University Press of Florida
Signals and Systems
Buy This Book ( Related Link )
CITATION PDF VIEWER
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/AA00011666/00001
 Material Information
Title: Signals and Systems
Physical Description: Book
Language: en-US
Creator: Baraniuk, Richard, Connexions, Rice University
 Subjects
Subjects / Keywords: Signals, Systems, Time Domain Analysis, Continuous Time Systems, Discrete Time Systems, Linear Algebra, Fourier Series, Fast Fourier Transform, Convergence, Sampling Theorem, Laplace Transform, Systems Design, Z-Transform, Digital Filtering, Hilbert Spaces, Orthogonal Expansions, OGT+ isbn: 9781616100681
Engineering,Physics
Science / Engineering, Science / Physics
 Notes
Abstract: This text deals with signals, systems, and transforms, from their theoretical mathematical foundations to practical implementation in circuits and computer algorithms. At its conclusion, learners will have a deep understanding of the mathematics and practical issues of signals in continuous and discrete time, linear time invariant systems, convolution, and Fourier transforms.
General Note: Expositive
General Note: Community College, Higher Education
General Note: http://www.ogtp-cart.com/product.aspx?ISBN=9781616100681
General Note: Adobe PDF Reader
General Note: Richard Baraniuk
General Note: Textbook
General Note: http://cnx.org/content/col10064/latest/
General Note: http://florida.theorangegrove.org/og/file/e66b88b8-2801-ee81-1d88-cacd33185dd7/1/Signals.pdf
 Record Information
Source Institution: University of Florida
Holding Location: University Press of Florida
Rights Management: Copyright © 2008 Richard Baraniuk. This selection and arrangement of content is licensed under the Creative Commons Attribution License: http://creativecommons.org/licenses/by/1.0
Resource Identifier: isbn - 9781616100681
System ID: AA00011666:00001

Downloads

This item is only available as the following downloads:

( PDF )


Full Text

PAGE 1

SignalsandSystems CollectionEditor: RichardBaraniuk

PAGE 3

SignalsandSystems CollectionEditor: RichardBaraniuk Authors: ThanosAntoulas RichardBaraniuk StevenCox BenjaminFite RoyHa MichaelHaag MatthewHutchinson DonJohnson RicardoRadaelli-Sanchez JustinRomberg PhilSchniter MelissaSelik JPSlavinsky Online: < http://cnx.org/content/col10064/1.11/ > CONNEXIONS RiceUniversity,Houston,Texas

PAGE 4

2008RichardBaraniuk ThisselectionandarrangementofcontentislicensedundertheCreativeCommonsAttributionLicense: http://creativecommons.org/licenses/by/1.0

PAGE 5

TableofContents 1Signals 1.1 SignalClassicationsandProperties.........................................................1 1.2 SizeofASignal:Norms.....................................................................9 1.3 SignalOperations..........................................................................19 1.4 UsefulSignals..............................................................................22 1.5 TheImpulseFunction......................................................................25 1.6 TheComplexExponential..................................................................28 1.7 Discrete-TimeSignals......................................................................31 Solutions........................................................................................35 2Systems 2.1 SystemClassicationsandProperties.......................................................37 2.2 PropertiesofSystems......................................................................41 Solutions........................................................................................ ?? 3TimeDomainAnalysisofContinuousTimeSystems 3.1 CTLinearSystemsandDierentialEquations..............................................47 3.2 Continuous-TimeConvolution..............................................................53 3.3 PropertiesofConvolution..................................................................59 3.4 BIBOStability.............................................................................65 Solutions........................................................................................ ?? 4TimeDomainAnalysisofDiscreteTimeSystems 4.1 Discrete-TimeSystemsintheTime-Domain................................................69 4.2 Discrete-TimeConvolution.................................................................72 4.3 CircularConvolutionandtheDFT........................................................157 4.4 LinearConstant-CoecientDierenceEquations...........................................82 4.5 SolvingLinearConstant-CoecientDierenceEquations...................................83 Solutions........................................................................................89 5LinearAlgebraOverview 5.1 LinearAlgebra:TheBasics.................................................................91 5.2 EigenvectorsandEigenvalues...............................................................96 5.3 MatrixDiagonalization....................................................................101 5.4 Eigen-stuinaNutshell...................................................................104 5.5 EigenfunctionsofLTISystems.............................................................105 5.6 FourierTransformProperties..............................................................108 Solutions.......................................................................................109 6ContinuousTimeFourierSeries 6.1 PeriodicSignals...........................................................................111 6.2 FourierSeries:EigenfunctionApproach....................................................112 6.3 DerivationofFourierCoecientsEquation................................................115 6.4 FourierSeriesinaNutshell................................................................116 6.5 FourierSeriesProperties..................................................................119 6.6 SymmetryPropertiesoftheFourierSeries.................................................122 6.7 CircularConvolutionPropertyofFourierSeries............................................126 6.8 FourierSeriesandLTISystems............................................................127 6.9 ConvergenceofFourierSeries..............................................................130 6.10 DirichletConditions......................................................................132

PAGE 6

iv 6.11 Gibbs'sPhenomena......................................................................134 6.12 FourierSeriesWrap-Up..................................................................137 Solutions.......................................................................................139 7DiscreteFourierTransform 7.1 FourierAnalysis...........................................................................141 7.2 FourierAnalysisinComplexSpaces.......................................................142 7.3 MatrixEquationfortheDTFS............................................................148 7.4 PeriodicExtensiontoDTFS...............................................................150 7.5 CircularShifts.............................................................................153 7.6 CircularConvolutionandtheDFT........................................................157 Solutions.......................................................................................162 8FastFourierTransformFFT 8.1 DFT:FastFourierTransform..............................................................163 8.2 TheFastFourierTransformFFT........................................................164 8.3 DerivingtheFastFourierTransform.......................................................165 Solutions.......................................................................................168 9Convergence 9.1 ConvergenceofSequences.................................................................169 9.2 ConvergenceofVectors....................................................................171 9.3 UniformConvergenceofFunctionSequences...............................................174 Solutions........................................................................................ ?? 10DiscreteTimeFourierTransformDTFT 10.1 DiscreteFourierTransformation..........................................................177 10.2 DiscreteFourierTransformDFT.......................................................179 10.3 TableofCommonFourierTransforms....................................................181 10.4 Discrete-TimeFourierTransformDTFT................................................182 10.5 Discrete-TimeFourierTransformProperties..............................................183 10.6 Discrete-TimeFourierTransformPair....................................................183 10.7 DTFTExamples.........................................................................184 Solutions.......................................................................................188 11ContinuousTimeFourierTransformCTFT 11.1 Continuous-TimeFourierTransformCTFT.............................................189 11.2 PropertiesoftheContinuous-TimeFourierTransform....................................190 Solutions.......................................................................................194 12SamplingTheorem 12.1 Sampling.................................................................................195 12.2 Reconstruction...........................................................................199 12.3 MoreonReconstruction..................................................................203 12.4 NyquistTheorem........................................................................205 12.5 Aliasing..................................................................................206 12.6 Anti-AliasingFilters.....................................................................210 12.7 DiscreteTimeProcessingofContinuousTimeSignals....................................211 Solutions.......................................................................................214 13LaplaceTransformandSystemDesign 13.1 TheLaplaceTransforms..................................................................217 13.2 PropertiesoftheLaplaceTransform......................................................219 13.3 TableofCommonLaplaceTransforms....................................................221 13.4 RegionofConvergencefortheLaplaceTransform........................................221

PAGE 7

v 13.5 TheInverseLaplaceTransform...........................................................223 13.6 PolesandZeros..........................................................................225 Solutions........................................................................................ ?? 14Z-TransformandDigitalFiltering 14.1 TheZTransform:Denition.............................................................229 14.2 TableofCommonz-Transforms..........................................................234 14.3 RegionofConvergencefortheZ-transform...............................................235 14.4 InverseZ-Transform......................................................................244 14.5 RationalFunctions.......................................................................247 14.6 DierenceEquation......................................................................249 14.7 UnderstandingPole/ZeroPlotsontheZ-Plane...........................................252 14.8 FilterDesignusingthePole/ZeroPlotofaZ-Transform..................................256 Solutions........................................................................................ ?? 15Appendix:HilbertSpacesandOrthogonalExpansions 15.1 VectorSpaces............................................................................261 15.2 Norms...................................................................................263 15.3 InnerProducts...........................................................................266 15.4 HilbertSpaces...........................................................................268 15.5 Cauchy-SchwarzInequality...............................................................268 15.6 CommonHilbertSpaces..................................................................275 15.7 TypesofBasis...........................................................................278 15.8 OrthonormalBasisExpansions...........................................................281 15.9 FunctionSpace...........................................................................285 15.10 HaarWaveletBasis.....................................................................286 15.11 OrthonormalBasesinRealandComplexSpaces........................................293 15.12 PlancharelandParseval'sTheorems.....................................................295 15.13 ApproximationandProjectionsinHilbertSpace........................................296 Solutions.......................................................................................299 16HomeworkSets 16.1 Homework1.............................................................................301 16.2 Homework1Solutions...................................................................305 Solutions........................................................................................ ?? 17ViewingEmbeddedLabVIEWContent .....................................................317 Glossary ............................................................................................318 Index ...............................................................................................323 Attributions ........................................................................................328

PAGE 8

vi

PAGE 9

Chapter1 Signals 1.1SignalClassicationsandProperties 1 1.1.1Introduction Thismodulewilllayoutsomeofthefundamentalsofsignalclassication.Thisisbasicallyalistofdenitions andpropertiesthatarefundamentaltothediscussionofsignalsandsystems.Itshouldbenotedthatsome discussionslikeenergysignalsvs.powersignals 2 havebeendesignatedtheirownmoduleforamorecomplete discussion,andwillnotbeincludedhere. 1.1.2ClassicationsofSignals Alongwiththeclassicationofsignalsbelow,itisalsoimportanttounderstandtheClassicationofSystems Section2.1. 1.1.2.1Continuous-Timevs.Discrete-Time Asthenamessuggest,thisclassicationisdeterminedbywhetherornotthetimeaxisx-axisis discrete countableor continuous Figure1.1.Acontinuous-timesignalwillcontainavalueforallrealnumbers alongthetimeaxis.Incontrasttothis,adiscrete-timesignalSection1.7isoftencreatedbyusingthe samplingtheorem 3 tosampleacontinuoussignal,soitwillonlyhavevaluesatequallyspacedintervalsalong thetimeaxis. 1 Thiscontentisavailableonlineat. 2 "SignalEnergyvs.SignalPower" 3 "TheSamplingTheorem" 1

PAGE 10

2 CHAPTER1.SIGNALS Figure1.1 1.1.2.2Analogvs.Digital Thedierencebetween analog and digital issimilartothedierencebetweencontinuous-timeanddiscretetime.Inthiscase,however,thedierenceiswithrespecttothevalueofthefunctiony-axisFigure1.2. Analogcorrespondstoacontinuousy-axis,whiledigitalcorrespondstoadiscretey-axis.Aneasyexample ofadigitalsignalisabinarysequence,wherethevaluesofthefunctioncanonlybeoneorzero. Figure1.2 1.1.2.3Periodicvs.Aperiodic PeriodicsignalsSection6.1repeatwithsome period T ,whileaperiodic,ornonperiodic,signalsdonot Figure1.3.Wecandeneaperiodicfunctionthroughthefollowingmathematicalexpression,where t can beanynumberand T isapositiveconstant: f t = f T + t .1 The fundamentalperiod ofourfunction, f t ,isthesmallestvalueof T thatthestillallows.1tobe true.

PAGE 11

3 a b Figure1.3: aAperiodicsignalwithperiod T 0 bAnaperiodicsignal 1.1.2.4Causalvs.Anticausalvs.Noncausal Causal signalsaresignalsthatarezeroforallnegativetime,while anticausal aresignalsthatarezerofor allpositivetime. Noncausal signalsaresignalsthathavenonzerovaluesinbothpositiveandnegativetime Figure1.4.

PAGE 12

4 CHAPTER1.SIGNALS a b c Figure1.4: aAcausalsignalbAnanticausalsignalcAnoncausalsignal 1.1.2.5Evenvs.Odd An evensignal isanysignal f suchthat f t = f )]TJ/F11 9.9626 Tf 7.749 0 Td [(t .Evensignalscanbeeasilyspottedasthey are symmetric aroundtheverticalaxis.An oddsignal ,ontheotherhand,isasignal f suchthat f t = )]TJ/F8 9.9626 Tf 9.409 0 Td [( f )]TJ/F11 9.9626 Tf 7.749 0 Td [(t Figure1.5.

PAGE 13

5 a b Figure1.5: aAnevensignalbAnoddsignal Usingthedenitionsofevenandoddsignals,wecanshowthatanysignalcanbewrittenasacombination ofanevenandoddsignal.Thatis,everysignalhasanodd-evendecomposition.Todemonstratethis,we havetolooknofurtherthanasingleequation. f t = 1 2 f t + f )]TJ/F11 9.9626 Tf 7.748 0 Td [(t + 1 2 f t )]TJ/F11 9.9626 Tf 9.963 0 Td [(f )]TJ/F11 9.9626 Tf 7.749 0 Td [(t .2 Bymultiplyingandaddingthisexpressionout,itcanbeshowntobetrue.Also,itcanbeshownthat f t + f )]TJ/F11 9.9626 Tf 7.749 0 Td [(t fulllstherequirementofanevenfunction,while f t )]TJ/F11 9.9626 Tf 10.111 0 Td [(f )]TJ/F11 9.9626 Tf 7.749 0 Td [(t fulllstherequirementofan oddfunctionFigure1.6. Example1.1

PAGE 14

6 CHAPTER1.SIGNALS a b c d Figure1.6: aThesignalwewilldecomposeusingodd-evendecompositionbEvenpart: e t = 1 2 f t + f )]TJ/F58 8.9664 Tf 7.167 0 Td [(t cOddpart: o t = 1 2 f t )]TJ/F58 8.9664 Tf 9.215 0 Td [(f )]TJ/F58 8.9664 Tf 7.168 0 Td [(t dCheck: e t + o t = f t

PAGE 15

7 1.1.2.6Deterministicvs.Random A deterministicsignal isasignalinwhicheachvalueofthesignalisxedandcanbedeterminedbya mathematicalexpression,rule,ortable.Becauseofthisthefuturevaluesofthesignalcanbecalculated frompastvalueswithcompletecondence.Ontheotherhand,a randomsignal 4 hasalotofuncertainty aboutitsbehavior.Thefuturevaluesofarandomsignalcannotbeaccuratelypredictedandcanusually onlybeguessedbasedontheaverages 5 ofsetsofsignalsFigure1.7. a b Figure1.7: aDeterministicSignalbRandomSignal 1.1.2.7Right-Handedvs.Left-Handed A right-handed signaland left-handed signalarethosesignalswhosevalueiszerobetweenagivenvariable andpositiveornegativeinnity.Mathematicallyspeaking,aright-handedsignalisdenedasanysignal where f t =0 for tt 1 > .SeeFigure1.8foranexample.Bothgures"begin"at t 1 andthenextendstopositiveor negativeinnitywithmainlynonzerovalues. 4 "IntroductiontoRandomSignalsandProcesses" 5 "RandomProcesses:MeanandVariance"

PAGE 16

8 CHAPTER1.SIGNALS a b Figure1.8: aRight-handedsignalbLeft-handedsignal 1.1.2.8Finitevs.InniteLength Asthenameapplies,signalscanbecharacterizedastowhethertheyhaveaniteorinnitelengthsetof values.Mostnitelengthsignalsareusedwhendealingwithdiscrete-timesignalsoragivensequenceof values.Mathematicallyspeaking, f t isa nite-lengthsignal ifitis nonzero overaniteinterval t 1 and t 2 < 1 .AnexamplecanbeseeninFigure1.9.Similarly,an innite-lengthsignal f t ,isdenedasnonzerooverallrealnumbers: 1 f t

PAGE 17

9 Figure1.9: Finite-LengthSignal.Notethatitonlyhasnonzerovaluesonaset,niteinterval. 1.2SizeofASignal:Norms 6 "Size"indicates largeness or strength .Wewillusethemathematicalconceptofthe norm toquantify thisnotionforbothcontinuous-timeanddiscrete-timesignals.Firstweconsiderawaytoquantifythesize ofasignalwhichmayalreadybefamiliar. 1.2.1Continuous-TimeEnergy Ourusualnotionoftheenergyofasignalistheareaunderthecurve j f t j 2 Figure1.10 E f = Z 1 j f t j 2 dt .3 6 Thiscontentisavailableonlineat.

PAGE 18

10 CHAPTER1.SIGNALS Example1.2 Calculate E f for Figure1.11 1.2.2GeneralizedEnergy:Norms Thenotionof"energy"canbegeneralizedthroughtheintroductionofthe L p norm: k f k p = Z j f t j p dt 1 p .4 where 1 p< 1 Example1.3 E f = k f k 2 2 Example1.4 Calculatethe L p normof Figure1.12

PAGE 19

11 Exercise1.1 Whathappensto k f k p = Z j f t j p dt 1 p as p !1 ? L 1 norm k f k 1 = esssup j f t j Figure1.13 1.2.3Discrete-TimeEnergy Figure1.14 E f = 1 X n = j f [ n ] j 2

PAGE 20

12 CHAPTER1.SIGNALS k f k p = Z j f t j p dt 1 p where 1 p< 1 k f k 1 =max n fj f [ n ] jg k f k p p = N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X n =0 j f [ n ] j p where 1 p< 1 k f k 1 =max n = 0,1,...,N-1 fj f [ n ] jg 1.2.4FiniteNormSignals Whataretheconditionsonasignalfor k f k p < 1 ?Lookatall4fundamentalsignalclasses 1.2.4.1Discrete-TimeandFiniteLength Figure1.15 Thisisalength N vector f = 0 B B B B B B B B @ f [0] f [1] f [2] f [ ::: ] f [ N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1] 1 C C C C C C C C A = 0 B B B B B B B B @ f 0 f 1 f 2 ... f N 1 1 C C C C C C C C A where f 2 C N ,or f 2 N N -dimensionalcomplexorrealEuclideanspace. Example1.5 N =3 f isa real signal.

