryinterestingandusefulway.Let'sconsideroursimpleattenuatorcircuitshowninthegureFigure3.17fromtheviewpointoftheoutputterminals.Wewanttondthev-irelationfortheoutputterminalpair,andthenndtheequivalentcircuitfortheboxedcircuit.Toperformthiscalculation,usethecircuitlawsandelementrelations,butdonotattachanythingtotheoutputterminals.Weseektherelationbetweenvandithatdescribesthekindofelementthatlurkswithinthedashedbox.Theresultisv=R1kR2i+R2 R1+R2vin.4 13Thiscontentisavailableonlineat.

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50CHAPTER3.ANALOGSIGNALPROCESSING Figure3.17 Ifthesourcewerezero,itcouldbereplacedbyashortcircuit,whichwouldconrmthatthecircuitdoesindeedfunctionasaparallelcombinationofresistors.However,thesource'spresencemeansthatthecircuitisnotwellmodeledasaresistor. Figure3.18:TheThveninequivalentcircuit. IfweconsiderthesimplecircuitofFigure3.18,wendithasthev-irelationatitsterminalsofv=Reqi+veq.5Comparingthetwov-irelations,wendthattheyhavethesameform.InthiscasetheThveninequivalentresistanceisReq=R1kR2andtheThveninequivalentsourcehasvoltageveq=R2 R1+R2vin.Thus,fromviewpointoftheterminals,youcannotdistinguishthetwocircuits.Becausetheequivalentcircuithasfewerelements,itiseasiertoanalyzeandunderstandthananyotheralternative.Foranycircuitcontainingresistorsandsources,thev-irelationwillbeoftheformv=Reqi+veq.6andtheThveninequivalentcircuitforanysuchcircuitisthatofFigure3.18.Thisequivalenceappliesnomatterhowmanysourcesorresistorsmaybepresentinthecircuit.IntheexampleExample3.2below,weknowthecircuit'sconstructionandelementvalues,andderivetheequivalentsourceandresistance.BecauseThvenin'stheoremappliesingeneral,weshouldbeabletomakemeasurementsorcalculationsonlyfromtheterminalstodeterminetheequivalentcircuit.

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51Tobemorespecic,considertheequivalentcircuitofthisgureFigure3.18.Lettheterminalsbeopen-circuited,whichhastheeectofsettingthecurrentitozero.Becausenocurrentowsthroughtheresistor,thevoltageacrossitiszeroremember,Ohm'sLawsaysthatv=Ri.Consequently,byapplyingKVLwehavethattheso-calledopen-circuitvoltagevocequalstheThveninequivalentvoltage.Nowconsiderthesituationwhenwesettheterminalvoltagetozeroshort-circuititandmeasuretheresultingcurrent.Referringtotheequivalentcircuit,thesourcevoltagenowappearsentirelyacrosstheresistor,leavingtheshort-circuitcurrenttobeisc=)]TJ/F1 9.9626 Tf 9.4091 11.0586 Td[(veq Req.Fromthisproperty,wecandeterminetheequivalentresistance.veq=voc.7Req=)]TJ/F1 9.9626 Tf 9.4091 14.0475 Td[(voc isc.8Exercise3.9Solutiononp.106.Usetheopen/short-circuitapproachtoderivetheThveninequivalentofthecircuitshowninFigure3.19. Figure3.19Example3.2 Figure3.20ForthecircuitdepictedinFigure3.20,let'sderiveitsThveninequivalenttwodierentways.Startingwiththeopen/short-circuitapproach,let'srstndtheopen-circuitvoltagevoc.WehaveacurrentdividerrelationshipasR1isinparallelwiththeseriescombinationofR2andR3.Thus,voc=iinR3R1 R1+R2+R3.Whenweshort-circuittheterminals,novoltageappearsacrossR3,andthusnocurrentowsthroughit.Inshort,R3doesnotaecttheshort-circuitcurrent,andcanbeeliminated.Weagainhaveacurrentdividerrelationship:isc=)]TJ/F1 9.9626 Tf 9.4091 11.0586 Td[(iinR1 R1+R2.Thus,theThveninequivalentresistanceisR3R1+R2 R1+R2+R3.

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52CHAPTER3.ANALOGSIGNALPROCESSINGToverify,let'sndtheequivalentresistancebyreachinginsidethecircuitandsettingthecurrentsourcetozero.Becausethecurrentisnowzero,wecanreplacethecurrentsourcebyanopencircuit.Fromtheviewpointoftheterminals,resistorR3isnowinparallelwiththeseriescombinationofR1andR2.Thus,Req=R3kR1+R2,andweobtainthesameresult. Figure3.21:Allcircuitscontainingsourcesandresistorscanbedescribedbysimplerequivalentcircuits.Choosingtheonetousedependsontheapplication,notonwhatisactuallyinsidethecircuit. Asyoumightexpect,equivalentcircuitscomeintwoforms:thevoltage-sourceorientedThveninequiv-alent14andthecurrent-sourceorientedMayer-NortonequivalentFigure3.21.Toderivethelatter,thev-irelationfortheThveninequivalentcanbewrittenasv=Reqi+veq.9ori=v Req)]TJ/F11 9.9626 Tf 9.9626 0 Td[(ieq.10whereieq=veq ReqistheMayer-Nortonequivalentsource.TheMayer-NortonequivalentshowninFigure3.21beeasilyshowntohavethisv-irelation.Notethatbothvariationshavethesameequivalentresistance.Theshort-circuitcurrentequalsthenegativeoftheMayer-Nortonequivalentsource.Exercise3.10Solutiononp.106.FindtheMayer-Nortonequivalentcircuitforthecircuitbelow. 14"FindingThveninEquivalentCircuits"

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53 Figure3.22Equivalentcircuitscanbeusedintwobasicways.TherstistosimplifytheanalysisofacomplicatedcircuitbyrealizingtheanyportionofacircuitcanbedescribedbyeitheraThveninorMayer-Nortonequivalent.WhichoneisuseddependsonwhetherwhatisattachedtotheterminalsisaseriescongurationmakingtheThveninequivalentthebestoraparallelonemakingMayer-Nortonthebest.Anotherapplicationismodeling.Whenwebuyaashlightbattery,eitherequivalentcircuitcanaccu-ratelydescribeit.Thesemodelshelpusunderstandthelimitationsofabattery.Sincebatteriesarelabeledwithavoltagespecication,theyshouldserveasvoltagesourcesandtheThveninequivalentservesasthenaturalchoice.IfaloadresistanceRLisplacedacrossitsterminals,thevoltageoutputcanbefoundusingvoltagedivider:v=veqRL RL+Req.Ifwehavealoadresistancemuchlargerthanthebattery'sequivalentresistance,then,toagoodapproximation,thebatterydoesserveasavoltagesource.Iftheloadresistanceismuchsmaller,wecertainlydon'thaveavoltagesourcetheoutputvoltagedependsdirectlyontheloadresistance.ConsidernowtheMayer-Nortonequivalent;thecurrentthroughtheloadresistanceisgivenbycurrentdivider,andequalsi=)]TJ/F1 9.9626 Tf 9.4091 11.0587 Td[(ieqReq RL+Req.Foracurrentthatdoesnotvarywiththeloadresistance,thisresistanceshouldbemuchsmallerthantheequivalentresistance.Iftheloadresistanceiscomparabletotheequivalentresistance,thebatteryservesneitherasavoltagesourceoracurrentcourse.Thus,whenyoubuyabattery,yougetavoltagesourceifitsequivalentresistanceismuchsmallerthantheequivalentresistanceofthecircuittowhichyouattachit.Ontheotherhand,ifyouattachittoacircuithavingasmallequivalentresistance,youboughtacurrentsource.LonCharlesThvenin:HewasanengineerwithFrance'sPostes,TlgrapheetTlphone.In1883,hepublishedtwice!aproofofwhatisnowcalledtheThveninequivalentwhiledevelopingwaysofteachingelectricalengineeringconceptsatthecolePolytechnique.HedidnotrealizethatthesameresulthadbeenpublishedbyHermannHelmholtz15,therenownednineteenthcenturyphysicist,thiryyearsearlier.HansFerdinandMayer:Afterearninghisdoctorateinphysicsin1920,heturnedtocom-municationsengineeringwhenhejoinedSiemens&Halskein1922.In1926,hepublishedinaGermantechnicaljournaltheMayer-Nortonequivalent.Duringhisinterestingcareer,herosetoleadSiemen'sCentralLaboratoryin1936,surruptiouslyleakedtotheBritishallheknewofGermanwarfarecapabilitiesamonthaftertheNazisinvadedPoland,wasarrestedbytheGestapoin1943forlisteningtoBBCradiobroadcasts,spenttwoyearsinNaziconcentrationcamps,andwenttotheUnitedStatesforfouryearsworkingfortheAirForceandCornellUniversitybeforereturningtoSiemensin1950.HerosetoapositiononSiemen'sBoardofDirectorsbeforeretiring.EdwardL.Norton:EdwardNorton16wasanelectricalengineerwhoworkedatBellLaboratoryfromitsinceptionin1922.InthesamemonthwhenMayer'spaperappeared,Nortonwroteinaninternaltechnicalmemorandumaparagraphdescribingthecurrent-sourceequivalent.NoevidencesuggestsNortonknewofMayer'spublication. 15http://www-gap.dcs.st-and.ac.uk/history/Mathematicians/Helmholtz.html16http://www.ece.rice.edu/dhj/norton

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54CHAPTER3.ANALOGSIGNALPROCESSING3.8CircuitswithCapacitorsandInductors17 Figure3.23:AsimpleRCcircuit. Let'sconsideracircuithavingsomethingotherthanresistorsandsources.BecauseofKVL,weknowthatvin=vR+vout.Thecurrentthroughthecapacitorisgivenbyi=Cd dtvout,andthiscurrentequalsthatpassingthroughtheresistor.SubstitutingvR=RiintotheKVLequationandusingthev-irelationforthecapacitor,wearriveatRCd dtvout+vout=vin.11Theinput-outputrelationforcircuitsinvolvingenergystorageelementstakestheformofanordinarydier-entialequation,whichwemustsolvetodeterminewhattheoutputvoltageisforagiveninput.Incontrasttoresistivecircuits,whereweobtainanexplicitinput-outputrelation,wenowhaveanimplicitrelationthatrequiresmoreworktoobtainanswers.Atthispoint,wecouldlearnhowtosolvedierentialequations.Noterstthatevenndingthedierentialequationrelatinganoutputvariabletoasourceisoftenverytedious.Theparallelandseriescombinationrulesthatapplytoresistorsdon'tdirectlyapplywhencapacitorsandinductorsoccur.Wewouldhavetoslogourwaythroughthecircuitequations,simplifyingthemuntilwenallyfoundtheequationthatrelatedthesourcestotheoutput.Attheturnofthetwentiethcentury,amethodwasdiscoveredthatnotonlymadendingthedierentialequationeasy,butalsosimpliedthesolutionprocessinthemostcommonsituation.Althoughnotoriginalwithhim,CharlesSteinmetz18presentedthekeypaperdescribingtheimpedanceapproachin1893.Itallowscircuitscontainingcapacitorsandinductorstobesolvedwiththesamemethodswehavelearnedtosolvedresistorcircuits.Touseimpedances,wemustmastercomplexnumbers.Thoughthearithmeticofcomplexnumbersismathematicallymorecomplicatedthanwithrealnumbers,theincreasedinsightintocircuitbehaviorandtheeasewithwhichcircuitsaresolvedwithimpedancesiswellworththediversion.Butmoreimportantly,theimpedanceconceptiscentraltoengineeringandphysics,havingareachfarbeyondjustcircuits.3.9TheImpedanceConcept19Ratherthansolvingthedierentialequationthatarisesincircuitscontainingcapacitorsandinductors,let'spretendthatallsourcesinthecircuitarecomplexexponentialshavingthesamefrequency.Althoughthispretensecanonlybemathematicallytrue,thisctionwillgreatlyeasesolvingthecircuitnomatterwhatthesourcereallyis. 17Thiscontentisavailableonlineat.18http://www.invent.org/hall_of_fame/139.html19Thiscontentisavailableonlineat.

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55 SimpleCircuit Figure3.24:AsimpleRCcircuit. Impedance a b cFigure3.25:aResistor:ZR=RbCapacitor:ZC=1 j2fCcInductor:ZL=j2fL FortheaboveexampleRCcircuitFigure3.24SimpleCircuit,letvin=Vinej2ft.ThecomplexamplitudeVindeterminesthesizeofthesourceanditsphase.Thecriticalconsequenceofassumingthatsourceshavethisformisthatallvoltagesandcurrentsinthecircuitarealsocomplexexponentials,havingamplitudesgovernedbyKVL,KCL,andthev-irelationsandthesamefrequencyasthesource.Toappreciatewhythisshouldbetrue,let'sinvestigatehoweachcircuitelementbehaveswheneitherthevoltageorcurrentisacomplexexponential.Fortheresistor,v=Ri.Whenv=Vej2ft;theni=V Rej2ft.Thus,iftheresistor'svoltageisacomplexexponential,soisthecurrent,withanamplitudeI=V Rdeterminedbytheresistor'sv-irelationandafrequencythesameasthevoltage.Clearly,ifthecurrentwereassumedtobeacomplexexponential,sowouldthevoltage.Foracapacitor,i=Cd dtv.Lettingthevoltagebeacomplexexponential,wehavei=CVj2fej2ft.TheamplitudeofthiscomplexexponentialisI=CVj2f.Finally,fortheinductor,wherev=Ld dti,assumingthecurrenttobeacomplexexponentialresultsinthevoltagehavingtheformv=LIj2fej2ft,makingitscomplexamplitudeV=LIj2f.ThemajorconsequenceofassumingcomplexexponentialvoltageandcurrentsisthattheratioZ=V Iforeachelementdoesnotdependontime,butdoesdependonsourcefrequency.Thisquantityisknownastheelement'simpedance.Theimpedanceis,ingeneral,acomplex-valued,frequency-dependentquantity.Forexample,themagni-tudeofthecapacitor'simpedanceisinverselyrelatedtofrequency,andhasaphaseof)]TJ/F1 9.9626 Tf 9.4092 8.0698 Td[()]TJ/F10 6.9738 Tf 5.7617 -4.1472 Td[( 2.Thisobservationmeansthatifthecurrentisacomplexexponentialandhasconstantamplitude,theamplitudeofthevoltagedecreaseswithfrequency.Let'sconsiderKircho'scircuitlaws.Whenvoltagesaroundaloopareallcomplexexponentialsofthe

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56CHAPTER3.ANALOGSIGNALPROCESSINGsamefrequency,wehavePnvn=Pn)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(Vnej2ft=0.12whichmeansXnVn=0.13thecomplexamplitudesofthevoltagesobeyKVL.WecaneasilyimaginethatthecomplexamplitudesofthecurrentsobeyKCL.Whatwehavediscoveredisthatsourcesequalingacomplexexponentialofthesamefrequencyforcesallcircuitvariablestobecomplexexponentialsofthesamefrequency.Consequently,theratioofvoltagetocurrentforeachelementequalstheratiooftheircomplexamplitudes,whichdependsonlyonthesource'sfrequencyandelementvalues.Thissituationoccursbecausethecircuitelementsarelinearandtime-invariant.Forexample,supposewehadacircuitelementwherethevoltageequaledthesquareofthecurrent:vt=Ki2t.Ifit=Iej2ft,vt=KI2ej22ft,meaningthatvoltageandcurrentnolongerhadthesamefrequencyandthattheirratiowastime-dependent.BecauseforlinearcircuitelementsthecomplexamplitudeofvoltageisproportionaltothecomplexamplitudeofcurrentV=ZIassumingcomplexexponentialsourcesmeanscircuitelementsbehaveasiftheywereresistors,whereinsteadofresistance,weuseimpedance.BecausecomplexamplitudesforvoltageandcurrentalsoobeyKircho'slaws,wecansolvecircuitsusingvoltageandcurrentdividerandtheseriesandparallelcombinationrulesbyconsideringtheelementstobeimpedances.3.10TimeandFrequencyDomains20Whenwendthedierentialequationrelatingthesourceandtheoutput,wearefacedwithsolvingthecircuitinwhatisknownasthetimedomain.Whatweemphasizehereisthatitisofteneasiertondtheoutputifweuseimpedances.Becauseimpedancesdependonlyonfrequency,wendourselvesinthefrequencydomain.Acommonerrorinusingimpedancesiskeepingthetime-dependentpart,thecomplexexponential,inthefray.Theentirepointofusingimpedancesistogetridoftimeandconcentrateonfrequency.Onlyafterwendtheresultinthefrequencydomaindowegobacktothetimedomainandputthingsbacktogetheragain.Toillustratehowthetimedomain,thefrequencydomainandimpedancesttogether,considerthetimedomainandfrequencydomaintobetwoworkrooms.Sinceyoucan'tbetwoplacesatthesametime,youarefacedwithsolvingyourcircuitprobleminoneofthetworoomsatanypointintime.Impedancesandcomplexexponentialsarethewayyougetbetweenthetworooms.Securityguardsmakesureyoudon'ttrytosneaktimedomainvariablesintothefrequencydomainroomandviceversa.Figure3.26TwoRoomsshowshowthisworks.Asweunfoldtheimpedancestory,we'llseethatthepowerfuluseofimpedancessuggestedbySteinmetz21greatlysimpliessolvingcircuits,alleviatesusfromsolvingdierentialequations,andsuggestsageneralwayofthinkingaboutcircuits.Becauseoftheimportanceofthisapproach,let'sgooverhowitworks.1.Eventhoughit'snot,pretendthesourceisacomplexexponential.Wedothisbecausetheimpedanceapproachsimpliesndinghowinputandoutputarerelated.Ifitwereavoltagesourcehavingvoltagevin=ptapulse,stillletvin=Vinej2ft.We'lllearnhowto"getthepulseback"later.2.Withasourceequalingacomplexexponential,allvariablesinalinearcircuitwillalsobecomplexexponentialshavingthesamefrequency.Thecircuit'sonlyremaining"mystery"iswhateachvariable'scomplexamplitudemightbe.Tondthese,weconsiderthesourcetobeacomplexnumberVinhereandtheelementstobeimpedances. 20Thiscontentisavailableonlineat.21http://www.invent.org/hall_of_fame/139.html

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57 TwoRooms Figure3.26:Thetimeandfrequencydomainsarelinkedbyassumingsignalsarecomplexexponentials.Inthetimedomain,signalscanhaveanyform.Passingintothefrequencydomain'workroom",signalsarerepresentedentirelybycomplexamplitudes. 3.Wecannowsolveusingseriesandparallelcombinationruleshowthecomplexamplitudeofanyvariablerelatestothesourcescomplexamplitude.Example3.3Toillustratetheimpedanceapproach,werefertotheRCcircuitFigure3.27SimpleCircuitsbelow,andweassumethatvin=Vinej2ft.Usingimpedances,thecomplexamplitudeoftheoutputvoltageVoutcanbefoundusingvoltagedivider:Vout=ZC ZC+ZRVinVout=1 j2fC 1 j2fC+RVinVout=1 j2fRC+1VinIfwerefertothedierentialequationforthiscircuitshowninCircuitswithCapacitorsandInductorsSection3.8tobeRCd dtvout+vout=vin,lettingtheoutputandinputvoltagesbecomplexexponentials,weobtainthesamerelationshipbetweentheircomplexamplitudes.Thus,usingimpedancesisequivalenttousingthedierentialequationandsolvingitwhenthesourceisacomplexexponential.

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58CHAPTER3.ANALOGSIGNALPROCESSINGSimpleCircuits a bFigure3.27:aAsimpleRCcircuit.bTheimpedancecounterpartfortheRCcircuit.Notethatthesourceandoutputvoltagearenowcomplexamplitudes.Infact,wecanndthedierentialequationdirectlyusingimpedances.Ifwecross-multiplytherelationbetweeninputandoutputamplitudes,Voutj2fRC+1=Vinandthenputthecomplexexponentialsbackin,wehaveRCj2fVoutej2ft+Voutej2ft=Vinej2ftIntheprocessofdeningimpedances,notethatthefactorj2farisesfromthederivativeofacomplexexponential.Wecanreversetheimpedanceprocess,andrevertbacktothedierentialequation.RCd dtvout+vout=vinThisisthesameequationthatwasderivedmuchmoretediouslyinCircuitswithCapacitorsandInductorsSection3.8.Findingthedierentialequationrelatingoutputtoinputisfarsimplerwhenweuseimpedancesthanwithanyothertechnique.Exercise3.11Solutiononp.106.Supposeyouhadanexpressionwhereacomplexamplitudewasdividedbyj2f.Whattime-domainoperationcorrespondstothisdivision?3.11PowerintheFrequencyDomain22Recallingthattheinstantaneouspowerconsumedbyacircuitelementoranequivalentcircuitthatrepresentsacollectionofelementsequalsthevoltagetimesthecurrententeringthepositive-voltageterminal,pt=vtit,whatistheequivalentexpressionusingimpedances?Theresultingcalculationrevealsmoreaboutpowerconsumptionincircuitsandtheintroductionoftheconceptofaveragepower.Whenallsourcesproducesinusoidsoffrequencyf,thevoltageandcurrentforanycircuitelementorcollectionofelementsaresinusoidsofthesamefrequency.vt=jVjcos2ft+it=jIjcos2ft+ 22Thiscontentisavailableonlineat.

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59Here,thecomplexamplitudeofthevoltageVequalsjVjejandthatofthecurrentisjIjej.WecanalsowritethevoltageandcurrentintermsoftheircomplexamplitudesusingEuler'sformulaSection2.1.2:Euler'sFormula.vt=1 2)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(Vej2ft+Ve)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2ftit=1 2)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(Iej2ft+Ie)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2ftMultiplyingthesetwoexpressionsandsimplifyinggivespt=1 4)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(VI+VI+VIej4ft+VIe)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j4ft=1 2ReVI+1 2Re)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(VIej4ft=1 2ReVI+1 2jVjjIjcos4ft++Wedene1 2VItobecomplexpower.Thereal-partofcomplexpoweristhersttermandsinceitdoesnotchangewithtime,itrepresentsthepowerconsistentlyconsumed/producedbythecircuit.Thesecondtermvarieswithtimeatafrequencytwicethatofthesource.Conceptually,thistermdetailshowpower"sloshes"backandforthinthecircuitbecauseofthesinusoidalsource.Fromanotherviewpoint,thereal-partofcomplexpowerrepresentslong-termenergyconsump-tion/production.Energyistheintegralofpowerand,astheintegrationintervalincreases,thersttermappreciateswhilethetime-varyingterm"sloshes."Consequently,themostconvenientdenitionoftheav-eragepowerconsumed/producedbyanycircuitisintermsofcomplexamplitudes.Pave=1 2ReVI.14Exercise3.12Solutiononp.107.Supposethecomplexamplitudesofthevoltageandcurrenthavexedmagnitudes.Whatphaserelationshipbetweenvoltageandcurrentmaximizestheaveragepower?Inotherwords,howareandrelatedformaximumpowerdissipation?Becausethecomplexamplitudesofthevoltageandcurrentarerelatedbytheequivalentimpedance,averagepowercanalsobewrittenasPave=1 2ReZjIj2=1 2Re1 ZjVj2Theseexpressionsgeneralizetheresults.3weobtainedforresistorcircuits.Wehavederivedafundamentalresult:Onlytherealpartofimpedancecontributestolong-termpowerdissipation.Ofthecircuitelements,onlytheresistordissipatespower.Capacitorsandinductorsdissipatenopowerinthelongterm.Itisimportanttorealizethatthesestatementsapplyonlyforsinusoidalsources.IfyouturnonaconstantvoltagesourceinanRC-circuit,chargingthecapacitordoesconsumepower.Exercise3.13Solutiononp.107.InanearlierproblemSection1.5.1:RMSValues,wefoundthatthermsvalueofasinusoidwasitsamplitudedividedbyp 2.WhatisaveragepowerexpressedintermsofthermsvaluesofthevoltageandcurrentVrmsandIrmsrespectively?3.12EquivalentCircuits:ImpedancesandSources23Whenwehavecircuitswithcapacitorsand/orinductorsaswellasresistorsandsources,ThveninandMayer-Nortonequivalentcircuitscanstillbedenedbyusingimpedancesandcomplexamplitudesforvoltageand 23Thiscontentisavailableonlineat.

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60CHAPTER3.ANALOGSIGNALPROCESSINGcurrents.Foranycircuitcontainingsources,resistors,capacitors,andinductors,theinput-outputrelationforthecomplexamplitudesoftheterminalvoltageandcurrentisV=ZeqI+VeqI=V Zeq)]TJ/F11 9.9626 Tf 9.9626 0 Td[(IeqwithVeq=ZeqIeq.Thus,wehaveThveninandMayer-NortonequivalentcircuitsasshowninFigure3.28EquivalentCircuits.Example3.4Let'sndtheThveninandMayer-NortonequivalentcircuitsforFigure3.29SimpleRCCircuit.Theopen-circuitvoltageandshort-circuitcurrenttechniquesstillwork,exceptweuseimpedancesandcomplexamplitudes.Theopen-circuitvoltagecorrespondstothetransferfunctionwehavealreadyfound.Whenweshorttheterminals,thecapacitornolongerhasanyeectonthecircuit,andtheshort-circuitcurrentIscequalsVout R.Theequivalentimpedancecanbefoundbysettingthesourcetozero,andndingtheimpedanceusingseriesandparallelcombinationrules.Inourcase,theresistorandcapacitorareinparalleloncethevoltagesourceisremovedsettingittozeroamountstoreplacingitwithashort-circuit.Thus,Zeq=Rk1 j2fC=R 1+j2fRC.Consequently,wehaveVeq=1 1+j2fRCVinIeq=1 RVinZeq=R 1+j2fRCAgain,weshouldchecktheunitsofouranswer.Noteinparticularthatj2fRCmustbedimen-sionless.Isit?3.13TransferFunctions24TheratiooftheoutputandinputamplitudesforFigure3.30SimpleCircuit,knownasthetransferfunctionorthefrequencyresponse,isgivenbyVout Vin=Hf=1 j2fRC+1.15Implicitinusingthetransferfunctionisthattheinputisacomplexexponential,andtheoutputisalsoacomplexexponentialhavingthesamefrequency.Thetransferfunctionrevealshowthecircuitmodiestheinputamplitudeincreatingtheoutputamplitude.Thus,thetransferfunctioncompletelydescribeshowthecircuitprocessestheinputcomplexexponentialtoproducetheoutputcomplexexponential.Thecircuit'sfunctionisthussummarizedbythetransferfunction.Infact,circuitsareoftendesignedtomeettransferfunctionspecications.Becausetransferfunctionsarecomplex-valued,frequency-dependentquantities,wecanbetterappreciateacircuit'sfunctionbyexaminingthemagnitudeandphaseofitstransferfunctionFigure3.31Magnitudeandphaseofthetransferfunction. 24Thiscontentisavailableonlineat.

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61 EquivalentCircuits aEquivalentcircuitswithresistors. bEquivalentcircuitswithimpedances.Figure3.28:Comparingtherst,simpler,gurewiththeslightlymorecomplicatedsecondgure,weseetwodierences.Firstofall,morecircuitsallthosecontaininglinearelementsinfacthaveequivalentcircuitsthatcontainequivalents.Secondly,theterminalandsourcevariablesarenowcomplexamplitudes,whichcarriestheimplicitassumptionthatthevoltagesandcurrentsaresinglecomplexexponentials,allhavingthesamefrequency.

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62CHAPTER3.ANALOGSIGNALPROCESSINGSimpleRCCircuit Figure3.29 SimpleCircuit Figure3.30:AsimpleRCcircuit.

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63 Magnitudeandphaseofthetransferfunction a bFigure3.31:MagnitudeandphaseofthetransferfunctionoftheRCcircuitshowninFigure3.30SimpleCircuitwhenRC=1.ajHfj=1 p fRC2+1bHf=)]TJ/F56 8.9664 Tf 8.7034 0 Td[(arctanfRC Thistransferfunctionhasmanyimportantpropertiesandprovides}alltheinsightsneededtodeterminehowthecircuitfunctions.Firstofall,notethatwecancomputethefrequencyresponseforbothpositiveandnegativefrequencies.Recallthatsinusoidsconsistofthesumoftwocomplexexponentials,onehavingthenegativefrequencyoftheother.Wewillconsiderhowthecircuitactsonasinusoidsoon.Donotethatthemagnitudehasevensymmetry:Thenegativefrequencyportionisamirrorimageofthepositivefrequencyportion:jH)]TJ/F11 9.9626 Tf 7.7487 0 Td[(fj=jHfj.Thephasehasoddsymmetry:H)]TJ/F11 9.9626 Tf 7.7488 0 Td[(f=)]TJ/F8 9.9626 Tf 9.4092 0 Td[(Hf.Thesepropertiesofthisspecicexampleapplyforalltransferfunctionsassociatedwithcircuits.Consequently,wedon'tneedtoplotthenegativefrequencycomponent;weknowwhatitisfromthepositivefrequencypart.Themagnitudeequals1 p 2ofitsmaximumgainatf=0when2fRC=1thetwotermsinthe

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64CHAPTER3.ANALOGSIGNALPROCESSINGdenominatorofthemagnitudeareequal.Thefrequencyfc=1 2RCdenestheboundarybetweentwooperatingranges.Forfrequenciesbelowthisfrequency,thecircuitdoesnotmuchaltertheamplitudeofthecomplexexponentialsource.Forfrequenciesgreaterthanfc,thecircuitstronglyattenuatestheamplitude.Thus,whenthesourcefrequencyisinthisrange,thecircuit'soutputhasamuchsmalleramplitudethanthatofthesource.Forthesereasons,thisfrequencyisknownasthecutofrequency.Inthiscircuitthecutofrequencydependsonlyontheproductoftheresistanceandthecapacitance.Thus,acutofrequencyof1kHzoccurswhen1 2RC=103orRC=10)]TJ/F6 4.9813 Tf 5.3965 0 Td[(3 2=1:5910)]TJ/F7 6.9738 Tf 6.2266 0 Td[(4.Thusresistance-capacitancecombinationsof1.59kand100nFor10and1.59Fresultinthesamecutofrequency.Thephaseshiftcausedbythecircuitatthecutofrequencypreciselyequals)]TJ/F1 9.9626 Tf 9.4091 8.0698 Td[()]TJ/F10 6.9738 Tf 5.7618 -4.1472 Td[( 4.Thus,belowthecutofrequency,phaseislittleaected,butathigherfrequencies,thephaseshiftcausedbythecircuitbecomes)]TJ/F1 9.9626 Tf 9.4091 8.0698 Td[()]TJ/F10 6.9738 Tf 5.7618 -4.1472 Td[( 2.Thisphaseshiftcorrespondstothedierencebetweenacosineandasine.Wecanusethetransferfunctiontondtheoutputwhentheinputvoltageisasinusoidfortworeasons.Firstofall,asinusoidisthesumoftwocomplexexponentials,eachhavingafrequencyequaltothenegativeoftheother.Secondly,becausethecircuitislinear,superpositionapplies.Ifthesourceisasinewave,weknowthatvint=Asinft=A 2j)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(ej2ft)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2ft.16Sincetheinputisthesumoftwocomplexexponentials,weknowthattheoutputisalsoasumoftwosimilarcomplexexponentials,theonlydierencebeingthatthecomplexamplitudeofeachismultipliedbythetransferfunctionevaluatedateachexponential'sfrequency.voutt=A 2jHfej2ft)]TJ/F11 9.9626 Tf 12.2493 6.7398 Td[(A 2jH)]TJ/F11 9.9626 Tf 7.7488 0 Td[(fe)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2ft.17Asnotedearlier,thetransferfunctionismostconvenientlyexpressedinpolarform:Hf=jHfjejHf.Furthermore,jH)]TJ/F11 9.9626 Tf 7.7487 0 Td[(fj=jHfjevensymmetryofthemagnitudeandH)]TJ/F11 9.9626 Tf 7.7488 0 Td[(f=)]TJ/F8 9.9626 Tf 9.4091 0 Td[(Hfoddsymmetryofthephase.Theoutputvoltageexpressionsimpliestovoutt=A 2jjHfjej2ft+Hf)]TJ/F10 6.9738 Tf 11.9979 3.9226 Td[(A 2jjHfje)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2ft)]TJ/F30 6.9738 Tf 6.2266 0 Td[(Hf=AjHfjsinft+Hf.18Thecircuit'soutputtoasinusoidalinputisalsoasinusoid,havingagainequaltothemagnitudeofthecircuit'stransferfunctionevaluatedatthesourcefrequencyandaphaseequaltothephaseofthetransferfunctionatthesourcefrequency.Itwillturnoutthatthisinput-outputrelationdescriptionappliestoanylinearcircuithavingasinusoidalsource.Exercise3.14Solutiononp.107.Thisinput-outputpropertyisaspecialcaseofamoregeneralresult.Showthatifthesourcecanbewrittenastheimaginarypartofacomplexexponentialvint=Im)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(Vej2fttheoutputisgivenbyvoutt=Im)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(VHfej2ft.Showthatasimilarresultalsoholdsfortherealpart.Thenotionofimpedanceariseswhenweassumethesourcesarecomplexexponentials.Thisassumptionmayseemrestrictive;whatwouldwedoifthesourcewereaunitstep?Whenweuseimpedancestondthetransferfunctionbetweenthesourceandtheoutputvariable,wecanderivefromitthedierentialequationthatrelatesinputandoutput.Thedierentialequationappliesnomatterwhatthesourcemaybe.Aswehaveargued,itisfarsimplertouseimpedancestondthedierentialequationbecausewecanuseseriesandparallelcombinationrulesthananyothermethod.Inthissense,wehavenotlostanythingbytemporarilypretendingthesourceisacomplexexponential.Infactwecanalsosolvethedierentialequationusingimpedances!Thus,despitetheapparentrestric-tivenessofimpedances,assumingcomplexexponentialsourcesisactuallyquitegeneral.

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65RLcircuit Figure3.323.14DesigningTransferFunctions25Ifthesourceconsistsoftwoormoresignals,weknowfromlinearsystemtheorythattheoutputvoltageequalsthesumoftheoutputsproducedbyeachsignalalone.Inshort,linearcircuitsareaspecialcaseoflinearsystems,andthereforesuperpositionapplies.Inparticular,supposethesecomponentsignalsarecomplexexponentials,eachofwhichhasafrequencydierentfromtheothers.Thetransferfunctionportrayshowthecircuitaectstheamplitudeandphaseofeachcomponent,allowingustounderstandhowthecircuitworksonacomplicatedsignal.Thosecomponentshavingafrequencylessthanthecutofrequencypassthroughthecircuitwithlittlemodicationwhilethosehavinghigherfrequenciesaresuppressed.Thecircuitissaidtoactasalter,lteringthesourcesignalbasedonthefrequencyofeachcomponentcomplexexponential.Becauselowfrequenciespassthroughthelter,wecallitalowpassltertoexpressmorepreciselyitsfunction.Wehavealsofoundtheeaseofcalculatingtheoutputforsinusoidalinputsthroughtheuseofthetransferfunction.Oncewendthetransferfunction,wecanwritetheoutputdirectlyasindicatedbytheoutputofacircuitforasinusoidalinput.18.Example3.5Let'sapplytheseresultstoanalexample,inwhichtheinputisavoltagesourceandtheoutputistheinductorcurrent.ThesourcevoltageequalsVin=2cos60t+3.Wewantthecircuittopassconstantosetvoltageessentiallyunalteredsaveforthefactthattheoutputisacurrentratherthanavoltageandremovethe60Hzterm.Becausetheinputisthesumoftwosinusoidsaconstantisazero-frequencycosineourapproachis1.ndthetransferfunctionusingimpedances;2.useittondtheoutputduetoeachinputcomponent;3.addtheresults;4.ndelementvaluesthataccomplishourdesigncriteria.Becausethecircuitisaseriescombinationofelements,let'susevoltagedividertondthetransferfunctionbetweenVinandV,thenusethev-irelationoftheinductortonditscurrent.Iout Vin=j2fL R+j2fL1 j2fL=1 j2fL+R=Hf.19 25Thiscontentisavailableonlineat.

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66CHAPTER3.ANALOGSIGNALPROCESSINGwherevoltagedivider=j2fL R+j2fLandinductoradmittance=1 j2fL[Dotheunitscheck?]Theformofthistransferfunctionshouldbefamiliar;itisalowpasslter,anditwillperformourdesiredfunctiononcewechooseelementvaluesproperly.Theconstanttermiseasiesttohandle.Theoutputisgivenby3jHj=3 R.Thus,thevaluewechoosefortheresistancewilldeterminethescalingfactorofhowvoltageisconvertedintocurrent.Forthe60Hzcomponentsignal,theoutputcurrentis2jHjcos260t+H.Thetotaloutputduetooursourceisiout=2jHjcos260t+H+3H.20Thecutofrequencyforthislteroccurswhentherealandimaginarypartsofthetransferfunction'sdenominatorequaleachother.Thus,2fcL=R,whichgivesfc=R 2L.Wewantthiscutofrequencytobemuchlessthan60Hz.Supposeweplaceitat,say,10Hz.ThisspecicationwouldrequirethecomponentvaluestoberelatedbyR L=20=62:8.Thetransferfunctionat60Hzwouldbe1 j260L+R=1 R1 6j+1=1 R1 p 370:161 R.21whichyieldsanattenuationrelativetothegainatzerofrequencyofabout1=6,andresultinanoutputamplitudeof0:3 Rrelativetotheconstantterm'samplitudeof3 R.Afactorof10relativesizebetweenthetwocomponentsseemsreasonable.Havinga100mHinductorwouldrequirea6.28resistor.Aneasilyavailableresistorvalueis6.8;thus,thischoiceresultsincheaplyandeasilypurchasedparts.Tomaketheresistancebiggerwouldrequireaproportionallylargerinductor.Unfortunately,evena1Hinductorisphysicallylarge;consequentlylowcutofrequenciesrequiresmall-valuedresistorsandlarge-valuedinductors.Thechoicemadehererepresentsonlyonecompromise.Thephaseofthe60Hzcomponentwillverynearlybe)]TJ/F1 9.9626 Tf 9.4091 8.0698 Td[()]TJ/F10 6.9738 Tf 5.7618 -4.1472 Td[( 2,leavingittobe0:3 Rcos)]TJ/F8 9.9626 Tf 4.5662 -8.0698 Td[(260t)]TJ/F10 6.9738 Tf 11.1581 3.9226 Td[( 2=0:3 Rsin60t.ThewaveformsfortheinputandoutputareshowninFig-ure3.33Waveforms.Notethatthesinusoid'sphasehasindeedshifted;thelowpasslternotonlyreducedthe60Hzsignal'samplitude,butalsoshifteditsphaseby90.3.15FormalCircuitMethods:NodeMethod26Insomecomplicatedcases,wecannotusethesimplicationtechniquessuchasparallelorseriescom-binationrulestosolveforacircuit'sinput-outputrelation.Inothermodules,wewrotev-irelationsandKircho'slawshaphazardly,solvingthemmoreonintuitionthanprocedure.Weneedaformalmethodthatproducesasmall,easysetofequationsthatleaddirectlytotheinput-outputrelationweseek.Onesuchtechniqueisthenodemethod. 26Thiscontentisavailableonlineat.

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67Waveforms Figure3.33:InputandoutputwaveformsfortheexampleRLcircuitwhentheelementvaluesareR=6:28andL=100mH. NodeVoltage Figure3.34 Thenodemethodbeginsbyndingallnodesplaceswherecircuitelementsattachtoeachotherinthecircuit.Wecalloneofthenodesthereferencenode;thechoiceofreferencenodeisarbitrary,butitisusuallychosentobeapointofsymmetryorthe"bottom"node.Fortheremainingnodes,wedenenodevoltagesenthatrepresentthevoltagebetweenthenodeandthereference.Thesenodevoltagesconstitute

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68CHAPTER3.ANALOGSIGNALPROCESSINGtheonlyunknowns;allweneedisasucientnumberofequationstosolveforthem.Inourexample,wehavetwonodevoltages.TheveryactofdeningnodevoltagesisequivalenttousingalltheKVLequationsatyourdisposal.Thereasonforthissimple,butastounding,factisthatanodevoltageisuniquelydenedregardlessofwhatpathistracedbetweenthenodeandthereference.Becausetwopathsbetweenanodeandreferencehavethesamevoltage,thesumofvoltagesaroundtheloopequalszero.Insomecases,anodevoltagecorrespondsexactlytothevoltageacrossavoltagesource.Insuchcases,thenodevoltageisspeciedbythesourceandisnotanunknown.Forexample,inourcircuit,e1=vin;thus,weneedonlytondonenodevoltage.TheequationsgoverningthenodevoltagesareobtainedbywritingKCLequationsateachnodehavinganunknownnodevoltage,usingthev-irelationsforeachelement.Inourexample,theonlycircuitequationise2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(vin R1+e2 R2+e2 R3=0.22AlittlereectionrevealsthatwhenwritingtheKCLequationsforthesumofcurrentsleavinganode,thatnode'svoltagewillalwaysappearwithaplussign,andallothernodevoltageswithaminussign.Systematicapplicationofthisproceduremakesiteasytowritenodeequationsandtocheckthembeforesolvingthem.Alsoremembertocheckunitsatthispoint:Everytermshouldhaveunitsofcurrent.Inourexample,solvingfortheunknownnodevoltageiseasy:e2=R2R3 R1R2+R1R2+R2R3vin.23Havewereallysolvedthecircuitwiththenodemethod?Alongtheway,wehaveusedKVL,KCL,andthev-irelations.Previously,weindicatedthatthesetofequationsresultingfromapplyingtheselawsisnecessaryandsucient.Thisresultguaranteesthatthenodemethodcanbeusedto"solve"anycircuit.Onefalloutofthisresultisthatwemustbeabletondanycircuitvariablegiventhenodevoltagesandsources.Allcircuitvariablescanbefoundusingthev-irelationsandvoltagedivider.Forexample,thecurrentthroughR3equalse2 R3. Figure3.35 Thepresenceofacurrentsourceinthecircuitdoesnotaectthenodemethodgreatly;justincludeitinwritingKCLequationsasacurrentleavingthenode.Thecircuithasthreenodes,requiringustodenetwonodevoltages.Thenodeequationsaree1 R1+e1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e2 R2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(iin=0Node1e2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e1 R2+e2 R3=0Node2

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69NotethatthenodevoltagecorrespondingtothenodethatwearewritingKCLforenterswithapositivesign,theotherswithanegativesign,andthattheunitsofeachtermisgiveninamperes.Rewritetheseequationsinthestandardset-of-linear-equationsform.e11 R1+1 R2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e21 R2=iin)]TJ/F11 9.9626 Tf 7.7487 0 Td[(e11 R2+e21 R2+1 R3=0Solvingtheseequationsgivese1=R2+R3 R3e2e2=R1R3 R1+R2+R3iinTondtheindicatedcurrent,wesimplyusei=e2 R3.Example3.6:NodeMethodExample Figure3.36InthiscircuitFigure3.36,wecannotusetheseries/parallelcombinationrules:Theverticalresistoratnode1keepsthetwohorizontal1resistorsfrombeinginseries,andthe2resistorpreventsthetwo1resistorsatnode2frombeinginseries.Wereallydoneedthenodemethodtosolvethiscircuit!Despitehavingsixelements,weneedonlydenetwonodevoltages.Thenodeequationsaree1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(vin 1+e1 1+e1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e2 1=0Node1e2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(vin 2+e2 1+e2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e1 1=0Node2Solvingtheseequationsyieldse1=2 5vinande2=5 13vin.Theoutputcurrentequalse2 1=5 13vin.Oneunfortunateconsequenceofusingtheelement'snumericvaluesfromtheoutsetisthatitbecomesimpossibletocheckunitswhilesettingupandsolvingequations.Exercise3.15Solutiononp.107.Whatistheequivalentresistanceseenbythevoltagesource?

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70CHAPTER3.ANALOGSIGNALPROCESSING NodeMethodandImpedances Figure3.37:ModicationofthecircuitshownonthelefttoillustratethenodemethodandtheeectofaddingtheresistorR2. ThenodemethodappliestoRLCcircuits,withoutsignicantmodicationfromthemethodsusedonsimpleresistivecircuits,ifweusecomplexamplitudes.WerelyonthefactthatcomplexamplitudessatisfyKVL,KCL,andimpedance-basedv-irelations.Intheexamplecircuit,wedenecomplexamplitudesfortheinputandoutputvariablesandforthenodevoltages.Weneedonlyonenodevoltagehere,anditsKCLequationisE)]TJ/F11 9.9626 Tf 9.9626 0 Td[(Vin R1+Ej2fC+E R2=0withtheresultE=R2 R1+R2+j2fR1R2CVinTondthetransferfunctionbetweeninputandoutputvoltages,wecomputetheratioE Vin.Thetransferfunction'smagnitudeandanglearejHfj=R2 q R1+R22+fR1R2C2Hf=)]TJ/F1 9.9626 Tf 9.4091 14.0475 Td[(arctan2fR1R2C R1+R2ThiscircuitdiersfromtheoneshownpreviouslyFigure3.30:SimpleCircuitinthattheresistorR2hasbeenaddedacrosstheoutput.Whateecthasithadonthetransferfunction,whichintheoriginalcircuitwasalowpasslterhavingcutofrequencyfc=1 2R1C?AsshowninFigure3.38TransferFunction,addingthesecondresistorhastwoeects:itlowersthegaininthepassbandtherangeoffrequenciesforwhichthelterhaslittleeectontheinputandincreasesthecutofrequency.

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71 TransferFunction Figure3.38:TransferfunctionsofthecircuitsshowninFigure3.37NodeMethodandImpedances.Here,R1=1,R2=1,andC=1. WhenR2=R1,asshownontheplot,thepassbandgainbecomeshalfoftheoriginal,andthecutofrequencyincreasesbythesamefactor.Thus,addingR2providesa'knob'bywhichwecantradepassbandgainforcutofrequency.Exercise3.16Solutiononp.107.Wecanchangethecutofrequencywithoutaectingpassbandgainbychangingtheresistanceintheoriginalcircuit.DoestheadditionoftheR2resistorhelpincircuitdesign?3.16PowerConservationinCircuits27Nowthatwehaveaformalmethodthenodemethodforsolvingcircuits,wecanuseittoproveapowerfulresult:KVLandKCLareallthatarerequiredtoshowthatallcircuitsconservepower,regardlessofwhatelementsareusedtobuildthecircuit. 27Thiscontentisavailableonlineat.

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72CHAPTER3.ANALOGSIGNALPROCESSING PartofageneralcircuittoproveConservationofPower Figure3.39 Firstofall,denenodevoltagesforallnodesinagivencircuit.Anynodechosenasthereferencewilldo.Forexample,intheportionofalargecircuitFigure3.39:PartofageneralcircuittoproveConservationofPowerdepictedhere,wedenenodevoltagesfornodesa,bandc.Withthesenodevoltages,wecanexpressthevoltageacrossanyelementintermsofthem.Forexample,thevoltageacrosselement1isgivenbyv1=eb)]TJ/F11 9.9626 Tf 9.9626 0 Td[(ea.Theinstantaneouspowerforelement1becomesv1i1=eb)]TJ/F11 9.9626 Tf 9.9626 0 Td[(eai1=ebi1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(eai1Writingthepowerfortheotherelements,wehavev2i2=eci2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(eai2v3i3=eci3)]TJ/F11 9.9626 Tf 9.9626 0 Td[(ebi3Whenweaddtogethertheelementpowerterms,wediscoverthatoncewecollecttermsinvolvingaparticularnodevoltage,itismultipliedbythesumofcurrentsleavingthenodeminusthesumofcurrentsentering.Forexample,fornodeb,wehaveebi3)]TJ/F11 9.9626 Tf 9.9626 0 Td[(i1.WeseethatthecurrentswillobeyKCLthatmultiplyeachnodevoltage.Consequently,weconcludethatthesumofelementpowersmustequalzeroinanycircuitregardlessoftheelementsusedtoconstructthecircuit.Xkvkik=0Thesimplicityandgeneralitywithwhichweprovedthisresultsgeneralizestoothersituationsaswell.Inparticular,notethatthecomplexamplitudesofvoltagesandcurrentsobeyKVLandKCL,respectively.Consequently,wehavethatPkVkIk=0.Furthermore,thecomplex-conjugateofcurrentsalsosatisesKCL,whichmeanswealsohavePkVkIk=0.Andnally,weknowthatevaluatingthereal-partofanexpressionislinear.Findingthereal-partofthispowerconservationgivestheresultthataveragepowerisalsoconservedinanycircuit.Xk1 2ReVkIk=0Note:Thisproofofpowerconservationcanbegeneralizedinanotherveryinterestingway.AllweneedisasetofvoltagesthatobeyKVLandasetofcurrentsthatobeyKCL.Thus,foragivencircuittopologythespecicwayelementsareinterconnected,thevoltagesandcurrentscanbemeasuredatdierenttimesandthesumofv-iproductsiszero.Xkvkt1ikt2=0

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73Evenmoreinterestingisthefactthattheelementsdon'tmatter.Wecantakeacircuitandmeasureallthevoltages.Wecanthenmakeelement-for-elementreplacementsand,ifthetopologyhasnotchanged,wecanmeasureasetofcurrents.Thesumoftheproductofelementvoltagesandcurrentswillalsobezero!3.17Electronics28Sofarwehaveanalyzedelectricalcircuits:Thesourcesignalhasmorepowerthantheoutputvariable,beitavoltageoracurrent.Powerhasnotbeenexplicitlydened,butnomatter.Resistors,inductors,andcapacitorsasindividualelementscertainlyprovidenopowergain,andcircuitsbuiltofthemwillnotmagicallydosoeither.Suchcircuitsaretermedelectricalindistinctiontothosethatdoprovidepowergain:electroniccircuits.Providingpowergain,suchasyourstereoreadingaCDandproducingsound,isaccomplishedbysemiconductorcircuitsthatcontaintransistors.Thebasicideaofthetransistoristolettheweakinputsignalmodulateastrongcurrentprovidedbyasourceofelectricalpowerthepowersupplytoproduceamorepowerfulsignal.Aphysicalanalogyisawaterfaucet:Byturningthefaucetbackandforth,thewaterowvariesaccordingly,andhasmuchmorepowerthanexpendedinturningthehandle.Thewaterpowerresultsfromthestaticpressureofthewaterinyourplumbingcreatedbythewaterutilitypumpingthewateruptoyourlocalwatertower.Thepowersupplyislikethewatertower,andthefaucetisthetransistor,withtheturningachievedbytheinputsignal.Justasinthisanalogy,apowersupplyisasourceofconstantvoltageasthewatertowerissupposedtoprovideaconstantwaterpressure.Adevicethatismuchmoreconvenientforprovidinggainandotherusefulfeaturesaswellthanthetransistoristheoperationalamplier,alsoknownastheop-amp.Anop-ampisanintegratedcircuitacomplicatedcircuitinvolvingseveraltransistorsconstructedonachipthatprovidesalargevoltagegainifyouattachthepowersupply.Wecanmodeltheop-ampwithanewcircuitelement:thedependentsource.3.18DependentSources29Adependentsourceiseitheravoltageorcurrentsourcewhosevalueisproportionaltosomeothervoltageorcurrentinthecircuit.Thus,therearefourdierentkindsofdependentsources;todescribeanop-amp,weneedavoltage-dependentvoltagesource.However,thestandardcircuit-theoreticalmodelforatransistor30containsacurrent-dependentcurrentsource.Dependentsourcesdonotserveasinputstoacircuitlikeindependentsources.Theyareusedtomodelactivecircuits:thosecontainingelectronicelements.TheRLCcircuitswehavebeenconsideringsofarareknownaspassivecircuits. 28Thiscontentisavailableonlineat.29Thiscontentisavailableonlineat.30"SmallSignalModelforBipolarTransistor"

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74CHAPTER3.ANALOGSIGNALPROCESSING dependentsources Figure3.40:Ofthefourpossibledependentsources,depictedisavoltage-dependentvoltagesourceinthecontextofagenericcircuit. Figure3.41op-ampshowsthecircuitsymbolfortheop-ampanditsequivalentcircuitintermsofavoltage-dependentvoltagesource. op-amp Figure3.41:Theop-amphasfourterminalstowhichconnectionscanbemade.Inputsattachtonodesaandb,andtheoutputisnodec.Asthecircuitmodelontherightshows,theop-ampservesasanamplierforthedierenceoftheinputnodevoltages. Here,theoutputvoltageequalsanampliedversionofthedierenceofnodevoltagesappearingacrossitsinputs.Thedependentsourcemodelportrayshowtheop-ampworksquitewell.Asinmostactivecircuitschematics,thepowersupplyisnotshown,butmustbepresentforthecircuitmodeltobeaccurate.Mostoperationalampliersrequirebothpositiveandnegativesupplyvoltagesforproperoperation.Becausedependentsourcescannotbedescribedasimpedances,andbecausethedependentvariablecannot"disappear"whenyouapplyparallel/seriescombiningrules,circuitsimplicationssuchascurrentandvoltagedividershouldnotbeappliedinmostcases.Analysisofcircuitscontainingdependentsourcesessentiallyrequiresuseofformalmethods,likethenodemethodSection3.15.Usingthenodemethodforsuchcircuitsisnotdicult,withnodevoltagesdenedacrossthesourcetreatedasiftheywereknownaswithindependentsources.ConsiderthecircuitshownonthetopinFigure3.42feedbackop-amp.

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75 feedbackop-amp Figure3.42:Thetopcircuitdepictsanop-ampinafeedbackamplierconguration.Onthebottomistheequivalentcircuit,andintegratestheop-ampcircuitmodelintothecircuit. Notethattheop-ampisplacedinthecircuit"upside-down,"withitsinvertinginputatthetopandservingastheonlyinput.Asweexploreop-ampsinmoredetailinthenextsection,thiscongurationwillappearagainandagainanditsusefulnessdemonstrated.Todeterminehowtheoutputvoltageisrelatedtotheinputvoltage,weapplythenodemethod.Onlytwonodevoltagesvandvoutneedbedened;theremainingnodesareacrosssourcesorserveasthereference.Thenodeequationsarev)]TJ/F11 9.9626 Tf 9.9626 0 Td[(vin R+v Rin+v)]TJ/F11 9.9626 Tf 9.9626 0 Td[(vout RF=0.24vout)]TJ/F8 9.9626 Tf 9.9626 0 Td[()]TJ/F11 9.9626 Tf 7.7487 0 Td[(Gv Rout+vout)]TJ/F11 9.9626 Tf 9.9626 0 Td[(v RF+vout RL=0.25Notethatnospecialconsiderationswereusedinapplyingthenodemethodtothisdependent-sourcecircuit.SolvingthesetolearnhowvoutrelatestovinyieldsRFRout Rout)]TJ/F11 9.9626 Tf 9.9626 0 Td[(GRF1 Rout+1 Rin+1 RL1 R+1 Rin+1 RF)]TJ/F8 9.9626 Tf 15.7613 6.7398 Td[(1 RFvout=1 Rvin.26Thisexpressionrepresentsthegeneralinput-outputrelationforthiscircuit,knownasthestandardfeed-backconguration.Oncewelearnmoreaboutop-ampsSection3.19,inparticularwhatitstypicalelementvaluesare,theexpressionwillsimplifygreatly.Donotethattheunitscheck,andthattheparame-terGofthedependentsourceisadimensionlessgain.

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76CHAPTER3.ANALOGSIGNALPROCESSING3.19OperationalAmpliers31 Op-Amp Figure3.43:Theop-amphasfourterminalstowhichconnectionscanbemade.Inputsattachtonodesaandb,andtheoutputisnodec.Asthecircuitmodelontherightshows,theop-ampservesasanamplierforthedierenceoftheinputnodevoltages. Op-ampsnotonlyhavethecircuitmodelshowninFigure3.43Op-Amp,buttheirelementvaluesareveryspecial.Theinputresistance,Rin,istypicallylarge,ontheorderof1M.Theoutputresistance,Rout,issmall,usuallylessthan100.Thevoltagegain,G,islarge,exceeding105.Thelargegaincatchestheeye;itsuggeststhatanop-ampcouldturna1mVinputsignalintoa100Vone.Ifyouweretobuildsuchacircuitattachingavoltagesourcetonodea,attachingnodebtothereference,andlookingattheoutputyouwouldbedisappointed.Indealingwithelectroniccomponents,youcannotforgettheunrepresentedbutneededpowersupply.Unmodeledlimitationsimposedbypowersupplies:Itisimpossibleforelectroniccompo-nentstoyieldvoltagesthatexceedthoseprovidedbythepowersupplyorforthemtoyieldcurrentsthatexceedthepowersupply'srating.Typicalpowersupplyvoltagesrequiredforop-ampcircuitsareV.Attachingthe1mvsignalnotonlywouldfailtoproducea100Vsignal,theresultingwaveformwouldbeseverelydistorted.Whileadesirableoutcomeifyouarearock&rollacionado,high-qualitystereosshouldnotdistortsignals.Anotherconsiderationindesigningcircuitswithop-ampsisthattheseelementvaluesaretypical:Carefulcontrolofthegaincanonlybeobtainedbychoosingacircuitsothatitselementvaluesdictatetheresultinggain,whichmustbesmallerthanthatprovidedbytheop-amp. 31Thiscontentisavailableonlineat.

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77 opamp Figure3.44:Thetopcircuitdepictsanop-ampinafeedbackamplierconguration.Onthebottomistheequivalentcircuit,andintegratestheop-ampcircuitmodelintothecircuit. 3.19.1InvertingAmplierThefeedbackcongurationshowninFigure3.44opampisthemostcommonop-ampcircuitforobtainingwhatisknownasaninvertingamplier.RFRout Rout)]TJ/F11 9.9626 Tf 9.9626 0 Td[(GRF1 Rout+1 Rin+1 RL1 R+1 Rin+1 RF)]TJ/F8 9.9626 Tf 15.7613 6.7398 Td[(1 RFvout=1 Rvin.27providestheexactinput-outputrelationship.Inchoosingelementvalueswithrespecttoop-ampcharacter-istics,wecansimplifytheexpressiondramatically.Maketheloadresistance,RL,muchlargerthanRout.Thissituationdropstheterm1 RLfromthesecondfactorof.27.Maketheresistor,R,smallerthanRin,whichmeansthatthe1 Rinterminthethirdfactorisnegligible.Withthesetwodesigncriteria,theexpression.27becomesRF Rout)]TJ/F11 9.9626 Tf 9.9626 0 Td[(GRF1 R+1 RF)]TJ/F8 9.9626 Tf 15.7613 6.7398 Td[(1 RFvout=1 Rvout.28BecausethegainislargeandtheresistanceRoutissmall,thersttermbecomes)]TJ/F1 9.9626 Tf 9.4091 8.0698 Td[()]TJ/F7 6.9738 Tf 6.8775 -4.1472 Td[(1 G,leavinguswith)]TJ/F1 9.9626 Tf 9.4091 14.0474 Td[(1 G1 R+1 RF)]TJ/F8 9.9626 Tf 15.7613 6.7398 Td[(1 RFvout=1 Rvin.29

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78CHAPTER3.ANALOGSIGNALPROCESSINGIfweselectthevaluesofRFandRsothatGRRF,thisfactorwillnolongerdependontheop-amp'sinherentgain,anditwillequal)]TJ/F1 9.9626 Tf 9.4092 11.0587 Td[(1 RF.Undertheseconditions,weobtaintheclassicinput-outputrelationshipfortheop-amp-basedinvertingam-plier.vout=)]TJ/F1 9.9626 Tf 9.4091 14.0475 Td[(RF Rvin.30Consequently,thegainprovidedbyourcircuitisentirelydeterminedbyourchoiceofthefeedbackresistorRFandtheinputresistorR.Itisalwaysnegative,andcanbelessthanoneorgreaterthanoneinmagnitude.Itcannotexceedtheop-amp'sinherentgainandshouldnotproducesuchlargeoutputsthatdistortionresultsrememberthepowersupply!.Interestingly,notethatthisrelationshipdoesnotdependontheloadresistance.Thiseectoccursbecauseweuseloadresistanceslargecomparedtotheop-amp'soutputresistance.Thusobservationmeansthat,ifcareful,wecanplaceop-ampcircuitsincascade,withoutincurringtheeectofsucceedingcircuitschangingthebehaviortransferfunctionofpreviousones;seethisproblemProblem3.37.3.19.2ActiveFiltersAslongasdesignrequirementsaremet,theinput-outputrelationfortheinvertingamplieralsoapplieswhenthefeedbackandinputcircuitelementsareimpedancesresistors,capacitors,andinductors. opamp Figure3.45:Vout Vin=)]TJ/F62 8.9664 Tf 8.7034 9.9329 Td[(ZF Z Example3.7Let'sdesignanop-ampcircuitthatfunctionsasalowpasslter.WewantthetransferfunctionbetweentheoutputandinputvoltagetobeHf=K 1+jf fcwhereKequalsthepassbandgainandfcisthecutofrequency.Let'sassumethattheinversionnegativegaindoesnotmatter.Withthetransferfunctionoftheaboveop-ampcircuitinmind,let'sconsidersomechoices.ZF=K,Z=1+jf fc.Thischoicemeansthefeedbackimpedanceisaresistorandthattheinputimpedanceisaseriescombinationofaninductorandaresistor.Incircuitdesign,wetrytoavoidinductorsbecausetheyarephysicallybulkierthancapacitors.

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79ZF=1 1+jf fc,Z=1 K.Considerthereciprocalofthefeedbackimpedanceitsadmittance:ZF)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1=1+jf fc.Sincethisadmittanceisasumofadmittances,thisexpressionsuggeststheparallelcombinationofaresistorvalue=1andacapacitorvalue=1 fcF.Wehavetherightidea,butthevalueslike1arenotright.ConsiderthegeneralRCparallelcombination;itsadmittanceis1 RF+j2fC.LettingtheinputresistanceequalR,thetransferfunctionoftheop-ampinvertingampliernowisHf=)]TJ/F1 9.9626 Tf 9.4091 14.0474 Td[(RF R 1+j2fRFCThus,wehavethegainequaltoRF Randthecutofrequency1 RFC.Creatingaspecictransferfunctionwithop-ampsdoesnothaveauniqueanswer.Asopposedtodesignwithpassivecircuits,electronicsismoreexibleacascadeofcircuitscanbebuiltsothateachhaslittleeectontheothers;seeProblem3.37andgainincreaseinpowerandamplitudecanresult.Tocompleteourexample,let'sassumewewantalowpasslterthatemulateswhatthetelephonecompaniesdo.Signalstransmittedoverthetelephonehaveanupperfrequencylimitofabout3kHz.Fortheseconddesignchoice,werequireRFC=3:310)]TJ/F7 6.9738 Tf 6.2267 0 Td[(4.Thus,manychoicesforresistanceandcapacitancevaluesarepossible.A1Fcapacitoranda330resistor,10nFand33k,and10pFand33Mwouldalltheoreticallywork.Let'salsodesireavoltagegainoften:RF R=10,whichmeansR=RF 10.RecallthatwemusthaveR
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80CHAPTER3.ANALOGSIGNALPROCESSING opamp Figure3.46 opamp Figure3.47 loadresistor.Becausethecurrentgoingintotheop-ampiszero,allofthecurrentowingthroughRowsthroughthefeedbackresistoriF=i!ThevoltageacrossthefeedbackresistorvequalsvinRF R.Becausetheleftendofthefeedbackresistorisessentiallyattachedtothereferencenode,thevoltageacrossitequalsthenegativeofthatacrosstheoutputresistor:vout=)]TJ/F11 9.9626 Tf 7.7487 0 Td[(v=)]TJ/F1 9.9626 Tf 9.4092 8.0698 Td[()]TJ/F10 6.9738 Tf 5.7617 -3.9796 Td[(vinRF R.Usingthisapproachmakesanalyzingnewop-ampcircuitsmucheasier.Whenusingthistechnique,checktomakesuretheresultsyouobtainareconsistentwiththeassumptionsofessentiallyzerocurrententeringtheop-ampandnearlyzerovoltageacrosstheop-amp'sinputs.Example3.8Let'strythisanalysistechniqueonasimpleextensionoftheinvertingampliercongurationshowninFigure3.48TwoSourceCircuit.Ifeitherofthesource-resistorcombinationswerenotpresent,theinvertingamplierremains,andweknowthattransferfunction.Bysuperposition,weknowthattheinput-outputrelationisvout=)]TJ/F1 9.9626 Tf 9.4091 14.0474 Td[(RF R1vin)]TJ/F11 9.9626 Tf 11.1581 6.7398 Td[(RF R2vin.31

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81TwoSourceCircuit Figure3.48:Two-source,single-outputop-ampcircuitexample. Diode Figure3.49:v-irelationandschematicsymbolforthediode.Here,thediodeparameterswereroomtemperatureandI0=1A. Whenwestartfromscratch,thenodejoiningthethreeresistorsisatthesamepotentialasthereference,e0,andthesumofcurrentsowingintothatnodeiszero.Thus,thecurrentiowingintheresistorRFequalsvin R1+vin R2.Becausethefeedbackresistorisessentiallyinparallelwiththeloadresistor,thevoltagesmustsatisfyv=)]TJ/F11 9.9626 Tf 7.7487 0 Td[(vout.Inthisway,weobtaintheinput-outputrelationgivenabove.Whatutilitydoesthiscircuithave?Canthebasicnotionofthecircuitbeextendedwithoutbound?3.20TheDiode32Theresistor,capacitor,andinductorarelinearcircuitelementsinthattheirv-irelationsarelinearinthemathematicalsense.Voltageandcurrentsourcesaretechnicallynonlineardevices:statedsimply,doublingthecurrentthroughavoltagesourcedoesnotdoublethevoltage.Amoreblatant,andveryuseful,nonlinear 32Thiscontentisavailableonlineat.

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82CHAPTER3.ANALOGSIGNALPROCESSINGcircuitelementisthediodelearnmore33.Itsinput-outputrelationhasanexponentialform.it=I0eq kTvt)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1.32Here,thequantityqrepresentsthechargeofasingleelectronincoulombs,kisBoltzmann'sconstant,andTisthediode'stemperatureinK.Atroomtemperature,theratiokT q=25mv.TheconstantI0istheleakagecurrent,andisusuallyverysmall.Viewingthisv-irelationinFigure3.49Diode,thenonlinearitybecomesobvious.Whenthevoltageispositive,currentowseasilythroughthediode.Thissituationisknownasforwardbiasing.Whenweapplyanegativevoltage,thecurrentisquitesmall,andequalsI0,knownastheleakageorreverse-biascurrent.Alessdetailedmodelforthediodehasanypositivecurrentowingthroughthediodewhenitisforwardbiased,andnocurrentwhennegativebiased.Notethatthediode'sschematicsymbollookslikeanarrowhead;thedirectionofcurrentowcorrespondstothedirectionthearrowheadpoints. diodecircuit Figure3.50 Becauseofthediode'snonlinearnature,wecannotuseimpedancesnorseries/parallelcombinationrulesanalyzecircuitscontainingthem.Thereliablenodemethodcanalwaysbeused;itonlyreliesonKVLforitsapplication,andKVLisastatementaboutvoltagedropsaroundaclosedpathregardlessofwhethertheelementsarelinearornot.Thus,forthissimplecircuitwehavevout R=I0eq kTvin)]TJ/F10 6.9738 Tf 6.2266 0 Td[(vout)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1.33Thisequationcannotbesolvedinclosedform.Wemustunderstandwhatisgoingonfrombasicprinciples,usingcomputationalandgraphicalaids.Asanapproximation,whenvinispositive,currentowsthroughthediodesolongasthevoltagevoutissmallerthanvinsothediodeisforwardbiased.Ifthesourceisnegativeorvout"tries"tobebiggerthanvin,thediodeisreverse-biased,andthereverse-biascurrentowsthroughthediode.Thus,atthislevelofanalysis,positiveinputvoltagesresultinpositiveoutputvoltageswithnegativeonesresultinginvout=)]TJ/F8 9.9626 Tf 9.4091 0 Td[(RI0. 33"P-NJunction:PartII"

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83 diodecircuit Figure3.51 Weneedtodetailtheexponentialnonlinearitytodeterminehowthecircuitdistortstheinputvoltagewaveform.WecanofcoursenumericallysolveFigure3.50diodecircuittodeterminetheoutputvoltagewhentheinputisasinusoid.Tolearnmore,let'sexpressthisequationgraphically.Weploteachtermasafunctionofvoutforvariousvaluesoftheinputvoltagevin;wheretheyintersectgivesustheoutputvoltage.Theleftside,thecurrentthroughtheoutputresistor,doesnotvaryitselfwithvin,andthuswehaveaxedstraightline.Asfortherightside,whichexpressesthediode'sv-irelation,thepointatwhichthecurvecrossesthevoutaxisgivesusthevalueofvin.Clearly,thetwocurveswillalwaysintersectjustonceforanyvalueofvin,andforpositivevintheintersectionoccursatavalueforvoutsmallerthanvin.Thisreductionissmallerifthestraightlinehasashallowerslope,whichcorrespondstousingabiggeroutputresistor.Fornegativevin,thediodeisreverse-biasedandtheoutputvoltageequals)]TJ/F8 9.9626 Tf 9.4091 0 Td[(RI0.Whatutilitymightthissimplecircuithave?Thediode'snonlinearitycannotbeescapedhere,andtheclearlyevidentdistortionmusthavesomepracticalapplicationifthecircuitweretobeuseful.Thiscircuit,knownasahalf-waverectier,ispresentinvirtuallyeveryAMradiotwiceandeachservesverydierentfunctions!We'lllearnwhatfunctionslater. diodecircuit Figure3.52 Hereisacircuitinvolvingadiodethatisactuallysimplertoanalyzethanthepreviousone.Weknowthat

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84CHAPTER3.ANALOGSIGNALPROCESSINGthecurrentthroughtheresistormustequalthatthroughthediode.Thus,thediode'scurrentisproportionaltotheinputvoltage.Asthevoltageacrossthediodeisrelatedtothelogarithmofitscurrent,weseethattheinput-outputrelationisvout=)]TJ/F1 9.9626 Tf 9.4091 14.0475 Td[(kT qlnvin RI0+1.34Clearly,thenamelogarithmicamplierisjustiedforthiscircuit.3.21AnalogSignalProcessingProblems34Problem3.1:SolvingSimpleCircuitsaWritethesetofequationsthatgovernCircuitA'sFigure3.53behavior.bSolvetheseequationsfori1:Inotherwords,expressthiscurrentintermsofelementandsourcevaluesbyeliminatingnon-sourcevoltagesandcurrents.cForCircuitB,ndthevalueforRLthatresultsinacurrentof5Apassingthroughit.dWhatisthepowerdissipatedbytheloadresistorRLinthiscase? aCircuitA bCircuitBFigure3.53 Problem3.2:EquivalentResistanceForeachofthefollowingcircuitsFigure3.54,ndtheequivalentresistanceusingseriesandparallelcombinationrules.Calculatetheconductanceseenattheterminalsforcircuitcintermsofeachelement'sconductance.Comparethisequivalentconductanceformulawiththeequivalentresistanceformulayoufoundforcircuitb.Howisthecircuitcderivedfromcircuitb?Problem3.3:SuperpositionPrincipleOneofthemostimportantconsequencesofcircuitlawsistheSuperpositionPrinciple:Thecurrentorvoltagedenedforanyelementequalsthesumofthecurrentsorvoltagesproducedintheelementbytheindependentsources.ThisPrinciplehasimportantconsequencesinsimplifyingthecalculationofciruitvariablesinmultiplesourcecircuits.aForthedepictedcircuitFigure3.55,ndtheindicatedcurrentusinganytechniqueyoulikeyoushouldusethesimplest. 34Thiscontentisavailableonlineat.