PAGE 21

13 Figure1.16 Denition1.1: l p [0 ;N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1]= n f 2 C N ; k f k p < 1 o butfrompreviousdiscussion l p [0 ;N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1]= C N 1.2.4.2Discrete-TimeandInniteLength Figure1.17 canstillinterpret f asan innite-lengthvector f = 0 B B B B B B B B B B @ f [ ::: ] f [ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1] f [0] f [1] f [2] f [ ::: ] 1 C C C C C C C C C C A

PAGE 22

14 CHAPTER1.SIGNALS but C 1 R 1 don'tmakesense. Denition1.2: l p z = n f; k f k p < 1 o k f k p p = 1 X n = j f [ n ] j p where 1 p< 1 k f k 1 =max n 2 z fj f [ n ] jg Whatdoesittakeforan f tobein l p z ? Example1.6 Sketchan f 2 l p z and f= 2 l p z Exercise1.2 Whatcharacteristicsdoes f 2 l p z haveandwhathappensaswecharge p ? 1.2.4.3Continuous-TimeandFinite-Length Figure1.18 Wewillstillreferto f t asavector;moreonthislater. Denition1.3: L p [ T 1 ;T 2 ]= n f [ T 1 ;T 2 ] ; k f k p < 1 o k f k p = Z T 2 T 1 j f t j p dt 1 p where 1 p< 1 k f k p = esssup j f t j where T 1 t T 2

PAGE 23

15 Exercise1.3 Whatdoesittakeforand f tobein L p [ T 1 ;T 2 ] ? 1.2.4.4Continuous-TimeandInnite-Length Figure1.19 Wewillstillreferto f t asa vector Denition1.4: L p R = n f; k f k p < 1 o k f k p = Z 1 j f [ n ] j p dt 1 p where 1 p< 1 k f k 1 = esssup j f t j where
PAGE 24

16 CHAPTER1.SIGNALS Figure1.20 Solution:Lookatthe" normperunittime ". ie -normoveroneperiod. ie -normofinnite-lengthsignalconvertedtonitelengthsignalbywindowing. Figure1.21 k f k p T isthemeasure. Units for p =2 ? L 2 Power = "energyperunittime" Usefulwhen E f = 1 timeaverageofenergy P f = lim T !1 Z T 2 )]TJ/F8 9.9626 Tf 6.226 -0.747 Td [( T 2 j f t j 2 dt

PAGE 25

17 Figure1.22 1.compute Energy T = k f k 2 2 T 2.lookat lim T !1 Energy T = k f k 2 2 T P f isoftencalledthemean-squarevalueof f p P f iscalledthe rootmeansquared RMSvalueof f Units? "Energysignals"havenitenormenergy E f < 1 "Powersignals"haveniteandnonzeropower P f < 1 P f 6 =0 ,and E f = 1 1.2.5.1Conclusions Energysignalsarenotpowersignals. Powersignalsarenotenergysignals. Why? Exercise1.5 Are all signalseitherenergyorpowersignals? Example1.9 f t = t

PAGE 26

18 CHAPTER1.SIGNALS Figure1.23 The4 fundamentalclasses ofsignalswewillstudydependontheindependenttimevariable.

PAGE 27

19 Figure1.24 1.3SignalOperations 7 Thismodulewilllookattwosignaloperations,timeshiftingandtimescaling.Signaloperationsareoperationsonthetimevariableofthesignal.Theseoperationsareverycommoncomponentstoreal-world systemsand,assuch,shouldbeunderstoodthoroughlywhenlearningaboutsignalsandsystems. 7 Thiscontentisavailableonlineat.

PAGE 28

20 CHAPTER1.SIGNALS 1.3.1TimeShifting Timeshiftingis,asthenamesuggests,theshiftingofasignalintime.Thisisdonebyaddingorsubtracting theamountoftheshifttothetimevariableinthefunction.Subtractingaxedamountfromthetime variablewillshiftthesignaltotherightdelaythatamount,whileaddingtothetimevariablewillshiftthe signaltotheleftadvance. Figure1.25: f t )]TJ/F58 8.9664 Tf 9.215 0 Td [(T movesdelays f totherightby T 1.3.2TimeScaling Timescalingcompressesanddilatesasignalbymultiplyingthetimevariablebysomeamount.Ifthat amountisgreaterthanone,thesignalbecomesnarrowerandtheoperationiscalledcompression,whileif theamountislessthanone,thesignalbecomeswiderandiscalleddilation.Itoftentakespeoplequitea whiletogetcomfortablewiththeseoperations,aspeople'sintuitionisoftenforthemultiplicationbyan amountgreaterthanonetodilateandlessthanonetocompress. Figure1.26: f at compresses f by a Example1.10 Actuallyplottingshiftedandscaledsignalscanbequitecounter-intuitive.Thisexamplewillshow afool-proofwaytopracticethisuntilyourproperintuitionisdeveloped. Given f t ,plot f )]TJ/F8 9.9626 Tf 9.41 0 Td [( at .

PAGE 29

21 a b c Figure1.27: aBeginwith f t bThenreplace t with at toget f at cFinally,replace t with t )]TJ/F59 5.9776 Tf 10.791 3.674 Td [(b a toget f )]TJ/F58 8.9664 Tf 4.224 -7.243 Td [(a )]TJ/F58 8.9664 Tf 4.224 -7.243 Td [(t )]TJ/F59 5.9776 Tf 10.791 3.674 Td [(b a = f at )]TJ/F58 8.9664 Tf 9.215 0 Td [(b 1.3.3TimeReversal Anaturalquestiontoconsiderwhenlearningabouttimescalingis:Whathappenswhenthetimevariable ismultipliedbyanegativenumber?Theanswertothisistimereversal.Thisoperationisthereversalof thetimeaxis,orippingthesignaloverthey-axis. Figure1.28: Reversethetimeaxis

PAGE 30

22 CHAPTER1.SIGNALS Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10125/latest/TDSignalOps.llb 1.4UsefulSignals 8 Beforelookingatthismodule,hopefullyyouhavesomebasicideaofwhatasignalisandwhatbasic classicationsandpropertiesSection1.1asignalcanhave.Toreview,asignalismerelyafunctiondened withrespecttoanindependentvariable.Thisvariableisoftentimebutcouldrepresentanindexofasequence oranynumberofthingsinanynumberofdimensions.Most,ifnotall,signalsthatyouwillencounterin yourstudiesandtherealworldwillbeabletobecreatedfromthebasicsignalswediscussbelow.Because ofthis,theseelementarysignalsareoftenreferredtoasthe buildingblocks forallothersignals. 1.4.1Sinusoids Probablythemostimportantelementalsignalthatyouwilldealwithisthereal-valuedsinusoid.Inits continuous-timeform,wewritethegeneralformas x t = A cos !t + .5 where A istheamplitude, isthefrequency,and representsthephase.Notethatitiscommontosee !t replacedwith 2 ft .Sincesinusoidalsignalsareperiodic,wecanexpresstheperiodofthese,oranyperiodic signal,as T = 2 .6 Figure1.29: Sinusoidwith A =2 w =2 ,and =0 8 Thiscontentisavailableonlineat.

PAGE 31

23 1.4.2ComplexExponentialFunction Maybeasimportantasthegeneralsinusoid,the complexexponential functionwillbecomeacriticalpart ofyourstudyofsignalsandsystems.Itsgeneralformiswrittenas f t = Be st .7 where s ,shownbelow,isacomplexnumberintermsof ,thephaseconstant,and thefrequency: s = + j! PleaselookatthecomplexexponentialmoduleSection1.6ortheotherelementalsignalspage 9 foramuch moreindepthlookatthisimportantsignal. 1.4.3RealExponentials Justasthenamesounds,realexponentialscontainnoimaginarynumbersandareexpressedsimplyas f t = Be t .8 whereboth B and arerealparameters.Unlikethecomplexexponentialthatoscillates,therealexponential eitherdecaysorgrowsdependingonthevalueof DecayingExponential ,when < 0 GrowingExponential ,when > 0 a b Figure1.30: ExamplesofRealExponentialsaDecayingExponentialbGrowingExponential 1.4.4UnitImpulseFunction The unitimpulseSection1.5 "function"or Diracdelta functionisasignalthathasinniteheight andinnitesimalwidth.However,becauseofthewayitisdened,itactuallyintegratestoone.While intheengineeringworld,thissignalisquiteniceandaidsintheunderstandingofmanyconcepts,some mathematicianshaveaproblemwithitbeingcalledafunction,sinceitisnotdenedat t =0 .Engineers 9 "ElementalSignals":SectionComplexExponentials

PAGE 32

24 CHAPTER1.SIGNALS reconcilethisproblembykeepingitaroundintegrals,inordertokeepitmorenicelydened.Theunit impulseismostcommonlydenotedas t Themostimportantpropertyoftheunit-impulseisshowninthefollowingintegral: Z 1 t dt =1 .9 1.4.5Unit-StepFunction Anotherverybasicsignalisthe unit-stepfunction thatisdenedas u t = 8 < : 0 if t< 0 1 if t 0 .10 a b Figure1.31: BasicStepFunctionsaContinuous-TimeUnit-StepFunctionbDiscrete-TimeUnitStepFunction Notethatthestepfunctionisdiscontinuousattheorigin;however,itdoesnotneedtobedenedhere asitdoesnotmatterinsignaltheory.Thestepfunctionisausefultoolfortestingandfordeningother signals.Forexample,whendierentshiftedversionsofthestepfunctionaremultipliedbyothersignals,one canselectacertainportionofthesignalandzeroouttherest. 1.4.6RampFunction Therampfunctioniscloselyrelatedtotheunit-stepdiscussedabove.Wheretheunit-stepgoesfromzeroto oneinstantaneously,therampfunctionbetterresemblesareal-worldsignal,wherethereissometimeneeded forthesignaltoincreasefromzerotoitssetvalue,oneinthiscase.Wedenearampfunctionasfollows r t = 8 > > < > > : 0 if t< 0 t t 0 if 0 t t 0 1 if t>t 0 .11

PAGE 33

25 Figure1.32: RampFunction 1.5TheImpulseFunction 10 Inengineering,weoftendealwiththeideaofanactionoccurringatapoint.Whetheritbeaforceata pointinspaceorasignalatapointintime,itbecomesworthwhiletodevelopsomewayofquantitatively deningthis.Thisleadsustotheideaofaunitimpulse,probablythesecondmostimportantfunction,next tothecomplexexponentialSection1.6,insystemsandsignalscourse. 1.5.1DiracDeltaFunction The DiracDeltafunction ,oftenreferredtoastheunitimpulseordeltafunction,isthefunctionthat denestheideaofaunitimpulse.Thisfunctionisonethatisinnitesimallynarrow,innitelytall,yet integratesto unity ,onesee.12below.Perhapsthesimplestwaytovisualizethisisasarectangular pulsefrom a )]TJ/F10 6.9738 Tf 11.056 3.922 Td [( 2 to a + 2 withaheightof 1 .Aswetakethelimitofthis, lim 0 0 ,weseethatthewidthtends tozeroandtheheighttendstoinnityasthetotalarearemainsconstantatone.Theimpulsefunctionis oftenwrittenas t Z 1 t dt =1 .12 10 Thiscontentisavailableonlineat.

PAGE 34

26 CHAPTER1.SIGNALS Figure1.33: ThisisonewaytovisualizetheDiracDeltaFunction. Figure1.34: Sinceitisquitediculttodrawsomethingthatisinnitelytall,werepresenttheDirac withanarrowcenteredatthepointitisapplied.Ifwewishtoscaleit,wemaywritethevalueitis scaledbynexttothepointofthearrow.Thisisaunitimpulsenoscaling. Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10059/latest/ImpulseFunction.llb 1.5.1.1TheSiftingPropertyoftheImpulse Therststeptounderstandingwhatthisunitimpulsefunctiongivesusistoexaminewhathappenswhen wemultiplyanotherfunctionbyit. f t t = f t .13 Sincetheimpulsefunctioniszeroeverywhereexcepttheorigin,weessentiallyjust"picko"thevalueof thefunctionwearemultiplyingbyevaluatedatzero.

PAGE 35

27 Atrstglancethismaynotappeartogiveusemuch,sincewealreadyknowthattheimpulseevaluated atzeroisinnity,andanythingtimesinnityisinnity.However,whathappensifweintegratethis? SiftingProperty R 1 f t t dt = R 1 f t dt = f R 1 t dt = f .14 Itquicklybecomesapparentthatwhatweendupwithissimplythefunctionevaluatedatzero.Hadwe used t )]TJ/F11 9.9626 Tf 9.962 0 Td [(T insteadof t ,wecouldhave"siftedout" f T .Thisiswhatwecallthe SiftingProperty oftheDiracfunction,whichisoftenusedtodenetheunitimpulse. TheSiftingPropertyisveryusefulindevelopingtheideaofconvolutionSection3.2whichisoneofthe fundamentalprinciplesofsignalprocessing.Byusingconvolutionandthesiftingpropertywecanrepresent anapproximationofanysystem'soutputifweknowthesystem'simpulseresponseandinput.Clickonthe convolutionlinkaboveformoreinformationonthis. 1.5.1.2OtherImpulseProperties Belowwewillbrieylistafewoftheotherpropertiesoftheunitimpulsewithoutgoingintodetailoftheir proofs-wewillleavethatuptoyoutoverifyasmostarestraightforward.Notethatthesepropertieshold forcontinuous and discretetime. UnitImpulseProperties t = 1 j j t t = )]TJ/F11 9.9626 Tf 7.749 0 Td [(t t = d dt u t ,where u t istheunitstep. 1.5.2Discrete-TimeImpulseUnitSample TheextensionoftheUnitImpulseFunctiontodiscrete-timebecomesquitetrivial.Allwereallyneedto realizeisthatintegrationincontinuous-timeequatestosummationindiscrete-time.Therefore,weare lookingforasignalthatsumstozeroandiszeroeverywhereexceptatzero. Discrete-TimeImpulse [ n ]= 8 < : 1 if n =0 0 otherwise .15 Figure1.35: Thegraphicalrepresentationofthediscrete-timeimpulsefunction

PAGE 36

28 CHAPTER1.SIGNALS Lookingatthediscrete-timeplotofanydiscretesignalonecannoticethatalldiscretesignalsarecomposed ofasetofscaled,time-shiftedunitsamples.Ifweletthevalueofasequenceateachinteger k bedenoted by s [ k ] andtheunitsampledelayedthatoccursat k tobewrittenas [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ] ,wecanwriteanysignalas thesumofdelayedunitsamplesthatarescaledbythesignalvalue,orweightedcoecients. s [ n ]= 1 X k = s [ k ] [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ] .16 Thisdecompositionisstrictlyapropertyofdiscrete-timesignalsandprovestobeaveryusefulproperty. note: Throughtheabovereasoning,wehaveformed.16,whichisthefundamentalconceptof discrete-timeconvolutionSection4.2. 1.5.3TheImpulseResponse The impulseresponse isexactlywhatitsnameimplies-theresponseofanLTIsystem,suchasalter, whenthesystem'sinputistheunitimpulseorunitsample.Asystemcanbecompleteddescribedbyits impulseresponseduetotheideamentionedabovethatallsignalscanberepresentedbyasuperpositionof signals.Animpulseresponsegivesanequivalentdescriptionofasystemasatransferfunction 11 ,sincethey areLaplaceTransformsSection13.1ofeachother. note: Mosttextsuse t and [ n ] todenotethecontinuous-timeanddiscrete-timeimpulse response,respectively. 1.6TheComplexExponential 12 1.6.1TheExponentialBasics The complexexponential isoneofthemostfundamentalandimportantsignalinsignalandsystem analysis.Itsimportancecomesfromitsfunctionsasabasisforperiodicsignalsaswellasbeingableto characterizelinear,time-invariantSection2.1signals.Beforeproceeding,youshouldbefamiliarwiththe ideasandfunctionsofcomplexnumbers 13 1.6.1.1BasicExponential Forallnumbers x ,weeasilyderiveanddenethe exponentialfunction fromtheTaylor'sseriesbelow: e x =1+ x 1 1! + x 2 2! + x 3 3! + ::: .17 e x = 1 X k =0 1 k x k .18 Wecanprove,usingtheratiotest,thatthisseriesdoesindeedconverge.Therefore,wecanstatethatthe exponentialfunctionshownaboveiscontinuousandeasilydened. Fromthisdenition,wecanprovethefollowingpropertyforexponentialsthatwillbeveryuseful, especiallyforthecomplexexponentialsdiscussedinthenextsection. e x 1 + x 2 = e x 1 e x 2 .19 11 "TransferFunctions" 12 Thiscontentisavailableonlineat. 13 "ComplexNumbers"

PAGE 37

29 1.6.1.2ComplexContinuous-TimeExponential Nowforallcomplexnumbers s ,wecandenethe complexcontinuous-timeexponentialsignal as f t = Ae st = Ae j!t .20 where A isaconstant, t isourindependentvariablefortime,andfor s imaginary, s = j! .Finally,from thisequationwecanrevealtheeverimportant Euler'sIdentity formoreinformationonEulerreadthis shortbiography 14 : Ae j!t = A cos !t + j A sin !t .21 FromEuler'sIdentitywecaneasilybreakthesignaldownintoitsrealandimaginarycomponents.Alsowe canseehowexponentialscanbecombinedtorepresentanyrealsignal.Bymodifyingtheirfrequencyand phase,wecanrepresentanysignalthroughasuperposityofmanysignals-allcapableofbeingrepresented byanexponential. Theaboveexpressionsdonotincludeanyinformationonphasehowever.Wecanfurthergeneralizeour aboveexpressionsfortheexponentialtogeneralizesinusoidswithanyphasebymakinganalsubstitution for s s = + j! ,whichleadsusto f t = Ae st = Ae + j! t = Ae t e j!t .22 wherewedene S asthe complexamplitude ,or phasor ,fromthersttwotermsoftheaboveequation as S = Ae t .23 GoingbacktoEuler'sIdentity,wecanrewritetheexponentialsassinusoids,wherethephasetermbecomes muchmoreapparent. f t = Ae t cos !t + j sin !t .24 Asstatedabovewecaneasilybreakthisformulaintoitsrealandimaginarypartasfollows: Re f t = Ae t cos !t .25 Im f t = Ae t sin !t .26 1.6.1.3ComplexDiscrete-TimeExponential Finallywehavereachedthelastformoftheexponentialsignalthatwewillbeinterestedin,the discretetimeexponentialsignal ,whichwewillnotgiveasmuchdetailaboutaswedidforitscontinuous-time counterpart,becausetheybothfollowthesamepropertiesandlogicdiscussedabove.Becauseitisdiscrete, thereisonlyaslightlydierentnotationusedtorepresentsitsdiscretenature f [ n ]= Be snT = Be j!nT .27 where nT representsthediscrete-timeinstantsofoursignal. 14 http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Euler.html

PAGE 38

30 CHAPTER1.SIGNALS 1.6.2Euler'sRelation AlongwithEuler'sIdentity,Euleralsodescribedawaytorepresentacomplexexponentialsignalinterms ofitsrealandimaginarypartsthrough Euler'sRelation : cos !t = e jwt + e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jwt 2 .28 sin !t = e jwt )]TJ/F11 9.9626 Tf 9.963 0 Td [(e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jwt 2 j .29 e jwt =cos !t + j sin !t .30 1.6.3DrawingtheComplexExponential Atthispoint,wehaveshownhowthecomplexexponentialcanbebrokenupintoitsrealpartandits imaginarypart.Itisnowworthlookingathowwecandraweachoftheseparts.Wecanseethatboth therealpartandtheimaginaryparthaveasinusoidtimesarealexponential.Wealsoknowthatsinusoids oscillatebetweenoneandnegativeone.Fromthisitbecomesapparentthattherealandimaginarypartsof thecomplexexponentialwilleachoscillatebetweenawindowdenedbytherealexponentialpart. a b c Figure1.36: Theshapespossiblefortherealpartofacomplexexponential.Noticethattheoscillations aretheresultofacosine,asthereisalocalmaximumat t =0 .aIf isnegative,wehavethecaseof adecayingexponentialwindow.bIf ispositive,wehavethecaseofagrowingexponentialwindow. cIf iszero,wehavethecaseofaconstantwindow.

PAGE 39

31 Whilethe determinestherateofdecay/growth,the partdeterminestherateoftheoscillations.This isapparentbynoticingthatthe ispartoftheargumenttothesinusoidalpart. Exercise1.6 Solutiononp.35. Whatdotheimaginarypartsofthecomplexexponentialsdrawnabovelooklike? Example1.11 Thefollowingdemonstrationallowsyoutoseehowtheargumentchangestheshapeofthecomplex exponential.Seehere 15 forinstructionsonhowtousethedemo. Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10060/latest/ComplexEXP.llb 1.6.4TheComplexPlane Itbecomesextremelyusefultoviewthecomplexvariable s asapointinthecomplexplane 16 the s-plane Figure1.37: Thisisthes-plane.Noticethatanytime s liesintherighthalfplane,thecomplex exponentialwillgrowthroughtime,whileanytimeitliesinthelefthalfplaneitwilldecay. 1.7Discrete-TimeSignals 17 Sofar,wehavetreatedwhatareknownas analog signalsandsystems.Mathematically,analogsignalsare functionshavingcontinuousquantitiesastheirindependentvariables,suchasspaceandtime.Discrete-time signals 18 arefunctionsdenedontheintegers;theyaresequences.Oneofthefundamentalresultsofsignal theory 19 willdetailconditionsunderwhichananalogsignalcanbeconvertedintoadiscrete-timeoneand 15 "HowtousetheLabVIEWdemos" 16 "TheComplexPlane" 17 Thiscontentisavailableonlineat. 18 "Discrete-TimeSignalsandSystems" 19 "TheSamplingTheorem"

PAGE 40

32 CHAPTER1.SIGNALS retrieved withouterror .Thisresultisimportantbecausediscrete-timesignalscanbemanipulatedby systemsinstantiatedascomputerprograms.Subsequentmodulesdescribehowvirtuallyallanalogsignal processingcanbeperformedwithsoftware. Asimportantassuchresultsare,discrete-timesignalsaremoregeneral,encompassingsignalsderived fromanalogones and signalsthataren't.Forexample,thecharactersformingatextleformasequence, whichisalsoadiscrete-timesignal.Wemustdealwithsuchsymbolicvalued 20 signalsandsystemsaswell. Aswithanalogsignals,weseekwaysofdecomposingreal-valueddiscrete-timesignalsintosimplercomponents.Withthisapproachleadingtoabetterunderstandingofsignalstructure,wecanexploitthat structuretorepresentinformationcreatewaysofrepresentinginformationwithsignalsandtoextractinformationretrievetheinformationthusrepresented.Forsymbolic-valuedsignals,theapproachisdierent: Wedevelopacommonrepresentationofallsymbolic-valuedsignalssothatwecanembodytheinformation theycontaininauniedway.Fromaninformationrepresentationperspective,themostimportantissue becomes,forbothreal-valuedandsymbolic-valuedsignals,eciency;Whatisthemostparsimoniousand compactwaytorepresentinformationsothatitcanbeextractedlater. 1.7.1Real-andComplex-valuedSignals Adiscrete-timesignalisrepresentedsymbolicallyas s n ,where n = f :::; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; 0 ; 1 ;::: g .Weusuallydraw discrete-timesignalsasstemplotstoemphasizethefacttheyarefunctionsdenedonlyontheintegers. Wecandelayadiscrete-timesignalbyanintegerjustaswithanalogones.Adelayedunitsamplehasthe expression n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m ,andequalsonewhen n = m Discrete-TimeCosineSignal Figure1.38: Thediscrete-timecosinesignalisplottedasastemplot.Canyoundtheformulafor thissignal? 1.7.2ComplexExponentials Themostimportantsignalis,ofcourse,the complexexponentialsequence s n = e j 2 fn .31 1.7.3Sinusoids Discrete-timesinusoidshavetheobviousform s n = A cos fn + .Asopposedtoanalogcomplex exponentialsandsinusoidsthatcanhavetheirfrequenciesbeanyrealvalue,frequenciesoftheirdiscrete20 "Discrete-TimeSignalsandSystems"

PAGE 41

33 timecounterpartsyielduniquewaveforms only when f liesintheinterval )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(1 2 ; 1 2 .Thispropertycanbe easilyunderstoodbynotingthataddinganintegertothefrequencyofthediscrete-timecomplexexponential hasnoeectonthesignal'svalue. e j 2 f + m n = e j 2 fn e j 2 mn = e j 2 fn .32 Thisderivationfollowsbecausethecomplexexponentialevaluatedatanintegermultipleof 2 equalsone. 1.7.4UnitSample Thesecond-mostimportantdiscrete-timesignalisthe unitsample ,whichisdenedtobe n = 8 < : 1 if n =0 0 otherwise .33 UnitSample Figure1.39: Theunitsample. Examinationofadiscrete-timesignal'splot,likethatofthecosinesignalshowninFigure1.38DiscreteTimeCosineSignal,revealsthatallsignalsconsistofasequenceofdelayedandscaledunitsamples.Because thevalueofasequenceateachinteger m isdenotedby s m andtheunitsampledelayedtooccurat m is written n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m ,wecandecompose any signalasasumofunitsamplesdelayedtotheappropriatelocation andscaledbythesignalvalue. s n = 1 X m = s m n )]TJ/F11 9.9626 Tf 9.962 0 Td [(m .34 Thiskindofdecompositionisuniquetodiscrete-timesignals,andwillproveusefulsubsequently. Discrete-timesystemscanactondiscrete-timesignalsinwayssimilartothosefoundinanalogsignals andsystems.Becauseoftheroleofsoftwareindiscrete-timesystems,manymoredierentsystemscan beenvisionedandconstructedwithprogramsthancanbewithanalogsignals.Infact,aspecialclassof analogsignalscanbeconvertedintodiscrete-timesignals,processedwithsoftware,andconvertedbackinto ananalogsignal,allwithouttheincursionoferror.Forsuchsignals,systemscanbeeasilyproducedin software,withequivalentanalogrealizationsdicult,ifnotimpossible,todesign. 1.7.5Symbolic-valuedSignals Anotherinterestingaspectofdiscrete-timesignalsisthattheirvaluesdonotneedtoberealnumbers.We dohavereal-valueddiscrete-timesignalslikethesinusoid,butwealsohavesignalsthatdenotethesequence

PAGE 42

34 CHAPTER1.SIGNALS ofcharacterstypedonthekeyboard.Suchcharacterscertainlyaren'trealnumbers,andasacollectionof possiblesignalvalues,theyhavelittlemathematicalstructureotherthanthattheyaremembersofaset. Moreformally,eachelementofthesymbolic-valuedsignal s n takesononeofthevalues f a 1 ;:::;a K g which comprisethe alphabet A .ThistechnicalterminologydoesnotmeanwerestrictsymbolstobeingmembersoftheEnglishorGreekalphabet.Theycouldrepresentkeyboardcharacters,bytes-bitquantities, integersthatconveydailytemperature.Whethercontrolledbysoftwareornot,discrete-timesystemsare ultimatelyconstructedfromdigitalcircuits,whichconsist entirely ofanalogcircuitelements.Furthermore, thetransmissionandreceptionofdiscrete-timesignals,likee-mail,isaccomplishedwithanalogsignalsand systems.Understandinghowdiscrete-timeandanalogsignalsandsystemsintertwineisperhapsthemain goalofthiscourse. [MediaObject] 21 21 ThismediaobjectisaLabVIEWVI.Pleaseviewordownloaditat

PAGE 43

35 SolutionstoExercisesinChapter1 SolutiontoExercise1.6p.31 Theylookthesameexcepttheoscillationisthatofasinusoidasopposedtoacosinusoidi.e.itpasses throughtheoriginratherthanbeingalocalmaximumat t =0 .

PAGE 44

36 CHAPTER1.SIGNALS

PAGE 45

Chapter2 Systems 2.1SystemClassicationsandProperties 1 2.1.1Introduction Inthismodulesomeofthebasicclassicationsofsystemswillbebrieyintroducedandthemostimportant propertiesofthesesystemsareexplained.Ascanbeseen,thepropertiesofasystemprovideaneasy waytoseparateonesystemfromanother.Understandingthesebasicdierence'sbetweensystems,and theirproperties,willbeafundamentalconceptusedinallsignalandsystemcourses,suchasdigitalsignal processingDSP.Onceasetofsystemscanbeidentiedassharingparticularproperties,onenolongerhas todealwithprovingacertaincharacteristicofasystemeachtime,butitcansimplybeaccepteddothe thesystemsclassication.Alsorememberthatthisclassicationpresentedhereisneitherexclusivesystems canbelongtoseveraldierentclassicationsnorisituniquethereareothermethodsofclassication 2 Examplesofsimplesystemscanbefoundhere 3 2.1.2ClassicationofSystems Alongwiththeclassicationofsystemsbelow,itisalsoimportanttounderstandotherClassicationof SignalsSection1.1. 2.1.2.1Continuousvs.Discrete Thismaybethesimplestclassicationtounderstandastheideaofdiscrete-timeandcontinuous-timeis oneofthemostfundamentalpropertiestoallofsignalsandsystem.Asystemwheretheinputandoutput signalsarecontinuousisa continuoussystem ,andonewheretheinputandoutputsignalsarediscreteis a discretesystem 2.1.2.2Linearvs.Nonlinear A linear systemisanysystemthatobeysthepropertiesofscalinghomogeneityandsuperpositionadditivity,whilea nonlinear systemisanysystemthatdoesnotobeyatleastoneofthese. Toshowthatasystem H obeysthescalingpropertyistoshowthat H kf t = kH f t .1 1 Thiscontentisavailableonlineat. 2 "IntroductiontoSystems" 3 "SimpleSystems" 37

PAGE 46

38 CHAPTER2.SYSTEMS Figure2.1: Ablockdiagramdemonstratingthescalingpropertyoflinearity Todemonstratethatasystem H obeysthesuperpositionpropertyoflinearityistoshowthat H f 1 t + f 2 t = H f 1 t + H f 2 t .2 Figure2.2: Ablockdiagramdemonstratingthesuperpositionpropertyoflinearity Itispossibletocheckasystemforlinearityinasinglethoughlargerstep.Todothis,simplycombine thersttwostepstoget H k 1 f 1 t + k 2 f 2 t = k 2 H f 1 t + k 2 H f 2 t .3 2.1.2.3TimeInvariantvs.TimeVariant A timeinvariant systemisonethatdoesnotdependonwhenitoccurs:theshapeoftheoutputdoesnot changewithadelayoftheinput.Thatistosaythatforasystem H where H f t = y t H istime invariantifforall T H f t )]TJ/F11 9.9626 Tf 9.963 0 Td [(T = y t )]TJ/F11 9.9626 Tf 9.963 0 Td [(T .4

PAGE 47

39 Figure2.3: Thisblockdiagramshowswhattheconditionfortimeinvariance.Theoutputisthesame whetherthedelayisputontheinputortheoutput. Whenthispropertydoesnotholdforasystem,thenitissaidtobe timevariant ,ortime-varying. 2.1.2.4Causalvs.Noncausal A causal systemisonethatis nonanticipative ;thatis,theoutputmaydependoncurrentandpastinputs, butnotfutureinputs.All"realtime"systemsmustbecausal,sincetheycannothavefutureinputsavailable tothem. Onemaythinktheideaoffutureinputsdoesnotseemtomakemuchphysicalsense;however,wehave onlybeendealingwithtimeasourdependentvariablesofar,whichisnotalwaysthecase.Imaginerather thatwewantedtodoimageprocessing.Thenthedependentvariablemightrepresentpixelstotheleftand rightthe"future"ofthecurrentpositionontheimage,andwewouldhavea noncausal system.

PAGE 48

40 CHAPTER2.SYSTEMS a b Figure2.4: aForatypicalsystemtobecausal...b...theoutputattime t 0 y t 0 ,canonlydepend ontheportionoftheinputsignalbefore t 0 2.1.2.5Stablevs.Unstable A stable systemisonewheretheoutputdoesnotdivergeaslongastheinputdoesnotdiverge.Thereare manywaystosaythatasignal"diverges";forexampleitcouldhaveinniteenergy.Oneparticularlyuseful denitionofdivergencerelatestowhetherthesignalisboundedornot.Thenasystemisreferredtoas boundedinput-boundedoutputBIBO stableif everypossible boundedinputproducesabounded output. Representingthisinamathematicalway,astablesystemmusthavethefollowingproperty,where x t istheinputand y t istheoutput.Theoutputmustsatisfythecondition j y t j M y < 1 .5 whenwehaveaninputtothesystemthatcanbedescribedas j x t j M x < 1 .6 M x and M y bothrepresentasetofnitepositivenumbersandtheserelationshipsholdforallof t Iftheseconditionsarenotmet, i.e. asystem'soutputgrowswithoutlimitdivergesfromabounded input,thenthesystemis unstable .NotethattheBIBOstabilityofalineartime-invariantsystemLTIis neatlydescribedintermsofwhetherornotitsimpulseresponseisabsolutelyintegrableSection3.4.

PAGE 49

41 2.2PropertiesofSystems 4 2.2.1LinearSystems Ifasystemislinear,thismeansthatwhenaninputtoagivensystemisscaledbyavalue,theoutputofthe systemisscaledbythesameamount. LinearScaling a b Figure2.5 InFigure2.5aabove,aninput x tothelinearsystem L givestheoutput y .If x isscaledbyavalue andpassedthroughthissamesystem,asinFigure2.5b,theoutputwillalsobescaledby Alinearsystemalsoobeystheprincipleofsuperposition.Thismeansthatiftwoinputsareadded togetherandpassedthroughalinearsystem,theoutputwillbethesumoftheindividualinputs'outputs. a b Figure2.6 4 Thiscontentisavailableonlineat.

PAGE 50

42 CHAPTER2.SYSTEMS SuperpositionPrinciple Figure2.7: IfFigure2.6istrue,thentheprincipleofsuperpositionsaysthatFigure2.7Superposition Principleistrueaswell.Thisholdsforlinearsystems. Thatis,ifFigure2.6istrue,thenFigure2.7SuperpositionPrincipleisalsotrueforalinearsystem. Thescalingpropertymentionedabovestillholdsinconjunctionwiththesuperpositionprinciple.Therefore, iftheinputsxandyarescaledbyfactors and ,respectively,thenthesumofthesescaledinputswill givethesumoftheindividualscaledoutputs: a b Figure2.8 SuperpositionPrinciplewithLinearScaling Figure2.9: GivenFigure2.8foralinearsystem,Figure2.9SuperpositionPrinciplewithLinear Scalingholdsaswell. 2.2.2Time-InvariantSystems Atime-invariantsystemhasthepropertythatacertaininputwillalwaysgivethesameoutput,without regardtowhentheinputwasappliedtothesystem.

PAGE 51

43 Time-InvariantSystems a b Figure2.10: Figure2.10ashowsaninputattime t whileFigure2.10bshowsthesameinput t 0 secondslater.Inatime-invariantsystembothoutputswouldbeidenticalexceptthattheonein Figure2.10bwouldbedelayedby t 0 Inthisgure, x t and x t )]TJ/F11 9.9626 Tf 9.963 0 Td [(t 0 arepassedthroughthesystemTI.BecausethesystemTIistimeinvariant,theinputs x t and x t )]TJ/F11 9.9626 Tf 9.962 0 Td [(t 0 producethesameoutput.Theonlydierenceisthattheoutput dueto x t )]TJ/F11 9.9626 Tf 9.963 0 Td [(t 0 isshiftedbyatime t 0 Whetherasystemistime-invariantortime-varyingcanbeseeninthedierentialequationordierence equationdescribingit. Time-invariantsystemsaremodeledwithconstantcoecientequations Aconstantcoecientdierentialordierenceequationmeansthattheparametersofthesystemare not changingovertimeandaninputnowwillgivethesameresultasthesameinputlater. 2.2.3LinearTime-InvariantLTISystems Certainsystemsarebothlinearandtime-invariant,andarethusreferredtoasLTIsystems. LinearTime-InvariantSystems a b Figure2.11: Thisisacombinationofthetwocasesabove.SincetheinputtoFigure2.11bisascaled, time-shiftedversionoftheinputinFigure2.11a,soistheoutput. AsLTIsystemsareasubsetoflinearsystems,theyobeytheprincipleofsuperposition.Inthegure below,weseetheeectofapplyingtime-invariancetothesuperpositiondenitioninthelinearsystems sectionabove.

PAGE 52

44 CHAPTER2.SYSTEMS a b Figure2.12 SuperpositioninLinearTime-InvariantSystems Figure2.13: TheprincipleofsuperpositionappliedtoLTIsystems 2.2.3.1LTISystemsinSeries IftwoormoreLTIsystemsareinserieswitheachother,theirordercanbeinterchangedwithoutaecting theoveralloutputofthesystem.Systemsinseriesarealsocalledcascadedsystems.

PAGE 53

45 CascadedLTISystems a b Figure2.14: TheorderofcascadedLTIsystemscanbeinterchangedwithoutchangingtheoverall eect. 2.2.3.2LTISystemsinParallel IftwoormoreLTIsystemsareinparallelwithoneanother,anequivalentsystemisonethatisdenedas thesumoftheseindividualsystems. ParallelLTISystems a b Figure2.15: Parallelsystemscanbecondensedintothesumofsystems.

PAGE 54

46 CHAPTER2.SYSTEMS 2.2.4Causality Asystemiscausalifitdoesnotdependonfuturevaluesoftheinputtodeterminetheoutput. Thismeansthatiftherstinputtoasystemcomesattime t 0 ,thenthesystemshouldnotgiveanyoutput untilthattime.Anexampleofanon-causalsystemwouldbeonethat"sensed"aninputcomingandgave anoutputbeforetheinputarrived: Non-causalSystem Figure2.16: Inthisnon-causalsystem,anoutputisproducedduetoaninputthatoccurslaterin time. Acausalsystemisalsocharacterizedbyanimpulseresponse h t thatiszerofor t< 0 .

PAGE 55

Chapter3 TimeDomainAnalysisofContinuous TimeSystems 3.1CTLinearSystemsandDierentialEquations 1 3.1.1Continuous-TimeLinearSystems Physicallyrealizable,lineartime-invariantsystemscanbedescribedbyasetoflineardierentialequations LDEs: Figure3.1: Graphicaldescriptionofabasiclineartime-invariantsystemwithaninput, f t andan output, y t d n dt n y t + a n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 d n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 dt n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 y t + + a 1 d dt y t + a 0 y t = b m d m dt m f t + + b 1 d dt f t + b 0 f t Equivalently, n X i =0 a i d i dt i y t = m X i =0 b i d i dt i f t .1 with a n =1 Itiseasytoshowthattheseequationsdeneasystemthatislinearandtimeinvariant.Anatural questiontoask,then,ishowtondthesystem'soutputresponse y t toaninput f t .Recallthatsucha solutioncanbewrittenas y t = y i t + y s t Wereferto y i t asthe zero-inputresponse thehomogeneoussolutiondueonlytotheinitialconditions ofthesystem.Wereferto y s t asthe zero-stateresponse theparticularsolutioninresponsetothe input f t .Wenowdiscusshowtosolveforeachofthesecomponentsofthesystem'sresponse. 1 Thiscontentisavailableonlineat. 47

PAGE 56

48 CHAPTER3.TIMEDOMAINANALYSISOFCONTINUOUSTIME SYSTEMS 3.1.1.1FindingtheZero-InputResponse Thezero-inputresponse, y i t ,isthesystemresponseduetoinitialconditionsonly. Example3.1:Zero-InputResponse ClosetheswitchinthecircuitpicturedinFigure3.2attimet=0andthenleaveeverythingelse alone.ThevoltageresponseisshowninFigure3.3. Figure3.2

PAGE 57

49 Figure3.3 Example3.2:Zero-InputResponse Imagineamassattachedtoaspring,asshowninFigure3.4.Whenyoupullthemassupandlet itgo,youhaveanexampleofazero-inputresponse.AplotofthisresponseisshowninFigure3.5.