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85 acircuita bcircuitb ccircuitc dcircuitdFigure3.54 Figure3.55 bYoushouldhavefoundthatthecurrentiisalinearcombinationofthetwosourcevalues:i=C1vin+C2iin.Thisresultmeansthatwecanthinkofthecurrentasasuperpositionoftwocomponents,eachofwhichisduetoasource.Wecanndeachcomponentbysettingtheothersourcestozero.Thus,tondthevoltagesourcecomponent,youcansetthecurrentsourcetozeroanopencircuitandusetheusualtricks.Tondthecurrentsourcecomponent,youwouldsetthevoltagesourcetozeroashortcircuitandndtheresultingcurrent.CalculatethetotalcurrentiusingtheSuperpositionPrinciple.IsapplyingtheSuperpositionPrincipleeasierthanthetechniqueyouusedinpart?Problem3.4:CurrentandVoltageDividerUsecurrentofvoltagedividerrulestocalculatetheindicatedcircuitvariablesinFigure3.56.

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86CHAPTER3.ANALOGSIGNALPROCESSING acircuita bcircuitc ccircuitbFigure3.56 Problem3.5:ThveninandMayer-NortonEquivalentsFindtheThveninandMayer-NortonequivalentcircuitsforthefollowingcircuitsFigure3.57. acircuita bcircuitb ccircuitcFigure3.57 Problem3.6:DetectiveWorkInthedepictedcircuitFigure3.58,thecircuitN1hasthev-irelationv1=3i1+7whenis=2.aFindtheThveninequivalentcircuitforcircuitN2.

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87bWithis=2,determineRsuchthati1=)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1. Figure3.58 Problem3.7:CartesiantoPolarConversionConvertthefollowingexpressionsintopolarform.Plottheirlocationinthecomplexplane35.a)]TJ/F8 9.9626 Tf 4.5663 -8.0698 Td[(1+p )]TJ/F8 9.9626 Tf 7.7487 0 Td[(32b3+j4c2)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j6 p 3 2+j6 p 3d)]TJ/F8 9.9626 Tf 4.5662 -8.0698 Td[(4)]TJ/F11 9.9626 Tf 9.9626 0 Td[(j3)]TJ/F8 9.9626 Tf 10.7929 -8.0698 Td[(1+j1 2e3ej+4ej 2f)]TJ/F14 9.9626 Tf 4.5662 0.1715 Td[(p 3+j2p 2e)]TJ/F8 9.9626 Tf 6.2266 -0.7472 Td[(j 4g3 1+j3Problem3.8:TheComplexPlaneThecomplexvariablezisrelatedtotherealvariableuaccordingtoz=1+ejuSketchthecontourofvaluesztakesoninthecomplexplane.Whatarethemaximumandminimumvaluesattainablebyjzj?Sketchthecontourtherationalfunctionz)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1 z+1tracesinthecomplexplane.Problem3.9:CoolCurvesInthefollowingexpressions,thevariablexrunsfromzerotoinnity.Whatgeometricshapesdothefollowingtraceinthecomplexplane?aejxb1+ejxce)]TJ/F10 6.9738 Tf 6.2266 0 Td[(xejxdejx+ejx+ 4Problem3.10:TrigonometricIdentitiesandComplexExponentialsShowthefollowingtrigonometricidentitiesusingcomplexexponentials.Inmanycases,theywerederivedusingthisapproach. 35"TheComplexPlane"

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88CHAPTER3.ANALOGSIGNALPROCESSINGasinu=2sinucosubcos2u=1+cos2u 2ccos2u+sin2u=1dd dusinu=cosuProblem3.11:TransferFunctionsFindthetransferfunctionrelatingthecomplexamplitudesoftheindicatedvariableandthesourceshowninFigure3.59.Plotthemagnitudeandphaseofthetransferfunction. acircuita bcircuitb ccircuitc dcircuitdFigure3.59 Problem3.12:UsingImpedancesFindthedierentialequationrelatingtheindicatedvariabletothesourcesusingimpedancesforeachcircuitshowninFigure3.60.Problem3.13:TransferFunctionsInthefollowingcircuitFigure3.61,thevoltagesourceequalsvint=10sin)]TJ/F10 6.9738 Tf 6.2426 -4.1472 Td[(t 2.aFindthetransferfunctionbetweenthesourceandtheindicatedoutputvoltage.bForthegivensource,ndtheoutputvoltage.Problem3.14:ASimpleCircuitYouaregiventhissimplecircuitFigure3.62.aWhatisthetransferfunctionbetweenthesourceandtheindicatedoutputcurrent?bIftheoutputcurrentismeasuredtobecos2t,whatwasthesource?Problem3.15:CircuitDesign

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89 acircuita bcircuitb ccircuitc dcircuitdFigure3.60 Figure3.61 Figure3.62

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90CHAPTER3.ANALOGSIGNALPROCESSING Figure3.63 aFindthetransferfunctionbetweentheinputandtheoutputvoltagesforthecircuitsshowninFig-ure3.63.bAtwhatfrequencydoesthetransferfunctionhaveaphaseshiftofzero?Whatisthecircuit'sgainatthisfrequency?cSpecicationsdemandthatthiscircuithaveanoutputimpedanceitsequivalentimpedancelessthan8forfrequenciesabove1kHz,thefrequencyatwhichthetransferfunctionismaximum.Findelementvaluesthatsatisfythiscriterion.Problem3.16:PowerTransmissionThenetworkshowninFigure3.64arepresentsasimplepowertransmissionsystem.Thegeneratorproduces60HzandismodeledbyasimpleThveninequivalent.Thetransmissionlineconsistsofalonglengthofcopperwireandcanbeaccuratelydescribedasa50resistor.aDeterminetheloadcurrentRLandtheaveragepowerthegeneratormustproducesothattheloadreceives1,000wattsofaveragepower.Whydoesthegeneratorneedtogeneratemorethan1,000wattsofaveragepowertomeetthisrequirement?bSupposetheloadischangedtothatshowninFigure3.64b.Nowhowmuchpowermustthegeneratorproducetomeetthesamepowerrequirement?Whyisitmorethanithadtoproducetomeettherequirementfortheresistiveload?cTheloadcanbecompensatedtohaveaunitypowerfactorseeexerciseExercise3.13sothatthevoltageandcurrentareinphaseformaximumpowereciency.Thecompensationtechniqueistoplaceacircuitinparalleltotheloadcircuit.Whatelementworksandwhatisitsvalue?dWiththiscompensatedcircuit,howmuchpowermustthegeneratorproducetodeliver1,000averagepowertotheload?

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91 aSimplepowertransmissionsystem bModiedloadcir-cuitFigure3.64 Problem3.17:OptimalPowerTransmissionThefollowinggureFigure3.65showsageneralmodelforpowertransmission.ThepowergeneratorisrepresentedbyaThvininequivalentandtheloadbyasimpleimpedance.Inmostapplications,thesourcecomponentsarexedwhilethereissomelatitudeinchoosingtheload.aSupposewewantedthemaximize"voltagetransmission:"makethevoltageacrosstheloadaslargeaspossible.Whatchoiceofloadimpedancecreatesthelargestloadvoltage?Whatisthelargestloadvoltage?bIfwewantedthemaximumcurrenttopassthroughtheload,whatwouldwechoosetheloadimpedancetobe?Whatisthislargestcurrent?cWhatchoicefortheloadimpedancemaximizestheaveragepowerdissipatedintheload?Whatismostpowerthegeneratorcandeliver?Note:Onewaytomaximizeafunctionofacomplexvariableistowritetheexpressionintermsofthevariable'srealandimaginaryparts,evaluatederivativeswithrespecttoeach,setbothderivativestozeroandsolvethetwoequationssimultaneously. Figure3.65 Problem3.18:SharingaChannelTwotransmitter-receiverpairswanttosharethesamedigitalcommunicationschannel.Thetransmittersignalswillbeaddedtogetherbythechannel.Receiverdesignisgreatlysimpliedifrstweremovetheunwantedtransmissionasmuchaspossible.Eachtransmittersignalhastheformxit=Asinfit;0tT

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92CHAPTER3.ANALOGSIGNALPROCESSINGwheretheamplitudeiseitherzeroorAandeachtransmitterusesitsownfrequencyfi.EachfrequencyisharmonicallyrelatedtothebitintervaldurationT,wherethetransmitter1usesthethefrequency1 T.Thedatarateis10Mbps.aDrawablockdiagramthatexpressesthiscommunicationscenario.bFindcircuitsthatthereceiverscouldemploytoseparateunwantedtransmissions.Assumethereceivedsignalisavoltageandtheoutputistobeavoltageaswell.cFindthesecondtransmitter'sfrequencysothatthereceiverscansuppresstheunwantedtransmissionbyatleastafactoroften.Problem3.19:CircuitDetectiveWorkInthelab,theopen-circuitvoltagemeasuredacrossanunknowncircuit'sterminalsequalssint.Whena1resistorisplaceacrosstheterminals,avoltageof1 p 2sin)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(t+ 4appears.aWhatistheThveninequivalentcircuit?bWhatvoltagewillappearifweplacea1Fcapacitoracrosstheterminals?Problem3.20:MoreCircuitDetectiveWorkTheleftterminalpairofatwoterminal-paircircuitisattachedtoatestingcircuit.ThetestsourcevintequalssintFigure3.66. Figure3.66 Wemakethefollowingmeasurements.Withnothingattachedtotheterminalsontheright,thevoltagevtequals1 p 2cos)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(t+ 4.Whenawireisplacedacrosstheterminalsontheright,thecurrentitwas)]TJ/F8 9.9626 Tf 9.4092 0 Td[(sint.aWhatistheimpedanceseenfromtheterminalsontheright?bFindthevoltagevtifacurrentsourceisattachedtotheterminalsontherightsothatit=sint.Problem3.21:Linear,Time-InvariantSystemsForasystemtobecompletelycharacterizedbyatransferfunction,itneedsnotonlybelinear,butalsotobetime-invariant.Asystemissaidtobetime-invariantifdelayingtheinputdelaystheoutputbythesameamount.Mathematically,ifSxt=yt,meaningytistheoutputofasystemSwhenxtistheinput,Sisthetime-invariantifSxt)]TJ/F11 9.9626 Tf 9.9626 0 Td[(=yt)]TJ/F11 9.9626 Tf 9.9626 0 Td[(foralldelaysandallinputsxt.Notethatbothlinearandnonlinearsystemshavethisproperty.Forexample,asystemthatsquaresitsinputistime-invariant.aShowthatifacircuithasxedcircuitelementstheirvaluesdon'tchangeovertime,itsinput-outputrelationshipistime-invariant.Hint:Considerthedierentialequationthatdescribesacircuit'sinput-outputrelationship.Whatisitsgeneralform?Examinethederivativesofdelayedsignals.bShowthatimpedancescannotcharacterizetime-varyingcircuitelementsR,L,andC.Consequently,showthatlinear,time-varyingsystemsdonothaveatransferfunction.

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93cDeterminethelinearityandtime-invarianceofthefollowing.Findthetransferfunctionofthelinear,time-invariantLTIones.idiodeiiyt=xtsinf0tiiiyt=xt)]TJ/F11 9.9626 Tf 9.9626 0 Td[(0ivyt=xt+NtProblem3.22:LongandSleeplessNightsSammywenttolabafteralong,sleeplessnight,andconstructedthecircuitshowninFigure3.67.Hecannotrememberwhatthecircuit,representedbytheimpedanceZ,was.Clearly,thisforgottencircuitisimportantastheoutputisthecurrentpassingthroughit.aWhatistheThveninequivalentcircuitseenbytheimpedance?bInsearchinghisnotes,SammyndsthatthecircuitistorealizethetransferfunctionHf=1 j10f+2FindtheimpedanceZaswellasvaluesfortheothercircuitelements. Figure3.67 Problem3.23:ATestingCircuitThesimplecircuithereFigure3.68wasgivenonatest. Figure3.68 Whenthevotlagesourceisp 5sint,thecurrentit=p 2cos)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(t)]TJ/F8 9.9626 Tf 9.9626 0 Td[(arctan)]TJ/F10 6.9738 Tf 11.1581 3.9226 Td[( 4.aWhatisvoltagevoutt?

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94CHAPTER3.ANALOGSIGNALPROCESSINGbWhatistheimpedanceZatthefrequencyofthesource?Problem3.24:MysteryCircuitYouaregivenacircuitFigure3.69thathastwoterminalsforattachingcircuitelements. Figure3.69 Whenyouattachavoltagesourceequalingsinttotheterminals,thecurrentthroughthesourceequals4sin)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(t+ 4)]TJ/F8 9.9626 Tf 10.9147 0 Td[(2sint.Whennosourceisattachedopen-circuitedterminals,thevoltageacrosstheterminalshastheformAsint+.aWhatwilltheterminalcurrentbewhenyoureplacethesourcebyashortcircuit?bIfyouweretobuildacircuitthatwasidenticalfromtheviewpointoftheterminalstothegivenone,whatwouldyourcircuitbe?cForyourcircuit,whatareAand?Problem3.25:MysteryCircuitSammymustdetermineasmuchashecanaboutamysterycircuitbyattachingelementstotheterminalandmeasuringtheresultingvoltage.Whenheattachesa1resistortothecircuit'sterminals,hemeasuresthevoltageacrosstheterminalstobe3sint.Whenheattachesa1Fcapacitoracrosstheterminals,thevoltageisnow3p 2sin)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(t)]TJ/F10 6.9738 Tf 11.1581 3.9226 Td[( 4.aWhatvoltageshouldhemeasurewhenheattachesnothingtothemysterycircuit?bWhatvoltageshouldSammymeasureifhedoubledthesizeofthecapacitorto2Fandattachedittothecircuit?Problem3.26:FindtheLoadImpedanceThedepictedcircuitFigure3.70hasatransferfunctionbetweentheoutputvoltageandthesourceequaltoHf=)]TJ/F8 9.9626 Tf 7.7487 0 Td[(82f2 )]TJ/F8 9.9626 Tf 7.7488 0 Td[(82f2+4+j6f.

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95 Figure3.70 aSketchthemagnitudeandphaseofthetransferfunction.bAtwhatfrequencydoesthephaseequal 2?cFindacircuitthatcorrespondstothisloadimpedance.Isyouranswerunique?Ifso,showittobeso;ifnot,giveanotherexample.Problem3.27:AnalogHumRejectionHumreferstocorruptionfromwallsocketpowerthatfrequentlysneaksintocircuits.Humgetsitsnamebecauseitsoundslikeapersistenthummingsound.Wewanttondacircuitthatwillremovehumfromanysignal.ARiceengineersuggestsusingasimplevoltagedividercircuitFigure3.71consistingoftwoseriesimpedances. Figure3.71 aTheimpedanceZ1isaresistor.TheRiceengineermustdecidebetweentwocircuitsFigure3.72fortheimpedanceZ2.Whichofthesewillwork?bPickingonecircuitthatworks,choosecircuitelementvaluesthatwillremovehum.cSketchthemagnitudeoftheresultingfrequencyresponse.

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96CHAPTER3.ANALOGSIGNALPROCESSING Figure3.72 Problem3.28:AnInterestingCircuit Figure3.73 aForthecircuitshowninFigure3.73,ndthetransferfunction.bWhatistheoutputvoltagewhentheinputhastheformiin=5sint?Problem3.29:ACircuitYouaregiventhedepictedcircuitFigure3.74. Figure3.74

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97aWhatisthetransferfunctionbetweenthesourceandtheoutputvoltage?bWhatwillthevoltagebewhenthesourceequalssint?cManyfunctiongeneratorsproduceaconstantosetinadditiontoasinusoid.Ifthesourceequals1+sint,whatistheoutputvoltage?Problem3.30:AnInterestingandUsefulCircuitThedepictedcircuitFigure3.75hasinterestingproperties,whichareexploitedinhigh-performanceoscilloscopes. Figure3.75 Theportionofthecircuitlabeled"Oscilloscope"representsthescope'sinputimpedance.R2=1MandC2=30pFnotethelabelunderthechannel1inputinthelab'soscilloscopes.Aprobeisadevicetoattachanoscilloscopetoacircuit,andithastheindicatedcircuitinsideit.aSupposeforamomentthattheprobeismerelyawireandthattheoscilloscopeisattachedtoacircuitthathasaresistiveThveninequivalentimpedance.Whatwouldbetheeectoftheoscilloscope'sinputimpedanceonmeasuredvoltages?bUsingthenodemethod,ndthetransferfunctionrelatingtheindicatedvoltagetothesourcewhentheprobeisused.cPlotthemagnitudeandphaseofthistransferfunctionwhenR1=9MandC1=2pF.dForaparticularrelationshipamongtheelementvalues,thetransferfunctionisquitesimple.Findthatrelationshipanddescribewhatissospecialaboutit.eThearrowthroughC1indicatesthatitsvaluecanbevaried.Selectthevalueforthiscapacitortomakethespecialrelationshipvalid.Whatistheimpedanceseenbythecircuitbeingmeasuredforthisspecialvalue?Problem3.31:ACircuitProblemYouaregiventhedepictedcircuitFigure3.76.aFindthedierentialequationrelatingtheoutputvoltagetothesource.bWhatistheimpedanceseenbythecapacitor?Problem3.32:AnalogComputersBecausethedierentialequationsarisingincircuitsresemblethosethatdescribemechanicalmotion,wecanusecircuitmodelstodescribemechanicalsystems.AnELEC241studentwantstounderstandthesuspensionsystemonhiscar.Withoutasuspension,thecar'sbodymovesinconcertwiththebumpsintheraod.Awell-designedsuspensionsystemwillsmoothoutbumpyroads,reducingthecar'sverticalmotion.

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98CHAPTER3.ANALOGSIGNALPROCESSING Figure3.76 Figure3.77 Ifthebumpsareverygradualthinkofahillasalargebutverygradualbump,thecar'sverticalmotionshouldfollowthatoftheroad.Thestudentwantstondasimplecircuitthatwillmodelthecar'smotion.HeistryingtodecidebetweentwocircuitmodelsFigure3.77.Here,roadandcardisplacementsarerepresentedbythevoltagesvroadtandvcart,respectively.aWhichcircuitwouldyoupick?Why?bForthecircuityoupicked,whatwillbetheamplitudeofthecar'smotioniftheroadhasadisplacementgivenbyvroadt=1+sint?Problem3.33:DependentSourcesFindthevoltagevoutineachofthedepictedcircuitsFigure3.78.Problem3.34:TransferFunctionsandCircuitsYouaregiventhedepictednetworkFigure3.79.

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99 acircuita bcircuitbFigure3.78 Figure3.79 aFindthetransferfunctionbetweenVinandVout.bSketchthemagnitudeandphaseofyourtransferfunction.Labelimportantfrequency,amplitudeandphasevalues.cFindvouttwhenvint=sin)]TJ/F10 6.9738 Tf 6.2426 -4.1472 Td[(t 2+ 4.Problem3.35:FunintheLabYouaregivenanunopenableboxthathastwoterminalsstickingout.Youassumetheboxcontainsacircuit.Youmeasurethevoltagesin)]TJ/F11 9.9626 Tf 4.5663 -8.0699 Td[(t+ 4acrosstheterminalswhennothingisconnectedtothemandthecurrentp 2costwhenyouplaceawireacrosstheterminals.aFindacircuitthathasthesecharacteristics.bYouattacha1Hinductoracrosstheterminals.Whatvoltagedoyoumeasure?Problem3.36:OperationalAmpliersFindthetransferfunctionbetweenthesourcevoltagesandtheindicatedoutputvoltageforthecircuitsshowninFigure3.80.

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100CHAPTER3.ANALOGSIGNALPROCESSING aop-ampa bop-ampb cop-ampc dop-ampdFigure3.80 Problem3.37:WhyOp-AmpsareUsefulThecircuitFigure3.81ofacascadeofop-ampcircuitsillustratethereasonwhyop-amprealizationsoftransferfunctionsaresouseful.aFindthetransferfunctionrelatingthecomplexamplitudeofthevoltagevoutttothesource.Showthatthistransferfunctionequalstheproductofeachstage'stransferfunction.bWhatistheloadimpedanceappearingacrosstherstop-amp'soutput?cFigure3.82illustratesthatsometimesdesignscangowrong.Findthetransferfunctionforthisop-ampcircuitFigure3.82,andthenshowthatitcan'twork!Whycan'tit?

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101 Figure3.81 Figure3.82 Problem3.38:OperationalAmpliersConsiderthedepictedcircuitFigure3.83.aFindthetransferfunctionrelatingthevoltagevoutttothesource.bInparticular,R1=530,C1=1F,R2=5:3k,C2=0:01F,andR3=R4=5:3k.Characterizetheresultingtransferfunctionanddeterminewhatusethiscircuitmighthave.Problem3.39:DesigningaBandpassFilterWewanttodesignabandpasslterthathastransferthefunctionHf=10j2f jf fl+1jf fh+1Here,flisthecutofrequencyofthelow-frequencyedgeofthepassbandandfhisthecutofrequencyofthehigh-frequencyedge.Wewantfl=1kHzandfh=10kHz.aPlotthemagnitudeandphaseofthisfrequencyresponse.Labelimportantamplitudeandphasevaluesandthefrequenciesatwhichtheyoccur.

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102CHAPTER3.ANALOGSIGNALPROCESSING Figure3.83 bDesignabandpasslterthatmeetsthesespecications.Specifycomponentvalues.Problem3.40:Pre-emphasisorDe-emphasis?Inaudioapplications,priortoanalog-to-digitalconversionsignalsarepassedthroughwhatisknownasapre-emphasiscircuitthatleavesthelowfrequenciesalonebutprovidesincreasinggainatincreasinglyhigherfrequenciesbeyondsomefrequencyf0.De-emphasiscircuitsdotheoppositeandareappliedafterdigital-to-analogconversion.Afterpre-emphasis,digitization,conversionbacktoanalogandde-emphasis,thesignal'sspectrumshouldbewhatitwas.Theop-ampcircuithereFigure3.84hasbeendesignedforpre-emphasisorde-emphasisSamanthacan'trecallwhich. Figure3.84 aIsthisapre-emphasisorde-emphasiscircuit?Findthefrequencyf0thatdenesthetransitionfromlowtohighfrequencies.bWhatisthecircuit'soutputwhentheinputvoltageissinft,withf=4kHz?cWhatcircuitcouldperformtheoppositefunctiontoyouranswerfortherstpart?

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103Problem3.41:ActiveFilterFindthetransferfunctionofthedepictedactivelterFigure3.85. Figure3.85 Problem3.42:Thisisalter?YouaregivenacircuitFigure3.86. Figure3.86 aWhatisthiscircuit'stransferfunction?Plotthemagnitudeandphase.bIftheinputsignalisthesinusoidsinf0t,whatwilltheoutputbewhenf0islargerthanthelter'scutofrequency?Problem3.43:OpticalReceiversInyouropticaltelephone,thereceivercircuithadtheformshownFigure3.87.

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104CHAPTER3.ANALOGSIGNALPROCESSING Figure3.87 Thiscircuitservedasatransducer,convertinglightenergyintoavoltagevout.Thephotodiodeactsasacurrentsource,producingacurrentproportionaltothelightintesityfallinguponit.Asisoftenthecaseinthiscrucialstage,thesignalsaresmallandnoisecanbeaproblem.Thus,theop-ampstageservestoboostthesignalandtolterout-of-bandnoise.aFindthetransferfunctionrelatinglightintensitytovout.bWhatshouldthecircuitrealizingthefeedbackimpedanceZfbesothatthetransduceractsasa5kHzlowpasslter?cAcleverengineersuggestsanalternativecircuitFigure3.88toaccomplishthesametask.Determinewhethertheideaworksornot.Ifitdoes,ndtheimpedanceZinthataccomplishesthelowpasslteringtask.Ifnot,showwhyitdoesnotwork. Figure3.88 Problem3.44:ReverseEngineeringThedepictedcircuitFigure3.89hasbeendevelopedbytheTBBGElectronicsdesigngroup.Theyaretryingtokeepitsusesecret;we,representingRUElectronics,havediscoveredtheschematicandwanttogureouttheintendedapplication.Assumethediodeisideal.

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105 Figure3.89 aAssumingthediodeisashort-circuitithasbeenremovedfromthecircuit,whatisthecircuit'stransferfunction?bWiththediodeinplace,whatisthecircuit'soutputwhentheinputvoltageissinf0t?cWhatfunctionmightthiscircuithave?

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106CHAPTER3.ANALOGSIGNALPROCESSINGSolutionstoExercisesinChapter3SolutiontoExercise3.1p.36Onekilowatt-hourequals360,000watt-seconds,whichindeeddirectlycorrespondsto360,000joules.SolutiontoExercise3.2p.41KCLsaysthatthesumofcurrentsenteringorleavinganodemustbezero.Ifweconsidertwonodestogetherasa"supernode",KCLappliesaswelltocurrentsenteringthecombination.Sincenocurrentsenteranentirecircuit,thesumofcurrentsmustbezero.Ifwehadatwo-nodecircuit,theKCLequationofonemustbethenegativeoftheother,Wecancombineallbutonenodeinacircuitintoasupernode;KCLforthesupernodemustbethenegativeoftheremainingnode'sKCLequation.Consequently,specifyingn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1KCLequationsalwaysspeciestheremainingone.SolutiontoExercise3.3p.43ThecircuitservesasanamplierhavingagainofR2 R1+R2.SolutiontoExercise3.4p.44ThepowerconsumedbytheresistorR1canbeexpressedasvin)]TJ/F11 9.9626 Tf 9.9626 0 Td[(voutiout=R1 R1+R22vin2SolutiontoExercise3.5p.441 R1+R2vin2=R1 R1+R22vin2++R2 R1+R22vin2SolutiontoExercise3.6p.46Replacingthecurrentsourcebyavoltagesourcedoesnotchangethefactthatthevoltagesareidentical.Consequently,vin=R2ioutoriout=vin R2.ThisresultdoesnotdependontheresistorR1,whichmeansthatwesimplyhavearesistorR2acrossavoltagesource.Thetwo-resistorcircuithasnoapparentuse.SolutiontoExercise3.7p.47Req=R2 1+R2 RL.Thus,a10%changemeansthattheratioR2 RLmustbelessthan0.1.A1%changemeansthatR2 RL<0:01.SolutiontoExercise3.8p.49Inaseriescombinationofresistors,thecurrentisthesameineach;inaparallelcombination,thevoltageisthesame.Foraseriescombination,theequivalentresistanceisthesumoftheresistances,whichwillbelargerthananycomponentresistor'svalue;foraparallelcombination,theequivalentconductanceisthesumofthecomponentconductances,whichislargerthananycomponentconductance.Theequivalentresistanceisthereforesmallerthananycomponentresistance.SolutiontoExercise3.9p.51voc=R2 R1+R2vinandisc=)]TJ/F1 9.9626 Tf 9.4091 11.0586 Td[(vin R1resistorR2isshortedoutinthiscase.Thus,veq=R2 R1+R2vinandReq=R1R2 R1+R2.SolutiontoExercise3.10p.52ieq=R1 R1+R2iinandReq=R3kR1+R2.SolutiontoExercise3.11p.58Divisionbyj2farisesfromintegratingacomplexexponential.Consequently,1 j2fV,ZVej2ftdt

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107SolutiontoExercise3.12p.59Formaximumpowerdissipation,theimaginarypartofcomplexpowershouldbezero.AsthecomplexpowerisgivenbyVI=jVjjIjej)]TJ/F10 6.9738 Tf 6.2267 0 Td[(,zeroimaginarypartoccurswhenthephasesofthevoltageandcurrentsagree.SolutiontoExercise3.13p.59Pave=VrmsIrmscos)]TJ/F11 9.9626 Tf 9.9626 0 Td[(.Thecosinetermisknownasthepowerfactor.SolutiontoExercise3.14p.64Thekeynotioniswritingtheimaginarypartasthedierencebetweenacomplexexponentialanditscomplexconjugate:Im)]TJ/F11 9.9626 Tf 4.5663 -8.0699 Td[(Vej2ft=Vej2ft)]TJ/F11 9.9626 Tf 9.9626 0 Td[(Ve)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2ft 2j.35TheresponsetoVej2ftisVHfej2ft,whichmeanstheresponsetoVe)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2ftisVH)]TJ/F11 9.9626 Tf 7.7487 0 Td[(fe)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2ft.AsH)]TJ/F11 9.9626 Tf 7.7487 0 Td[(f=)]TJ/F11 9.9626 Tf 4.5662 -8.0699 Td[(Hf,theSuperpositionPrinciplesaysthattheoutputtotheimaginarypartisIm)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(VHfej2ft.Thesameargumentholdsfortherealpart:Re)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(Vej2ft!Re)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(VHfej2ft.SolutiontoExercise3.15p.69Tondtheequivalentresistance,weneedtondthecurrentowingthroughthevoltagesource.Thiscurrentequalsthecurrentwehavejustfoundplusthecurrentowingthroughtheothervertical1resistor.Thiscurrentequalse1 1=6 13vin,makingthetotalcurrentthroughthevoltagesourceowingoutofit11 13vin.Thus,theequivalentresistanceis13 11.SolutiontoExercise3.16p.71Notnecessarily,especiallyifwedesireindividualknobsforadjustingthegainandthecutofrequency.SolutiontoExercise3.17p.79Theratiobetweenadjacentvaluesisaboutp 2.

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108CHAPTER3.ANALOGSIGNALPROCESSING

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Chapter4FrequencyDomain4.1IntroductiontotheFrequencyDomain1Indevelopingwaysofanalyzinglinearcircuits,weinventedtheimpedancemethodbecauseitmadesolvingcircuitseasier.Alongtheway,wedevelopedthenotionofacircuit'sfrequencyresponseortransferfunction.Thisnotion,whichalsoappliestoalllinear,time-invariantsystems,describeshowthecircuitrespondstoasinusoidalinputwhenweexpressitintermsofacomplexexponential.WealsolearnedtheSuperpositionPrincipleforlinearsystems:Thesystem'soutputtoaninputconsistingofasumoftwosignalsisthesumofthesystem'soutputstoeachindividualcomponent.Thestudyofthefrequencydomaincombinesthesetwonotionsasystem'ssinusoidalresponseiseasytondandalinearsystem'soutputtoasumofinputsisthesumoftheindividualoutputstodevelopthecrucialideaofasignal'sspectrum.Webeginbyndingthatthosesignalsthatcanberepresentedasasumofsinusoidsisverylarge.Infact,allsignalscanbeexpressedasasuperpositionofsinusoids.Asthisstoryunfolds,we'llseethatinformationsystemsrelyheavilyonspectralideas.Forexample,radio,television,andcellulartelephonestransmitoverdierentportionsofthespectrum.Infact,spectrumissoimportantthatcommunicationssystemsareregulatedastowhichportionsofthespectrumtheycanusebytheFederalCommunicationsCommissionintheUnitedStatesandbyInternationalTreatyfortheworldseeFrequencyAllocationsSection7.3.Calculatingthespectrumiseasy:TheFouriertransformdeneshowwecanndasignal'sspectrum.4.2ComplexFourierSeries2InanearliermoduleExercise2.4,weshowedthatasquarewavecouldbeexpressedasasuperpositionofpulses.Asusefulasthisdecompositionwasinthisexample,itdoesnotgeneralizewelltootherperiodicsignals:Howcanasuperpositionofpulsesequalasmoothsignallikeasinusoid?Becauseoftheimportanceofsinusoidstolinearsystems,youmightwonderwhethertheycouldbeaddedtogethertorepresentalargenumberofperiodicsignals.Youwouldberightandingoodcompanyaswell.Euler3andGauss4inparticularworriedaboutthisproblem,andJeanBaptisteFourier5gotthecrediteventhoughtoughmathematicalissueswerenotsettleduntillater.TheyworkedonwhatisnowknownastheFourierseries:representinganyperiodicsignalasasuperpositionofsinusoids.ButtheFourierseriesgoeswellbeyondbeinganothersignaldecompositionmethod.Rather,theFourierseriesbeginsourjourneytoappreciatehowasignalcanbedescribedineitherthetime-domainorthe 1Thiscontentisavailableonlineat.2Thiscontentisavailableonlineat.3http://www-groups.dcs.st-and.ac.uk/history/Mathematicians/Euler.html4http://www-groups.dcs.st-and.ac.uk/history/Mathematicians/Guass.html5http://www-groups.dcs.st-and.ac.uk/history/Mathematicians/Fourier.html109

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110CHAPTER4.FREQUENCYDOMAINfrequency-domainwithnocompromise.LetstbeaperiodicsignalwithperiodT.Wewanttoshowthatperiodicsignals,eventhosethathaveconstant-valuedsegmentslikeasquarewave,canbeexpressedassumofharmonicallyrelatedsinewaves:sinusoidshavingfrequenciesthatareintegermultiplesofthefundamentalfrequency.BecausethesignalhasperiodT,thefundamentalfrequencyis1 T.ThecomplexFourierseriesexpressesthesignalasasuperpositionofcomplexexponentialshavingfrequenciesk T,k=f:::;)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1;0;1;:::g.st=1Xk=ckej2kt T.1withck=1 2ak+)]TJ/F8 9.9626 Tf 9.4091 0 Td[(jbk.TherealandimaginarypartsoftheFouriercoecientsckarewritteninthisunusualwayforconvenienceindeningtheclassicFourierseries.Thezerothcoecientequalsthesignal'saveragevalueandisreal-valuedforreal-valuedsignals:c0=a0.Thefamilyoffunctionsnej2kt ToarecalledbasisfunctionsandformthefoundationoftheFourierseries.Nomatterwhattheperiodicsignalmightbe,thesefunctionsarealwayspresentandformtherepresentation'sbuildingblocks.TheydependonthesignalperiodT,andareindexedbyk.Keypoint:Assumingweknowtheperiod,knowingtheFouriercoecientsisequivalenttoknowingthesignal.Thus,itmakesnotdierenceifwehaveatime-domainorafrequency-domaincharacterizationofthesignal.Exercise4.1Solutiononp.152.WhatisthecomplexFourierseriesforasinusoid?TondtheFouriercoecients,wenotetheorthogonalitypropertyZT0ej2kt Te)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2lt Tdt=8<:Tifk=l0ifk6=l.2AssumingforthemomentthatthecomplexFourierseries"works,"wecanndasignal'scomplexFouriercoecients,itsspectrum,byexploitingtheorthogonalitypropertiesofharmonicallyrelatedcomplexexpo-nentials.Simplymultiplyeachsideof.1bye)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2ltandintegrateovertheinterval[0;T].ck=1 TRT0ste)]TJ/F8 9.9626 Tf 6.2266 -0.7472 Td[(j2kt Tdtc0=1 TRT0stdt.3Example4.1FindingtheFourierseriescoecientsforthesquarewaveisverysimple.sqTt.Mathematically,thissignalcanbeexpressedassqTt=8<:1if0t
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111Thus,thecomplexFourierseriesforthesquarewaveissqt=Xk2f:::;)]TJ/F7 6.9738 Tf 6.2267 0 Td[(3;)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1;1;3;:::g2 jke+j2kt T.6Consequently,thesquarewaveequalsasumofcomplexexponentials,butonlythosehavingfre-quenciesequaltooddmultiplesofthefundamentalfrequency1 T.Thecoecientsdecayslowlyasthefrequencyindexkincreases.Thisindexcorrespondstothek-thharmonicofthesignal'speriod.Asignal'sFourierseriesspectrumckhasinterestingproperties.Property4.1:Ifstisreal,c)]TJ/F10 6.9738 Tf 6.2267 0 Td[(k=ckreal-valuedperiodicsignalshaveconjugate-symmetricspectra.Thisresultfollowsfromtheintegralthatcalculatestheckfromthesignal.Furthermore,thisresultmeansthatReck=Rec)]TJ/F10 6.9738 Tf 6.2267 0 Td[(k:TherealpartoftheFouriercoecientsforreal-valuedsignalsiseven.Similarly,Imck=)]TJ/F8 9.9626 Tf 9.4091 0 Td[(Imc)]TJ/F10 6.9738 Tf 6.2267 0 Td[(k:TheimaginarypartsoftheFouriercoecientshaveoddsymmetry.Consequently,ifyouaregiventheFouriercoecientsforpositiveindicesandzeroandaretoldthesignalisreal-valued,youcanndthenegative-indexedcoecients,hencetheentirespectrum.Thiskindofsymmetry,ck=ck,isknownasconjugatesymmetry.Property4.2:Ifs)]TJ/F11 9.9626 Tf 7.7487 0 Td[(t=st,whichsaysthesignalhasevensymmetryabouttheorigin,c)]TJ/F10 6.9738 Tf 6.2267 0 Td[(k=ck.Giventhepreviouspropertyforreal-valuedsignals,theFouriercoecientsofevensignalsarereal-valued.Areal-valuedFourierexpansionamountstoanexpansionintermsofonlycosines,whichisthesimplestexampleofanevensignal.Property4.3:Ifs)]TJ/F11 9.9626 Tf 7.7487 0 Td[(t=)]TJ/F8 9.9626 Tf 9.4091 0 Td[(st,whichsaysthesignalhasoddsymmetry,c)]TJ/F10 6.9738 Tf 6.2266 0 Td[(k=)]TJ/F11 9.9626 Tf 7.7488 0 Td[(ck.Therefore,theFouriercoecientsarepurelyimaginary.Thesquarewaveisagreatexampleofanodd-symmetricsignal.Property4.4:Thespectralcoecientsforaperiodicsignaldelayedbyst)]TJ/F11 9.9626 Tf 9.9626 0 Td[(arecke)]TJ/F8 9.9626 Tf 6.2266 -0.7472 Td[(j2k T,whereckdenotesthespectrumofst.Delayingasignalbysecondsresultsinaspectrumhavingalinearphaseshiftof)]TJ/F1 9.9626 Tf 9.4092 8.0698 Td[()]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(2k Tincomparisontothespectrumoftheundelayedsignal.Notethatthespectralmagnitudeisunaected.Showingthispropertyiseasy.Proof:1 TZT0st)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2kt Tdt=1 TZT)]TJ/F10 6.9738 Tf 6.2266 0 Td[()]TJ/F10 6.9738 Tf 6.2267 0 Td[(ste)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2kt+ Tdt=1 Te)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2k TZT)]TJ/F10 6.9738 Tf 6.2266 0 Td[()]TJ/F10 6.9738 Tf 6.2267 0 Td[(ste)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2kt Tdt.7Notethattherangeofintegrationextendsoveraperiodoftheintegrand.Consequently,itshouldnotmatterhowweintegrateoveraperiod,whichmeansthatRT)]TJ/F10 6.9738 Tf 6.2267 0 Td[()]TJ/F10 6.9738 Tf 6.2266 0 Td[(dt=RT0dt,andwehaveourresult.ThecomplexFourierseriesobeysParseval'sTheorem,oneofthemostimportantresultsinsignalanalysis.Thisgeneralmathematicalresultsaysyoucancalculateasignal'spowerineitherthetimedomainorthefrequencydomain.Theorem4.1:Parseval'sTheoremAveragepowercalculatedinthetimedomainequalsthepowercalculatedinthefrequencydomain.1 TZT0s2tdt=1Xk=jckj2.8

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112CHAPTER4.FREQUENCYDOMAINThisresultisasimplerre-expressionofhowtocalculateasignal'spowerthanwiththereal-valuedFourierseriesexpressionforpower.22.Let'scalculatetheFouriercoecientsoftheperiodicpulsesignalshownhereFigure4.1. Figure4.1:Periodicpulsesignal.Thepulsewidthis,theperiodT,andtheamplitudeA.ThecomplexFourierspectrumofthissignalisgivenbyck=1 TZ0Ae)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(j2k Tdt=)]TJ/F1 9.9626 Tf 9.4091 14.0474 Td[(A j2ke)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(j2k T)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1Atthispoint,simplifyingthisexpressionrequiresknowinganinterestingproperty.1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j=e)]TJ/F8 9.9626 Tf 6.2266 -0.7472 Td[(j 2e+j 2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(j 2=e)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(j 22jsin 2Armedwiththisresult,wecansimplyexpresstheFourierseriescoecientsforourpulsesequence.ck=Ae)]TJ/F8 9.9626 Tf 6.2266 -0.7472 Td[(jk Tsin)]TJ/F10 6.9738 Tf 5.7617 -4.1472 Td[(k T k.9Becausethissignalisreal-valued,wendthatthecoecientsdoindeedhaveconjugatesymmetry:ck=c)]TJ/F10 6.9738 Tf 6.2267 0 Td[(k.Theperiodicpulsesignalhasneitherevennoroddsymmetry;consequently,noadditionalsymmetryexistsinthespectrum.Becausethespectrumiscomplexvalued,toplotitweneedtocalculateitsmagnitudeandphase.jckj=Ajsin)]TJ/F10 6.9738 Tf 5.7618 -4.1472 Td[(k T kj.10ck=)]TJ/F1 9.9626 Tf 9.4091 14.0475 Td[(k T+negsin)]TJ/F10 6.9738 Tf 5.7618 -4.1472 Td[(k T k!signkThefunctionnegequals)]TJ/F8 9.9626 Tf 7.7488 0 Td[(1ifitsargumentisnegativeandzerootherwise.Thesomewhatcomplicatedexpressionforthephaseresultsbecausethesinetermcanbenegative;magnitudesmustbepositive,leavingtheoccasionalnegativevaluestobeaccountedforasaphaseshiftof.

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113 PeriodicPulseSequence Figure4.2:Themagnitudeandphaseoftheperiodicpulsesequence'sspectrumisshownforpositive-frequencyindices.Here T=0:2andA=1. Alsonotethepresenceofalinearphasetermtherstterminckisproportionaltofrequencyk T.Comparingthistermwiththatpredictedfromdelayingasignal,adelayof 2ispresentinoursignal.Advancingthesignalbythisamountcentersthepulseabouttheorigin,leavinganevensignal,whichinturnmeansthatitsspectrumisreal-valued.Thus,ourcalculatedspectrumisconsistentwiththepropertiesoftheFourierspectrum.Exercise4.2Solutiononp.152.Whatisthevalueofc0?Recallingthatthisspectralcoecientcorrespondstothesignal'saveragevalue,doesyouranswermakesense?ThephaseplotshowninFigure4.2PeriodicPulseSequencerequiressomeexplanationasitdoesnotseemtoagreewithwhat.10suggests.There,thephasehasalinearcomponent,withajumpofeverytimethesinusoidaltermchangessign.Wemustrealizethatanyintegermultipleof2canbeaddedtoaphaseateachfrequencywithoutaectingthevalueofthecomplexspectrum.Weseethatatfrequencyindex4thephaseisnearly)]TJ/F11 9.9626 Tf 7.7487 0 Td[(.Thephaseatindex5isundenedbecausethemagnitudeiszerointhisexample.Atindex6,theformulasuggeststhatthephaseofthelineartermshouldbelessthanmorenegativethan)]TJ/F11 9.9626 Tf 7.7488 0 Td[(.Inaddition,weexpectashiftof)]TJ/F11 9.9626 Tf 7.7487 0 Td[(inthephasebetweenindices4and6.Thus,thephasevaluepredictedbytheformulaisalittlelessthan)]TJ/F8 9.9626 Tf 9.4092 0 Td[(.Becausewecanadd2withoutaectingthevalueofthespectrumatindex6,theresultisaslightlynegativenumberasshown.Thus,theformulaandtheplotdoagree.InphasecalculationslikethosemadeinMATLAB,valuesareusuallyconnedtotherange[)]TJ/F11 9.9626 Tf 7.7488 0 Td[(;byaddingsomepossiblynegativemultipleof2toeachphasevalue.

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114CHAPTER4.FREQUENCYDOMAIN4.3ClassicFourierSeries6TheclassicFourierseriesasderivedoriginallyexpressedaperiodicsignalperiodTintermsofharmonicallyrelatedsinesandcosines.st=a0+1Xk=1akcos2kt T+1Xk=1bksin2kt T.11ThecomplexFourierseriesandthesine-cosineseriesareidentical,eachrepresentingasignal'sspectrum.TheFouriercoecients,akandbk,expresstherealandimaginarypartsrespectivelyofthespectrumwhilethecoecientsckofthecomplexFourierseriesexpressthespectrumasamagnitudeandphase.EquatingtheclassicFourierseries.11tothecomplexFourierseries.1,anextrafactoroftwoandcomplexconjugatebecomenecessarytorelatetheFouriercoecientsineach.ck=1 2ak)]TJ/F11 9.9626 Tf 9.9626 0 Td[(jbkExercise4.3Solutiononp.152.DerivethisrelationshipbetweenthecoecientsofthetwoFourierseries.JustaswiththecomplexFourierseries,wecanndtheFouriercoecientsusingtheorthogonalitypropertiesofsinusoids.Notethatthecosineandsineofharmonicallyrelatedfrequencies,eventhesamefrequency,areorthogonal.ZT0sin2kt Tcos2lt Tdt=0;k2Zl2Z.12ZT0sin2kt Tsin2lt Tdt=8<:T 2ifk=landk6=0andl6=00ifk6=lork=0=lZT0cos2kt Tcos2lt Tdt=8>><>>:T 2ifk=landk6=0andl6=0Tifk=0=l0ifk6=lTheseorthogonalityrelationsfollowfromthefollowingimportanttrigonometricidentities.sinsin=1 2cos)]TJ/F11 9.9626 Tf 9.9626 0 Td[()]TJ/F8 9.9626 Tf 9.9626 0 Td[(cos+coscos=1 2cos++cos)]TJ/F11 9.9626 Tf 9.9626 0 Td[(sincos=1 2sin++sin)]TJ/F11 9.9626 Tf 9.9626 0 Td[(.13Theseidentitiesallowyoutosubstituteasumofsinesand/orcosinesforaproductofthem.Eachterminthesumcanbeintegratingbynoticingoneoftwoimportantpropertiesofsinusoids.Theintegralofasinusoidoveranintegernumberofperiodsequalszero.Theintegralofthesquareofaunit-amplitudesinusoidoveraperiodTequalsT 2. 6Thiscontentisavailableonlineat.

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115Tousethese,let's,forexample,multiplytheFourierseriesforasignalbythecosineofthelthharmoniccos)]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(2lt Tandintegrate.Theideaisthat,becauseintegrationislinear,theintegrationwillsiftoutallbuttheterminvolvingal.RT0stcos)]TJ/F72 7.9701 Tf 6.675 -4.9766 Td[(2lt Tdt=RT0a0cos)]TJ/F72 7.9701 Tf 6.675 -4.9766 Td[(2lt Tdt+P1k=1akRT0cos)]TJ/F72 7.9701 Tf 6.675 -4.9766 Td[(2kt Tcos)]TJ/F72 7.9701 Tf 6.675 -4.9766 Td[(2lt Tdt+P1k=1bkRT0sin)]TJ/F72 7.9701 Tf 6.675 -4.9766 Td[(2kt Tcos)]TJ/F72 7.9701 Tf 6.675 -4.9767 Td[(2lt Tdt.14Therstandthirdtermsarezero;inthesecond,theonlynon-zeroterminthesumresultswhentheindiceskandlareequalbutnotzero,inwhichcaseweobtainalT 2.Ifk=0=l,weobtaina0T.Consequently,al=2 TZT0stcos2lt Tdt;l6=0AlloftheFouriercoecientscanbefoundsimilarly.a0=1 TZT0stdtak=2 TZT0stcos2kt Tdt;k6=0bk=2 TZT0stsin2kt Tdt.15Exercise4.4Solutiononp.152.Theexpressionfora0isreferredtoastheaveragevalueofst.Why?Exercise4.5Solutiononp.152.WhatistheFourierseriesforaunit-amplitudesquarewave?Example4.2Let'sndtheFourierseriesrepresentationforthehalf-waverectiedsinusoid.st=8<:sin)]TJ/F7 6.9738 Tf 5.7618 -4.1472 Td[(2t Tif0t
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116CHAPTER4.FREQUENCYDOMAINOntothecosineterms.Theaveragevalue,whichcorrespondstoa0,equals1 .Theremainderofthecosinecoecientsareeasytond,butyieldthecomplicatedresultak=8<:)]TJ/F1 9.9626 Tf 9.4091 11.0586 Td[(2 1 k2)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1ifk2f2;4;:::g0ifkodd.19Thus,theFourierseriesforthehalf-waverectiedsinusoidhasnon-zerotermsfortheaverage,thefundamental,andtheevenharmonics.4.4ASignal'sSpectrum7Aperiodicsignal,suchasthehalf-waverectiedsinusoid,consistsofasumofelementalsinusoids.AplotoftheFouriercoecientsasafunctionofthefrequencyindex,suchasshowninFigure4.3FourierSeriesspectrumofahalf-waverectiedsinewave,displaysthesignal'sspectrum.Theword"spectrum"impliesthattheindependentvariable,herek,correspondssomehowtofrequency.Eachcoecientisdirectlyrelatedtoasinusoidhavingafrequencyofk T.Thus,ifwehalf-waverectieda1kHzsinusoid,k=1correspondsto1kHz,k=2to2kHz,etc. FourierSeriesspectrumofahalf-waverectiedsinewave Figure4.3:TheFourierseriesspectrumofahalf-waverectiedsinusoidisshown.Theindexindicatesthemultipleofthefundamentalfrequencyatwhichthesignalhasenergy. Asubtle,butveryimportant,aspectoftheFourierspectrumisitsuniqueness:Youcanunambiguouslyndthespectrumfromthesignaldecomposition.15andthesignalfromthespectrumcomposition.Thus,anyaspectofthesignalcanbefoundfromthespectrumandviceversa.Asignal'sfrequencydomainexpressionisitsspectrum.Aperiodicsignalcanbedenedeitherinthetimedomainasafunctionorinthefrequencydomainasaspectrum. 7Thiscontentisavailableonlineat.

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117Afundamentalaspectofsolvingelectricalengineeringproblemsiswhetherthetimeorfrequencydomainprovidesthemostunderstandingofasignal'spropertiesandthesimplestwayofmanipulatingit.Theuniquenesspropertysaysthateitherdomaincanprovidetherightanswer.Asasimpleexample,supposewewanttoknowtheperiodicsignal'smaximumvalue.Clearlythetimedomainprovidestheanswerdirectly.Touseafrequencydomainapproachwouldrequireustondthespectrum,formthesignalfromthespectrumandcalculatethemaximum;we'rebackinthetimedomain!Anotherfeatureofasignalisitsaveragepower.Asignal'sinstantaneouspowerisdenedtobeitssquare.Theaveragepoweristheaverageoftheinstantaneouspoweroversometimeinterval.Foraperiodicsignal,thenaturaltimeintervalisclearlyitsperiod;fornonperiodicsignals,abetterchoicewouldbeentiretimeortimefromonset.Foraperiodicsignal,theaveragepoweristhesquareofitsroot-mean-squaredrmsvalue.Wedenethermsvalueofaperiodicsignaltobermss=s 1 TZT0s2tdt.20andthusitsaveragepowerispowers=rms2s=1 TRT0s2tdt.21Exercise4.6Solutiononp.152.Whatisthermsvalueofthehalf-waverectiedsinusoid?Tondtheaveragepowerinthefrequencydomain,weneedtosubstitutethespectralrepresentationofthesignalintothisexpression.powers=1 TZT0a0+1Xk=1akcos2kt T+1Xk=1bksin2kt T!2dtThesquareinsidetheintegralwillcontainallpossiblepairwiseproducts.However,theorthogonalityproper-ties.12saythatmostofthesecrosstermsintegratetozero.Thesurvivorsleavearathersimpleexpressionforthepowerweseek.powers=a02+1 21Xk=1)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ak2+bk2.22Itcouldwellbethatcomputingthissumiseasierthanintegratingthesignal'ssquare.Furthermore,thecontributionofeachtermintheFourierseriestowardrepresentingthesignalcanbemeasuredbyitscontributiontothesignal'saveragepower.Thus,thepowercontainedinasignalatitskthharmonicisak2+bk2 2.Thepowerspectrum,Psk,suchasshowninFigure4.4PowerSpectrumofaHalf-WaveRectiedSinusoid,plotseachharmonic'scontributiontothetotalpower.Exercise4.7Solutiononp.152.Inhigh-endaudio,deviationofasinewavefromtheidealismeasuredbythetotalharmonicdistortion,whichequalsthetotalpowerintheharmonicshigherthantherstcomparedtopowerinthefundamental.Findanexpressionforthetotalharmonicdistortionforanyperiodicsignal.Isthiscalculationmosteasilyperformedinthetimeorfrequencydomain?4.5FourierSeriesApproximationofSignals8ItisinterestingtoconsiderthesequenceofsignalsthatweobtainasweincorporatemoretermsintotheFourierseriesapproximationofthehalf-waverectiedsinewaveExample4.2.DenesKttobethe 8Thiscontentisavailableonlineat.

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118CHAPTER4.FREQUENCYDOMAIN PowerSpectrumofaHalf-WaveRectiedSinusoid Figure4.4:Powerspectrumofahalf-waverectiedsinusoid. signalcontainingK+1Fourierterms.sKt=a0+KXk=1akcos2kt T+KXk=1bksin2kt T.23Figure4.5FourierSeriesspectrumofahalf-waverectiedsinewaveshowshowthissequenceofsignalsportraysthesignalmoreaccuratelyasmoretermsareadded.WeneedtoassessquantitativelytheaccuracyoftheFourierseriesapproximationsothatwecanjudgehowrapidlytheseriesapproachesthesignal.WhenweuseaK+1-termseries,theerrorthedierencebetweenthesignalandtheK+1-termseriescorrespondstotheunusedtermsfromtheseries.Kt=1Xk=K+1akcos2kt T+1Xk=K+1bksin2kt T.24Tondthermserror,wemustsquarethisexpressionandintegrateitoveraperiod.Again,theintegralofmostcross-termsiszero,leavingrmsK=vuut 1 21Xk=K+1)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ak2+bk2.25Figure4.6Approximationerrorforahalf-waverectiedsinusoidshowshowtheerrorintheFourierseriesforthehalf-waverectiedsinusoiddecreasesasmoretermsareincorporated.Inparticular,theuseoffourterms,asshowninthebottomplotofFigure4.5FourierSeriesspectrumofahalf-waverectiedsinewave,hasarmserrorrelativetothermsvalueofthesignalofabout3%.TheFourierseriesinthiscaseconvergesquicklytothesignal.

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119 FourierSeriesspectrumofahalf-waverectiedsinewave a bFigure4.5:TheFourierseriesspectrumofahalf-waverectiedsinusoidisshownintheupperportion.Theindexindicatesthemultipleofthefundamentalfrequencyatwhichthesignalhasenergy.ThecumulativeeectofaddingtermstotheFourierseriesforthehalf-waverectiedsinewaveisshowninthebottomportion.Thedashedlineistheactualsignal,withthesolidlineshowingtheniteseriesapproximationtotheindicatednumberofterms,K+1.

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120CHAPTER4.FREQUENCYDOMAIN Approximationerrorforahalf-waverectiedsinusoid Figure4.6:Thermserrorcalculatedaccordingto.25isshownasafunctionofthenumberoftermsintheseriesforthehalf-waverectiedsinusoid.Theerrorhasbeennormalizedbythermsvalueofthesignal. WecanlookatFigure4.7Powerspectrumandapproximationerrorforasquarewavetoseethepowerspectrumandthermsapproximationerrorforthesquarewave.BecausetheFouriercoecientsdecaymoreslowlyherethanforthehalf-waverectiedsinusoid,thermserrorisnotdecreasingquickly.Saidanotherway,thesquare-wave'sspectrumcontainsmorepowerathigherfrequenciesthandoesthehalf-wave-rectiedsinusoid.ThisdierencebetweenthetwoFourierseriesresultsbecausethehalf-waverectiedsinusoid'sFouriercoecientsareproportionalto1 k2whilethoseofthesquarewaveareproportionalto1 k.Iffact,after99termsofthesquarewave'sapproximation,theerrorisbiggerthan10termsoftheapproximationforthehalf-waverectiedsinusoid.Mathematicianshaveshownthatnosignalhasanrmsapproximationerrorthatdecaysmoreslowlythanitdoesforthesquarewave.Exercise4.8Solutiononp.153.Calculatetheharmonicdistortionforthesquarewave.Morethanjustdecayingslowly,FourierseriesapproximationshowninFigure4.8Fourierseriesapproxi-mationofasquarewaveexhibitsinterestingbehavior.Althoughthesquarewave'sFourierseriesrequiresmoretermsforagivenrepresentationaccuracy,whencomparingplotsitisnotclearthatthetwoareequal.DoestheFourierseriesreallyequalthesquarewaveatallvaluesoft?Inparticular,ateachstep-changeinthesquarewave,theFourierseriesexhibitsapeakfollowedbyrapidoscillations.Asmoretermsareaddedtotheseries,theoscillationsseemtobecomemorerapidandsmaller,butthepeaksarenotdecreasing.FortheFourierseriesapproximationforthehalf-waverectiedsinusoidFigure4.5:FourierSeriesspectrumofahalf-waverectiedsinewave,nosuchbehavioroccurs.Whatishappening?Considerthismathematicalquestionintuitively:Canadiscontinuousfunction,likethesquarewave,beexpressedasasum,evenaninniteone,ofcontinuoussignals?Oneshouldatleastbesuspicious,andinfact,itcan'tbethusexpressed.ThisissuebroughtFourier9muchcriticismfromtheFrenchAcademyofScienceLaplace,Lagrange,MongeandLaCroixcomprisedthereviewcommitteeforseveralyearsafteritspresentationon1807.Itwasnotresolvedforalsoacentury,anditsresolutionisinterestingandimportanttounderstandfromapracticalviewpoint.Theextraneouspeaksinthesquarewave'sFourierseriesneverdisappear;theyaretermedGibb's 9http://www-groups.dcs.st-and.ac.uk/history/Mathematicians/Fourier.html

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121Powerspectrumandapproximationerrorforasquarewave Figure4.7:Theupperplotshowsthepowerspectrumofthesquarewave,andthelowerplotthermserrorofthenite-lengthFourierseriesapproximationtothesquarewave.TheasteriskdenotesthermserrorwhenthenumberoftermsKintheFourierseriesequals99.phenomenonaftertheAmericanphysicistJosiahWillardGibbs.Theyoccurwheneverthesignalisdis-continuous,andwillalwaysbepresentwheneverthesignalhasjumps.Let'sreturntothequestionofequality;howcantheequalsigninthedenitionoftheFourierseriesbejustied?Thepartialansweristhatpointwiseeachandeveryvalueoftequalityisnotguaranteed.However,mathematicianslaterinthenineteenthcenturyshowedthatthermserroroftheFourierserieswasalwayszero.limK!1rmsK=0WhatthismeansisthattheerrorbetweenasignalanditsFourierseriesapproximationmaynotbezero,butthatitsrmsvaluewillbezero!Itisthroughtheeyesofthermsvaluethatweredeneequality:Theusualdenitionofequalityiscalledpointwiseequality:Twosignalss1t,s2taresaidtobeequalpointwiseifs1t=s2tforallvaluesoft.Anewdenitionofequalityismean-squareequality:Twosignalsaresaidtobeequalinthemeansquareifrmss1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(s2=0.ForFourierseries,Gibb'sphenomenonpeakshaveniteheightandzerowidth.Theerrordiersfromzeroonlyatisolatedpointswhenevertheperiodicsignalcontainsdiscontinuitiesandequalsabout9%ofthesizeofthediscontinuity.Thevalueofafunctionatanitesetofpointsdoesnotaectitsintegral.Thiseectunderliesthereasonwhydeningthevalueofadiscontinuousfunction,likewerefrainedfromdoingindeningthestepfunctionSection2.2.4:UnitStep,atitsdiscontinuityismeaningless.Whateveryoupickforavaluehasnopracticalrelevanceforeitherthesignal'sspectrumorforhowasystemrespondstothesignal.TheFourierseriesvalue"at"thediscontinuityistheaverageofthevaluesoneithersideofthejump.

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122CHAPTER4.FREQUENCYDOMAINFourierseriesapproximationofasquarewave Figure4.8:Fourierseriesapproximationtosqt.ThenumberoftermsintheFouriersumisindicatedineachplot,andthesquarewaveisshownasadashedlineovertwoperiods.4.6EncodingInformationintheFrequencyDomain10Toemphasizethefactthateveryperiodicsignalhasbothatimeandfrequencydomainrepresentation,wecanexploitbothtoencodeinformationintoasignal.RefertotheFundamentalModelofCommunicationFigure1.4:Fundamentalmodelofcommunication.Wehaveaninformationsource,andwanttoconstructatransmitterthatproducesasignalxt.Forthesource,let'sassumewehaveinformationtoencodeeveryTseconds.Forexample,wewanttorepresenttypedlettersproducedbyanextremelygoodtypistakeyisstruckeveryTseconds.Let'sconsiderthecomplexFourierseriesformulainthelightoftryingtoencodeinformation.xt=KXk=)]TJ/F10 6.9738 Tf 6.2267 0 Td[(Kckej2kt T.26Weuseanitesumheremerelyforsimplicityfewerparameterstodetermine.Animportantaspectofthespectrumisthateachfrequencycomponentckcanbemanipulatedseparately:Insteadofndingthe 10Thiscontentisavailableonlineat.

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123Fourierspectrumfromatime-domainspecication,let'sconstructitinthefrequencydomainbyselectingtheckaccordingtosomerulethatrelatescoecientvaluestothealphabet.Indeningthisrule,wewanttoalwayscreateareal-valuedsignalxt.BecauseoftheFourierspectrum'spropertiesProperty4.1,p.111,thespectrummusthaveconjugatesymmetry.Thisrequirementmeansthatwecanonlyassignpositive-indexedcoecientspositivefrequencies,withnegative-indexedonesequalingthecomplexconjugateofthecorrespondingpositive-indexedones.AssumewehaveNletterstoencode:fa1;:::;aNg.OnesimpleencodingrulecouldbetomakeasingleFouriercoecientbenon-zeroandallotherszeroforeachletter.Forexample,ifanoccurs,wemakecn=1andck=0,k6=n.Inthisway,thenthharmonicofthefrequency1 Tisusedtorepresentaletter.NotethatthebandwidththerangeoffrequenciesrequiredfortheencodingequalsN T.Anotherpossibilityistoconsiderthebinaryrepresentationoftheletter'sindex.Forexample,ifthelettera13occurs,converting13toitsbase2representation,wehave13=11012.WecanusethepatternofzerosandonestorepresentdirectlywhichFouriercoecientswe"turnon"setequaltooneandwhichwe"turno."Exercise4.9Solutiononp.153.ComparethebandwidthrequiredforthedirectencodingschemeonenonzeroFouriercoecientforeachlettertothebinarynumberscheme.Comparethebandwidthsfora128-letteralphabet.Sincebothschemesrepresentinformationwithoutlosswecandeterminethetypedletteruniquelyfromthesignal'sspectrumbothareviable.Whichmakesmoreecientuseofbandwidthandthusmightbepreferred?Exercise4.10Solutiononp.153.Canyouthinkofaninformation-encodingschemethatmakesevenmoreecientuseofthespec-trum?Inparticular,canweuseonlyoneFouriercoecienttorepresentNlettersuniquely?Wecancreateanencodingschemeinthefrequencydomainp.123torepresentanalphabetofletters.But,asthisinformation-encodingschemestands,wecanrepresentoneletterforalltime.However,wenotethattheFouriercoecientsdependonlyonthesignal'scharacteristicsoverasingleperiod.Wecouldchangethesignal'sspectrumeveryTaseachletteristyped.Inthisway,weturnspectralcoecientsonandoaslettersaretyped,therebyencodingtheentiretypeddocument.ForthereceiverseetheFundamentalModelofCommunicationFigure1.4:Fundamentalmodelofcommunicationtoretrievethetypedletter,itwouldsimplyusetheFourierformulaforthecomplexFourierspectrum11foreachT-secondintervaltodeterminewhateachtypedletterwas.Figure4.9EncodingSignalsshowssuchasignalinthetime-domain. 11"ComplexFourierSeriesandTheirProperties",

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124CHAPTER4.FREQUENCYDOMAIN EncodingSignals Figure4.9:TheencodingofsignalsviatheFourierspectrumisshownoverthree"periods."Inthisex-ample,onlythethirdandfourthharmonicsareused,asshownbythespectralmagnitudescorrespondingtoeachT-secondintervalplottedbelowthewaveforms.Canyoudeterminethephaseoftheharmonicsfromthewaveform? InthisFourier-seriesencodingscheme,wehaveusedthefactthatspectralcoecientscanbeindepen-dentlyspeciedandthattheycanbeuniquelyrecoveredfromthetime-domainsignaloverone"period."Donotethatthesignalrepresentingtheentiredocumentisnolongerperiodic.ByunderstandingtheFourierse-ries'propertiesinparticularthatcoecientsaredeterminedonlyoveraT-secondinterval,wecanconstructacommunicationssystem.Thisapproachrepresentsasimplicationofhowmodernmodemsrepresenttextthattheytransmitovertelephonelines.4.7FilteringPeriodicSignals12TheFourierseriesrepresentationofaperiodicsignalmakesiteasytodeterminehowalinear,time-invariantlterreshapessuchsignalsingeneral.Thefundamentalpropertyofalinearsystemisthatitsinput-outputrelationobeyssuperposition:La1s1t+a2s2t=a1Ls1t+a2Ls2t.BecausetheFourierseriesrepresentsaperiodicsignalasalinearcombinationofcomplexexponentials,wecanexploitthesuperpositionproperty.Furthermore,wefoundforlinearcircuitsthattheiroutputtoacomplexexponentialinputisjustthefrequencyresponseevaluatedatthesignal'sfrequencytimesthecomplexexponential.Saidmathematically,ifxt=ej2kt T,thentheoutputyt=H)]TJ/F10 6.9738 Tf 6.4489 -4.1472 Td[(k Tej2kt Tbecausef=k T.Thus,ifxtisperiodictherebyhavingaFourierseries,alinearcircuit'soutputtothissignalwillbethesuperpositionoftheoutputtoeachcomponent.yt=1Xk=ckHk Tej2kt T.27 12Thiscontentisavailableonlineat.

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125Thus,theoutputhasaFourierseries,whichmeansthatittooisperiodic.ItsFouriercoecientsequalckH)]TJ/F10 6.9738 Tf 6.4489 -4.1472 Td[(k T.Toobtainthespectrumoftheoutput,wesimplymultiplytheinputspectrumbythefrequencyresponse.ThecircuitmodiesthemagnitudeandphaseofeachFouriercoecient.NoteespeciallythatwhiletheFouriercoecientsdonotdependonthesignal'speriod,thecircuit'stransferfunctiondoesdependonfrequency,whichmeansthatthecircuit'soutputwilldierastheperiodvaries. Filteringaperiodsignal a bFigure4.10:Aperiodicpulsesignal,suchasshownontheleftpart T=0:2,servesastheinputtoanRClowpasslter.Theinput'speriodwas1msmillisecond.Thelter'scutofrequencywassettothevariousvaluesindicatedinthetoprow,whichdisplaytheoutputsignal'sspectrumandthelter'stransferfunction.ThebottomrowshowstheoutputsignalderivedfromtheFourierseriescoecientsshowninthetoprow.aPeriodicpulsesignalbTopplotsshowthepulsesignal'sspectrumforvariouscutofrequencies.Bottomplotsshowthelter'soutputsignals.