PAGE 58

50 CHAPTER3.TIMEDOMAINANALYSISOFCONTINUOUSTIME SYSTEMS Figure3.4

PAGE 59

51 Figure3.5 Thereisnoinput,sowesolvefor y 0 t suchthat n X i =0 a i d i dt i y 0 t =0 ;a n =1 .2 If D isthederivativeoperator,wecanwritethepreviousequationas: )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(D n + a n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 D n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + + a 0 y 0 t =0 .3 Sinceweneedtheweightedsumofabunchof y 0 t 'sderivativestobe 0 forall t ,then n y 0 t ; d dt y 0 t ; d 2 dt 2 y 0 t ;::: o mustallbeofthesameform. Onlytheexponential, e st where s 2 C ,hasthispropertyseeyourDierentialEquation'stextbookfor details.Sowemustassumethat, y 0 t = ce st ;c 6 =0 .4 forsome c and s Since d dt y 0 t = cse st d 2 dt 2 y 0 t = cs 2 e st ::: wehave )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(D n + a n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 D n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + + a 0 y 0 t =0

PAGE 60

52 CHAPTER3.TIMEDOMAINANALYSISOFCONTINUOUSTIME SYSTEMS c )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(s n + a n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 s n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + + a 1 s + a 0 e st =0 .5 .5holdsforall t onlywhen s n + a n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 s n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 + + a 1 s + a 0 =0 .6 Wherethisequationisreferredtoasthe characteristicequation ofthesystem.Thepossiblevaluesof s aretherootsofthispolynomial f s 1 ;s 2 ;:::;s n g s )]TJ/F11 9.9626 Tf 9.962 0 Td [(s 1 s )]TJ/F11 9.9626 Tf 9.962 0 Td [(s 2 s )]TJ/F11 9.9626 Tf 9.962 0 Td [(s 3 ::: s )]TJ/F11 9.9626 Tf 9.963 0 Td [(s n =0 i.e. possiblesolutionsare c 1 e s 1 t c 2 e s 2 t ::: c n e s n t .Sincethesystemis linear ,thegeneralsolutionifofthe form: y 0 t = c 1 e s 1 t + c 2 e s 2 t + + c n e s n t .7 Then,solveforthe f c 1 ;:::;c n g usingtheinitialconditions. Example3.3 SeeLathip.108foragoodexample! WegenerallyassumethattheIC'sofasystemarezero,whichimplies y i t =0 .However,the method of solvingfor y i t willproveusefullateron. 3.1.1.2FindingtheZero-StateResponse Solvingalineardierentialequation n X i =0 a i d i dt i y t = m X i =0 b i d i dt i f t .8 givenaspecicinput f t isadiculttaskingeneral.Moreimportantly,themethoddependsentirelyon thenatureof f t ;ifwechangetheinputsignal,wemustcompletelyre-solvethesystemofequationsto ndthesystemresponse. ConvolutionSection3.2helpstobypassthesediculties.Insection2,weexplainhowconvolutionhelps todeterminethesystem'soutput,givenonlytheinput f t andthesystem'simpulseresponseSection1.5, h t Beforederivingtheconvolutionprocedure,weshowthatasystem'simpulseresponseiseasilyderived fromitslinear,dierentialequationLDE.WewillshowthederivationfortheLDEbelow,where m
PAGE 61

53 3.2Continuous-TimeConvolution 2 3.2.1Motivation Convolutionhelpstodeterminetheeectasystemhasonaninputsignal.Itcanbeshownthatalinear, time-invariantsystemSection2.1iscompletelycharacterizedbyitsimpulseresponse.Atrstglance,this mayappeartobeoflittleuse,sinceimpulsefunctionsarenotwelldenedinrealapplications.However, thesiftingpropertyofimpulsesSection1.5.1.1:TheSiftingPropertyoftheImpulsetellsusthatasignal canbedecomposedintoaninnitesumintegralofscaledandshiftedimpulses.Byknowinghowasystem aectsasingleimpulse,andbyunderstandingthewayasignaliscomprisedofscaledandsummedimpulses, itseemsreasonablethatitshouldbepossibletoscaleandsumtheimpulseresponsesofasysteminorderto determinewhatoutputsignalwillresultsfromaparticularinput.Thisispreciselywhatconvolutiondoes convolutiondeterminesthesystem'soutputfromknowledgeoftheinputandthesystem's impulseresponse Intherestofthismodule,wewillexamineexactlyhowconvolutionisdenedfromthereasoningabove. ThiswillresultintheconvolutionintegralseethenextsectionanditspropertiesSection3.3.These conceptsareveryimportantinElectricalEngineeringandwillmakeanyengineer'slifealoteasierifthe timeisspentnowtotrulyunderstandwhatisgoingon. Inordertofullyunderstandconvolution,youmaynditusefultolookatthediscrete-timeconvolution Section4.2aswell.Itwillalsobehelpfultoexperimentwiththeapplets 3 availableontheinternet.These resourceswilloerdierentapproachestothiscrucialconcept. 3.2.2ConvolutionIntegral Asmentionedabove,theconvolutionintegralprovidesaneasymathematicalwaytoexpresstheoutputofan LTIsystembasedonanarbitrarysignal, x t ,andthesystem'simpulseresponse, h t .The convolution integral isexpressedas y t = Z 1 x h t )]TJ/F11 9.9626 Tf 9.963 0 Td [( d .13 Convolutionissuchanimportanttoolthatitisrepresentedbythesymbol ,andcanbewrittenas y t = x t h t .14 Bymakingasimplechangeofvariablesintotheconvolutionintegral, = t )]TJ/F11 9.9626 Tf 10.257 0 Td [( ,wecaneasilyshowthat convolutionis commutative : x t h t = h t x t .15 Formoreinformationonthecharacteristicsoftheconvolutionintegral,readaboutthePropertiesofConvolutionSection3.3. Wenowpresenttwodistinctapproachesforderivingtheconvolutionintegral.Thesederivations,along withabasicexample,willhelptobuildintuitionaboutconvolution. 3.2.3DerivationI:TheShortApproach ThederivationusedherecloselyfollowstheonediscussedintheMotivationSection3.2.1:Motivation sectionabove.Tobeginthis,itisnecessarytostatetheassumptionswewillbemaking.Inthisinstance, theonlyconstraintsonoursystemarethatitbelinearandtime-invariant. BriefOverviewofDerivationSteps: 1.Animpulseinputleadstoanimpulseresponseoutput. 2 Thiscontentisavailableonlineat. 3 http://www.jhu.edu/ signals

PAGE 62

54 CHAPTER3.TIMEDOMAINANALYSISOFCONTINUOUSTIME SYSTEMS 2.Ashiftedimpulseinputleadstoashiftedimpulseresponseoutput.Thisisduetothetime-invariance ofthesystem. 3.Wenowscaletheimpulseinputtogetascaledimpulseoutput.Thisisusingthescalarmultiplication propertyoflinearity. 4.Wecannow"sumup"aninnitenumberofthesescaledimpulsestogetasumofaninnitenumber ofscaledimpulseresponses.Thisisusingtheadditivityattributeoflinearity. 5.Nowwerecognizethatthisinnitesumisnothingmorethananintegral,soweconvertbothsidesinto integrals. 6.Recognizingthattheinputisthefunction f t ,wealsorecognizethattheoutputisexactlythe convolutionintegral. Figure3.6: Webeginwithasystemdenedbyitsimpulseresponse, h t Figure3.7: Wethenconsiderashiftedversionoftheinputimpulse.Duetothetimeinvarianceofthe system,weobtainashiftedversionoftheoutputimpulseresponse.

PAGE 63

55 Figure3.8: Nowweusethescalingpartoflinearitybyscalingthesystembyavalue, f ,thatis constantwithrespecttothesystemvariable, t Figure3.9: Wecannowusetheadditivityaspectoflinearitytoaddaninnitenumberofthese,one foreachpossible .Sinceaninnitesumisexactlyanintegral,weendupwiththeintegrationknownas theConvolutionIntegral.Usingthesiftingproperty,werecognizetheleft-handsidesimplyastheinput f t 3.2.4DerivationII:TheLongApproach Thisderivationisreallynottoodierentfromtheoneabove.Itis,however,alittlemorerigorousanda littlelonger.Hopefully,ifyouthinkyou"kindof"getthederivationabove,thiswillhelpyougainamore completeunderstandingofconvolution. TherststepinthisderivationistodeneaparticularrealizationoftheunitimpulsefunctionSection1.5.Forthis,wewilluse t = 8 < : 1 if )]TJ/F1 9.9626 Tf 9.963 8.069 Td [()]TJ/F7 6.9738 Tf 5.762 -4.147 Td [( 2
PAGE 64

56 CHAPTER3.TIMEDOMAINANALYSISOFCONTINUOUSTIME SYSTEMS Figure3.10: Therealizationoftheunitimpulsefunctionthatwewilluseforthisderivation. Afterdeningourrealizationoftheunitimpulseresponse,wecanderiveourconvolutionintegralfrom thefollowingstepsfoundinthetablebelow.Notethattheleftcolumnrepresentstheinputandtheright columnisthesystem'soutputgiventhatinput. DerivationIIofConvolutionIntegral Input Output lim 0 t h lim 0 h t lim 0 t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n h lim 0 h t )]TJ/F11 9.9626 Tf 9.962 0 Td [(n lim 0 f n t )]TJ/F11 9.9626 Tf 9.962 0 Td [(n h lim 0 f n h t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n lim 0 P n f n t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n h lim 0 P n f n h t )]TJ/F11 9.9626 Tf 9.962 0 Td [(n R 1 f t )]TJ/F11 9.9626 Tf 9.963 0 Td [( d h R 1 f h t )]TJ/F11 9.9626 Tf 9.962 0 Td [( d f t h y t = R 1 f h t )]TJ/F11 9.9626 Tf 9.963 0 Td [( d Table3.1 3.2.5ImplementationofConvolution Takingacloserlookattheconvolutionintegral,wendthatwearemultiplyingtheinputsignalbythe time-reversedimpulseresponseandintegrating.Thiswillgiveusthevalueoftheoutputatonegivenvalue of t .Ifwethenshiftthetime-reversedimpulseresponsebyasmallamount,wegettheoutputforanother valueof t .Repeatingthisforeverypossiblevalueof t ,yieldsthetotaloutputfunction.Whilewewould neveractuallydothiscomputationbyhandinthisfashion,itdoesprovideuswithsomeinsightintowhat isactuallyhappening.Wendthatweareessentiallyreversingtheimpulseresponsefunctionandsliding itacrosstheinputfunction,integratingaswego.Thismethod,referredtoasthe graphicalmethod providesuswithamuchsimplerwaytosolvefortheoutputforsimplecontrivedsignals,whileimproving

PAGE 65

57 ourintuitionforthemorecomplexcaseswherewerelyoncomputers.InfactTexasInstruments 4 develops DigitalSignalProcessors 5 whichhavespecialinstructionsetsforcomputationssuchasconvolution. Example3.4 Thisdemonstrationillustratesthegraphicalmethodforconvolution.Seehere 6 forinstructionson howtousethedemo. ThismediaobjectisaLabVIEWVI.Pleaseviewordownloaditat 3.2.6BasicExample Letuslookatabasiccontinuous-timeconvolutionexampletohelpexpresssomeoftheideasmentioned abovethroughashortexample.Wewillconvolvetogethertwounitpulses, x t and h t a b Figure3.11: Herearethetwobasicsignalsthatwewillconvolvetogether. 3.2.6.1ReectandShift Nowwewilltakeoneofthefunctionsandreectitaroundthey-axis.Thenwemustshiftthefunction,such thattheorigin,thepointofthefunctionthatwasoriginallyontheorigin,islabeledaspoint .Thisstepis showninthegurebelow, h t )]TJ/F11 9.9626 Tf 9.963 0 Td [( .Sinceconvolutioniscommutativeitwillnevermatterwhichfunctionis reectedandshifted;however,asthefunctionsbecomemorecomplicatedreectingandshiftingthe"right one"willoftenmaketheproblemmucheasier. 4 http://www.ti.com 5 http://dspvillage.ti.com/docs/toolssoftwarehome.jhtml 6 "HowtousetheLabVIEWdemos"

PAGE 66

58 CHAPTER3.TIMEDOMAINANALYSISOFCONTINUOUSTIME SYSTEMS Figure3.12: Thereectedandshiftedunitpulse. 3.2.6.2RegionsofIntegration Next,wewanttolookatthefunctionsanddividethespanofthefunctionsintodierentlimitsofintegration. Thesedierentregionscanbeunderstoodbythinkingabouthowweslide h t )]TJ/F11 9.9626 Tf 9.963 0 Td [( overtheotherfunction. Theselimitscomefromthedierentregionsofoverlapthatoccurbetweenthetwofunctions.Ifthefunction weremorecomplex,thenwewouldneedtohavemorelimitssothatthatoverlappingpartsofbothfunction couldbeexpressedinasingle,linearintegral.Forthisproblemwewillhavethefollowingfourregions. Comparetheselimitsofintegrationtothesketchesof h t )]TJ/F11 9.9626 Tf 9.962 0 Td [( and x t toseeifyoucanunderstandwhywe havethefourregions.Notethatthe t inthelimitsofintegrationreferstotheright-handsideof h t )]TJ/F11 9.9626 Tf 9.962 0 Td [( 's function,labeledas t betweenzeroandoneontheplot. FourLimitsofIntegration 1. t< 0 2. 0 t< 1 3. 1 t< 2 4. t 2 3.2.6.3UsingtheConvolutionIntegral Finallywearereadyforalittlemath.Usingtheconvolutionintegral,letusintegratetheproductof x t h t )]TJ/F11 9.9626 Tf 9.963 0 Td [( .Forourrstandfourthregionthiswillbetrivialasitwillalwaysbe 0 .Thesecondregion, 0 t< 1 ,willrequirethefollowingmath: y t = R t 0 1 d = t .16 Thethirdregion, 1 t< 2 ,issolvedinmuchthesamemanner.Takenoteofthechangesinourintegration though.Aswemove h t )]TJ/F11 9.9626 Tf 9.963 0 Td [( acrossourotherfunction,theleft-handedgeofthefunction, t )]TJ/F8 9.9626 Tf 10.127 0 Td [(1 ,becomes

PAGE 67

59 ourlowlimitfortheintegral.Thisisshownthroughourconvolutionintegralas y t = R 1 t )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 1 d =1 )]TJ/F8 9.9626 Tf 9.962 0 Td [( t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 =2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(t .17 Theaboveformulasshowthemethodforcalculatingconvolution;however,donotletthesimplicityofthis exampleconfuseyouwhenyouworkonotherproblems.Themethodwillbethesame,youwilljusthaveto dealwithmoremathinmorecomplicatedintegrals. 3.2.6.4ConvolutionResults Thus,wehavethefollowingresultsforourfourregions: y t = 8 > > > > > < > > > > > : 0 if t< 0 t if 0 t< 1 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(t if 1 t< 2 0 if t 2 .18 Nowthatwehavefoundtheresultingfunctionforeachofthefourregions,wecancombinethemtogether andgraphtheconvolutionof x t h t Figure3.13: Showsthesystem'sresponsetotheinput, x t 3.3PropertiesofConvolution 7 Inthismodulewewillstudyseveralofthemostprevalentpropertiesofconvolution.Notethatthesepropertiesapplytobothcontinuous-timeconvolutionSection3.2anddiscrete-timeconvolutionSection4.2. 7 Thiscontentisavailableonlineat.

PAGE 68

60 CHAPTER3.TIMEDOMAINANALYSISOFCONTINUOUSTIME SYSTEMS Referbacktothesetwomodulesifyouneedareviewofconvolution.Also,fortheproofsofsomeofthe properties,wewillbeusingcontinuous-timeintegrals,butwecouldprovethemthesamewayusingthe discrete-timesummations. 3.3.1Associativity Theorem3.1: AssociativeLaw f 1 t f 2 t f 3 t = f 1 t f 2 t f 3 t .19 Figure3.14: Graphicalimplicationoftheassociativepropertyofconvolution. 3.3.2Commutativity Theorem3.2: CommutativeLaw y t = f t h t = h t f t .20 Proof: Toprove.20,allweneedtodoismakeasimplechangeofvariablesinourconvolutionintegral orsum, y t = Z 1 f h t )]TJ/F11 9.9626 Tf 9.963 0 Td [( d .21 Byletting = t )]TJ/F11 9.9626 Tf 9.962 0 Td [( ,wecaneasilyshowthatconvolutionis commutative : y t = R 1 f t )]TJ/F11 9.9626 Tf 9.963 0 Td [( h d = R 1 h f t )]TJ/F11 9.9626 Tf 9.962 0 Td [( d .22 f t h t = h t f t .23

PAGE 69

61 Figure3.15: Thegureshowsthateitherfunctioncanberegardedasthesystem'sinputwhilethe otheristheimpulseresponse. 3.3.3Distribution Theorem3.3: DistributiveLaw f 1 t f 2 t + f 3 t = f 1 t f 2 t + f 1 t f 3 t .24 Proof: Theproofofthistheoremcanbetakendirectlyfromthedenitionofconvolutionandbyusing thelinearityoftheintegral. Figure3.16

PAGE 70

62 CHAPTER3.TIMEDOMAINANALYSISOFCONTINUOUSTIME SYSTEMS 3.3.4TimeShift Theorem3.4: ShiftProperty For c t = f t h t ,then c t )]TJ/F11 9.9626 Tf 9.963 0 Td [(T = f t )]TJ/F11 9.9626 Tf 9.962 0 Td [(T h t .25 and c t )]TJ/F11 9.9626 Tf 9.963 0 Td [(T = f t h t )]TJ/F11 9.9626 Tf 9.962 0 Td [(T .26 a b c Figure3.17: Graphicaldemonstrationoftheshiftproperty. 3.3.5ConvolutionwithanImpulse Theorem3.5: ConvolvingwithUnitImpulse f t t = f t .27 Proof: Forthisproof,wewilllet t betheunitimpulselocatedattheorigin.Usingthedenitionof convolutionwestartwiththeconvolutionintegral f t t = Z 1 f t )]TJ/F11 9.9626 Tf 9.963 0 Td [( d .28

PAGE 71

63 Fromthedenitionoftheunitimpulse,weknowthat =0 whenever 6 =0 .Weusethisfact toreducetheaboveequationtothefollowing: f t t = R 1 f t d = f t R 1 d .29 Theintegralof willonlyhaveavaluewhen =0 fromthedenitionoftheunitimpulse, thereforeitsintegralwillequalone.Thuswecansimplifytheequationtoourtheorem: f t t = f t .30 a b Figure3.18: Thegures,andequationabove,revealtheidentityfunctionoftheunitimpulse. 3.3.6Width Incontinuoustime,if Duration f 1 = T 1 and Duration f 2 = T 2 ,then Duration f 1 f 2 = T 1 + T 2 .31

PAGE 72

64 CHAPTER3.TIMEDOMAINANALYSISOFCONTINUOUSTIME SYSTEMS a b c Figure3.19: Incontinuous-time,thedurationoftheconvolutionresultequalsthesumofthelengths ofeachofthetwosignalsthatareconvolved. Indiscretetime,if Duration f 1 = N 1 and Duration f 2 = N 2 ,then Duration f 1 f 2 = N 1 + N 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 .32

PAGE 73

65 3.3.7Causality If f and h arebothcausal,then f h isalsocausal. Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10088/latest/ConvTIMEDOM.llb 3.4BIBOStability 8 BIBOstandsforboundedinput,boundedoutput.BIBOstableisaconditionsuchthatanyboundedinput yieldsaboundedoutput.Thisistosaythataslongasweinputastablesignal,weareguaranteedtohave astableoutput. Inordertounderstandthisconcept,wemustrstlookmorecloselyintoexactlywhatwemeanby bounded.Aboundedsignalisanysignalsuchthatthereexistsavaluesuchthattheabsolutevalueofthe signalisnevergreaterthansomevalue.Sincethisvalueisarbitrary,whatwemeanisthatatnopointcan thesignaltendtoinnity. Figure3.20: Aboundedsignalisasignalforwhichthereexistsaconstant A suchthat j f t j .

PAGE 74

66 CHAPTER3.TIMEDOMAINANALYSISOFCONTINUOUSTIME SYSTEMS Thisistosaythatthetransferfunctionis absolutelyintegrable Toextendthisconcepttodiscrete-time,wemakethestandardtransitionfromintegrationtosummation andgetthatthetransferfunction, h n ,mustbe absolutelysummable .Thatis Discrete-TimeConditionforBIBOStability 1 X n = j h n j < 1 .34 3.4.1StabilityandLaplace Stabilityisveryeasytoinferfromthepole-zeroplotSection13.6ofatransferfunction.Theonlycondition necessarytodemonstratestabilityistoshowthatthe j! -axisisintheregionofconvergence. a b Figure3.21: aExampleofapole-zeroplotforastablecontinuous-timesystem.bExampleofa pole-zeroplotforanunstablecontinuous-timesystem. 3.4.2StabilityandtheZ-Transform Stabilityfordiscrete-timesignalsSection1.1inthez-domainSection14.1isaboutaseasytodemonstrate asitisforcontinuous-timesignalsintheLaplacedomain.However,insteadoftheregionofconvergence needingtocontainthe j! -axis,theROCmustcontaintheunitcircle.

PAGE 75

67 a b Figure3.22: aAstablediscrete-timesystem.bAnunstablediscrete-timesystem. Thisisanunsupportedmediatype.Toview,pleaseseehttp://cnx.org/content/m10113/latest/BIBO.llb

PAGE 76

68 CHAPTER3.TIMEDOMAINANALYSISOFCONTINUOUSTIME SYSTEMS

PAGE 77

Chapter4 TimeDomainAnalysisofDiscreteTime Systems 4.1Discrete-TimeSystemsintheTime-Domain 1 Adiscrete-timesignal s n is delayed by n 0 sampleswhenwewrite s n )]TJ/F11 9.9626 Tf 9.963 0 Td [(n 0 ,with n 0 > 0 .Choosing n 0 tobenegativeadvancesthesignalalongtheintegers.Asopposedtoanalogdelays 2 ,discrete-timedelays can only beintegervalued.Inthefrequencydomain,delayingasignalcorrespondstoalinearphaseshift ofthesignal'sdiscrete-timeFouriertransform: )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(s n )]TJ/F11 9.9626 Tf 9.962 0 Td [(n 0 $ e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j 2 fn 0 S )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 f Linear discrete-timesystemshavethesuperpositionproperty. S a 1 x 1 n + a 2 x 2 n = a 1 S x 1 n + a 2 S x 2 n .1 Adiscrete-timesystemiscalled shift-invariant analogoustotime-invariantanalogsystems 3 ifdelaying theinputdelaysthecorrespondingoutput.If S x n = y n ,thenashift-invariantsystemhastheproperty S x n )]TJ/F11 9.9626 Tf 9.963 0 Td [(n 0 = y n )]TJ/F11 9.9626 Tf 9.963 0 Td [(n 0 .2 Weusethetermshift-invarianttoemphasizethatdelayscanonlyhaveintegervaluesindiscrete-time,while inanalogsignals,delayscanbearbitrarilyvalued. Wewanttoconcentrateonsystemsthatarebothlinearandshift-invariant.Itwillbethesethatallowus thefullpoweroffrequency-domainanalysisandimplementations.Becausewehavenophysicalconstraints in"constructing"suchsystems,weneedonlyamathematicalspecication.Inanalogsystems,thedierentialequationspeciestheinput-outputrelationshipinthetime-domain.Thecorrespondingdiscrete-time specicationisthe dierenceequation y n = a 1 y n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1+ + a p y n )]TJ/F11 9.9626 Tf 9.963 0 Td [(p + b 0 x n + b 1 x n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1+ + b q x n )]TJ/F11 9.9626 Tf 9.963 0 Td [(q .3 Here,theoutputsignal y n isrelatedtoits past values y n )]TJ/F11 9.9626 Tf 9.963 0 Td [(l l = f 1 ;:::;p g ,andtothecurrentand pastvaluesoftheinputsignal x n .Thesystem'scharacteristicsaredeterminedbythechoicesforthe numberofcoecients p and q andthecoecients'values f a 1 ;:::;a p g and f b 0 ;b 1 ;:::;b q g aside: Thereisanasymmetryinthecoecients:whereis a 0 ?Thiscoecientwouldmultiply the y n termin.3.Wehaveessentiallydividedtheequationbyit,whichdoesnotchangethe input-outputrelationship.Wehavethuscreatedtheconventionthat a 0 isalwaysone. 1 Thiscontentisavailableonlineat. 2 "SimpleSystems":SectionDelay 3 "SimpleSystems" 69

PAGE 78

70 CHAPTER4.TIMEDOMAINANALYSISOFDISCRETETIMESYSTEMS Asopposedtodierentialequations,whichonlyprovidean implicit descriptionofasystemwemust somehowsolvethedierentialequation,dierenceequationsprovidean explicit wayofcomputingthe outputforanyinput.Wesimplyexpressthedierenceequationbyaprogramthatcalculateseachoutput fromthepreviousoutputvalues,andthecurrentandpreviousinputs. Dierenceequationsareusuallyexpressedinsoftwarewith for loops.AMATLABprogramthatwould computetherst1000valuesoftheoutputhastheform forn=1:1000 yn=suma.*yn-1:-1:n-p+sumb.*xn:-1:n-q; end Animportantdetailemergeswhenweconsidermakingthisprogramwork;infact,aswrittenithasatleast twobugs.Whatinputandoutputvaluesenterintothecomputationof y ?Weneedvaluesfor y y )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 ,...,valueswehavenotyetcomputed.Tocomputethem,wewouldneedmorepreviousvaluesofthe output,whichwehavenotyetcomputed.Tocomputethesevalues,wewouldneedevenearliervalues,ad innitum.Thewayoutofthispredicamentistospecifythesystem's initialconditions :wemustprovide the p outputvaluesthatoccurredbeforetheinputstarted.Thesevaluescanbearbitrary,butthechoice doesimpacthowthesystemrespondstoagiveninput. One choicegivesrisetoalinearsystem:Makethe initialconditionszero.Thereasonliesinthedenitionofalinearsystem 4 :Theonlywaythattheoutput toasumofsignalscanbethesumoftheindividualoutputsoccurswhentheinitialconditionsineachcase arezero. Exercise4.1 Solutiononp.89. Theinitialconditionissueresolvesmakingsenseofthedierenceequationforinputsthatstartat someindex.However,theprogramwillnotworkbecauseofaprogramming,notconceptual,error. Whatisit?Howcanitbe"xed?" Example4.1 Let'sconsiderthesimplesystemhaving p =1 and q =0 y n = ay n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1+ bx n .4 Tocomputetheoutputatsomeindex,thisdierenceequationsaysweneedtoknowwhatthe previousoutput y n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 andwhattheinputsignalisatthatmomentoftime.Inmoredetail,let's computethissystem'soutputtoaunit-sampleinput: x n = n .Becausetheinputiszerofor negativeindices,westartbytryingtocomputetheoutputat n =0 y = ay )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+ b .5 Whatisthevalueof y )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ?Becausewehaveusedaninputthatiszeroforallnegativeindices,it isreasonabletoassumethattheoutputisalsozero.Certainly,thedierenceequationwouldnot describealinearsystem 5 iftheinputthatiszerofor all timedidnotproduceazerooutput.With thisassumption, y )]TJ/F8 9.9626 Tf 7.749 0 Td [(1=0 ,leaving y = b .For n> 0 ,theinputunit-sampleiszero,which leavesuswiththedierenceequation y n = ay n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ;n> 0 .Wecanenvisionhowthelter respondstothisinputbymakingatable. y n = ay n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1+ b n .6 4 "SimpleSystems":SectionLinearSystems 5 "SimpleSystems":SectionLinearSystems

PAGE 79

71 n x n y n )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 0 0 1 b 1 0 ba 2 0 ba 2 : 0 : n 0 ba n Table4.1 Coecientvaluesdeterminehowtheoutputbehaves.Theparameter b canbeanyvalue,and servesasagain.Theeectoftheparameter a ismorecomplicatedTable4.1.Ifitequalszero, theoutputsimplyequalstheinputtimesthegain b .Forallnon-zerovaluesof a ,theoutput lastsforever;suchsystemsaresaidtobe IIR I nnite I mpulse R esponse.Thereasonforthis terminologyisthattheunitsamplealsoknownastheimpulseespeciallyinanalogsituations,and thesystem'sresponsetothe"impulse"lastsforever.If a ispositiveandlessthanone,theoutput isadecayingexponential.When a =1 ,theoutputisaunitstep.If a isnegativeandgreater than )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ,theoutputoscillateswhiledecayingexponentially.When a = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ,theoutputchanges signforever,alternatingbetween b and )]TJ/F11 9.9626 Tf 7.749 0 Td [(b .Moredramaticeectswhen j a j > 1 ;whetherpositive ornegative,theoutputsignalbecomeslargerandlarger, growing exponentially. Figure4.1: Theinputtothesimpleexamplesystem,aunitsample,isshownatthetop,withthe outputsforseveralsystemparametervaluesshownbelow. Positivevaluesof a areusedinpopulationmodelstodescribehowpopulationsizeincreases overtime.Here, n mightcorrespondtogeneration.Thedierenceequationsaysthatthenumber inthenextgenerationissomemultipleofthepreviousone.Ifthismultipleislessthanone,the populationbecomesextinct;ifgreaterthanone,thepopulationourishes.Thesamedierence equationalsodescribestheeectofcompoundinterestondeposits.Here, n indexesthetimesat whichcompoundingoccursdaily,monthly,etc., a equalsthecompoundinterestrateplusone,and b =1 thebankprovidesnogain.Insignalprocessingapplications,wetypicallyrequirethatthe

PAGE 80

72 CHAPTER4.TIMEDOMAINANALYSISOFDISCRETETIMESYSTEMS outputremainboundedforanyinput.Forourexample,thatmeansthatwerestrict j a j =1 and chosevaluesforitandthegainaccordingtotheapplication. Exercise4.2 Solutiononp.89. Notethatthedierenceequation.3, y n = a 1 y n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1+ + a p y n )]TJ/F11 9.9626 Tf 9.963 0 Td [(p + b 0 x n + b 1 x n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1+ + b q x n )]TJ/F11 9.9626 Tf 9.963 0 Td [(q doesnotinvolvetermslike y n +1 or x n +1 ontheequation'srightside.Cansuchtermsalso beincluded?Whyorwhynot? Figure4.2: Theplotshowstheunit-sampleresponseofalength-5boxcarlter. Example4.2 Asomewhatdierentsystemhasno" a "coecients.Considerthedierenceequation y n = 1 q x n + + x n )]TJ/F11 9.9626 Tf 9.963 0 Td [(q +1 .7 Becausethissystem'soutputdependsonlyoncurrentandpreviousinputvalues,weneednot beconcernedwithinitialconditions.Whentheinputisaunit-sample,theoutputequals 1 q for n = f 0 ;:::;q )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 g ,thenequalszerothereafter.Suchsystemsaresaidtobe FIR F inite I mpulse R esponsebecausetheirunitsampleresponseshaveniteduration.PlottingthisresponseFigure4.2showsthattheunit-sampleresponseisapulseofwidth q andheight 1 q .Thiswaveform isalsoknownasaboxcar,hencethename boxcarlter giventothissystem.We'llderiveits frequencyresponseanddevelopitslteringinterpretationinthenextsection.Fornow,notethat thedierenceequationsaysthateachoutputvalueequalsthe average oftheinput'scurrentand previousvalues.Thus,theoutputequalstherunningaverageofinput'sprevious q values.Sucha systemcouldbeusedtoproducetheaverageweeklytemperature q =7 thatcouldbeupdated daily. [MediaObject] 6 4.2Discrete-TimeConvolution 7 4.2.1Overview Convolutionisaconceptthatextendstoallsystemsthatareboth linearandtime-invariantSection2.1 LTI .Theideaof discrete-timeconvolution isexactlythesameasthatofcontinuous-timeconvolution 6 ThismediaobjectisaLabVIEWVI.Pleaseviewordownloaditat 7 Thiscontentisavailableonlineat.

PAGE 81

73 Section3.2.Forthisreason,itmaybeusefultolookatbothversionstohelpyourunderstandingofthis extremelyimportantconcept.Recallthatconvolutionisaverypowerfultoolindeterminingasystem's outputfromknowledgeofanarbitraryinputandthesystem'simpulseresponse.Itwillalsobehelpfulto seeconvolutiongraphicallywithyourowneyesandtoplayaroundwithitsome,soexperimentwiththe applets 8 availableontheinternet.Theseresourceswilloerdierentapproachestothiscrucialconcept. 4.2.2ConvolutionSum Asmentionedabove,theconvolutionsumprovidesaconcise,mathematicalwaytoexpresstheoutputofan LTIsystembasedonanarbitrarydiscrete-timeinputsignalandthesystem'sresponse.The convolution sum isexpressedas y [ n ]= 1 X k = x [ k ] h [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ] .8 Aswithcontinuous-time,convolutionisrepresentedbythesymbol*,andcanbewrittenas y [ n ]= x [ n ] h [ n ] .9 Bymakingasimplechangeofvariablesintotheconvolutionsum, k = n )]TJ/F11 9.9626 Tf 10.641 0 Td [(k ,wecaneasilyshowthat convolutionis commutative : x [ n ] h [ n ]= h [ n ] x [ n ] .10 Formoreinformationonthecharacteristicsofconvolution,readaboutthePropertiesofConvolutionSection3.3. 4.2.3Derivation Weknowthatanydiscrete-timesignalcanberepresentedbyasummationofscaledandshifteddiscrete-time impulses.Sinceweareassumingthesystemtobelinearandtime-invariant,itwouldseemtoreasonthat aninputsignalcomprisedofthesumofscaledandshiftedimpulseswouldgiverisetoanoutputcomprised ofasumofscaledandshiftedimpulseresponses.Thisisexactlywhatoccursin convolution .Belowwe presentamorerigorousandmathematicallookatthederivation: Letting H beaDTLTIsystem,westartwiththefollowingequationandworkourwaydownthe convolutionsum! y [ n ]= H [ x [ n ]] = H P 1 k = x [ k ] [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ] = P 1 k = H [ x [ k ] [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k ]] = P 1 k = x [ k ] H [ [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k ]] = P 1 k = x [ k ] h [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ] .11 Letustakeaquicklookatthestepstakenintheabovederivation.Afterourinitialequation,weusingthe DTsiftingpropertySection1.5.1.1:TheSiftingPropertyoftheImpulsetorewritethefunction, x [ n ] ,asa sumofthefunctiontimestheunitimpulse.Next,wecanmovearoundthe H operatorandthesummation because H [ ] isalinear,DTsystem.Becauseofthislinearityandthefactthat x [ k ] isaconstant,wecan pullthepreviousmentionedconstantoutandsimplymultiplyitby H [ ] .Finally,weusethefactthat H [ ] istimeinvariantinordertoreachournalstate-theconvolutionsum! Aquickgraphicalexamplemayhelpindemonstratingwhyconvolutionworks. 8 http://www.jhu.edu/ signals

PAGE 82

74 CHAPTER4.TIMEDOMAINANALYSISOFDISCRETETIMESYSTEMS Figure4.3: Asingleimpulseinputyieldsthesystem'simpulseresponse. Figure4.4: Ascaledimpulseinputyieldsascaledresponse,duetothescalingpropertyofthesystem's linearity.

PAGE 83

75 Figure4.5: Wenowusethetime-invariancepropertyofthesystemtoshowthatadelayedinput resultsinanoutputofthesameshape,onlydelayedbythesameamountastheinput. Figure4.6: Wenowusetheadditivityportionofthelinearitypropertyofthesystemtocompletethe picture.Sinceanydiscrete-timesignalisjustasumofscaledandshifteddiscrete-timeimpulses,wecan ndtheoutputfromknowingtheinputandtheimpulseresponse. 4.2.4ConvolutionThroughTimeAGraphicalApproach Inthissectionwewilldevelopasecondgraphicalinterpretationofdiscrete-timeconvolution.Wewillbegin thisbywritingtheconvolutionsumallowing x tobeacausal,lengthm signaland h tobeacausal,lengthk ,

PAGE 84

76 CHAPTER4.TIMEDOMAINANALYSISOFDISCRETETIMESYSTEMS LTIsystem.Thisgivesusthenitesummation, y [ n ]= m )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X l =0 x [ l ] h [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(l ] .12 Noticethatforanygiven n wehaveasumoftheproductsof x l andatime-delayed h )]TJ/F10 6.9738 Tf 6.227 0 Td [(l .Thisistosaythat wemultiplythetermsof x bythetermsofatime-reversed h andaddthemup. Goingbacktothepreviousexample: Figure4.7: Thisistheendresultthatwearelookingtond. Figure4.8: Herewereversetheimpulseresponse, h ,andbeginitstraverseattime 0

PAGE 85

77 Figure4.9: Wecontinuethetraverse.Seethatattime 1 ,wearemultiplyingtwoelementsofthe inputsignalbytwoelementsoftheimpulseresponse. Figure4.10

PAGE 86

78 CHAPTER4.TIMEDOMAINANALYSISOFDISCRETETIMESYSTEMS Figure4.11: Ifwefollowthisthroughtoonemorestep, n =4 ,thenwecanseethatweproducethe sameoutputaswesawintheinitialexample. Whatwearedoingintheabovedemonstrationisreversingtheimpulseresponseintimeand"walking itacross"theinputsignal.Clearly,thisyieldsthesameresultasscaling,shiftingandsummingimpulse responses. Thisapproachoftime-reversing,andslidingacrossisacommonapproachtopresentingconvolution, sinceitdemonstrateshowconvolutionbuildsupanoutputthroughtime. 4.3CircularConvolutionandtheDFT 9 4.3.1Introduction YoushouldbefamiliarwithDiscrete-TimeConvolutionSection4.2,whichtellsusthatgiventwodiscretetimesignals x [ n ] ,thesystem'sinput,and h [ n ] ,thesystem'sresponse,wedenetheoutputofthesystem as y [ n ]= x [ n ] h [ n ] = P 1 k = x [ k ] h [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ] .13 WhenwearegiventwoDFTsnite-lengthsequencesusuallyoflength N ,wecannotjustmultiplythem togetheraswedointheaboveconvolutionformula,oftenreferredtoas linearconvolution .Becausethe DFTsareperiodic,theyhavenonzerovaluesfor n N andthusthemultiplicationofthesetwoDFTswillbe nonzerofor n N .Weneedtodeneanewtypeofconvolutionoperationthatwillresultinourconvolved signalbeingzerooutsideoftherange n = f 0 ; 1 ;:::;N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 g .Thisidealedtothedevelopmentof circular convolution ,alsocalledcyclicorperiodicconvolution. 9 Thiscontentisavailableonlineat.