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126CHAPTER4.FREQUENCYDOMAINExample4.3TheperiodicpulsesignalshownontheleftaboveservesastheinputtoaRC-circuitthathasthetransferfunctioncalculatedelsewhereFigure3.31:MagnitudeandphaseofthetransferfunctionHf=1 1+j2fRC.28Figure4.10Filteringaperiodsignalshowstheoutputchangesaswevarythelter'scutofrequency.Notehowthesignal'sspectrumextendswellaboveitsfundamentalfrequency.Havingacutofrequencytentimeshigherthanthefundamentaldoesperceptiblychangetheoutputwaveform,roundingtheleadingandtrailingedges.Asthecutofrequencydecreasescenter,thenleft,theroundingbecomesmoreprominent,withtheleftmostwaveformshowingasmallripple.Exercise4.11Solutiononp.153.Whatistheaveragevalueofeachoutputwaveform?Thecorrectanswermaysurpriseyou.Thisexamplealsoillustratestheimpactalowpassltercanhaveonawaveform.ThesimpleRClterusedherehasarathergradualfrequencyresponse,whichmeansthathigherharmonicsaresmoothlysuppressed.Later,wewilldescribeltersthathavemuchmorerapidlyvaryingfrequencyresponses,allowingamuchmoredramaticselectionoftheinput'sFouriercoecients.Moreimportantly,wehavecalculatedtheoutputofacircuittoaperiodicinputwithoutwriting,muchlesssolving,thedierentialequationgoverningthecircuit'sbehavior.Furthermore,wemadethesecalculationsentirelyinthefrequencydomain.UsingFourierseries,wecancalculatehowanylinearcircuitwillrespondtoaperiodicinput.4.8DerivationoftheFourierTransform13Fourierseriesclearlyopenthefrequencydomainasaninterestingandusefulwayofdetermininghowcircuitsandsystemsrespondtoperiodicinputsignals.Canweusesimilartechniquesfornonperiodicsignals?Whatistheresponseoftheltertoasinglepulse?AddressingtheseissuesrequiresustondtheFourierspectrumofallsignals,bothperiodicandnonperiodicones.WeneedadenitionfortheFourierspectrumofasignal,periodicornot.ThisspectrumiscalculatedbywhatisknownastheFouriertransform.LetsTtbeaperiodicsignalhavingperiodT.Wewanttoconsiderwhathappenstothissignal'sspectrumaswelettheperiodbecomelongerandlonger.WedenotethespectrumforanyassumedvalueoftheperiodbyckT.WecalculatethespectrumaccordingtothefamiliarformulackT=1 TZT 2)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(T 2sTte)]TJ/F8 9.9626 Tf 6.2266 -0.7472 Td[(j2kt Tdt.29wherewehaveusedasymmetricplacementoftheintegrationintervalabouttheoriginforsubsequentderivationalconvenience.Letfbeaxedfrequencyequalingk T;wevarythefrequencyindexkproportionallyasweincreasetheperiod.DeneSTfTckT=ZT 2)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(T 2sTte)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2ftdt.30makingthecorrespondingFourierseriessTt=1Xk=STfej2ft1 T.31 13Thiscontentisavailableonlineat.

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127Astheperiodincreases,thespectrallinesbecomeclosertogether,becomingacontinuum.Therefore,limT!1sTtst=Z1Sfej2ftdf.32withSf=Z1ste)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2ftdt.33SfistheFouriertransformofsttheFouriertransformissymbolicallydenotedbytheuppercaseversionofthesignal'ssymbolandisdenedforanysignalforwhichtheintegral.33converges.Example4.4Let'scalculatetheFouriertransformofthepulsesignalSection2.2.5:Pulse,pt.Pf=Z1pte)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2ftdt=Z0e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2ftdt=1 )]TJ/F8 9.9626 Tf 9.4091 0 Td[(j2fe)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2f)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1Pf=e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2fsinf fNotehowcloselythisresultresemblestheexpressionforFourierseriescoecientsoftheperiodicpulsesignal4.10. Spectrum Figure4.11:TheupperplotshowsthemagnitudeoftheFourierseriesspectrumforthecaseofT=1withtheFouriertransformofptshownasadashedline.Forthebottompanel,weexpandedtheperiodtoT=5,keepingthepulse'sdurationxedat0.2,andcomputeditsFourierseriescoecients.

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128CHAPTER4.FREQUENCYDOMAINFigure4.11Spectrumshowshowincreasingtheperioddoesindeedleadtoacontinuumofcoecients,andthattheFouriertransformdoescorrespondtowhatthecontinuumbecomes.Thequantitysint thasaspecialname,thesincpronounced"sink"function,andisdenotedbysinct.Thus,themagnitudeofthepulse'sFouriertransformequalsjsincfj.TheFouriertransformrelatesasignal'stimeandfrequencydomainrepresentationstoeachother.ThedirectFouriertransformorsimplytheFouriertransformcalculatesasignal'sfrequencydomainrepresen-tationfromitstime-domainvariant.34.TheinverseFouriertransform.35ndsthetime-domainrepresentationfromthefrequencydomain.Ratherthanexplicitlywritingtherequiredintegral,weoftensymbolicallyexpressthesetransformcalculationsasFsandF)]TJ/F7 6.9738 Tf 6.2266 0 Td[(1S,respectively.Fs=Sf=R1ste)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2ftdt.34F)]TJ/F7 6.9738 Tf 6.2266 0 Td[(1S=st=R1Sfe+j2ftdf.35Wemusthavest=F)]TJ/F7 6.9738 Tf 6.2266 0 Td[(1FstandSf=F)]TJ/F14 9.9626 Tf 4.5663 -8.0698 Td[(F)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1Sf,andtheseresultsareindeedvalidwithminorexceptions.Note:RecallthattheFourierseriesforasquarewavegivesavalueforthesignalatthedisconti-nuitiesequaltotheaveragevalueofthejump.Thisvaluemaydierfromhowthesignalisdenedinthetimedomain,butbeingunequalatapointisindeedminor.Showingthatyou"getbacktowhereyoustarted"isdicultfromananalyticviewpoint,andwewon'ttryhere.Notethatthedirectandinversetransformsdieronlyinthesignoftheexponent.Exercise4.12Solutiononp.153.Thedieringexponentsignsmeansthatsomecuriousresultsoccurwhenweusethewrongsign.WhatisFSf?Inotherwords,usethewrongexponentsigninevaluatingtheinverseFouriertransform.PropertiesoftheFouriertransformandsomeusefultransformpairsareprovidedintheaccompanyingtablesShortTableofFourierTransformPairs,p.129andFourierTransformProperties,p.129.EspeciallyimportantamongthesepropertiesisParseval'sTheorem,whichstatesthatpowercomputedineitherdomainequalsthepowerintheother.Z1s2tdt=Z1jSfj2df.36Ofpracticalimportanceistheconjugatesymmetryproperty:Whenstisreal-valued,thespectrumatnegativefrequenciesequalsthecomplexconjugateofthespectrumatthecorrespondingpositivefrequencies.Consequently,weneedonlyplotthepositivefrequencyportionofthespectrumwecaneasilydeterminetheremainderofthespectrum.Exercise4.13Solutiononp.153.HowmanyFouriertransformoperationsneedtobeappliedtogettheoriginalsignalback:FFs=st?Notethatthemathematicalrelationshipsbetweenthetimedomainandfrequencydomainversionsofthesamesignalaretermedtransforms.Wearetransforminginthenontechnicalmeaningofthewordasignalfromonerepresentationtoanother.WeexpressFouriertransformpairsasst$Sf.Asignal'stimeandfrequencydomainrepresentationsareuniquelyrelatedtoeachother.Asignalthus"exists"inboththetimeandfrequencydomains,withtheFouriertransformbridgingbetweenthetwo.Wecandeneaninformationcarryingsignalineitherthetimeorfrequencydomains;itbehoovesthewiseengineertousethesimplerofthetwo.

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129Acommonmisunderstandingisthatwhileasignalexistsinboththetimeandfrequencydomains,asingleformulaexpressingasignalmustcontainonlytimeorfrequency:Bothcannotbepresentsimultaneously.Thissituationmirrorswhathappenswithcomplexamplitudesincircuits:Aswerevealhowcommunicationssystemsworkandaredesigned,wewilldenesignalsentirelyinthefrequencydomainwithoutexplicitlyndingtheirtimedomainvariants.ThisideaisshowninanothermoduleSection4.6wherewedeneFourierseriescoecientsaccordingtolettertobetransmitted.Thus,asignal,thoughmostfamiliarlydenedinthetime-domain,reallycanbedenedequallyaswellandsometimesmoreeasilyinthefrequencydomain.Forexample,impedancesdependonfrequencyandthetimevariablecannotappear.WewilllearnSection4.9thatndingalinear,time-invariantsystem'soutputinthetimedomaincanbemosteasilycalculatedbydeterminingtheinputsignal'sspectrum,performingasimplecalculationinthefrequencydomain,andinversetransformingtheresult.Furthermore,understandingcommunicationsandinformationprocessingsystemsrequiresathoroughunderstandingofsignalstructureandofhowsystemsworkinboththetimeandfrequencydomains.TheonlydicultyincalculatingtheFouriertransformofanysignaloccurswhenwehaveperiodicsignalsineitherdomain.RealizingthattheFourierseriesisaspecialcaseoftheFouriertransform,wesimplycalculatetheFourierseriescoecientsinstead,andplotthemalongwiththespectraofnonperiodicsignalsonthesamefrequencyaxis.ShortTableofFourierTransformPairs st Sf e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(atut 1 j2f+a e)]TJ/F10 6.9738 Tf 6.2267 0 Td[(ajtj 2a 42f2+a2 pt=8<:1ifjtj< 20ifjtj> 2 sinf f sinWt t Sf=8<:1ifjfjW

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130CHAPTER4.FREQUENCYDOMAINFourierTransformProperties Time-Domain FrequencyDomain Linearity a1s1t+a2s2t a1S1f+a2S2f ConjugateSymmetry st2R Sf=S)]TJ/F11 9.9626 Tf 7.7487 0 Td[(f EvenSymmetry st=s)]TJ/F11 9.9626 Tf 7.7487 0 Td[(t Sf=S)]TJ/F11 9.9626 Tf 7.7487 0 Td[(f OddSymmetry st=)]TJ/F8 9.9626 Tf 9.4091 0 Td[(s)]TJ/F11 9.9626 Tf 7.7487 0 Td[(t Sf=)]TJ/F8 9.9626 Tf 9.4091 0 Td[(S)]TJ/F11 9.9626 Tf 7.7487 0 Td[(f ScaleChange sat 1 jajSf a TimeDelay st)]TJ/F11 9.9626 Tf 9.9626 0 Td[( e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2fSf ComplexModulation ej2f0tst Sf)]TJ/F11 9.9626 Tf 9.9626 0 Td[(f0 AmplitudeModulationbyCosine stcosf0t Sf)]TJ/F10 6.9738 Tf 6.2266 0 Td[(f0+Sf+f0 2 AmplitudeModulationbySine stsinf0t Sf)]TJ/F10 6.9738 Tf 6.2266 0 Td[(f0)]TJ/F10 6.9738 Tf 6.2267 0 Td[(Sf+f0 2j Dierentiation d dtst j2fSf Integration Rtsd 1 j2fSfifS=0 Multiplicationbyt tst 1 )]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2d dfSf Area R1stdt S ValueatOrigin s R1Sfdf Parseval'sTheorem R1jstj2dt R1jSfj2df Example4.5Incommunications,averyimportantoperationonasignalstistoamplitudemodulateit.Usingthisoperationmoreasanexampleratherthanelaboratingthecommunicationsaspectshere,wewanttocomputetheFouriertransformthespectrumof+stcosfctThus,+stcosfct=cosfct+stcosfctForthespectrumofcos2fct,weusetheFourierseries.Itsperiodis1 fc,anditsonlynonzeroFouriercoecientsarec1=1 2.Thesecondtermisnotperiodicunlesssthasthesameperiodasthesinusoid.UsingEuler'srelation,thespectrumofthesecondtermcanbederivedasstcosfct=Z1Sfej2ftdfcos2fctUsingEuler'srelationforthecosine,stcosfct=1 2Z1Sfej2f+fctdf+1 2Z1Sfej2f)]TJ/F10 6.9738 Tf 6.2267 0 Td[(fctdfstcosfct=1 2Z1Sf)]TJ/F11 9.9626 Tf 9.9626 0 Td[(fcej2ftdf+1 2Z1Sf+fcej2ftdfstcosfct=Z1Sf)]TJ/F11 9.9626 Tf 9.9626 0 Td[(fc+Sf+fc 2ej2ftdf

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131ExploitingtheuniquenesspropertyoftheFouriertransform,wehaveFstcosfct=Sf)]TJ/F11 9.9626 Tf 9.9626 0 Td[(fc+Sf+fc 2.37Thiscomponentofthespectrumconsistsoftheoriginalsignal'sspectrumdelayedandadvancedinfrequency.ThespectrumoftheamplitudemodulatedsignalisshowninFigure4.12. Figure4.12:Asignalwhichhasatriangularshapedspectrumisshowninthetopplot.ItshighestfrequencythelargestfrequencycontainingpowerisWHz.Onceamplitudemodulated,theresultingspectrumhas"lines"correspondingtotheFourierseriescomponentsatfcandtheoriginaltriangularspectrumshiftedtocomponentsatfcandscaledby1 2.Notehowinthisgurethesignalstisdenedinthefrequencydomain.Tonditstimedomainrepresentation,wesimplyusetheinverseFouriertransform.Exercise4.14Solutiononp.153.WhatisthesignalstthatcorrespondstothespectrumshownintheupperpanelofFigure4.12?Exercise4.15Solutiononp.153.Whatisthepowerinxt,theamplitude-modulatedsignal?Trythecalculationinboththetimeandfrequencydomains.Inthisexample,wecallthesignalstabasebandsignalbecauseitspoweriscontainedatlowfrequencies.SignalssuchasspeechandtheDowJonesaveragesarebasebandsignals.Thebasebandsignal'sbandwidthequalsW,thehighestfrequencyatwhichithaspower.Sincext'sspectrumisconnedtoafrequencybandnotclosetotheoriginweassumefcW,wehaveabandpasssignal.Thebandwidthofabandpasssignalisnotitshighestfrequency,buttherangeofpositivefrequencieswherethesignalhaspower.Thus,inthisexample,thebandwidthis2WHz.Whyasignal'sbandwidthshoulddependonitsspectralshapewillbecomeclearoncewedevelopcommunicationssystems.4.9LinearTimeInvariantSystems14Whenweapplyaperiodicinputtoalinear,time-invariantsystem,theoutputisperiodicandhasFourierseriescoecientsequaltotheproductofthesystem'sfrequencyresponseandtheinput'sFouriercoecientsFilteringPeriodicSignals4.27.Thewaywederivedthespectrumofnon-periodicsignalfromperiodiconesmakesitclearthatthesamekindofresultworkswhentheinputisnotperiodic:Ifxtservesas 14Thiscontentisavailableonlineat.

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132CHAPTER4.FREQUENCYDOMAINtheinputtoalinear,time-invariantsystemhavingfrequencyresponseHf,thespectrumoftheoutputisXfHf.Example4.6Let'susethisfrequency-domaininput-outputrelationshipforlinear,time-invariantsystemstondaformulafortheRC-circuit'sresponsetoapulseinput.Wehaveexpressionsfortheinput'sspectrumandthesystem'sfrequencyresponse.Pf=e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(jfsinf f.38Hf=1 1+j2fRC.39Thus,theoutput'sFouriertransformequalsYf=e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(jfsinf f1 1+j2fRC.40Youwon'tndthisFouriertransforminourtable,andtherequiredintegralisdiculttoevaluateastheexpressionstands.ThissituationrequiresclevernessandanunderstandingoftheFouriertransform'sproperties.Inparticular,recallEuler'srelationforthesinusoidaltermandnotethefactthatmultiplicationbyacomplexexponentialinthefrequencydomainamountstoatimedelay.Let'smomentarilymaketheexpressionforYfmorecomplicated.e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(jfsinf f=e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(jfejf)]TJ/F10 6.9738 Tf 6.2267 0 Td[(e)]TJ/F6 4.9813 Tf 5.3965 0 Td[(jf j2f=1 j2f)]TJ/F8 9.9626 Tf 4.5662 -8.0698 Td[(1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2f.41Consequently,Yf=1 j2f1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(jf1 1+j2fRC.42ThetableofFouriertransformpropertiesFourierTransformProperties,p.129suggeststhinkingaboutthisexpressionasaproductofterms.Multiplicationby1 j2fmeansintegration.Multiplicationbythecomplexexponentiale)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2fmeansdelaybysecondsinthetimedomain.Theterm1)]TJ/F11 9.9626 Tf 9.5889 0 Td[(e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2fmeans,inthetimedomain,subtractthetime-delayedsignalfromitsoriginal.Theinversetransformofthefrequencyresponseis1 RCe)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(t RCut.Wecantranslateeachofthesefrequency-domainproductsintotime-domainoperationsinanyorderwelikebecausetheorderinwhichmultiplicationsoccurdoesn'taecttheresult.Let'sstartwiththeproductof1 j2fintegrationinthetimedomainandthetransferfunction:1 j2f1 1+j2fRC$1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(t RCut.43ThemiddletermintheexpressionforYfconsistsofthedierenceoftwoterms:theconstant1andthecomplexexponentiale)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2f.BecauseoftheFouriertransform'slinearity,wesimplysubtracttheresults.Yf$1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(t RCut)]TJ/F1 9.9626 Tf 9.9626 11.0586 Td[(1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e)]TJ/F8 9.9626 Tf 6.2266 -0.7472 Td[(t)]TJ/F6 4.9813 Tf 5.3965 0 Td[( RCut)]TJ/F8 9.9626 Tf 9.9626 0 Td[(.44

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133Notethatindelayingthesignalhowwecarefullyincludedtheunitstep.Thesecondterminthisresultdoesnotbeginuntilt=.Thus,thewaveformsshownintheFilteringPeriodicSignalsFigure4.10:Filteringaperiodsignalexamplementionedaboveareexponentials.Wesaythatthetimeconstantofanexponentiallydecayingsignalequalsthetimeittakestodecreaseby1 eofitsoriginalvalue.Thus,thetime-constantoftherisingandfallingportionsoftheoutputequaltheproductofthecircuit'sresistanceandcapacitance.Exercise4.16Solutiononp.153.Derivethelter'soutputbyconsideringthetermsin.41intheordergiven.Integratelastratherthanrst.Youshouldgetthesameanswer.Inthisexample,weusedthetableextensivelytondtheinverseFouriertransform,relyingmostlyonwhatmultiplicationbycertainfactors,like1 j2fande)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2f,meant.Weessentiallytreatedmultiplicationbythesefactorsasiftheyweretransferfunctionsofsomectitiouscircuit.Thetransferfunction1 j2fcorrespondedtoacircuitthatintegrated,ande)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2ftoonethatdelayed.Weevenimplicitlyinterpretedthecircuit'stransferfunctionastheinput'sspectrum!Thisapproachtondinginversetransformsbreakingdownacomplicatedexpressionintoproductsandsumsofsimplecomponentsistheengineer'swayofbreakingdowntheproblemintoseveralsubproblemsthataremucheasiertosolveandthengluingtheresultstogether.Alongthewaywemaymakethesystemserveastheinput,butintheruleYf=XfHf,whichtermistheinputandwhichisthetransferfunctionismerelyanotationalmatterwelabeledonefactorwithanXandtheotherwithanH.4.9.1TransferFunctionsThenotionofatransferfunctionapplieswellbeyondlinearcircuits.Althoughwedon'thaveallweneedtodemonstratetheresultasyet,alllinear,time-invariantsystemshaveafrequency-domaininput-outputrelationgivenbytheproductoftheinput'sFouriertransformandthesystem'stransferfunction.Thus,linearcircuitsareaspecialcaseoflinear,time-invariantsystems.Aswetacklemoresophisticatedproblemsintransmitting,manipulating,andreceivinginformation,wewillassumelinearsystemshavingcertainpropertiestransferfunctionswithoutworryingaboutwhatcircuithasthedesiredproperty.Atthispoint,youmaybeconcernedthatthisapproachisglib,andrightlyso.Laterwe'llshowthatbyinvolvingsoftwarethatwereallydon'tneedtobeconcernedaboutconstructingatransferfunctionfromcircuitelementsandop-amps.4.9.2CommutativeTransferFunctionsAnotherinterestingnotionarisesfromthecommutativepropertyofmultiplicationexploitedinanexampleaboveExample4.6:Wecanratherarbitrarilychoseanorderinwhichtoapplyeachproduct.Consideracascadeoftwolinear,time-invariantsystems.BecausetheFouriertransformoftherstsystem'soutputisXfH1fanditservesasthesecondsystem'sinput,thecascade'soutputspectrumisXfH1fH2f.BecausethisproductalsoequalsXfH2fH1f,thecascadehavingthelinearsystemsintheoppositeorderyieldsthesameresult.Furthermore,thecascadeactslikeasinglelinearsystem,havingtransferfunctionH1fH2f.Thisresultappliestoothercongurationsoflinear,time-invariantsystemsaswell;seethisFrequencyDomainProblemProblem4.12.Engineersexploitthispropertybydeterminingwhattransferfunctiontheywant,thenbreakingitdownintocomponentsarrangedaccordingtostandardcongu-rations.Usingthefactthatop-ampcircuitscanbeconnectedincascadewiththetransferfunctionequalingtheproductofitscomponent'stransferfunctionseethisanalogsignalprocessingproblemProblem3.37,wendareadywayofrealizingdesigns.Wenowunderstandwhyop-ampimplementationsoftransferfunctionsaresoimportant.

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134CHAPTER4.FREQUENCYDOMAIN4.10ModelingtheSpeechSignal15 VocalTract Figure4.13:Thevocaltractisshownincross-section.Airpressureproducedbythelungsforcesairthroughthevocalcordsthat,whenundertension,producepusofairthatexciteresonancesinthevocalandnasalcavities.Whatarenotshownarethebrainandthemusculaturethatcontroltheentirespeechproductionprocess. 15Thiscontentisavailableonlineat.

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135 ModeloftheVocalTract Figure4.14:Thesystemsmodelforthevocaltract.Thesignalslt,pTt,andstaretheairpressureprovidedbythelungs,theperiodicpulseoutputprovidedbythevocalcords,andthespeechoutputrespectively.Controlsignalsfromthebrainareshownasenteringthesystemsfromthetop.Clearly,thesecomefromthesamesource,butformodelingpurposeswedescribethemseparatelysincetheycontroldierentaspectsofthespeechsignal. Theinformationcontainedinthespokenwordisconveyedbythespeechsignal.Becauseweshallanalyzeseveralspeechtransmissionandprocessingschemes,weneedtounderstandthespeechsignal'sstructurewhat'sspecialaboutthespeechsignalandhowwecandescribeandmodelspeechproduction.Thismodelingeortconsistsofndingasystem'sdescriptionofhowrelativelyunstructuredsignals,arisingfromsimplesources,aregivenstructurebypassingthemthroughaninterconnectionofsystemstoyieldspeech.Forspeechandformanyothersituations,systemchoiceisgovernedbythephysicsunderlyingtheactualproductionprocess.Becausethefundamentalequationofacousticsthewaveequationapplieshereandislinear,wecanuselinearsystemsinourmodelwithafairamountofaccuracy.Thenaturalnessoflinearsystemmodelsforspeechdoesnotextendtoothersituations.Inmanycases,theunderlyingmathematicsgovernedbythephysics,biology,and/orchemistryoftheproblemarenonlinear,leavinglinearsystemsmodelsasapproximations.Nonlinearmodelsarefarmoredicultatthecurrentstateofknowledgetounderstand,andinformationengineersfrequentlypreferlinearmodelsbecausetheyprovideagreaterlevelofcomfort,butnotnecessarilyasucientlevelofaccuracy.Figure4.13VocalTractshowstheactualspeechproductionsystemandFigure4.14ModeloftheVocalTractshowsthemodelspeechproductionsystem.Thecharacteristicsofthemodeldependsonwhetheryouaresayingavoweloraconsonant.Weconcentraterstonthevowelproductionmechanism.Whenthevocalcordsareplacedundertensionbythesurroundingmusculature,airpressurefromthelungscausesthevocalcordstovibrate.Tovisualizethiseect,takearubberbandandholditinfrontofyourlips.Ifheldopenwhenyoublowthroughit,theairpassesthroughmoreorlessfreely;thissituationcorrespondsto"breathingmode".Ifheldtautlyandclosetogether,blowingthroughtheopeningcausesthesidesoftherubberbandtovibrate.Thiseectworksbestwithawiderubberband.Youcanimaginewhattheairowislikeontheoppositesideoftherubberbandorthevocalcords.Yourlungpoweristhesimplesourcereferredtoearlier;itcanbemodeledasaconstantsupplyofairpressure.Thevocalcordsrespondtothisinputbyvibrating,whichmeanstheoutputofthissystemissomeperiodicfunction.Exercise4.17Solutiononp.153.Notethatthevocalcordsystemtakesaconstantinputandproducesaperiodicairowthatcorrespondstoitsoutputsignal.Isthissystemlinearornonlinear?Justifyyouranswer.Singersmodifyvocalcordtensiontochangethepitchtoproducethedesiredmusicalnote.Vocalcordtensionisgovernedbyacontrolinputtothemusculature;insystem'smodelswerepresentcontrolinputsassignalscomingintothetoporbottomofthesystem.Certainlyinthecaseofspeechandinmanyothercasesaswell,itisthecontrolinputthatcarriesinformation,impressingitonthesystem'soutput.Thechangeofsignalstructureresultingfromvaryingthecontrolinputenablesinformationtobeconveyedbythesignal,aprocessgenericallyknownasmodulation.Insinging,musicalityislargelyconveyedbypitch;inwesternspeech,pitchismuchlessimportant.Asentencecanbereadinamonotonefashionwithoutcompletely

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136CHAPTER4.FREQUENCYDOMAINdestroyingtheinformationexpressedbythesentence.However,thedierencebetweenastatementandaquestionisfrequentlyexpressedbypitchchanges.Forexample,notethesounddierencesbetween"Let'sgotothepark."and"Let'sgotothepark?";Forsomeconsonants,thevocalcordsvibratejustasinvowels.Forexample,theso-callednasalsounds"n"and"m"havethisproperty.Forothers,thevocalcordsdonotproduceaperiodicoutput.Goingbacktomechanism,whenconsonantssuchas"f"areproduced,thevocalcordsareplacedundermuchlesstension,whichresultsinturbulentow.Theresultingoutputairowisquiteerratic,somuchsothatwedescribeitasbeingnoise.Wedenenoisecarefullylaterwhenwedelveintocommunicationproblems.Thevocalcords'periodicoutputcanbewelldescribedbytheperiodicpulsetrainpTtasshownintheperiodicpulsesignalFigure4.1,withTdenotingthepitchperiod.Thespectrumofthissignal.9containsharmonicsofthefrequency1 T,whatisknownasthepitchfrequencyorthefundamentalfrequencyF0.Theprimarydierencebetweenadultmaleandfemale/prepubescentspeechispitch.Beforepuberty,pitchfrequencyfornormalspeechrangesbetween150-400Hzforbothmalesandfemales.Afterpuberty,thevocalcordsofmalesundergoaphysicalchange,whichhastheeectofloweringtheirpitchfrequencytotherange80-160Hz.Ifwecouldexaminethevocalcordoutput,wecouldprobablydiscernwhetherthespeakerwasmaleorfemale.Thisdierenceisalsoreadilyapparentinthespeechsignalitself.Tosimplifyourspeechmodelingeort,weshallassumethatthepitchperiodisconstant.Withthissimplication,wecollapsethevocal-cord-lungsystemasasimplesourcethatproducestheperiodicpulsesignalFigure4.14ModeloftheVocalTract.Thesoundpressuresignalthusproducedentersthemouthbehindthetongue,createsacousticdisturbances,andexitsprimarilythroughthelipsandtosomeextentthroughthenose.Speechspecialiststendtonamethemouth,tongue,teeth,lips,andnasalcavitythevocaltract.Thephysicsgoverningthesounddisturbancesproducedinthevocaltractandthoseofanorganpipearequitesimilar.Whereastheorganpipehasthesimplephysicalstructureofastraighttube,thecross-sectionofthevocaltract"tube"variesalongitslengthbecauseofthepositionsofthetongue,teeth,andlips.Itisthesepositionsthatarecontrolledbythebraintoproducethevowelsounds.Spreadingthelips,bringingtheteethtogether,andbringingthetonguetowardthefrontportionoftheroofofthemouthproducesthesound"ee."Roundingthelips,spreadingtheteeth,andpositioningthetonguetowardthebackoftheoralcavityproducesthesound"oh."Thesevariationsresultinalinear,time-invariantsystemthathasafrequencyresponsetypiedbyseveralpeaks,asshowninFigure4.15SpeechSpectrum.Thesepeaksareknownasformants.Thus,speechsignalprocessorswouldsaythatthesound"oh"hasahigherrstformantfrequencythanthesound"ee,"withF2beingmuchhigherduring"ee."F2andF3thesecondandthirdformantshavemoreenergyin"ee"thanin"oh."Ratherthanservingasalter,rejectinghighorlowfrequencies,thevocaltractservestoshapethespectrumofthevocalcords.Inthetimedomain,wehaveaperiodicsignal,thepitch,servingastheinputtoalinearsystem.Weknowthattheoutputthespeechsignalweutterandthatisheardbyothersandourselveswillalsobeperiodic.Exampletime-domainspeechsignalsareshowninFigure4.15SpeechSpectrum,wheretheperiodicityisquiteapparent.Exercise4.18Solutiononp.153.FromthewaveformplotsshowninFigure4.15SpeechSpectrum,determinethepitchperiodandthepitchfrequency.Sincespeechsignalsareperiodic,speechhasaFourierseriesrepresentationgivenbyalinearcircuit'sresponsetoaperiodicsignal.27.Becausetheacousticsofthevocaltractarelinear,weknowthatthespectrumoftheoutputequalstheproductofthepitchsignal'sspectrumandthevocaltract'sfrequencyresponse.Wethusobtainthefundamentalmodelofspeechproduction.Sf=PTfHVf.45Here,HVfisthetransferfunctionofthevocaltractsystem.TheFourierseriesforthevocalcords'output,derivedinthisequationp.112,isck=Ae)]TJ/F8 9.9626 Tf 6.2266 -0.7472 Td[(jk Tsin)]TJ/F10 6.9738 Tf 5.7617 -4.1472 Td[(k T k.46

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137 SpeechSpectrum Figure4.15:Theidealfrequencyresponseofthevocaltractasitproducesthesounds"oh"and"ee"areshownonthetopleftandtopright,respectively.Thespectralpeaksareknownasformants,andarenumberedconsecutivelyfromlowtohighfrequency.Thebottomplotsshowspeechwaveformscorrespondingtothesesounds. andisplottedonthetopinFigure4.16voicespectrum.Ifwehad,forexample,amalespeakerwithabouta110HzpitchT9:1mssayingthevowel"oh",thespectrumofhisspeechpredictedbyourmodelisshowninFigure4.16bvoicespectrum.Themodelspectrumidealizesthemeasuredspectrum,andcapturesalltheimportantfeatures.Themeasuredspectrumcertainlydemonstrateswhatareknownaspitchlines,andwerealizefromourmodelthattheyareduetothevocalcord'speriodicexcitationofthevocaltract.Thevocaltract'sshapingofthelinespectrumisclearlyevident,butdiculttodiscernexactly,especiallyatthehigherfrequencies.Themodeltransferfunctionforthevocaltractmakestheformantsmuchmorereadilyevident.Exercise4.19Solutiononp.153.TheFourierseriescoecientsforspeecharerelatedtothevocaltract'stransferfunctiononlyatthefrequenciesk T,k2f1;2;:::g;seepreviousresult.9.Wouldmaleorfemalespeechtendtohaveamoreclearlyidentiableformantstructurewhenitsspectrumiscomputed?Consider,forexample,howthespectrumshownontherightinFigure4.16voicespectrumwouldchangeifthepitchweretwiceashigh300Hz.

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138CHAPTER4.FREQUENCYDOMAIN voicespectrum apulse bvoicespectrumFigure4.16:Thevocaltract'stransferfunction,shownasthethin,smoothline,issuperimposedonthespectrumofactualmalespeechcorrespondingtothesound"oh."Thepitchlinescorrespondingtoharmonicsofthepitchfrequencyareindicated.aThevocalcords'outputspectrumPTf.bThevocaltract'stransferfunction,HVfandthespeechspectrum.

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139spectrogram Figure4.17:Displayedisthespectrogramoftheauthorsaying"RiceUniversity."Blueindicateslowenergyportionofthespectrum,withredindicatingthemostenergeticportions.Belowthespectrogramisthetime-domainspeechsignal,wheretheperiodicitiescanbeseen.Whenwespeak,pitchandthevocaltract'stransferfunctionarenotstatic;theychangeaccordingtotheircontrolsignalstoproducespeech.EngineerstypicallydisplayhowthespeechspectrumchangesovertimewithwhatisknownasaspectrogramSection5.10Figure4.17spectrogram.Notehowthelinespectrum,whichindicateshowthepitchchanges,isvisibleduringthevowels,butnotduringtheconsonantslikethecein"Rice".Thefundamentalmodelforspeechindicateshowengineersusethephysicsunderlyingthesignalgen-erationprocessandexploititsstructuretoproduceasystemsmodelthatsuppressesthephysicswhileemphasizinghowthesignalis"constructed."Fromeverydaylife,weknowthatspeechcontainsawealthofinformation.Wewanttodeterminehowtotransmitandreceiveit.Ecientandeectivespeechtransmis-sionrequiresustoknowthesignal'spropertiesanditsstructureasexpressedbythefundamentalmodelofspeechproduction.WeseefromFigure4.17spectrogram,forexample,thatspeechcontainssignicantenergyfromzerofrequencyuptoaround5kHz.Eectivespeechtransmissionsystemsmustbeabletocopewithsignalshavingthisbandwidth.Itisinterestingthatonesystemthatdoesnotsupportthis5kHzbandwidthisthetelephone:Telephonesystemsactlikeabandpasslterpassingenergybetweenabout200Hzand3.2kHz.Themostimportant

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140CHAPTER4.FREQUENCYDOMAINconsequenceofthislteringistheremovalofhighfrequencyenergy.Inoursampleutterance,the"ce"soundin"Rice""containsmostofitsenergyabove3.2kHz;thislteringeectiswhyitisextremelydiculttodistinguishthesounds"s"and"f"overthetelephone.Trythisyourself:Callafriendanddetermineiftheycandistinguishbetweenthewords"six"and"x".Ifyousaythesewordsinisolationsothatnocontextprovidesahintaboutwhichwordyouaresaying,yourfriendwillnotbeabletotellthemapart.RadiodoessupportthisbandwidthseemoreaboutAMandFMradiosystemsSection6.11.Ecientspeechtransmissionsystemsexploitthespeechsignal'sspecialstructure:Whatmakesspeechspeech?Youcanconjuremanysignalsthatspanthesamefrequenciesasspeechcarenginesounds,violinmusic,dogbarksbutdon'tsoundatalllikespeech.Weshalllearnlaterthattransmissionofany5kHzbandwidthsignalrequiresabout80kbpsthousandsofbitspersecondtotransmitdigitally.Speechsignalscanbetransmittedusinglessthan1kbpsbecauseofitsspecialstructure.Toreducethe"digitalbandwidth"sodrasticallymeansthatengineersspentmanyyearstodevelopsignalprocessingandcodingmethodsthatcouldcapturethespecialcharacteristicsofspeechwithoutdestroyinghowitsounds.Ifyouusedaspeechtransmissionsystemtosendaviolinsound,itwouldarrivehorriblydistorted;speechtransmittedthesamewaywouldsoundne.Exploitingthespecialstructureofspeechrequiresgoingbeyondthecapabilitiesofanalogsignalprocessingsystems.Manyspeechtransmissionsystemsworkbyndingthespeaker'spitchandtheformantfrequencies.Fundamentally,weneedtodomorethanlteringtodeterminethespeechsignal'sstructure;weneedtomanipulatesignalsinmorewaysthanarepossiblewithanalogsystems.Suchexibilityisachievablebutnotwithoutsomelosswithprogrammabledigitalsystems.4.11FrequencyDomainProblems16Problem4.1:SimpleFourierSeriesFindthecomplexFourierseriesrepresentationsofthefollowingsignalswithoutexplicitlycalculatingFourierintegrals.Whatisthesignal'speriodineachcase?ast=sintbst=sin2tcst=cost+2costdst=costcostest=cos)]TJ/F8 9.9626 Tf 4.5662 -8.0698 Td[(10t+ 6+costfstgivenbythedepictedwaveformFigure4.18. Figure4.18Problem4.2:FourierSeriesFindtheFourierseriesrepresentationforthefollowingperiodicsignalsFigure4.19.Forthethirdsignal, 16Thiscontentisavailableonlineat.

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141 a b cFigure4.19 Figure4.20 ndthecomplexFourierseriesforthetrianglewavewithoutperformingtheusualFourierintegrals.Hint:Howisthissignalrelatedtooneforwhichyoualreadyhavetheseries?Problem4.3:PhaseDistortionWecanlearnaboutphasedistortionbyreturningtocircuitsandinvestigatethefollowingcircuitFig-ure4.20.aFindthislter'stransferfunction.bFindthemagnitudeandphaseofthistransferfunction.Howwouldyoucharacterizethiscircuit?cLetvintbeasquare-waveofperiodT.WhatistheFourierseriesfortheoutputvoltage?dUseMatlabtondtheoutput'swaveformforthecasesT=0:01andT=2.WhatvalueofTdelineatesthetwokindsofresultsyoufound?Thesoftwareinfourier2.mmightbeuseful.eInsteadofthedepictedcircuit,thesquarewaveispassedthroughasystemthatdelaysitsinput,which

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142CHAPTER4.FREQUENCYDOMAINappliesalinearphaseshifttothesignal'sspectrum.LetthedelaybeT 4.UsethetransferfunctionofadelaytocomputeusingMatlabtheFourierseriesoftheoutput.Showthatthesquarewaveisindeeddelayed.Problem4.4:ApproximatingPeriodicSignalsOften,wewanttoapproximateareferencesignalbyasomewhatsimplersignal.Toassessthequalityofanapproximation,themostfrequentlyusederrormeasureisthemean-squarederror.Foraperiodicsignalst,2=1 TZT0st)]TJ/F8 9.9626 Tf 10.3605 0 Td[(~st2dtwherestisthereferencesignaland~stitsapproximation.OneconvenientwayofndingapproximationsforperiodicsignalsistotruncatetheirFourierseries.~st=KXk=)]TJ/F10 6.9738 Tf 6.2267 0 Td[(Kckej2k TtThepointofthisproblemistoanalyzewhetherthisapproachisthebesti.e.,alwaysminimizesthemean-squarederror.aFindafrequency-domainexpressionfortheapproximationerrorwhenweusethetruncatedFourierseriesastheapproximation.bInsteadoftruncatingtheseries,let'sgeneralizethenatureoftheapproximationtoincludinganysetof2K+1terms:We'llalwaysincludethec0andthenegativeindexedtermcorrespondingtock.Whatselectionoftermsminimizesthemean-squarederror?Findanexpressionforthemean-squarederrorresultingfromyourchoice.cFindtheFourierseriesforthedepictedsignalFigure4.21.UseMatlabtondthetruncatedapprox-imationandbestapproximationinvolvingtwoterms.Plotthemean-squarederrorasafunctionofKforbothapproximations. Figure4.21 Problem4.5:Long,HotDaysThedailytemperatureisaconsequenceofseveraleects,oneofthembeingthesun'sheating.Ifthiswerethedominanteect,thendailytemperatureswouldbeproportionaltothenumberofdaylighthours.TheplotFigure4.22showsthattheaveragedailyhightemperaturedoesnotbehavethatway.

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143 Figure4.22 Inthisproblem,wewanttounderstandthetemperaturecomponentofourenvironmentusingFourierseriesandlinearsystemtheory.Theletemperature.matcontainsthesedatadaylighthoursintherstrow,correspondingaveragedailyhighsinthesecondforHouston,Texas.aLetthelengthofdayserveasthesoleinputtoasystemhavinganoutputequaltotheaveragedailytemperature.Examiningtheplotsofinputandoutput,wouldyousaythatthesystemislinearornot?Howdidyoureachyouconclusion?bFindtherstvetermsc0,...,c4ofthecomplexFourierseriesforeachsignal.cWhatistheharmonicdistortioninthetwosignals?Excludec0fromthiscalculation.dBecausetheharmonicdistortionissmall,let'sconcentrateonlyontherstharmonic.Whatisthephaseshiftbetweeninputandoutputsignals?eFindthetransferfunctionofthesimplestpossiblelinearmodelthatwoulddescribethedata.Char-acterizeandinterpretthestructureofthismodel.Inparticular,giveaphysicalexplanationforthephaseshift.fPredictwhattheoutputwouldbeifthemodelhadnophaseshift.Woulddaysbehotter?Ifso,byhowmuch?Problem4.6:FourierTransformPairsFindtheFourierorinverseFouriertransformofthefollowing.axt=e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(ajtj;bxt=te)]TJ/F7 6.9738 Tf 6.2267 0 Td[(atutcXf=8<:1ifjfjWdxt=e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(atcos2f0tutProblem4.7:DualityinFourierTransforms"Duality"meansthattheFouriertransformandtheinverseFouriertransformareverysimilar.Conse-quently,thewaveformstinthetimedomainandthespectrumsfhaveaFouriertransformandaninverseFouriertransform,respectively,thatareverysimilar.aCalculatetheFouriertransformofthesignalshownbelowFigure4.23a.bCalculatetheinverseFouriertransformofthespectrumshownbelowFigure4.23b.cHowaretheseanswersrelated?WhatisthegeneralrelationshipbetweentheFouriertransformofstandtheinversetransformofsf?

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144CHAPTER4.FREQUENCYDOMAIN a bFigure4.23 Problem4.8:SpectraofPulseSequencesPulsesequencesoccuroftenindigitalcommunicationandinothereldsaswell.Whataretheirspectralproperties?aCalculatetheFouriertransformofthesinglepulseshownbelowFigure4.24a.bCalculatetheFouriertransformofthetwo-pulsesequenceshownbelowFigure4.24b.cCalculatetheFouriertransformfortheten-pulsesequenceshowninbelowFigure4.24c.Youshouldlookforageneralexpressionthatholdsforsequencesofanylength.dUsingMatlab,plotthemagnitudesofthethreespectra.Describehowthespectrachangeasthenumberofrepeatedpulsesincreases. a b cFigure4.24 Problem4.9:LowpassFilteringaSquareWaveLetasquarewaveperiodTserveastheinputtoarst-orderlowpasssystemconstructedasaRClter.Wewanttoderiveanexpressionforthetime-domainresponseoftheltertothisinput.aFirst,considertheresponseoftheltertoasimplepulse,havingunitamplitudeandwidthT 2.Deriveanexpressionforthelter'soutputtothispulse.bNotingthatthesquarewaveisasuperpositionofasequenceofthesepulses,whatisthelter'sresponsetothesquarewave?cThenatureofthisresponseshouldchangeastherelationbetweenthesquarewave'speriodandthelter'scutofrequencychange.Howlongmusttheperiodbesothattheresponsedoesnotachievearelativelyconstantvaluebetweentransitionsinthesquarewave?Whatistherelationofthelter'scutofrequencytothesquarewave'sspectruminthiscase?Problem4.10:MathematicswithCircuitsSimplecircuitscanimplementsimplemathematicaloperations,suchasintegrationanddierentiation.Wewanttodevelopanactivecircuititcontainsanop-amphavinganoutputthatisproportionaltothe

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145integralofitsinput.Forexample,youcoulduseanintegratorinacartodeterminedistancetraveledfromthespeedometer.aWhatisthetransferfunctionofanintegrator?bFindanop-ampcircuitsothatitsvoltageoutputisproportionaltotheintegralofitsinputforallsignals.Problem4.11:Whereisthatsoundcomingfrom?Wedeterminewheresoundiscomingfrombecausewehavetwoearsandabrain.Soundtravelsatarelativelyslowspeedandourbrainusesthefactthatsoundwillarriveatoneearbeforetheother.AsshownhereFigure4.25,asoundcomingfromtherightarrivesattheleftearsecondsafteritarrivesattherightear. Figure4.25 Oncethebrainndsthispropagationdelay,itcandeterminethesounddirection.Inanattempttomodelwhatthebrainmightdo,RUsignalprocessorswanttodesignanoptimalsystemthatdelayseachear'ssignalbysomeamountthenaddsthemtogether.landrarethedelaysappliedtotheleftandrightsignalsrespectively.Theideaistodeterminethedelayvaluesaccordingtosomecriterionthatisbasedonwhatismeasuredbythetwoears.aWhatisthetransferfunctionbetweenthesoundsignalstandtheprocessoroutputyt?bOnewayofdeterminingthedelayistochooselandrtomaximizethepowerinyt.Howarethesemaximum-powerprocessingdelaysrelatedto?Problem4.12:ArrangementsofSystemsArchitectingasystemofmodularcomponentsmeansarrangingtheminvariouscongurationstoachievesomeoverallinput-outputrelation.ForeachofthefollowingFigure4.26,determinetheoveralltransferfunctionbetweenxtandyt.Theoveralltransferfunctionforthecascaderstdepictedsystemisparticularlyinteresting.Whatdoesitsayabouttheeectoftheorderingoflinear,time-invariantsystemsinacascade?Problem4.13:FilteringLetthesignalst=sint tbetheinputtoalinear,time-invariantlterhavingthetransferfunctionshownbelowFigure4.27.Findtheexpressionforyt,thelter'soutput.

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146CHAPTER4.FREQUENCYDOMAIN asystema bsystemb csystemcFigure4.26 Problem4.14:CircuitsFilter!Aunit-amplitudepulsewithdurationofonesecondservesastheinputtoanRC-circuithavingtransferfunctionHf=j2f 4+j2faHowwouldyoucategorizethistransferfunction:lowpass,highpass,bandpass,other?bFindacircuitthatcorrespondstothistransferfunction.cFindanexpressionforthelter'soutput.Problem4.15:ReverberationReverberationcorrespondstoaddingtoasignalitsdelayedversion.aAssumingrepresentsthedelay,whatistheinput-outputrelationforareverberationsystem?Isthesystemlinearandtime-invariant?Ifso,ndthetransferfunction;ifnot,whatlinearityortime-invariancecriteriondoesreverberationviolate.bAmusicgroupknownastheROwlsishavingtroublesellingitsrecordings.Therecordcompany'sengineergetstheideaofapplyingdierentdelaytothelowandhighfrequenciesandaddingtheresulttocreateanewmusicaleect.Thus,theROwls'audiowouldbeseparatedintotwopartsoneless

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147 Figure4.27 thanthefrequencyf0,theothergreaterthanf0,thesewouldbedelayedbylandhrespectively,andtheresultingsignalsadded.Drawablockdiagramforthisnewaudioprocessingsystem,showingitsvariouscomponents.cHowdoesthemagnitudeofthesystem'stransferfunctiondependonthetwodelays?Problem4.16:EchoesinTelephoneSystemsAfrequentlyencounteredproblemintelephonesisecho.Here,becauseofacousticcouplingbetweentheearpieceandmicrophoneinthehandset,whatyouhearisalsosenttothepersontalking.Thatpersonthusnotonlyhearsyou,butalsohearsherownspeechdelayedbecauseofpropagationdelayoverthetelephonenetworkandattenuatedtheacousticcouplinggainislessthanone.Furthermore,thesameproblemappliestoyouaswell:Theacousticcouplingoccursinherhandsetaswellasyours.aDevelopablockdiagramthatdescribesthissituation.bFindthetransferfunctionbetweenyourvoiceandwhatthelistenerhears.cEachtelephonecontainsasystemforreducingechoesusingelectricalmeans.Whatsimplesystemcouldnulltheechoes?Problem4.17:DemodulatinganAMSignalLetmtdenotethesignalthathasbeenamplitudemodulated.xt=A+mtsinfctRadiostationstrytorestricttheamplitudeofthesignalmtsothatitislessthanoneinmagnitude.Thefrequencyfcisverylargecomparedtothefrequencycontentofthesignal.Whatweareconcernedabouthereisnottransmission,butreception.aTheso-calledcoherentdemodulatorsimplymultipliesthesignalxtbyasinusoidhavingthesamefrequencyasthecarrierandlowpasslterstheresult.Analyzethisreceiverandshowthatitworks.Assumethelowpasslterisideal.bOneissueincoherentreceptionisthephaseofthesinusoidusedbythereceiverrelativetothatusedbythetransmitter.Assumingthatthesinusoidofthereceiverhasaphase,howdoestheoutputdependon?Whatistheworstpossiblevalueforthisphase?cTheincoherentreceiverismorecommonlyusedbecauseofthephasesensitivityprobleminherentincoherentreception.Here,thereceiverfull-waverectiesthereceivedsignalandlowpasslterstheresultagainideally.Analyzethisreceiver.Doesitsoutputdierfromthatofthecoherentreceiverinasignicantway?Problem4.18:UnusualAmplitudeModulationWewanttosendaband-limitedsignalhavingthedepictedspectrumFigure4.28awithamplitudemod-ulationintheusualway.I.B.Dierentsuggestsusingthesquare-wavecarriershownbelowFigure4.28b.Well,itisdierent,buthisfriendswonderifanytechniquecandemodulateit.aFindanexpressionforXf,theFouriertransformofthemodulatedsignal.bSketchthemagnitudeofXf,beingcarefultolabelimportantmagnitudesandfrequencies.

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148CHAPTER4.FREQUENCYDOMAIN a bFigure4.28 Figure4.29 cWhatdemodulationtechniqueobviouslyworks?dI.B.challengesthreeofhisfriendstodemodulatextsomeotherway.Onefriendsuggestsmodulatingxtwithcos)]TJ/F10 6.9738 Tf 5.7618 -4.1472 Td[(t 2,anotherwantstotrymodulatingwithcostandthethirdthinkscos)]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(3t 2willwork.SketchthemagnitudeoftheFouriertransformofthesignaleachstudent'sapproachproduces.Whichstudentcomesclosesttorecoveringtheoriginalsignal?Why?Problem4.19:SammyFallsAsleep...WhilesittinginELEC241class,hefallsasleepduringacriticaltimewhenanAMreceiverisbeingdescribed.Thereceivedsignalhastheformrt=A+mtcosfct+wherethephaseisunknown.Themessagesignalismt;ithasabandwidthofWHzandamagnitudelessthan1jmtj<1.Thephaseisunknown.TheinstructordrewadiagramFigure4.29forareceiverontheboard;Sammysleptthroughthedescriptionofwhattheunknownsystemswhere.aWhatarethesignalsxctandxst?bWhatwouldyouputinfortheunknownsystemsthatwouldguaranteethatthenaloutputcontainedthemessageregardlessofthephase?hint:Thinkofatrigonometricidentitythatwouldproveuseful.

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149 Figure4.30 Figure4.31 cSammymayhavebeenasleep,buthecanthinkofafarsimplerreceiver.Whatisit?Problem4.20:JammingSidRichardsoncollegedecidestosetupitsownAMradiostationKSRR.Theresidentelectricalengineerdecidesthatshecanchooseanycarrierfrequencyandmessagebandwidthforthestation.Arivalcollegedecidestojamitstransmissionsbytransmittingahigh-powersignalthatinterfereswithradiosthattrytoreceiveKSRR.ThejammingsignaljamtiswhatisknownasasawtoothwavedepictedinthefollowinggureFigure4.30havingaperiodknowntoKSRR'sengineer.aFindthespectrumofthejammingsignal.bCanKSRRentirelycircumventtheattempttojamitbycarefullychoosingitscarrierfrequencyandtransmissionbandwidth?Ifso,ndthestation'scarrierfrequencyandtransmissionbandwidthintermsofT,theperiodofthejammingsignal;ifnot,showwhynot.Problem4.21:AMStereoAstereophonicsignalconsistsofa"left"signalltanda"right"signalrtthatconveyssoundscomingfromanorchestra'sleftandrightsides,respectively.Totransmitthesetwosignalssimultaneously,thetransmitterrstformsthesumsignals+t=lt+rtandthedierencesignals)]TJ/F8 9.9626 Tf 8.3852 1.4944 Td[(t=lt)]TJ/F11 9.9626 Tf 10.5464 0 Td[(rt.Then,thetransmitteramplitude-modulatesthedierencesignalwithasinusoidhavingfrequency2W,whereWisthebandwidthoftheleftandrightsignals.Thesumsignalandthemodulateddierencesignalareadded,thesumamplitude-modulatedtotheradiostation'scarrierfrequencyfc,andtransmitted.AssumethespectraoftheleftandrightsignalsareasshownFigure4.31.aWhatistheexpressionforthetransmittedsignal?Sketchitsspectrum.bShowtheblockdiagramofastereoAMreceiverthatcanyieldtheleftandrightsignalsasseparateoutputs.cWhatsignalwouldbeproducedbyaconventionalcoherentAMreceiverthatexpectstoreceiveastandardAMsignalconveyingamessagesignalhavingbandwidthW?

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150CHAPTER4.FREQUENCYDOMAIN Figure4.32Problem4.22:NovelAMStereoMethodAcleverengineerhassubmittedapatentforanewmethodfortransmittingtwosignalssimultaneouslyinthesametransmissionbandwidthascommercialAMradio.AsshownFigure4.32,herapproachistomodulatethepositiveportionofthecarrierwithonesignalandthenegativeportionwithasecond.Indetailthetwomessagesignalsm1tandm2tarebandlimitedtoWHzandhavemaximalamplitudesequalto1.ThecarrierhasafrequencyfcmuchgreaterthanW.Thetransmittedsignalxtisgivenbyxt=8<:A+am1tsinfctifsinfct0A+am2tsinfctifsinfct<0Inallcases,0
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151 Figure4.33 aSketchthespectrumofthedemodulatedsignalproducedbyacoherentdemodulatortunedtofcHz.bWillthisdemodulatedsignalbeascrambledversionoftheoriginal?Ifso,howso;ifnot,whynot?cCanyoudevelopareceiverthatcandemodulatethemessagewithoutknowingtheosetfrequencyfc?Problem4.25:SignalScramblingAnexcitedinventorannouncesthediscoveryofawayofusinganalogtechnologytorendermusicunlistenablewithoutknowingthesecretrecoverymethod.Theideaistomodulatethebandlimitedmessagemtbyaspecialperiodicsignalstthatiszeroduringhalfofitsperiod,whichrendersthemessageunlistenableandsupercially,atleast,unrecoverableFigure4.33.aWhatistheFourierseriesfortheperiodicsignal?bWhataretherestrictionsontheperiodTsothatthemessagesignalcanberecoveredfrommtst?cELEC241studentsthinktheyhave"broken"theinventor'sschemeandaregoingtoannounceittotheworld.Howwouldtheyrecovertheoriginalmessagewithouthavingdetailedknowledgeofthemodulatingsignal?

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152CHAPTER4.FREQUENCYDOMAINSolutionstoExercisesinChapter4SolutiontoExercise4.1p.110BecauseofEuler'srelation,sinft=1 2je+j2ft)]TJ/F8 9.9626 Tf 13.4946 6.7398 Td[(1 2je)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2ft.47Thus,c1=1 2j,c)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1=)]TJ/F1 9.9626 Tf 9.4092 11.0586 Td[(1 2j,andtheothercoecientsarezero.SolutiontoExercise4.2p.113c0=A T.Thisquantityclearlycorrespondstotheperiodicpulsesignal'saveragevalue.SolutiontoExercise4.3p.114WritethecoecientsofthecomplexFourierseriesinCartesianformasck=Ak+jBkandsubstituteintotheexpressionforthecomplexFourierseries.1Xk=ckej2kt T=1Xk=Ak+jBkej2kt TSimplifyingeachterminthesumusingEuler'sformula,Ak+jBkej2kt T=Ak+jBk)]TJ/F8 9.9626 Tf 4.5662 -8.0698 Td[(cos)]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(2kt T+jsin)]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(2kt T=Akcos)]TJ/F7 6.9738 Tf 5.7618 -4.1472 Td[(2kt T)]TJ/F11 9.9626 Tf 9.9626 0 Td[(Bksin)]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(2kt T+j)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(Aksin)]TJ/F7 6.9738 Tf 5.7618 -4.1472 Td[(2kt T+Bkcos)]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(2kt TWenowcombinetermsthathavethesamefrequencyindexinmagnitude.Becausethesignalisreal-valued,thecoecientsofthecomplexFourierserieshaveconjugatesymmetry:c)]TJ/F10 6.9738 Tf 6.2267 0 Td[(k=ckorA)]TJ/F10 6.9738 Tf 6.2267 0 Td[(k=AkandB)]TJ/F10 6.9738 Tf 6.2267 0 Td[(k=)]TJ/F11 9.9626 Tf 7.7487 0 Td[(Bk.Afterweaddthepositive-indexedandnegative-indexedterms,eachtermintheFourierseriesbecomes2Akcos)]TJ/F7 6.9738 Tf 5.7618 -4.1472 Td[(2kt T)]TJ/F8 9.9626 Tf 9.588 0 Td[(2Bksin)]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(2kt T.ToobtaintheclassicFourierseries.11,wemusthave2Ak=akand2Bk=)]TJ/F11 9.9626 Tf 7.7487 0 Td[(bk.SolutiontoExercise4.4p.115Theaverageofasetofnumbersisthesumdividedbythenumberofterms.ViewingsignalintegrationasthelimitofaRiemannsum,theintegralcorrespondstotheaverage.SolutiontoExercise4.5p.115WefoundthatthecomplexFourierseriescoecientsaregivenbyck=2 jk.Thecoecientsarepureimaginary,whichmeansak=0.Thecoecientsofthesinetermsaregivenbybk=)]TJ/F8 9.9626 Tf 9.4091 0 Td[(Imcksothatbk=8<:4 kifkodd0ifkevenThus,theFourierseriesforthesquarewaveissqt=Xk2f1;3;:::g4 ksin2kt T.48SolutiontoExercise4.6p.117Thermsvalueofasinusoidequalsitsamplitudedividedbyp 2.Asahalf-waverectiedsinewaveiszeroduringhalfoftheperiod,itsrmsvalueisA 2p 2.SolutiontoExercise4.7p.117TotalharmonicdistortionequalsP1k=2ak2+bk2 a12+b12.Clearly,thisquantityismosteasilycomputedinthefrequencydomain.However,thenumeratorequalsthesquareofthesignal'srmsvalueminusthepowerintheaverageandthepowerintherstharmonic.

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153SolutiontoExercise4.8p.120Totalharmonicdistortioninthesquarewaveis1)]TJ/F7 6.9738 Tf 11.1581 3.9226 Td[(1 2)]TJ/F7 6.9738 Tf 6.2306 -4.1472 Td[(4 2=20%.SolutiontoExercise4.9p.123NsignalsdirectlyencodedrequireabandwidthofN T.Usingabinaryrepresentation,weneedlog2N T.ForN=128,thebinary-encodingschemehasafactorof7 128=0:05smallerbandwidth.Clearly,binaryencodingissuperior.SolutiontoExercise4.10p.123WecanuseNdierentamplitudevaluesatonlyonefrequencytorepresentthevariousletters.SolutiontoExercise4.11p.126Becausethelter'sgainatzerofrequencyequalsone,theaverageoutputvaluesequaltherespectiveaverageinputvalues.SolutiontoExercise4.12p.128FSf=Z1Sfe)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2ftdf=Z1Sfe+j2f)]TJ/F10 6.9738 Tf 6.2267 0 Td[(tdf=s)]TJ/F11 9.9626 Tf 7.7487 0 Td[(tSolutiontoExercise4.13p.128FFFFst=st.WeknowthatFSf=R1Sfe)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2ftdf=R1Sfe+j2f)]TJ/F10 6.9738 Tf 6.2267 0 Td[(tdf=s)]TJ/F11 9.9626 Tf 7.7487 0 Td[(t.Therefore,twoFouriertransformsappliedtostyieldss)]TJ/F11 9.9626 Tf 7.7487 0 Td[(t.Weneedtwomoretogetusbackwherewestarted.SolutiontoExercise4.14p.131ThesignalistheinverseFouriertransformofthetriangularlyshapedspectrum,andequalsst=WsinWt Wt2SolutiontoExercise4.15p.131Theresultismosteasilyfoundinthespectrum'sformula:thepowerinthesignal-relatedpartofxtishalfthepowerofthesignalst.SolutiontoExercise4.16p.133Theinversetransformofthefrequencyresponseis1 RCe)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(t RCut.Multiplyingthefrequencyresponseby1)]TJ/F11 9.9626 Tf 10.2557 0 Td[(e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2fmeanssubtractfromtheoriginalsignalitstime-delayedversion.Delayingthefrequencyresponse'stime-domainversionbyresultsin1 RCe)]TJ/F6 4.9813 Tf 5.3965 0 Td[(t)]TJ/F6 4.9813 Tf 5.3965 0 Td[( RCut)]TJ/F8 9.9626 Tf 9.9626 0 Td[(.Subtractingfromtheundelayedsignalyields1 RCe)]TJ/F9 4.9813 Tf 5.3965 0 Td[(t RCut)]TJ/F7 6.9738 Tf 13.3341 3.9226 Td[(1 RCe)]TJ/F6 4.9813 Tf 5.3965 0 Td[(t)]TJ/F6 4.9813 Tf 5.3965 0 Td[( RCut)]TJ/F8 9.9626 Tf 9.9626 0 Td[(.Nowweintegratethissum.Becausetheintegralofasumequalsthesumofthecomponentintegralsintegrationislinear,wecanconsidereachseparately.Becauseintegrationandsignal-delayarelinear,theintegralofadelayedsignalequalsthedelayedversionoftheintegral.Theintegralisprovidedintheexample.44.SolutiontoExercise4.17p.135Iftheglottiswerelinear,aconstantinputazero-frequencysinusoidshouldyieldaconstantoutput.Theperiodicoutputindicatesnonlinearbehavior.SolutiontoExercise4.18p.136Inthebottom-leftpanel,theperiodisabout0.009s,whichequalsafrequencyof111Hz.Thebottom-rightpanelhasaperiodofabout0.0065s,afrequencyof154Hz.SolutiontoExercise4.19p.137Becausemaleshavealowerpitchfrequency,thespacingbetweenspectrallinesissmaller.Thiscloserspacingmoreaccuratelyrevealstheformantstructure.Doublingthepitchfrequencyto300HzforFigure4.16voicespectrumwouldamounttoremovingeveryotherspectralline.

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154CHAPTER4.FREQUENCYDOMAIN

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Chapter5DigitalSignalProcessing5.1IntroductiontoDigitalSignalProcessing1Notonlydowehaveanalogsignalssignalsthatarereal-orcomplex-valuedfunctionsofacontinuousvariablesuchastimeorspacewecandenedigitalonesaswell.Digitalsignalsaresequences,functionsdenedonlyfortheintegers.Wethususethenotationsntodenoteadiscrete-timeone-dimensionalsignalsuchasadigitalmusicrecordingandsm;nforadiscrete-"time"two-dimensionalsignallikeaphototakenwithadigitalcamera.Sequencesarefundamentallydierentthancontinuous-timesignals.Forexample,continuityhasnomeaningforsequences.Despitesuchfundamentaldierences,thetheoryunderlyingdigitalsignalprocessingmirrorsthatforana-logsignals:Fouriertransforms,linearltering,andlinearsystemsparallelwhatpreviouschaptersdescribed.Thesesimilaritiesmakeiteasytounderstandthedenitionsandwhyweneedthem,butthesimilaritiesshouldnotbeconstruedas"analogwannabes."Wewilldiscoverthatdigitalsignalprocessingisnotanapproximationtoanalogprocessing.Wemustexplicitlyworryaboutthedelityofconvertinganalogsignalsintodigitalones.ThemusicstoredonCDs,thespeechsentoverdigitalcellulartelephones,andthevideocarriedbydigitaltelevisionallevidencethatanalogsignalscanbeaccuratelyconvertedtodigitalonesandbackagain.Thekeyreasonwhydigitalsignalprocessingsystemshaveatechnologicaladvantagetodayisthecom-puter:computations,liketheFouriertransform,canbeperformedquicklyenoughtobecalculatedasthesignalisproduced,2andprogrammabilitymeansthatthesignalprocessingsystemcanbeeasilychanged.Thisexibilityhasobviousappeal,andhasbeenwidelyacceptedinthemarketplace.Programmabilitymeansthatwecanperformsignalprocessingoperationsimpossiblewithanalogsystemscircuits.Wewillalsodiscoverthatdigitalsystemsenjoyanalgorithmicadvantagethatcontributestorapidprocessingspeeds:Computationscanberestructuredinnon-obviouswaystospeedtheprocessing.Thisexibilitycomesataprice,aconsequenceofhowcomputerswork.Howdocomputersperformsignalprocessing?5.2IntroductiontoComputerOrganization35.2.1ComputerArchitectureTounderstanddigitalsignalprocessingsystems,wemustunderstandalittleabouthowcomputerscompute.Themoderndenitionofacomputerisanelectronicdevicethatperformscalculationsondata,presenting 1Thiscontentisavailableonlineat.2Takingasystemsviewpointforthemoment,asystemthatproducesitsoutputasrapidlyastheinputarisesissaidtobeareal-timesystem.Allanalogsystemsoperateinrealtime;digitalonesthatdependonacomputertoperformsystemcomputationsmayormaynotworkinrealtime.Clearly,weneedreal-timesignalprocessingsystems.Onlyrecentlyhavecomputersbecomefastenoughtomeetreal-timerequirementswhileperformingnon-trivialsignalprocessing.3Thiscontentisavailableonlineat.155

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156CHAPTER5.DIGITALSIGNALPROCESSINGtheresultstohumansorothercomputersinavarietyofhopefullyusefulways. OrganizationofaSimpleComputer Figure5.1:Genericcomputerhardwareorganization. Thegenericcomputercontainsinputdeviceskeyboard,mouse,A/Danalog-to-digitalconverter,etc.,acomputationalunit,andoutputdevicesmonitors,printers,D/Aconverters.Thecomputationalunitisthecomputer'sheart,andusuallyconsistsofacentralprocessingunitCPU,amemory,andaninput/outputI/Ointerface.WhatI/Odevicesmightbepresentonagivencomputervarygreatly.Asimplecomputeroperatesfundamentallyindiscretetime.Computersareclockeddevices,inwhichcomputationalstepsoccurperiodicallyaccordingtoticksofaclock.Thisdescriptionbeliesclockspeed:Whenyousay"Ihavea1GHzcomputer,"youmeanthatyourcomputertakes1nanosecondtoperformeachstep.Thatisincrediblyfast!A"step"doesnot,unfortunately,necessarilymeanacomputationlikeanaddition;computersbreaksuchcomputationsdownintoseveralstages,whichmeansthattheclockspeedneednotexpressthecomputationalspeed.Computationalspeedisexpressedinunitsofmillionsofinstructions/secondMips.Your1GHzcomputerclockspeedmayhaveacomputationalspeedof200Mips.Computersperformintegerdiscrete-valuedcomputations.Computercalculationscanbenumericobeyingthelawsofarithmetic,logicalobeyingthelawsofanalgebra,orsymbolicobeyinganylawyoulike.4Eachcomputerinstructionthatperformsanelementarynumericcalculationanaddition,amultiplication,oradivisiondoessoonlyforintegers.Thesumorproductoftwointegersisalsoaninteger,butthequotientoftwointegersislikelytonotbeaninteger.Howdoesacomputerdealwithnumbersthathavedigitstotherightofthedecimalpoint?Thisproblemisaddressedbyusingtheso-calledoating-pointrepresentationofrealnumbers.Atitsheart,however,thisrepresentationreliesoninteger-valuedcomputations.5.2.2RepresentingNumbersFocusingonnumbers,allnumberscanrepresentedbythepositionalnotationsystem.5Theb-arypositionalrepresentationsystemusesthepositionofdigitsrangingfrom0tob-1todenoteanumber.Thequantitybis 4Anexampleofasymboliccomputationissortingalistofnames.5Alternativenumberrepresentationsystemsexist.Forexample,wecouldusestickgurecountingorRomannumerals.Thesewereusefulinancienttimes,butverylimitingwhenitcomestoarithmeticcalculations:evertriedtodividetwoRomannumerals?

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157knownasthebaseofthenumbersystem.Mathematically,positionalsystemsrepresentthepositiveintegernasn=1Xk=0)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(dkbk;dk2f0;:::;b)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1g.1andwesuccinctlyexpressninbase-basnb=dNdN)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1:::d0.Thenumber25inbase10equals2101+5100,sothatthedigitsrepresentingthisnumberared0=5,d1=2,andallotherdkequalzero.Thissamenumberinbinarybase2equals11001124+123+022+021+120and19inhexadecimalbase16.Fractionsbetweenzeroandonearerepresentedthesameway.f=)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1Xk=)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(dkbk;dk2f0;:::;b)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1g.2Allnumberscanberepresentedbytheirsign,integerandfractionalparts.ComplexnumbersSection2.1canbethoughtofastworealnumbersthatobeyspecialrulestomanipulatethem.Humansusebase10,commonlyassumedtobeduetoushavingtenngers.Digitalcomputersusethebase2orbinarynumberrepresentation,eachdigitofwhichisknownasabitbinarydigit.Numberrepresentationsoncomputers Figure5.2:Thevariouswaysnumbersarerepresentedinbinaryareillustrated.Thenumberofbytesfortheexponentandmantissacomponentsofoatingpointnumbersvaries.Here,eachbitisrepresentedasavoltagethatiseither"high"or"low,"therebyrepresenting"1"or"0,"respectively.Torepresentsignedvalues,wetackonaspecialbitthesignbittoexpressthesign.Thecomputer'smemoryconsistsofanorderedsequenceofbytes,acollectionofeightbits.Abytecanthereforerepresentanunsignednumberrangingfrom0to255.Ifwetakeoneofthebitsandmakeitthesignbit,wecanmakethesamebytetorepresentnumbersrangingfrom)]TJ/F8 9.9626 Tf 7.7487 0 Td[(128to127.Butacomputercannotrepresentallpossiblerealnumbers.Thefaultisnotwiththebinarynumbersystem;ratherhavingonlyanitenumberofbytesistheproblem.Whileagigabyteofmemorymayseemtobealot,ittakesaninnitenumberofbitstorepresent.Sincewewanttostoremanynumbersinacomputer'smemory,wearerestrictedtothosethathaveanitebinaryrepresentation.Largeintegerscanberepresentedbyanorderedsequenceofbytes.Commonlengths,usuallyexpressedintermsofthenumberofbits,are16,32,and64.Thus,anunsigned32-bitnumbercanrepresentintegersrangingbetween0and232)]TJ/F8 9.9626 Tf 10.2562 0 Td[(1,294,967,295,anumberalmostbigenoughtoenumerateeveryhumanintheworld!6Exercise5.1Solutiononp.204.Forboth32-bitand64-bitintegerrepresentations,whatarethelargestnumbersthatcanberepresentedifasignbitmustalsobeincluded. 6Youneedonemorebittodothat.

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158CHAPTER5.DIGITALSIGNALPROCESSINGWhilethissystemrepresentsintegerswell,howaboutnumbershavingnonzerodigitstotherightofthedecimalpoint?Inotherwords,howarenumbersthathavefractionalpartsrepresented?Forsuchnumbers,thebinaryrepresentationsystemisused,butwithalittlemorecomplexity.Theoating-pointsystemusesanumberofbytes-typically4or8-torepresentthenumber,butwithonebytesometimestwobytesreservedtorepresenttheexponenteofapower-of-twomultiplierforthenumber-themantissam-expressedbytheremainingbytes.x=m2e.3Themantissaisusuallytakentobeabinaryfractionhavingamagnitudeintherange1 2;1,whichmeansthatthebinaryrepresentationissuchthatd)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1=1.7Thenumberzeroisanexceptiontothisrule,anditistheonlyoatingpointnumberhavingazerofraction.Thesignofthemantissarepresentsthesignofthenumberandtheexponentcanbeasignedinteger.Acomputer'srepresentationofintegersiseitherperfectoronlyapproximate,thelattersituationoccurringwhentheintegerexceedstherangeofnumbersthatalimitedsetofbytescanrepresent.Floatingpointrepresentationshavesimilarrepresentationproblems:ifthenumberxcanbemultiplied/dividedbyenoughpowersoftwotoyieldafractionlyingbetween1/2and1thathasanitebinary-fractionrepresentation,thenumberisrepresentedexactlyinoatingpoint.Otherwise,wecanonlyrepresentthenumberapproximately,notcatastrophicallyinerroraswithintegers.Forexample,thenumber2.5equals0:62522,thefractionalpartofwhichhasanexactbinaryrepresentation.8However,thenumber2:6doesnothaveanexactbinaryrepresentation,andonlyberepresentedapproximatelyinoatingpoint.Insingleprecisionoatingpointnumbers,whichrequire32bitsonebytefortheexponentandtheremaining24bitsforthemantissa,thenumber2.6willberepresentedas2:600000079:::.Notethatthisapproximationhasamuchlongerdecimalexpansion.Thislevelofaccuracymaynotsuceinnumericalcalculations.Doubleprecisionoatingpointnumbersconsume8bytes,andquadrupleprecision16bytes.Themorebitsusedinthemantissa,thegreatertheaccuracy.Thisincreasingaccuracymeansthatmorenumberscanberepresentedexactly,buttherearealwayssomethatcannot.Suchinexactnumbershaveaninnitebinaryrepresentation.9Realizingthatrealnumberscanbeonlyrepresentedapproximatelyisquiteimportant,andunderliestheentireeldofnumericalanalysis,whichseekstopredictthenumericalaccuracyofanycomputation.Exercise5.2Solutiononp.204.Whatarethelargestandsmallestnumbersthatcanberepresentedin32-bitoatingpoint?in64-bitoatingpointthathassixteenbitsallocatedtotheexponent?Notethatbothexponentandmantissarequireasignbit.Solongastheintegersaren'ttoolarge,theycanberepresentedexactlyinacomputerusingthebinarypositionalnotation.Electroniccircuitsthatmakeupthephysicalcomputercanaddandsubtractintegerswithouterror.Thisstatementisn'tquitetrue;whendoesadditioncauseproblems? 7Insomecomputers,thisnormalizationistakentoanextreme:theleadingbinarydigitisnotexplicitlyexpressed,providinganextrabittorepresentthemantissaalittlemoreaccurately.Thisconventionisknownasthehidden-onesnotation.8Seeifyoucanndthisrepresentation.9Notethattherewillalwaysbenumbersthathaveaninniterepresentationinanychosenpositionalsystem.Thechoiceofbasedeneswhichdoandwhichdon't.Ifyouwerethinkingthatbase10numberswouldsolvethisinaccuracy,notethat1=3=0:333333::::hasaninniterepresentationindecimalandbinaryforthatmatter,buthasniterepresentationinbase3.

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1595.2.3ComputerArithmeticandLogicThebinaryadditionandmultiplicationtablesare0BBBBBBBBBBBBBBBBBBB@0+0=00+1=11+1=101+0=100=001=011=110=01CCCCCCCCCCCCCCCCCCCA.4Notethatifcarriesareignored,10subtractionoftwosingle-digitbinarynumbersyieldsthesamebitasaddition.Computersusehighandlowvoltagevaluestoexpressabit,andanarrayofsuchvoltagesexpressnumbersakintopositionalnotation.Logiccircuitsperformarithmeticoperations.Exercise5.3Solutiononp.204.Addtwenty-veandseveninbase2.Notethecarriesthatmightoccur.Whyistheresult"nice"?AlsonotethatthelogicaloperationsofANDandORareequivalenttobinaryadditionagainifcarriesareignored.Thevariablesoflogicindicatetruthorfalsehood.ATB,theANDofAandB,representsastatementthatbothAandBmustbetrueforthestatementtobetrue.YouusethiskindofstatementtotellsearchenginesthatyouwanttorestricthitstocaseswherebothoftheeventsAandBoccur.ASB,theORofAandB,yieldsavalueoftruthifeitheristrue.Notethatifwerepresenttruthbya"1"andfalsehoodbya"0,"binarymultiplicationcorrespondstoANDandadditionignoringcarriestoOR.TheIrishmathematicianGeorgeBoolediscoveredthisequivalenceinthemid-nineteenthcentury.ItlaidthefoundationforwhatwenowcallBooleanalgebra,whichexpressesasequationslogicalstatements.Moreimportantly,anycomputerusingbase-2representationsandarithmeticcanalsoeasilyevaluatelogicalstatements.Thisfactmakesaninteger-basedcomputationaldevicemuchmorepowerfulthanmightbeapparent.5.3TheSamplingTheorem115.3.1Analog-to-DigitalConversionBecauseofthewaycomputersareorganized,signalmustberepresentedbyanitenumberofbytes.Thisrestrictionmeansthatboththetimeaxisandtheamplitudeaxismustbequantized:Theymusteachbeamultipleoftheintegers.12Quitesurprisingly,theSamplingTheoremallowsustoquantizethetimeaxiswithouterrorforsomesignals.Thesignalsthatcanbesampledwithoutintroducingerrorareinteresting,andasdescribedinthenextsection,wecanmakeasignal"samplable"byltering.Incontrast,noonehasfoundawayofperformingtheamplitudequantizationstepwithoutintroducinganunrecoverableerror.Thus,asignal'svaluecannolongerbeanyrealnumber.Signalsprocessedbydigitalcomputersmustbediscrete-valued:theirvaluesmustbeproportionaltotheintegers.Consequently,analog-to-digitalconversionintroduceserror. 10Acarrymeansthatacomputationperformedatagivenpositionaectsotherpositionsaswell.Here,1+1=10isanexampleofacomputationthatinvolvesacarry.11Thiscontentisavailableonlineat.12Weassumethatwedonotuseoating-pointA/Dconverters.

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160CHAPTER5.DIGITALSIGNALPROCESSING5.3.2TheSamplingTheoremDigitaltransmissionofinformationanddigitalsignalprocessingallrequiresignalstorstbe"acquired"byacomputer.Oneofthemostamazingandusefulresultsinelectricalengineeringisthatsignalscanbeconvertedfromafunctionoftimeintoasequenceofnumberswithouterror:Wecanconvertthenumbersbackintothesignalwiththeoreticallynoerror.HaroldNyquist,aBellLaboratoriesengineer,rstderivedthisresult,knownastheSamplingTheorem,inthe1920s.Itfoundnorealapplicationbackthen.ClaudeShannon13,alsoatBellLaboratories,revivedtheresultoncecomputersweremadepublicafterWorldWarII.ThesampledversionoftheanalogsignalstissnTs,withTsknownasthesamplinginterval.Clearly,thevalueoftheoriginalsignalatthesamplingtimesispreserved;theissueishowthesignalvaluesbetweenthesamplescanbereconstructedsincetheyarelostinthesamplingprocess.Tocharacterizesampling,weapproximateitastheproductxt=stPTst,withPTstbeingtheperiodicpulsesignal.Theresultingsignal,asshowninFigure5.3SampledSignal,hasnonzerovaluesonlyduringthetimeintervals)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(nTs)]TJ/F7 6.9738 Tf 11.1581 3.9226 Td[( 2;nTs+ 2,n2f:::;)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1;0;1;:::g.SampledSignal Figure5.3:Thewaveformofanexamplesignalisshowninthetopplotanditssampledversioninthebottom.Forourpurposeshere,wecentertheperiodicpulsesignalabouttheoriginsothatitsFourierseriescoecientsarerealthesignaliseven.PTst=1Xk=ckej2kt Ts.5whereck=sink Ts k.6Ifthepropertiesofstandtheperiodicpulsesignalarechosenproperly,wecanrecoverstfromxtbyltering. 13http://www.lucent.com/minds/infotheory/

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161Tounderstandhowsignalvaluesbetweenthesamplescanbe"lled"in,weneedtocalculatethesampledsignal'sspectrum.UsingtheFourierseriesrepresentationoftheperiodicsamplingsignal,xt=1Xk=ckej2kt Tsst.7Consideringeachterminthesumseparately,weneedtoknowthespectrumoftheproductofthecomplexexponentialandthesignal.Evaluatingthistransformdirectlyisquiteeasy.Z1stej2kt Tse)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2ftdt=Z1ste)]TJ/F8 9.9626 Tf 6.2266 -0.7472 Td[(j2f)]TJ/F9 4.9813 Tf 9.4848 2.6773 Td[(k Tstdt=Sf)]TJ/F11 9.9626 Tf 13.4482 6.7398 Td[(k Ts.8Thus,thespectrumofthesampledsignalconsistsofweightedbythecoecientsckanddelayedversionsofthesignal'sspectrumFigure5.4aliasing.Xf=1Xk=ckSf)]TJ/F11 9.9626 Tf 13.4482 6.7398 Td[(k Ts.9Ingeneral,thetermsinthissumoverlapeachotherinthefrequencydomain,renderingrecoveryoftheoriginalsignalimpossible.Thisunpleasantphenomenonisknownasaliasing.aliasing Figure5.4:ThespectrumofsomebandlimitedtoWHzsignalisshowninthetopplot.IfthesamplingintervalTsischosentoolargerelativetothebandwidthW,aliasingwilloccur.Inthebottomplot,thesamplingintervalischosensucientlysmalltoavoidaliasing.Notethatifthesignalwerenotbandlimited,thecomponentspectrawouldalwaysoverlap.If,however,wesatisfytwoconditions:ThesignalstisbandlimitedhaspowerinarestrictedfrequencyrangetoWHz,andthesamplingintervalTsissmallenoughsothattheindividualcomponentsinthesumdonotoverlapTs<1=2W,

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162CHAPTER5.DIGITALSIGNALPROCESSINGaliasingwillnotoccur.Inthisdelightfulcase,wecanrecovertheoriginalsignalbylowpasslteringxtwithalterhavingacutofrequencyequaltoWHz.Thesetwoconditionsensuretheabilitytorecoverabandlimitedsignalfromitssampledversion:WethushavetheSamplingTheorem.Exercise5.4Solutiononp.204.TheSamplingTheoremasstateddoesnotmentionthepulsewidth.WhatistheeectofthisparameteronourabilitytorecoverasignalfromitssamplesassumingtheSamplingTheorem'stwoconditionsaremet?Thefrequency1 2Ts,knowntodayastheNyquistfrequencyandtheShannonsamplingfrequency,correspondstothehighestfrequencyatwhichasignalcancontainenergyandremaincompatiblewiththeSamplingTheorem.High-qualitysamplingsystemsensurethatnoaliasingoccursbyunceremoniouslylowpasslteringthesignalcutofrequencybeingslightlylowerthantheNyquistfrequencybeforesampling.Suchsystemsthereforevarytheanti-aliasinglter'scutofrequencyasthesamplingratevaries.Becausesuchqualityfeaturescostmoney,manysoundcardsdonothaveanti-aliasingltersor,forthatmatter,post-samplinglters.Theysampleathighfrequencies,44.1kHzforexample,andhopethesignalcontainsnofrequenciesabovetheNyquistfrequency.05kHzinourexample.If,however,thesignalcontainsfrequenciesbeyondthesoundcard'sNyquistfrequency,theresultingaliasingcanbeimpossibletoremove.Exercise5.5Solutiononp.204.Togainabetterappreciationofaliasing,sketchthespectrumofasampledsquarewave.Forsimplicityconsideronlythespectralrepetitionscenteredat)]TJ/F1 9.9626 Tf 9.4092 11.0587 Td[(1 Ts,0,1 Ts.LetthesamplingintervalTsbe1;considertwovaluesforthesquarewave'speriod:3.5and4.Noteinparticularwherethespectrallinesgoastheperioddecreases;somewillmovetotheleftandsometotheright.Whatpropertycharacterizestheonesgoingthesamedirection?IfwesatisfytheSamplingTheorem'sconditions,thesignalwillchangeonlyslightlyduringeachpulse.Aswenarrowthepulse,makingsmallerandsmaller,thenonzerovaluesofthesignalstpTstwillsimplybesnTs,thesignal'ssamples.IfindeedtheNyquistfrequencyequalsthesignal'shighestfrequency,atleasttwosampleswilloccurwithintheperiodofthesignal'shighestfrequencysinusoid.Intheseways,thesamplingsignalcapturesthesampledsignal'stemporalvariationsinawaythatleavesalltheoriginalsignal'sstructureintact.Exercise5.6Solutiononp.204.Whatisthesimplestbandlimitedsignal?Usingthissignal,convinceyourselfthatlessthantwosamples/periodwillnotsucetospecifyit.Ifthesamplingrate1 Tsisnothighenough,whatsignalwouldyourresultingundersampledsignalbecome?5.4AmplitudeQuantization14TheSamplingTheoremsaysthatifwesampleabandlimitedsignalstfastenough,itcanberecoveredwithouterrorfromitssamplessnTs,n2f:::;)]TJ/F8 9.9626 Tf 7.7488 0 Td[(1;0;1;:::g.Samplingisonlytherstphaseofacquiringdataintoacomputer:Computationalprocessingfurtherrequiresthatthesamplesbequantized:analogvaluesareconvertedintodigitalSection1.2.2:DigitalSignalsform.Inshort,wewillhaveperformedanalog-to-digitalA/Dconversion. 14Thiscontentisavailableonlineat.

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163 a bFigure5.5:Athree-bitA/Dconverterassignsvoltageintherange[)]TJ/F56 8.9664 Tf 7.1675 0 Td[(1;1]tooneofeightintegersbetween0and7.Forexample,allinputshavingvalueslyingbetween0.5and0.75areassignedtheintegervaluesixand,uponconversionbacktoananalogvalue,theyallbecome0.625.Thewidthofasinglequantizationintervalequals2 2B.Thebottompanelshowsasignalgoingthroughtheanalog-to-digital,whereBisthenumberofbitsusedintheA/Dconversionprocessinthecasedepictedhere.Firstitissampled,thenamplitude-quantizedtothreebits.Notehowthesampledsignalwaveformbecomesdistortedafteramplitudequantization.Forexamplethetwosignalvaluesbetween0.5and0.75become0.625.Thisdistortionisirreversible;itcanbereducedbutnoteliminatedbyusingmorebitsintheA/Dconverter. Aphenomenonreminiscentoftheerrorsincurredinrepresentingnumbersonacomputerpreventssignalamplitudesfrombeingconvertedwithnoerrorintoabinarynumberrepresentation.Inanalog-to-digitalconversion,thesignalisassumedtoliewithinapredenedrange.Assumingwecanscalethesignalwithoutaectingtheinformationitexpresses,we'lldenethisrangetobe[)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1;1].Furthermore,theA/Dconverterassignsamplitudevaluesinthisrangetoasetofintegers.AB-bitconverterproducesoneoftheintegers0;1;:::;2B)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1foreachsampledinput.Figure5.5showshowathree-bitA/Dconverterassignsinputvaluestotheintegers.Wedeneaquantizationintervaltobetherangeofvaluesassignedtothesameinteger.Thus,forourexamplethree-bitA/Dconverter,thequantizationintervalis0:25;ingeneral,itis2 2B.

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164CHAPTER5.DIGITALSIGNALPROCESSINGExercise5.7Solutiononp.204.RecallingtheplotofaveragedailyhighsinthisfrequencydomainproblemProblem4.5,whyisthisplotsojagged?Interpretthiseectintermsofanalog-to-digitalconversion.Becausevalueslyinganywherewithinaquantizationintervalareassignedthesamevalueforcomputerprocessing,theoriginalamplitudevaluecannotberecoveredwithouterror.Typically,theD/Aconverter,thedevicethatconvertsintegerstoamplitudes,assignsanamplitudeequaltothevaluelyinghalfwayinthequantizationinterval.Theinteger6wouldbeassignedtotheamplitude0.625inthisscheme.Theerrorintroducedbyconvertingasignalfromanalogtodigitalformbysamplingandamplitudequantizationthenbackagainwouldbehalfthequantizationintervalforeachamplitudevalue.Thus,theso-calledA/Derrorequalshalfthewidthofaquantizationinterval:1 2B.Aswehavexedtheinput-amplituderange,themorebitsavailableintheA/Dconverter,thesmallerthequantizationerror.Toanalyzetheamplitudequantizationerrormoredeeply,weneedtocomputethesignal-to-noiseratio,whichequalstheratioofthesignalpowerandthequantizationerrorpower.Assumingthesignalisasinusoid,thesignalpoweristhesquareofthermsamplitude:powers=1 p 22=1 2.TheillustrationFigure5.6detailsasinglequantizationinterval. Figure5.6:Asinglequantizationintervalisshown,alongwithatypicalsignal'svaluebeforeamplitudequantizationsnTsandafterQsnTs.denotestheerrorthusincurred.Itswidthisandthequantizationerrorisdenotedby.Tondthepowerinthequantizationerror,wenotethatnomatterintowhichquantizationintervalthesignal'svaluefalls,theerrorwillhavethesamecharacteristics.Tocalculatethermsvalue,wemustsquaretheerrorandaverageitovertheinterval.rms=r 1 R 2)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[( 22d=2 121 2.10SincethequantizationintervalwidthforaB-bitconverterequals2 2B=2)]TJ/F7 6.9738 Tf 6.2267 0 Td[(B)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1,wendthatthesignal-to-noiseratiofortheanalog-to-digitalconversionprocessequalsSNR=1 2 2)]TJ/F6 4.9813 Tf 5.3965 0 Td[(B)]TJ/F6 4.9813 Tf 5.3965 0 Td[(1 12=3 222B=6B+10log101:5dB.11Thus,everybitincreaseintheA/Dconverteryieldsa6dBincreaseinthesignal-to-noiseratio.Exercise5.8Solutiononp.205.Thisderivationassumedthesignal'samplitudelayintherange[)]TJ/F8 9.9626 Tf 7.7488 0 Td[(1;1].Whatwouldtheamplitudequantizationsignal-to-noiseratiobeifitlayintherange[)]TJ/F11 9.9626 Tf 7.7488 0 Td[(A;A]?Exercise5.9Solutiononp.205.HowmanybitswouldberequiredintheA/Dconvertertoensurethatthemaximumamplitudequantizationerrorwaslessthan60dbsmallerthanthesignal'speakvalue?Exercise5.10Solutiononp.205.MusiconaCDisstoredto16-bitaccuracy.Towhatsignal-to-noiseratiodoesthiscorrespond?

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165OncewehaveacquiredsignalswithanA/Dconverter,wecanprocessthemusingdigitalhardwareorsoftware.Itcanbeshownthatifthecomputerprocessingislinear,theresultofsampling,computerprocessing,andunsamplingisequivalenttosomeanaloglinearsystem.Whygotoallthebotherifthesamefunctioncanbeaccomplishedusinganalogtechniques?Knowingwhendigitalprocessingexcelsandwhenitdoesnotisanimportantissue.5.5Discrete-TimeSignalsandSystems15Mathematically,analogsignalsarefunctionshavingastheirindependentvariablescontinuousquantities,suchasspaceandtime.Discrete-timesignalsarefunctionsdenedontheintegers;theyaresequences.Aswithanalogsignals,weseekwaysofdecomposingdiscrete-timesignalsintosimplercomponents.Becausethisapproachleadingtoabetterunderstandingofsignalstructure,wecanexploitthatstructuretorepresentinformationcreatewaysofrepresentinginformationwithsignalsandtoextractinformationretrievetheinformationthusrepresented.Forsymbolic-valuedsignals,theapproachisdierent:Wedevelopacommonrepresentationofallsymbolic-valuedsignalssothatwecanembodytheinformationtheycontaininauniedway.Fromaninformationrepresentationperspective,themostimportantissuebecomes,forbothreal-valuedandsymbolic-valuedsignals,eciency:whatisthemostparsimoniousandcompactwaytorepresentinformationsothatitcanbeextractedlater.5.5.1Real-andComplex-valuedSignalsAdiscrete-timesignalisrepresentedsymbolicallyassn,wheren=f:::;)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1;0;1;:::g.Cosine Figure5.7:Thediscrete-timecosinesignalisplottedasastemplot.Canyoundtheformulaforthissignal?Weusuallydrawdiscrete-timesignalsasstemplotstoemphasizethefacttheyarefunctionsdenedonlyontheintegers.Wecandelayadiscrete-timesignalbyanintegerjustaswithanalogones.Asignaldelayedbymsampleshastheexpressionsn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(m.5.5.2ComplexExponentialsThemostimportantsignalis,ofcourse,thecomplexexponentialsequence.sn=ej2fn.12 15Thiscontentisavailableonlineat.

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166CHAPTER5.DIGITALSIGNALPROCESSINGNotethatthefrequencyvariablefisdimensionlessandthataddinganintegertothefrequencyofthediscrete-timecomplexexponentialhasnoeectonthesignal'svalue.ej2f+mn=ej2fnej2mn=ej2fn.13Thisderivationfollowsbecausethecomplexexponentialevaluatedatanintegermultipleof2equalsone.Thus,theperiodofadiscrete-timecomplexexponentialequalsone.5.5.3SinusoidsDiscrete-timesinusoidshavetheobviousformsn=Acos2fn+.Asopposedtoanalogcomplexexponentialsandsinusoidsthatcanhavetheirfrequenciesbeanyrealvalue,frequenciesoftheirdiscrete-timecounterpartsyielduniquewaveformsonlywhenfliesintheinterval)]TJ/F14 9.9626 Tf 4.5662 -8.0698 Td[()]TJ/F1 9.9626 Tf 9.4092 8.0698 Td[()]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(1 2;1 2.Fromthepropertiesofthecomplexexponential,thesinusoid'speriodisalwaysone;thischoiceoffrequencyintervalwillbecomeevidentlater.5.5.4UnitSampleThesecond-mostimportantdiscrete-timesignalistheunitsample,whichisdenedtoben=8<:1ifn=00otherwise.14Unitsample Figure5.8:Theunitsample.Examinationofadiscrete-timesignal'splot,likethatofthecosinesignalshowninFigure5.7Cosine,revealsthatallsignalsconsistofasequenceofdelayedandscaledunitsamples.Becausethevalueofasequenceateachintegermisdenotedbysmandtheunitsampledelayedtooccuratmiswrittenn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(m,wecandecomposeanysignalasasumofunitsamplesdelayedtotheappropriatelocationandscaledbythesignalvalue.sn=1Xm=smn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(m.15Thiskindofdecompositionisuniquetodiscrete-timesignals,andwillproveusefulsubsequently.5.5.5UnitStepTheunitsampleindiscrete-timeiswell-denedattheorigin,asopposedtothesituationwithanalogsignals.un=8<:1ifn00ifn<0.16

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1675.5.6SymbolicSignalsAninterestingaspectofdiscrete-timesignalsisthattheirvaluesdonotneedtoberealnumbers.Wedohavereal-valueddiscrete-timesignalslikethesinusoid,butwealsohavesignalsthatdenotethesequenceofcharacterstypedonthekeyboard.Suchcharacterscertainlyaren'trealnumbers,andasacollectionofpossiblesignalvalues,theyhavelittlemathematicalstructureotherthanthattheyaremembersofaset.Moreformally,eachelementofthesymbolic-valuedsignalsntakesononeofthevaluesfa1;:::;aKgwhichcomprisethealphabetA.Thistechnicalterminologydoesnotmeanwerestrictsymbolstobeingmem-bersoftheEnglishorGreekalphabet.Theycouldrepresentkeyboardcharacters,bytes8-bitquantities,integersthatconveydailytemperature.Whethercontrolledbysoftwareornot,discrete-timesystemsareultimatelyconstructedfromdigitalcircuits,whichconsistentirelyofanalogcircuitelements.Furthermore,thetransmissionandreceptionofdiscrete-timesignals,likee-mail,isaccomplishedwithanalogsignalsandsystems.Understandinghowdiscrete-timeandanalogsignalsandsystemsintertwineisperhapsthemaingoalofthiscourse.5.5.7Discrete-TimeSystemsDiscrete-timesystemscanactondiscrete-timesignalsinwayssimilartothosefoundinanalogsignalsandsystems.Becauseoftheroleofsoftwareindiscrete-timesystems,manymoredierentsystemscanbeenvisionedand"constructed"withprogramsthancanbewithanalogsignals.Infact,aspecialclassofanalogsignalscanbeconvertedintodiscrete-timesignals,processedwithsoftware,andconvertedbackintoananalogsignal,allwithouttheincursionoferror.Forsuchsignals,systemscanbeeasilyproducedinsoftware,withequivalentanalogrealizationsdicult,ifnotimpossible,todesign.5.6Discrete-TimeFourierTransformDTFT16TheFouriertransformofthediscrete-timesignalsnisdenedtobeS)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f=1Xn=sne)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2fn.17Frequencyherehasnounits.Asshouldbeexpected,thisdenitionislinear,withthetransformofasumofsignalsequalingthesumoftheirtransforms.Real-valuedsignalshaveconjugate-symmetricspectra:S)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2f=S)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(ej2f.Exercise5.11Solutiononp.205.Aspecialpropertyofthediscrete-timeFouriertransformisthatitisperiodicwithperiodone:S)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f+1=S)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f.DerivethispropertyfromthedenitionoftheDTFT.Becauseofthisperiodicity,weneedonlyplotthespectrumoveroneperiodtounderstandcompletelythespectrum'sstructure;typically,weplotthespectrumoverthefrequencyrange)]TJ/F1 9.9626 Tf 9.4091 8.0698 Td[()]TJ/F7 6.9738 Tf 5.7618 -4.1472 Td[(1 2;1 2.Whenthesignalisreal-valued,wecanfurthersimplifyourplottingchoresbyshowingthespectrumonlyover0;1 2;thespectrumatnegativefrequenciescanbederivedfrompositive-frequencyspectralvalues.Whenweobtainthediscrete-timesignalviasamplingananalogsignal,theNyquistfrequencyp.162correspondstothediscrete-timefrequency1 2.Toshowthis,notethatasinusoidhavingafrequencyequaltotheNyquistfrequency1 2Tshasasampledwaveformthatequalscos21 2TsnTs=cosn=)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1n 16Thiscontentisavailableonlineat.

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168CHAPTER5.DIGITALSIGNALPROCESSINGTheexponentialintheDTFTatfrequency1 2equalse)]TJ/F8 9.9626 Tf 6.2266 -0.7472 Td[(j2n 2=e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(jn=)]TJ/F8 9.9626 Tf 7.7488 0 Td[(1n,meaningthatdiscrete-timefrequencyequalsanalogfrequencymultipliedbythesamplingintervalfD=fATs.18fDandfArepresentdiscrete-timeandanalogfrequencyvariables,respectively.ThealiasinggureFig-ure5.4:aliasingprovidesanotherwayofderivingthisresult.AsthedurationofeachpulseintheperiodicsamplingsignalpTstnarrows,theamplitudesofthesignal'sspectralrepetitions,whicharegovernedbytheFourierseriescoecients.10ofpTst,becomeincreasinglyequal.ExaminationoftheperiodicpulsesignalFigure4.1revealsthatasdecreases,thevalueofc0,thelargestFouriercoecient,decreasestozero:jc0j=A Ts.Thus,tomaintainamathematicallyviableSamplingTheorem,theamplitudeAmustincreaseas1 ,becominginnitelylargeasthepulsedurationdecreases.Practicalsystemsuseasmallvalueof,say0:1Tsanduseamplierstorescalethesignal.Thus,thesampledsignal'sspectrumbecomesperiodicwithperiod1 Ts.Thus,theNyquistfrequency1 2Tscorrespondstothefrequency1 2.Example5.1Let'scomputethediscrete-timeFouriertransformoftheexponentiallydecayingsequencesn=anun,whereunistheunit-stepsequence.Simplypluggingthesignal'sexpressionintotheFouriertransformformula,S)]TJ/F11 9.9626 Tf 4.5663 -8.0699 Td[(ej2f=1Xn=anune)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2fn=1Xn=0ae)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2fn.19Thissumisaspecialcaseofthegeometricseries.1Xn=0n=1 1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(;jj<1.20Thus,aslongasjaj<1,wehaveourFouriertransform.S)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f=1 1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(ae)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2f.21UsingEuler'srelation,wecanexpressthemagnitudeandphaseofthisspectrum.jS)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2fj=1 q )]TJ/F11 9.9626 Tf 9.9626 0 Td[(acos2f2+a2sin2f.22)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(S)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f=)]TJ/F1 9.9626 Tf 9.4091 14.0475 Td[(tan)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1asinf 1)]TJ/F11 9.9626 Tf 9.9627 0 Td[(acos2f.23Nomatterwhatvalueofawechoose,theaboveformulaeclearlydemonstratetheperiodicnatureofthespectraofdiscrete-timesignals.Figure5.9Spectrumofexponentialsignalshowsindeedthatthespectrumisaperiodicfunction.Weneedonlyconsiderthespectrumbetween)]TJ/F1 9.9626 Tf 9.4091 8.0699 Td[()]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(1 2and1 2tounambiguouslydeneit.Whena>0,wehavealowpassspectrumthespectrumdiminishesasfrequencyincreasesfrom0to1 2withincreasingaleadingtoagreaterlowfrequencycontent;fora<0,wehaveahighpassspectrumFigure5.10Spectraofexponentialsignals.

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169 Spectrumofexponentialsignal Figure5.9:Thespectrumoftheexponentialsignala=0:5isshownoverthefrequencyrange[-2,2],clearlydemonstratingtheperiodicityofalldiscrete-timespectra.Theanglehasunitsofdegrees. Spectraofexponentialsignals Figure5.10:Thespectraofseveralexponentialsignalsareshown.Whatistheapparentrelationshipbetweenthespectrafora=0:5anda=)]TJ/F56 8.9664 Tf 7.1675 0 Td[(0:5?

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170CHAPTER5.DIGITALSIGNALPROCESSINGExample5.2Analogoustotheanalogpulsesignal,let'sndthespectrumofthelength-Npulsesequence.sn=8<:1if0nN)]TJ/F8 9.9626 Tf 9.9626 0 Td[(10otherwise.24TheFouriertransformofthissequencehastheformofatruncatedgeometricseries.S)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f=N)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1Xn=0e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2fn.25Fortheso-callednitegeometricseries,weknowthatN+n0)]TJ/F7 6.9738 Tf 6.2266 0 Td[(1Xn=n0n=n01)]TJ/F11 9.9626 Tf 9.9626 0 Td[(N 1)]TJ/F11 9.9626 Tf 9.9627 0 Td[(.26forallvaluesof.Exercise5.12Solutiononp.205.Derivethisformulaforthenitegeometricseriessum.The"trick"istoconsiderthedierencebetweentheseries'sumandthesumoftheseriesmultipliedby.ApplyingthisresultyieldsFigure5.11Spectrumoflength-tenpulse.S)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f=1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2fN 1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2f=e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(jfN)]TJ/F7 6.9738 Tf 6.2266 0 Td[(1sinfN sinf.27TheratioofsinefunctionshasthegenericformofsinNx sinx,whichisknownasthediscrete-timesincfunctiondsincx.Thus,ourtransformcanbeconciselyexpressedasS)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f=e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(jfN)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1dsincf.Thediscrete-timepulse'sspectrumcontainsmanyripples,thenumberofwhichincreasewithN,thepulse'sduration.Theinversediscrete-timeFouriertransformiseasilyderivedfromthefollowingrelationship:Z1 2)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(1 2e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2fmej2fndf=8<:1ifm=n0ifm6=n.28Therefore,wendthatZ1 2)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(1 2S)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2fej2fndf=Z1 2)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(1 2Xmsme)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2fmej2fndf=XmsmZ1 2)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(1 2e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2fm)]TJ/F10 6.9738 Tf 6.2267 0 Td[(ndf!=sn.29TheFouriertransformpairsindiscrete-timeareS)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f=1Xn=sne)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2fn.30

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171 Spectrumoflength-tenpulse Figure5.11:Thespectrumofalength-tenpulseisshown.Canyouexplaintherathercomplicatedappearanceofthephase? sn=Z1 2)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(1 2S)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(ej2fej2fndf.31Thepropertiesofthediscrete-timeFouriertransformmirrorthoseoftheanalogFouriertransform.TheDTFTpropertiestable17showssimilaritiesanddierences.OneimportantcommonpropertyisParseval'sTheorem.1Xn=jsnj2=Z1 2)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(1 2)]TJ/F14 9.9626 Tf 4.5663 -8.0698 Td[(jS)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(ej2fj2df.32Toshowthisimportantproperty,wesimplysubstitutetheFouriertransformexpressionintothefrequency-domainexpressionforpower.Z1 2)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(1 2)]TJ/F14 9.9626 Tf 4.5662 -8.0698 Td[(jS)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2fj2df=Z1 2)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(1 2Xnsne)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2fn!Xm)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(snej2fmdf=Xn;msnsnZ1 2)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(1 2ej2fm)]TJ/F10 6.9738 Tf 6.2267 0 Td[(ndf!.33Usingtheorthogonalityrelation.28,theintegralequalsm)]TJ/F11 9.9626 Tf 9.9626 0 Td[(n,wherenistheunitsampleFig-ure5.8:Unitsample.Thus,thedoublesumcollapsesintoasinglesumbecausenonzerovaluesoccuronlywhenn=m,givingParseval'sTheoremasaresult.WetermPn)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(s2ntheenergyinthediscrete-timesignalsninspiteofthefactthatdiscrete-timesignalsdon'tconsumeorproduceforthatmatterenergy.Thisterminologyisacarry-overfromtheanalogworld. 17"Discrete-TimeFourierTransformProperties"

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172CHAPTER5.DIGITALSIGNALPROCESSINGExercise5.13Solutiononp.205.Supposeweobtainedourdiscrete-timesignalfromvaluesoftheproductstpTst,wherethedurationofthecomponentpulsesinpTstis.Howisthediscrete-timesignalenergyrelatedtothetotalenergycontainedinst?AssumethesignalisbandlimitedandthatthesamplingratewaschosenappropriatetotheSamplingTheorem'sconditions.5.7DiscreteFourierTransformsDFT18Thediscrete-timeFouriertransformandthecontinuous-timetransformaswellcanbeevaluatedwhenwehaveananalyticexpressionforthesignal.Supposewejusthaveasignal,suchasthespeechsignalusedinthepreviouschapter,forwhichthereisnoformula.Howthenwouldyoucomputethespectrum?Forexample,howdidwecomputeaspectrogramsuchastheoneshowninthespeechsignalexampleFigure4.17:spectrogram?TheDiscreteFourierTransformDFTallowsthecomputationofspectrafromdiscrete-timedata.Whileindiscrete-timewecanexactlycalculatespectra,foranalogsignalsnosimilarexactspectrumcomputationexists.Foranalog-signalspectra,usemustbuildspecialdevices,whichturnoutinmostcasestoconsistofA/Dconvertersanddiscrete-timecomputations.Certainlydiscrete-timespectralanalysisismoreexiblethancontinuous-timespectralanalysis.TheformulafortheDTFT.17isasum,whichconceptuallycanbeeasilycomputedsavefortwoissues.Signalduration.Thesumextendsoverthesignal'sduration,whichmustbenitetocomputethesignal'sspectrum.Itisexceedinglydiculttostoreaninnite-lengthsignalinanycase,sowe'llassumethatthesignalextendsover[0;N)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1].Continuousfrequency.Subtlerthanthesignaldurationissueisthefactthatthefrequencyvariableiscontinuous:Itmayonlyneedtospanoneperiod,like)]TJ/F1 9.9626 Tf 9.4091 8.0698 Td[()]TJ/F7 6.9738 Tf 5.7618 -4.1472 Td[(1 2;1 2or[0;1],buttheDTFTformulaasitstandsrequiresevaluatingthespectraatallfrequencieswithinaperiod.Let'scomputethespectrumatafewfrequencies;themostobviousonesaretheequallyspacedonesf=k K,k2f0;:::;K)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1g.WethusdenethediscreteFouriertransformDFTtobeSk=N)]TJ/F7 6.9738 Tf 6.2266 0 Td[(1Xn=0sne)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(j2nk K;k2f0;:::;K)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1g.34Here,SkisshorthandforSej2k K.Wecancomputethespectrumatasmanyequallyspacedfrequenciesaswelike.Notethatyoucanthinkaboutthiscomputationallymotivatedchoiceassamplingthespectrum;moreaboutthisinterpretationlater.Theissuenowishowmanyfrequenciesareenoughtocapturehowthespectrumchangeswithfrequency.OnewayofansweringthisquestionisdetermininganinversediscreteFouriertransformformula:givenSk,k=f0;:::;K)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1ghowdowendsn,n=f0;:::;N)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1g?Presumably,theformulawillbeoftheformsn=PK)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1k=0Skej2nk K.SubstitutingtheDFTformulainthisprototypeinversetransformyieldssn=K)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1Xk=0N)]TJ/F7 6.9738 Tf 6.2266 0 Td[(1Xm=0sme)]TJ/F8 9.9626 Tf 6.2266 -0.7472 Td[(j2mk Kej2nk K!.35Notethattheorthogonalityrelationweusesooftenhasadierentcharacternow.K)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1Xk=0e)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(j2km Kej2kn K=8<:Kifm=fn;nK;n2K;:::g0otherwise.36 18Thiscontentisavailableonlineat.

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173WeobtainnonzerovaluewheneverthetwoindicesdierbymultiplesofK.WecanexpressthisresultasKPlm)]TJ/F11 9.9626 Tf 9.9626 0 Td[(n)]TJ/F11 9.9626 Tf 9.9626 0 Td[(lK.Thus,ourformulabecomessn=N)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1Xm=0smK1Xl=m)]TJ/F11 9.9626 Tf 9.9626 0 Td[(n)]TJ/F11 9.9626 Tf 9.9626 0 Td[(lK!.37Theintegersnandmbothrangeoverf0;:::;N)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1g.Tohaveaninversetransform,weneedthesumtobeasingleunitsampleform,ninthisrange.Ifitdidnot,thensnwouldequalasumofvalues,andwewouldnothaveavalidtransform:Oncegoingintothefrequencydomain,wecouldnotgetbackunambiguously!Clearly,theterml=0alwaysprovidesaunitsamplewe'lltakecareofthefactorofKsoon.Ifweevaluatethespectrumatfewerfrequenciesthanthesignal'sduration,thetermcorrespondingtom=n+Kwillalsoappearforsomevaluesofm,n=f0;:::;N)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1g.Thissituationmeansthatourprototypetransformequalssn+sn+Kforsomevaluesofn.TheonlywaytoeliminatethisproblemistorequireKN:Wemusthaveatleastasmanyfrequencysamplesasthesignal'sduration.Inthisway,wecanreturnfromthefrequencydomainweenteredviatheDFT.Exercise5.14Solutiononp.205.Whenwehavefewerfrequencysamplesthanthesignal'sduration,somediscrete-timesignalvaluesequalthesumoftheoriginalsignalvalues.Giventhesamplinginterpretationofthespectrum,characterizethiseectadierentway.Anotherwaytounderstandthisrequirementistousethetheoryoflinearequations.IfwewriteouttheexpressionfortheDFTasasetoflinearequations,s+s++sN)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1=S.38s+se)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2 K++sN)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1e)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2N)]TJ/F6 4.9813 Tf 5.3965 0 Td[(1 K=S...s+se)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2K)]TJ/F6 4.9813 Tf 5.3965 0 Td[(1 K++sN)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1e)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2N)]TJ/F6 4.9813 Tf 5.3965 0 Td[(1K)]TJ/F6 4.9813 Tf 5.3965 0 Td[(1 K=SK)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1wehaveKequationsinNunknownsifwewanttondthesignalfromitssampledspectrum.Thisrequire-mentisimpossibletofulllifK
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174CHAPTER5.DIGITALSIGNALPROCESSINGExample5.4UsethisdemonstrationtosynthesizeasignalfromaDFTsequence. Thisisanunsupportedmediatype.Toview,pleaseseehttp://cnx.org/content/m10249/latest/DFT_Component_Manipulation.llb 5.8DFT:ComputationalComplexity19Wenowhaveawayofcomputingthespectrumforanarbitrarysignal:TheDiscreteFourierTransformDFT5.34computesthespectrumatNequallyspacedfrequenciesfromalength-Nsequence.Anissuethatneverarisesinanalog"computation,"likethatperformedbyacircuit,ishowmuchworkittakestoperformthesignalprocessingoperationsuchasltering.Incomputation,thisconsiderationtranslatestothenumberofbasiccomputationalstepsrequiredtoperformtheneededprocessing.Thenumberofsteps,knownasthecomplexity,becomesequivalenttohowlongthecomputationtakeshowlongmustwewaitforananswer.Complexityisnotsomuchtiedtospeciccomputersorprogramminglanguagesbuttohowmanystepsarerequiredonanycomputer.Thus,aprocedure'sstatedcomplexitysaysthatthetimetakenwillbeproportionaltosomefunctionoftheamountofdatausedinthecomputationandtheamountdemanded.Forexample,considertheformulaforthediscreteFouriertransform.Foreachfrequencywechose,wemustmultiplyeachsignalvaluebyacomplexnumberandaddtogethertheresults.Forareal-valuedsignal,eachreal-times-complexmultiplicationrequirestworealmultiplications,meaningwehave2Nmultiplicationstoperform.Toaddtheresultstogether,wemustkeeptherealandimaginarypartsseparate.AddingNnumbersrequiresN)]TJ/F8 9.9626 Tf 10.2724 0 Td[(1additions.Consequently,eachfrequencyrequires2N+2N)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1=4N)]TJ/F8 9.9626 Tf 10.2724 0 Td[(2basiccomputationalsteps.AswehaveNfrequencies,thetotalnumberofcomputationsisNN)]TJ/F8 9.9626 Tf 9.9626 0 Td[(2.Incomplexitycalculations,weonlyworryaboutwhathappensasthedatalengthsincrease,andtakethedominanttermherethe4N2termasreectinghowmuchworkisinvolvedinmakingthecomputation.Asmultiplicativeconstantsdon'tmattersincewearemakinga"proportionalto"evaluation,wendtheDFTisanO)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(N2computationalprocedure.Thisnotationisread"orderN-squared".Thus,ifwedoublethelengthofthedata,wewouldexpectthatthecomputationtimetoapproximatelyquadruple.Exercise5.15Solutiononp.205.InmakingthecomplexityevaluationfortheDFT,weassumedthedatatobereal.Threequestionsemerge.Firstofall,thespectraofsuchsignalshaveconjugatesymmetry,meaningthatnegativefrequencycomponentsk=N 2+1;:::;N+1intheDFT.34canbecomputedfromthecorrespondingpositivefrequencycomponents.DoesthissymmetrychangetheDFT'scomplexity?Secondly,supposethedataarecomplex-valued;whatistheDFT'scomplexitynow?Finally,alessimportantbutinterestingquestionissupposewewantKfrequencyvaluesinsteadofN;nowwhatisthecomplexity?5.9FastFourierTransformFFT20OnewondersiftheDFTcanbecomputedfaster:Doesanothercomputationalprocedureanalgorithmexistthatcancomputethesamequantity,butmoreeciently.Wecouldseekmethodsthatreducethe 19Thiscontentisavailableonlineat.20Thiscontentisavailableonlineat.

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175constantofproportionality,butdonotchangetheDFT'scomplexityO)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(N2.Here,wehavesomethingmoredramaticinmind:Canthecomputationsberestructuredsothatasmallercomplexityresults?In1965,IBMresearcherJimCooleyandPrincetonfacultymemberJohnTukeydevelopedwhatisnowknownastheFastFourierTransformFFT.ItisanalgorithmforcomputingthatDFTthathasorderONlogNforcertainlengthinputs.Nowwhenthelengthofdatadoubles,thespectralcomputationaltimewillnotquadrupleaswiththeDFTalgorithm;instead,itapproximatelydoubles.LaterresearchshowedthatnoalgorithmforcomputingtheDFTcouldhaveasmallercomplexitythantheFFT.Surprisingly,historicalworkhasshownthatGauss21intheearlynineteenthcenturydevelopedthesamealgorithm,butdidnotpublishit!AftertheFFT'srediscovery,notonlywasthecomputationofasignal'sspectrumgreatlyspeeded,butalsotheaddedfeatureofalgorithmmeantthatcomputationshadexibilitynotavailabletoanalogimplementations.Exercise5.16Solutiononp.205.BeforedevelopingtheFFT,let'strytoappreciatethealgorithm'simpact.Supposeashort-lengthtransformtakes1ms.Wewanttocalculateatransformofasignalthatis10timeslonger.ComparehowmuchlongerastraightforwardimplementationoftheDFTwouldtakeincomparisontoanFFT,bothofwhichcomputeexactlythesamequantity.ToderivetheFFT,weassumethatthesignal'sdurationisapoweroftwo:N=2L.Considerwhathappenstotheeven-numberedandodd-numberedelementsofthesequenceintheDFTcalculation.Sk=s+se)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j22k N++sN)]TJ/F8 9.9626 Tf 9.9626 0 Td[(2e)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2N)]TJ/F6 4.9813 Tf 5.3965 0 Td[(2k N+se)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2k N+se)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2+1k N++sN)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1e)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2N)]TJ/F6 4.9813 Tf 5.3965 0 Td[()]TJ/F6 4.9813 Tf 5.3965 0 Td[(1k N=24s+se)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2k N 2++sN)]TJ/F8 9.9626 Tf 9.9626 0 Td[(2e)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2N 2)]TJ/F6 4.9813 Tf 5.3965 0 Td[(1k N 235+24s+se)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2 ++sN)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1e)]TJ/F10 6.9738 Tf 6.2267 0 Td[(j2N 2)]TJ/F6 4.9813 Tf 5.3965 0 Td[(1k N 235e)]TJ/F6 4.9813 Tf 5.3965 0 Td[(j2k N.41EachterminsquarebracketshastheformofaN 2-lengthDFT.TherstoneisaDFToftheeven-numberedelements,andthesecondoftheodd-numberedelements.TherstDFTiscombinedwiththesecondmultipliedbythecomplexexponentiale)]TJ/F8 9.9626 Tf 6.2267 -0.7472 Td[(j2k N.Thehalf-lengthtransformsareeachevaluatedatfrequencyindicesk=0,:::,N)]TJ/F8 9.9626 Tf 8.6584 0 Td[(1.Normally,thenumberoffrequencyindicesinaDFTcalculationrangebetweenzeroandthetransformlengthminusone.ThecomputationaladvantageoftheFFTcomesfromrecognizingtheperiodicnatureofthediscreteFouriertransform.TheFFTsimplyreusesthecomputationsmadeinthehalf-lengthtransformsandcombinesthemthroughadditionsandthemultiplicationbye)]TJ/F8 9.9626 Tf 6.2266 -0.7472 Td[(j2k N,whichisnotperiodicoverN 2,torewritethelength-NDFT.Figure5.12Length-8DFTdecompositionillustratesthisdecomposition.Asitstands,wenowcomputetwolength-N 2transformscomplexity2ON2 4,multiplyoneofthembythecomplexexponentialcomplexityON,andaddtheresultscomplexityON.Atthispoint,thetotalcomplexityisstilldominatedbythehalf-lengthDFTcalculations,buttheproportionalitycoecienthasbeenreduced.Nowforthefun.BecauseN=2L,eachofthehalf-lengthtransformscanbereducedtotwoquarter-lengthtransforms,eachofthesetotwoeighth-lengthones,etc.Thisdecompositioncontinuesuntilweareleftwithlength-2transforms.Thistransformisquitesimple,involvingonlyadditions.Thus,therststageoftheFFThasN 2length-2transformsseethebottompartofFigure5.12Length-8DFTdecomposition.Pairsofthesetransformsarecombinedbyaddingonetotheothermultipliedbyacomplexexponential.Eachpairrequires4additionsand4multiplications,givingatotalnumberofcomputationsequaling8N 4=N 2.Thisnumberofcomputationsdoesnotchangefromstagetostage.Becausethenumberofstages,thenumberoftimesthelengthcanbedividedbytwo,equalslog2N,thecomplexityoftheFFTisONlog2N. 21http://www-groups.dcs.st-and.ac.uk/history/Mathematicians/Gauss.html

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176CHAPTER5.DIGITALSIGNALPROCESSING Length-8DFTdecomposition a bFigure5.12:Theinitialdecompositionofalength-8DFTintothetermsusingeven-andodd-indexedinputsmarkstherstphaseofdevelopingtheFFTalgorithm.Whenthesehalf-lengthtransformsaresuccessivelydecomposed,weareleftwiththediagramshowninthebottompanelthatdepictsthelength-8FFTcomputation. Doinganexamplewillmakecomputationalsavingsmoreobvious.Let'slookatthedetailsofalength-8DFT.AsshownonFigure5.13Buttery,werstdecomposetheDFTintotwolength-4DFTs,withtheoutputsaddedandsubtractedtogetherinpairs.ConsideringFigure5.13Butteryasthefrequencyindexgoesfrom0through7,werecyclevaluesfromthelength-4DFTsintothenalcalculationbecauseoftheperiodicityoftheDFToutput.Examininghowpairsofoutputsarecollectedtogether,wecreatethebasiccomputationalelementknownasabutteryFigure5.13Buttery.

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177Buttery Figure5.13:ThebasiccomputationalelementofthefastFouriertransformisthebuttery.Ittakestwocomplexnumbers,representedbyaandb,andformsthequantitiesshown.Eachbutteryrequiresonecomplexmultiplicationandtwocomplexadditions.Byconsideringtogetherthecomputationsinvolvingcommonoutputfrequenciesfromthetwohalf-lengthDFTs,weseethatthetwocomplexmultipliesarerelatedtoeachother,andwecanreduceourcomputationalworkevenfurther.Byfurtherdecomposingthelength-4DFTsintotwolength-2DFTsandcombiningtheiroutputs,wearriveatthediagramsummarizingthelength-8fastFouriertransformFigure5.12Length-8DFTdecomposition.Althoughmostofthecomplexmultipliesarequitesimplemultiplyingbye)]TJ/F7 6.9738 Tf 6.2267 0 Td[(jmeansnegatingrealandimaginaryparts,let'scountthoseforpurposesofevaluatingthecomplexityasfullcomplexmultiplies.WehaveN 2=4complexmultipliesand2N=16additionsforeachstageandlog2N=3stages,makingthenumberofbasiccomputations3N 2log2Naspredicted.Exercise5.17Solutiononp.205.NotethattheorderingoftheinputsequenceinthetwopartsofFigure5.12Length-8DFTdecompositionaren'tquitethesame.Whynot?Howistheorderingdetermined?Other"fast"algorithmswerediscovered,allofwhichmakeuseofhowmanycommonfactorsthetransformlengthNhas.Innumbertheory,thenumberofprimefactorsagivenintegerhasmeasureshowcompositeitis.Thenumbers16and81arehighlycompositeequaling24and34respectively,thenumber18islessso2132,and17notatallit'sprime.InoverthirtyyearsofFouriertransformalgorithmdevelopment,theoriginalCooley-Tukeyalgorithmisfarandawaythemostfrequentlyused.Itissocomputationallyecientthatpower-of-twotransformlengthsarefrequentlyusedregardlessofwhattheactuallengthofthedata.Exercise5.18Solutiononp.205.Supposethelengthofthesignalwere500?HowwouldyoucomputethespectrumofthissignalusingtheCooley-Tukeyalgorithm?WhatwouldthelengthNofthetransformbe?5.10Spectrograms22Weknowhowtoacquireanalogsignalsfordigitalprocessingpre-lteringSection5.3,samplingSec-tion5.3,andA/DconversionSection5.4andtocomputespectraofdiscrete-timesignalsusingtheFFTalgorithmSection5.9,let'sputthesevariouscomponentstogethertolearnhowthespectrogramshowninFigure5.14speechspectrogram,whichisusedtoanalyzespeechSection4.10,iscalculated.Thespeechwassampledatarateof11.025kHzandpassedthrougha16-bitA/Dconverter.pointofinterest:MusiccompactdiscsCDsencodetheirsignalsatasamplingrateof44.1kHz.We'lllearntherationaleforthisnumberlater.The11.025kHzsamplingrateforthespeechis1/4oftheCDsamplingrate,andwasthelowestavailablesamplingratecommensuratewithspeechsignalbandwidthsavailableonmycomputer. 22Thiscontentisavailableonlineat.

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178CHAPTER5.DIGITALSIGNALPROCESSINGExercise5.19Solutiononp.205.LookingatFigure5.14speechspectrogramthesignallastedalittleover1.2seconds.Howlongwasthesampledsignalintermsofsamples?Whatwasthedatarateduringthesamplingprocessinbpsbitspersecond?Assumingthecomputerstorageisorganizedintermsofbytes-bitquantities,howmanybytesofcomputermemorydoesthespeechconsume? speechspectrogram Figure5.14 Theresultingdiscrete-timesignal,showninthebottomofFigure5.14speechspectrogram,clearlychangesitscharacterwithtime.Todisplaythesespectralchanges,thelongsignalwassectionedintoframes:comparativelyshort,contiguousgroupsofsamples.Conceptually,aFouriertransformofeachframeiscalculatedusingtheFFT.Eachframeisnotsolongthatsignicantsignalvariationsareretainedwithinaframe,butnotsoshortthatwelosethesignal'sspectralcharacter.Roughlyspeaking,thespeechsignal'sspectrumisevaluatedoversuccessivetimesegmentsandstackedsidebysidesothatthex-axiscorrespondstotimeandthey-axisfrequency,withcolorindicatingthespectralamplitude.

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179AnimportantdetailemergeswhenweexamineeachframedsignalFigure5.15SpectrogramHanningvs.Rectangular.SpectrogramHanningvs.Rectangular Figure5.15:Thetopwaveformisasegment1024sampleslongtakenfromthebeginningofthe"RiceUniversity"phrase.ComputingFigure5.14speechspectrograminvolvedcreatingframes,heredemarkedbytheverticallines,thatwere256sampleslongandndingthespectrumofeach.Ifarectangularwindowisappliedcorrespondingtoextractingaframefromthesignal,oscillationsappearinthespectrummiddleofbottomrow.ApplyingaHanningwindowgracefullytapersthesignaltowardframeedges,therebyyieldingamoreaccuratecomputationofthesignal'sspectrumatthatmomentoftime.Attheframe'sedges,thesignalmaychangeveryabruptly,afeaturenotpresentintheoriginalsignal.Atransformofsuchasegmentrevealsacuriousoscillationinthespectrum,anartifactdirectlyrelatedtothissharpamplitudechange.Abetterwaytoframesignalsforspectrogramsistoapplyawindow:Shapethesignalvalueswithinaframesothatthesignaldecaysgracefullyasitnearstheedges.Thisshapingisaccomplishedbymultiplyingtheframedsignalbythesequencewn.Insectioningthesignal,weessentiallyappliedarectangularwindow:wn=1,0nN)]TJ/F8 9.9626 Tf 10.1653 0 Td[(1.AmuchmoregracefulwindowistheHanningwindow;ithasthecosineshapewn=1 2)]TJ/F8 9.9626 Tf 4.5662 -8.0698 Td[(1)]TJ/F8 9.9626 Tf 9.9626 0 Td[(cos)]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(2n N.AsshowninFigure5.15SpectrogramHanningvs.Rectangular,thisshapinggreatlyreducesspuriousoscillationsineachframe'sspectrum.ConsideringthespectrumoftheHanningwindowedframe,wendthattheoscillationsresultingfromapplyingtherectangularwindowobscuredaformanttheonelocatedatalittlemorethanhalftheNyquistfrequency.Exercise5.20Solutiononp.206.Whatmightbethesourceoftheseoscillations?Togainsomeinsight,whatisthelength-2NdiscreteFouriertransformofalength-Npulse?Thepulseemulatestherectangularwindow,andcertainlyhasedges.Compareyouranswerwiththelength-2Ntransformofalength-NHanningwindow.

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180CHAPTER5.DIGITALSIGNALPROCESSING Hanningspeech Figure5.16:Incomparisonwiththeoriginalspeechsegmentshownintheupperplot,thenon-overlappedHanningwindowedversionshownbelowitisveryragged.Clearly,spectralinformationextractedfromthebottomplotcouldwellmissimportantfeaturespresentintheoriginal. Ifyouexaminethewindowedsignalsectionsinsequencetoexaminewindowing'saectonsignalampli-tude,weseethatwehavemanagedtoamplitude-modulatethesignalwiththeperiodicallyrepeatedwindowFigure5.16Hanningspeech.Toalleviatethisproblem,framesareoverlappedtypicallybyhalfaframeduration.ThissolutionrequiresmoreFouriertransformcalculationsthanneededbyrectangularwindowing,butthespectraaremuchbetterbehavedandspectralchangesaremuchbettercaptured.Thespeechsignal,suchasshowninthespeechspectrogramFigure5.14:speechspectrogram,issec-tionedintooverlapping,equal-lengthframes,withaHanningwindowappliedtoeachframe.Thespectraofeachoftheseiscalculated,anddisplayedinspectrogramswithfrequencyextendingvertically,windowtimelocationrunninghorizontally,andspectralmagnitudecolor-coded.Figure5.17Hanningwindowsillustratesthesecomputations.Exercise5.21Solutiononp.206.Whythespecicvaluesof256forNand512forK?Anotherissueishowwasthelength-512transformofeachlength-256windowedframecomputed?5.11Discrete-TimeSystems23Whenwedevelopedanalogsystems,interconnectingthecircuitelementsprovidedanaturalstartingplaceforconstructingusefuldevices.Indiscrete-timesignalprocessing,wearenotlimitedbyhardwareconsiderationsbutbywhatcanbeconstructedinsoftware.Exercise5.22Solutiononp.206.OneoftherstanalogsystemswedescribedwastheamplierSection2.6.2:Ampliers.Wefoundthatimplementinganamplierwasdicultinanalogsystems,requiringanop-ampatleast.Whatisthediscrete-timeimplementationofanamplier?Isthisespeciallyhardoreasy?Infact,wewilldiscoverthatfrequency-domainimplementationofsystems,whereinwemultiplytheinputsignal'sFouriertransformbyafrequencyresponse,isnotonlyaviablealternative,butalsoacomputationallyecientone.Webeginwithdiscussingtheunderlyingmathematicalstructureoflinear,shift-invariantsystems,anddevisehowsoftwarelterscanbeconstructed. 23Thiscontentisavailableonlineat.

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181 Hanningwindows Figure5.17:TheoriginalspeechsegmentandthesequenceofoverlappingHanningwindowsappliedtoitareshownintheupperportion.Frameswere256sampleslongandaHanningwindowwasappliedwithahalf-frameoverlap.Alength-512FFTofeachframewascomputed,withthemagnitudeoftherst257FFTvaluesdisplayedvertically,withspectralamplitudevaluescolor-coded. 5.12Discrete-TimeSystemsintheTime-Domain24Adiscrete-timesignalsnisdelayedbyn0sampleswhenwewritesn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(n0,withn0>0.Choosingn0tobenegativeadvancesthesignalalongtheintegers.AsopposedtoanalogdelaysSection2.6.3:Delay,discrete-timedelayscanonlybeintegervalued.Inthefrequencydomain,delayingasignalcorrespondstoalinearphaseshiftofthesignal'sdiscrete-timeFouriertransform:)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(sn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(n0$e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2fn0S)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f.Lineardiscrete-timesystemshavethesuperpositionproperty.Sa1x1n+a2x2n=a1Sx1n+a2Sx2n.42Adiscrete-timesystemiscalledshift-invariantanalogoustotime-invariantanalogsystemsp.28ifdelayingtheinputdelaysthecorrespondingoutput.IfSxn=yn,thenashift-invariantsystemhasthepropertySxn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(n0=yn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(n0.43Weusethetermshift-invarianttoemphasizethatdelayscanonlyhaveintegervaluesindiscrete-time,whileinanalogsignals,delayscanbearbitrarilyvalued.Wewanttoconcentrateonsystemsthatarebothlinearandshift-invariant.Itwillbethesethatallowusthefullpoweroffrequency-domainanalysisandimplementations.Becausewehavenophysicalconstraintsin"constructing"suchsystems,weneedonlyamathematicalspecication.Inanalogsystems,thedier-entialequationspeciestheinput-outputrelationshipinthetime-domain.Thecorrespondingdiscrete-time 24Thiscontentisavailableonlineat.

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182CHAPTER5.DIGITALSIGNALPROCESSINGspecicationisthedierenceequation.yn=a1yn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1++apyn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(p+b0xn+b1xn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1++bqxn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(q.44Here,theoutputsignalynisrelatedtoitspastvaluesyn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(l,l=f1;:::;pg,andtothecurrentandpastvaluesoftheinputsignalxn.Thesystem'scharacteristicsaredeterminedbythechoicesforthenumberofcoecientspandqandthecoecients'valuesfa1;:::;apgandfb0;b1;:::;bqg.aside:Thereisanasymmetryinthecoecients:whereisa0?Thiscoecientwouldmultiplytheyntermin.44.Wehaveessentiallydividedtheequationbyit,whichdoesnotchangetheinput-outputrelationship.Wehavethuscreatedtheconventionthata0isalwaysone.Asopposedtodierentialequations,whichonlyprovideanimplicitdescriptionofasystemwemustsomehowsolvethedierentialequation,dierenceequationsprovideanexplicitwayofcomputingtheoutputforanyinput.Wesimplyexpressthedierenceequationbyaprogramthatcalculateseachoutputfromthepreviousoutputvalues,andthecurrentandpreviousinputs.Dierenceequationsareusuallyexpressedinsoftwarewithforloops.AMATLABprogramthatwouldcomputetherst1000valuesoftheoutputhastheformforn=1:1000yn=suma.*yn-1:-1:n-p+sumb.*xn:-1:n-q;endAnimportantdetailemergeswhenweconsidermakingthisprogramwork;infact,aswrittenithasatleasttwobugs.Whatinputandoutputvaluesenterintothecomputationofy?Weneedvaluesfory,y)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1,...,valueswehavenotyetcomputed.Tocomputethem,wewouldneedmorepreviousvaluesoftheoutput,whichwehavenotyetcomputed.Tocomputethesevalues,wewouldneedevenearliervalues,adinnitum.Thewayoutofthispredicamentistospecifythesystem'sinitialconditions:wemustprovidethepoutputvaluesthatoccurredbeforetheinputstarted.Thesevaluescanbearbitrary,butthechoicedoesimpacthowthesystemrespondstoagiveninput.Onechoicegivesrisetoalinearsystem:Maketheinitialconditionszero.ThereasonliesinthedenitionofalinearsystemSection2.6.6:LinearSystems:Theonlywaythattheoutputtoasumofsignalscanbethesumoftheindividualoutputsoccurswhentheinitialconditionsineachcasearezero.Exercise5.23Solutiononp.206.Theinitialconditionissueresolvesmakingsenseofthedierenceequationforinputsthatstartatsomeindex.However,theprogramwillnotworkbecauseofaprogramming,notconceptual,error.Whatisit?Howcanitbe"xed?"Example5.5Let'sconsiderthesimplesystemhavingp=1andq=0.yn=ayn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1+bxn.45Tocomputetheoutputatsomeindex,thisdierenceequationsaysweneedtoknowwhatthepreviousoutputyn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1andwhattheinputsignalisatthatmomentoftime.Inmoredetail,let'scomputethissystem'soutputtoaunit-sampleinput:xn=n.Becausetheinputiszerofornegativeindices,westartbytryingtocomputetheoutputatn=0.y=ay)]TJ/F8 9.9626 Tf 7.7488 0 Td[(1+b.46Whatisthevalueofy)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1?Becausewehaveusedaninputthatiszeroforallnegativeindices,itisreasonabletoassumethattheoutputisalsozero.Certainly,thedierenceequationwouldnotdescribealinearsystemSection2.6.6:LinearSystemsiftheinputthatiszeroforalltimedidnot

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183produceazerooutput.Withthisassumption,y)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1=0,leavingy=b.Forn>0,theinputunit-sampleiszero,whichleavesuswiththedierenceequationyn=ayn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1;n>0.Wecanenvisionhowthelterrespondstothisinputbymakingatable.yn=ayn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1+bn.47 n xn yn )]TJ/F8 9.9626 Tf 7.7487 0 Td[(1 0 0 0 1 b 1 0 ba 2 0 ba2 : 0 : n 0 ban Figure5.18Coecientvaluesdeterminehowtheoutputbehaves.Theparameterbcanbeanyvalue,andservesasagain.TheeectoftheparameteraismorecomplicatedFigure5.18.Ifitequalszero,theoutputsimplyequalstheinputtimesthegainb.Forallnon-zerovaluesofa,theoutputlastsforever;suchsystemsaresaidtobeIIRInniteImpulseResponse.Thereasonforthisterminologyisthattheunitsamplealsoknownastheimpulseespeciallyinanalogsituations,andthesystem'sresponsetothe"impulse"lastsforever.Ifaispositiveandlessthanone,theoutputisadecayingexponential.Whena=1,theoutputisaunitstep.Ifaisnegativeandgreaterthan)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1,theoutputoscillateswhiledecayingexponentially.Whena=)]TJ/F8 9.9626 Tf 7.7488 0 Td[(1,theoutputchangessignforever,alternatingbetweenband)]TJ/F11 9.9626 Tf 7.7487 0 Td[(b.Moredramaticeectswhenjaj>1;whetherpositiveornegative,theoutputsignalbecomeslargerandlarger,growingexponentially. Figure5.19:Theinputtothesimpleexamplesystem,aunitsample,isshownatthetop,withtheoutputsforseveralsystemparametervaluesshownbelow.