PAGE 87

79 4.3.2CircularConvolutionFormula WhathappenswhenwemultiplytwoDFT'stogether,where Y [ k ] istheDFTof y [ n ] ? Y [ k ]= F [ k ] H [ k ] .14 when 0 k N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 UsingtheDFTsynthesisformulafor y [ n ] y [ n ]= 1 N N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 F [ k ] H [ k ] e j 2 N kn .15 Andthenapplyingtheanalysisformula F [ k ]= P N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m =0 f [ m ] e )]TJ/F10 6.9738 Tf 6.226 0 Td [(j 2 N kn y [ n ]= 1 N P N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k =0 P N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m =0 f [ m ] e )]TJ/F10 6.9738 Tf 6.227 0 Td [(j 2 N kn H [ k ] e j 2 N kn = P N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 m =0 f [ m ] 1 N P N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k =0 H [ k ] e j 2 N k n )]TJ/F10 6.9738 Tf 6.227 0 Td [(m .16 wherewecanreducethesecondsummationfoundintheaboveequationinto h [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m N ]= 1 N P N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k =0 H [ k ] e j 2 N k n )]TJ/F10 6.9738 Tf 6.226 0 Td [(m y [ n ]= N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X m =0 f [ m ] h [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m N ] whichequalscircularconvolution!Whenwehave 0 n N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 intheabove,thenweget: y [ n ] f [ n ] ~ h [ n ] .17 note: Thenotation ~ representscyclicconvolution"modN". 4.3.2.1StepsforCyclicConvolution Stepsforcyclicconvolutionarethesameastheusualconvo,exceptallindexcalculationsaredone"modN" ="onthewheel" StepsforCyclicConvolution Step1:"Plot" f [ m ] and h [ )]TJ/F11 9.9626 Tf 7.748 0 Td [(m N ] a b Figure4.12: Step1

PAGE 88

80 CHAPTER4.TIMEDOMAINANALYSISOFDISCRETETIMESYSTEMS Step2:"Spin" h [ )]TJ/F11 9.9626 Tf 7.749 0 Td [(m N ] n notchesACWcounter-clockwisetoget h [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m N ] i.e.Simply rotatethesequence, h [ n ] ,clockwiseby n steps. Figure4.13: Step2 Step3:Pointwisemultiplythe f [ m ] wheelandthe h [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(m N ] wheel. sum= y [ n ] Step4:Repeatforall 0 n N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 Example4.3:Convolven=4 a b Figure4.14: Twodiscrete-timesignalstobeconvolved. h [ )]TJ/F11 9.9626 Tf 7.749 0 Td [(m N ] Figure4.15 Multiply f [ m ] and sum toyield: y [0]=3

PAGE 89

81 h [ )]TJ/F11 9.9626 Tf 9.962 0 Td [(m N ] Figure4.16 Multiply f [ m ] and sum toyield: y [1]=5 h [ )]TJ/F11 9.9626 Tf 9.962 0 Td [(m N ] Figure4.17 Multiply f [ m ] and sum toyield: y [2]=3 h [ )]TJ/F11 9.9626 Tf 9.962 0 Td [(m N ] Figure4.18 Multiply f [ m ] and sum toyield: y [3]=1 Example4.4 Thefollowingdemonstrationallowsyoutoexplorethisalgorithmforcircularconvolution.See here 10 forinstructionsonhowtousethedemo. 10 "HowtousetheLabVIEWdemos"

PAGE 90

82 CHAPTER4.TIMEDOMAINANALYSISOFDISCRETETIMESYSTEMS Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10786/latest/DTCircularConvolution.llb 4.3.2.2AlternativeAlgorithm AlternativeCircularConvolutionAlgorithm Step1:CalculatetheDFTof f [ n ] whichyields F [ k ] andcalculatetheDFTof h [ n ] whichyields H [ k ] Step2:Pointwisemultiply Y [ k ]= F [ k ] H [ k ] Step3:InverseDFT Y [ k ] whichyields y [ n ] Seemslikearoundaboutwayofdoingthings, but itturnsoutthatthereare extremely fastwaysto calculatetheDFTofasequence. Tocircularilyconvolve 2 N -pointsequences: y [ n ]= N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X m =0 f [ m ] h [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m N ] Foreach n : N multiples, N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 additions N pointsimplies N 2 multiplications, N N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 additionsimplies O )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(N 2 complexity. 4.4LinearConstant-CoecientDierenceEquations 11 rememberlineardierentialequations? d dt y t )]TJ/F11 9.9626 Tf 9.963 0 Td [(y t = x t A dierenceequation isthediscrete-timeanalogueofa dierentailequation .Wesimplyuse dierences x [ n ] )]TJ/F11 9.9626 Tf 9.962 0 Td [(x [ n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1] ratherthanderivatives d dt x t Animportant subclass oflinearsystemsconsistsofthosewhoseinput x [ n ] andoutput x [ n ] obeyan N -thorderLCCDE: N X K =0 a K y [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(K ]= M X K =0 b K x [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(K ] .18 Example4.5:Movingaveragesystem y [ n ]= 1 M 1 + M 2 +1 M 2 X K = M 1 x [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(K ] whereset M 1 =0 and M 2 = M a K = 8 < : 1 if K =0 0 otherwise b K = 8 < : 1 M +1 if 0 K M 0 otherwise 11 Thiscontentisavailableonlineat.

PAGE 91

83 4.4.1HowtoImplement? code/hardware M =2 : y [ n ]= 1 3 P 2 K =0 x [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(K ] IIR?FIR? Example4.6:RecursiveSystem y [ n ]= N X K =1 K y [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(K ]+ x [ n ] N X K =0 a K y [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(K ]= x [ n ] Where a K = 8 > > < > > : 1 if K =0 )]TJ/F11 9.9626 Tf 7.749 0 Td [( K if 1 K N 0 otherwise 4.4.1HowtoImplement? N =2 : y [ n ]= P 2 K =1 K y [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(K ]+ x [ n ] IIR?FIR? FIR MOVINGAVERAGE FEEDFORWARD IIR RECURSIVE FEEDBACK The SOLUTION ofadierenceequationissimilartoadierentialequation. Inparticular,notethatasingleinput-outputpair x [ n ] y p [ n ] thatsolvestheDEisnotenoughto characterizethesolution. N X k =0 a k y p [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ]= M X k =0 b k x [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k ] Addinzerotogetthehomogenousequation: P N k =0 a k y h [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ]=0 N X k =0 a k y p [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k ]+ y h [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ]= M X k =0 b k x [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ] wheretheparticularsolution"forced" y p [ n ] andhomogeneoussolution"unforced" y h [ n ] GeneralSolution y n = y p n + y h n .19 4.5SolvingLinearConstant-CoecientDierenceEquations 12 Step1 -Giventheinput x [ n ] ,ndasolutionto N X k =0 a k y p [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ]= M X k =0 b k x [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k ] note: Justanyoldsolutionwilldo! y p [ n ] -particularsolution. 12 Thiscontentisavailableonlineat.

PAGE 92

84 CHAPTER4.TIMEDOMAINANALYSISOFDISCRETETIMESYSTEMS Solvethehomogeneousequation N X k =0 a k y h [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k ]=0 for y h [ n ] -homogeneoussolution. Completesolutiongivenby y [ n ]= y p [ n ]+ y h [ n ] 4.5.1SolvingTheHomogeneousEquation Whatdoesitmean? Figure4.19 Clearly y h [ n ] dependsonthe INITIALCONDITIONS ofthesystem T Linearity Time-Invariance Causality willeachdependontheseconditions. Inthiscourse,wewillemphasizethesimplestcase,when T is"initiallyatrest"with"zeroinitial conditions." wewillgetLTIandcausalsolutions.althoughpossiblyattheexpenseof stability Example4.7 Solve y [ n ] )]TJ/F11 9.9626 Tf 9.963 0 Td [(ay [ n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1]= x [ n ] where j a j < 1 for x [ n ]= [ n ] Step1:ParticularSolution -Assume n 0 and"zeroinitialconditions" y p [0]= [0] 1+ a y p [ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1] 0=1 y p [1]= [1] 0+ a y p [0] 1= a y p [2]= [2] 0+ a y p [1] a = a 2 y p [ n ]= a n .20 where n 0

PAGE 93

85 Figure4.20 Step2:HomogeneousSolution -If x [ n ]=0 ,then y h [ n ] )]TJ/F11 9.9626 Tf 9.963 0 Td [(ay h [ n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1]=0 y h [ n ]= ay h [ n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1] Asolutionisgivenby y h [ n ]= ca n .21 forall n Figure4.21 Step3:Reconcile y [ n ]= y n [ p ]+ y h [ n ] = a n u [ n ]+ ca n Howtopick c ?Needauxilliaryconditions.

PAGE 94

86 CHAPTER4.TIMEDOMAINANALYSISOFDISCRETETIMESYSTEMS Figure4.22 Ifwedesirea causalsystem ,then c =0 and y [ n ]= a n u [ n ] Figure4.23 Ifwedesirean anticausalsystem thenchoose c = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 ,so y [ n ]= )]TJ/F8 9.9626 Tf 9.409 0 Td [( a n u [ )]TJ/F11 9.9626 Tf 7.749 0 Td [(n ] Thisdoesnotassume"systeminitiallyatrest!"

PAGE 95

87 Figure4.24 Notes 1.Solution1p.86wascausalandstable. 2.Solution2p.86wasanticausalandunstable. Ingeneral,linearity,time-invariance,andcausalityofasystemimplementedasaDEwilldependonthe auxilliaryconditions. note: Ifweassumethatthesystemisinitiallyat rest "zeroinitialconditions",thenitwillbe LINEAR,TIME-INVARIANT,andCAUSAL Note: Settinginput x = 0 impulseandsettinginitialconditionsall =0 andsolvingfor y p yields y p = h astheimpulseresponseofthisLSIsystem. Example4.8:FrequencyResponseofa"wire" Figure4.25

PAGE 96

88 CHAPTER4.TIMEDOMAINANALYSISOFDISCRETETIMESYSTEMS ImpulseResponse: Figure4.26 soFrequencyResponse: H = F H 0 = 1 p N N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X n =0 h [ n ] e )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( 2 N kn = 1 p N N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X n =0 0 [ n ] e )]TJ/F8 9.9626 Tf 6.226 -0.747 Td [( 2 N kn 0 [ n ]= 8 < : 1 if n =0 0 otherwise = 1 p N Flat Figure4.27

PAGE 97

89 SolutionstoExercisesinChapter4 SolutiontoExercise4.1p.70 Theindicescanbenegative,andthisconditionisnotallowedinMATLAB.Toxit,wemuststartthe signalslaterinthearray. SolutiontoExercise4.2p.72 Suchtermswouldrequirethesystemtoknowwhatfutureinputoroutputvalueswouldbebeforethe currentvaluewascomputed.Thus,suchtermscancausediculties.

PAGE 98

90 CHAPTER4.TIMEDOMAINANALYSISOFDISCRETETIMESYSTEMS

PAGE 99

Chapter5 LinearAlgebraOverview 5.1LinearAlgebra:TheBasics 1 Thisbrieftutorialonsomekeytermsinlinearalgebraisnotmeanttoreplaceorbeveryhelpfultothoseof youtryingtogainadeepinsightintolinearalgebra.Rather,thisbriefintroductiontosomeofthetermsand ideasoflinearalgebraismeanttoprovidealittlebackgroundtothosetryingtogetabetterunderstanding orlearnabouteigenvectorsandeigenfunctions,whichplayabigroleinderivingafewimportantideason SignalsandSystems.Thegoaloftheseconceptswillbetoprovideabackgroundforsignaldecomposition andtoleaduptothederivationoftheFourierSeriesSection6.2. 5.1.1LinearIndependence Asetofvectors f x 1 ;x 2 ;:::;x k g ;x i 2 C n are linearlyindependent ifnoneofthemcanbewrittenas alinearcombinationoftheothers. Denition5.1:LinearlyIndependent Foragivensetofvectors, f x 1 ;x 2 ;:::;x n g ,theyarelinearlyindependentif c 1 x 1 + c 2 x 2 + + c n x n =0 onlywhen c 1 = c 2 = = c n =0 Example Wearegiventhefollowingtwovectors: x 1 = 0 @ 3 2 1 A x 2 = 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 1 A Theseare notlinearlyindependent asprovenbythefollowingstatement,which,byinspection, canbeseentonotadheretothedenitionoflinearindependencestatedabove. x 2 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x 1 2 x 1 + x 2 =0 Anotherapproachtorevealavectorsindependenceisbygraphingthevectors.Lookingatthese twovectorsgeometricallyasinFigure5.1,onecanagainprovethatthesevectorsare not linearly independent. 1 Thiscontentisavailableonlineat. 91

PAGE 100

92 CHAPTER5.LINEARALGEBRAOVERVIEW Figure5.1: Graphicalrepresentationoftwovectorsthatarenotlinearlyindependent. Example5.1 Wearegiventhefollowingtwovectors: x 1 = 0 @ 3 2 1 A x 2 = 0 @ 1 2 1 A Theseare linearlyindependent since c 1 x 1 = )]TJ/F8 9.9626 Tf 9.409 0 Td [( c 2 x 2 onlyif c 1 = c 2 =0 .Basedonthedenition,thisproofshowsthatthesevectorsareindeedlinearly independent.Again,wecouldalsographthesetwovectorsseeFigure5.2tocheckforlinear independence. Figure5.2: Graphicalrepresentationoftwovectorsthatarelinearlyindependent. Exercise5.1 Solutiononp.109. Are f x 1 ;x 2 ;x 3 g linearlyindependent? x 1 = 0 @ 3 2 1 A

PAGE 101

93 x 2 = 0 @ 1 2 1 A x 3 = 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 1 A Aswehaveseeninthetwoaboveexamples,oftentimestheindependenceofvectorscanbeeasilyseen throughagraph.Howeverthismaynotbeaseasywhenwearegiventhreeormorevectors.Canyoueasily tellwhetherornotthesevectorsareindependentfromFigure5.3.Probablynot,whichiswhythemethod usedintheabovesolutionbecomesimportant. Figure5.3: Plotofthethreevectors.Canbeshownthatalinearcombinationexistsamongthethree, andthereforetheyare not linearindependent. note: Asetof m vectorsin C n cannotbelinearlyindependentif m>n 5.1.2Span Denition5.2:Span Thespan 2 ofasetofvectors f x 1 ;x 2 ;:::;x k g isthesetofvectorsthatcanbewrittenasalinear combinationof f x 1 ;x 2 ;:::;x k g span f x 1 ;:::;x k g = f 1 x 1 + 2 x 2 + + k x k ; i 2 C n g Example Giventhevector x 1 = 0 @ 3 2 1 A thespanof x 1 isa line Example Giventhevectors x 1 = 0 @ 3 2 1 A 2 "Subspaces",Denition2:"Span"

PAGE 102

94 CHAPTER5.LINEARALGEBRAOVERVIEW x 2 = 0 @ 1 2 1 A thespanofthesevectorsis C 2 5.1.3Basis Denition5.3:Basis Abasisfor C n isasetofvectorsthat:spans C n and islinearlyindependent. Clearly,anysetof n linearlyindependentvectorsisa basis for C n Example5.2 Wearegiventhefollowingvector e i = 0 B B B B B B B B B B B B B B @ 0 . 0 1 0 . 0 1 C C C C C C C C C C C C C C A wherethe 1 isalwaysinthe i thplaceandtheremainingvaluesarezero.Thenthe basis for C n is f e i ;i =[1 ; 2 ;:::;n ] g note: f e i ;i =[1 ; 2 ;:::;n ] g iscalledthe standardbasis Example5.3 h 1 = 0 @ 1 1 1 A h 2 = 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 A f h 1 ;h 2 g isabasisfor C 2 .

PAGE 103

95 Figure5.4: Plotofbasisfor C 2 If f b 1 ;:::;b 2 g isabasisfor C n ,thenwecanexpress any x 2 C n asalinearcombinationofthe b i 's: x = 1 b 1 + 2 b 2 + + n b n ; i 2 C Example5.4 Giventhefollowingvector, x = 0 @ 1 2 1 A writing x intermsof f e 1 ;e 2 g givesus x = e 1 +2 e 2 Exercise5.2 Solutiononp.109. Tryandwrite x intermsof f h 1 ;h 2 g denedinthepreviousexample. Inthetwobasisexamplesabove, x isthesamevectorinbothcases,butwecanexpressitinmanydierent wayswegiveonlytwooutofmany,manypossibilities.Youcantakethisevenfurtherbyextendingthis ideaofabasisto functionspaces note: Asmentionedintheintroduction,theseconceptsoflinearalgebrawillhelpprepareyouto understandtheFourierSeriesSection6.2,whichtellsusthatwecanexpressperiodicfunctions, f t ,intermsoftheirbasisfunctions, e j! 0 nt Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10734/latest/LinearAlgebraCalc3.llb Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10734/latest/LinearTransform.llb

PAGE 104

96 CHAPTER5.LINEARALGEBRAOVERVIEW 5.2EigenvectorsandEigenvalues 3 Inthissection,ourlinearsystemswillben nmatricesofcomplexnumbers.Foralittlebackgroundinto someoftheconceptsthatthismoduleisbasedon,refertothebasicsoflinearalgebraSection5.1. 5.2.1EigenvectorsandEigenvalues Let A beann nmatrix,where A isalinearoperatoronvectorsin C n A x = b .1 where x and b aren 1vectorsFigure5.5. a b Figure5.5: Illustrationoflinearsystemandvectors. Denition5.4:eigenvector Aneigenvectorof A isavector v 2 C n suchthat A v = v .2 where iscalledthecorresponding eigenvalue A onlychangesthe length of v ,notitsdirection. 5.2.1.1GraphicalModel ThroughFigure5.6andFigure5.7,letuslookatthedierencebetween.1and.2. 3 Thiscontentisavailableonlineat.

PAGE 105

97 Figure5.6: Represents.1, A x = b If v isaneigenvectorof A ,thenonlyitslengthchanges.SeeFigure5.7andnoticehowourvector's lengthissimplyscaledbyourvariable, ,calledthe eigenvalue : Figure5.7: Represents.2, A v = v note: Whendealingwithamatrix A ,eigenvectorsarethe simplest possiblevectorstooperate on. 5.2.1.2Examples Exercise5.3 Solutiononp.109. Frominspectionandunderstandingofeigenvectors,ndthetwoeigenvectors, v 1 and v 2 ,of A = 0 @ 30 0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 1 A Also,whatarethecorrespondingeigenvalues, 1 and 2 ?Donotworryifyouarehavingproblems seeingthesevaluesfromtheinformationgivensofar,wewilllookatmorerigorouswaystond thesevaluessoon. Exercise5.4 Solutiononp.109. Showthatthesetwovectors, v 1 = 0 @ 1 1 1 A

PAGE 106

98 CHAPTER5.LINEARALGEBRAOVERVIEW v 2 = 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 A areeigenvectorsof A ,where A = 0 @ 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(13 1 A .Also,ndthecorrespondingeigenvalues. 5.2.2CalculatingEigenvaluesandEigenvectors Intheaboveexamples,wereliedonyourunderstandingofthedenitionandonsomebasicobservationsto ndandprovethevaluesoftheeigenvectorsandeigenvalues.However,asyoucanprobablytell,nding thesevalueswillnotalwaysbethateasy.Below,wewalkthrougharigorousandmathematicalapproachat calculatingtheeigenvaluesandeigenvectorsofamatrix. 5.2.2.1FindingEigenvalues Find 2 C suchthat v 6 = 0 ,where 0 isthe"zerovector."Wewillstartwith.2,andthenworkourway downuntilwendawaytoexplicitlycalculate A v = v A v )]TJ/F11 9.9626 Tf 9.962 0 Td [( v =0 A )]TJ/F11 9.9626 Tf 9.962 0 Td [(I v =0 Inthepreviousstep,weusedthefactthat v = I v where I istheidentitymatrix. I = 0 B B B B B @ 10 ::: 0 01 ::: 0 00 . . . 0 :::::: 1 1 C C C C C A So, A )]TJ/F11 9.9626 Tf 9.963 0 Td [(I isjustanewmatrix. Example5.5 Giventhefollowingmatrix, A ,thenwecanndournewmatrix, A )]TJ/F11 9.9626 Tf 9.963 0 Td [(I A = 0 @ a 11 a 12 a 21 a 22 1 A A )]TJ/F11 9.9626 Tf 9.962 0 Td [(I = 0 @ a 11 )]TJ/F11 9.9626 Tf 9.962 0 Td [(a 12 a 21 a 22 )]TJ/F11 9.9626 Tf 9.963 0 Td [( 1 A If A )]TJ/F11 9.9626 Tf 9.962 0 Td [(I v =0 forsome v 6 =0 ,then A )]TJ/F11 9.9626 Tf 9.963 0 Td [(I is notinvertible .Thismeans: det A )]TJ/F11 9.9626 Tf 9.963 0 Td [(I =0

PAGE 107

99 Thisdeterminantshowndirectlyaboveturnsouttobeapolynomialexpressionoforder n .Lookatthe examplesbelowtoseewhatthismeans. Example5.6 Startingwithmatrix A shownbelow,wewillndthepolynomialexpression,whereoureigenvalueswillbethedependentvariable. A = 0 @ 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(13 1 A A )]TJ/F11 9.9626 Tf 9.963 0 Td [(I = 0 @ 3 )]TJ/F11 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(13 )]TJ/F11 9.9626 Tf 9.962 0 Td [( 1 A det A )]TJ/F11 9.9626 Tf 9.962 0 Td [(I = )]TJ/F11 9.9626 Tf 9.962 0 Td [( 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 2 = 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 +8 = f 2 ; 4 g Example5.7 Startingwithmatrix A shownbelow,wewillndthepolynomialexpression,whereoureigenvalueswillbethedependentvariable. A = 0 @ a 11 a 12 a 21 a 22 1 A A )]TJ/F11 9.9626 Tf 9.962 0 Td [(I = 0 @ a 11 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 12 a 21 a 22 )]TJ/F11 9.9626 Tf 9.963 0 Td [( 1 A det A )]TJ/F11 9.9626 Tf 9.962 0 Td [(I = 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [( a 11 + a 22 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 21 a 12 + a 11 a 22 Ifyouhavenotalreadynoticedit,calculatingtheeigenvaluesisequivalenttocalculatingtherootsof det A )]TJ/F11 9.9626 Tf 9.962 0 Td [(I = c n n + c n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + + c 1 + c 0 =0 note: Therefore,bysimplyusingcalculustosolvefortherootsofourpolynomialwecaneasily ndtheeigenvaluesofourmatrix. 5.2.2.2FindingEigenvectors Givenaneigenvalue, i ,theassociatedeigenvectorsaregivenby A v = i v A 0 B B B @ v 1 . v n 1 C C C A = 0 B B B @ 1 v 1 . n v n 1 C C C A setof n equationswith n unknowns.Simply solvethe n equations tondtheeigenvectors.

PAGE 108

100 CHAPTER5.LINEARALGEBRAOVERVIEW 5.2.3MainPoint Saytheeigenvectorsof A f v 1 ;v 2 ;:::;v n g ,spanSection5.1.2:Span C n ,meaning f v 1 ;v 2 ;:::;v n g are linearlyindependentSection5.1.1:LinearIndependenceandwecanwriteany x 2 C n as x = 1 v 1 + 2 v 2 + + n v n .3 where f 1 ; 2 ;:::; n g2 C .Allthatwearedoingisrewriting x intermsofeigenvectorsof A .Then, A x = A 1 v 1 + 2 v 2 + + n v n A x = 1 Av 1 + 2 Av 2 + + n Av n A x = 1 1 v 1 + 2 2 v 2 + + n n v n = b Thereforewecanwrite, x = X i i v i andthisleadsustothefollowingdepictedsystem: Figure5.8: Depictionofsystemwherewebreakourvector, x ,intoasumofitseigenvectors. whereinFigure5.8wehave, b = X i i i v i note: Bybreakingupavector, x ,intoacombinationofeigenvectors,thecalculationof A x is brokeninto"easytoswallow"pieces. 5.2.4PracticeProblem Exercise5.5 Solutiononp.109. Forthefollowingmatrix, A andvector, x ,solvefortheirproduct.Trysolvingitusingtwodierent methods:directlyandusingeigenvectors. A = 0 @ 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(13 1 A x = 0 @ 5 3 1 A Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10736/latest/LinearAlgebraCalc3.llb

PAGE 109

101 5.3MatrixDiagonalization 4 FromourunderstandingofeigenvaluesandeigenvectorsSection5.2wehavediscoveredseveralthingsabout ouroperatormatrix, A .Weknowthatiftheeigenvectorsof A span C n and weknowhowtoexpressany vector x intermsof f v 1 ;v 2 ;:::;v n g ,thenwehavetheoperator A allguredout.Ifwehave A actingon x thenthisisequalto A actingonthecombinationsofeigenvectors.Whichweknowprovestobefairlyeasy! Wearestillleftwithtwoquestionsthatneedtobeaddressed: 1.Whendotheeigenvectors f v 1 ;v 2 ;:::;v n g of A span C n assuming f v 1 ;v 2 ;:::;v n g arelinearlyindependent? 2.Howdoweexpressagivenvector x intermsof f v 1 ;v 2 ;:::;v n g ? 5.3.1AnswertoQuestion#1 note: Whendotheeigenvectors f v 1 ;v 2 ;:::;v n g of A span C n ? If A has n distinct eigenvalues i 6 = j ;i 6 = j where i and j areintegers,then A has n linearlyindependenteigenvectors f v 1 ;v 2 ;:::;v n g whichthenspan C n aside: Theproofofthisstatementisnotveryhard,butisnotreallyinterestingenoughtoinclude here.Ifyouwishtoresearchthisideafurther,readStrang,G.,"LinearAlgebraanditsApplication" fortheproof. Furthermore, n distincteigenvaluesmeans det A )]TJ/F11 9.9626 Tf 9.962 0 Td [(I = c n n + c n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 + + c 1 + c 0 =0 has n distinctroots. 5.3.2AnswertoQuestion#2 note: Howdoweexpressagivenvector x intermsof f v 1 ;v 2 ;:::;v n g ? Wewanttond f 1 ; 2 ;:::; n g2 C suchthat x = 1 v 1 + 2 v 2 + + n v n .4 Inordertondthissetofvariables,wewillbeginbycollectingthevectors f v 1 ;v 2 ;:::;v n g ascolumnsina n nmatrix V V = 0 B B B @ . . . . v 1 v 2 :::v n . . . . 1 C C C A Now.4becomes x = 0 B B B @ . . . . v 1 v 2 :::v n . . . . 1 C C C A 0 B B B @ 1 . n 1 C C C A 4 Thiscontentisavailableonlineat.

PAGE 110

102 CHAPTER5.LINEARALGEBRAOVERVIEW or x = V whichgivesusaneasyformtosolveforourvariablesinquestion, : = V )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x Notethat V isinvertiblesinceithas n linearlyindependentcolumns. 5.3.2.1Aside Letusrecallourknowledgeoffunctionsandtherebasisandexaminetheroleof V x = V 0 B B B @ x 1 . x n 1 C C C A = V 0 B B B @ 1 . n 1 C C C A where isjust x expressedinadierentbasisSection5.1.3:Basis: x = x 1 0 B B B B B @ 1 0 . 0 1 C C C C C A + x 2 0 B B B B B @ 0 1 . 0 1 C C C C C A + + x n 0 B B B B B @ 0 0 . 1 1 C C C C C A x = 1 0 B B B @ . v 1 . 1 C C C A + 2 0 B B B @ . v 2 . 1 C C C A + + n 0 B B B @ . v n . 1 C C C A V transforms x fromthestandardbasistothebasis f v 1 ;v 2 ;:::;v n g 5.3.3MatrixDiagonalizationandOutput Wecanalsousethevectors f v 1 ;v 2 ;:::;v n g torepresenttheoutput, b ,ofasystem: b = A x = A 1 v 1 + 2 v 2 + + n v n A x = 1 1 v 1 + 2 2 v 2 + + n n v n = b A x = 0 B B B @ . . . . v 1 v 2 :::v n . . . . 1 C C C A 0 B B B @ 1 1 . 1 n 1 C C C A A x = V

PAGE 111

103 A x = V V )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x where isthematrixwiththeeigenvaluesdownthediagonal: = 0 B B B B B @ 1 0 ::: 0 0 2 ::: 0 . . . . . . 00 ::: n 1 C C C C C A Finally,wecancanceloutthe x andareleftwithanalequationfor A : A = V V )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 5.3.3.1Interpretation Forourinterpretation,recallourkeyformulas: = V )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x b = X i i i v i Wecaninterpretoperatingon x with A as: 0 B B B @ x 1 . x n 1 C C C A 0 B B B @ 1 . n 1 C C C A 0 B B B @ 1 1 . 1 n 1 C C C A 0 B B B @ b 1 . b n 1 C C C A wherethethreestepsarrowsintheaboveillustrationrepresentthefollowingthreeoperations: 1.Transform x using V )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ,whichyields 2.Multiplicationby 3.Inversetransformusing V ,whichgivesus b ThisistheparadigmwewilluseforLTIsystems! Figure5.9: SimpleillustrationofLTIsystem! Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10738/latest/LinearAlgebraCalc3.llb

PAGE 112

104 CHAPTER5.LINEARALGEBRAOVERVIEW 5.4Eigen-stuinaNutshell 5 5.4.1AMatrixanditsEigenvector ThereasonwearestressingeigenvectorsSection5.2andtheirimportanceisbecausetheactionofamatrix A ononeofitseigenvectors v is 1.extremelyeasyandfasttocalculate A v = v .5 just multiply v by 2.easytointerpret: A just scales v ,keepingitsdirectionconstantandonlyalteringthevector'slength. Ifonlyeveryvectorwereaneigenvectorof A .... 5.4.2UsingEigenvectors'Span Ofcourse,noteveryvectorcanbe...BUT...Forcertainmatricesincludingoneswithdistincteigenvalues, 's,theireigenvectorsspanSection5.1.2:Span C n ,meaningthatfor any x 2 C n ,wecannd f 1 ; 2 ; n g2 C suchthat: x = 1 v 1 + 2 v 2 + + n v n .6 Given.6,wecanrewrite A x = b .ThisequationismodeledinourLTIsystempicturedbelow: Figure5.10: LTISystem. x = X i i v i b = X i i i v i TheLTIsystemaboverepresentsour.5.Belowisanillustrationofthestepstakentogofrom x to b x )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [( = V )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [( V )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x V V )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x = b wherethethreestepsarrowsintheaboveillustrationrepresentthefollowingthreeoperations: 1.Transform x using V )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 -yields 2.Actionof A innewbasis-amultiplicationby 3.Translatebacktooldbasis-inversetransformusingamultiplicationby V ,whichgivesus b 5 Thiscontentisavailableonlineat.

PAGE 113

105 5.5EigenfunctionsofLTISystems 6 5.5.1Introduction Hopefullyyouarefamiliarwiththenotionoftheeigenvectorsofa"matrixsystem,"ifnottheydoaquick reviewofeigen-stuSection5.4.WecandevelopthesameideasforLTIsystemsactingonsignals.Alinear timeinvariantLTIsystem 7 H operatingonacontinuousinput f t toproducecontinuoustimeoutput y t H [ f t ]= y t .7 Figure5.11: H [ f t ]= y t f and t arecontinuoustimeCTsignalsand H isanLTIoperator. ismathematicallyanalogoustoan N x N matrix A operatingonavector x 2 C N toproduceanother vector b 2 C N seeMatricesandLTISystemsforanoverview. Ax = b .8 Figure5.12: Ax = b where x and b arein C N and A isan N x N matrix. JustasaneigenvectorSection5.2of A isa v 2 C N suchthat Av = v 2 C 6 Thiscontentisavailableonlineat. 7 "IntroductiontoSystems"

PAGE 114

106 CHAPTER5.LINEARALGEBRAOVERVIEW Figure5.13: Av = v where v 2 C N isaneigenvectorof A wecandenean eigenfunction or eigensignal ofanLTIsystem H tobeasignal f t suchthat H [ f t ]= f t ; 2 C .9 Figure5.14: H [ f t ]= f t where f isaneigenfunctionof H Eigenfunctionsarethe simplest possiblesignalsfor H tooperateon:tocalculatetheoutput,wesimply multiplytheinputbyacomplexnumber 5.5.2EigenfunctionsofanyLTISystem TheclassofLTIsystemshasasetofeigenfunctionsincommon:thecomplexexponentialsSection1.6 e st s 2 C areeigenfunctionsfor all LTIsystems. H e st = s e st .10

PAGE 115

107 Figure5.15: H e st = s e st where H isanLTIsystem. note: While f e st ;s 2 C g arealwayseigenfunctionsofanLTIsystem,theyarenotnecessarily the only eigenfunctions. Wecanprove.10byexpressingtheoutputasaconvolutionSection3.2oftheinput e st andthe impulseresponseSection1.5 h t of H : H [ e st ]= R 1 h e s t )]TJ/F10 6.9738 Tf 6.227 0 Td [( d = R 1 h e st e )]TJ/F7 6.9738 Tf 6.226 0 Td [( s d = e st R 1 h e )]TJ/F7 6.9738 Tf 6.227 0 Td [( s d .11 Sincetheexpressionontherighthandsidedoesnotdependon t ,itisaconstant, s .Therefore H e st = s e st .12 Theeigenvalue s isacomplexnumberthatdependsontheexponent s and,ofcourse,thesystem H .To makethesedependenciesexplicit,wewillusethenotation H s s Figure5.16: e st istheeigenfunctionand H s aretheeigenvalues. SincetheactionofanLTIoperatoronitseigenfunctions e st iseasytocalculateandinterpret,itis convenienttorepresentanarbitrarysignal f t asalinearcombinationofcomplexexponentials.TheFourier seriesSection6.2givesusthisrepresentationforperiodiccontinuoustimesignals,whiletheslightlymore complicatedFouriertransform 8 letsusexpandarbitrarycontinuoustimesignals. 8 "DerivationoftheFourierTransform"

PAGE 116

108 CHAPTER5.LINEARALGEBRAOVERVIEW 5.6FourierTransformProperties 9 ShortTableofFourierTransformPairs st Sf e )]TJ/F7 6.9738 Tf 6.227 0 Td [( at u t 1 j 2 f + a e )]TJ/F10 6.9738 Tf 6.227 0 Td [(a j t j 2 a 4 2 f 2 + a 2 p t = 8 < : 1 if j t j < 2 0 if j t j > 2 sin f f sin Wt t S f = 8 < : 1 if j f j W Table5.1 FourierTransformProperties Time-Domain FrequencyDomain Linearity a 1 s 1 t + a 2 s 2 t a 1 S 1 f + a 2 S 2 f ConjugateSymmetry s t 2 R S f = S )]TJ/F11 9.9626 Tf 7.749 0 Td [(f EvenSymmetry s t = s )]TJ/F11 9.9626 Tf 7.749 0 Td [(t S f = S )]TJ/F11 9.9626 Tf 7.749 0 Td [(f OddSymmetry s t = )]TJ/F8 9.9626 Tf 9.409 0 Td [( s )]TJ/F11 9.9626 Tf 7.749 0 Td [(t S f = )]TJ/F8 9.9626 Tf 9.409 0 Td [( S )]TJ/F11 9.9626 Tf 7.748 0 Td [(f ScaleChange s at 1 j a j S f a TimeDelay s t )]TJ/F11 9.9626 Tf 9.963 0 Td [( e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j 2 f S f ComplexModulation e j 2 f 0 t s t S f )]TJ/F11 9.9626 Tf 9.963 0 Td [(f 0 AmplitudeModulationbyCosine s t cos f 0 t S f )]TJ/F10 6.9738 Tf 6.227 0 Td [(f 0 + S f + f 0 2 AmplitudeModulationbySine s t sin f 0 t S f )]TJ/F10 6.9738 Tf 6.227 0 Td [(f 0 )]TJ/F10 6.9738 Tf 6.227 0 Td [(S f + f 0 2 j Dierentiation d dt s t j 2 fS f Integration R t s d 1 j 2 f S f if S =0 Multiplicationby t ts t 1 )]TJ/F7 6.9738 Tf 6.227 0 Td [( j 2 d df S f Area R 1 s t dt S ValueatOrigin s R 1 S f df Parseval'sTheorem R 1 j s t j 2 dt R 1 j S f j 2 df Table5.2 9 Thiscontentisavailableonlineat.

PAGE 117

109 SolutionstoExercisesinChapter5 SolutiontoExercise5.1p.92 Byplayingaroundwiththevectorsanddoingalittletrialanderror,wewilldiscoverthefollowingrelationship: x 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 2 +2 x 3 =0 Thuswehavefoundalinearcombinationofthesethreevectorsthatequalszerowithoutsettingthecoecients equaltozero.Therefore,thesevectorsare notlinearlyindependent SolutiontoExercise5.2p.95 x = 3 2 h 1 + )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 2 h 2 SolutiontoExercise5.3p.97 Theeigenvectorsyoufoundshouldbe: v 1 = 0 @ 1 0 1 A v 2 = 0 @ 0 1 1 A Andthecorrespondingeigenvaluesare 1 =3 2 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise5.4p.97 Inordertoprovethatthesetwovectorsareeigenvectors,wewillshowthatthesestatementsmeetthe requirementsstatedinthedenitionDenition:"eigenvector",p.96. Av 1 = 0 @ 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(13 1 A 0 @ 1 1 1 A = 0 @ 2 2 1 A Av 2 = 0 @ 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(13 1 A 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 A = 0 @ 4 )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 1 A Theseresultsshowusthat A onlyscalesthetwovectors i.e. changestheirlengthandthusitprovesthat .2holdstrueforthefollowingtwoeigenvaluesthatyouwereaskedtond: 1 =2 2 =4 Ifyouneedmoreconvincing,thenonecouldalsoeasilygraphthevectorsandtheircorrespondingproduct with A toseethattheresultsaremerelyscaledversionsofouroriginalvectors, v 1 and v 2 SolutiontoExercise5.5p.100 DirectMethod usebasicmatrixmultiplication A x = 0 @ 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(13 1 A 0 @ 5 3 1 A = 0 @ 12 4 1 A

PAGE 118

110 CHAPTER5.LINEARALGEBRAOVERVIEW Eigenvectors usetheeigenvectorsandeigenvalueswefoundearlierforthissamematrix v 1 = 0 @ 1 1 1 A v 2 = 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 A 1 =2 2 =4 Asshownin.3,wewanttorepresent x asasumofitsscaledeigenvectors.Forthiscase,wehave: x =4 v 1 + v 2 x = 0 @ 5 3 1 A =4 0 @ 1 1 1 A + 0 @ 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 1 A A x = A v 1 + v 2 = i v 1 + v 2 Therefore,wehave A x =4 2 0 @ 1 1 1 A +4 0 @ 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 1 A = 0 @ 12 4 1 A Noticethatthismethodusingeigenvectorsrequired no matrixmultiplication.Thismayhaveseemedmore complicatedhere,butjustimagine A beingreallybig,orevenjustafewdimensionslarger!