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184CHAPTER5.DIGITALSIGNALPROCESSINGPositivevaluesofaareusedinpopulationmodelstodescribehowpopulationsizeincreasesovertime.Here,nmightcorrespondtogeneration.Thedierenceequationsaysthatthenumberinthenextgenerationissomemultipleofthepreviousone.Ifthismultipleislessthanone,thepopulationbecomesextinct;ifgreaterthanone,thepopulationourishes.Thesamedierenceequationalsodescribestheeectofcompoundinterestondeposits.Here,nindexesthetimesatwhichcompoundingoccursdaily,monthly,etc.,aequalsthecompoundinterestrateplusone,andb=1thebankprovidesnogain.Insignalprocessingapplications,wetypicallyrequirethattheoutputremainboundedforanyinput.Forourexample,thatmeansthatwerestrictjaj=1andchosevaluesforitandthegainaccordingtotheapplication.Exercise5.24Solutiononp.206.Notethatthedierenceequation5.44,yn=a1yn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1++apyn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(p+b0xn+b1xn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1++bqxn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(qdoesnotinvolvetermslikeyn+1orxn+1ontheequation'srightside.Cansuchtermsalsobeincluded?Whyorwhynot? Figure5.20:Theplotshowstheunit-sampleresponseofalength-5boxcarlter. Example5.6Asomewhatdierentsystemhasno"a"coecients.Considerthedierenceequationyn=1 qxn++xn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(q+1.48Becausethissystem'soutputdependsonlyoncurrentandpreviousinputvalues,weneednotbeconcernedwithinitialconditions.Whentheinputisaunit-sample,theoutputequals1 qforn=f0;:::;q)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1g,thenequalszerothereafter.SuchsystemsaresaidtobeFIRFiniteImpulseResponsebecausetheirunitsampleresponseshaveniteduration.PlottingthisresponseFig-ure5.20showsthattheunit-sampleresponseisapulseofwidthqandheight1 q.Thiswaveformisalsoknownasaboxcar,hencethenameboxcarltergiventothissystem.We'llderiveitsfrequencyresponseanddevelopitslteringinterpretationinthenextsection.Fornow,notethatthedierenceequationsaysthateachoutputvalueequalstheaverageoftheinput'scurrentandpreviousvalues.Thus,theoutputequalstherunningaverageofinput'spreviousqvalues.Suchasystemcouldbeusedtoproducetheaverageweeklytemperatureq=7thatcouldbeupdateddaily. Thisisanunsupportedmediatype.Toview,pleaseseehttp://cnx.org/content/m10251/latest/DiscreteTimeSys.llb

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1855.13Discrete-TimeSystemsintheFrequencyDomain25Aswithanaloglinearsystems,weneedtondthefrequencyresponseofdiscrete-timesystems.Weusedimpedancestoderivedirectlyfromthecircuit'sstructurethefrequencyresponse.Theonlystructurewehavesofarforadiscrete-timesystemisthedierenceequation.Weproceedaswhenweusedimpedances:lettheinputbeacomplexexponentialsignal.Whenwehavealinear,shift-invariantsystem,theoutputshouldalsobeacomplexexponentialofthesamefrequency,changedinamplitudeandphase.Theseamplitudeandphasechangescomprisethefrequencyresponseweseek.Thecomplexexponentialinputsignalisxn=Xej2fn.Notethatthisinputoccursforallvaluesofn.Noneedtoworryaboutinitialconditionshere.Assumetheoutputhasasimilarform:yn=Yej2fn.Pluggingthesesignalsintothefundamentaldierenceequation.44,wehaveYej2fn=a1Yej2fn)]TJ/F7 6.9738 Tf 6.2266 0 Td[(1++apYej2fn)]TJ/F10 6.9738 Tf 6.2266 0 Td[(p+b0Xej2fn+b1Xej2fn)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1++bqXej2fn)]TJ/F10 6.9738 Tf 6.2266 0 Td[(q.49TheassumedoutputdoesindeedsatisfythedierenceequationiftheoutputcomplexamplitudeisrelatedtotheinputamplitudebyY=b0+b1e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2f++bqe)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2qf 1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(a1e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2f)-222()]TJ/F11 9.9626 Tf 33.7621 0 Td[(ape)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2pfXThisrelationshipcorrespondstothesystem'sfrequencyresponseor,byanothername,itstransferfunction.Wendthatanydiscrete-timesystemdenedbyadierenceequationhasatransferfunctiongivenbyH)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f=b0+b1e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2f++bqe)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2qf 1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(a1e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2f)-222()]TJ/F11 9.9626 Tf 33.7621 0 Td[(ape)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2pf.50Furthermore,becauseanydiscrete-timesignalcanbeexpressedasasuperpositionofcomplexexponentialsignalsandbecauselineardiscrete-timesystemsobeytheSuperpositionPrinciple,thetransferfunctionrelatesthediscrete-timeFouriertransformofthesystem'soutputtotheinput'sFouriertransform.Y)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f=X)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2fH)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f.51Example5.7ThefrequencyresponseofthesimpleIIRsystemdierenceequationgiveninapreviousexampleExample5.5isgivenbyH)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f=b 1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(ae)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2f.52ThisFouriertransformoccurredinapreviousexample;theexponentialsignalspectrumFig-ure5.10:Spectraofexponentialsignalsportraysthemagnitudeandphaseofthistransferfunction.Whentheltercoecientaispositive,wehavealowpasslter;negativearesultsinahighpasslter.Thelargerthecoecientinmagnitude,themorepronouncedthelowpassorhighpasslter-ing.Example5.8Thelength-qboxcarlterdierenceequationfoundinapreviousexampleExample5.6hasthefrequencyresponseH)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f=1 qq)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1Xm=0e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2fm.53ThisexpressionamountstotheFouriertransformoftheboxcarsignalFigure5.20.Therewefoundthatthisfrequencyresponsehasamagnitudeequaltotheabsolutevalueofdsincf;seethelength-10lter'sfrequencyresponseFigure5.11:Spectrumoflength-tenpulse.Weseethatboxcarlterslength-qsignalaveragershavealowpassbehavior,havingacutofrequencyof1 q. 25Thiscontentisavailableonlineat.

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186CHAPTER5.DIGITALSIGNALPROCESSINGExercise5.25Solutiononp.206.Supposewemultiplytheboxcarlter'scoecientsbyasinusoid:bm=1 qcos2f0mUseFouriertransformpropertiestodeterminethetransferfunction.Howwouldyoucharacterizethissystem:Doesitactlikealter?Ifso,whatkindoflterandhowdoyoucontrolitscharacteristicswiththelter'scoecients?Theseexamplesillustratethepointthatsystemsdescribedandimplementedbydierenceequationsserveasltersfordiscrete-timesignals.Thelter'sorderisgivenbythenumberpofdenominatorcoecientsinthetransferfunctionifthesystemisIIRorbythenumberqofnumeratorcoecientsifthelterisFIR.Whenasystem'stransferfunctionhasbothterms,thesystemisusuallyIIR,anditsorderequalspregardlessofq.Byselectingthecoecientsandltertype,ltershavingvirtuallyanyfrequencyresponsedesiredcanbedesigned.Thisdesignexibilitycan'tbefoundinanalogsystems.Inthenextsection,wedetailhowanalogsignalscanbelteredbycomputers,oeringamuchgreaterrangeoflteringpossibilitiesthanispossiblewithcircuits.5.14FilteringintheFrequencyDomain26Becauseweareinterestedinactualcomputationsratherthananalyticcalculations,wemustconsiderthedetailsofthediscreteFouriertransform.Tocomputethelength-NDFT,weassumethatthesignalhasadurationlessthanorequaltoN.Becausefrequencyresponseshaveanexplicitfrequency-domainspecication.49intermsofltercoecients,wedon'thaveadirecthandleonwhichsignalhasaFouriertransformequalingagivenfrequencyresponse.Findingthissignalisquiteeasy.Firstofall,notethatthediscrete-timeFouriertransformofaunitsampleequalsoneforallfrequencies.Becauseoftheinputandoutputoflinear,shift-invariantsystemsarerelatedtoeachotherbyY)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f=H)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2fX)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f,aunit-sampleinput,whichhasX)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(ej2f=1,resultsintheoutput'sFouriertransformequalingthesystem'stransferfunction.Exercise5.26Solutiononp.206.Thisstatementisaveryimportantresult.Deriveityourself.Inthetime-domain,theoutputforaunit-sampleinputisknownasthesystem'sunit-sampleresponse,andisdenotedbyhn.Combiningthefrequency-domainandtime-domaininterpretationsofalinear,shift-invariantsystem'sunit-sampleresponse,wehavethathnandthetransferfunctionareFouriertransformpairsintermsofthediscrete-timeFouriertransform.)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(hn$H)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f.54ReturningtotheissueofhowtousetheDFTtoperformltering,wecananalyticallyspecifythefrequencyresponse,andderivethecorrespondinglength-NDFTbysamplingthefrequencyresponse.Hk=Hej2k N;k=f0;:::;N)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1g.55ComputingtheinverseDFTyieldsalength-Nsignalnomatterwhattheactualdurationoftheunit-sampleresponsemightbe.Iftheunit-sampleresponsehasadurationlessthanorequaltoNit'saFIRlter,computingtheinverseDFTofthesampledfrequencyresponseindeedyieldstheunit-sampleresponse.If,however,thedurationexceedsN,errorsareencountered.ThenatureoftheseerrorsiseasilyexplainedbyappealingtotheSamplingTheorem.Bysamplinginthefrequencydomain,wehavethepotentialforaliasinginthetimedomainsamplinginonedomain,beittimeorfrequency,canresultinaliasingintheotherunlesswesamplefastenough.Here,thedurationoftheunit-sampleresponsedeterminestheminimalsamplingratethatpreventsaliasing.ForFIRsystemstheybydenitionhavenite-durationunitsampleresponsesthenumberofrequiredDFTsamplesequalstheunit-sampleresponse'sduration:Nq. 26Thiscontentisavailableonlineat.

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187Exercise5.27Solutiononp.206.DerivetheminimalDFTlengthforalength-qunit-sampleresponseusingtheSamplingTheorem.Becausesamplinginthefrequencydomaincausesrepetitionsoftheunit-sampleresponseinthetimedomain,sketchthetime-domainresultforvariouschoicesoftheDFTlengthN.Exercise5.28Solutiononp.206.Expresstheunit-sampleresponseofaFIRlterintermsofdierenceequationcoecients.NotethatthecorrespondingquestionforIIRltersisfarmorediculttoanswer:ConsidertheexampleExample5.5.ForIIRsystems,wecannotusetheDFTtondthesystem'sunit-sampleresponse:aliasingoftheunit-sampleresponsewillalwaysoccur.Consequently,wecanonlyimplementanIIRlteraccuratelyinthetimedomainwiththesystem'sdierenceequation.Frequency-domainimplementationsarerestrictedtoFIRlters.Anotherissuearisesinfrequency-domainlteringthatisrelatedtotime-domainaliasing,thistimewhenweconsidertheoutput.AssumewehaveaninputsignalhavingdurationNxthatwepassthroughaFIRlterhavingalength-q+1unit-sampleresponse.Whatisthedurationoftheoutputsignal?Thedierenceequationforthislterisyn=b0xn++bqxn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(q.56Thisequationsaysthattheoutputdependsoncurrentandpastinputvalues,withtheinputvalueqsamplespreviousdeningtheextentofthelter'smemoryofpastinputvalues.Forexample,theoutputatindexNxdependsonxNxwhichequalszero,xNx)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1,throughxNx)]TJ/F11 9.9626 Tf 9.9626 0 Td[(q.Thus,theoutputreturnstozeroonlyafterthelastinputvaluepassesthroughthelter'smemory.Astheinputsignal'slastvalueoccursatindexNx)]TJ/F8 9.9626 Tf 9.1459 0 Td[(1,thelastnonzerooutputvalueoccurswhenn)]TJ/F11 9.9626 Tf 9.1459 0 Td[(q=Nx)]TJ/F8 9.9626 Tf 9.1459 0 Td[(1orn=q+Nx)]TJ/F8 9.9626 Tf 9.1459 0 Td[(1.Thus,theoutputsignal'sdurationequalsq+Nx.Exercise5.29Solutiononp.206.Inwords,weexpressthisresultas"Theoutput'sdurationequalstheinput'sdurationplusthelter'sdurationminusone.".Demonstratetheaccuracyofthisstatement.Themainthemeofthisresultisthatalter'soutputextendslongerthaneitheritsinputoritsunit-sampleresponse.Thus,toavoidaliasingwhenweuseDFTs,thedominantfactorisnotthedurationofinputoroftheunit-sampleresponse,butoftheoutput.Thus,thenumberofvaluesatwhichwemustevaluatethefrequencyresponse'sDFTmustbeatleastq+NxandwemustcomputethesamelengthDFToftheinput.ToaccommodateashortersignalthanDFTlength,wesimplyzero-padtheinput:Ensurethatforindicesextendingbeyondthesignal'sdurationthatthesignaliszero.Frequency-domainltering,diagrammedinFigure5.21,isaccomplishedbystoringthelter'sfrequencyresponseastheDFTHk,computingtheinput'sDFTXk,multiplyingthemtocreatetheoutput'sDFTYk=HkXk,andcomputingtheinverseDFToftheresulttoyieldyn. Figure5.21:Tolterasignalinthefrequencydomain,rstcomputetheDFToftheinput,multiplytheresultbythesampledfrequencyresponse,andnallycomputetheinverseDFToftheproduct.TheDFT'slengthmustbeatleastthesumoftheinput'sandunit-sampleresponse'sdurationminusone.WecalculatethesediscreteFouriertransformsusingthefastFouriertransformalgorithm,ofcourse.

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188CHAPTER5.DIGITALSIGNALPROCESSINGBeforedetailingthisprocedure,let'sclarifywhysomanynewissuesaroseintryingtodevelopafrequency-domainimplementationoflinearltering.Thefrequency-domainrelationshipbetweenalter'sinputandoutputisalwaystrue:Y)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(ej2f=H)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2fX)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f.ThisFouriertransformsinthisresultarediscrete-timeFouriertransforms;forexample,X)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(ej2f=Pn)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(xne)]TJ/F7 6.9738 Tf 6.2266 0 Td[(j2fn.Unfortunately,usingthisrelation-shiptoperformlteringisrestrictedtothesituationwhenwehaveanalyticformulasforthefrequencyresponseandtheinputsignal.Thereasonwhywehadto"invent"thediscreteFouriertransformDFThasthesameorigin:Thespectrumresultingfromthediscrete-timeFouriertransformdependsonthecontinuousfrequencyvariablef.That'sneforanalyticcalculation,butcomputationallywewouldhavetomakeanuncountablyinnitenumberofcomputations.note:Didyouknowthattwokindsofinnitiescanbemeaningfullydened?Acountablyinnitequantitymeansthatitcanbeassociatedwithalimitingprocessassociatedwithintegers.Anuncountablyinnitequantitycannotbesoassociated.Thenumberofrationalnumbersiscountablyinnitethenumeratoranddenominatorcorrespondtolocatingtherationalbyrowandcolumn;thetotalnumberso-locatedcanbecounted,voila!;thenumberofirrationalnumbersisuncountablyinnite.Guesswhichis"bigger?"TheDFTcomputestheFouriertransformatanitesetoffrequenciessamplesthetruespectrumwhichcanleadtoaliasinginthetime-domainunlesswesamplesucientlyfast.Thesamplingintervalhereis1 Kforalength-KDFT:fastersamplingtoavoidaliasingthusrequiresalongertransformcalculation.Sincethelongestsignalamongtheinput,unit-sampleresponseandoutputistheoutput,itisthatsignal'sdurationthatdeterminesthetransformlength.Wesimplyextendtheothertwosignalswithzeroszero-padtocomputetheirDFTs.Example5.9Supposewewanttoaveragedailystockpricestakenoverlastyeartoyieldarunningweeklyaverageaverageovervetradingsessions.Thelterwewantisalength-5averagerasshownintheunit-sampleresponseFigure5.20,andtheinput'sdurationis253calendardaysminusweekenddaysandholidays.Theoutputdurationwillbe253+5)]TJ/F8 9.9626 Tf 8.7099 0 Td[(1=257,andthisdeterminesthetransformlengthweneedtouse.BecausewewanttousetheFFT,wearerestrictedtopower-of-twotransformlengths.WeneedtochooseanyFFTlengththatexceedstherequiredDFTlength.Asitturnsout,256isapoweroftwo28=256,andthislengthjustundershootsourrequiredlength.Tousefrequencydomaintechniques,wemustuselength-512fastFouriertransforms.Figure5.22showstheinputandthelteredoutput.TheMATLABprogramsthatcomputethelteredoutputinthetimeandfrequencydomainsareTimeDomainh=[11111]/5;y=filterh,1,[djiazeros,4];FrequencyDomainh=[11111]/5;DJIA=fftdjia,512;H=ffth,512;Y=H.*X;y=ifftY;note:Thefilterprogramhasthefeaturethatthelengthofitsoutputequalsthelengthofitsinput.Toforceittoproduceasignalhavingtheproperlength,theprogramzero-padstheinputappropriately.

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189 Figure5.22:ThebluelineshowstheDowJonesIndustrialAveragefrom1997,andtheredonethelength-5boxcar-lteredresultthatprovidesarunningweeklyofthismarketindex.Notethe"edge"eectsinthelteredoutput.MATLAB'sfftfunctionautomaticallyzero-padsitsinputifthespeciedtransformlengthitssecondargumentexceedsthesignal'slength.Thefrequencydomainresultwillhaveasmallimaginarycomponentlargestvalueis2:210)]TJ/F7 6.9738 Tf 6.2267 0 Td[(11becauseoftheinherentniteprecisionnatureofcomputerarithmetic.BecauseoftheunfortunatemistbetweensignallengthsandfavoredFFTlengths,thenumberofarithmeticoperationsinthetime-domainimplementationisfarlessthanthoserequiredbythefrequencydomainversion:514versus62,271.Iftheinputsignalhadbeenonesampleshorter,thefrequency-domaincomputationswouldhavebeenmorethanafactoroftwoless28,696,butfarmorethaninthetime-domainimplementation.Aninterestingsignalprocessingaspectofthisexampleisdemonstratedatthebeginningandendoftheoutput.Therampingupanddownthatoccurscanbetracedtoassumingtheinputiszerobeforeitbeginsandafteritends.Thelter"sees"theseinitialandnalvaluesasthedierenceequationpassesovertheinput.Theseartifactscanbehandledintwoways:wecanjustignoretheedgeeectsorthedatafrompreviousandsucceedingyears'lastandrstweek,respectively,canbeplacedattheends.5.15EciencyofFrequency-DomainFiltering27Todetermineforwhatsignalandlterdurationsatime-orfrequency-domainimplementationwouldbethemostecient,weneedonlycountthecomputationsrequiredbyeach.Forthetime-domain,dierence-equationapproach,weneedNxq+1.Thefrequency-domainapproachrequiresthreeFouriertransforms,eachrequiring)]TJ/F10 6.9738 Tf 5.7617 -4.1472 Td[(K 2log2Kcomputationsforalength-KFFT,andthemultiplicationoftwospectra6Kcomputations.Theoutput-signal-duration-determinedlengthmustbeatleastNx+q.Thus,wemustcompareNxq+1$6Nx+q+3 2Nx+qlog2Nx+q 27Thiscontentisavailableonlineat.

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190CHAPTER5.DIGITALSIGNALPROCESSINGExactanalyticevaluationofthiscomparisonisquitedicultwehaveatranscendentalequationtosolve.InsightintothiscomparisonisbestobtainedbydividingbyNx.2q+1$61+q Nx+3 21+q Nxlog2Nx+qWiththismanipulation,weareevaluatingthenumberofcomputationspersample.Foranygivenvalueofthelter'sorderq,therightside,thenumberoffrequency-domaincomputations,willexceedtheleftifthesignal'sdurationislongenough.However,forlterdurationsgreaterthanabout10,aslongastheinputisatleast10samples,thefrequency-domainapproachisfastersolongastheFFT'spower-of-twoconstraintisadvantageous.Thefrequency-domainapproachisnotyetviable;whatwillwedowhentheinputsignalisinnitelylong?ThedierenceequationscenariotsperfectlywiththeenvisioneddigitallteringstructureFigure5.25,butsofarwehaverequiredtheinputtohavelimiteddurationsothatwecouldcalculateitsFouriertransform.Thesolutiontothisproblemisquitesimple:Sectiontheinputintoframes,ltereach,andaddtheresultstogether.Tosectionasignalmeansexpressingitasalinearcombinationoflength-Nxnon-overlapping"chunks."Becausethelterislinear,lteringasumoftermsisequivalenttosummingtheresultsoflteringeachterm.xn=1Xm=xn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(mNxyn=1Xm=yn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(mNx.57AsillustratedinFigure5.23,notethateachlteredsectionhasadurationlongerthantheinput.Conse-quently,wemustliterallyaddthelteredsectionstogether,notjustbuttthemtogether.Computationalconsiderationsrevealasubstantialadvantageforafrequency-domainimplementationoveratime-domainone.Thenumberofcomputationsforatime-domainimplementationessentiallyremainsconstantwhetherwesectiontheinputornot.Thus,thenumberofcomputationsforeachoutputis2q+1.Inthefrequency-domainapproach,computationcountingchangesbecauseweneedonlycomputethelter'sfrequencyresponseHkonce,whichamountstoaxedoverhead.WeneedonlycomputetwoDFTsandmultiplythemtolterasection.LettingNxdenoteasection'slength,thenumberofcomputationsforasectionamountstoNx+qlog2Nx+q+6Nx+q.Inaddition,wemustaddthelteredoutputstogether;thenumberoftermstoaddcorrespondstotheexcessdurationoftheoutputcomparedwiththeinputq.Thefrequency-domainapproachthusrequires1+q Nxlog2Nx+q+7q Nx+6computationsperoutputvalue.Forevenmodestlterorders,thefrequency-domainapproachismuchfaster.Exercise5.30Solutiononp.206.Showthatasthesectionlengthincreases,thefrequencydomainapproachbecomesincreasinglymoreecient.Notethatthechoiceofsectiondurationisarbitrary.Oncethelterischosen,weshouldsectionsothattherequiredFFTlengthispreciselyapoweroftwo:ChooseNxsothatNx+q=2L.ImplementingthedigitalltershownintheA/DblockdiagramFigure5.25withafrequency-domainimplementationrequiressomeadditionalsignalmanagementnotrequiredbytime-domainimplementations.Conceptually,areal-time,time-domainltercouldaccepteachsampleasitbecomesavailable,calculatethedierenceequation,andproducetheoutputvalue,allinlessthatthesamplingintervalTs.Frequency-domainapproachesdon'toperateonasample-by-samplebasis;instead,theyoperateonsections.TheylterinrealtimebyproducingNxoutputsforthesamenumberofinputsfasterthanNxTs.Becausetheygenerallytakelongertoproduceanoutputsectionthanthesamplingintervalduration,wemustlteronesectionwhileacceptingintomemorythenextsectiontobeltered.Inprogramming,theoperationofbuildingupsectionswhilecomputingonpreviousonesisknownasbuering.Bueringcanalsobeusedintime-domainltersaswellbutisn'trequired.Example5.10Wewanttolowpasslterasignalthatcontainsasinusoidandasignicantamountofnoise.TheexampleshowninFigure5.23showsaportionofthenoisysignal'swaveform.Ifitweren'tforthe

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191 Figure5.23:Thenoisyinputsignalissectionedintolength-48frames,eachofwhichislteredusingfrequency-domaintechniques.Eachlteredsectionisaddedtootheroutputsthatoverlaptocreatethesignalequivalenttohavinglteredtheentireinput.Thesinusoidalcomponentofthesignalisshownasthereddashedline. overlaidsinusoid,discerningthesinewaveinthesignalisvirtuallyimpossible.Oneoftheprimaryapplicationsoflinearltersisnoiseremoval:preservethesignalbymatchinglter'spassbandwiththesignal'sspectrumandgreatlyreduceallotherfrequencycomponentsthatmaybepresentinthenoisysignal.AsmartRiceengineerhasselectedaFIRlterhavingaunit-sampleresponsecorrespondingaperiod-17sinusoid:hn=1 17)]TJ/F8 9.9626 Tf 4.5662 -8.0698 Td[(1)]TJ/F8 9.9626 Tf 9.9626 0 Td[(cos)]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(2n 17,n=f0;:::;16g,whichmakesq=16.ItsfrequencyresponsedeterminedbycomputingthediscreteFouriertransformisshowninFigure5.24.Toapply,wecanselectthelengthofeachsectionsothatthefrequency-domainlteringapproachismaximallyecient:ChoosethesectionlengthNxsothatNx+qisapoweroftwo.Tousealength-64FFT,eachsectionmustbe48sampleslong.Filteringwiththedierenceequationwouldrequire33computationsperoutputwhilethefrequencydomainrequiresalittleover16;thisfrequency-domainimplementationisovertwiceasfast!Figure5.23showshowfrequency-domainlteringworks.

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192CHAPTER5.DIGITALSIGNALPROCESSING Figure5.24:Thegureshowstheunit-sampleresponseofalength-17Hanninglterontheleftandthefrequencyresponseontheright.Thislterfunctionsasalowpasslterhavingacutofrequencyofabout0.1.Wenotethatthenoisehasbeendramaticallyreduced,withasinusoidnowclearlyvisibleinthelteredoutput.Someresidualnoiseremainsbecausenoisecomponentswithinthelter'spassbandappearintheoutputaswellasthesignal.Exercise5.31Solutiononp.207.Notethatwhencomparedtotheinputsignal'ssinusoidalcomponent,theoutput'ssinusoidalcomponentseemstobedelayed.Whatisthesourceofthisdelay?Canitberemoved?5.16Discrete-TimeFilteringofAnalogSignals28BecauseoftheSamplingTheoremSection5.3.2:TheSamplingTheorem,wecanprocess,inparticularlter,analogsignals"withacomputer"byconstructingthesystemshowninFigure5.25.Tousethissystem,weareassumingthattheinputsignalhasalowpassspectrumandcanbebandlimitedwithoutaectingimportantsignalaspects.Bandpasssignalscanalsobeltereddigitally,butrequireamorecomplicatedsystem.Highpasssignalscannotbeltereddigitally.Notethattheinputandoutputltersmustbeanaloglters;tryingtooperatewithoutthemcanleadtopotentiallyveryinaccuratedigitization.Anotherimplicitassumptionisthatthedigitalltercanoperateinrealtime:Thecomputerandthelteringalgorithmmustbesucientlyfastsothatoutputsarecomputedfasterthaninputvaluesarrive.Thesamplinginterval,whichisdeterminedbytheanalogsignal'sbandwidth,thusdetermineshowlongourprogramhastocomputeeachoutputyn.Thecomputationalcomplexityforcalculatingeachoutputwithadierenceequation.44isOp+q.Frequencydomainimplementationofthelterisalsopossible.TheideabeginsbycomputingtheFouriertransformofalength-Nportionoftheinputxn,multiplyingitbythelter'stransferfunction,andcomputingtheinversetransformoftheresult.Thisapproachseemsoverlycomplexandpotentiallyinecient.Detailingthecomplexity,however,wehaveONlogNforthetwotransformscomputedusingtheFFTalgorithmandONforthemultiplicationbythetransferfunction,whichmakesthetotalcomplexityONlogNforNinputvalues.AfrequencydomainimplementationthusrequiresOlogNcomputationalcomplexityforeachoutputvalue.Thecomplexitiesoftime-domainandfrequency-domainimplementationsdependondierentaspectsoftheltering:Thetime-domainimplemen-tationdependsonthecombinedordersofthelterwhilethefrequency-domainimplementationdependsonthelogarithmoftheFouriertransform'slength. 28Thiscontentisavailableonlineat.

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193 Figure5.25:Toprocessananalogsignaldigitally,thesignalxtmustbelteredwithananti-aliasingltertoensureabandlimitedsignalbeforeA/Dconversion.ThislowpasslterLPFhasacutofrequencyofWHz,whichdeterminesallowablesamplingintervalsTs.ThegreaterthenumberofbitsintheamplitudequantizationportionQ[]oftheA/Dconverter,thegreatertheaccuracyoftheentiresystem.Theresultingdigitalsignalxncannowbelteredinthetime-domainwithadierenceequationorinthefrequencydomainwithFouriertransforms.TheresultingoutputynthendrivesaD/Aconverterandasecondanti-aliasinglterhavingthesamebandwidthastherstone. Itcouldwellbethatinsomeproblemsthetime-domainversionismoreecientmoreeasilysatisestherealtimerequirement,whileinothersthefrequencydomainapproachisfaster.Inthelattersituations,itistheFFTalgorithmforcomputingtheFouriertransformsthatenablesthesuperiorityoffrequency-domainimplementations.Becausecomplexityconsiderationsonlyexpresshowalgorithmrunning-timeincreaseswithsystemparameterchoices,weneedtodetailbothimplementationstodeterminewhichwillbemoresuitableforanygivenlteringproblem.Filteringwithadierenceequationisstraightforward,andthenumberofcomputationsthatmustbemadeforeachoutputvalueis2p+q.Exercise5.32Solutiononp.207.Derivethisvalueforthenumberofcomputationsforthegeneraldierenceequation.44.5.17DigitalSignalProcessingProblems29Problem5.1:SamplingandFilteringThesignalstisbandlimitedto4kHz.Wewanttosampleit,butithasbeensubjectedtovarioussignalprocessingmanipulations.aWhatsamplingfrequencyifanyworkscanbeusedtosampletheresultofpassingstthroughanRChighpasslterwithR=10kandC=8nF?bWhatsamplingfrequencyifanyworkscanbeusedtosamplethederivativeofst?cThesignalsthasbeenmodulatedbyan8kHzsinusoidhavinganunknownphase:theresultingsignalisstsinf0t+,withf0=8kHzand=?Canthemodulatedsignalbesampledsothattheoriginalsignalcanberecoveredfromthemodulatedsignalregardlessofthephasevalue?Ifso,showhowandndthesmallestsamplingratethatcanbeused;ifnot,showwhynot.Problem5.2:Non-StandardSamplingUsingthepropertiesoftheFourierseriescaneasendingasignal'sspectrum.aSupposeasignalstisperiodicwithperiodT.Ifckrepresentsthesignal'sFourierseriescoecients,whataretheFourierseriescoecientsofs)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(t)]TJ/F10 6.9738 Tf 11.1581 3.9226 Td[(T 2?bFindtheFourierseriesofthesignalptshowninFigure5.26PulseSignal. 29Thiscontentisavailableonlineat.

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194CHAPTER5.DIGITALSIGNALPROCESSINGcSupposethissignalisusedtosampleasignalbandlimitedto1 THz.Findanexpressionforandsketchthespectrumofthesampledsignal.dDoesaliasingoccur?Ifso,canachangeinsamplingratepreventaliasing;ifnot,showhowthesignalcanberecoveredfromthesesamples. PulseSignal Figure5.26 Problem5.3:ADierentSamplingSchemeAsignalprocessingengineerfromTexasA&Mclaimstohavedevelopedanimprovedsamplingscheme.HemultipliesthebandlimitedsignalbythedepictedperiodicpulsesignaltoperformsamplingFigure5.27. Figure5.27 aFindtheFourierspectrumofthissignal.bWillthisschemework?Ifso,howshouldTSberelatedtothesignal'sbandwidth?Ifnot,whynot?

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195Problem5.4:BandpassSamplingThesignalsthastheindicatedspectrum. Figure5.28 aWhatistheminimumsamplingrateforthissignalsuggestedbytheSamplingTheorem?bBecauseoftheparticularstructureofthisspectrum,onewonderswhetheralowersamplingratecouldbeused.Showthatthisisindeedthecase,andndthesystemthatreconstructsstfromitssamples.Problem5.5:SamplingSignalsIfasignalisbandlimitedtoWHz,wecansampleitatanyrate1 Ts>2Wandrecoverthewaveformexactly.ThisstatementoftheSamplingTheoremcanbetakentomeanthatallinformationabouttheoriginalsignalcanbeextractedfromthesamples.Whiletrueinprinciple,youdohavetobecarefulhowyoudoso.Inadditiontothermsvalueofasignal,animportantaspectofasignalisitspeakvalue,whichequalsmaxfjstjg.aLetstbeasinusoidhavingfrequencyWHz.IfwesampleitatpreciselytheNyquistrate,howaccuratelydothesamplesconveythesinusoid'samplitude?Inotherwords,ndtheworstcaseexample.bHowfastwouldyouneedtosamplefortheamplitudeestimatetobewithin5%ofthetruevalue?cAnotherissueinsamplingistheinherentamplitudequantizationproducedbyA/Dconverters.AssumethemaximumvoltageallowedbytheconverterisVmaxvoltsandthatitquantizesamplitudestobbits.WecanexpressthequantizedsampleQsnTsassnTs+t,wheretrepresentsthequantizationerroratthenthsample.Assumingtheconverterrounds,howlargeismaximumquantizationerror?dWecandescribethequantizationerrorasnoise,withapowerproportionaltothesquareofthemaximumerror.Whatisthesignal-to-noiseratioofthequantizationerrorforafull-rangesinusoid?Expressyourresultindecibels.Problem5.6:HardwareErrorAnA/Dconverterhasacurioushardwareproblem:EveryothersamplingpulseishalfitsnormalamplitudeFigure5.29.aFindtheFourierseriesforthissignal.bCanthissignalbeusedtosampleabandlimitedsignalhavinghighestfrequencyW=1 2T?Problem5.7:SimpleD/AConverterCommercialdigital-to-analogconvertersdon'tworkthisway,butasimplecircuitillustrateshowtheywork.Let'sassumewehaveaB-bitconverter.Thus,wewanttoconvertnumbershavingaB-bitrepresentation

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196CHAPTER5.DIGITALSIGNALPROCESSING Figure5.29 Figure5.30 intoavoltageproportionaltothatnumber.TherststeptakenbyoursimpleconverteristorepresentthenumberbyasequenceofBpulsesoccurringatmultiplesofatimeintervalT.Thepresenceofapulseindicatesainthecorrespondingbitposition,andpulseabsencemeansaoccurred.Fora4-bitconverter,thenumber13hasthebinaryrepresentation11011310=123+122+021+120andwouldberepresentedbythedepictedpulsesequence.Notethatthepulsesequenceisbackwardsfromthebinaryrepresentation.We'llseewhythatis.ThissignalFigure5.30servesastheinputtoarst-orderRClowpasslter.WewanttodesignthelterandtheparametersandTsothattheoutputvoltageattime4Tfora4-bitconverterisproportionaltothenumber.ThiscombinationofpulsecreationandlteringconstitutesoursimpleD/Aconverter.TherequirementsareThevoltageattimet=4Tshoulddiminishbyafactorof2thefurtherthepulseoccursfromthistime.Inotherwords,thevoltageduetoapulseat3Tshouldbetwicethatofapulseproducedat2T,whichinturnistwicethatofapulseatT,etc.The4-bitD/Aconvertermustsupporta10kHzsamplingrate.Showthecircuitthatworks.Howdotheconverter'sparameterschangewithsamplingrateandnumberofbitsintheconverter?Problem5.8:Discrete-TimeFourierTransformsFindtheFouriertransformsofthefollowingsequences,wheresnissomesequencehavingFouriertransformS)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f.

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197a)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1nsnbsncosf0ncxn=8<:s)]TJ/F10 6.9738 Tf 5.7617 -4.1472 Td[(n 2ifneven0ifnodddnsnProblem5.9:SpectraofFinite-DurationSignalsFindtheindicatedspectraforthefollowingsignals.aThediscrete-timeFouriertransformofsn=8<:cos2)]TJ/F10 6.9738 Tf 5.7617 -4.1472 Td[( 4nifn=f)]TJ/F8 9.9626 Tf 12.7301 0 Td[(1;0;1g0ifotherwisebThediscrete-timeFouriertransformofsn=8<:nifn=f)]TJ/F8 9.9626 Tf 12.7301 0 Td[(2;)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1;0;1;)]TJ/F8 9.9626 Tf 7.7488 0 Td[(2g0ifotherwisecThediscrete-timeFouriertransformofsn=8<:sin)]TJ/F10 6.9738 Tf 5.7618 -4.1472 Td[( 4nifn=f0;:::;7g0ifotherwisedThelength-8DFToftheprevioussignal.Problem5.10:JustWhistlin'Sammylovestowhistleanddecidestorecordandanalyzehiswhistlinginlab.Heisaverygoodwhistler;hiswhistleisapuresinusoidthatcanbedescribedbysat=sint.Toanalyzethespectrum,hesampleshisrecordedwhistlewithasamplingintervalofTS=2:510)]TJ/F7 6.9738 Tf 6.2267 0 Td[(4toobtainsn=sanTS.Sammywiselydecidestoanalyzeafewsamplesatatime,sohegrabs30consecutive,butarbitrarilychosen,samples.Hecallsthissequencexnandrealizeshecanwriteitasxn=sinnTS+,n=f0;:::;29gaDidSammyunder-orover-samplehiswhistle?bWhatisthediscrete-timeFouriertransformofxnandhowdoesitdependon?cHowdoesthe32-pointDFTofxndependon?Problem5.11:Discrete-TimeFilteringWecanndtheinput-outputrelationforadiscrete-timeltermuchmoreeasilythanforanaloglters.Thekeyideaisthatasequencecanbewrittenasaweightedlinearcombinationofunitsamples.aShowthatxn=Pixin)]TJ/F11 9.9626 Tf 9.9626 0 Td[(iwherenistheunit-sample.n=8<:1ifn=00otherwisebIfhndenotestheunit-sampleresponsetheoutputofadiscrete-timelinear,shift-invariantltertoaunit-sampleinputndanexpressionfortheoutput.cInparticular,assumeourlterisFIR,withtheunit-sampleresponsehavingdurationq+1.IftheinputhasdurationN,whatisthedurationofthelter'soutputtothissignal?dLetthelterbeaboxcaraverager:hn=1 q+1forn=f0;:::;qgandzerootherwise.LettheinputbeapulseofunitheightanddurationN.Findthelter'soutputwhenN=q+1 2,qanoddinteger.

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198CHAPTER5.DIGITALSIGNALPROCESSINGProblem5.12:ADigitalFilterAdigitallterhasthedepictedFigure5.31unit-samplereponse. Figure5.31 aWhatisthedierenceequationthatdenesthislter'sinput-outputrelationship?bWhatisthislter'stransferfunction?cWhatisthelter'soutputwhentheinputissin)]TJ/F10 6.9738 Tf 5.7618 -4.1472 Td[(n 4?Problem5.13:ASpecialDiscrete-TimeFilterConsideraFIRltergovernedbythedierenceequationyn=1 3xn+2+2 3xn+1+xn+2 3xn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1+1 3xn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(2aFindthislter'sunit-sampleresponse.bFindthislter'stransferfunction.Characterizethistransferfunctioni.e.,whatclassicltercategorydoesitfallinto.cSupposewetakeasequenceandstretchitoutbyafactorofthree.xn=8<:s)]TJ/F10 6.9738 Tf 5.7617 -4.1472 Td[(n 3ifn=3m;m=f:::;)]TJ/F8 9.9626 Tf 7.7488 0 Td[(1;0;1;:::g0otherwiseSketchthesequencexnforsomeexamplesn.Whatisthelter'soutputtothisinput?Inparticular,whatistheoutputattheindiceswheretheinputxnisintentionallyzero?Nowhowwouldyoucharacterizethissystem?Problem5.14:SimulatingtheRealWorldMuchofphysicsisgovernedbydierntialequations,andwewanttousesignalprocessingmethodstosimulatephysicalproblems.Theideaistoreplacethederivativewithadiscrete-timeapproximationandsolvetheresultingdierentialequation.Forexample,supposewehavethedierentialequationd dtyt+ayt=xtandweapproximatethederivativebyd dtytjt=nTynT)]TJ/F11 9.9626 Tf 9.9626 0 Td[(yn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1T TwhereTessentiallyamountstoasamplinginterval.

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199aWhatisthedierenceequationthatmustbesolvedtoapproximatethedierentialequation?bWhenxt=ut,theunitstep,whatwillbethesimulatedoutput?cAssumingxtisasinusoid,howshouldthesamplingintervalTbechosensothattheapproximationworkswell?Problem5.15:TheDFTLet'sexploretheDFTanditsproperties.aWhatisthelength-KDFToflength-Nboxcarsequence,whereN
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200CHAPTER5.DIGITALSIGNALPROCESSINGcomputingtwotransformsatonetimebycomputingthetransformofsn=s1n+js2n,wheres1nands2naretworeal-valuedsignalsofwhichheneedstocomputethespectra.TheissueiswhetherhecanretrievetheindividualDFTsfromtheresultornot.aWhatwillbetheDFTSkofthiscomplex-valuedsignalintermsofS1kandS2k,theDFTsoftheoriginalsignals?bSammy'sfriend,anAggiewhoknowssomesignalprocessing,saysthatretrievingthewantedDFTsiseasy:JustndtherealandimaginarypartsofSk.Showthatthisapproachistoosimplistic.cWhilehisfriend'sideaisnotcorrect,itdoesgivehimanidea.Whatapproachwillwork?Hint:UsethesymmetrypropertiesoftheDFT.dHowdoesthenumberofcomputationschangewiththisapproach?WillSammy'sideaultimatelyleadtoafastercomputationoftherequiredDFTs?Problem5.18:DiscreteCosineTransformDCTThediscretecosinetransformofalength-NsequenceisdenedtobeSck=N)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1Xn=0sncos2nk 2NNotethatthenumberoffrequencytermsis2N)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1:k=f0;:::;2N)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1g.aFindtheinverseDCT.bDoesaParseval'sTheoremholdfortheDCT?cYouchoosetotransmitinformationaboutthesignalsnaccordingtotheDCTcoecients.Youcouldonlysendone,whichonewouldyousend?Problem5.19:ADigitalFilterAdigitallterisdescribedbythefollowingdierenceequation:yn=ayn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1+axn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(xn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1,a=1 p 2aWhatisthislter'sunitsampleresponse?bWhatisthislter'stransferfunction?cWhatisthislter'soutputwhentheinputissin)]TJ/F10 6.9738 Tf 5.7618 -4.1472 Td[(n 4?Problem5.20:AnotherDigitalFilterAdigitallterisdeterminedbythefollowingdierenceequation.yn=yn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1+xn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(xn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(4aFindthislter'sunitsampleresponse.bWhatisthelter'stransferfunction?cFindthelter'soutputwhentheinputisthesinusoidsin)]TJ/F10 6.9738 Tf 5.7617 -4.1472 Td[(n 2.Problem5.21:YetAnotherDigitalFilterAlterhasaninput-outputrelationshipgivenbythedierenceequationyn=1 4xn+1 2xn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1+1 4xn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(2.

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201aWhatisthelter'stransferfunction?Howwouldyoucharacterizeit?bWhatisthelter'soutputwhentheinputequalscos)]TJ/F10 6.9738 Tf 5.7618 -4.1472 Td[(n 2?cWhatisthelter'soutputwhentheinputisthedepicteddiscrete-timesquare-waveFigure5.33? Figure5.33 Problem5.22:ADigitalFilterintheFrequencyDomainWehavealterwiththetransferfunctionH)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2f=e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2fcos2foperatingontheinputsignalxn=n)]TJ/F11 9.9626 Tf 9.9626 0 Td[(n)]TJ/F8 9.9626 Tf 9.9626 0 Td[(2thatyieldstheoutputyn.Whatisthelter'sunit-sampleresponse?Whatisthediscrete-Fouriertransformoftheoutput?Whatisthetime-domainexpressionfortheoutput?Problem5.23:DigitalFiltersAdiscrete-timesystemisgovernedbythedierenceequationyn=yn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1+xn+xn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1 2aFindthetransferfunctionforthissystem.bWhatisthissystem'soutputwhentheinputissin)]TJ/F10 6.9738 Tf 5.7618 -4.1472 Td[(n 2?cIftheoutputisobservedtobeyn=n+n)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1,thenwhatistheinput?Problem5.24:DigitalFilteringAdigitallterhasaninput-outputrelationshipexpressedbythedierenceequationyn=xn+xn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1+xn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(2+xn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(3 4.aPlotthemagnitudeandphaseofthislter'stransferfunction.bWhatisthislter'soutputwhenxn=cos)]TJ/F10 6.9738 Tf 5.7617 -4.1472 Td[(n 2+2sin)]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(2n 3?Problem5.25:DetectiveWorkThesignalxnequalsn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(n)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1.

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202CHAPTER5.DIGITALSIGNALPROCESSINGaFindthelength-8DFTdiscreteFouriertransformofthissignal.bYouaretoldthatwhenxnservedastheinputtoalinearFIRniteimpulseresponselter,theoutputwasyn=n)]TJ/F11 9.9626 Tf 10.0522 0 Td[(n)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1+2n)]TJ/F8 9.9626 Tf 9.9626 0 Td[(2.Isthisstatementtrue?Ifso,indicatewhyandndthesystem'sunitsampleresponse;ifnot,showwhynot.Problem5.26:Adiscrete-time,shiftinvariant,linearsystemproducesanoutputyn=f1;)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1;0;0;:::gwhenitsinputxnequalsaunitsample.aFindthedierenceequationgoverningthesystem.bFindtheoutputwhenxn=cosf0n.cHowwouldyoudescribethissystem'sfunction?Problem5.27:TimeReversalhasUsesAdiscrete-timesystemhastransferfunctionH)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(ej2f.Asignalxnispassedthroughthissystemtoyieldthesignalwn.Thetime-reversedsignalw)]TJ/F11 9.9626 Tf 7.7487 0 Td[(nisthenpassedthroughthesystemtoyieldthetime-reversedoutputy)]TJ/F11 9.9626 Tf 7.7487 0 Td[(n.Whatisthetransferfunctionbetweenxnandyn?Problem5.28:RemovingHumTheslangwordhumrepresentspowerlinewaveformsthatcreepintosignalsbecauseofpoorcircuitconstruction.Usually,the60Hzsignalanditsharmonicsareaddedtothedesiredsignal.Whatweseekareltersthatcanremovehum.Inthisproblem,thesignalandtheaccompanyinghumhavebeensampled;wewanttodesignadigitallterforhumremoval.aFindltercoecientsforthelength-3FIRlterthatcanremoveasinusoidhavingdigitalfrequencyf0fromitsinput.bAssumingthesamplingrateisfstowhatanalogfrequencydoesf0correspond?cAmoregeneralapproachistodesignalterhavingafrequencyresponsemagnitudeproportionaltotheabsolutevalueofacosine:jH)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(ej2fj/jcosfNj.Inthisway,notonlycanthefundamentalbutalsoitsrstfewharmonicsberemoved.SelecttheparameterNandthesamplingratesothatthefrequenciesatwhichthecosineequalszerocorrespondto60Hzanditsoddharmonicsthroughthefth.dFindthedierenceequationthatdenesthislter.Problem5.29:DigitalAMReceiverThinkingthatdigitalimplementationsarealwaysbetter,ourcleverengineerwantstodesignadigitalAMreceiver.Thereceiverwouldbandpassthereceivedsignal,passtheresultthroughanA/Dconverter,performallthedemodulationwithdigitalsignalprocessingsystems,andendwithaD/Aconvertertoproducetheanalogmessagesignal.Assumeinthisproblemthatthecarrierfrequencyisalwaysalargeevenmultipleofthemessagesignal'sbandwidthW.aWhatisthesmallestsamplingratethatwouldbeneeded?bShowtheblockdiagramoftheleastcomplexdigitalAMreceiver.cAssumingthechanneladdswhitenoiseandthatab-bitA/Dconverterisused,whatistheoutput'ssignal-to-noiseratio?Problem5.30:DFTsAproblemonSamantha'shomeworkasksforthe8-pointDFTofthediscrete-timesignaln)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1+n)]TJ/F8 9.9626 Tf 9.9626 0 Td[(7.aWhatanswershouldSamanthaobtain?bAsacheck,hergrouppartnerSammysaysthathecomputedtheinverseDFTofheranswerandgotn+1+n)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1.DoesSammy'sresultmeanthatSamantha'sansweriswrong?cThehomeworkproblemsaystolowpass-lterthesequencebymultiplyingitsDFTbyHk=8<:1ifk=f0;1;7g0otherwise

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203andthencomputingtheinverseDFT.Willthislteringalgorithmwork?Ifso,ndthelteredoutput;ifnot,whynot?Problem5.31:StockMarketDataProcessingBecauseatradingweeklastsvedays,stockmarketsfrequentlycomputerunningaverageseachdayoverthepreviousvetradingdaystosmoothpriceuctuations.ThetechnicalstockanalystattheBuy-LoSell-HibrokeragermhasheardthatFFTlteringtechniquesworkbetterthananyothersintermsofproducingmoreaccurateaverages.aWhatisthedierenceequationgoverningtheve-dayaveragerfordailystockprices?bDesignanecientFFT-basedlteringalgorithmforthebroker.Howmuchdatashouldbeprocessedatoncetoproduceanecientalgorithm?Whatlengthtransformshouldbeused?cIstheanalyst'sinformationcorrectthatFFTtechniquesproducemoreaccurateaveragesthananyothers?Whyorwhynot?Problem5.32:DigitalFilteringofAnalogSignalsRUElectronicswantstodevelopalterthatwouldbeusedinanalogapplications,butthatisimplementeddigitally.Thelteristooperateonsignalsthathavea10kHzbandwidth,andwillserveasalowpasslter.aWhatistheblockdiagramforyourlterimplementation?Explicitlydenotewhichcomponentsareanalog,whicharedigitalacomputerperformsthetask,andwhichinterfacebetweenanaloganddigitalworlds.bWhatsamplingratemustbeusedandhowmanybitsmustbeusedintheA/Dconverterfortheacquiredsignal'ssignal-to-noiseratiotobeatleast60dB?Forthiscalculation,assumethesignalisasinusoid.cIfthelterisalength-128FIRlterthedurationofthelter'sunit-sampleresponseequals128,shoulditbeimplementedinthetimeorfrequencydomain?dAssumingH)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(ej2fisthetransferfunctionofthedigitallter,whatisthetransferfunctionofyoursystem?Problem5.33:SignalCompressionBecauseoftheslownessoftheInternet,lossysignalcompressionbecomesimportantifyouwantsignalstobereceivedquickly.Anenterprising241studenthasproposedaschemebasedonfrequency-domainprocessing.Firstofall,hewouldsectionthesignalintolength-Nblocks,andcomputeitsN-pointDFT.Hethenwoulddiscardzerothespectrumathalfofthefrequencies,quantizethemtob-bits,andsendtheseoverthenetwork.ThereceiverwouldassemblethetransmittedspectrumandcomputetheinverseDFT,thusreconstitutinganN-pointblock.aAtwhatfrequenciesshouldthespectrumbezeroedtominimizetheerrorinthislossycompressionscheme?bThenominalwaytorepresentasignaldigitallyistousesimpleb-bitquantizationofthetime-domainwaveform.Howlongshouldasectionbeintheproposedschemesothattherequirednumberofbits/sampleissmallerthanthatnominallyrequired?cAssumingthateectivecompressioncanbeachieved,wouldtheproposedschemeyieldsatisfactoryresults?

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204CHAPTER5.DIGITALSIGNALPROCESSINGSolutionstoExercisesinChapter5SolutiontoExercise5.1p.157Forb-bitsignedintegers,thelargestnumberis2b)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1)]TJ/F8 9.9626 Tf 9.8623 0 Td[(1.Forb=32,wehave2,147,483,647andforb=64,wehave9,223,372,036,854,775,807orabout9:21018.SolutiontoExercise5.2p.158Inoatingpoint,thenumberofbitsintheexponentdeterminesthelargestandsmallestrepresentablenumbers.For32-bitoatingpoint,thelargestsmallestnumbersare2127=1:710385:910)]TJ/F7 6.9738 Tf 6.2267 0 Td[(39.For64-bitoatingpoint,thelargestnumberisabout109863.SolutiontoExercise5.3p.15925=110112and7=1112.Wendthat110012+1112=1000002=32.SolutiontoExercise5.4p.162Theonlyeectofpulsedurationistounequallyweightthespectralrepetitions.Becauseweareonlyconcernedwiththerepetitioncenteredabouttheorigin,thepulsedurationhasnosignicanteectonrecoveringasignalfromitssamples.SolutiontoExercise5.5p.162 Figure5.34Thesquarewave'sspectrumisshownbytheboldersetoflinescenteredabouttheorigin.ThedashedlinescorrespondtothefrequenciesaboutwhichthespectralrepetitionsduetosamplingwithTs=1occur.Asthesquarewave'sperioddecreases,thenegativefrequencylinesmovetotheleftandthepositivefrequencyonestotheright.SolutiontoExercise5.6p.162Thesimplestbandlimitedsignalisthesinewave.AttheNyquistfrequency,exactlytwosamples/periodwouldoccur.Reducingthesamplingratewouldresultinfewersamples/period,andthesesampleswouldappeartohavearisenfromalowerfrequencysinusoid.SolutiontoExercise5.7p.164Theplottedtemperatureswerequantizedtothenearestdegree.Thus,thehightemperature'samplitudewasquantizedasaformofA/Dconversion.

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205SolutiontoExercise5.8p.164Thesignal-to-noiseratiodoesnotdependonthesignalamplitude.WithanA/Drangeof[)]TJ/F11 9.9626 Tf 7.7488 0 Td[(A;A],thequantizationinterval=2A 2Bandthesignal'srmsvalueagainassumingitisasinusoidisA p 2.SolutiontoExercise5.9p.164Solving2)]TJ/F10 6.9738 Tf 6.2267 0 Td[(B=:001resultsinB=10bits.SolutiontoExercise5.10p.164A16-bitA/DconverteryieldsaSNRof616+10log101:5=97:8dB.SolutiontoExercise5.11p.167S)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(ej2f+1=P1n=)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(sne)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2f+1n=P1n=)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2nsne)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2fn=P1n=)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(sne)]TJ/F7 6.9738 Tf 6.2267 0 Td[(j2fn=S)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(ej2f.58SolutiontoExercise5.12p.170N+n0)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1Xn=n0n)]TJ/F10 6.9738 Tf 9.9626 12.5641 Td[(N+n0)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1Xn=n0n=N+n0)]TJ/F11 9.9626 Tf 9.9626 0 Td[(n0which,aftermanipulation,yieldsthegeometricsumformula.SolutiontoExercise5.13p.171IfthesamplingfrequencyexceedstheNyquistfrequency,thespectrumofthesamplesequalstheanalogspectrum,butoverthenormalizedanalogfrequencyfT.Thus,theenergyinthesampledsignalequalstheoriginalsignal'senergymultipliedbyT.SolutiontoExercise5.14p.173Thissituationamountstoaliasinginthetime-domain.SolutiontoExercise5.15p.174Whenthesignalisreal-valued,wemayonlyneedhalfthespectralvalues,butthecomplexityremainsunchanged.Ifthedataarecomplex-valued,whichdemandsretainingallfrequencyvalues,thecomplexityisagainthesame.WhenonlyKfrequenciesareneeded,thecomplexityisOKN.SolutiontoExercise5.16p.175IfaDFTrequired1mstocompute,andsignalhavingtentimesthedurationwouldrequire100mstocompute.UsingtheFFT,a1mscomputingtimewouldincreasebyafactorofaboutlog210=3:3,afactorof30lessthantheDFTwouldhaveneeded.SolutiontoExercise5.17p.177TheupperpanelhasnotusedtheFFTalgorithmtocomputethelength-4DFTswhiletheloweronehas.Theorderingisdeterminedbythealgorithm.SolutiontoExercise5.18p.177Thetransformcanhaveanygreaterthanorequaltotheactualdurationofthesignal.Wesimplypadthesignalwithzero-valuedsamplesuntilacomputationallyadvantageoussignallengthresults.RecallthattheFFTisanalgorithmtocomputetheDFTSection5.7.Extendingthelengthofthesignalthiswaymerelymeanswearesamplingthefrequencyaxismorenelythanrequired.TousetheCooley-Tukeyalgorithm,thelengthoftheresultingzero-paddedsignalcanbe512,1024,etc.sampleslong.SolutiontoExercise5.19p.178Numberofsamplesequals1:211025=13230.Thedatarateis1102516=176:4kbps.Thestoragerequiredwouldbe26460bytes.

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206CHAPTER5.DIGITALSIGNALPROCESSINGSolutiontoExercise5.20p.179Theoscillationsareduetotheboxcarwindow'sFouriertransform,whichequalsthesincfunction.SolutiontoExercise5.21p.180Thesenumbersarepowers-of-two,andtheFFTalgorithmcanbeexploitedwiththeselengths.Tocomputealongertransformthantheinputsignal'sduration,wesimplyzero-padthesignal.SolutiontoExercise5.22p.180Indiscrete-timesignalprocessing,anamplieramountstoamultiplication,averyeasyoperationtoperform.SolutiontoExercise5.23p.182Theindicescanbenegative,andthisconditionisnotallowedinMATLAB.Toxit,wemuststartthesignalslaterinthearray.SolutiontoExercise5.24p.184Suchtermswouldrequirethesystemtoknowwhatfutureinputoroutputvalueswouldbebeforethecurrentvaluewascomputed.Thus,suchtermscancausediculties.SolutiontoExercise5.25p.186Itnowactslikeabandpasslterwithacenterfrequencyoff0andabandwidthequaltotwiceoftheoriginallowpasslter.SolutiontoExercise5.26p.186TheDTFToftheunitsampleequalsaconstantequaling1.Thus,theFouriertransformoftheoutputequalsthetransferfunction.SolutiontoExercise5.27p.186Insamplingadiscrete-timesignal'sFouriertransformLtimesequallyover[0;2toformtheDFT,thecorrespondingsignalequalstheperiodicrepetitionoftheoriginalsignal.Sk$1Xi=sn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(iL!.59Toavoidaliasinginthetimedomain,thetransformlengthmustequalorexceedthesignal'sduration.SolutiontoExercise5.28p.187ThedierenceequationforanFIRlterhastheformyn=qXm=0bmxn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(m.60Theunit-sampleresponseequalshn=qXm=0bmn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(m.61whichcorrespondstotherepresentationdescribedinaproblemExample5.6ofalength-qboxcarlter.SolutiontoExercise5.29p.187Theunit-sampleresponse'sdurationisq+1andthesignal'sNx.Thusthestatementiscorrect.SolutiontoExercise5.30p.190LetNdenotetheinput'stotalduration.Thetime-domainimplementationrequiresatotalofNq+1computations,or2q+1computationsperinputvalue.Inthefrequencydomain,wesplittheinputintoN Nxsections,eachofwhichrequires1+q Nxlog2Nx+q+7q Nx+6perinputinthesection.BecausewedivideagainbyNxtondthenumberofcomputationsperinputvalueintheentireinput,thisquantitydecreasesasNxincreases.Forthetime-domainimplementation,itstaysconstant.

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207SolutiontoExercise5.31p.192Thedelayisnotcomputationaldelayheretheplotshowstherstoutputvalueisalignedwiththelter'srstinputalthoughinrealsystemsthisisanimportantconsideration.Rather,thedelayisduetothelter'sphaseshift:Aphase-shiftedsinusoidisequivalenttoatime-delayedone:cos2fn)]TJ/F11 9.9626 Tf 9.9626 0 Td[(=cos2fn)]TJ/F10 6.9738 Tf 15.5364 4.4444 Td[( 2f.Allltershavephaseshifts.Thisdelaycouldberemovedifthelterintroducednophaseshift.Suchltersdonotexistinanalogform,butdigitalonescanbeprogrammed,butnotinrealtime.Doingsowouldrequiretheoutputtoemergebeforetheinputarrives!SolutiontoExercise5.32p.193Wehavep+q+1multiplicationsandp+q)]TJ/F8 9.9626 Tf 9.6475 0 Td[(1additions.Thus,thetotalnumberofarithmeticoperationsequals2p+q.

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208CHAPTER5.DIGITALSIGNALPROCESSING

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Chapter6InformationCommunication6.1InformationCommunication1Asfarasacommunicationsengineerisconcerned,signalsexpressinformation.Becausesystemsmanipulatesignals,theyalsoaecttheinformationcontent.Informationcomesneatlypackagedinbothanaloganddigitalforms.Speech,forexample,isclearlyananalogsignal,andcomputerlesconsistofasequenceofbytes,aformof"discrete-time"signaldespitethefactthattheindexsequencesbyteposition,nottimesample.Communicationsystemsendeavornottomanipulateinformation,buttotransmititfromoneplacetoanother,so-calledpoint-to-pointcommunication,fromoneplacetomanyothers,broadcastcommunication,orfrommanytomany,likeatelephoneconferencecallorachatroom.Communicationsystemscanbefundamentallyanalog,likeradio,ordigital,likecomputernetworks.Thischapterdevelopsacommontheorythatunderlieshowsuchsystemswork.Wedescribeandanalyzeseveralsuchsystems,someoldlikeAMradio,somenewlikecomputernetworks.Thequestionastowhichisbetter,analogordigitalcommunication,hasbeenanswered,becauseofClaudeShannon's2fundamentalworkonatheoryofinformationpublishedin1948,thedevelopmentofcheap,high-performancecomputers,andthecreationofhigh-bandwidthcommunicationsystems.Theansweristouseadigitalcommunicationstrategy.Inmostcases,youshouldconvertallinformation-bearingsignalsintodiscrete-time,amplitude-quantizedsignals.Fundamentallydigitalsignals,likecomputerleswhichareaspecialcaseofsymbolicsignals,areintheproperform.BecauseoftheSamplingTheorem,weknowhowtoconvertanalogsignalsintodigitalones.Shannonshowedthatonceinthisform,aproperlyengineeredsystemcancommunicatedigitalinformationwithnoerrordespitethefactthatthecommunicationchannelthrustsnoiseontoalltransmissions.Thisstartlingresulthasnocounterpartinanalogsystems;AMradiowillremainnoisy.Theconvergenceofthesetheoreticalandengineeringresultsoncommunicationssystemshashadimportantconsequencesinotherarenas.TheaudiocompactdiscCDandthedigitalvideodiskDVDarenowconsidereddigitalcommunicationssystems,withcommunicationdesignconsiderationsusedthroughout.GobacktothefundamentalmodelofcommunicationFigure1.4:Fundamentalmodelofcommunication.Communicationsdesignbeginswithtwofundamentalconsiderations.1.Whatisthenatureoftheinformationsource,andtowhatextentcanthereceivertolerateerrorsinthereceivedinformation?2.Whatarethechannel'scharacteristicsandhowdotheyaectthetransmittedsignal?Inshort,whatarewegoingtosendandhowarewegoingtosendit?Interestingly,digitalaswellasanalogtransmissionareaccomplishedusinganalogsignals,likevoltagesinEthernetanexampleofwirelinecommunicationsandelectromagneticradiationwirelessincellulartelephone. 1Thiscontentisavailableonlineat.2http://www.lucent.com/minds/infotheory/209

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210CHAPTER6.INFORMATIONCOMMUNICATION6.2TypesofCommunicationChannels3Electricalcommunicationschannelsareeitherwirelineorwirelesschannels.Wirelinechannelsphysicallyconnecttransmittertoreceiverwitha"wire"whichcouldbeatwistedpair,coaxialcableoropticber.Consequently,wirelinechannelsaremoreprivateandmuchlesspronetointerference.Simplewirelinechannelsconnectasingletransmittertoasinglereceiver:apoint-to-pointconnectionaswiththetelephone.Listeninginonaconversationrequiresthatthewirebetappedandthevoltagemeasured.Somewirelinechannelsoperateinbroadcastmodes:oneormoretransmitterisconnectedtoseveralreceivers.Onesimpleexampleofthissituationiscabletelevision.Computernetworkscanbefoundthatoperateinpoint-to-pointorinbroadcastmodes.Wirelesschannelsaremuchmorepublic,withatransmitter'santennaradiatingasignalthatcanbereceivedbyanyantennasucientlycloseenough.Incontrasttowirelinechannelswherethereceivertakesinonlythetransmitter'ssignal,thereceiver'santennawillreacttoelectromagneticradiationcomingfromanysource.Thisfeaturehastwofaces:Thesmileyfacesaysthatareceivercantakeintransmissionsfromanysource,lettingreceiverelectronicsselectwantedsignalsanddisregardingothers,therebyallowingportabletransmissionandreception,whilethefrownyfacesaysthatinterferenceandnoisearemuchmoreprevalentthaninwirelinesituations.Anoisierchannelsubjecttointerferencecompromisestheexibilityofwirelesscommunication.PointofInterest:Youwillhearthetermtetherlessnetworkingappliedtocompletelywirelesscomputernetworks.Maxwell'sequationsneatlysummarizethephysicsofallelectromagneticphenomena,includingcir-cuits,radio,andopticbertransmission.rE=)]TJ/F1 9.9626 Tf 9.4091 14.0474 Td[(@ @tHdivE=rH=E+@ @tEdivH=0.1whereEistheelectriceld,Hthemagneticeld,dielectricpermittivity,magneticpermeability,electricalconductivity,andisthechargedensity.Kircho'sLawsrepresentspecialcasesoftheseequationsforcircuits.WearenotgoingtosolveMaxwell'sequationshere;dobearinmindthatafundamentalunderstandingofcommunicationschannelsultimatelydependsonuencywithMaxwell'sequations.Perhapsthemostimportantaspectofthemisthattheyarelinearwithrespecttotheelectricalandmagneticelds.Thus,theeldsandthereforethevoltagesandcurrentsresultingfromtwoormoresourceswilladd.PointofInterest:Nonlinearelectromagneticmediadoexist.Theequationsaswrittenherearesimplerversionsthatapplytofree-spacepropagationandconductioninmetals.Nonlinearmediaarebecomingincreasinglyimportantinopticbercommunications,whicharealsogovernedbyMaxwell'sequations.6.3WirelineChannels4Wirelinechannelsweretherstusedforelectricalcommunicationsinthemid-nineteenthcenturyforthetelegraph.Here,thechannelisoneofseveralwiresconnectingtransmittertoreceiver.Thetransmittersimplycreatesavoltagerelatedtothemessagesignalandappliesittothewires.Wemusthaveacircuitaclosedpaththatsupportscurrentow.Inthecaseofsingle-wirecommunications,theearthisusedasthe 3Thiscontentisavailableonlineat.4Thiscontentisavailableonlineat.

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211current'sreturnpath.Infact,thetermgroundforthereferencenodeincircuitsoriginatedinsingle-wiretelegraphs.Youcanimaginethattheearth'selectricalcharacteristicsarehighlyvariable,andtheyare.Single-wiremetallicchannelscannotsupporthigh-qualitysignaltransmissionhavingabandwidthbeyondafewhundredHertzoveranyappreciabledistance. CoaxialCableCross-section Figure6.1:Coaxialcableconsistsofoneconductorwrappedaroundthecentralconductor.Thistypeofcablesupportsbroaderbandwidthsignalsthantwistedpair,andndsuseincabletelevisionandEthernet. Consequently,mostwirelinechannelstodayessentiallyconsistofpairsofconductingwiresFigure6.1CoaxialCableCross-section,andthetransmitterappliesamessage-relatedvoltageacrossthepair.Howthesepairsofwiresarephysicallyconguredgreatlyaectstheirtransmissioncharacteristics.Oneexampleistwistedpair,whereinthewiresarewrappedabouteachother.Telephonecablesareoneexampleofatwistedpairchannel.Anotheriscoaxialcable,whereaconcentricconductorsurroundsacentralwirewithadielectricmaterialinbetween.Coaxialcable,fondlycalled"co-ax"byengineers,iswhatEthernetusesasitschannel.Ineithercase,wirelinechannelsformadedicatedcircuitbetweentransmitterandreceiver.Asweshallndsubsequently,severaltransmissionscansharethecircuitbyamplitudemodulationtechniques;commercialcableTVisanexample.Theseinformation-carryingcircuitsaredesignedsothatinterferencefromnearbyelectromagneticsourcesisminimized.Thus,bythetimesignalsarriveatthereceiver,theyarerelativelyinterference-andnoise-free.Bothtwistedpairandco-axareexamplesoftransmissionlines,whichallhavethecircuitmodelshowninFigure6.2CircuitModelforaTransmissionLineforaninnitesimallysmalllength.ThiscircuitmodelarisesfromsolvingMaxwell'sequationsfortheparticulartransmissionlinegeometry.CircuitModelforaTransmissionLine Figure6.2:Theso-calleddistributedparametermodelfortwo-wirecableshasthedepictedcircuitmodelstructure.Elementvaluesdependongeometryandthepropertiesofmaterialsusedtoconstructthetransmissionline.

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212CHAPTER6.INFORMATIONCOMMUNICATIONTheseriesresistancecomesfromtheconductorusedinthewiresandfromtheconductor'sgeometry.Theinductanceandthecapacitancederivefromtransmissionlinegeometry,andtheparallelconductancefromthemediumbetweenthewirepair.Notethatallthecircuitelementshavevaluesexpressedbytheproductofaconstanttimesalength;thisnotationrepresentsthatelementvaluesherehaveper-unit-lengthunits.Forexample,theseriesresistanceRhasunitsofohms/meter.Forcoaxialcable,theelementvaluesdependontheinnerconductor'sradiusri,theouterradiusofthedielectricrd,theconductivityoftheconductors,andtheconductivityd,dielectricconstantd,andmagneticpermittivitydofthedielectricasR=1 21 rd+1 ri.2C=2d lnrd riG=2d lnrd riL=d 2lnrd riFortwistedpair,havingaseparationdbetweentheconductorsthathaveconductivityandcommonradiusrandthatareimmersedinamediumhavingdielectricandmagneticproperties,theelementvaluesarethenR=1 r.3C= arccosh)]TJ/F10 6.9738 Tf 7.6305 -4.1472 Td[(d 2rG= arccosh)]TJ/F10 6.9738 Tf 7.6305 -4.1472 Td[(d 2rL= 2r+arccoshd 2rThevoltagebetweenthetwoconductorsandthecurrentowingthroughthemwilldependondistancexalongthetransmissionlineaswellastime.Weexpressthisdependenceasvx;tandix;t.Whenweplaceasinusoidalsourceatoneendofthetransmissionline,thesevoltagesandcurrentswillalsobesinusoidalbecausethetransmissionlinemodelconsistsoflinearcircuitelements.Asiscustomaryinanalyzinglinearcircuits,weexpressvoltagesandcurrentsastherealpartofcomplexexponentialsignals,andwritecircuitvariablesasacomplexamplitudeheredependentondistancetimesacomplexexponential:vx;t=Re)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(Vxej2ftandix;t=Re)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(Ixej2ft.Usingthetransmissionlinecircuitmodel,wendfromKCL,KVL,andv-irelationstheequationsgoverningthecomplexamplitudes.KCLatCenterNodeIx=Ix)]TJ/F8 9.9626 Tf 9.9626 0 Td[(x)]TJ/F11 9.9626 Tf 9.9626 0 Td[(VxG+j2fCx.4V-IrelationforRLseriesVx)]TJ/F11 9.9626 Tf 9.9626 0 Td[(Vx+x=IxR+j2fLx.5Rearrangingandtakingthelimitx!0yieldstheso-calledtransmissionlineequations.d dxIx=)]TJ/F1 9.9626 Tf 9.4092 11.0587 Td[(G+j2fCVx.6

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213d dxVx=)]TJ/F1 9.9626 Tf 9.4092 11.0586 Td[(R+j2fLIxBycombiningtheseequations,wecanobtainasingleequationthatgovernshowthevoltage'sorthecurrent'scomplexamplitudechangeswithpositionalongthetransmissionline.Takingthederivativeofthesecondequationandpluggingtherstequationintotheresultyieldstheequationgoverningthevoltage.d2 dx2Vx=G+j2fCR+j2fLVx.7Thisequation'ssolutionisVx=V+e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(x+V)]TJ/F11 9.9626 Tf 6.7248 1.4944 Td[(ex.8Calculatingitssecondderivativeandcomparingtheresultwithourequationforthevoltagecancheckthissolution.d2 dx2Vx=2)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(V+e)]TJ/F7 6.9738 Tf 6.2266 0 Td[(x+V)]TJ/F11 9.9626 Tf 6.7248 1.4944 Td[(ex=2Vx.9Oursolutionworkssolongasthequantitysatises=r G+j2fCR+j2fL=af+jbf.10Thus,dependsonfrequency,andweexpressitintermsofrealandimaginarypartsasindicated.ThequantitiesV+andV)]TJ/F15 9.9626 Tf 10.0235 1.4944 Td[(areconstantsdeterminedbythesourceandphysicalconsiderations.Forexample,letthespatialoriginbethemiddleofthetransmissionlinemodelFigure6.2CircuitModelforaTransmissionLine.Becausethecircuitmodelcontainssimplecircuitelements,physicallypossiblesolutionsforvoltageamplitudecannotincreasewithdistancealongthetransmissionline.Expressingintermsofitsrealandimaginarypartsinoursolutionshowsthatsuchincreasesareamathematicalpossibility.Vx=V+e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(a+jbx+V)]TJ/F11 9.9626 Tf 6.7248 1.4944 Td[(ea+jbxThevoltagecannotincreasewithoutlimit;becauseafisalwayspositive,wemustsegregatethesolutionfornegativeandpositivex.Thersttermwillincreaseexponentiallyforx<0unlessV+=0inthisregion;asimilarresultappliestoV)]TJ/F15 9.9626 Tf 10.3503 1.4944 Td[(forx>0.Thesephysicalconstraintsgiveusacleanersolution.Vx=8<:V+e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(a+jbxifx>0V)]TJ/F11 9.9626 Tf 6.7248 1.4944 Td[(ea+jbxifx<0.11Thissolutionsuggeststhatvoltagesandcurrentstoowilldecreaseexponentiallyalongatransmissionline.Thespaceconstant,alsoknownastheattenuationconstant,isthedistanceoverwhichthevoltagedecreasesbyafactorof1 e.Itequalsthereciprocalofaf,whichdependsonfrequency,andisexpressedbymanufacturersinunitsofdB/m.Thepresenceoftheimaginarypartof,bf,alsoprovidesinsightintohowtransmissionlineswork.Becausethesolutionforx>0isproportionaltoe)]TJ/F7 6.9738 Tf 6.2267 0 Td[(jbx,weknowthatthevoltage'scomplexamplitudewillvarysinusoidallyinspace.Thecompletesolutionforthevoltagehastheformvx;t=ReV+e)]TJ/F7 6.9738 Tf 6.2267 0 Td[(axejft)]TJ/F10 6.9738 Tf 6.2266 0 Td[(bx.12Thecomplexexponentialportionhastheformofapropagatingwave.Ifwecouldtakeasnapshotofthevoltagetakeitspictureatt=t1,wewouldseeasinusoidallyvaryingwaveformalongthetransmissionline.Oneperiodofthisvariation,knownasthewavelength,equals=2 b.Ifweweretotakeasecondpictureatsomelatertimet=t2,wewouldalsoseeasinusoidalvoltage.Because2ft2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(bx=2ft1+t2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(t1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(bx=2ft1)]TJ/F11 9.9626 Tf 9.9626 0 Td[(bx)]TJ/F8 9.9626 Tf 11.1581 6.7398 Td[(2f bt2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(t1

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214CHAPTER6.INFORMATIONCOMMUNICATIONthesecondwaveformappearstobetherstone,butdelayedshiftedtotherightinspace.Thus,thevolt-ageappearedtomovetotherightwithaspeedequalto2f bassumingb>0.Wedenotethispropagationspeedbyc,anditequalsc=2f Imr G+j2fCR+j2fL.13Inthehigh-frequencyregionwherej2fLRandj2fCG,thequantityundertheradicalsimpliesto)]TJ/F8 9.9626 Tf 7.7487 0 Td[(42f2LC,andwendthepropagationspeedtobelimf!1c=1 q LC.14Fortypicalcoaxialcable,thispropagationspeedisafractionone-thirdtotwo-thirdsofthespeedoflight.Exercise6.1Solutiononp.272.Findthepropagationspeedintermsofphysicalparametersforboththecoaxialcableandtwistedpairexamples.Byusingthesecondofthetransmissionlineequation6.6,wecansolveforthecurrent'scomplexamplitude.Consideringthespatialregionx>0,forexample,wendthatd dxVx=)]TJ/F8 9.9626 Tf 9.4091 0 Td[(Vx=)]TJ/F1 9.9626 Tf 9.4091 11.0586 Td[(R+j2fLIxwhichmeansthattheratioofvoltageandcurrentcomplexamplitudesdoesnotdependondistance.Vx Ix=vuut R+j2fL G+j2fC=Z0.15ThequantityZ0isknownasthetransmissionline'scharacteristicimpedance.Notethatwhenthesignalfrequencyissucientlyhigh,thecharacteristicimpedanceisreal,whichmeansthetransmissionlineappearsresistiveinthishigh-frequencyregime.limf!1Z0=vuut L C.16Typicalvaluesforcharacteristicimpedanceare50and75.Arelatedtransmissionlineistheopticber.Here,theelectromagneticeldislight,anditpropagatesdownacylinderofglass.Inthissituation,wedon'thavetwoconductorsinfactwehavenoneandtheenergyispropagatinginwhatcorrespondstothedielectricmaterialofthecoaxialcable.Opticbercom-municationhasexactlythesamepropertiesasothertransmissionlines:Signalstrengthdecaysexponentiallyaccordingtotheber'sspaceconstantandpropagatesatsomespeedlessthanlightwouldinfreespace.FromtheencompassingviewofMaxwell'sequations,theonlydierenceistheelectromagneticsignal'sfre-quency.Becausenoelectricconductorsarepresentandtheberisprotectedbyanopaqueinsulator,opticbertransmissionisinterference-free.Exercise6.2Solutiononp.272.Fromtablesofphysicalconstants,ndthefrequencyofasinusoidinthemiddleofthevisiblelightrange.Comparethisfrequencywiththatofamid-frequencycabletelevisionsignal.Tosummarize,weusetransmissionlinesforhigh-frequencywirelinesignalcommunication.Inwirelinecommunication,wehaveadirect,physicalconnectionacircuitbetweentransmitterandreceiver.When

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215weselectthetransmissionlinecharacteristicsandthetransmissionfrequencysothatweoperateinthehigh-frequencyregime,signalsarenotlteredastheypropagatealongthetransmissionline:Thecharacteristicimpedanceisreal-valuedthetranmissionline'sequivalentimpedanceisaresistorandallthesignal'scomponentsatvariousfrequenciespropagateatthesamespeed.Transmittedsignalamplitudedoesdecayexponentiallyalongthetransmissionline.Notethatinthehigh-frequencyregimethatthespaceconstantisapproximatelyzero,whichmeanstheattenuationisquitesmall.Exercise6.3Solutiononp.272.Whatisthelimitingvalueofthespaceconstantinthehighfrequencyregime?6.4WirelessChannels5WirelesschannelsexploitthepredictionmadebyMaxwell'sequationthatelectromagneticeldspropagateinfreespacelikelight.Whenavoltageisappliedtoanantenna,itcreatesanelectromagneticeldthatpropagatesinalldirectionsalthoughantennageometryaectshowmuchpowerowsinanygivendirectionthatinduceselectriccurrentsinthereceiver'santenna.Antennageometrydetermineshowenergeticaeldavoltageofagivenfrequencycreates.Ingeneralterms,thedominantfactoristherelationoftheantenna'ssizetotheeld'swavelength.Thefundamentalequationrelatingfrequencyandwavelengthforapropagatingwaveisf=cThus,wavelengthandfrequencyareinverselyrelated:Highfrequencycorrespondstosmallwavelengths.Forexample,a1MHzelectromagneticeldhasawavelengthof300m.Antennashavingasizeordistancefromthegroundcomparabletothewavelengthradiateeldsmosteciently.Consequently,thelowerthefrequencythebiggertheantennamustbe.Becausemostinformationsignalsarebasebandsignals,havingspectralenergyatlowfrequencies,theymustbemodulatedtohigherfrequenciestobetransmittedoverwirelesschannels.Formostantenna-basedwirelesssystems,howthesignaldiminishesasthereceivermovesfurtherfromthetransmitterderivesbyconsideringhowradiatedpowerchangeswithdistancefromthetransmittingantenna.Anantennaradiatesagivenamountofpowerintofreespace,andideallythispowerpropagateswithoutlossinalldirections.Consideringaspherecenteredatthetransmitter,thetotalpower,whichisfoundbyintegratingtheradiatedpoweroverthesurfaceofthesphere,mustbeconstantregardlessofthesphere'sradius.Thisrequirementresultsfromtheconservationofenergy.Thus,ifpdrepresentsthepowerintegratedwithrespecttodirectionatadistancedfromtheantenna,thetotalpowerwillbepd4d2.Forthisquantitytobeaconstant,wemusthavepd/1 d2whichmeansthatthereceivedsignalamplitudeARmustbeproportionaltothetransmitter'samplitudeATandinverselyrelatedtodistancefromthetransmitter.AR=kAT d.17forsomevalueoftheconstantk.Thus,thefurtherfromthetransmitterthereceiverislocated,theweakerthereceivedsignal.Whereastheattenuationfoundinwirelinechannelscanbecontrolledbyphysicalparametersandchoiceoftransmissionfrequency,theinverse-distanceattenuationfoundinwirelesschannelspersistsacrossallfrequencies. 5Thiscontentisavailableonlineat.

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216CHAPTER6.INFORMATIONCOMMUNICATIONExercise6.4Solutiononp.272.Whydon'tsignalsattenuateaccordingtotheinverse-squarelawinaconductor?Whatisthedierencebetweenthewirelineandwirelesscases?Thespeedofpropagationisgovernedbythedielectricconstant0andmagneticpermeability0offreespace.c=1 p 00=3108m/s.18Knownfamiliarlyasthespeedoflight,itsetsanupperlimitonhowfastsignalscanpropagatefromoneplacetoanother.Becausesignalstravelatanitespeed,areceiversensesatransmittedsignalonlyafteratimedelaydirectlyrelatedtothepropagationspeed:t=d cAtthespeedoflight,asignaltravelsacrosstheUnitedStatesin16ms,areasonablysmalltimedelay.IfalosslesszerospaceconstantcoaxialcableconnectedtheEastandWestcoasts,thisdelaywouldbetwotothreetimeslongerbecauseoftheslowerpropagationspeed.6.5Line-of-SightTransmission6Long-distancetransmissionovereitherkindofchannelencountersattenuationproblems.LossesinwirelinechannelsareexploredintheCircuitModelsmoduleSection6.3,whererepeaterscanextendthedistancebetweentransmitterandreceiverbeyondwhatpassivelossesthewirelinechannelimposes.Inwirelesschannels,notonlydoesradiationlossoccurp.215,butalsooneantennamaynot"see"anotherbecauseoftheearth'scurvature. Figure6.3:Twoantennaeareshowneachhavingthesameheight.Line-of-sighttransmissionmeansthetransmittingandreceivingantennaecan"see"eachotherasshown.Themaximumdistanceatwhichtheycanseeeachother,dLOS,occurswhenthesightinglinejustgrazestheearth'ssurface. Attheusualradiofrequencies,propagatingelectromagneticenergydoesnotfollowtheearth'ssurface.Line-of-sightcommunicationhasthetransmitterandreceiverantennasinvisualcontactwitheachother. 6Thiscontentisavailableonlineat.

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217Assumingbothantennashaveheighthabovetheearth'ssurface,maximumline-of-sightdistanceisdLOS=2p 2hR+h22p 2Rh.19whereRistheearth'sradius6:38106m.Exercise6.5Solutiononp.272.Derivetheexpressionofline-of-sightdistanceusingonlythePythagoreanTheorem.Generalizeittothecasewheretheantennashavedierentheightsasisthecasewithcommercialradioandcellulartelephone.Whatistherangeofcellulartelephonewherethehandsetantennahasessentiallyzeroheight?Exercise6.6Solutiononp.273.Canyouimagineasituationwhereinglobalwirelesscommunicationispossiblewithonlyonetransmittingantenna?Inparticular,whathappenstowavelengthwhencarrierfrequencydecreases?Usinga100mantennawouldprovideline-of-sighttransmissionoveradistanceof71.4km.Usingsuchverytallantennaswouldprovidewirelesscommunicationwithinatownorbetweencloselyspacedpopulationcenters.Consequently,networksofantennassprinklethecountrysideeachlocatedonthehighesthillpossibletoprovidelong-distancewirelesscommunications:Eachantennareceivesenergyfromoneantennaandretransmitstoanother.Thiskindofnetworkisknownasarelaynetwork.6.6TheIonosphereandCommunications7Ifwewerelimitedtoline-of-sightcommunications,longdistancewirelesscommunication,likeship-to-shorecommunication,wouldbeimpossible.Attheturnofthecentury,Marconi,theinventorofwirelesstelegraphy,boldlytriedsuchlongdistancecommunicationwithoutanyevidenceeitherempiricalortheoreticalthatitwaspossible.Whentheexperimentworked,butonlyatnight,physicistsscrambledtodeterminewhyusingMaxwell'sequations,ofcourse.ItwasOliverHeaviside,amathematicalphysicistwithstrongengineeringinterests,whohypothesizedthataninvisibleelectromagnetic"mirror"surroundedtheearth.Whathemeantwasthatatopticalfrequenciesandothersasitturnedout,themirrorwastransparent,butatthefrequenciesMarconiused,itreectedelectromagneticradiationbacktoearth.Hehadpredictedtheexistenceoftheionosphere,aplasmathatencompassestheearthataltitudeshibetween80and180kmthatreactstosolarradiation:ItbecomestransparentatMarconi'sfrequenciesduringtheday,butbecomesamirroratnightwhensolarradiationdiminishes.Themaximumdistancealongtheearth'ssurfacethatcanbereachedbyasingleionosphericreectionis2RarccosR R+hi,whichrangesbetween2,010and3,000kmwhenwesubstituteminimumandmaximumionosphericaltitudes.ThisdistancedoesnotspantheUnitedStatesorcrosstheAtlantic;fortransatlanticcommunication,atleasttworeectionswouldberequired.Thecommunicationdelayencounteredwithasinglereectioninthischannelis2p 2Rhi+hi2 c,whichrangesbetween6.8and10ms,againasmalltimeinterval.6.7CommunicationwithSatellites8Globalwirelesscommunicationreliesonsatellites.Here,groundstationstransmittoorbitingsatellitesthatamplifythesignalandretransmititbacktoearth.Satelliteswillmoveacrosstheskyunlesstheyareingeosynchronousorbits,wherethetimeforonerevolutionabouttheequatorexactlymatchestheearth'srotationtimeofoneday.TVsatelliteswouldrequirethehomeownertocontinuallyadjusthisorherantennaifthesatelliteweren'tingeosynchronousorbit.Newton'sequationsappliedtoorbitingbodiespredictthat 7Thiscontentisavailableonlineat.8Thiscontentisavailableonlineat.

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218CHAPTER6.INFORMATIONCOMMUNICATIONthetimeTforoneorbitisrelatedtodistancefromtheearth'scenterRasR=3r GMT2 42.20whereGisthegravitationalconstantandMtheearth'smass.CalculationsyieldR=42200km,whichcorrespondstoanaltitudeof35700km.Thisaltitudegreatlyexceedsthatoftheionosphere,requiringsatellitetransmitterstousefrequenciesthatpassthroughit.Ofgreatimportanceinsatellitecommunicationsisthetransmissiondelay.Thetimeforelectromagneticeldstopropagatetoageosynchronoussatelliteandreturnis0.24s,asignicantdelay.Exercise6.7Solutiononp.273.Inadditiontodelay,thepropagationattenuationencounteredinsatellitecommunicationfarex-ceedswhatoccursinionospheric-mirrorbasedcommunication.Calculatetheattenuationincurredbyradiationgoingtothesatelliteone-waylosswiththatencounteredbyMarconitotalgoingupanddown.Notethattheattenuationcalculationintheionosphericcase,assumingtheionosphereactslikeaperfectmirror,isnotastraightforwardapplicationofthepropagationlossformulap.215.6.8NoiseandInterference9Wehavementionedthatcommunicationsare,tovaryingdegrees,subjecttointerferenceandnoise.It'stimetobemorepreciseaboutwhatthesequantitiesareandhowtheydier.Interferencerepresentsman-madesignals.Telephonelinesaresubjecttopower-lineinterferenceintheUnitedStatesadistorted60Hzsinusoid.Cellulartelephonechannelsaresubjecttoadjacent-cellphoneconversationsusingthesamesignalfrequency.Theproblemwithsuchinterferenceisthatitoccupiesthesamefrequencybandasthedesiredcommunicationsignal,andhasasimilarstructure.Exercise6.8Solutiononp.273.Supposeinterferenceoccupiedadierentfrequencyband;howwouldthereceiverremoveit?Weusethenotationittorepresentinterference.Becauseinterferencehasman-madestructure,wecanwriteanexplicitexpressionforitthatmaycontainsomeunknownaspectshowlargeitis,forexample.Noisesignalshavelittlestructureandarisefrombothhumanandnaturalsources.Satellitechannelsaresubjecttodeepspacenoisearisingfromelectromagneticradiationpervasiveinthegalaxy.Thermalnoiseplaguesallelectroniccircuitsthatcontainresistors.Thus,inreceivingsmallamplitudesignals,receiveramplierswillmostcertainlyaddnoiseastheyboostthesignal'samplitude.Allchannelsaresubjecttonoise,andweneedawayofdescribingsuchsignalsdespitethefactwecan'twriteaformulaforthenoisesignallikewecanforinterference.Themostwidelyusednoisemodeliswhitenoise.Itisdenedentirelybyitsfrequency-domaincharacteristics.Whitenoisehasconstantpoweratallfrequencies.Ateachfrequency,thephaseofthenoisespectrumistotallyuncertain:Itcanbeanyvalueinbetween0and2,anditsvalueatanyfrequencyisunrelatedtothephaseatanyotherfrequency.Whennoisesignalsarisingfromtwodierentsourcesadd,theresultantnoisesignalhasapowerequaltothesumofthecomponentpowers.Becauseoftheemphasishereonfrequency-domainpower,weareleadtodenethepowerspectrum.BecauseofParseval'sTheorem10,wedenethepowerspectrumPsfofanon-noisesignalsttobethemagnitude-squaredofitsFouriertransform.PsfjSfj2.21 9Thiscontentisavailableonlineat.10"Parseval'sTheorem",

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219Integratingthepowerspectrumoveranyrangeoffrequenciesequalsthepowerthesignalcontainsinthatband.Becausesignalsmusthavenegativefrequencycomponentsthatmirrorpositivefrequencyones,weroutinelycalculatethepowerinaspectralbandastheintegraloverpositivefrequenciesmultipliedbytwo.Powerin[f1;f2]=2Zf2f1Psfdf.22Usingthenotationnttorepresentanoisesignal'swaveform,wedenenoiseintermsofitspowerspectrum.Forwhitenoise,thepowerspectrumequalstheconstantN0 2.Withthisdenition,thepowerinafrequencybandequalsN0f2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(f1.Whenwepassasignalthroughalinear,time-invariantsystem,theoutput'sspectrumequalstheproductp.131ofthesystem'sfrequencyresponseandtheinput'sspectrum.Thus,thepowerspectrumofthesystem'soutputisgivenbyPyf=jHfj2Pxf.23Thisresultappliestonoisesignalsaswell.Whenwepasswhitenoisethroughalter,theoutputisalsoanoisesignalbutwithpowerspectrumjHfj2N0 2.6.9ChannelModels11Bothwirelineandwirelesschannelssharecharacteristics,allowingustouseacommonmodelforhowthechannelaectstransmittedsignals.Thetransmittedsignalisusuallynotlteredbythechannel.Thesignalcanbeattenuated.Thesignalpropagatesthroughthechannelataspeedequaltoorlessthanthespeedoflight,whichmeansthatthechanneldelaysthetransmission.Thechannelmayintroduceadditiveinterferenceand/ornoise.Lettingrepresenttheattenuationintroducedbythechannel,thereceiver'sinputsignalisrelatedtothetransmittedonebyrt=xt)]TJ/F11 9.9626 Tf 9.9626 0 Td[(+it+nt.24ThisexpressioncorrespondstothesystemmodelforthechannelshowninFigure6.4.Inthisbook,weshallassumethatthenoiseiswhite. Figure6.4:ThechannelcomponentofthefundamentalmodelofcommunicationFigure1.4:Fun-damentalmodelofcommunicationhasthedepictedform.Theattenuationisduetopropagationloss.AddingtheinterferenceandnoiseisjustiedbythelinearitypropertyofMaxwell'sequations. 11Thiscontentisavailableonlineat.

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220CHAPTER6.INFORMATIONCOMMUNICATIONExercise6.9Solutiononp.273.Isthismodelforthechannellinear?Asexpected,thesignalthatemergesfromthechanneliscorrupted,butdoescontainthetransmittedsignal.Communicationsystemdesignbeginswithdetailingthechannelmodel,thendevelopingthetransmitterandreceiverthatbestcompensateforthechannel'scorruptingbehavior.Wecharacterizethechannel'squalitybythesignal-to-interferenceratioSIRandthesignal-to-noiseratioSNR.Theratiosarecomputedaccordingtotherelativepowerofeachwithinthetransmittedsignal'sbandwidth.Assumingthesignalxt'sspectrumspansthefrequencyinterval[fl;fu],theseratioscanbeexpressedintermsofpowerspectra.SIR=22R10Pxfdf 2RfuflPifdf.25SNR=22R10Pxfdf N0fu)]TJ/F11 9.9626 Tf 9.9626 0 Td[(fl.26Inmostcases,theinterferenceandnoisepowersdonotvaryforagivenreceiver.Variationsinsignal-to-interferenceandsignal-to-noiseratiosarisefromtheattenuationbecauseoftransmitter-to-receiverdistancevariations.6.10BasebandCommunication12Weuseanalogcommunicationtechniquesforanalogmessagesignals,likemusic,speech,andtelevision.Transmissionandreceptionofanalogsignalsusinganalogresultsinaninherentlynoisyreceivedsignalassumingthechanneladdsnoise,whichitalmostcertainlydoes.Thesimplestformofanalogcommunicationisbasebandcommunication.PointofInterest:Weuseanalogcommunicationtechniquesforanalogmessagesignals,likemusic,speech,andtelevision.Transmissionandreceptionofanalogsignalsusinganalogresultsinaninherentlynoisyreceivedsignalassumingthechanneladdsnoise,whichitalmostcertainlydoes.Here,thetransmittedsignalequalsthemessagetimesatransmittergain.xt=Gmt.27Anexample,whichissomewhatoutofdate,isthewirelinetelephonesystem.Youdon'tusebasebandcommunicationinwirelesssystemssimplybecauselow-frequencysignalsdonotradiatewell.Thereceiverinabasebandsystemcan'tdomuchmorethanlterthereceivedsignaltoremoveout-of-bandnoiseinterferenceissmallinwirelinechannels.AssumingthesignaloccupiesabandwidthofWHzthesignal'sspectrumextendsfromzerotoW,thereceiverappliesalowpasslterhavingthesamebandwidth,asshowninFigure6.5. Figure6.5:Thereceiverforbasebandcommunicationsystemsisquitesimple:alowpasslterhavingthesamebandwidthasthesignal. 12Thiscontentisavailableonlineat.

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221Weusethesignal-to-noiseratioofthereceiver'soutput^mttoevaluateanyanalog-messagecommu-nicationsystem.AssumethatthechannelintroducesanattenuationandwhitenoiseofspectralheightN0 2.Thelterdoesnotaectthesignalcomponentweassumeitsgainisunitybutdoeslterthenoise,removingfrequencycomponentsaboveWHz.Inthelter'soutput,thereceivedsignalpowerequals2G2powermandthenoisepowerN0W,whichgivesasignal-to-noiseratioofSNRbaseband=2G2powerm N0W.28ThesignalpowerpowermwillbeproportionaltothebandwidthW;thus,inbasebandcommunicationthesignal-to-noiseratiovariesonlywithtransmittergainandchannelattenuationandnoiselevel.6.11ModulatedCommunication13Especiallyforwirelesschannels,likecommercialradioandtelevision,butalsoforwirelinesystemslikecabletelevision,ananalogmessagesignalmustbemodulated:Thetransmittedsignal'sspectrumoccursatmuchhigherfrequenciesthanthoseoccupiedbythesignal.PointofInterest:Weuseanalogcommunicationtechniquesforanalogmessagesignals,likemusic,speech,andtelevision.Transmissionandreceptionofanalogsignalsusinganalogresultsinaninherentlynoisyreceivedsignalassumingthechanneladdsnoise,whichitalmostcertainlydoes.Thekeyideaofmodulationistoaecttheamplitude,frequencyorphaseofwhatisknownasthecarriersinusoid.FrequencymodulationFMandlessfrequentlyusedphasemodulationPMarenotdiscussedhere;wefocusonamplitudemodulationAM.Theamplitudemodulatedmessagesignalhastheformxt=Ac+mtcosfct.29wherefcisthecarrierfrequencyandActhecarrieramplitude.Also,thesignal'samplitudeisassumedtobelessthanone:jmtj<1.FromourpreviousexposuretoamplitudemodulationseetheFourierTransformexampleExample4.5,weknowthatthetransmittedsignal'sspectrumoccupiesthefrequencyrange[fc)]TJ/F11 9.9626 Tf 9.9626 0 Td[(W;fc+W],assumingthesignal'sbandwidthisWHzseethegureFigure6.6.Thecarrierfrequencyisusuallymuchlargerthanthesignal'shighestfrequency:fcW,whichmeansthatthetransmitterantennaandcarrierfrequencyarechosenjointlyduringthedesignprocess. Figure6.6:TheAMcoherentreceiveralongwiththespectraofkeysignalsisshownforthecaseofatriangular-shapedsignalspectrum.Thedashedlineindicatesthewhitenoiselevel.Notethatthelters'characteristicscutofrequencyandcenterfrequencyforthebandpassltermustbematchtothemodulationandmessageparameters. 13Thiscontentisavailableonlineat.

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222CHAPTER6.INFORMATIONCOMMUNICATIONIgnoringtheattenuationandnoiseintroducedbythechannelforthemoment,receptionofanamplitudemodulatedsignalisquiteeasyseeProblem4.17.Theso-calledcoherentreceivermultipliestheinputsignalbyasinusoidandlowpass-lterstheresultFigure6.6.^mt=LPFxtcosfct=LPF)]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(Ac+mtcos2fct.30Becauseofourtrigonometricidentities,weknowthatcos2fct=1 2+cos2fct.31Atthispoint,themessagesignalismultipliedbyaconstantandasinusoidattwicethecarrierfrequency.Multiplicationbytheconstanttermreturnsthemessagesignaltobasebandwherewewantittobe!whilemultiplicationbythedouble-frequencytermyieldsaveryhighfrequencysignal.Thelowpasslterremovesthishigh-frequencysignal,leavingonlythebasebandsignal.Thus,thereceivedsignalis^mt=Ac 2+mt.32Exercise6.10Solutiononp.273.Thisderivationreliessolelyonthetimedomain;derivethesameresultinthefrequencydomain.Youwon'tneedthetrigonometricidentitywiththisapproach.Becauseitissoeasytoremovetheconstanttermbyelectricalmeansweinsertacapacitorinserieswiththereceiver'soutputwetypicallyignoreitandconcentrateonthesignalportionofthereceiver'soutputwhencalculatingsignal-to-noiseratio.6.12Signal-to-NoiseRatioofanAmplitude-ModulatedSignal14Whenweconsiderthemuchmorerealisticsituationwhenwehaveachannelthatintroducesattenuationandnoise,wecanmakeuseofthejust-describedreceiver'slinearnaturetodirectlyderivethereceiver'soutput.Theattenuationaectstheoutputinthesamewayasthetransmittedsignal:Itscalestheoutputsignalbythesameamount.Thewhitenoise,ontheotherhand,shouldbelteredfromthereceivedsignalbeforedemodulation.Wemustthusinsertabandpasslterhavingbandwidth2Wandcenterfrequencyfc:Thislterhasnoeectonthereceivedsignal-relatedcomponent,butdoesremoveout-of-bandnoisepower.Asshowninthetriangular-shapedsignalspectrumFigure6.6,weapplycoherentreceivertothislteredsignal,withtheresultthatthedemodulatedoutputcontainsnoisethatcannotberemoved:Itliesinthesamespectralbandasthesignal.Aswederivethesignal-to-noiseratiointhedemodulatedsignal,let'salsocalculatethesignal-to-noiseratioofthebandpasslter'soutput~rt.Thesignalcomponentof~rtequalsAcmtcosfct.Thissignal'sFouriertransformequalsAc 2Mf+fc+Mf)]TJ/F11 9.9626 Tf 9.9626 0 Td[(fc.33makingthepowerspectrum,2Ac2 4jMf+fcj2+jMf)]TJ/F11 9.9626 Tf 9.9626 0 Td[(fcj2.34Exercise6.11Solutiononp.273.Ifyoucalculatethemagnitude-squaredoftherstequation,youdon'tobtainthesecondunlessyoumakeanassumption.Whatisit? 14Thiscontentisavailableonlineat.

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223Thus,thetotalsignal-relatedpowerin~rtis2Ac2 2powerm.Thenoisepowerequalstheintegralofthenoisepowerspectrum;becausethepowerspectrumisconstantoverthetransmissionband,thisintegralequalsthenoiseamplitudeN0timesthelter'sbandwidth2W.Theso-calledreceivedsignal-to-noiseratiothesignal-to-noiseratioafterthederigeurfront-endbandpasslterandbeforedemodulationequalsSNRr=2Ac2powerm 4N0W.35Thedemodulatedsignal^mt=Acmt 2+noutt.Clearly,thesignalpowerequals2Ac2powerm 4.Todeterminethenoisepower,wemustunderstandhowthecoherentdemodulatoraectsthebandpassnoisefoundin~rt.Becauseweareconcernedwithnoise,wemustdealwiththepowerspectrumsincewedon'thavetheFouriertransformavailabletous.LettingPfdenotethepowerspectrumof~rt'snoisecomponent,thepowerspectrumaftermultiplicationbythecarrierhastheformPf+fc+Pf)]TJ/F11 9.9626 Tf 9.9626 0 Td[(fc 4.36Thedelayandadvanceinfrequencyindicatedhereresultsintwospectralnoisebandsfallinginthelow-frequencyregionoflowpasslter'spassband.Thus,thetotalnoisepowerinthislter'soutputequals)]TJ/F8 9.9626 Tf 4.5662 -8.0698 Td[(2N0 2W21 4=N0W 2.Thesignal-to-noiseratioofthereceiver'soutputthusequalsSNR^m=2Ac2powerm 2N0W=2SNRr.37Let'sbreakdownthecomponentsofthissignal-to-noiseratiotobetterappreciatehowthechannelandthetransmitterparametersaectcommunicationsperformance.Betterperformance,asmeasuredbytheSNR,occursasitincreases.MoretransmitterpowerincreasingAcincreasesthesignal-to-noiseratioproportionally.ThecarrierfrequencyfchasnoeectonSNR,butwehaveassumedthatfcW.ThesignalbandwidthWentersthesignal-to-noiseexpressionintwoplaces:implicitlythroughthesignalpowerandexplicitlyintheexpression'sdenominator.Ifthesignalspectrumhadaconstantamplitudeasweincreasedthebandwidth,signalpowerwouldincreaseproportionally.Ontheotherhand,ourtransmitterenforcedthecriterionthatsignalamplitudewasconstantSection6.7.Signalamplitudeessentiallyequalstheintegralofthemagnitudeofthesignal'sspectrum.Note:Thisresultisn'texact,butwedoknowthatm=R1Mfdf.Enforcingthesignalamplitudespecicationmeansthatasthesignal'sbandwidthincreaseswemustde-creasethespectralamplitude,withtheresultthatthesignalpowerremainsconstant.Thus,increasingsignalbandwidthdoesindeeddecreasethesignal-to-noiseratioofthereceiver'soutput.Increasingchannelattenuationmovingthereceiverfartherfromthetransmitterdecreasesthesignal-to-noiseratioasthesquare.Thus,signal-to-noiseratiodecreasesasdistance-squaredbetweentransmitterandreceiver.Noiseaddedbythechanneladverselyaectsthesignal-to-noiseratio.Insummary,amplitudemodulationprovidesaneectivemeansforsendingabandlimitedsignalfromoneplacetoanother.Forwirelinechannels,usingbasebandoramplitudemodulationmakeslittledierenceintermsofsignal-to-noiseratio.Forwirelesschannels,amplitudemodulationistheonlyalternative.TheoneAMparameterthatdoesnotaectsignal-to-noiseratioisthecarrierfrequencyfc:Wecanchooseanyvaluewewantsolongasthetransmitterandreceiverusethesamevalue.However,supposesomeoneelsewantstouseAMandchoosesthesamecarrierfrequency.Thetworesultingtransmissionswilladd,andbothreceiverswillproducethesumofthetwosignals.Whatweclearlyneedtodoistalktotheotherparty,andagreetouseseparatecarrierfrequencies.Asmoreandmoreuserswishtouseradio,weneedaforumforagreeingon

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224CHAPTER6.INFORMATIONCOMMUNICATIONcarrierfrequenciesandonsignalbandwidth.Onearth,thisforumisthegovernment.IntheUnitedStates,theFederalCommunicationsCommissionFCCstrictlycontrolstheuseoftheelectromagneticspectrumforcommunications.SeparatefrequencybandsareallocatedforcommercialAM,FM,cellulartelephonetheanalogversionofwhichisAM,shortwavealsoAM,andsatellitecommunications.Exercise6.12Solutiononp.273.Supposeallusersagreetousethesamesignalbandwidth.Howcloselycanthecarrierfrequenciesbewhileavoidingcommunicationscrosstalk?WhatisthesignalbandwidthforcommercialAM?Howdoesthisbandwidthcomparetothespeechbandwidth?6.13DigitalCommunication15Eective,error-freetransmissionofasequenceofbitsabitstreamfb;b;:::gisthegoalhere.Wefoundthatanalogschemes,asrepresentedbyamplitudemodulation,alwaysyieldareceivedsignalcontainingnoiseaswellasthemessagesignalwhenthechanneladdsnoise.Digitalcommunicationschemesareverydierent.Oncewedecidehowtorepresentbitsbyanalogsignalsthatcanbetransmittedoverwirelinelikeacomputernetworkorwirelesslikedigitalcellulartelephonechannels,wewillthendevelopawayoftackingoncommunicationbitstothemessagebitsthatwillreducechannel-inducederrorsgreatly.Intheory,digitalcommunicationerrorscanbezero,eventhoughthechanneladdsnoise!Werepresentabitbyassociatingoneoftwospecicanalogsignalswiththebit'svalue.Thus,ifbn=0,wetransmitthesignals0t;ifbn=1,sends1t.Thesetwosignalscomprisethesignalsetfordigitalcommunicationandaredesignedwiththechannelandbitstreaminmind.Invirtuallyeverycase,thesesignalshaveanitedurationTcommontobothsignals;thisdurationisknownasthebitinterval.Exactlywhatsignalsweuseultimatelyaectshowwellthebitscanbereceived.Interestingly,basebandandmodulatedsignalsetscanyieldthesameperformance.Otherconsiderationsdeterminehowsignalsetchoiceaectsdigitalcommunicationperformance.Exercise6.13Solutiononp.273.Whatistheexpressionforthesignalarisingfromadigitaltransmittersendingthebitstreambn,n=f:::;)]TJ/F8 9.9626 Tf 7.7488 0 Td[(1;0;1;:::gusingthesignalsets0t,s1t,eachsignalofwhichhasdurationT?6.14BinaryPhaseShiftKeying16AcommonlyusedexampleofasignalsetconsistsofpulsesthatarenegativesofeachotherFigure6.7.s0t=ApTts1t=)]TJ/F8 9.9626 Tf 9.4091 0 Td[(ApTt.38 Figure6.7 15Thiscontentisavailableonlineat.16Thiscontentisavailableonlineat.

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225Here,wehaveabasebandsignalsetsuitableforwirelinetransmission.Theentirebitstreambnisrepresentedbyasequenceofthesesignals.Mathematically,thetransmittedsignalhastheformxt=Xn)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1bnApTt)]TJ/F11 9.9626 Tf 9.9626 0 Td[(nT.39andgraphicallyFigure6.8showswhatatypicaltransmittedsignalmightbe. a bFigure6.8:Theupperplotshowshowabasebandsignalsetfortransmittingthebitsequence0110.Theloweroneshowsanamplitude-modulatedvariantsuitableforwirelesschannels. ThiswayofrepresentingabitstreamchangingthebitchangesthesignofthetransmittedsignalisknownasbinaryphaseshiftkeyingandabbreviatedBPSK.Thenamecomesfromconciselyexpressingthispopularwayofcommunicatingdigitalinformation.Theword"binary"isclearenoughonebinary-valuedquantityistransmittedduringabitinterval.Changingthesignofsinusoidamountstochangingshiftingthephasebyalthoughwedon'thaveasinusoidyet.Theword"keying"reectsbacktotherstelectricalcommunicationsystem,whichhappenedtobedigitalaswell:thetelegraph.ThedatarateRofadigitalcommunicationsystemishowfrequentlyaninformationbitistransmitted.Inthisexampleitequalsthereciprocalofthebitinterval:R=1 T.Thus,fora1Mbpsmegabitpersecondtransmission,wemusthaveT=1s.Thechoiceofsignalstorepresentbitvaluesisarbitrarytosomedegree.Clearly,wedonotwanttochoosesignalsetmemberstobethesame;wecouldn'tdistinguishbitsifwedidso.Wecouldalsohavemadethenegative-amplitudepulserepresenta0andthepositiveonea1.Thischoiceisindeedarbitraryandwillhavenoeectonperformanceassumingthereceiverknowswhichsignalrepresentswhichbit.Asinallcommunicationsystems,wedesigntransmitterandreceivertogether.Asimplesignalsetforbothwirelessandwirelinechannelsamountstoamplitudemodulatingabasebandsignalsetmoreappropriateforawirelinechannelbyacarrierhavingafrequencyharmonicwiththebitinterval.s0t=ApTtsin2kt Ts1t=)]TJ/F1 9.9626 Tf 9.4091 14.0474 Td[(ApTtsin2kt T.40

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226CHAPTER6.INFORMATIONCOMMUNICATION Figure6.9 Exercise6.14Solutiononp.273.Whatisthevalueofkinthisexample?ThissignalsetisalsoknownasaBPSKsignalset.We'llshowlaterthatindeedbothsignalsetsprovideidenticalperformancelevelswhenthesignal-to-noiseratiosareequal.Exercise6.15Solutiononp.273.Writeaformula,inthestyleofthebasebandsignalset,forthetransmittedsignalasshownintheplotofthebasebandsignalset17thatemergeswhenweusethismodulatedsignal.Whatisthetransmissionbandwidthofthesesignalsets?Weneedonlyconsiderthebasebandversionasthesecondisanamplitude-modulatedversionoftherst.Thebandwidthisdeterminedbythebitsequence.Ifthebitsequenceisconstantalways0oralways1thetransmittedsignalisaconstant,whichhaszerobandwidth.Theworst-casebandwidthconsumingbitsequenceisthealternatingoneshowninFigure6.10.Inthiscase,thetransmittedsignalisasquarewavehavingaperiodof2T. Figure6.10:Hereweshowthetransmittedwaveformcorrespondingtoanalternatingbitsequence. FromourworkinFourierseries,weknowthatthissignal'sspectrumcontainsodd-harmonicsofthefundamental,whichhereequals1 2T.Thus,strictlyspeaking,thesignal'sbandwidthisinnite.Inpracticalterms,weusethe90%-powerbandwidthtoassesstheeectiverangeoffrequenciesconsumedbythesignal.Therstandthirdharmonicscontainthatfractionofthetotalpower,meaningthattheeectivebandwidthofourbasebandsignalis3 2Tor,expressingthisquantityintermsofthedatarate,3R 2.Thus,adigitalcommunicationssignalrequiresmorebandwidththanthedatarate:a1Mbpsbasebandsystemrequiresabandwidthofatleast1.5MHz.Listencarefullywhensomeonedescribesthetransmissionbandwidthofdigitalcommunicationsystems:Didtheysay"megabits"or"megahertz"?Exercise6.16Solutiononp.274.Showthatindeedtherstandthirdharmonicscontain90%ofthetransmittedpower.Ifthereceiverusesafront-endlterofbandwidth3 2T,whatisthetotalharmonicdistortionofthereceivedsignal?Exercise6.17Solutiononp.274.Whatisthe90%transmissionbandwidthofthemodulatedsignalset? 17"SignalSets",Figure2

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2276.15FrequencyShiftKeying18Infrequency-shiftkeyingFSK,thebitaectsthefrequencyofacarriersinusoid.s0t=ApTtsinf0ts1t=ApTtsinf1t.41 Figure6.11 Thefrequenciesf0,f1areusuallyharmonicallyrelatedtothebitinterval.Inthedepictedexample,f0=3 Tandf1=4 T.AscanbeseenfromthetransmittedsignalforourexamplebitstreamFigure6.12,thetransitionsatbitintervalboundariesaresmootherthanthoseofBPSK. Figure6.12:ThisplotshowstheFSKwaveformforsamebitstreamusedintheBPSKexampleFigure6.8. Todeterminethebandwidthrequiredbythissignalset,weagainconsiderthealternatingbitstream.Thinkofitastwosignalsaddedtogether:Therstcomprisedofthesignals0t,thezerosignal,s0t,zero,etc.,andthesecondhavingthesamestructurebutinterleavedwiththerstandcontainings1tFigure6.13.Eachcomponentcanbethoughtofasaxed-frequencysinusoidmultipliedbyasquarewaveofperiod2Tthatalternatesbetweenoneandzero.ThisbasebandsquarewavehasthesameFourierspectrumasourBPSKexample,butwiththeadditionoftheconstanttermc0.Thisquantity'spresencechangesthenumberofFourierseriestermsrequiredforthe90%bandwidth:Nowweneedonlyincludethezeroandrstharmonicstoachieveit.Thebandwidththusequals,withf0k0,thebandwidthequalsk1+)]TJ/F10 6.9738 Tf 6.2267 0 Td[(k0+1 T.Ifthedierencebetweenharmonicnumbersis1,thentheFSKbandwidthis 18Thiscontentisavailableonlineat.

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228CHAPTER6.INFORMATIONCOMMUNICATION Figure6.13:ThedepicteddecompositionoftheFSK-modulatedalternatingbitstreamintoitsfrequencycomponentssimpliesthecalculationofitsbandwidth. smallerthantheBPSKbandwidth.Ifthedierenceis2,thebandwidthsareequalandlargerdierencesproduceatransmissionbandwidthlargerthanthatresultingfromusingaBPSKsignalset.6.16DigitalCommunicationReceivers19Thereceiverinterestedinthetransmittedbitstreammustperformtwotaskswhenreceivedwaveformrtbegins.Itmustdeterminewhenbitboundariesoccur:Thereceiverneedstosynchronizewiththetransmittedsignal.Becausetransmitterandreceiveraredesignedinconcert,bothusethesamevalueforthebitintervalT.Synchronizationcanoccurbecausethetransmitterbeginssendingwithareferencebitsequence,knownasthepreamble.Thisreferencebitsequenceisusuallythealternatingsequenceasshowninthesquarewaveexample20andintheFSKexampleFigure6.13.Thereceiverknowswhatthepreamblebitsequenceisandusesittodeterminewhenbitboundariesoccur.Thisprocedureamountstowhatindigitalhardwareasself-clockingsignaling:Thereceiverofabitstreammustderivetheclockwhenbitboundariesoccurfromitsinputsignal.Becausethereceiverusuallydoesnotdeterminewhichbitwassentuntilsynchronizationoccurs,itdoesnotknowwhenduringthepreambleitobtainedsynchronization.Thetransmittersignalstheendofthepreamblebyswitchingtoasecondbitsequence.Thesecondpreamblephaseinformsthereceiverthatdatabitsareabouttocomeandthatthepreambleisalmostover.Oncesynchronizedanddatabitsaretransmitted,thereceivermustthendetermineeveryTsecondswhatbitwastransmittedduringthepreviousbitinterval.Wefocusonthisaspectofthedigitalreceiverbecausethisstrategyisalsousedinsynchronization.Thereceiverfordigitalcommunicationisknownasamatchedlter. 19Thiscontentisavailableonlineat.20"TransmissionBandwidth",Figure1

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229Optimalreceiverstructure Figure6.14:Theoptimalreceiverstructurefordigitalcommunicationfacedwithadditivewhitenoisechannelsisthedepictedmatchedlter.Thisreceiver,showninFigure6.14Optimalreceiverstructure,multipliesthereceivedsignalbyeachofthepossiblemembersofthetransmittersignalset,integratestheproductoverthebitinterval,andcomparestheresults.Whicheverpaththroughthereceiveryieldsthelargestvaluecorrespondstothereceiver'sdecisionastowhatbitwassentduringthepreviousbitinterval.Forthenextbitinterval,themultiplicationandintegrationbeginsagain,withthenextbitdecisionmadeattheendofthebitinterval.Mathematically,thereceivedvalueofbn,whichwelabel^bn,isgivenby^bn=argmaxiZn+1TnTrtsitdt.42Youmaynothaveseentheargmaxinotationbefore.maxifgyieldsthemaximumvalueofitsargumentwithrespecttotheindexi.argmaxiequalsthevalueoftheindexthatyieldsthemaximum.Notethattheprecisenumericalvalueoftheintegrator'soutputdoesnotmatter;whatdoesmatterisitsvaluerelativetotheotherintegrator'soutput.Let'sassumeaperfectchannelforthemoment:Thereceivedsignalequalsthetransmittedone.Ifbit0weresentusingthebasebandBPSKsignalset,theintegratoroutputswouldbeZn+1TnTrts0tdt=A2T.43Zn+1TnTrts1tdt=)]TJ/F1 9.9626 Tf 9.4091 8.0698 Td[()]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(A2TIfbit1weresent,Zn+1TnTrts0tdt=)]TJ/F1 9.9626 Tf 9.4091 8.0698 Td[()]TJ/F11 9.9626 Tf 4.5662 -8.0698 Td[(A2T.44Zn+1TnTrts1tdt=A2TExercise6.18Solutiononp.274.CanyoudevelopareceiverforBPSKsignalsetsthatrequiresonlyonemultiplier-integratorcombination?Exercise6.19Solutiononp.274.Whatisthecorrespondingresultwhentheamplitude-modulatedBPSKsignalsetisused?Clearly,thisreceiverwouldalwayschoosethebitcorrectly.Channelattenuationwouldnotaectthiscorrectness;itwouldonlymakethevaluessmaller,butallthatmattersiswhichislargest.

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230CHAPTER6.INFORMATIONCOMMUNICATION6.17DigitalCommunicationinthePresenceofNoise21Whenweincorporateadditivenoiseintoourchannelmodel,sothatrt=sit+nt,errorscancreepin.Ifthetransmittersentbit0usingaBPSKsignalsetSection6.14,theintegrators'outputsinthematchedlterreceiverFigure6.14:Optimalreceiverstructurewouldbe:Zn+1TnTrts0tdt=A2T+Zn+1TnTnts0tdt.45Zn+1TnTrts1tdt=)]TJ/F11 9.9626 Tf 7.7488 0 Td[(A2T+Zn+1TnTnts1tdtItisthequantitiescontainingthenoisetermsthatcauseerrorsinthereceiver'sdecision-makingprocess.Becausetheyinvolvenoise,thevaluesoftheseintegralsarerandomquantitiesdrawnfromsomeprobabilitydistributionthatvaryerraticallyfrombitintervaltobitinterval.Becausethenoisehaszeroaveragevalueandhasanequalamountofpowerinallfrequencybands,thevaluesoftheintegralswillhoveraboutzero.Whatisimportantishowmuchtheyvary.IfthenoiseissuchthatitsintegraltermismorenegativethanA2T,thenthereceiverwillmakeanerror,decidingthatthetransmittedzero-valuedbitwasindeedaone.Theprobabilitythatthissituationoccursdependsonthreefactors:SignalSetChoiceThedierencebetweenthesignal-dependenttermsintheintegrators'outputsequations.45deneshowlargethenoisetermmustbeforanincorrectreceiverdecisiontoresult.Whataectstheprobabilityofsucherrorsoccurringisthesquareofthisdierenceincomparisontothenoiseterm'svariability.ForourBPSKbasebandsignalset,thesignal-relatedvalueis42A4T2.VariabilityoftheNoiseTermWequantifyvariabilitybytheaveragevalueofitssquare,whichisessentiallythenoiseterm'spower.ThiscalculationisbestperformedinthefrequencydomainandequalspowerZn+1TnTnts0tdt!=Z1N0 2jS0fj2dfBecauseofParseval'sTheorem,weknowthatR1jS0fj2df=R1s02tdt,whichforthebase-bandsignalsetequalsA2T.Thus,thenoiseterm'spowerisN0A2T 2.ProbabilityDistributionoftheNoiseTermThevalueofthenoisetermsrelativetothesignaltermsandtheprobabilityoftheiroccurrencedirectlyaectthelikelihoodthatareceivererrorwilloccur.Forthewhitenoisewehavebeenconsidering,theunderlyingdistributionsareGaussian.Theprobabilitythereceivermakesanerroronanybittransmissionequals:pe=Q1 2s squareofsignaldierence noisetermpower!.46pe=Q0@s 22A2T N01AHereQistheintegralQx=1 p 2R1xe)]TJ/F27 6.9738 Tf 6.2266 7.6813 Td[(2 2d.Thisintegralhasnoclosedformexpression,butitcanbeaccuratelycomputed.AsFigure6.15illustrates,Qisadecreasing,verynonlinearfunction. 21Thiscontentisavailableonlineat.

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231 Figure6.15:ThefunctionQxisplottedinsemilogarithmiccoordinates.Notethatitdecreasesveryrapidlyforsmallincreasesinitsarguments.Forexample,whenxincreasesfrom4to5,Qxdecreasesbyafactorof100. ThetermA2Tequalstheenergyexpendedbythetransmitterinsendingthebit;welabelthistermEb.Wearriveataconciseexpressionfortheprobabilitythematchedlterreceivermakesabit-receptionerror.pe=Q0@s 22Eb N01A.47Figure6.16showshowthereceiver'serrorratevarieswiththesignal-to-noiseratio2Eb N0.Exercise6.20Solutiononp.274.DerivetheprobabilityoferrorexpressionforthemodulatedBPSKsignalset,andshowthatitsperformanceidenticallyequalsthatofthebasebandBPSKsignalset.6.18DigitalCommunicationSystemProperties22ResultsfromtheReceiverErrormoduleSection6.17revealsseveralpropertiesaboutdigitalcommunicationsystems.Asthereceivedsignalbecomesincreasinglynoisy,whetherduetoincreaseddistancefromthetransmit-tersmallerortoincreasednoiseinthechannellargerN0,theprobabilitythereceivermakesanerrorapproaches1=2.Insuchsituations,thereceiverperformsonlyslightlybetterthanthe"receiver"thatignoreswhatwastransmittedandmerelyguesseswhatbitwastransmitted.Consequently,itbecomesalmostimpossibletocommunicateinformationwhendigitalchannelsbecomenoisy. 22Thiscontentisavailableonlineat.

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232CHAPTER6.INFORMATIONCOMMUNICATION Figure6.16:Theprobabilitythatthematched-lterreceivermakesanerroronanybittransmissionisplottedagainstthesignal-to-noiseratioofthereceivedsignal.TheuppercurveshowstheperformanceoftheFSKsignalset,thelowerandthereforebetteronetheBPSKsignalset. Asthesignal-to-noiseratioincreases,performancegainssmallerprobabilityoferrorpecanbeeasilyobtained.Atasignal-to-noiseratioof12dB,theprobabilitythereceivermakesanerrorequals10)]TJ/F7 6.9738 Tf 6.2267 0 Td[(8.Inwords,oneoutofonehundredmillionbitswill,ontheaverage,beinerror.Oncethesignal-to-noiseratioexceedsabout5dB,theerrorprobabilitydecreasesdramatically.Adding1dBimprovementinsignal-to-noiseratiocanresultinafactorof10smallerpe.Signalsetchoicecanmakeasignicantdierenceinperformance.AllBPSKsignalsets,basebandormodulated,yieldthesameperformanceforthesamebitenergy.TheBPSKsignalsetdoesperformmuchbetterthantheFSKsignalsetoncethesignal-to-noiseratioexceedsabout5dB.Exercise6.21Solutiononp.274.DerivetheexpressionfortheprobabilityoferrorthatwouldresultiftheFSKsignalsetwereused.Thematched-lterreceiverprovidesimpressiveperformanceonceadequatesignal-to-noiseratiosoccur.Youmightwonderwhetheranotherreceivermightbebetter.Theansweristhatthematched-lterreceiverisoptimal:NootherreceivercanprovideasmallerprobabilityoferrorthanthematchedlterregardlessoftheSNR.Furthermore,nosignalsetcanprovidebetterperformancethantheBPSKsignalset,wherethesignalrepresentingabitisthenegativeofthesignalrepresentingtheotherbit.Thereasonforthisresultrestsinthedependenceofprobabilityoferrorpeonthedierencebetweenthenoise-freeintegratoroutputs:ForagivenEb,noothersignalsetprovidesagreaterdierence.Howsmallshouldtheerrorprobabilitybe?OutofNtransmittedbits,ontheaverageNpebitswillbereceivedinerror.Donotethephrase"ontheaverage"here:Errorsoccurrandomlybecauseofthenoiseintroducedbythechannel,andwecanonlypredicttheprobabilityofoccurrence.SincebitsaretransmittedatarateR,errorsoccuratanaveragefrequencyofRpe.Supposetheerrorprobabilityisanimpressivelysmallnumberlike10)]TJ/F7 6.9738 Tf 6.2267 0 Td[(6.DataonacomputernetworklikeEthernetistransmittedatarateR=100Mbps,whichmeansthaterrorswouldoccurroughly100persecond.Thiserrorrateisveryhigh,requiringamuchsmallerpetoachieveamoreacceptableaverageoccurrencerateforerrorsoccurring.BecauseEthernetisawirelinechannel,whichmeansthechannelnoiseissmallandtheattenuationlow,obtainingverysmallerrorprobabilitiesisnotdicult.Wedohavesometricksupoursleeves,however,thatcanessentiallyreducethe

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233errorratetozerowithoutresortingtoexpendingalargeamountofenergyatthetransmitter.WeneedtounderstanddigitalchannelsSection6.19andShannon'sNoisyChannelCodingTheoremSection6.30.6.19DigitalChannels23Let'sreviewhowdigitalcommunicationsystemsworkwithintheFundamentalModelofCommunicationFigure1.4:Fundamentalmodelofcommunication.AsshowninFigure6.17DigMC,themessageisasinglebit.Theentireanalogtransmission/receptionsystem,whichisdiscussedinDigitalCommunicationSection6.13,SignalSets24,BPSKSignalSet25,TransmissionBandwidth26,FrequencyShiftKeyingSec-tion6.15,DigitalCommunicationReceiversSection6.16,FactorsinReceiverErrorSection6.17,DigitalCommunicationSystemProperties27,andErrorProbability28,canbelumpedintoasinglesystemknownasthedigitalchannel. DigMC Figure6.17:Thestepsintransmittingdigitalinformationareshownintheuppersystem,theFundamentalModelofCommunication.Thesymbolic-valuedsignalsmformsthemessage,anditisencodedintoabitsequencebn.Theindicesdierbecausemorethanonebit/symbolisusuallyrequiredtorepresentthemessagebyabitstream.Eachbitisrepresentedbyananalogsignal,transmittedthroughtheunfriendlychannel,andreceivedbyamatched-lterreceiver.Fromthereceivedbitstream^bnthereceivedsymbolic-valuedsignal^smisderived.Thelowerblockdiagramshowsanequivalentsystemwhereintheanalogportionsarecombinedandmodeledbyatransitiondiagram,whichshowshoweachtransmittedbitcouldbereceived.Forexample,transmittinga0resultsinthereceptionofa1withprobabilitypeanerrorora0withprobability1)]TJ/F58 8.9664 Tf 9.2154 0 Td[(penoerror. Digitalchannelsaredescribedbytransitiondiagrams,whichindicatetheoutputalphabetsymbolsthatresultforeachpossibletransmittedsymbolandtheprobabilitiesofthevariousreceptionpossibilities.Theprobabilitiesontransitionscomingfromthesamesymbolmustsumtoone.Forthematched-lterreceiverandthesignalsetswehaveseen,thedepictedtransitiondiagram,knownasabinarysymmetricchannel,captureshowtransmittedbitsarereceived.Theprobabilityoferrorpeisthesoleparameterofthedigitalchannel,anditencapsulatessignalsetchoice,channelproperties,andthematched-lterreceiver.Withthissimplebutentirelyaccuratemodel,wecanconcentrateonhowbitsarereceived. 23Thiscontentisavailableonlineat.24"SignalSets"25"BPSKsignalset"26"TransmissionBandwidth"27"DigitalCommuncationSystemProperties"28"ErrorProbability"

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234CHAPTER6.INFORMATIONCOMMUNICATION6.20Entropy29Communicationtheoryhasbeenformulatedbestforsymbolic-valuedsignals.ClaudeShannon30publishedin1948TheMathematicalTheoryofCommunication,whichbecamethecornerstoneofdigitalcommunication.Heshowedthepowerofprobabilisticmodelsforsymbolic-valuedsignals,whichallowedhimtoquantifytheinformationpresentinasignal.Inthesimplestsignalmodel,eachsymbolcanoccuratindexnwithaprobabilityPr[ak],k=f0;:::;K)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1g.WhatthismodelsaysisthatforeachsignalvalueaK-sidedcoinisippednotethatthecoinneednotbefair.Forthismodeltomakesense,theprobabilitiesmustbenumbersbetweenzeroandoneandmustsumtoone.0Pr[ak]1.48KXk=1Pr[ak]=1.49Thiscoin-ippingmodelassumesthatsymbolsoccurwithoutregardtowhatprecedingorsucceedingsymbolswere,afalseassumptionfortypedtext.Despitethisprobabilisticmodel'sover-simplicity,theideaswedevelopherealsoworkwhenmoreaccurate,butstillprobabilistic,modelsareused.Thekeyquantitythatcharacterizesasymbolic-valuedsignalistheentropyofitsalphabet.HA=)]TJ/F1 9.9626 Tf 9.4091 17.0363 Td[(XkPr[ak]log2Pr[ak]!.50Becauseweusethebase-2logarithm,entropyhasunitsofbits.Forthisdenitiontomakesense,wemusttakespecialnoteofsymbolshavingprobabilityzeroofoccurring.Azero-probabilitysymbolneveroccurs;thus,wedene0log20=0sothatsuchsymbolsdonotaecttheentropy.Themaximumvalueattainablebyanalphabet'sentropyoccurswhenthesymbolsareequallylikelyPr[ak]=Pr[al].Inthiscase,theentropyequalslog2K.Theminimumvalueoccurswhenonlyonesymboloccurs;ithasprobabilityoneofoccurringandtheresthaveprobabilityzero.Exercise6.22Solutiononp.274.Derivethemaximum-entropyresults,boththenumericaspectentropyequalslog2Kandthetheoreticaloneequallylikelysymbolsmaximizeentropy.Derivethevalueoftheminimumentropyalphabet.Example6.1Afour-symbolalphabethasthefollowingprobabilities.Pr[a0]=1 2Pr[a1]=1 4Pr[a2]=1 8Pr[a3]=1 8Notethattheseprobabilitiessumtooneastheyshould.As1 2=2)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1,log2)]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(1 2=)]TJ/F8 9.9626 Tf 7.7488 0 Td[(1.TheentropyofthisalphabetequalsHA=)]TJ/F1 9.9626 Tf 9.4091 8.0699 Td[()]TJ/F7 6.9738 Tf 5.7618 -4.1472 Td[(1 2log2)]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(1 2+1 4log2)]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(1 4+1 8log2)]TJ/F7 6.9738 Tf 5.7618 -4.1472 Td[(1 8+1 8log2)]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(1 8=)]TJ/F1 9.9626 Tf 9.4092 8.0698 Td[()]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(1 2)]TJ/F8 9.9626 Tf 7.7488 0 Td[(1+1 4)]TJ/F8 9.9626 Tf 7.7487 0 Td[(2+1 8)]TJ/F8 9.9626 Tf 7.7487 0 Td[(3+1 8)]TJ/F8 9.9626 Tf 7.7488 0 Td[(3=1:75bits.51 29Thiscontentisavailableonlineat.30http://www.lucent.com/minds/infotheory/

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2356.21SourceCodingTheorem31Thesignicanceofanalphabet'sentropyrestsinhowwecanrepresentitwithasequenceofbits.Bitsequencesformthe"coinoftherealm"indigitalcommunications:theyaretheuniversalwayofrepresentingsymbolic-valuedsignals.Weconvertbackandforthbetweensymbolstobit-sequenceswithwhatisknownasacodebook:atablethatassociatessymbolstobitsequences.Increatingthistable,wemustbeabletoassignauniquebitsequencetoeachsymbolsothatwecangobetweensymbolandbitsequenceswithouterror.PointofInterest:Youmaybeconjuringthenotionofhidinginformationfromotherswhenweusethenamecodebookforthesymbol-to-bit-sequencetable.Thereisnorelationtocryptol-ogy,whichcomprisesmathematicallyprovablemethodsofsecuringinformation.ThecodebookterminologywasdevelopedduringthebeginningsofinformationtheoryjustafterWorldWarII.Asweshallexploreinsomedetailelsewhere,digitalcommunicationSection6.13isthetransmissionofsymbolic-valuedsignalsfromoneplacetoanother.Whenfacedwiththeproblem,forexample,ofsendingaleacrosstheInternet,wemustrstrepresenteachcharacterbyabitsequence.Becausewewanttosendthelequickly,wewanttouseasfewbitsaspossible.However,wedon'twanttousesofewbitsthatthereceivercannotdeterminewhateachcharacterwasfromthebitsequence.Forexample,wecoulduseonebitforeverycharacter:Filetransmissionwouldbefastbutuselessbecausethecodebookcreateserrors.Shannon32provedinhismonumentalworkwhatwecalltodaytheSourceCodingTheorem.LetBakdenotethenumberofbitsusedtorepresentthesymbolak.Theaveragenumberofbits BArequiredtorepresenttheentirealphabetequalsPKk=1BakPr[ak].TheSourceCodingTheoremstatesthattheaveragenumberofbitsneededtoaccuratelyrepresentthealphabetneedonlytosatisfyHA BA.32http://www.lucent.com/minds/infotheory/

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236CHAPTER6.INFORMATIONCOMMUNICATIONNow BA=11 2+21 4+31 8+31 8=1:75.Wecanreachtheentropylimit!Thesimplebinarycodeis,inthiscase,lessecientthantheunequal-lengthcode.Usingtheecientcode,wecantransmitthesymbolic-valuedsignalhavingthisalphabet12.5%faster.Furthermore,weknowthatnomoreecientcodebookcanbefoundbecauseofShannon'sTheorem.6.22CompressionandtheHumanCode33Shannon'sSourceCodingTheorem.52hasadditionalapplicationsindatacompression.Here,wehaveasymbolic-valuedsignalsource,likeacomputerleoranimage,thatwewanttorepresentwithasfewbitsaspossible.CompressionschemesthatassignsymbolstobitsequencesareknownaslosslessiftheyobeytheSourceCodingTheorem;theyarelossyiftheyusefewerbitsthanthealphabet'sentropy.Usingalossycompressionschememeansthatyoucannotrecoverasymbolic-valuedsignalfromitscompressedversionwithoutincurringsomeerror.Youmightbewonderingwhyanyonewouldwanttointentionallycreateerrors,butlossycompressionschemesarefrequentlyusedwheretheeciencygainedinrepresentingthesignaloutweighsthesignicanceoftheerrors.Shannon'sSourceCodingTheoremstatesthatsymbolic-valuedsignalsrequireontheaverageatleastHAnumberofbitstorepresenteachofitsvalues,whicharesymbolsdrawnfromthealphabetA.InthemoduleontheSourceCodingTheoremSection6.21wendthatusingaso-calledxedratesourcecoder,onethatproducesaxednumberofbits/symbol,maynotbethemostecientwayofencodingsymbolsintobits.Whatisnotdiscussedthereisaprocedurefordesigninganecientsourcecoder:oneguaranteedtoproducethefewestbits/symbolontheaverage.Thatsourcecoderisnotunique,andoneapproachthatdoesachievethatlimitistheHumansourcecodingalgorithm.PointofInterest:Intheearlyyearsofinformationtheory,theracewasontobethersttondaprovablymaximallyecientsourcecodingalgorithm.TheracewaswonbythenMITgraduatestudentDavidHumanin1954,whoworkedontheproblemasaprojectinhisinformationtheorycourse.We'reprettysurehereceivedanA.Createaverticaltableforthesymbols,thebestorderingbeingindecreasingorderofprobability.Formabinarytreetotherightofthetable.Abinarytreealwayshastwobranchesateachnode.Buildthetreebymergingthetwolowestprobabilitysymbolsateachlevel,makingtheprobabilityofthenodeequaltothesumofthemergednodes'probabilities.Ifmorethantwonodes/symbolssharethelowestprobabilityatagivenlevel,pickanytwo;yourchoicewon'taect BA.Ateachnode,labeleachoftheemanatingbrancheswithabinarynumber.Thebitsequenceobtainedfrompassingfromthetree'sroottothesymbolisitsHumancode.Example6.3Thesimplefour-symbolalphabetusedintheEntropyExample6.1andSourceCodingExam-ple6.2moduleshasafour-symbolalphabetwiththefollowingprobabilities,Pr[a0]=1 2Pr[a1]=1 4Pr[a2]=1 8Pr[a3]=1 8andanentropyof1.75bitsExample6.1.ThisalphabethastheHumancodingtreeshowninFigure6.18HumanCodingTree. 33Thiscontentisavailableonlineat.

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237HumanCodingTree Figure6.18:WeformaHumancodeforafour-letteralphabethavingtheindicatedprobabilitiesofoccurrence.Thebinarytreecreatedbythealgorithmextendstotheright,withtherootnodetheoneatwhichthetreebeginsdeningthecodewords.Thebitsequenceobtainedbytraversingthetreefromtheroottothesymboldenesthatsymbol'sbinarycode.Thecodethusobtainedisnotuniqueaswecouldhavelabeledthebranchescomingoutofeachnodedierently.Theaveragenumberofbitsrequiredtorepresentthisalphabetequals1:75bits,whichistheShannonentropylimitforthissourcealphabet.Ifwehadthesymbolic-valuedsignalsm=fa2;a3;a1;a4;a1;a2;:::g,ourHumancodewouldproducethebitstreambn=101100111010:::.Ifthealphabetprobabilitiesweredierent,clearlyadierenttree,andthereforedierentcode,couldwellresult.Furthermore,wemaynotbeabletoachievetheentropylimit.IfoursymbolshadtheprobabilitiesPr[a1]=1 2,Pr[a2]=1 4,Pr[a3]=1 5,andPr[a4]=1 20,theaveragenumberofbits/symbolresultingfromtheHumancodingalgorithmwouldequal1:75bits.However,theentropylimitis1.68bits.TheHumancodedoessatisfytheSourceCodingTheoremitsaveragelengthiswithinonebitofthealphabet'sentropybutyoumightwonderifabettercodeexisted.DavidHumanshowedmathematicallythatnoothercodecouldachieveashorteraveragecodethanhis.Wecan'tdobetter.Exercise6.23Solutiononp.274.DerivetheHumancodeforthissecondsetofprobabilities,andverifytheclaimedaveragecodelengthandalphabetentropy.6.23SubtliesofCoding34IntheHumancode,thebitsequencesthatrepresentindividualsymbolscanhavedieringlengthssothebitstreamindexmdoesnotincreaseinlockstepwiththesymbol-valuedsignal'sindexn.Tocapturehowoftenbitsmustbetransmittedtokeepupwiththesource'sproductionofsymbols,wecanonlycomputeaverages.Ifoursourcecodeaverages BAbits/symbolandsymbolsareproducedatarateR,theaveragebitrateequals BAR,andthisquantitydeterminesthebitintervaldurationT.Exercise6.24Solutiononp.274.CalculatewhattherelationbetweenTandtheaveragebitrate BARis.Asubtletyofsourcecodingiswhetherweneed"commas"inthebitstream.Whenweuseanunequalnumberofbitstorepresentsymbols,howdoesthereceiverdeterminewhensymbolsbeginandend?Ifyoucreatedasourcecodethatrequiredaseparationmarkerinthebitstreambetweensymbols,itwouldbeveryinecientsinceyouareessentiallyrequiringanextrasymbolinthetransmissionstream. 34Thiscontentisavailableonlineat.

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238CHAPTER6.INFORMATIONCOMMUNICATIONpointofinterest:AgoodexampleofthisneedistheMorseCode:Betweeneachletter,thetelegrapherneedstoinsertapausetoinformthereceiverwhenletterboundariesoccur.AsshowninthisexampleExample6.3,nocommasareplacedinthebitstream,butyoucanunambiguouslydecodethesequenceofsymbolsfromthebitstream.Humanshowedthathismaximallyecientcodehadtheprexproperty:Nocodeforasymbolbegananothersymbol'scode.Onceyouhavetheprexproperty,thebitstreamispartiallyself-synchronizing:Oncethereceiverknowswherethebitstreamstarts,wecanassignauniqueandcorrectsymbolsequencetothebitstream.Exercise6.25Solutiononp.274.Sketchanargumentthatprexcoding,whetherderivedfromaHumancodeornot,willprovideuniquedecodingwhenanunequalnumberofbits/symbolareusedinthecode.However,havingaprexcodedoesnotguaranteetotalsynchronization:Afterhoppingintothemiddleofabitstream,canwealwaysndthecorrectsymbolboundaries?Theself-synchronizationissuedoesmitigatetheuseofecientsourcecodingalgorithms.Exercise6.26Solutiononp.274.ShowbyexamplethatabitstreamproducedbyaHumancodeisnotnecessarilyself-synchronizing.Arexed-lengthcodesselfsynchronizing?Anotherissueisbiterrorsinducedbythedigitalchannel;iftheyoccurandtheywill,synchronizationcaneasilybelostevenifthereceiverstarted"insynch"withthesource.Despitethesmallprobabilitiesoferroroeredbygoodsignalsetdesignandthematchedlter,aninfrequenterrorcandevastatetheabilitytotranslateabitstreamintoasymbolicsignal.Weneedwaysofreducingreceptionerrorswithoutdemandingthatpebesmaller.Example6.4Therstelectricalcommunicationssystemthetelegraphwasdigital.Whenrstdeployedin1844,itcommunicatedtextoverwirelineconnectionsusingabinarycodetheMorsecodetorepresentindividualletters.Tosendamessagefromoneplacetoanother,telegraphoperatorswouldtapthemessageusingatelegraphkeytoanotheroperator,whowouldrelaythemessageontothenextoperator,presumablygettingthemessageclosertoitsdestination.Inshort,thetelegraphreliedonanetworknotunlikethebasicsofmoderncomputernetworks.Tosayitpresagedmoderncommunicationswouldbeanunderstatement.Itwasalsofaraheadofsomeneededtechnologies,namelytheSourceCodingTheorem.TheMorsecode,showninFigure6.19,wasnotaprexcode.Toseparatecodesforeachletter,Morsecoderequiredthataspaceapausebeinsertedbetweeneachletter.Ininformationtheory,thatspacecountsasanothercodeletter,whichmeansthattheMorsecodeencodedtextwithathree-lettersourcecode:dots,dashesandspace.Theresultingsourcecodeisnotwithinabitofentropy,andisgrosslyinecientabout25%.Figure6.19showsaHumancodeforEnglishtext,whichasweknowisecient.6.24ChannelCoding35Wecan,tosomeextent,correcterrorsmadebythereceiverwithonlytheerror-lledbitstreamemergingfromthedigitalchannelavailabletous.Theideaisforthetransmittertosendnotonlythesymbol-derivedbitsemergingfromthesourcecoderbutalsoadditionalbitsderivedfromthecoder'sbitstream.Theseadditionalbits,theerrorcorrectingbits,helpthereceiverdetermineifanerrorhasoccurredinthedatabitstheimportantbitsorintheerror-correctionbits.InsteadofthecommunicationmodelFigure6.17:DigMCshownpreviously,thetransmitterinsertsachannelcoderbeforeanalogmodulation,andthereceiverthecorrespondingchanneldecoderFigure6.20.ThisblockdiagramshownthereformstheFundamentalModelofDigitalCommunication. 35Thiscontentisavailableonlineat.

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239MorseandHumanCodeTable % MorseCode HumanCode A 6.22 .1011 B 1.32 -... 010100 C 3.11 -.-. 10101 D 2.97 -.. 01011 E 10.53 001 F 1.68 ..-. 110001 G 1.65 110000 H 3.63 .... 11001 I 6.14 .. 1001 J 0.06 01010111011 K 0.31 -.01010110 L 3.07 .-.. 10100 M 2.48 00011 N 5.73 -. 0100 O 6.06 1000 P 1.87 .. 00000 Q 0.10 .0101011100 R 5.87 .-. 0111 S 5.81 ... 0110 T 7.68 1101 U 2.27 ..00010 V 0.70 ...0101010 W 1.13 000011 X 0.25 -..010101111 Y 1.07 -. 000010 Z 0.06 .. 0101011101011 Figure6.19:MorseandHumanCodesforAmerican-RomanAlphabet.The%columnindicatestheaverageprobabilityexpressedinpercentoftheletteroccurringinEnglish.TheentropyHAofthethissourceis4.14bits.TheaverageMorsecodewordlengthis2.5symbols.Addingonemoresymbolfortheletterseparatorandconvertingtobitsyieldsanaveragecodewordlengthof5.56bits.TheaverageHumancodewordlengthis4.35bits.

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240CHAPTER6.INFORMATIONCOMMUNICATION Figure6.20:Tocorrecterrorsthatoccurinthedigitalchannel,achannelcoderanddecoderareaddedtothecommunicationsystem.Properlydesignedchannelcodingcangreatlyreducetheprobabilityfromtheuncodedvalueofpethatadatabitbnisreceivedincorrectlyevenwhentheprobabilityofclbereceivedinerrorremainspeorbecomeslarger.ThissystemformstheFundamentalModelofDigitalCommunication. Shannon'sNoisyChannelCodingTheoremSection6.30saysthatifthedataaren'ttransmittedtooquickly,thaterrorcorrectioncodesexistthatcancorrectallthebiterrorsintroducedbythechannel.Unfortunately,Shannondidnotdemonstrateanerrorcorrectingcodethatwouldachievethisremarkablefeat;infact,noonehasfoundsuchacode.Shannon'sresultprovesitexists;seemslikethereisalwaysmoreworktodo.Inanycase,thatshouldnotpreventusfromstudyingcommonlyusederrorcorrectingcodesthatnotonlyndtheirwayintoalldigitalcommunicationsystems,butalsointoCDsandbarcodesusedonmerchandise.6.25RepetitionCodes36Perhapsthesimplesterrorcorrectingcodeistherepetitioncode.RepetitionCode Figure6.21:TheupperportiondepictstheresultofdirectlymodulatingthebitstreambnintoatransmittedsignalxtusingabasebandBPSKsignalset.R'isthedatarateproducedbythesourcecoder.Ifthatbitstreampassesthrougha,1channelcodertoyieldthebitstreamcl,theresultingtransmittedsignalrequiresabitintervalTthreetimessmallerthantheuncodedversion.Thisreductioninthebitintervalmeansthatthetransmittedenergy/bitdecreasesbyafactorofthree,whichresultsinanincreasederrorprobabilityinthereceiver.Here,thetransmittersendsthedatabitseveraltimes,anoddnumberoftimesinfact.Becausethe 36Thiscontentisavailableonlineat.

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241errorprobabilitypeisalwayslessthan1 2,weknowthatmoreofthebitsshouldbecorrectratherthaninerror.Simplemajorityvotingofthereceivedbitshencethereasonfortheoddnumberdeterminesthetransmittedbitmoreaccuratelythansendingitalone.Forexample,let'sconsiderthethree-foldrepetitioncode:foreverybitbnemergingfromthesourcecoder,thechannelcoderproducesthree.Thus,thebitstreamemergingfromthechannelcoderclhasadataratethreetimeshigherthanthatoftheoriginalbitstreambn.Thecodingtableillustrateswhenerrorscanbecorrectedandwhentheycan'tbythemajority-votedecoder. CodingTable Code Probability Bit 000 )]TJ/F11 9.9626 Tf 9.9626 0 Td[(pe3 0 001 pe)]TJ/F11 9.9626 Tf 9.9626 0 Td[(pe2 0 010 pe)]TJ/F11 9.9626 Tf 9.9626 0 Td[(pe2 0 011 pe2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(pe 1 100 pe)]TJ/F11 9.9626 Tf 9.9626 0 Td[(pe2 0 101 pe2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(pe 1 110 pe2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(pe 1 111 pe3 1 Figure6.22:Inthisexample,thetransmitterencodes0as000.Thechannelcreatesanerrorchanginga0intoa1thatwithprobabilitype.Therstcolumnlistsallpossiblereceiveddatawordsandthesecondtheprobabilityofeachdatawordbeingreceived.Thelastcolumnshowstheresultsofthemajority-votedecoder.Whenthedecoderproduces0,itsuccessfullycorrectedtheerrorsintroducedbythechanneliftherewereany;thetoprowcorrespondstothecaseinwhichnoerrorsoccurred.Theerrorprobabilityofthedecodersisthesumoftheprobabilitieswhenthedecoderproduces1. Thus,ifonebitofthethreebitsisreceivedinerror,thereceivercancorrecttheerror;ifmorethanoneerroroccurs,thechanneldecoderannouncesthebitis1insteadoftransmittedvalueof0.Usingthisrepetitioncode,theprobabilityof^bn6=0equals3pe2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(pe+pe3.Thisprobabilityofadecodingerrorisalwayslessthanpe,theuncodedvalue,solongaspe<1 2.Exercise6.27Solutiononp.275.Demonstratemathematicallythatthisclaimisindeedtrue.Is3pe2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(pe+pe3pe?6.26BlockChannelCoding37Becauseofthehigherdatarateimposedbythechannelcoder,theprobabilityofbiterroroccurringinthedigitalchannelincreasesrelativetothevalueobtainedwhennochannelcodingisused.ThebitintervaldurationmustbereducedbyK Nincomparisontotheno-channel-codingsituation,whichmeanstheenergy 37Thiscontentisavailableonlineat.

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242CHAPTER6.INFORMATIONCOMMUNICATIONperbitEbgoesdownbythesameamount.Thebitintervalmustdecreasebyafactorofthreeifthetransmitteristokeepupwiththedatastream,asillustratedhereFigure6.21:RepetitionCode.PointofInterest:Itisunlikelythatthetransmitter'spowercouldbeincreasedtocompensate.Suchisthesometimes-unfriendlynatureoftherealworld.Becauseofthisreduction,theerrorprobabilitypeofthedigitalchannelgoesup.Thequestionthusbecomesdoeschannelcodingreallyhelp:Istheeectiveerrorprobabilitylowerwithchannelcodingeventhoughtheerrorprobabilityforeachtransmittedbitislarger?Theanswerisno:Usingarepetitioncodeforchannelcodingcannotultimatelyreducetheprobabilitythatadatabitisreceivedinerror.Theultimatereasonistherepetitioncode'sineciency:transmittingonedatabitforeverythreetransmittedistooinecientfortheamountoferrorcorrectionprovided.Exercise6.28Solutiononp.275.UsingMATLAB,calculatetheprobabilityabitisreceivedincorrectlywithathree-foldrepetitioncode.ShowthatwhentheenergyperbitEbisreducedby1=3thatthisprobabilityislargerthantheno-codingprobabilityoferror.Therepetitioncodep.240representsaspecialcaseofwhatisknownasblockchannelcoding.ForeveryKbitsthatentertheblockchannelcoder,itinsertsanadditionalN)]TJ/F11 9.9626 Tf 10.6514 0 Td[(Kerror-correctionbitstoproduceablockofNbitsfortransmission.WeusethenotationN,Ktorepresentagivenblockcode'sparameters.Inthethree-foldrepetitioncodep.240,K=1andN=3.Ablockcode'scodingeciencyEequalstheratioK N,andquantiestheoverheadintroducedbychannelcoding.Therateatwhichbitsmustbetransmittedagainchanges:So-calleddatabitsbnemergefromthesourcecoderatanaveragerate BAandexitthechannelatarate1 Ehigher.Werepresentthefactthatthebitssentthroughthedigitalchanneloperateatadierentratebyusingtheindexlforthechannel-codedbitstreamcl.Notethattheblockingframingimposedbythechannelcoderdoesnotcorrespondtosymbolboundariesinthebitstreambn,especiallywhenweemployvariable-lengthsourcecodes.Doesanyerror-correctingcodereducecommunicationerrorswhenreal-worldconstraintsaretakenintoaccount?Theanswernowisyes.Tounderstandchannelcoding,weneedtodeveloprstageneralframeworkforchannelcoding,anddiscoverwhatittakesforacodetobemaximallyecient:CorrectasmanyerrorsaspossibleusingthefewesterrorcorrectionbitsaspossiblemakingtheeciencyK Naslargeaspossible.6.27Error-CorrectingCodes:HammingDistance38So-calledlinearcodescreateerror-correctionbitsbycombiningthedatabitslinearly.Thephrase"linearcombination"meansheresingle-bitbinaryarithmetic. 0=0 1=0 1=1 0=1 0=0 1=1 1=0 0=0 Figure6.23Forexample,let'sconsiderthespecic,1errorcorrectioncodedescribedbythefollowingcodingtableand,moreconcisely,bythesucceedingmatrixexpression.c=bc=bc=b 38Thiscontentisavailableonlineat.

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243orc=GbwhereG=0BB@1111CCAc=0BB@ccc1CCAb=bThelength-KinthissimpleexampleK=1blockofdatabitsisrepresentedbythevectorb,andthelength-Noutputblockofthechannelcoder,knownasacodeword,byc.ThegeneratormatrixGdenesallblock-orientedlinearchannelcoders.Asweconsiderotherblockcodes,thesimpleideaofthedecodertakingamajorityvoteofthereceivedbitswon'tgeneralizeeasily.Weneedabroaderviewthattakesintoaccountthedistancebetweencodewords.Alength-Ncodewordmeansthatthereceivermustdecideamongthe2Npossibledatawordstoselectwhichofthe2Kcodewordswasactuallytransmitted.AsshowninFigure6.24,wecanthinkofthedatawordsgeometrically.WedenetheHammingdistancebetweenbinarydatawordsc1andc2,denotedbydc1;c2tobetheminimumnumberofbitsthatmustbe"ipped"togofromonewordtotheother.Forexample,thedistancebetweencodewordsis3bits.Inourtableofbinaryarithmetic,weseethataddinga1correspondstoippingabit.Furthermore,subtractionandadditionareequivalent.WecanexpresstheHammingdistanceasdc1;c2=sumc1c2.55Exercise6.29Solutiononp.275.Showthataddingtheerrorvectorcol[1,0,...,0]toacodewordipsthecodeword'sleadingbitandleavestherestunaected.TheprobabilityofonebitbeingippedanywhereinacodewordisNpe)]TJ/F11 9.9626 Tf 9.9626 0 Td[(peN)]TJ/F7 6.9738 Tf 6.2266 0 Td[(1.Thenumberoferrorsthechannelintroducesequalsthenumberofonesine;theprobabilityofanyparticularerrorvectordecreaseswiththenumberoferrors.Toperformdecodingwhenerrorsoccur,wewanttondthecodewordoneofthelledcirclesinFig-ure6.24thathasthehighestprobabilityofoccurring:theoneclosesttotheonereceived.Notethatifadatawordliesadistanceof1fromtwocodewords,itisimpossibletodeterminewhichcodewordwasactuallysent.Thiscriterionmeansthatifanytwocodewordsaretwobitsapart,thenthecodecannotcorrectthechannel-inducederror.Thus,tohaveacodethatcancorrectallsingle-biterrors,codewordsmusthaveaminimumseparationofthree.Ourrepetitioncodehasthisproperty.Introducingcodebitsincreasestheprobabilitythatanybitarrivesinerrorbecausebitintervaldurationsdecrease.However,usingawell-designederror-correctingcodecorrectsbitreceptionerrors.Dowewinorlosebyusinganerror-correctingcode?Theansweristhatwecanwinifthecodeiswell-designed.The,1repetitioncodedemonstratesthatwecanloseExercise6.28.Todevelopgoodchannelcoding,weneedtodeveloprstageneralframeworkforchannelcodesanddiscoverwhatittakesforacodetobemaximallyecient:CorrectasmanyerrorsaspossibleusingthefewesterrorcorrectionbitsaspossiblemakingtheeciencyK Naslargeaspossible.Wealsoneedasystematicwayofndingthecodewordclosesttoany

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244CHAPTER6.INFORMATIONCOMMUNICATION Figure6.24:Ina,1repetitioncode,only2ofthepossible8three-bitdatablocksarecodewords.Wecanrepresentthesebitpatternsgeometricallywiththeaxesbeingbitpositionsinthedatablock.Intheleftplot,thelledcirclesrepresentthecodewords[000]and[111],theonlypossiblecodewords.Theunlledonescorrespondtothetransmission.Thecenterplotshowsthatthedistancebetweencodewordsis3.Becausedistancecorrespondstoippingabit,calculatingtheHammingdistancegeometricallymeansfollowingtheaxesratherthangoing"asthecrowies".Therightplotshowsthedatawordsthatresultwhenoneerroroccursasthecodewordgoesthroughthechannel.Thethreedatawordsareunitdistancefromtheoriginalcodeword.Notethatthereceiveddatawordgroupsdonotoverlap,whichmeansthecodecancorrectallsingle-biterrors. receiveddataword.Amuchbettercodethanour,1repetitioncodeisthefollowing,4code.c=bc=bc=bc=bc=bbbc=bbbc=bbbwherethegeneratormatrixisG=0BBBBBBBBBBBBB@10000100001000011110011111011CCCCCCCCCCCCCAInthis,4code,24=16ofthe27=128possibleblocksatthechanneldecodercorrespondtoerror-freetransmissionandreception.Errorcorrectionamountstosearchingforthecodewordcclosesttothereceivedblock^cintermsoftheHammingdistancebetweenthetwo.Theerrorcorrectioncapabilityofachannelcodeislimitedbyhowclosetogetheranytwoerror-freeblocksare.Badcodeswouldproduceblocksclosetogether,whichwouldresultinambiguitywhenassigningablockofdatabitstoareceivedblock.Thequantitytoexamine,therefore,in

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245designingcodeerrorcorrectioncodesistheminimumdistancebetweencodewords.dmin=mindci;cj;ci6=cj.56Tohaveachannelcodethatcancorrectallsingle-biterrors,dmin3.Exercise6.30Solutiononp.276.Supposewewantachannelcodetohaveanerror-correctioncapabilityofnbits.WhatmusttheminimumHammingdistancebetweencodewordsdminbe?Howdowecalculatetheminimumdistancebetweencodewords?Becausewehave2Kcodewords,thenumberofpossibleuniquepairsequals2K)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1)]TJ/F8 9.9626 Tf 4.5663 -8.0698 Td[(2K)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1,whichcanbealargenumber.Recallthatourchannelcodingprocedureislinear,withc=Gb.Thereforecicj=Gbibj.Becausebibjalwaysyieldsanotherblockofdatabits,wendthatthedierencebetweenanytwocodewordsisanothercodeword!Thus,tonddminweneedonlycomputethenumberofonesthatcompriseallnon-zerocodewords.Findingthesecodewordsiseasyonceweexaminethecoder'sgeneratormatrix.NotethatthecolumnsofGarecodewordswhyisthis?,andthatallcodewordscanbefoundbyallpossiblepairwisesumsofthecolumns.Tonddmin,weneedonlycountthenumberofbitsineachcolumnandsumsofcolumns.Forourexample,4,G'srstcolumnhasthreeones,thenextonefour,andthelasttwothree.Consideringsumsofcolumnpairsnext,notethatbecausetheupperportionofGisanidentitymatrix,thecorrespondingupperportionofallcolumnsumsmusthaveexactlytwobits.Becausethebottomportionofeachcolumndiersfromtheothercolumnsinatleastoneplace,thebottomportionofasumofcolumnsmusthaveatleastonebit.TriplesumswillhaveatleastthreebitsbecausetheupperportionofGisanidentitymatrix.Thus,nosumofcolumnshasfewerthanthreebits,whichmeansthatdmin=3,andwehaveachannelcoderthatcancorrectalloccurrencesofoneerrorwithinareceived7-bitblock.6.28Error-CorrectingCodes:ChannelDecoding39Becausetheideaofchannelcodinghasmeritsolongasthecodeisecient,let'sdevelopasystematicprocedureforperformingchanneldecoding.Onewayofcheckingforerrorsistotryrecreatingtheerrorcorrectionbitsfromthedataportionofthereceivedblock^c.Usingmatrixnotation,wemakethiscalculationbymultiplyingthereceivedblock^cbythematrixHknownastheparitycheckmatrix.ItisformedfromthegeneratormatrixGbytakingthebottom,error-correctionportionofGandattachingtoitanidentitymatrix.Forour,4code,H=266666664111001111101| {z }LowerportionofG100010001| {z }Identity377777775.57TheparitycheckmatrixthushassizeN)]TJ/F11 9.9626 Tf 9.9626 0 Td[(KN,andtheresultofmultiplyingthismatrixwithareceivedwordisalength-N)]TJ/F11 9.9626 Tf 9.9626 0 Td[(Kbinaryvector.Ifnodigitalchannelerrorsoccurwereceiveacodewordsothat^c=cthenH^c=0.Forexample,therstcolumnofG,;0;0;0;1;0;1T,isacodeword.SimplecalculationsshowthatmultiplyingthisvectorbyHresultsinalength-N)]TJ/F11 9.9626 Tf 9.9626 0 Td[(Kzero-valuedvector.Exercise6.31Solutiononp.276.ShowthatHc=0forallthecolumnsofG.Inotherwords,showthatHG=0anN)]TJ/F11 9.9626 Tf 9.9626 0 Td[(KKmatrixofzeroes.DoesthispropertyguaranteethatallcodewordsalsosatisfyHc=0?Whenthereceivedbits^cdonotformacodeword,H^cdoesnotequalzero,indicatingthepresenceofoneormoreerrorsinducedbythedigitalchannel.Becausethepresenceofanerrorcanbemathematicallywrittenas^c=ce,witheavectorofbinaryvalueshavinga1inthosepositionswhereabiterroroccurred. 39Thiscontentisavailableonlineat.

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246CHAPTER6.INFORMATIONCOMMUNICATIONExercise6.32Solutiononp.276.Showthataddingtheerrorvector;0;:::;0Ttoacodewordipsthecodeword'sleadingbitandleavestherestunaected.Consequently,H^c=Hce=He.Becausetheresultoftheproductisalength-N)]TJ/F11 9.9626 Tf 9.9626 0 Td[(Kvectorofbinaryvalues,wecanhave2N)]TJ/F10 6.9738 Tf 6.2267 0 Td[(K)]TJ/F8 9.9626 Tf 10.6581 0 Td[(1non-zerovaluesthatcorrespondtonon-zeroerrorpatternse.Toperformourchanneldecoding,1.computeconceptuallyatleastH^c;2.ifthisresultiszero,nodetectableorcorrectableerroroccurred;3.ifnon-zero,consultatableoflength-N)]TJ/F11 9.9626 Tf 9.9626 0 Td[(Kbinaryvectorstoassociatethemwiththeminimalerrorpatternthatcouldhaveresultedinthenon-zeroresult;then4.addtheerrorvectorthusobtainedtothereceivedvector^ctocorrecttheerrorbecausecee=c.5.Selectthedatabitsfromthecorrectedwordtoproducethereceivedbitsequence^bn.Thephraseminimalinthethirditemraisesthepointthatadoubleortripleorquadruple:::erroroccurringduringthetransmission/receptionofonecodewordcancreatethesamereceivedwordasasingle-biterrorornoerrorinanothercodeword.Forexample,;0;0;0;1;0;1Tand;1;0;0;1;1;1Tarebothcodewordsintheexample,4code.Thesecondresultswhentherstoneexperiencesthreebiterrorsrst,second,andsixthbits.Suchanerrorpatterncannotbedetectedbyourcodingstrategy,butsuchmultipleerrorpatternsareveryunlikelytooccur.Ourreceiverusestheprincipleofmaximumprobability:Anerror-freetransmissionismuchmorelikelythanonewiththreeerrorsifthebit-errorprobabilitypeissmallenough.Exercise6.33Solutiononp.276.Howsmallmustpebesothatasingle-biterrorismorelikelytooccurthanatriple-biterror?6.29Error-CorrectingCodes:HammingCodes40Forthe,4example,wehave2N)]TJ/F10 6.9738 Tf 6.2266 0 Td[(K)]TJ/F8 9.9626 Tf 9.6681 0 Td[(1=7errorpatternsthatcanbecorrected.Westartwithsingle-biterrorpatterns,andmultiplythembytheparitycheckmatrix.Ifweobtainuniqueanswers,wearedone;iftwoormoreerrorpatternsyieldthesameresult,wecantrydouble-biterrorpatterns.Inourcase,single-biterrorpatternsgiveauniqueresult.ParityCheckMatrix e He 1000000 101 0100000 111 0010000 110 0001000 011 0000100 100 0000010 010 0000001 001 Thiscorrespondstoourdecodingtable:Weassociatetheparitycheckmatrixmultiplicationresultwiththeerrorpatternandaddthistothereceivedword.Ifmorethanoneerroroccursunlikelythoughitmaybe,this"errorcorrection"strategyusuallymakestheerrorworseinthesensethatmorebitsarechangedfromwhatwastransmitted. 40Thiscontentisavailableonlineat.

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247Aswiththerepetitioncode,wemustquestionwhetherour,4code'serrorcorrectioncapabilitycom-pensatesfortheincreasederrorprobabilityduetothenecessitatedreductioninbitenergy.Figure6.25Probabilityoferroroccurringshowsthatifthesignal-to-noiseratioislargeenoughchannelcodingyieldsasmallererrorprobability.Becausethebitstreamemergingfromthesourcedecoderissegmentedintofour-bitblocks,thefairwayofcomparingcodedanduncodedtransmissionistocomputetheprobabilityofblockerror:theprobabilitythatanybitinablockremainsinerrordespiteerrorcorrectionandregardlessofwhethertheerroroccursinthedataorincodingbuts.Clearly,our,4channelcodedoesyieldsmallererrorrates,andisworththeadditionalsystemsrequiredtomakeitwork. Probabilityoferroroccurring Figure6.25:TheprobabilityofanerroroccurringintransmittedK=4databitsequals1)]TJ/F56 8.9664 Tf 8.4827 0 Td[()]TJ/F58 8.9664 Tf 9.2154 0 Td[(pe4as)]TJ/F58 8.9664 Tf 9.2154 0 Td[(pe4equalstheprobabilitythatthefourbitsarereceivedwithouterror.Theuppercurvedisplayshowthisprobabilityofanerroranywhereinthefour-bitblockvarieswiththesignal-to-noiseratio.Whena,4single-biterrorcorrectingcodeisused,thetransmitterreducedtheenergyitexpendsduringasingle-bittransmissionby4/7,appendingthreeextrabitsforerrorcorrection.Nowtheprobabilityofanybitintheseven-bitblockbeinginerroraftererrorcorrectionequals1)]TJ/F56 8.9664 Tf 9.1893 0 Td[()]TJ/F58 8.9664 Tf 9.2154 0 Td[(pe7)]TJ/F56 8.9664 Tf 9.1893 0 Td[(p0e)]TJ/F58 8.9664 Tf 9.2154 0 Td[(p0e6,wherep0eistheprobabilityofabiterroroccurringinthechannelwhenchannelcodingoccurs.Herep0e)]TJ/F58 8.9664 Tf 9.2154 0 Td[(p0e6equalstheprobabilityofexactlyoninsevenbitsemergingfromthechannelinerror;Thechanneldecodercorrectsthistypeoferror,andalldatabitsintheblockarereceivedcorrectly. Notethatour,4codehasthelengthandnumberofdatabitsthatperfectlytscorrectingsinglebiterrors.Thispleasantpropertyarisesbecausethenumberoferrorpatternsthatcanbecorrected,2N)]TJ/F10 6.9738 Tf 6.2267 0 Td[(K)]TJ/F8 9.9626 Tf 9.4448 0 Td[(1,equalsthecodewordlengthN.Codesthathave2N)]TJ/F10 6.9738 Tf 6.2267 0 Td[(K)]TJ/F8 9.9626 Tf 10.7255 0 Td[(1=NareknownasHammingcodes,andthefollowingtableHammingCodes,p.247providestheparametersofthesecodes.Hammingcodesarethesimplestsingle-biterrorcorrectioncodes,andthegenerator/paritycheckmatrixformalismforchannelcodinganddecodingworksforthem.

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248CHAPTER6.INFORMATIONCOMMUNICATIONHammingCodes N K Eeciency 3 1 0.33 7 4 0.57 15 11 0.73 31 26 0.84 63 57 0.90 127 120 0.94 Unfortunately,forsuchlargeblocks,theprobabilityofmultiple-biterrorscanexceedthenumberofsingle-biterrorsunlessthechannelsingle-biterrorprobabilitypeisverysmall.Consequently,weneedtoenhancethecode'serrorcorrectingcapabilitybyaddingdoubleaswellassingle-biterrorcorrection.Exercise6.34Solutiononp.276.WhatmusttherelationbetweenNandKbeforacodetocorrectallsingle-anddouble-biterrorswitha"perfectt"?6.30NoisyChannelCodingTheorem41Astheblocklengthbecomeslarger,moreerrorcorrectionwillbeneeded.Docodesexistthatcancorrectallerrors?PerhapsthecrowningachievementofClaudeShannon's42creationofinformationtheoryanswersthisquestion.Hisresultcomesintwocomplementaryforms:theNoisyChannelCodingTheoremanditsconverse.6.30.1NoisyChannelCodingTheoremLetEdenotetheeciencyofanerror-correctingcode:theratioofthenumberofdatabitstothetotalnumberofbitsusedtorepresentthem.Iftheeciencyislessthanthecapacityofthedigitalchannel,anerror-correctingcodeexiststhathasthepropertythatasthelengthofthecodeincreases,theprobabilityofanerroroccurringinthedecodedblockapproacheszero.limN!1Pr[blockerror]=0;EC,theprobabilityofanerrorinadecodedblockmustapproachoneregardlessofthecodethatmightbechosen.limN!1Pr[blockerror]=1.59Theseresultsmeanthatitispossibletotransmitdigitalinformationoveranoisychannelonethatin-troduceserrorsandreceivetheinformationwithouterrorifthecodeissucientlyinecientcomparedtothechannel'scharacteristics.Generally,achannel'scapacitychangeswiththesignal-to-noiseratio:Asoneincreasesordecreases,sodoestheother.Thecapacitymeasurestheoverallerrorcharacteristicsofa 41Thiscontentisavailableonlineat.42http://www.lucent.com/minds/infotheory/

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249channelthesmallerthecapacitythemorefrequentlyerrorsoccurandanoverlyecienterror-correctingcodewillnotbuildinenougherrorcorrectioncapabilitytocounteractchannelerrors.ThisresultastoundedcommunicationengineerswhenShannonpublisheditin1948.Analogcommuni-cationalwaysyieldsanoisyversionofthetransmittedsignal;indigitalcommunication,errorcorrectioncanbepowerfulenoughtocorrectallerrorsastheblocklengthincreases.Thekeyforthiscapabilitytoexististhatthecode'seciencybelessthanthechannel'scapacity.Forabinarysymmetricchannel,thecapacityisgivenbyC=1+pelog2pe+)]TJ/F11 9.9626 Tf 9.9626 0 Td[(pelog2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(pebits=transmission.60Figure6.26capacityofachannelshowshowcapacityvarieswitherrorprobability.Forexample,our,4Hammingcodehasaneciencyof0:57,andcodeshavingthesameeciencybutlongerblocksizescanbeusedonadditivenoisechannelswherethesignal-to-noiseratioexceeds0dB. capacityofachannel Figure6.26:Thecapacitypertransmissionthroughabinarysymmetricchannelisplottedasafunctionofthedigitalchannel'serrorprobabilityupperandasafunctionofthesignal-to-noiseratioforaBPSKsignalsetlower. 6.31CapacityofaChannel43InadditiontotheNoisyChannelCodingTheoremanditsconverseSection6.30,ShannonalsoderivedthecapacityforabandlimitedtoWHzadditivewhitenoisechannel.Forthiscase,thesignalsetisunrestricted,eventothepointthatmorethanonebitcanbetransmittedeach"bitinterval."Insteadofconstrainingchannelcodeeciency,therevisedNoisyChannelCodingTheoremstatesthatsomeerror-correctingcodeexistssuchthatastheblocklengthincreases,error-freetransmissionispossibleifthesourcecoder'sdatarate, BAR,islessthancapacity.C=Wlog2+SNRbits/s.61 43Thiscontentisavailableonlineat.

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250CHAPTER6.INFORMATIONCOMMUNICATIONThisresultsetsthemaximumdatarateofthesourcecoder'soutputthatcanbetransmittedthroughthebandlimitedchannelwithnoerror.44Shannon'sproofofhistheoremwasveryclever,anddidnotindicatewhatthiscodemightbe;ithasneverbeenfound.CodessuchastheHammingcodeworkquitewellinpracticetokeeperrorrateslow,buttheyremaingreaterthanzero.Untilthe"magic"codeisfound,moreimportantincommunicationsystemdesignistheconverse.Itstatesthatifyourdatarateexceedscapacity,errorswilloverwhelmyounomatterwhatchannelcodingyouuse.Forthisreason,capacitycalculationsaremadetounderstandthefundamentallimitsontransmissionrates.Exercise6.35Solutiononp.276.Therstdenitionofcapacityappliesonlyforbinarysymmetricchannels,andrepresentsthenumberofbits/transmission.Thesecondresultstatescapacitymoregenerally,havingunitsofbits/second.Howwouldyouconverttherstdenition'sresultintounitsofbits/second?Example6.5Thetelephonechannelhasabandwidthof3kHzandasignal-to-noiseratioexceeding30dBatleasttheypromisethismuch.Themaximumdatarateamodemcanproduceforthiswirelinechannelandhopethaterrorswillnotbecomerampantisthecapacity.C=3103log2)]TJ/F8 9.9626 Tf 4.5662 -8.0698 Td[(1+103=29.901kbps.62Thus,theso-called33kbpsmodemsoperaterightatthecapacitylimit.Notethatthedatarateallowedbythecapacitycanexceedthebandwidthwhenthesignal-to-noiseratioexceeds0dB.OurresultsforBPSKandFSKindicatedthebandwidththeyrequireexceeds1 T.Whatkindofsignalsetsmightbeusedtoachievecapacity?Modemsignalsetssendmorethanonebit/transmissionusinganumber,oneofthemostpopularofwhichismulti-levelsignaling.Here,wecantransmitseveralbitsduringonetransmissionintervalbyrepresentingbitbysomesignal'samplitude.Forexample,twobitscanbesentwithasignalsetcomprisedofasinusoidwithamplitudesofAand)]TJ/F10 6.9738 Tf 5.7617 -4.1472 Td[(A 2.6.32ComparisonofAnalogandDigitalCommunication45Analogcommunicationsystems,amplitudemodulationAMradiobeingatypifyingexample,caninexpen-sivelycommunicateabandlimitedanalogsignalfromonelocationtoanotherpoint-to-pointcommunicationorfromonepointtomanybroadcast.Althoughitisnotshownhere,thecoherentreceiverFigure6.6providesthelargestpossiblesignal-to-noiseratioforthedemodulatedmessage.AnanalysisSection6.12ofthisreceiverthusindicatesthatsomeresidualerrorwillalwaysbepresentinananalogsystem'soutput.Althoughanalogsystemsarelessexpensiveinmanycasesthandigitalonesforthesameapplication,digitalsystemsoermuchmoreeciency,betterperformance,andmuchgreaterexibility.Eciency:TheSourceCodingTheoremallowsquanticationofjusthowcomplexagivenmessagesourceisandallowsustoexploitthatcomplexitybysourcecodingcompression.Inanalogcommu-nication,theonlyparametersofinterestaremessagebandwidthandamplitude.Wecannotexploitsignalstructuretoachieveamoreecientcommunicationsystem.Performance:BecauseoftheNoisyChannelCodingTheorem,wehaveaspeciccriterionbywhichtoformulateerror-correctingcodesthatcanbringusasclosetoerror-freetransmissionaswemightwant.Eventhoughwemaysendinformationbywayofanoisychannel,digitalschemesarecapableoferror-freetransmissionwhileanalogonescannotovercomechanneldisturbances;seethisproblemProblem6.15foracomparison. 44Thebandwidthrestrictionarisesnotsomuchfromchannelproperties,butfromspectralregulation,especiallyforwirelesschannels.45Thiscontentisavailableonlineat.

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251Flexibility:Digitalcommunicationsystemscantransmitreal-valueddiscrete-timesignals,whichcouldbeanalogonesobtainedbyanalog-to-digitalconversion,andsymbolic-valuedonescomputerdata,forexample.Anysignalthatcanbetransmittedbyanalogmeanscanbesentbydigitalmeans,withtheonlyissuebeingthenumberofbitsusedinA/Dconversionhowaccuratelydoweneedtorepresentsignalamplitude.Imagescanbesentbyanalogmeanscommercialtelevision,butbettercommunicationperformanceoccurswhenweusedigitalsystemsHDTV.Inadditiontodigitalcommunication'sabilitytotransmitawidervarietyofsignalsthananalogsystems,point-to-pointdigitalsystemscanbeorganizedintoglobalandbeyondaswellsystemsthatprovideecientandexibleinformationtransmission.Computernetworks,exploredinthenextsection,arewhatwecallsuchsystemstoday.Evenanalog-basednetworks,suchasthetelephonesystem,employmoderncomputernetworkingideasratherthanthepurelyanalogsystemsofthepast.Consequently,withtheincreasedspeedofdigitalcomputers,thedevelopmentofincreasinglyecientalgo-rithms,andtheabilitytointerconnectcomputerstoformacommunicationsinfrastructure,digitalcommu-nicationisnowthebestchoiceformanysituations.6.33CommunicationNetworks46CommunicationnetworkselaboratetheFundamentalModelofCommunicationsFigure1.4:Fundamentalmodelofcommunication.ThemodelshowninFigure6.27describespoint-to-pointcommunicationswell,whereinthelinkbetweentransmitterandreceiverisstraightforward,andtheyhavethechanneltothemselves.Onemodernexampleofthiscommunicationsmodeisthemodemthatconnectsapersonalcomputerwithaninformationserverviaatelephoneline.Thekeyaspect,somewouldsayaw,ofthismodelisthatthechannelisdedicated:Onlyonecommunicationslinkthroughthechannelisallowedforalltime.Regardlesswhetherwehaveawirelineorwirelesschannel,communicationbandwidthisprecious,andifitcouldbesharedwithoutsignicantdegradationincommunicationsperformancemeasuredbysignal-to-noiseratioforanalogsignaltransmissionandbybit-errorprobabilityfordigitaltransmissionsomuchthebetter. Figure6.27:Theprototypicalcommunicationsnetworkwhetheritbethepostalservice,cellulartelephone,ortheInternetconsistsofnodesinterconnectedbylinks.Messagesformedbythesourcearetransmittedwithinthenetworkbydynamicrouting.Tworoutesareshown.Thelongeronewouldbeusedifthedirectlinkweredisabledorcongested. Theideaofanetworkrstemergedwithperhapstheoldestformoforganizedcommunication:thepostalservice.Mostcommunicationnetworks,evenmodernones,sharemanyofitsaspects.Auserwritesaletter,servinginthecommunicationscontextasthemessagesource.Thismessageissenttothenetworkbydeliverytooneofthenetwork'spublicentrypoints.Entrypointsinthepostalcasearemailboxes,postoces,oryourfriendlymailmanormailwomanpickinguptheletter. 46Thiscontentisavailableonlineat.

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252CHAPTER6.INFORMATIONCOMMUNICATIONThecommunicationsnetworkdeliversthemessageinthemostecienttimelywaypossible,tryingnottocorruptthemessagewhiledoingso.Themessagearrivesatoneofthenetwork'sexitpoints,andisdeliveredtotherecipientwhatwehavetermedthemessagesink.Exercise6.36Solutiononp.276.Developthenetworkmodelforthetelephonesystem,makingitasanalogousaspossiblewiththepostalservice-communicationsnetworkmetaphor.Whatismostinterestingaboutthenetworksystemistheambivalenceofthemessagesourceandsinkabouthowthecommunicationslinkismade.Whattheydocareaboutismessageintegrityandcommunicationseciency.Furthermore,today'snetworksuseheterogeneouslinks.CommunicationpathsthatformtheInternetusewireline,opticalber,andsatellitecommunicationlinks.Therstelectricalcommunicationsnetworkwasthetelegraph.HerethenetworkconsistedoftelegraphoperatorswhotransmittedthemessageecientlyusingMorsecodeandroutedthemessagesothatittooktheshortestpossiblepathtoitsdestinationwhiletakingintoaccountinternalnetworkfailuresdownedlines,drunkenoperators.Fromtoday'sperspective,thefactthatthisnineteenthcenturysystemhandleddigitalcommunicationsisastounding.Morsecode,whichassignedasequenceofdotsanddashestoeachletterofthealphabet,servedasthesourcecodingalgorithm.Thesignalsetconsistedofashortandalongpulse.Ratherthanamatchedlter,thereceiverwastheoperator'sear,andhewrotethemessagetranslatingfromreceivedbitstosymbols.Note:Becauseoftheneedforacommabetweendot-dashsequencestodenelettersymbolboundaries,theaveragenumberofbits/symbol,asdescribedinSubtletiesofCodingExample6.4,exceededtheSourceCodingTheorem'supperbound.Internally,communicationnetworksdohavepoint-to-pointcommunicationlinksbetweennetworknodeswelldescribedbytheFundamentalModelofCommunications.However,manymessagessharethecom-municationschannelbetweennodesusingwhatwecalltime-domainmultiplexing:RatherthanthecontinuouscommunicationsmodeimpliedintheModelaspresented,messagesequencesaresent,sharingintimethechannel'scapacity.Atagranderviewpoint,thenetworkmustroutemessagesdecidewhatnodesandlinkstousebasedondestinationinformationtheaddressthatisusuallyseparatefromthemessageinformation.Routinginnetworksisnecessarilydynamic:Thecompleteroutetakenbymessagesisformedasthenetworkhandlesthemessage,withnodesrelayingthemessagehavingsomenotionofthebestpossiblepathatthetimeoftransmission.Notethatnoomnipotentrouterviewsthenetworkasawholeandpre-determineseverymessage'sroute.Certainlyinthecaseofthepostalsystemdynamicroutingoccurs,andcanconsiderissueslikeinoperativeandoverlybusylinks.Inthetelephonesystem,routingtakesplacewhenyouplacethecall;therouteisxedoncethephonestartsringing.Moderncommunicationnetworksstrivetoachievethemostecienttimelyandmostreliableinformationdeliverysystempossible.6.34MessageRouting47Focusingonelectricalnetworks,mostanalogonesmakeinecientuseofcommunicationlinksbecausetrulydynamicroutingisdicult,ifnotimpossible,toobtain.Inradionetworks,suchascommercialtelevision,eachstationhasadedicatedportionoftheelectromagneticspectrum,andthisspectrumcannotbesharedwithotherstationsorusedinanyotherthantheregulatedway.Thetelephonenetworkismoredynamic,butonceitestablishesacallthepaththroughthenetworkisxed.Theusersofthatpathcontrolitsuse,andmaynotmakeecientuseofitlongpauseswhileonepersonthinks,forexample.Telephonenetworkcustomerswouldbequiteupsetifthetelephonecompanymomentarilydisconnectedthepathsothatsomeoneelsecoulduseit.Thiskindofconnectionthroughanetworkxedforthedurationofthecommunicationsessionisknownasacircuit-switchedconnection. 47Thiscontentisavailableonlineat.

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253Duringthe1960s,itwasbecomingclearthatnotonlywasdigitalcommunicationtechnicallysuperior,butalsothatthewidevarietyofcommunicationmodescomputerlogin,letransfer,andelectronicmailneededadierentapproachthanpoint-to-point.Thenotionofcomputernetworkswasbornthen,andwhatwasthencalledtheARPANET,nowcalledtheInternet,wasborn.ComputernetworkselaboratethebasicnetworkmodelbysubdividingmessagesintosmallerchunkscalledpacketsFigure6.28.Therationaleforthenetworkenforcingsmallertransmissionswasthatlargeletransferswouldconsumenetworkresourcesallalongtheroute,and,becauseofthelongtransmissiontime,acommunicationfailuremightrequireretransmissionoftheentirele.Bycreatingpackets,eachofwhichhasitsownaddressandisroutedindependentlyofothers,thenetworkcanbettermanagecongestion.Theanalogyisthatthepostalservice,ratherthansendingalongletterintheenvelopeyouprovide,openstheenvelope,placeseachpageinaseparateenvelope,andusingtheaddressonyourenvelope,addresseseachpage'senvelopeaccordingly,andmailsthemseparately.Thenetworkdoesneedtomakesurepacketsequencepagenumberingismaintained,andthenetworkexitpointmustreassembletheoriginalmessageaccordingly. Figure6.28:Longmessages,suchasles,arebrokenintoseparatepackets,thentransmittedovercomputernetworks.Apacket,likealetter,containsthedestinationaddress,thereturnaddresstrans-mitteraddress,andthedata.Thedataincludesthemessagepartandasequencenumberidentifyingitsorderinthetransmittedmessage. Communicationsnetworksarenowcategorizedaccordingtowhethertheyusepacketsornot.Asystemlikethetelephonenetworkissaidtobecircuitswitched:Thenetworkestablishesaxedroutethatlaststheentiredurationofthemessage.Circuitswitchinghastheadvantagethatoncetherouteisdetermined,theuserscanusethecapacityprovidedthemhowevertheylike.Itsmaindisadvantageisthattheusersmaynotusetheircapacityeciently,cloggingnetworklinksandnodesalongtheway.Packet-switchednetworkscontinuouslymonitornetworkutilization,androutemessagesaccordingly.Thus,messagescan,ontheaverage,bedeliveredeciently,butthenetworkcannotguaranteeaspecicamountofcapacitytotheusers.6.35Networkarchitecturesandinterconnection48ThenetworkstructureitsarchitectureFigure6.27typieswhatareknownaswideareanetworksWANs.Thenodes,andusersforthatmatter,arespreadgeographicallyoverlongdistances."Long"hasnoprecisedenition,andisintendedtosuggestthatthecommunicationlinksvarywidely.TheInternetiscertainlythelargestWAN,spanningtheentireearthandbeyond.Localareanetworks,LANs,employasinglecommunicationlinkandspecialrouting.PerhapsthebestknownLANisEthernet49.LANscon48Thiscontentisavailableonlineat.49"Ethernet"

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254CHAPTER6.INFORMATIONCOMMUNICATIONnecttootherLANsandtowideareanetworksthroughspecialnodesknownasgatewaysFigure6.29.IntheInternet,acomputer'saddressconsistsofafourbytesequence,whichisknownasitsIPaddressInternetProtocoladdress.Anexampleaddressis128.42.4.32:eachbyteisseparatedbyaperiod.Thersttwobytesspecifythecomputer'sdomainhereRiceUniversity.Computersarealsoaddressedbyamorehuman-readableform:asequenceofalphabeticabbreviationsrepresentinginstitution,typeofinsti-tution,andcomputername.Agivencomputerhasbothnames128.42.4.32isthesameassoma.rice.edu.DatatransmissionontheInternetrequiresthenumericalform.So-callednameserverstranslatebetweenalphabeticandnumericalforms,andthetransmittingcomputerrequeststhistranslationbeforethemessageissenttothenetwork. Figure6.29:ThegatewayservesasaninterfacebetweenlocalareanetworksandtheInternet.ThetwoshownheretranslatebetweenLANandWANprotocols;oneofthesealsointerfacesbetweentwoLANs,presumablybecausetogetherthetwoLANswouldbegeographicallytoodispersed. 6.36Ethernet50 Figure6.30:TheEthernetarchitectureconsistsofasinglecoaxialcableterminatedateitherendbyaresistorhavingavalueequaltothecable'scharacteristicimpedance.ComputersattachtotheEthernetthroughaninterfaceknownasatransceiverbecauseitsendsaswellasreceivesbitstreamsrepresentedasanalogvoltages. 50Thiscontentisavailableonlineat.

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255EthernetusesasitscommunicationmediumasinglelengthofcoaxialcableFigure6.30.Thiscableservesasthe"ether",throughwhichalldigitaldatatravel.Electrically,computersinterfacetothecoaxialcableFigure6.30throughadeviceknownasatransceiver.Thisdeviceiscapableofmonitoringthevoltageappearingbetweenthecoreconductorandtheshieldaswellasapplyingavoltagetoit.Conceptuallyitconsistsoftwoop-amps,oneapplyingavoltagecorrespondingtoabitstreamtransmittingdataandanotherservingasanamplierofEthernetvoltagesignalsreceivingdata.ThesignalsetforEthernetresemblesthatshowninBPSKSignalSets,withonesignalthenegativeoftheother.Computersareattachedinparallel,resultinginthecircuitmodelforEthernetshowninFigure6.31.Exercise6.37Solutiononp.276.Fromtheviewpointofatransceiver'ssendingop-amp,whatistheloaditseesandwhatisthetransferfunctionbetweenthisoutputvoltageandsomeothertransceiver'sreceivingcircuit?WhyshouldtheoutputresistorRoutbelarge? Figure6.31:Thetopcircuitexpressesasimpliedcircuitmodelforatransceiver.TheoutputresistanceRoutmustbemuchlargerthanZ0sothatthesumofthevarioustransmittervoltagesaddtocreatetheEthernetconductor-to-shieldvoltagethatservesasthereceivedsignalrtforalltransceivers.Inthiscase,theequivalentcircuitshowninthebottomcircuitapplies. Noonecomputerhasmoreauthoritythananyothertocontrolwhenandhowmessagesaresent.With-outschedulingauthority,youmightwellwonderhowonecomputersendstoanotherwithoutthelargeinterferencethattheothercomputerswouldproduceiftheytransmittedatthesametime.TheinnovationofEthernetisthatcomputersschedulethemselvesbyarandom-accessmethod.Thismethodreliesonthefactthatallpacketstransmittedoverthecoaxialcablecanbereceivedbyalltransceivers,regardlessofwhichcomputermightactuallybetheintendedrecipient.Incommunicationsterminology,Ethernetdirectlysupportsbroadcast.Eachcomputergoesthroughthefollowingstepstosendapacket.1.Thecomputersensesthevoltageacrossthecabletodetermineifsomeothercomputeristransmitting.

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256CHAPTER6.INFORMATIONCOMMUNICATION2.Ifanothercomputeristransmitting,waituntilthetransmissionsnishandgobacktotherststep.Ifthecablehasnotransmissions,begintransmittingthepacket.3.Ifthereceiverportionofthetransceiverdeterminesthatnoothercomputerisalsosendingapacket,continuetransmittingthepacketuntilcompletion.4.Ontheotherhand,ifthereceiversensesinterferencefromanothercomputer'stransmissions,immedi-atelyceasetransmission,waitingarandomamountoftimetoattemptthetransmissionagaingotostep1untilonlyonecomputertransmitsandtheothersdefer.Theconditionwhereintwoormorecomputers'transmissionsinterferewithothersisknownasacollision.Thereasontwocomputerswaitingtotransmitmaynotsensetheother'stransmissionimmediatelyarisesbecauseofthenitepropagationspeedofvoltagesignalsthroughthecoaxialcable.Thelongesttimeanycomputermustwaittodetermineifitstransmissionsdonotencounterinterferenceis2L c,whereListhecoaxialcable'slength.Themaximum-length-specicationforEthernetis1km.Assumingapropagationspeedof2/3thespeedoflight,thistimeintervalismorethan10s.AsanalyzedinProblem22Prob-lem6.30,thenumberofthesetimeintervalsrequiredtoresolvethecollisionis,ontheaverage,lessthantwo!Exercise6.38Solutiononp.276.Whydoesthefactoroftwoenterintothisequation?Considertheworst-casesituationoftwotransmittingcomputerslocatedattheEthernet'sends.Thus,despitenothavingseparatecommunicationpathsamongthecomputerstocoordinatetheirtransmis-sions,theEthernetrandomaccessprotocolallowscomputerstocommunicatewithoutonlyaslightdegra-dationineciency,asmeasuredbythetimetakentoresolvecollisionsrelativetothetimetheEthernetisusedtotransmitinformation.AsubtleconsiderationinEthernetistheminimumpacketsizePmin.ThetimerequiredtotransmitsuchpacketsequalsPmin C,whereCistheEthernet'scapacityinbps.Ethernetnowcomesintwodierenttypes,eachwithindividualspecications,themostdistinguishingofwhichiscapacity:10Mbpsand100Mbps.IftheminimumtransmissiontimeissuchthatthebeginningofthepackethasnotpropagatedthefulllengthoftheEthernetbeforetheend-of-transmission,itispossiblethattwocomputerswillbegintransmissionatthesametimeand,bythetimetheirtransmissionscease,theother'spacketwillnothavepropagatedtotheother.Inthiscase,computersin-betweenthetwowillsenseacollision,whichrendersbothcomputer'stransmissionssenselesstothem,withoutthetwotransmittingcomputersknowingacollisionhasoccurredatall!ForEthernettosucceed,wemusthavetheminimumpackettransmissiontimeexceedtwicethevoltagepropagationtime:Pmin C>2L corPmin>2LC c.63Thus,forthe10MbpsEthernethavinga1kmmaximumlengthspecication,theminimumpacketsizeis200bits.Exercise6.39Solutiononp.277.The100MbpsEthernetwasdesignedmorerecentlythanthe10Mbpsalternative.Tomaintainthesameminimumpacketsizeastheearlier,slowerversion,whatshoulditslengthspecicationbe?Whyshouldtheminimumpacketsizeremainthesame?6.37CommunicationProtocols51Thecomplexityofinformationtransmissioninacomputernetworkreliabletransmissionofbitsacrossachannel,routing,anddirectinginformationtothecorrectdestinationwithinthedestinationcomputersoperatingsystemdemandsanoverarchingconceptofhowtoorganizeinformationdelivery.Nouniquesetof 51Thiscontentisavailableonlineat.

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257rulessatisesthevariousconstraintscommunicationchannelsandnetworkorganizationplaceoninformationtransmission.Forexample,randomaccessissuesinEthernetarenotpresentinwide-areanetworkssuchastheInternet.Aprotocolisasetofrulesthatgovernshowinformationisdelivered.Forexample,tousethetelephonenetwork,theprotocolistopickupthephone,listenforadialtone,dialanumberhavingaspecicnumberofdigits,waitforthephonetoring,andsayhello.Inradio,thestationusesamplitudeorfrequencymodulationwithaspeciccarrierfrequencyandtransmissionbandwidth,andyouknowtoturnontheradioandtuneinthestation.Intechnicalterms,nooneprotocolorsetofprotocolscanbeusedforanycommunicationsituation.Bethatasitmay,communicationengineershavefoundthatacommonthreadrunsthroughtheorganizationofthevariousprotocols.Thisgranddesignofinformationtransmissionorganizationrunsthroughallmodernnetworkstoday.Whathasbeendenedasanetworkingstandardisalayered,hierarchicalprotocolorganization.AsshowninFigure6.32ProtocolPicture,protocolsareorganizedbyfunctionandlevelofdetail.ProtocolPicture Figure6.32:Protocolsareorganizedaccordingtothelevelofdetailrequiredforinformationtransmis-sion.Protocolsatthelowerlevelsshowntowardthebottomconcernreliablebittransmission.Higherlevelprotocolsconcernhowbitsareorganizedtorepresentinformation,whatkindofinformationisde-nedbybitsequences,whatsoftwareneedstheinformation,andhowtheinformationistobeinterpreted.BodiessuchastheIEEEInstituteforElectronicsandElectricalEngineersandtheISOInternationalStandardsOrganizationdenestandardssuchasthis.Despitebeingastandard,itdoesnotconstrainprotocolimplementationsomuchthatinnovationandcompetitiveindividualityareruledout.Segregationofinformationtransmission,manipulation,andinterpretationintothesecategoriesdirectlyaectshowcommunicationsystemsareorganized,andwhatrolessoftwaresystemsfulll.Althoughnotthoughtaboutinthiswayinearliertimes,thisorganizationalstructuregovernsthewaycommunicationengineersthinkaboutallcommunicationsystems,fromradiototheInternet.Exercise6.40Solutiononp.277.Howdothevariousaspectsofestablishingandmaintainingatelephoneconversationtintothislayeredprotocolorganization?Wenowexplicitlystatewhetherweareworkinginthephysicallayersignalsetdesign,forexample,thedatalinklayersourceandchannelcoding,oranyotherlayer.IPabbreviatesInternetprotocol,andgovernsgatewayshowinformationistransmittedbetweennetworkshavingdierentinternalorganizations.TCPtransmissioncontrolprotocolgovernshowpacketsaretransmittedthroughawide-areanetworksuchastheInternet.Telnetisaprotocolthatconcernshowapersonatonecomputerlogsontoanothercomputeracrossanetwork.Amoderatelyhighlevelprotocolsuchastelnet,isnotconcernedwithwhatdatalinks

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258CHAPTER6.INFORMATIONCOMMUNICATIONwirelineorwirelessmighthavebeenusedbythenetworkorhowpacketsarerouted.Rather,itestablishesconnectionsbetweencomputersanddirectseachbytepresumedtorepresentatypedcharactertotheappropriateoperationsystemcomponentateachend.Itisnotconcernedwithwhatthecharactersmeanorwhatprogramsthepersonistypingto.Thataspectofinformationtransmissionislefttoprotocolsathigherlayers.Recently,animportantsetofprotocolscreatedtheWorldWideWeb.TheseprotocolsexistindependentlyoftheInternet.TheInternetinsuresthatmessagesaretransmittedecientlyandintact;theInternetisnotconcernedtodatewithwhatmessagescontain.HTTPhypertexttransferprotocolframewhatmessagescontainandwhatshouldbedonewiththedata.TheextremelyrapiddevelopmentoftheWebontopofanessentiallystagnantInternetisbutoneexampleofthepoweroforganizinghowinformationtransmissionoccurswithoutoverlyconstrainingthedetails.6.38InformationCommunicationProblems52Problem6.1:SignalsonTransmissionLinesAmodulatedsignalneedstobesentoveratransmissionlinehavingacharacteristicimpedanceofZ0=50.Sothatthesignaldoesnotinterferewithsignalsothersmaybetransmitting,itmustbebandpasslteredsothatitsbandwidthis1MHzandcenteredat3.5MHz.Thelter'sgainshouldbeoneinmagnitude.Anop-amplterFigure6.33isproposed. Figure6.33 aWhatisthetransferfunctionbetweentheinputvoltageandthevoltageacrossthetransmissionline?bFindvaluesfortheresistorsandcapacitorssothatdesigngoalsaremet.Problem6.2:NoiseinAMSystemsThesignal^stemergingfromanAMcommunicationsystemconsistsoftwoparts:themessagesignal,st,andadditivenoise.TheplotFigure6.34showsthemessagespectrumSfandnoisepowerspectrumPNf.Thenoisepowerspectrumliescompletelywithinthesignal'sband,andhasaconstantvaluethereofN0 2. 52Thiscontentisavailableonlineat.

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259 Figure6.34 aWhatisthemessagesignal'spower?Whatisthesignal-to-noiseratio?bBecausethepowerinthemessagedecreaseswithfrequency,thesignal-to-noiseratioisnotconstantwithinsubbands.Whatisthesignal-to-noiseratiointheupperhalfofthefrequencyband?cAclever241studentsuggestslteringthemessagebeforethetransmittermodulatesitsothatthesignalspectrumisbalancedconstantacrossfrequency.Realizingthatthislteringaectsthemessagesignal,thestudentrealizesthatthereceivermustalsocompensateforthemessagetoarriveintact.Drawablockdiagramofthiscommunicationsystem.Howdoesthissystem'ssignal-to-noiseratiocomparewiththatoftheusualAMradio?Problem6.3:ComplementaryFiltersComplementaryltersusuallyhaveoppositelteringcharacteristicslikealowpassandahighpassandhavetransferfunctionsthataddtoone.Mathematically,H1fandH2farecomplementaryifH1f+H2f=1Wecanusecomplementarylterstoseparateasignalintotwopartsbypassingitthrougheachlter.Eachoutputcanthenbetransmittedseparatelyandtheoriginalsignalreconstructedatthereceiver.Let'sassumethemesageisbandlimitedtoWHzandthatH1f=a a+j2f.aWhatcircuitswouldbeusedtoproducethecomplementarylters?bSketchablockdiagramforacommunicationsystemtransmitterandreceiverthatemployscomple-mentarysignaltransmissiontosendamessagemt.cWhatisthereceiver'ssignal-to-noiseratio?Howdoesitcomparetothestandardsystemthatsendsthesignalbysimpleamplitudemodulation?Problem6.4:PhaseModulationAmessagesignalmtphasemodulatesacarrierifthetransmittedsignalequalsxt=Asinfct+dmtwheredisknownasthephasedeviation.Inthisproblem,thephasedeviationissmall.Aswithallanalogmodulationschemes,assumethatjmtj<1,themessageisbandlimitedtoWHz,andthecarrierfrequencyfcismuchlargerthanW.aWhatisthetransmissionbandwidth?bFindareceiverforthismodulationscheme.cWhatisthesignal-to-noiseratioofthereceivedsignal?hint:Usethefactsthatcosx1andsinxxforsmallx.

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260CHAPTER6.INFORMATIONCOMMUNICATIONProblem6.5:DigitalAmplitudeModulationTwoELEC241studentsdisagreeaboutahomeworkproblem.Theissueconcernsthediscrete-timesignalsncosf0n,wherethesignalsnhasnospecialcharacteristicsandthemodulationfrequencyf0isknown.Sammysaysthathecanrecoversnfromitsamplitude-modulatedversionbythesameapproachusedinanalogcommunications.Samanthasaysthatapproachwon'twork.aWhatisthespectrumofthemodulatedsignal?bWhoiscorrect?Why?cTheteachingassistantdoesnotwanttotakesides.Hetellsthemthatifsncosf0nandsnsinf0nwerebothavailable,sncanberecovered.Whatdoeshehaveinmind?Problem6.6:Anti-JammingOnewayforsomeonetokeeppeoplefromreceivinganAMtransmissionistotransmitnoiseatthesamecar-rierfrequency.Thus,ifthecarrierfrequencyisfcsothatthetransmittedsignalisAT+mtsinfctthejammerwouldtransmitAJntsinfct+.Thenoisenthasaconstantpowerdensityspectrumoverthebandwidthofthemessagemt.ThechanneladdswhitenoiseofspectralheightN0 2.aWhatwouldbetheoutputofatraditionalAMreceivertunedtothecarrierfrequencyfc?bRUElectronicsproposestocounteractjammingbyusingadierentmodulationscheme.Thescheme'stransmittedsignalhastheformAT+mtctwherectisaperiodiccarriersignalperiod1 fchavingtheindicatedwaveformFigure6.35.Whatisthespectrumofthetransmittedsignalwiththeproposedscheme?AssumethemessagebandwidthWismuchlessthanthefundamentalcarrierfrequencyfc.cThejammer,unawareofthechange,istransmittingwithacarrierfrequencyoffc,whilethereceivertunesastandardAMreceivertoaharmonicofthecarrierfrequency.Whatisthesignal-to-noiseratioofthereceivertunedtotheharmonichavingthelargestpowerthatdoesnotcontainthejammer? Figure6.35 Problem6.7:SecretComunicationsAsystemforhidingAMtransmissionshasthetransmitterrandomlyswitchingbetweentwocarrierfre-quenciesf1andf2."Randomswitching"meansthatonecarrierfrequencyisusedforsomeperiodoftime,switchestotheotherforsomeotherperiodoftime,backtotherst,etc.Thereceiverknowswhatthecarrierfrequenciesarebutnotwhencarrierfrequencyswitchesoccur.Consequently,thereceivermustbedesignedtoreceivethetransmissionsregardlessofwhichcarrierfrequencyisused.AssumethemessagesignalhasbandwidthW.ThechanneladdswhitenoiseofspectralheightN0 2.aHowdierentshouldthecarrierfrequenciesbesothatthemessagecouldbereceived?bWhatreceiverwouldyoudesign?cWhatsignal-to-noiseratioforthedemodulatedsignaldoesyourreceiveryield?

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261Problem6.8:AMStereoStereophonicradiotransmitstwosignalssimultaneouslythatcorrespondtowhatcomesoutoftheleftandrightspeakersofthereceivingradio.WhileFMstereoiscommonplace,AMstereoisnot,butismuchsimplertounderstandandanalyze.AnamazingaspectofAMstereoisthatbothsignalsaretransmittedwithinthesamebandwidthasusedtotransmitjustone.AssumetheleftandrightsignalsarebandlimitedtoWHz.xt=A+mltcosfct+AmrtsinfctaFindtheFouriertransformofxt.WhatisthetransmissionbandwidthandhowdoesitcomparewiththatofstandardAM?bLetususeacoherentdemodulatorasthereceiver,showninFigure6.36.Showthatthisreceiverindeedworks:Itproducestheleftandrightsignalsseparately.cAssumethechanneladdswhitenoisetothetransmittedsignal.Findthesignal-to-noiseratioofeachsignal. Figure6.36 Problem6.9:ANovelCommunicationSystemAcleversystemdesignerclaimsthatthedepictedtransmitterFigure6.37has,despiteitscomplexity,advantagesovertheusualamplitudemodulationsystem.ThemessagesignalmtisbandlimitedtoWHz,andthecarrierfrequencyfcW.ThechannelattenuatesthetransmittedsignalxtandaddswhitenoiseofspectralheightN0 2. Figure6.37

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262CHAPTER6.INFORMATIONCOMMUNICATIONThetransferfunctionHfisgivenbyHf=8<:jiff<0)]TJ/F11 9.9626 Tf 7.7487 0 Td[(jiff>0aFindanexpressionforthespectrumofxt.Sketchyouranswer.bShowthattheusualcoherentreceiverdemodulatesthissignal.cFindthesignal-to-noiseratiothatresultswhenthisreceiverisused.dFindasuperiorreceiveronethatyieldsabettersignal-to-noiseratio,andanalyzeitsperformance.Problem6.10:Multi-ToneDigitalCommunicationInaso-calledmulti-tonesystem,severalbitsaregatheredtogetherandtransmittedsimultaneouslyondierentcarrierfrequenciesduringaTsecondinterval.Forexample,Bbitswouldbetransmittedaccordingtoxt=AB)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1Xk=0bksink+1f0t;0t
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263bAssumedis1km.Findandsketchthemagnitudeofthetransferfunctionforthemultipathcomponentofthechannel.Howwouldyoucharacterizethistransferfunction?cWouldthemultipathaectAMradio?Ifnot,whynot;ifso,howso?Wouldanalogcellulartelephone,whichoperatesatmuchhighercarrierfrequenciesMHzvs.1MHzforradio,beaectedornot?Analogcellulartelephoneusesamplitudemodulationtotransmitvoice.dHowwouldtheusualAMreceiverbemodiedtominimizemultipatheects?Expressyourmodiedreceiverasablockdiagram.Problem6.12:DownlinkSignalSetsIndigitalcellulartelephonesystems,thebasestationtransmitterneedstorelaydierentvoicesignalstoseveraltelephonesatthesametime.Ratherthansendsignalsatdierentfrequencies,acleverRiceengineersuggestsusingadierentsignalsetforeachdatastream.Forexample,fortwosimultaneousdatastreams,shesuggestsBPSKsignalsetsthathavethedepictedbasicsignalsFigure6.39. Figure6.39 Thus,bitsarerepresentedindatastream1bys1tand)]TJ/F8 9.9626 Tf 9.4091 0 Td[(s1tandindatastream2bys2tand)]TJ/F8 9.9626 Tf 9.4091 0 Td[(s2t,eachofwhicharemodulatedby900MHzcarrier.Thetransmittersendsthetwodatastreamssothattheirbitintervalsalign.Eachreceiverusesamatchedlterforitsreceiver.Therequirementisthateachreceivernotreceivetheother'sbitstream.aWhatistheblockdiagramdescribingtheproposedsystem?bWhatisthetransmissionbandwidthrequiredbytheproposedsystem?cWilltheproposalwork?Doesthefactthatthetwodatastreamsaretransmittedinthesamebandwidthatthesametimemeanthateachreceiver'sperformanceisaected?Caneachbitstreambereceivedwithoutinterferencefromtheother?Problem6.13:MixedAnalogandDigitalTransmissionAsignalmtistransmittedusingamplitudemodulationintheusualway.ThesignalhasbandwidthWHz,andthecarrierfrequencyisfc.Inadditiontosendingthisanalogsignal,thetransmitteralsowantstosendASCIItextinanauxiliarybandthatliesslightlyabovetheanalogtransmissionband.Usingan8-bitrepresentationofthecharactersandasimplebasebandBPSKsignalsettheconstantsignal+1correspondstoa0,theconstant-1toa1,thedatasignaldtrepresentingthetextistransmittedasthesametimeastheanalogsignalmt.ThetransmissionsignalspectrumisasshownFigure6.40,andhasatotalbandwidthB.aWriteanexpressionforthetime-domainversionofthetransmittedsignalintermsofmtandthedigitalsignaldt.bWhatisthemaximumdataratetheschemecanprovideintermsoftheavailablebandwidth?cFindareceiverthatyieldsboththeanalogsignalandthebitstream.

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264CHAPTER6.INFORMATIONCOMMUNICATION Figure6.40 Problem6.14:DigitalStereoJustaswithanalogcommunication,itshouldbepossibletosendtwosignalssimultaneouslyoveradigitalchannel.AssumeyouhavetwoCD-qualitysignalseachsampledat44.1kHzwith16bits/sample.OnesuggestedtransmissionschemeistouseaquadratureBPSKscheme.Ifbnandbneachrepresentabitstream,thetransmittedsignalhastheformxt=AXbnsinfct)]TJ/F11 9.9626 Tf 9.9626 0 Td[(nTpt)]TJ/F11 9.9626 Tf 9.9626 0 Td[(nT+bncosfct)]TJ/F11 9.9626 Tf 9.9626 0 Td[(nTpt)]TJ/F11 9.9626 Tf 9.9626 0 Td[(nTwhereptisaunit-amplitudepulsehavingdurationTandbn,bnequaleither+1or-1accordingtothebitbeingtransmittedforeachsignal.Thechanneladdswhitenoiseandattenuatesthetransmittedsignal.aWhatvaluewouldyouchooseforthecarrierfrequencyfc?bWhatisthetransmissionbandwidth?cWhatreceiverwouldyoudesignthatwouldyieldbothbitstreams?Problem6.15:DigitalandAnalogSpeechCommunicationSupposewetransmitspeechsignalsovercomparabledigitalandanalogchannels.Wewanttocomparetheresultingqualityofthereceivedsignals.Assumethetransmittersusethesamepower,andthechannelsintroducethesameattenuationandadditivewhitenoise.Assumethespeechsignalhasa4kHzbandwidthand,inthedigitalcase,issampledatan8kHzratewitheight-bitA/Dconversion.AssumesimplebinarysourcecodingandamodulatedBPSKtransmissionscheme.aWhatisthetransmissionbandwidthoftheanalogAManddigitalschemes?bAssumethespeechsignal'samplitudehasamagnitudelessthanone.WhatismaximumamplitudequantizationerrorintroducedbytheA/Dconverter?cInthedigitalcase,eachbitinquantizedspeechsampleisreceivedinerrorwithprobabilitypethatdependsonsignal-to-noiseratioEb N0.However,errorsineachbithaveadierentimpactontheerrorinthereconstructedspeechsample.Findthemean-squarederrorbetweenthetransmittedandreceivedamplitude.dInthedigitalcase,therecoveredspeechsignalcanbeconsideredtohavetwonoisesourcesaddedtoeachsample'struevalue:OneistheA/Damplitudequantizationnoiseandthesecondisduetochannelerrors.Becausetheseareseparate,thetotalnoisepowerequalsthesumofthesetwo.Whatisthesignal-to-noiseratioofthereceivedspeechsignalasafunctionofpe?eComputeandplotthereceivedsignal'ssignal-to-noiseratioforthetwotransmissionschemesforafewvaluesofchannelsignal-to-noiseratios.fCompareandevaluatethesesystems.

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265Problem6.16:SourceCompressionConsiderthefollowing5-lettersource. Letter Probability a 0.5 b 0.25 c 0.125 d 0.0625 e 0.0625 aFindthissource'sentropy.bShowthatthesimplebinarycodingisinecient.cFindanunequal-lengthcodebookforthissequencethatsatisestheSourceCodingTheorem.Doesyourcodeachievetheentropylimit?dHowmuchmoreecientisthiscodethanthesimplebinarycode?Problem6.17:SourceCompressionConsiderthefollowing5-lettersource. Letter Probability a 0.4 b 0.2 c 0.15 d 0.15 e 0.1 aFindthissource'sentropy.bShowthatthesimplebinarycodingisinecient.cFindtheHumancodeforthissource.Whatisitsaveragecodelength?Problem6.18:SpeechCompressionWhenwesampleasignal,suchasspeech,wequantizethesignal'samplitudetoasetofintegers.Forab-bitconverter,signalamplitudesarerepresentedby2bintegers.Althoughtheseintegerscouldberepresentedbyabinarycodefordigitaltransmission,weshouldconsiderwhetheraHumancodingwouldbemoreecient.aLoadintoMatlabthesegmentofspeechcontainediny.mat.Itssampledvalueslieintheinterval-1,1.Tosimulatea3-bitconverter,weuseMatlab'sroundfunctiontocreatequantizedamplitudescorrespondingtotheintegers[01234567].y_quant=round.5*y+3.5;Findtherelativefrequencyofoccurrenceofquantizedamplitudevalues.ThefollowingMatlabprogramcomputesthenumberoftimeseachquantizedvalueoccurs.forn=0:7;countn+1=sumy_quant==n;end;Findtheentropyofthissource.

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266CHAPTER6.INFORMATIONCOMMUNICATIONbFindtheHumancodeforthissource.Howwouldyoucharacterizethissourcecodeinwords?cHowmanyfewerbitswouldbeusedintransmittingthisspeechsegmentwithyourHumancodeincomparisontosimplebinarycoding?Problem6.19:DigitalCommunicationInadigitalcellularsystem,asignalbandlimitedto5kHzissampledwithatwo-bitA/DconverteratitsNyquistfrequency.Thesamplevaluesarefoundtohavetheshownrelativefrequencies. SampleValue Probability 0 0.15 1 0.35 2 0.3 3 0.2 WesendthebitstreamconsistingofHuman-codedsamplesusingoneofthetwodepictedsignalsetsFigure6.41. Figure6.41 aWhatisthedatarateofthecompressedsource?bWhichchoiceofsignalsetmaximizesthecommunicationsystem'sperformance?cWithnoerror-correctingcoding,whatsignal-to-noiseratiowouldbeneededforyourchosensignalsettoguaranteethatthebiterrorprobabilitywillnotexceed10)]TJ/F7 6.9738 Tf 6.2266 0 Td[(3?Ifthereceivermovestwiceasfarfromthetransmitterrelativetothedistanceatwhichthe10)]TJ/F7 6.9738 Tf 6.2267 0 Td[(3errorratewasobtained,howdoestheperformancechange?Problem6.20:SignalCompressionLettersdrawnfromafour-symbolalphabethavetheindicatedprobabilities. Letter Probability a 1/3 b 1/3 c 1/4 d 1/12

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267aWhatistheaveragenumberofbitsnecessarytorepresentthisalphabet?bUsingasimplebinarycodeforthisalphabet,atwo-bitblockofdatabitsnaturallyemerges.Findanerrorcorrectingcodefortwo-bitdatablocksthatcorrectsallsingle-biterrors.cHowwouldyoumodifyyourcodesothattheprobabilityoftheletterabeingconfusedwiththeletterdisminimized?Ifso,whatisyournewcode;ifnot,demonstratethatthisgoalcannotbeachieved.Problem6.21:UniversalProductCodeTheUniversalProductCodeUPC,oftenknownasabarcode,labelsvirtuallyeverysoldgood.AnexampleFigure6.42ofaportionofthecodeisshown. Figure6.42 Hereasequenceofblackandwhitebars,eachhavingwidthd,presentsan11-digitnumberconsistingofdecimaldigitsthatuniquelyidentiestheproduct.Inretailstores,laserscannersreadthiscode,andafteraccessingadatabaseofprices,enterthepriceintothecashregister.aHowmanybarsmustbeusedtorepresentasingledigit?bAcomplicationofthelaserscanningsystemisthatthebarcodemustbereadeitherforwardsorbackwards.Nowhowmanybarsareneededtorepresenteachdigit?cWhatistheprobabilitythatthe11-digitcodeisreadcorrectlyiftheprobabilityofreadingasinglebitincorrectlyispe?dHowmanyerrorcorrectingbarswouldneedtobepresentsothatanysinglebarerroroccurringinthe11-digitcodecanbecorrected?Problem6.22:ErrorCorrectingCodesAcodemapspairsofinformationbitsintocodewordsoflength5asfollows. Data Codeword 00 00000 01 01101 10 10111 11 11010 aWhatisthiscode'seciency?bFindthegeneratormatrixGandparity-checkmatrixHforthiscode.cGivethedecodingtableforthiscode.Howmanypatternsof1,2,and3errorsarecorrectlydecoded?dWhatistheblockerrorprobabilitytheprobabilityofanynumberoferrorsoccurringinthedecodedcodeword?

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268CHAPTER6.INFORMATIONCOMMUNICATIONProblem6.23:OverlyDesignedErrorCorrectionCodesAnAggieengineerwantsnotonlytohavecodewordsforhisdata,butalsotohidetheinformationfromRiceengineersnofearoftheUTengineers.Hedecidestorepresent3-bitdatawith6-bitcodewordsinwhichnoneofthedatabitsappearexplicitly.c1=d1d2c2=d2d3c3=d1d3c4=d1d2d3c5=d1d2c6=d1d2d3aFindthegeneratormatrixGandparity-checkmatrixHforthiscode.bFinda36matrixthatrecoversthedatabitsfromthecodeword.cWhatistheerrorcorrectingcapabilityofthecode?Problem6.24:ErrorCorrection?Itisimportanttorealizethatwhenmoretransmissionerrorsthancanbecorrected,errorcorrectionalgorithmsbelievethatasmallernumberoferrorshaveoccurredandcorrectaccordingly.Forexample,considera,4HammingCodehavingthegeneratormatrixG=0BBBBBBBBBBBBB@10000100001000011110011110111CCCCCCCCCCCCCAThiscodecorrectsallsingle-biterror,butifadoublebiterroroccurs,itcorrectsusingasingle-biterrorcorrectionapproach.aHowmanydouble-biterrorscanoccurinacodeword?bForeachdouble-biterrorpattern,whatistheresultofchanneldecoding?Expressyourresultasabinaryerrorsequenceforthedatabits.Problem6.25:SelectiveErrorCorrectionWehavefoundthatdigitaltransmissionerrorsoccurwithaprobabilitythatremainsconstantnomatterhow"important"thebitmaybe.Forexample,intransmittingdigitizedsignals,errorsoccurasfrequentlyforthemostsignicantbitastheydofortheleastsignicantbit.Yet,theformererrorshaveamuchlargerimpactontheoverallsignal-to-noiseratiothanthelatter.Ratherthanapplyingerrorcorrectiontoeachsamplevalue,whynotconcentratetheerrorcorrectiononthemostimportantbits?Assumethatwesamplean8kHzsignalwithan8-bitA/Dconverter.Weusesingle-biterrorcorrectiononthemostsignicantfourbitsandnoneontheleastsignicantfour.BitsaretransmittedusingamodulatedBPSKsignalsetoveranadditivewhitenoisechannel.aHowmanyerrorcorrectionbitsmustbeaddedtoprovidesingle-biterrorcorrectiononthemostsignicantbits?bHowlargemustthesignal-to-noiseratioofthereceivedsignalbetoinsurereliablecommunication?

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269cAssumethatonceerrorcorrectionisapplied,onlytheleastsignicant4bitscanbereceivedinerror.Howmuchwouldtheoutputsignal-to-noiseratioimproveusingthiserrorcorrectionscheme?Problem6.26:CompactDiskErrorsoccurinreadingaudiocompactdisks.Veryfewerrorsareduetonoiseinthecompactdiskplayer;mostoccurbecauseofdustandscratchesonthedisksurface.Becausescratchesspanseveralbits,asingle-biterrorisrare;severalconsecutivebitsinerroraremuchmorecommon.Assumethatscratchanddust-inducederrorsarefourorfewerconsecutivebitslong.TheaudioCDstandardrequires16-bit,44.1kHzanalog-to-digitalconversionofeachchannelofthestereoanalogsignal.aHowmanyerror-correctionbitsarerequiredtocorrectscratch-inducederrorsforeach16-bitsample?bRatherthanuseacodethatcancorrectseveralerrorsinacodeword,aclever241engineerproposesinterleavingconsecutivecodedsamples.AsthecartoonFigure6.43shows,thebitsrepresentingcodedsamplesareinterpersedbeforetheyarewrittenontheCD.TheCDplayerde-interleavesthecodeddata,thenperformserror-correction.Now,evaluatethisproposedschemewithrespecttothenon-interleavedone. Figure6.43 Problem6.27:CommunicationSystemDesignRUCommunicationSystemshasbeenaskedtodesignacommunicationsystemthatmeetsthefollowingrequirements.Thebasebandmessagesignalhasabandwidthof10kHz.TheRUCSengineersndthattheentropyHofthesampledmessagesignaldependsonhowmanybitsbareusedintheA/Dconverterseetablebelow.Thesignalistobesentthroughanoisychannelhavingabandwidthof25kHzchannelcenteredat2MHzandasignal-to-noiserationwithinthatbandof10dB.Oncereceived,themessagesignalmusthaveasignal-to-noiseratioofatleast20dB. b H 3 2.19 4 3.25 5 4.28 6 5.35 Canthesespecicationsbemet?Justifyyouranswer.Problem6.28:HDTVAsHDTVhigh-denitiontelevisionwasbeingdeveloped,theFCCrestrictedthisdigitalsystemtouseinthesamebandwidth6MHzasitsanalogAMcounterpart.HDTVvideoissampledona10351840rasterat30imagespersecondforeachofthethreecolors.Theleast-acceptablepicturereceivedbytelevisionsetslocatedatananalogstation'sbroadcastperimeterhasasignal-to-noiseratioofabout10dB.

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270CHAPTER6.INFORMATIONCOMMUNICATIONaUsingsignal-to-noiseratioasthecriterion,howmanybitspersamplemustbeusedtoguaranteethatahigh-qualitypicture,whichachievesasignal-to-noiseratioof20dB,canbereceivedbyanyHDTVsetwithinthesamebroadcastregion?bAssumingthedigitaltelevisionchannelhasthesamecharacteristicsasananalogone,howmuchcompressionmustHDTVsystemsemploy?Problem6.29:DigitalCellularTelephonesIndesigningadigitalversionofawirelesstelephone,youmustrstconsidercertainfundamentals.Firstofall,thequalityofthereceivedsignal,asmeasuredbythesignal-to-noiseratio,mustbeatleastasgoodasthatprovidedbywirelinetelephones30dBandthemessagebandwidthmustbethesameaswirelinetelephone.Thesignal-to-noiseratiooftheallocatedwirelsschannel,whichhasa5kHzbandwidth,measured100metersfromthetoweris70dB.Thedesiredrangeforacellis1km.Canadigitalcellphonesystembedesignedaccordingtothesecriteria?Problem6.30:OptimialEthernetRandomAccessProtocolsAssumeapopulationofNcomputerswanttotransmitinformationonarandomaccesschannel.Theaccessalgorithmworksasfollows.Beforetransmitting,ipacointhathasprobabilitypofcomingupheadsIfonlyoneoftheNcomputer'scoinscomesupheads,itstransmissionoccurssuccessfully,andtheothersmustwaituntilthattransmissioniscompleteandthenresumethealgorithm.Ifnoneormorethanoneheadcomesup,theNcomputerswilleitherremainsilentnoheadsoracollisionwilloccurmorethanonehead.Thisunsuccessfultransmissionsituationwillbedetectedbyallcomputersoncethesignalshavepropagatedthelengthofthecable,andthealgorithmresumesreturntothebeginning.aWhatistheoptimalprobabilitytouseforippingthecoin?Inotherwords,whatshouldpbetomaximizetheprobabilitythatexactlyonecomputertransmits?bWhatistheprobabilityofonecomputertransmittingwhenthisoptimalvalueofpisusedasthenumberofcomputersgrowstoinnity?cUsingthisoptimalprobability,whatistheaveragenumberofcoinipsthatwillbenecessarytoresolvetheaccesssothatonecomputersuccessfullytransmits?dEvaluatethisalgorithm.Isitrealistic?Isitecient?Problem6.31:RepeatersBecausesignalsattenuatewithdistancefromthetransmitter,repeatersarefrequentlyemployedforbothanaloganddigitalcommunication.Forexample,let'sassumethatthetransmitterandreceiverareDmapart,andarepeaterispositionedhalfwaybetweenthemFigure6.44.Whattherepaterdoesisamplifyitsreceivedsignaltoexactlycanceltheattenuationencounteredalongtherstlegandtore-transmitthesignaltotheultimatereceiver.However,thesignaltherepeaterreceivescontainswhitenoiseaswellasthetransmittedsignal.Thereceiverexperiencesthesameamountofwhitenoiseastherepeater. Figure6.44

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271aWhatistheblockdiagramforthissystem?bForanamplitude-modulationcommunicationsystem,whatisthesignal-to-noiseratioofthedemodu-latedsignalatthereceiver?Isthisbetterorworsethanthesignal-to-noiseratiowhennorepeaterispresent?cFordigitalcommunication,wemustconsiderthesystem'scapacity.Isthecapacitylargerwiththerepeatersystemthanwithoutit?Ifso,when;ifnot,whynot?

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272CHAPTER6.INFORMATIONCOMMUNICATIONSolutionstoExercisesinChapter6SolutiontoExercise6.1p.214Inbothcases,theanswerdependslessongeometrythanonmaterialproperties.Forcoaxialcable,c=1 p dd.Fortwistedpair,c=1 p r arccoshd 2r 2r+arccoshd 2r.SolutiontoExercise6.2p.214YoucanndthesefrequenciesfromthespectrumallocationchartSection7.3.Lightinthemiddleofthevisiblebandhasawavelengthofabout600nm,whichcorrespondstoafrequencyof51014Hz.Cabletelevisiontransmitswithinthesamefrequencybandasbroadcasttelevisionabout200MHzor2108Hz.Thus,thevisibleelectromagneticfrequenciesareoversixordersofmagnitudehigher!SolutiontoExercise6.3p.215Asfrequencyincreases,2fCGand2fLR.Inthishigh-frequencyregion,=j2fq LCvuut 1+G j2fC!1+R j2fL!.65j2fq LC1+1 21 j2fG C+R L!!j2fq LC+1 20B@Gvuut L C+Rvuut C L1CAThus,theattenuationspaceconstantequalstherealpartofthisexpression,andequalsaf=GZ0+R Z0 2.SolutiontoExercise6.4p.215Asshownpreviously.11,voltagesandcurrentsinawirelinechannel,whichismodeledasatransmissionlinehavingresistance,capacitanceandinductance,decayexponentiallywithdistance.Theinverse-squarelawgovernsfree-spacepropagationbecausesuchpropagationislossless,withtheinverse-squarelawaconsequenceoftheconservationofpower.Theexponentialdecayofwirelinechannelsoccursbecausetheyhavelossesandsomeltering.SolutiontoExercise6.5p.217 Figure6.45UsethePythagoreanTheorem,h+R2=R2+d2,wherehistheantennaheight,disthedistancefromthetopoftheearthtoatangencypointwiththeearth'ssurface,andRtheearth'sradius.Theline-of-sightdistancebetweentwoearth-basedantennaeequalsdLOS=q 2h1R+h12+q 2h2R+h22.66

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273Astheearth'sradiusismuchlargerthantheantennaheight,wehavetoagoodapproximationthatdLOS=p 2h1R+p 2h2R.Ifoneantennaisatgroundelevation,sayh2=0,theotherantenna'srangeisp 2h1R.SolutiontoExercise6.6p.217Asfrequencydecreases,wavelengthincreasesandcanapproachthedistancebetweentheearth'ssurfaceandtheionosphere.Assumingadistancebetweenthetwoof80km,therelationf=cgivesacorrespondingfrequencyof3.75kHz.Suchlowcarrierfrequencieswouldbelimitedtolowbandwidthanalogcommunicationandtolowdataratedigitalcommunications.TheUSNavydidusesuchacommunicationschemetoreachallofitssubmarinesatonce.SolutiontoExercise6.7p.218Transmissiontothesatellite,knownastheuplink,encountersinverse-squarelawpowerlosses.Reectingotheionospherenotonlyencountersthesameloss,buttwice.Reectionisthesameastransmittingexactlywhatarrives,whichmeansthatthetotallossistheproductoftheuplinkanddownlinklosses.Thegeosynchronousorbitliesatanaltitudeof35700km.Theionospherebeginsatanaltitudeofabout50km.Theamplitudelossinthesatellitecaseisproportionalto2:810)]TJ/F7 6.9738 Tf 6.2267 0 Td[(8;forMarconi,itwasproportionalto4:410)]TJ/F7 6.9738 Tf 6.2267 0 Td[(10.Marconiwasverylucky.SolutiontoExercise6.8p.218Iftheinterferer'sspectrumdoesnotoverlapthatofourcommunicationschanneltheinterfererisout-of-bandweneedonlyuseabandpasslterthatselectsourtransmissionbandandremovesotherportionsofthespectrum.SolutiontoExercise6.9p.219Theadditive-noisechannelisnotlinearbecauseitdoesnothavethezero-input-zero-outputpropertyeventhoughwemighttransmitnothing,thereceiver'sinputconsistsofnoise.SolutiontoExercise6.10p.222Thesignal-relatedportionofthetransmittedspectrumisgivenbyXf=1 2Mf)]TJ/F11 9.9626 Tf 9.9626 0 Td[(fc+1 2Mf+fc.Multiplyingatthereceiverbythecarriershiftsthisspectrumto+fcandto)]TJ/F11 9.9626 Tf 7.7487 0 Td[(fc,andscalestheresultbyhalf.1 2Xf)]TJ/F11 9.9626 Tf 9.9626 0 Td[(fc+1 2Xf+fc=1 4Mf)]TJ/F8 9.9626 Tf 9.9626 0 Td[(2fc+Mf+1 4Mf+2fc+Mf=1 4Mf)]TJ/F8 9.9626 Tf 9.9626 0 Td[(2fc+1 2Mf+1 4Mf+2fc.67Thesignalcomponentscenteredattwicethecarrierfrequencyareremovedbythelowpasslter,whilethebasebandsignalMfemerges.SolutiontoExercise6.11p.222ThekeyhereisthatthetwospectraMf)]TJ/F11 9.9626 Tf 9.9626 0 Td[(fc,Mf+fcdonotoverlapbecausewehaveassumedthatthecarrierfrequencyfcismuchgreaterthanthesignal'shighestfrequency.Consequently,thetermMf)]TJ/F11 9.9626 Tf 9.9626 0 Td[(fcMf+fcnormallyobtainedincomputingthemagnitude-squaredequalszero.SolutiontoExercise6.12p.224Separationis2W.CommercialAMsignalbandwidthis5kHz.Speechiswellcontainedinthisbandwidth,muchbetterthaninthetelephone!SolutiontoExercise6.13p.224xt=P1n=)]TJ/F11 9.9626 Tf 4.5663 -8.0698 Td[(sbnt)]TJ/F11 9.9626 Tf 9.9626 0 Td[(nT.SolutiontoExercise6.14p.226k=4.SolutiontoExercise6.15p.226xt=Xn)]TJ/F8 9.9626 Tf 7.7487 0 Td[(1bnApTt)]TJ/F11 9.9626 Tf 9.9626 0 Td[(nTsin2kt T

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274CHAPTER6.INFORMATIONCOMMUNICATIONSolutiontoExercise6.16p.226Theharmonicdistortionis10%.SolutiontoExercise6.17p.226Twicethebasebandbandwidthbecausebothpositiveandnegativefrequenciesareshiftedtothecarrierbythemodulation:3R.SolutiontoExercise6.18p.229InBPSK,thesignalsarenegativesofeachother:s1t=)]TJ/F8 9.9626 Tf 9.4091 0 Td[(s0t.Consequently,theoutputofeachmultiplier-integratorcombinationisthenegativeoftheother.Choosingthelargestthereforeamountstochoosingwhichoneispositive.Weonlyneedtocalculateoneofthese.Ifitispositive,wearedone.Ifitisnegative,wechoosetheothersignal.SolutiontoExercise6.19p.229ThematchedlteroutputsareA2T 2becausethesinusoidhaslesspowerthanapulsehavingthesameamplitude.SolutiontoExercise6.20p.231Thenoise-freeintegratoroutputsdierbyA2T,thefactoroftwosmallervaluethaninthebasebandcasearisingbecausethesinusoidalsignalshavelessenergyforthesameamplitude.StatedintermsofEb,thedierenceequals2Ebjustasinthebasebandcase.SolutiontoExercise6.21p.232Thenoise-freeintegratoroutputdierencenowequalsA2T=Eb 2.ThenoisepowerremainsthesameasintheBPSKcase,whichfromtheprobabilityoferrorequation.46yieldspe=Qq 2Eb N0.SolutiontoExercise6.22p.234Equallylikelysymbolseachhaveaprobabilityof1 K.Thus,HA=)]TJ/F1 9.9626 Tf 9.4091 8.0698 Td[()]TJ 4.5663 -0.5977 Td[(Pk)]TJ/F7 6.9738 Tf 7.3675 -4.1472 Td[(1 Klog2)]TJ/F7 6.9738 Tf 7.3675 -4.1472 Td[(1 K=log2K.Toprovethatthisisthemaximum-entropyprobabilityassignment,wemustexplic-itlytakeintoaccountthatprobabilitiessumtoone.Focusonaparticularsymbol,saytherst.Pr[a0]appearstwiceintheentropyformula:thetermsPr[a0]log2Pr[a0]and)]TJ/F8 9.9626 Tf 9.9626 0 Td[(Pr[a0]++Pr[aK)]TJ/F7 6.9738 Tf 6.2266 0 Td[(2]log2)]TJ/F8 9.9626 Tf 9.9626 0 Td[(Pr[a0]++Pr[aK)]TJ/F7 6.9738 Tf 6.2266 0 Td[(2].Thederivativewithrespecttothisprobabilityandalltheothersmustbezero.Thederivativeequalslog2Pr[a0])]TJ/F8 9.9626 Tf -460.2513 -11.9552 Td[(log2)]TJ/F8 9.9626 Tf 9.9626 0 Td[(Pr[a0]++Pr[aK)]TJ/F7 6.9738 Tf 6.2267 0 Td[(2],andallotherderivativeshavethesameformjustsubstituteyourletter'sindex.Thus,eachprobabilitymustequaltheothers,andwearedone.Fortheminimumentropyanswer,onetermis1log21=0,andtheothersare0log20,whichwedenetobezeroalso.Theminimumvalueofentropyiszero.SolutiontoExercise6.23p.237TheHumancodingtreeforthesecondsetofprobabilitiesisidenticaltothatfortherstFigure6.18HumanCodingTree.Theaveragecodelengthis1 21+1 42+1 53+1 203=1:75bits.Theentropycalculationisstraightforward:HA=)]TJ/F1 9.9626 Tf 9.4091 8.0698 Td[()]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(1 2log)]TJ/F7 6.9738 Tf 5.7617 -4.1472 Td[(1 2+1 4log)]TJ/F7 6.9738 Tf 5.7618 -4.1472 Td[(1 4+1 5log)]TJ/F7 6.9738 Tf 5.7618 -4.1472 Td[(1 5+1 20log)]TJ/F7 6.9738 Tf 7.7474 -4.1472 Td[(1 20,whichequals1:68bits.SolutiontoExercise6.24p.237T=1 BAR.SolutiontoExercise6.25p.238Becausenocodewordbeginswithanother'scodeword,therstcodewordencounteredinabitstreammustbetherightone.Notethatwemuststartatthebeginningofthebitstream;jumpingintothemiddledoesnotguaranteeperfectdecoding.Theendofonecodewordandthebeginningofanothercouldbeacodeword,andwewouldgetlost.SolutiontoExercise6.26p.238Considerthebitstream:::0110111:::takenfromthebitstream0|10|110|110|111|:::.Wewoulddecodetheinitialpartincorrectly,thenwouldsynchronize.Ifwehadaxed-lengthcodesay00,01,10,11,thesituationismuchworse.Jumpingintothemiddleleadstonosynchronizationatall!

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275SolutiontoExercise6.27p.241Thisquestionisequivalentto3pe)]TJ/F11 9.9626 Tf 9.9626 0 Td[(pe+pe21or2pe2+)]TJ/F8 9.9626 Tf 7.7488 0 Td[(3pe+10.Becausethisisanupward-goingparabola,weneedonlycheckwhereitsrootsare.Usingthequadraticformula,wendthattheyarelocatedat1 2and1.Consequentlyintherange0pe1 2theerrorrateproducedbycodingissmaller.SolutiontoExercise6.28p.242Withnocoding,theaveragebit-errorprobabilitypeisgivenbytheprobabilityoferrorequation.47:pe=Qq 22Eb N0.Withathreefoldrepetitioncode,thebit-errorprobabilityisgivenby3p0e2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(p0e+p0e3,wherep0e=Qq 22Eb 3N0.Plottingthisrevealsthattheincreaseinbit-errorprobabilityoutofthechannelbecauseoftheenergyreductionisnotcompensatedbytherepetitioncoding. Figure6.46SolutiontoExercise6.29p.243InbinaryarithmeticseeFigure6.23,adding0toabinaryvalueresultsinthatbinaryvaluewhileadding1resultsintheoppositebinaryvalue.

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276CHAPTER6.INFORMATIONCOMMUNICATIONSolutiontoExercise6.30p.245dmin=2n+1SolutiontoExercise6.31p.245Whenwemultiplytheparity-checkmatrixtimesanycodewordequaltoacolumnofG,theresultconsistsofthesumofanentryfromthelowerportionofGanditselfthat,bythelawsofbinaryarithmetic,isalwayszero.BecausethecodeislinearsumofanytwocodewordsisacodewordwecangenerateallcodewordsassumsofcolumnsofG.SincemultiplyingbyHisalsolinear,Hc=0.SolutiontoExercise6.32p.246Inbinaryarithmeticseethistable53,adding0toabinaryvalueresultsinthatbinaryvaluewhileadding1resultsintheoppositebinaryvalue.SolutiontoExercise6.33p.246Theprobabilityofasingle-biterrorinalength-NblockisNpe)]TJ/F11 9.9626 Tf 9.9626 0 Td[(peN)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1andatriple-biterrorhasprobability)]TJ/F10 6.9738 Tf 4.5662 -3.6491 Td[(N3pe3)]TJ/F11 9.9626 Tf 9.9626 0 Td[(peN)]TJ/F7 6.9738 Tf 6.2266 0 Td[(3.Forthersttobegreaterthanthesecond,wemusthavepe<1 q N)]TJ/F7 6.9738 Tf 6.2266 0 Td[(1N)]TJ/F7 6.9738 Tf 6.2267 0 Td[(2 6+1ForN=7,pe<0:31.SolutiontoExercise6.34p.248Inalength-Nblock,Nsingle-bitandNN)]TJ/F7 6.9738 Tf 6.2266 0 Td[(1 2double-biterrorscanoccur.Thenumberofnon-zerovectorsresultingfromH^cmustequalorexceedthesumofthesetwonumbers.2N)]TJ/F10 6.9738 Tf 6.2267 0 Td[(K)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1N+NN)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1 2or2N)]TJ/F10 6.9738 Tf 6.2267 0 Td[(KN2+N+2 2.68Thersttwosolutionsthatattainequalityare,1and,78codes.However,noperfectcodeexistsotherthanthesingle-biterrorcorrectingHammingcode.Perfectcodessatisfyrelationslike.68withequality.SolutiontoExercise6.35p.250Toconverttobits/second,wedividethecapacitystatedinbits/transmissionbythebitintervaldurationT.SolutiontoExercise6.36p.252Thenetworkentrypointisthetelephonehandset,whichconnectsyoutotheneareststation.Dialingthetelephonenumberinformsthenetworkofwhowillbethemessagerecipient.Thetelephonesystemformsanelectricalcircuitbetweenyourhandsetandyourfriend'shandset.Yourfriendreceivesthemessageviathesamedevicethehandsetthatservedasthenetworkentrypoint.SolutiontoExercise6.37p.255Thetransmittingop-ampseesaloadorRout+Z0kRout=N,whereNisthenumberoftransceiversotherthanthisoneattachedtothecoaxialcable.Thetransferfunctiontosomeothertransceiver'sreceivercircuitisRoutdividedbythisload.SolutiontoExercise6.38p.256Theworst-casesituationoccurswhenonecomputerbeginstotransmitjustbeforetheother'spacketarrives.Transmittersmustsenseacollisionbeforepackettransmissionends.Thetimetakenforonecomputer'spackettotraveltheEthernet'slengthandfortheothercomputer'stransmissiontoarriveequalstheround-trip,notone-way,propagationtime. 53"ErrorCorrection"

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277SolutiontoExercise6.39p.256Thecablemustbeafactoroftenshorter:Itcannotexceed100m.Dierentminimumpacketsizesmeansdierentpacketformats,makingconnectingoldandnewsystemstogethermorecomplexthanneedbe.SolutiontoExercise6.40p.257Whenyoupickupthetelephone,youinitiateadialogwithyournetworkinterfacebydialingthenumber.Thenetworklooksupwherethedestinationcorrespondingtothatnumberislocated,androutesthecallaccordingly.Therouteremainsxedaslongasthecallpersists.Whatyousayamountstohigh-levelprotocolwhileestablishingtheconnectionandmaintainingitcorrespondstolow-levelprotocol.

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278CHAPTER6.INFORMATIONCOMMUNICATION

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Chapter7Appendix7.1Decibels1Thedecibelscaleexpressesamplitudesandpowervalueslogarithmically.Thedenitionsforthesedier,butareconsistentwitheachother.powers;indecibels=10log10powers powers0.1amplitudes;indecibels=20log10amplitudes amplitudes0Herepowers0andamplitudes0representareferencepowerandamplitude,respectively.Quantifyingpoweroramplitudeindecibelsessentiallymeansthatwearecomparingquantitiestoastandardorthatwewanttoexpresshowtheychanged.Youwillhearstatementslike"Thesignalwentdownby3dB"and"Thelter'sgaininthestopbandis)]TJ/F8 9.9626 Tf 7.7487 0 Td[(60"DecibelsisabbreviateddB..Exercise7.1Solutiononp.283.Theprex"deci"impliesatenth;adecibelisatenthofaBel.Whoisthismeasurenamedfor?Theconsistencyofthesetwodenitionsarisesbecausepowerisproportionaltothesquareofamplitude:)]TJ/F8 9.9626 Tf 4.5663 -8.0698 Td[(powers/amplitude2s.2Pluggingthisexpressionintothedenitionfordecibels,wendthat10log10powers powers0=10log10amplitude2s amplitude2s0=20log10amplitudes amplitudes0.3Becauseofthisconsistency,statingrelativechangeintermsofdecibelsisunambiguous.Afactorof10increaseinamplitudecorrespondstoa20dBincreaseinbothamplitudeandpower!Theaccompanyingtableprovides"nice"decibelvalues.Convertingdecibelvaluesbackandforthisfun,andtestsyourabilitytothinkofdecibelvaluesassumsand/ordierencesofthewell-knownvaluesandofratiosasproductsand/orquotients.Thisconversionrestsonthelogarithmicnatureofthedecibelscale.Forexample,tondthedecibelvalueforp 2,wehalvethedecibelvaluefor2;26dBequals10+10+6dBthatcorrespondstoaratioof10104=400.Decibelquantitiesadd;ratiovaluesmultiply.Onereasondecibelsareusedsomuchisthefrequency-domaininput-outputrelationforlinearsystems:Yf=XfHf.Becausethetransferfunctionmultipliestheinputsignal'sspectrum,tondtheoutputamplitudeatagivenfrequencywesimplyaddthelter'sgainindecibelsrelativetoareferenceofonetotheinputamplitudeatthatfrequency.Thiscalculationisonereasonthatweplottransferfunctionmagnitudeonalogarithmicverticalscaleexpressedindecibels. 1Thiscontentisavailableonlineat.279

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280CHAPTER7.APPENDIX Decibeltable PowerRatio dB 1 0 p 2 1:5 2 3 p 10 5 4 6 5 7 8 9 10 10 0:1 )]TJ/F8 9.9626 Tf 7.7487 0 Td[(10 Figure7.1:Commonvaluesforthedecibel.Thedecibelvaluesforallbutthepowersoftenareapproximate,butareaccuratetoadecimalplace. 7.2PermutationsandCombinations27.2.1PermutationsandCombinationsThelottery"game"consistsofpickingknumbersfromapoolofn.Forexample,youselect6numbersoutof60.Towin,theorderinwhichyoupickthenumbersdoesn'tmatter;youonlyhavetochoosetherightsetof6numbers.Thechancesofwinningequalthenumberofdierentlength-ksequencesthatcanbechosen.Arelated,butdierent,problemisselectingthebattinglineupforabaseballteam.Nowtheordermatters,andmanymorechoicesarepossiblethanwhenorderdoesnotmatter.Answeringsuchquestionsoccursinmanyapplicationsbeyondgames.Indigitalcommunications,forexample,youmightaskhowmanypossibledouble-biterrorscanoccurinacodeword.Numberingthebitpositionsfrom1toN,theansweristhesameasthelotteryproblemwithk=6.Solvingthesekindofproblemsamountstounderstandingpermutations-thenumberofwaysofchoosingthingswhenordermattersasinbaseballlineups-andcombinations-thenumberofwaysofchoosingthingswhenorderdoesnotmatterasinlotteriesandbiterrors.Calculatingpermutationsistheeasiest.Ifwearetopickknumbersfromapoolofn,wehavenchoicesfortherstone.Forthesecondchoice,wehaven)]TJ/F8 9.9626 Tf 10.5574 0 Td[(1.Thenumberoflength-twoorderedsequencesisthereforebenn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1.Continuingtochooseuntilwemakekchoicesmeansthenumberofpermutationsisnn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1n)]TJ/F8 9.9626 Tf 9.9626 0 Td[(2:::n)]TJ/F11 9.9626 Tf 9.9626 0 Td[(k+1.Thisresultcanbewrittenintermsoffactorialsasn! n)]TJ/F10 6.9738 Tf 6.2266 0 Td[(k!,withn!=nn)]TJ/F8 9.9626 Tf 9.9626 0 Td[(1n)]TJ/F8 9.9626 Tf 9.9626 0 Td[(2:::1.Formathematicalconvenience,wedene0!=1.Whenorderdoesnotmatter,thenumberofcombinationsequalsthenumberofpermutationsdividedbythenumberoforderings.Thenumberofwaysapoolofkthingscanbeorderedequalsk!.Thus,oncewechoosetheninestartersforourbaseballgame,wehave9!=362;880dierentlineups!Thesymbolforthecombinationofkthingsdrawnfromapoolofnis)]TJ/F10 6.9738 Tf 4.5663 -3.6491 Td[(nkandequalsn! n)]TJ/F10 6.9738 Tf 6.2266 0 Td[(k!k!. 2Thiscontentisavailableonlineat.

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281Exercise7.2Solutiononp.283.Whatarethechancesofwinningthelottery?Assumeyoupick6numbersfromthenumbers1-60.Combinatorialsoccurininterestingplaces.Forexample,Newtonderivedthatthen-thpowerofasumobeyedtheformulax+yn=)]TJ/F10 6.9738 Tf 4.5662 -3.6491 Td[(n0xn+)]TJ/F10 6.9738 Tf 4.5662 -3.6491 Td[(n1xn)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1y+)]TJ/F10 6.9738 Tf 4.5663 -3.6491 Td[(n2xn)]TJ/F7 6.9738 Tf 6.2266 0 Td[(2y2++)]TJ/F10 6.9738 Tf 4.5662 -3.6491 Td[(nnyn.Exercise7.3Solutiononp.283.Whatdoesthesumofbinomialcoecientsequal?Inotherwords,whatisnXk=0nkArelatedproblemiscalculatingtheprobabilitythatanytwobitsareinerrorinalength-ncodewordwhenpistheprobabilityofanybitbeinginerror.Theprobabilityofanyparticulartwo-biterrorsequenceisp2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(pn)]TJ/F7 6.9738 Tf 6.2267 0 Td[(2.Theprobabilityofatwo-biterroroccurringanywhereequalsthisprobabilitytimesthenumberofcombinations:)]TJ/F10 6.9738 Tf 4.5662 -3.6491 Td[(n2p2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(pn)]TJ/F7 6.9738 Tf 6.2267 0 Td[(2.Notethattheprobabilitythatzerooroneortwo,etc.errorsoccurringmustbeone;inotherwords,somethingmusthappentothecodeword!Thatmeansthatwemusthave)]TJ/F10 6.9738 Tf 4.5663 -3.6491 Td[(n0)]TJ/F11 9.9626 Tf 9.9626 0 Td[(pn+)]TJ/F10 6.9738 Tf 4.5662 -3.6491 Td[(n1p)]TJ/F11 9.9626 Tf 9.9626 0 Td[(pn)]TJ/F7 6.9738 Tf 6.2266 0 Td[(1+)]TJ/F10 6.9738 Tf 4.5662 -3.6491 Td[(n2p2)]TJ/F11 9.9626 Tf 9.9626 0 Td[(pn)]TJ/F7 6.9738 Tf 6.2267 0 Td[(2++)]TJ/F10 6.9738 Tf 4.5662 -3.6491 Td[(nnpn=1.Canyouprovethis?7.3FrequencyAllocations3Topreventradiostationsfromtransmittingsignals"ontopofeachother,";theUnitedStatesandothernationalgovernmentsinthe1930sbeganregulatingthecarrierfrequenciesandpoweroutputsstationscoulduse.Withincreaseduseoftheradiospectrumforbothpublicandprivateuse,thisregulationhasbecomeincreasinglyimportant.Thisistheso-calledFrequencyAllocationChart,whichshowswhatkindsofbroadcastingcanoccurinwhichfrequencybands.Detailedradiocarrierfrequencyassignmentsaremuchtoodetailedtopresenthere. 3Thiscontentisavailableonlineat.

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282CHAPTER7.APPENDIX FrequencyAllocationChart Figure7.2

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283SolutionstoExercisesinChapter7SolutiontoExercise7.1p.279AlexanderGrahamBell.Hedevelopeditbecauseweseemtoperceivephysicalquantitieslikeloudnessandbrightnesslogarithmically.Inotherwords,percentage,notabsolutedierences,mattertous.Weusedecibelstodaybecausecommonvaluesaresmallintegers.IfweusedBels,theywouldbedecimalfractions,whicharen'taselegant.SolutiontoExercise7.2p.280)]TJ/F7 6.9738 Tf 4.5662 -3.6491 Td[(606=60! 54!6!=50;063;860.SolutiontoExercise7.3p.281BecauseofNewton'sbinomialtheorem,thesumequals+1n=2n.

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284INDEXIndexofKeywordsandTermsKeywordsarelistedbythesectionwiththatkeywordpagenumbersareinparentheses.Keywordsdonotnecessarilyappearinthetextofthepage.Theyaremerelyassociatedwiththatsection.Ex.apples,1.1Termsarereferencedbythepagetheyappearon.Ex.apples,1Aactivecircuits,73address,252algorithm,174,175aliasing,161alphabet,2.421,23,167AM,6.11Ampere,2.2,3.1amplier,2.6amplitude,1.4,7,2.2,7.1amplitudemodulate,130amplitudemodulation,6.11,6.12analog,1,21,3.21,5.6,5.14,5.16,6.32analogcommunication,6.100,6.11,6.12,6.32analogproblem,3.21analogsignal,1.2,6.10220analogsignals,5.14,5.16analog-to-digitalA/Dconversion,5.4162,162angle,15angleofcomplexnumber,2.1ARPANET,6.34ASP,3.21attenuation,2.6,6.9,6.12attenuationconstant,213auxiliaryband,263averagepower,58,59Bbandlimited,6.31bandpasslter,139bandpasssignal,131bandwidth,4.6,123,131,6.9,6.12,6.14,6.15basebandcommunication,6.10,220basebandsignal,131,6.100basisfunctions,110binaryphaseshiftkeying,6.14,225binarysymmetricchannel,6.193,233bit,157bitinterval,224,6.14bitstream,6.13,224bit-receptionerror,6.17bits,6.21235,235block,247blockchannelcoding,6.26,242blockdiagram,1.3,6,2.5booleanarithmetic,5.2155boxcarlter,184,5.13BPSK,6.14,6.19broadcast,6.1,210,6.36broadcastcommunication,209broadcastmode,6.2buering,5.15,190buttery,5.9,176bytes,157Ccapacitor,3.2,3.8capacity,6.30,248,6.31,6.36carrier,1.47,8,221carrieramplitude,221carrierfrequency,6.11,221Cartesianform,2.1Cartesianformofz,13cascade,2.5channel,1.3,7,6.9,6.30,6.31channelcoder,238,6.25channelcoding,6.25,6.26,6.27channeldecoding,6.28characteristicimpedance,214charge,3.1circuit,3.135,3.2,3.4,39,3.8,3.9,3.20circuitmodel,6.2circuitswitched,253circuit-switched,6.34,252circuits,35clockspeed,5.2closedcircuit,3.2coaxialcable,6.3,211codebook,6.21,235

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INDEX285codeword,6.27,243codeworderror,7.2coding,6.23codingeciency,6.26,242coherent,222coherentreceiver,6.11collision,6.36,256combination,7.2combinations,280combinatorial,7.2communication,1.3,6.32communicationchannel,6.2communicationchannels,6.5,6.6,6.7,6.8218,6.9communicationnetwork,6.33,6.35,6.37communicationnetworks,6.34,6.36communicationprotocol,6.37communicationsystems,6.1communicationtheory,6.20Complementarylters,259complex,2.3,2.4complexamplitude,17complexamplitudes,60complexconjugate,14complexexponential,13,2.2complexexponentialsequence,22,165complexFourierseries,4.2complexfrequency,18complexnumber,2.1,13complexnumbers,54complexplane,13complexpower,59complex-valued,2.4complexity,2.3,174component,2.2compression,6.20,6.21,6.22,6.23computationaladvantage,175computationalcomplexity,5.8,5.9,5.16computernetwork,6.376Computernetworks,251computerorganization,5.2conductance,3.2,37conductor,3.1conjugatesymmetry,4.2,111Cooley-Tukeyalgorithm,5.9cosine,1.4countablyinnite,188current,3.1,35,3.2currentdivider,3.6,46cutofrequency,64Ddatacompression,6.22,236datarate,6.14,225,6.31De-emphasiscircuits,102decibel,7.1279decode,6.29decoding,6.28decompose,2.4decomposition,2.320dedicated,251dependentsource,73deviceelectronic,3.1DFT,5.7172,5.9,5.10,5.14dierenceequation,5.12,182,5.14digital,1,5.6,6.32,6.33digitalcommunication,6.1,6.13,6.14,6.15,6.16,6.17,6.18,6.19,6.20,6.21,6.22,6.23,6.25,6.26,6.27,6.28,6.29,6.30,6.31,6.32digitalcommunicationreceiver,6.19digitalcommunicationreceivers,6.17digitalcommunicationsystems,6.19digitallter,5.14,5.16digitalsignal,1.2digitalsignalprocessing,5.6,5.10,5.11,5.12,5.14,5.15,5.16digitalsources,6.20,6.21235,6.22,6.23diode,3.2081DiscreteFourierTransform,5.7,172,5.8,5.9,5.10,5.14discrete-time,2.4,5.6,5.14,5.15,5.16discrete-timeltering,5.14,5.15,5.16discrete-timeFouriertransform,5.6discrete-timesincfunction,170Discrete-TimeSystems,5.11,5.12,5.13discrete-valued,159domain,254doubleprecisionoatingpoint,5.2double-bit,6.29

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286INDEXDSP,5.1,5.6,5.10177,5.11,5.12,5.14,5.15,5.16Eeciency,6.308,250elec241problems,5.17electrical,252electricalengineering,1.1electron,3.1electroniccircuits,73electronics,3.17,3.19element,3.1,3.2elementalsignals,2.2energy,3.1,36,3.11entropy,6.20,234,6.21equivalentcircuit,49,3.12EquivalentCircuits,3.7error,6.29,6.30errorcorrectingcode,6.29,6.30errorcorrectingcodes,6.25,6.26,6.27errorcorrection,6.25,6.26241,6.27,6.28,6.31errorprobability,6.19error-correctingcode,6.31error-correctingcodes,6.28ethernet,6.36Euler,2.2,4.3Eulerrelations,4.2Euler'srelation,2.2Euler'srelations,15exponential,2.4,3.9Ffarad,3.2Faraday,3.2fastFouriertransform,5.9,5.15feedback,2.5FFT,5.9,5.15lter,65ltering,4.7124,5.14,5.15,5.16FIR,184xed,253xedrate,236Flexibility,251oatingpoint,5.2ux,3.2form,175formalcircuitmethod,3.15formants,136forwardbias,3.20forwardbiasing,82Fouriercoecients,4.2,110,4.3,114Fourierseries,4.2,109,4.3,4.8,4.9fourierspectrum,4.6Fouriertransform,4.1,109,4.8,126,5.6,5.7,5.10,5.14frames,178frequency,1.4,7,2.2,4.3,6.4frequencyallocationchart,7.3,281frequencydomain,3.10,56,4.1,5.13,5.15frequencyresponse,60,5.13frequencyshiftkeying,6.157frequency-shiftkeying,227FSK,6.15,6.19functional,24fundamentalassumption,38fundamentalfrequency,110,136fundamentalmodelofcommunication,6.19,6.33FundamentalModelofDigitalCommunication,238,6.25fundamentalmodelofspeechproduction,136Ggain,2.6,26gateway,6.35gateways,254Gauss,4.3114generatormatrix,6.27,243,6.29geometricseries,168geosynchronousorbits,6.7,217Gibb'sphenomenon,121ground,211Hhalfwaverectiedsinusoid,4.46half-waverectier,83Hamming,6.29Hammingcode,6.29Hammingcodes,247Hammingdistance,6.27,243Hanningwindow,5.10,179harmonically,110Heaviside,6.6HeinrichHertz,1.1Henry,3.2hidden-onesnotation,158historyofelectricalengineering,1.11hole,3.1holes,35

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INDEX287Human,6.22,6.23HumanCode,6.22,6.23Humansourcecodingalgorithm,6.22,236Ii,2.217IIR,183imaginary,2.2imaginarynumber,2.1,13imaginarypart,2.1,13impedance,54,3.9,55,3.10,3.11inductor,3.2,3.8information,1.1,1,1.2informationcommunication,6.1,6.5,6.6,6.7217,6.8,6.9,6.10,6.11,6.12,6.13,6.14,6.15,6.16,6.17,6.18,6.20,6.21,6.22,6.23,6.25,6.26,6.27,6.28,6.29,6.30,6.31,6.32,6.33,6.34,6.35,6.36informationtheory,1.11initialconditions,182input,2.5inputresistance,76input-outputrelationship,3.6,44instantaneouspower,36integratedcircuit,3.17integrator,2.6interference,6.8,218internet,6.34,6.37internetprotocoladdress,6.35inverseFouriertransform,4.8invertingamplier,77ionosphere,6.6IPaddress,6.35,254Jj,2.2jam,149JamesMaxwell,1.1joules,36KKCL,43,3.15,3.16Kircho,3.4,3.15Kircho'sLaws,41KVL,43,3.15,3.1671LLAN,6.35leakage,82leakagecurrent,3.20line-of-sight,6.5,216linear,3.2,36,5.14,210linearcircuit,4.7linearcodes,6.27,242linearphaseshift,111linearsystems,2.6load,3.6,47localareanetwork,6.35253Localareanetworks,253logarithmicamplier,84logarithmically,279long-distance,6.5long-distancecommunication,6.7long-distancetransmission,6.7lossless,236lossy,236lottery,7.2lowpasslter,65,4.7Mmagnitude,15magnitudeofcomplexnumber,2.1Marconi,6.6matchedlter,6.16,228Maxwell'sequations,6.2,210Mayer-Norton,3.7Mayer-Nortonequivalent,52mean-squareequality,121message,6.34messagerouting,6.34modelofcommunication,1.3modelsandreality,3.3modem,1.5modulate,8modulated,221modulatedcommunication,6.12modulation,1.4,135,6.11Morsecode,6.23multi-levelsignaling,250Nnameserver,6.35nameservers,254negative,3.1nerve,3.1network,238,6.33,6.36networkarchitecture,6.35,6.36networks,217node,3.439nodemethod,3.15,66nodevoltages,67nodes,41,252noise,136,6.1,6.8,218,

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288INDEX6.12,6.17,6.30noiseremoval,191noisychannelcodingtheorem,6.30nonlinear,3.20Norton,3.749,3.12numbersonacomputer,5.2Nyquistfrequency,162,5.6Oohm,3.2OliverHeaviside,1.1op-amp,3.17,73,3.19opencircuit,3.2,37operationalamplier,3.17,73,3.19orthogonality,4.2109,4.3,114output,2.5outputresistance,76outputspectrum,4.9Ppacket,6.34,6.36packetsize,6.364packet-switched,6.34,253packets,253parallel,2.5,3.6,45,47parity,6.29paritycheck,6.29paritycheckmatrix,245Parseval'stheorem,4.2109,111,4.86,128,5.6passivecircuits,73Performance,250period,18periodicsignal,4.2permutation,7.2permutations,280phase,1.4,7,2.2phasemodulates,259phasor,2.2,17physical,3.1pitchfrequency,136pitchlines,137pointtopoint,6.33pointtopointcommunication,6.33point-to-point,6.2210,210,251,6.34point-to-pointcommunication,6.1,209pointwiseequality,121polarform,2.1,14,14positionalnotation,5.2positive,3.135postalservice,6.33Power,1,3.1,36,3.5,43,3.11,3.16,4.2109,117,7.1powerfactor,90,107powerspectrum,117,6.8,218pre-emphasiscircuit,102preamble,228prex,238probabilisticmodels,234probability,7.2probabilityoferror,6.18problems,3.21propagatingwave,213propagationspeed,214proportional,174protocol,6.36,6.37,257pulse,2.2,4.2,4.8Qquadrupleprecisionoatingpoint,5.2quantizationinterval,5.4,163quantized,159,5.4,162Rrandomaccess,6.36random-access,255real,2.2realpart,13real-valued,2.4receivedsignal-to-noise-ratio,6.12receiver,1.3,6.16rectication,4.4reference,279referencenode,67relaynetwork,217relaynetworks,6.56repeaters,270repetitioncode,6.25,240resistance,3.2,37resistivity,43resistor,3.2,3.5reversebias,3.20reverse-bias,82rms,1.5,117route,252routing,6.34Ssamples,162sampling,172samplinginterval,160SamplingTheorem,162satellite,6.7satellitecommunication,6.7sawtooth,149self-clockingsignaling,6.16,228self-synchronization,6.23

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INDEX289sequences,2.4series,3.6,44Shannon,1.3,6.1,6.20,6.21,6.30,6.31Shannonsamplingfrequency,162shift-invariant,181,5.14shift-invariantsystems,5.12shortcircuit,37Siemens,3.2signbit,5.2,157signal,1.2,2,1.3,2.5signaldecomposition,20signalset,224,6.14,6.17,6.19signalspectrum,4.4signal-to-noise,5.4162,164signal-to-noiseratio,6.10,221,6.31signal-to-noise-ratio,6.12signal-to-noise-ration,6.181,6.27signals,1,2.4,3.21simplebinarycode,235sinc,128sine,1.4,2.2,2.4single-bit,6.29246sink,1.36,6,7sinusoid,1.4,7,2.217,2.4,4.1,4.3SIR,6.9SNR,6.9,6.10,6.12,6.18,6.27,6.31source,1.3,6,2.6,3.2sourcecodingtheorem,6.21,235,6.22spaceconstant,213spectrograms,5.10spectrum,4.1,109,4.2,110,116speechmodel,4.10squarewave,2.2,4.2,4.3standardfeedbackconguration,75Steinmetz,2.2superposition,2.320,4.1109,5.12181SuperpositionPrinciple,84symbolic-valuedsignals,2.4synchronization,6.16synchronize,228system,6systemtheory,2.5systems,2.4Ttelegraph,6.33telephone,6.33tetherlessnetworking,210themesofelectricalengineering,1.1Thevenin,3.7,3.12Thveninequivalentcircuit,50timeconstant,2.2,18,133timedelay,2.6timedomain,3.10,56,5.12timeinvariant,4.9timereversal,2.6time-domainmultiplexing,6.33,252time-invariant,2.6totalharmonicdistortion,117transatlanticcommunication,6.6transceiver,6.36,255transferfunction,3.13,60,3.14,5.13,5.14transforms,128transitiondiagrams,6.19,233transmission,6.5,6.6transmissionbandwidth,6.14,6.19transmissionerror,6.17transmissionline,6.3transmissionlineequations,212transmissionlines,211transmitter,1.3,6twistedpair,6.3,211Uuncountablyinnite,188unitsample,2.4,23,166,166unitstep,2.2unit-sampleresponse,186Vvocaltract,4.10,136Volta,3.1voltage,3.1,35,3.2voltagedivider,3.6,44voltagegain,76WWAN,6.35watts,36wavelength,213,6.4whitenoise,6.8,218,6.31wideareanetwork,6.35wideareanetworks,253window,179wireless,209,210,6.5,6.6wirelesschannel,6.2,6.4wireline,209,210,6.5wirelinechannel,6.2,6.3WorldWideWeb,6.37Zzero-pad,187

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290ATTRIBUTIONSAttributionsCollection:FundamentalsofElectricalEngineeringIEditedby:DonJohnsonURL:http://cnx.org/content/col10040/1.9/License:http://creativecommons.org/licenses/by/1.0Module:"Themes"By:DonJohnsonURL:http://cnx.org/content/m0000/2.18/Pages:1-2Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"SignalsRepresentInformation"By:DonJohnsonURL:http://cnx.org/content/m0001/2.26/Pages:2-5Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"StructureofCommunicationSystems"By:DonJohnsonURL:http://cnx.org/content/m0002/2.16/Pages:6-7Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"TheFundamentalSignal"By:DonJohnsonURL:http://cnx.org/content/m0003/2.15/Pages:7-8Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"IntroductionProblems"By:DonJohnsonURL:http://cnx.org/content/m10353/2.16/Pages:8-10Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"ComplexNumbers"By:DonJohnsonURL:http://cnx.org/content/m0081/2.27/Pages:13-17Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0

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ATTRIBUTIONS291Module:"ElementalSignals"By:DonJohnsonURL:http://cnx.org/content/m0004/2.26/Pages:17-20Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"SignalDecomposition"By:DonJohnsonURL:http://cnx.org/content/m0008/2.12/Pages:20-21Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Discrete-TimeSignals"By:DonJohnsonURL:http://cnx.org/content/m0009/2.23/Pages:21-23Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"IntroductiontoSystems"By:DonJohnsonURL:http://cnx.org/content/m0005/2.18/Pages:24-26Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"SimpleSystems"By:DonJohnsonURL:http://cnx.org/content/m0006/2.23/Pages:26-29Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"SignalsandSystemsProblems"By:DonJohnsonURL:http://cnx.org/content/m10348/2.24/Pages:29-33Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Voltage,Current,andGenericCircuitElements"By:DonJohnsonURL:http://cnx.org/content/m0011/2.12/Pages:35-36Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"IdealCircuitElements"By:DonJohnsonURL:http://cnx.org/content/m0012/2.19/Pages:36-39Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0

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292ATTRIBUTIONSModule:"IdealandReal-WorldCircuitElements"By:DonJohnsonURL:http://cnx.org/content/m0013/2.9/Pages:39-39Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"ElectricCircuitsandInterconnectionLaws"By:DonJohnsonURL:http://cnx.org/content/m0014/2.26/Pages:39-43Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"PowerDissipationinResistorCircuits"By:DonJohnsonURL:http://cnx.org/content/m17305/1.5/Pages:43-44Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/2.0/Module:"SeriesandParallelCircuits"By:DonJohnsonURL:http://cnx.org/content/m10674/2.7/Pages:44-49Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"EquivalentCircuits:ResistorsandSources"By:DonJohnsonURL:http://cnx.org/content/m0020/2.22/Pages:49-53Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"CircuitswithCapacitorsandInductors"By:DonJohnsonURL:http://cnx.org/content/m0023/2.11/Pages:54-54Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"TheImpedanceConcept"By:DonJohnsonURL:http://cnx.org/content/m0024/2.22/Pages:54-56Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"TimeandFrequencyDomains"By:DonJohnsonURL:http://cnx.org/content/m10708/2.6/Pages:56-58Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0

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ATTRIBUTIONS293Module:"PowerintheFrequencyDomain"By:DonJohnsonURL:http://cnx.org/content/m17308/1.2/Pages:58-59Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/2.0/Module:"EquivalentCircuits:ImpedancesandSources"By:DonJohnsonURL:http://cnx.org/content/m0030/2.19/Pages:59-60Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"TransferFunctions"By:DonJohnsonURL:http://cnx.org/content/m0028/2.18/Pages:60-64Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"DesigningTransferFunctions"By:DonJohnsonURL:http://cnx.org/content/m0031/2.20/Pages:65-66Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"FormalCircuitMethods:NodeMethod"By:DonJohnsonURL:http://cnx.org/content/m0032/2.18/Pages:66-71Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"PowerConservationinCircuits"By:DonJohnsonURL:http://cnx.org/content/m17317/1.1/Pages:71-73Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/2.0/Module:"Electronics"By:DonJohnsonURL:http://cnx.org/content/m0035/2.8/Pages:73-73Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"DependentSources"By:DonJohnsonURL:http://cnx.org/content/m0053/2.13/Pages:73-75Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0

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294ATTRIBUTIONSModule:"OperationalAmpliers"By:DonJohnsonURL:http://cnx.org/content/m0036/2.28/Pages:76-81Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"TheDiode"By:DonJohnsonURL:http://cnx.org/content/m0037/2.14/Pages:81-84Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"AnalogSignalProcessingProblems"By:DonJohnsonURL:http://cnx.org/content/m10349/2.37/Pages:84-105Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"IntroductiontotheFrequencyDomain"By:DonJohnsonURL:http://cnx.org/content/m0038/2.10/Pages:109-109Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"ComplexFourierSeries"By:DonJohnsonURL:http://cnx.org/content/m0042/2.24/Pages:109-113Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"ClassicFourierSeries"By:DonJohnsonURL:http://cnx.org/content/m0039/2.22/Pages:114-116Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"ASignal'sSpectrum"By:DonJohnsonURL:http://cnx.org/content/m0040/2.19/Pages:116-117Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"FourierSeriesApproximationofSignals"By:DonJohnsonURL:http://cnx.org/content/m10687/2.8/Pages:117-121Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0

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ATTRIBUTIONS295Module:"EncodingInformationintheFrequencyDomain"By:DonJohnsonURL:http://cnx.org/content/m0043/2.15/Pages:122-124Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"FilteringPeriodicSignals"By:DonJohnsonURL:http://cnx.org/content/m0044/2.9/Pages:124-126Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"DerivationoftheFourierTransform"By:DonJohnsonURL:http://cnx.org/content/m0046/2.19/Pages:126-131Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"LinearTimeInvariantSystems"By:DonJohnsonURL:http://cnx.org/content/m0048/2.18/Pages:131-133Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"ModelingtheSpeechSignal"By:DonJohnsonURL:http://cnx.org/content/m0049/2.25/Pages:134-140Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"FrequencyDomainProblems"By:DonJohnsonURL:http://cnx.org/content/m10350/2.32/Pages:140-151Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"IntroductiontoDigitalSignalProcessing"By:DonJohnsonURL:http://cnx.org/content/m10781/2.3/Pages:155-155Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"IntroductiontoComputerOrganization"By:DonJohnsonURL:http://cnx.org/content/m10263/2.27/Pages:155-159Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0

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296ATTRIBUTIONSModule:"TheSamplingTheorem"By:DonJohnsonURL:http://cnx.org/content/m0050/2.18/Pages:159-162Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"AmplitudeQuantization"By:DonJohnsonURL:http://cnx.org/content/m0051/2.21/Pages:162-165Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Discrete-TimeSignalsandSystems"By:DonJohnsonURL:http://cnx.org/content/m10342/2.13/Pages:165-167Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Discrete-TimeFourierTransformDTFT"By:DonJohnsonURL:http://cnx.org/content/m10247/2.28/Pages:167-172Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"DiscreteFourierTransformDFT"Usedhereas:"DiscreteFourierTransformsDFT"By:DonJohnsonURL:http://cnx.org/content/m10249/2.26/Pages:172-174Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"DFT:ComputationalComplexity"By:DonJohnsonURL:http://cnx.org/content/m0503/2.11/Pages:174-174Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"FastFourierTransformFFT"By:DonJohnsonURL:http://cnx.org/content/m10250/2.15/Pages:174-177Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0

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ATTRIBUTIONS297Module:"Spectrograms"By:DonJohnsonURL:http://cnx.org/content/m0505/2.18/Pages:177-180Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Discrete-TimeSystems"By:DonJohnsonURL:http://cnx.org/content/m0507/2.5/Pages:180-180Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Discrete-TimeSystemsintheTime-Domain"By:DonJohnsonURL:http://cnx.org/content/m10251/2.22/Pages:181-184Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Discrete-TimeSystemsintheFrequencyDomain"By:DonJohnsonURL:http://cnx.org/content/m0510/2.14/Pages:185-186Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"FilteringintheFrequencyDomain"By:DonJohnsonURL:http://cnx.org/content/m10257/2.16/Pages:186-189Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"EciencyofFrequency-DomainFiltering"By:DonJohnsonURL:http://cnx.org/content/m10279/2.14/Pages:189-192Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Discrete-TimeFilteringofAnalogSignals"By:DonJohnsonURL:http://cnx.org/content/m0511/2.20/Pages:192-193Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"DigitalSignalProcessingProblems"By:DonJohnsonURL:http://cnx.org/content/m10351/2.33/Pages:193-203Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0

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298ATTRIBUTIONSModule:"InformationCommunication"By:DonJohnsonURL:http://cnx.org/content/m0513/2.8/Pages:209-209Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"TypesofCommunicationChannels"By:DonJohnsonURL:http://cnx.org/content/m0099/2.13/Pages:210-210Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"WirelineChannels"By:DonJohnsonURL:http://cnx.org/content/m0100/2.27/Pages:210-215Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"WirelessChannels"By:DonJohnsonURL:http://cnx.org/content/m0101/2.14/Pages:215-216Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Line-of-SightTransmission"By:DonJohnsonURL:http://cnx.org/content/m0538/2.13/Pages:216-217Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"TheIonosphereandCommunications"By:DonJohnsonURL:http://cnx.org/content/m0539/2.10/Pages:217-217Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"CommunicationwithSatellites"By:DonJohnsonURL:http://cnx.org/content/m0540/2.10/Pages:217-218Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"NoiseandInterference"By:DonJohnsonURL:http://cnx.org/content/m0515/2.17/Pages:218-219Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0

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ATTRIBUTIONS299Module:"ChannelModels"By:DonJohnsonURL:http://cnx.org/content/m0516/2.10/Pages:219-220Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"BasebandCommunication"By:DonJohnsonURL:http://cnx.org/content/m0517/2.17/Pages:220-221Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"ModulatedCommunication"By:DonJohnsonURL:http://cnx.org/content/m0518/2.24/Pages:221-222Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Signal-to-NoiseRatioofanAmplitude-ModulatedSignal"By:DonJohnsonURL:http://cnx.org/content/m0541/2.16/Pages:222-224Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"DigitalCommunication"By:DonJohnsonURL:http://cnx.org/content/m0519/2.10/Pages:224-224Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"BinaryPhaseShiftKeying"By:DonJohnsonURL:http://cnx.org/content/m10280/2.13/Pages:224-227Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"FrequencyShiftKeying"By:DonJohnsonURL:http://cnx.org/content/m0545/2.11/Pages:227-228Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"DigitalCommunicationReceivers"By:DonJohnsonURL:http://cnx.org/content/m0520/2.17/Pages:228-229Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0

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300ATTRIBUTIONSModule:"DigitalCommunicationinthePresenceofNoise"By:DonJohnsonURL:http://cnx.org/content/m0546/2.12/Pages:230-231Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"DigitalCommunicationSystemProperties"By:DonJohnsonURL:http://cnx.org/content/m10282/2.9/Pages:231-233Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"DigitalChannels"By:DonJohnsonURL:http://cnx.org/content/m0102/2.13/Pages:233-233Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Entropy"By:DonJohnsonURL:http://cnx.org/content/m0070/2.13/Pages:234-234Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"SourceCodingTheorem"By:DonJohnsonURL:http://cnx.org/content/m0091/2.13/Pages:235-236Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"CompressionandtheHumanCode"By:DonJohnsonURL:http://cnx.org/content/m0092/2.17/Pages:236-237Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"SubtletiesofSourceCoding"Usedhereas:"SubtliesofCoding"By:DonJohnsonURL:http://cnx.org/content/m0093/2.16/Pages:237-238Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0

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ATTRIBUTIONS301Module:"ChannelCoding"By:DonJohnsonURL:http://cnx.org/content/m10782/2.4/Pages:238-240Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"RepetitionCodes"By:DonJohnsonURL:http://cnx.org/content/m0071/2.21/Pages:240-241Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"BlockChannelCoding"By:DonJohnsonURL:http://cnx.org/content/m0094/2.14/Pages:241-242Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Error-CorrectingCodes:HammingDistance"By:DonJohnsonURL:http://cnx.org/content/m10283/2.28/Pages:242-245Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Error-CorrectingCodes:ChannelDecoding"By:DonJohnsonURL:http://cnx.org/content/m0072/2.20/Pages:245-246Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Error-CorrectingCodes:HammingCodes"By:DonJohnsonURL:http://cnx.org/content/m0097/2.24/Pages:246-248Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"NoisyChannelCodingTheorem"By:DonJohnsonURL:http://cnx.org/content/m0073/2.11/Pages:248-249Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"CapacityofaChannel"By:DonJohnsonURL:http://cnx.org/content/m0098/2.13/Pages:249-250Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0

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302ATTRIBUTIONSModule:"ComparisonofAnalogandDigitalCommunication"By:DonJohnsonURL:http://cnx.org/content/m0074/2.11/Pages:250-251Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"CommunicationNetworks"By:DonJohnsonURL:http://cnx.org/content/m0075/2.10/Pages:251-252Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"MessageRouting"By:DonJohnsonURL:http://cnx.org/content/m0076/2.8/Pages:252-253Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Networkarchitecturesandinterconnection"By:DonJohnsonURL:http://cnx.org/content/m0077/2.9/Pages:253-254Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Ethernet"By:DonJohnsonURL:http://cnx.org/content/m10284/2.12/Pages:254-256Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"CommunicationProtocols"By:DonJohnsonURL:http://cnx.org/content/m0080/2.18/Pages:256-258Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"InformationCommunicationProblems"By:DonJohnsonURL:http://cnx.org/content/m10352/2.21/Pages:258-271Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"Decibels"By:DonJohnsonURL:http://cnx.org/content/m0082/2.16/Pages:279-279Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0

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ATTRIBUTIONS303Module:"PermutationsandCombinations"By:DonJohnsonURL:http://cnx.org/content/m10262/2.12/Pages:280-281Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0Module:"FrequencyAllocations"By:DonJohnsonURL:http://cnx.org/content/m0083/2.10/Pages:281-283Copyright:DonJohnsonLicense:http://creativecommons.org/licenses/by/1.0

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FundamentalsofElectricalEngineeringIThecoursefocusesonthecreation,manipulation,transmission,andreceptionofinformationbyelectronicmeans.Elementarysignaltheory;time-andfrequency-domainanalysis;SamplingTheorem.Digitalinfor-mationtheory;digitaltransmissionofanalogsignals;error-correctingcodes.AboutConnexionsSince1999,Connexionshasbeenpioneeringaglobalsystemwhereanyonecancreatecoursematerialsandmakethemfullyaccessibleandeasilyreusablefreeofcharge.WeareaWeb-basedauthoring,teachingandlearningenvironmentopentoanyoneinterestedineducation,includingstudents,teachers,professorsandlifelonglearners.Weconnectideasandfacilitateeducationalcommunities.Connexions'smodular,interactivecoursesareinuseworldwidebyuniversities,communitycolleges,K-12schools,distancelearners,andlifelonglearners.Connexionsmaterialsareinmanylanguages,includingEnglish,Spanish,Chinese,Japanese,Italian,Vietnamese,French,Portuguese,andThai.ConnexionsispartofanexcitingnewinformationdistributionsystemthatallowsforPrintonDemandBooks.Connexionshaspartneredwithinnovativeon-demandpublisherQOOPtoacceleratethedeliveryofprintedcoursematerialsandtextbooksintoclassroomsworldwideatlowerpricesthantraditionalacademicpublishers.