PAGE 119

Chapter6 ContinuousTimeFourierSeries 6.1PeriodicSignals 1 Recallthataperiodicfunctionisafunctionthatrepeatsitselfexactlyaftersomegivenperiod,orcycle.We representthedenitionofa periodicfunction mathematicallyas: f t = f t + mT m 2 Z .1 where T> 0 representsthe period .Becauseofthis,youmayalsoseeasignalreferredtoasaT-periodic signal.Anyfunctionthatsatisesthisequationisperiodic. Wecanthinkofperiodicfunctionswithperiod T twodierentways: #1asfunctionson all of R Figure6.1: Functionoverallof R where f t 0 = f t 0 + T #2or,wecancutoutalloftheredundancy,andthinkofthemasfunctionsonaninterval [0 ;T ] or, moregenerally, [ a;a + T ] .IfweknowthesignalisT-periodicthenalltheinformationofthesignalis capturedbytheaboveinterval. 1 Thiscontentisavailableonlineat. 111

PAGE 120

112 CHAPTER6.CONTINUOUSTIMEFOURIERSERIES Figure6.2: Removetheredundancyoftheperiodfunctionsothat f t isundenedoutside [0 ;T ] An aperiodic CTfunction f t doesnotrepeatfor any T 2 R ; i.e. thereexistsno T s.t.thisequation .1holds. Question:DTdenitions? 6.1.1Continuous-Time 6.1.2Discrete-Time Note:Circularvs.Line Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10744/latest/PhaseShift.llb 6.2FourierSeries:EigenfunctionApproach 2 6.2.1Introduction SincecomplexexponentialsSection1.6areeigenfunctionsoflineartime-invariantLTIsystemsSection5.5,calculatingtheoutputofanLTIsystem H given e st asaninputamountstosimplemultiplcation, where H s 2 C isaconstantthatdependsons.InthegureFigure6.3belowwehaveasimple exponentialinputthatyieldsthefollowingoutput: y t = H s e st .2 Figure6.3: SimpleLTIsystem. 2 Thiscontentisavailableonlineat.

PAGE 121

113 Usingthisandthefactthat H islinear,calculating y t forcombinationsofcomplexexponentialsisalso straightforward.Thislinearitypropertyisdepictedinthetwoequationsbelow-showingtheinputtothe linearsystem H ontheleftsideandtheoutput, y t ,ontheright: 1. c 1 e s 1 t + c 2 e s 2 t c 1 H s 1 e s 1 t + c 2 H s 2 e s 2 t 2. X n )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c n e s n t X n )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(c n H s n e s n t Theactionof H onaninputsuchasthoseinthetwoequationsaboveiseasytoexplain: H independentlyscales eachexponentialcomponent e s n t byadierentcomplexnumber H s n 2 C .Assuch,ifwe canwriteafunction f t asacombinationofcomplexexponentialsitallowsusto: easilycalculatetheoutputof H given f t asaninputprovidedweknowtheeigenvalues H s interprethow H manipulates f t 6.2.2FourierSeries JosephFourier 3 demonstratedthatanarbitraryT-periodicfunctionSection6.1 f t canbewrittenasa linearcombinationofharmoniccomplexsinusoids f t = 1 X n = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c n e j! 0 nt .3 where 0 = 2 T isthefundamentalfrequency.Foralmostall f t ofpracticalinterest,thereexists c n tomake .3true.If f t isniteenergy f t 2 L 2 [0 ;T ] ,thentheequalityin.3holdsinthesenseofenergy convergence;if f t iscontinuous,then.3holdspointwise.Also,if f t meetssomemildconditionsthe Dirichletconditions,then.3holdspointwiseeverywhereexceptatpointsofdiscontinuity. The c n -calledthe Fouriercoecients -tellus"howmuch"ofthesinusoid e j! 0 nt isin f t ..3 essentiallybreaksdown f t intopieces,eachofwhichiseasilyprocessedbyanLTIsystemsinceitis aneigenfunctionof every LTIsystem.Mathematically,.3tellsusthatthesetofharmoniccomplex exponentials e j! 0 nt ;n 2 Z formabasisforthespaceofT-periodiccontinuoustimefunctions.Below areafewexamplesthatareintendedtohelpyouthinkaboutagivensignalorfunction, f t ,intermsof itsexponentialbasisfunctions. 6.2.2.1Examples Foreachofthegivenfunctionsbelow,breakitdownintoits"simpler"partsandnditsfouriercoecients. Clicktoseethesolution. Exercise6.1 Solutiononp.139. f t =cos 0 t Exercise6.2 Solutiononp.139. f t =sin 0 t Exercise6.3 Solutiononp.139. f t =3+4cos 0 t +2cos 0 t 3 http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Fourier.html

PAGE 122

114 CHAPTER6.CONTINUOUSTIMEFOURIERSERIES 6.2.3FourierCoecients Ingeneral f t ,theFouriercoecientscanbecalculatedfrom.3bysolvingfor c n ,whichrequiresalittle algebraicmanipulationforthecompletederivationseetheFouriercoecientsderivationSection6.3.The endresultswillyieldthefollowinggeneralequationforthefouriercoecients: c n = 1 T Z T 0 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt .4 Thesequenceofcomplexnumbers f c n ;n 2 Z g isjustanalternaterepresentationofthefunction f t KnowingtheFouriercoecients c n isthesameasknowing f t explicitlyandviceversa.Givenaperiodic function,wecan transform itintoitFourierseriesrepresentationusing.4.Likewise,wecan inverse transform agivensequenceofcomplexnumbers, c n ,using.3toreconstructthefunction f t AlongwithbeinganaturalrepresentationforsignalsbeingmanipulatedbyLTIsystems,theFourier seriesprovidesadescriptionofperiodicsignalsthatisconvenientinmanyways.BylookingattheFourier seriesofasignal f t ,wecaninfermathematicalpropertiesof f t suchassmoothness,existenceofcertain symmetries,aswellasthephysicallymeaningfulfrequencycontent. 6.2.3.1Example:UsingFourierCoecientEquation Herewewilllookatarathersimpleexamplethatalmostrequirestheuseof.4tosolveforthefourier coecients.Onceyouunderstandtheformula,thesolutionbecomesastraightforwardcalculusproblem. Findthefouriercoecientsforthefollowingequation: Exercise6.4 Solutiononp.139. f t = 8 < : 1 if j t j T 0 otherwise 6.2.4Summary:FourierSeriesEquations Ourrstequation.3isthe synthesis equation,whichbuildsourfunction, f t ,bycombiningsinusoids. Synthesis f t = 1 X n = )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(c n e j! 0 nt .5 Andoursecondequation.4,termedthe analysis equation,revealshowmuchofeachsinusoidisin f t Analysis c n = 1 T Z T 0 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt .6 wherewehavestatedthat 0 = 2 T note: Understandthatourintervalofintegrationdoesnothavetobe [0 ;T ] inourAnalysis Equation.Wecoulduseanyinterval [ a;a + T ] oflength T Example6.1 Thisdemonstrationletsyousynthesizeasignalbycombiningsinusoids,similartothesynthesis equationfortheFourierseries.Seehere 4 forinstructionsonhowtousethedemo. 4 "HowtousetheLabVIEWdemos"

PAGE 123

115 Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10496/latest/FourierCompManip.llb 6.3DerivationofFourierCoecientsEquation 5 6.3.1Introduction YoushouldalreadybefamiliarwiththeexistenceofthegeneralFourierSeriesequationSection6.2.2: FourierSeries,whichiswrittenas: f t = 1 X n = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c n e j! 0 nt .7 WhatweareinterestedinhereishowtodeterminetheFouriercoecients, c n ,givenafunction f t .Below wewillwalkthroughthestepsofderivingthegeneralequationfortheFouriercoecientsofagivenfunction. 6.3.2Derivation Tosolve.7for c n ,wehavetodoalittlealgebraicmanipulation.Firstofallwewillmultiplybothsides of.7by e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 kt ,where k 2 Z f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 kt = 1 X n = c n e j! 0 nt e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 kt .8 Nowintegratebothsidesoveragivenperiod, T : Z T 0 f t e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 kt dt = Z T 0 1 X n = c n e j! 0 nt e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 kt dt .9 Ontheright-handsidewecanswitchthesummationandintegralalongwithpullingouttheconstantout oftheintegral. Z T 0 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 kt dt = 1 X n = c n Z T 0 e j! 0 n )]TJ/F10 6.9738 Tf 6.226 0 Td [(k t dt .10 Nowthatwehavemadethisseeminglymorecomplicated,letusfocusonjusttheintegral, R T 0 e j! 0 n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k t dt ontheright-handsideoftheaboveequation.Forthisintegralwewillneedtoconsidertwocases: n = k and n 6 = k .For n = k wewillhave: Z T 0 e j! 0 n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k t dt = T;n = k .11 For n 6 = k ,wewillhave: Z T 0 e j! 0 n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k t dt = Z T 0 cos 0 n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t dt + j Z T 0 sin 0 n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t dt;n 6 = k .12 5 Thiscontentisavailableonlineat.

PAGE 124

116 CHAPTER6.CONTINUOUSTIMEFOURIERSERIES But cos 0 n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t hasanintegernumberofperiods, n )]TJ/F11 9.9626 Tf 10.27 0 Td [(k ,between 0 and T .Imagineagraphofthe cosine;becauseithasanintegernumberofperiods,thereareequalareasaboveandbelowthex-axisofthe graph.Thisstatementholdstruefor sin 0 n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k t aswell.Whatthismeansis Z T 0 cos 0 n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t dt =0 .13 aswellastheintegralinvolvingthesinefunction.Therefore,weconcludethefollowingaboutourintegral ofinterest: Z T 0 e j! 0 n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k t dt = 8 < : T if n = k 0 otherwise .14 Nowletusreturnourattentiontoourcomplicatedequation,.10,toseeifwecannishndingan equationforourFouriercoecients.Usingthefactsthatwehavejustprovenabove,wecanseethatthe onlytime.10willhaveanonzeroresultiswhen k and n areequal: Z T 0 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt = Tc n ;n = k .15 Finally,wehaveourgeneralequationfortheFouriercoecients: c n = 1 T Z T 0 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt .16 6.3.2.1FindingFourierCoecientsSteps TondtheFouriercoecientsofperiodic f t : 1.Foragiven k ,multiply f t by e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 kt ,andtaketheareaunderthecurvedividingby T 2.Repeatstepforall k 2 Z 6.4FourierSeriesinaNutshell 6 6.4.1Introduction TheconvolutionintegralSection3.2isthefundamentalexpressionrelatingtheinputandoutputofanLTI system.However,ithasthreeshortcomings: 1.Itcanbetedioustocalculate. 2.Itoersonlylimitedphysicalinterpretationofwhatthesystemisactuallydoing. 3.Itgiveslittleinsightonhowtodesignsystemstoaccomplishcertaintasks. TheFourierSeriesSection6.2,alongwiththeFourierTransformandLaplaceTransofrm,providesaway toaddressthesethreepoints.CentraltoallofthesemethodsistheconceptofaneigenfunctionSection5.5 oreigenvectorSection5.3.Wewilllookathowwecanrewriteanygivensignal, f t ,intermsofcomplex exponentialsSection1.6. Infact,bymakingournotionsofsignalsandlinearsystemsmoremathematicallyabstract,wewillbe abletodrawenlighteningparallelsbetweensignalsandsystemsandlinearalgebraSection5.1. 6 Thiscontentisavailableonlineat.

PAGE 125

117 6.4.2EigenfunctionsandLTISystems TheactionofaLTIsystem H [ ::: ] ononeofitseigenfunctions e st is 1.extremelyeasyandfasttocalculate H [ st ]= H [ s ] e st .17 2.easytointerpret: H [ ::: ] just scales e st ,keepingitsfrequencyconstant. Ifonlyeveryfunctionwereaneigenfunctionof H [ ::: ] ... 6.4.2.1LTISystem ...ofcourse,noteveryfunctioncanbe,butforLTIsystems,theireigenfunctionsspanSection5.1.2:Span thespaceofperiodicfunctionsSection6.1,meaningthatforalmostanyperiodicfunction f t wecan nd f c n g where n 2 Z and c i 2 C suchthat: f t = 1 X n = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c n e j! 0 nt .18 Given.18,wecanrewrite H [ t ]= y t asthefollowingsystem Figure6.4: TransferFunctionsmodeledasLTISystem. wherewehave: f t = X n )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c n e j! 0 nt y t = X n )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c n H j! 0 n e j! 0 nt Thistransformationfrom f t to y t canalsobeillustratedthroughtheprocessbelow.Notethateach arrowindicatesanoperationonoursignalorcoecients. f t !f c n g!f c n H j! 0 n g! y t .19 wherethethreestepsarrowsintheaboveillustrationrepresentthefollowingthreeoperations: 1.TransformwithanalysisFourierCoecientSection6.3equation: c n = 1 T Z T 0 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt 2.Actionof H ontheFourierseriesSection6.2-equalsamultiplicationby H j! 0 n 3.Translatebacktooldbasis-inversetransformusingoursynthesisequationfromtheFourierseries: y t = 1 X n = )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(c n e j! 0 nt

PAGE 126

118 CHAPTER6.CONTINUOUSTIMEFOURIERSERIES 6.4.3PhysicalInterpretationofFourierSeries TheFourierseries f c n g ofasignal f t ,denedin.18,alsohasaveryimportantphysicalinterpretation. Coecient c n tellsus"howmuch"offrequency 0 n isinthesignal. Signalsthatvaryslowlyovertimesmoothsignals -havelarge c n forsmall n a b Figure6.5: Webeginwithoursmoothsignal f t ontheleft,andthenusetheFourierseriestond ourFouriercoecients-showninthegureontheright. Signalsthatvaryquicklywithtimeedgy or noisysignals -willhavelarge c n forlarge n a b Figure6.6: Webeginwithournoisysignal f t ontheleft,andthenusetheFourierseriestond ourFouriercoecients-showninthegureontheright. Example6.2:PeriodicPulse Wehavethefollowingpulsefunction, f t ,overtheinterval )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F10 6.9738 Tf 5.762 -4.147 Td [(T 2 ; T 2 : Figure6.7: PeriodicSignal f t

PAGE 127

119 UsingourformulafortheFouriercoecients, c n = 1 T Z T 0 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt .20 wecaneasilycalculateour c n .Wewillleavethecalculationasanexerciseforyou!Aftersolving thetheequationforour f t ,youwillgetthefollowingresults: c n = 8 < : 2 T 1 T if n =0 2sin 0 nT 1 n if n 6 =0 .21 For T 1 = T 8 ,seethegurebelowforourresults: Figure6.8: OurFouriercoecientswhen T 1 = T 8 Oursignal f t isatexceptfortwoedgesdiscontinuities.Becauseofthis, c n around n =0 arelargeand c n getssmalleras n approachesinnity. note: Whydoes c n =0 for n = f :::; )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 ; 4 ; 8 ; 16 ;::: g ?Whatpartof e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 nt liesoverthepulse forthesevaluesof n ? 6.5FourierSeriesProperties 7 WewillbeginbyrefreshingyourmemoryofourbasicFourierseriesSection6.2equations: f t = 1 X n = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c n e j! 0 nt .22 c n = 1 T Z T 0 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt .23 Let Ffg denotethetransformationfrom f t totheFouriercoecients Ff f t g = c n ;n 2 Z Ffg mapscomplexvaluedfunctionstosequencesofcomplexnumbers 8 7 Thiscontentisavailableonlineat. 8 "ComplexNumbers"

PAGE 128

120 CHAPTER6.CONTINUOUSTIMEFOURIERSERIES 6.5.1Linearity Ffg isa lineartransformation Theorem6.1: If Ff f t g = c n and Ff g t g = d n .Then Ff f t g = c n ; 2 C and Ff f t + g t g = c n + d n Proof: Easy.Justlinearityofintegral. Ff f t + g t g = R T 0 f t + g t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt;n 2 Z = 1 T R T 0 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt + 1 T R T 0 g t e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 nt dt;n 2 Z = c n + d n ;n 2 Z = c n + d n .24 6.5.2Shifting ShiftingintimeequalsaphaseshiftofFouriercoecientsSection6.3 Theorem6.2: Ff f t )]TJ/F11 9.9626 Tf 9.963 0 Td [(t 0 g = e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt 0 c n if c n = j c n j e j c n ,then j e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 nt 0 c n j = j e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt 0 jj c n j = j c n j e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 t 0 n = c n )]TJ/F11 9.9626 Tf 9.963 0 Td [(! 0 t 0 n Proof: Ff f t )]TJ/F11 9.9626 Tf 9.962 0 Td [(t 0 g = 1 T R T 0 f t )]TJ/F11 9.9626 Tf 9.963 0 Td [(t 0 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt;n 2 Z = 1 T R T )]TJ/F10 6.9738 Tf 6.227 0 Td [(t 0 )]TJ/F10 6.9738 Tf 6.227 0 Td [(t 0 f t )]TJ/F11 9.9626 Tf 9.962 0 Td [(t 0 e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 n t )]TJ/F10 6.9738 Tf 6.227 0 Td [(t 0 e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 nt 0 dt;n 2 Z = 1 T R T )]TJ/F10 6.9738 Tf 6.227 0 Td [(t 0 )]TJ/F10 6.9738 Tf 6.227 0 Td [(t 0 f t e )]TJ/F27 6.9738 Tf 6.227 7.681 Td [( j! 0 n t e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 nt 0 dt;n 2 Z = e )]TJ/F27 6.9738 Tf 6.227 7.681 Td [( j! 0 n t c n ;n 2 Z .25 6.5.3Parseval'sRelation Z T 0 j f t j 2 dt = T 1 X n = j c n j 2 .26 Parseval'srelationallowsustocalculatetheenergyofasignalfromitsFourierseries. note: ParsevaltellsusthattheFourierseries maps L 2 [0 ;T ] to l 2 Z .

PAGE 129

121 Figure6.9 Exercise6.5 Solutiononp.140. For f t tohave"niteenergy,"whatdothe c n doas n !1 ? Exercise6.6 Solutiononp.140. If c n = 1 n ; j n j > 0 ,is f 2 L 2 [0 ;T ] ? Exercise6.7 Solutiononp.140. Now,if c n = 1 p n ; j n j > 0 ,is f 2 L 2 [0 ;T ] ? TherateofdecayoftheFourierseriesdeterminesif f t has niteenergy 6.5.4DierentiationinFourierDomain Ff f t g = c n F d dt f t = jn! 0 c n .27 Since f t = 1 X n = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c n e j! 0 nt .28 then d dt f t = P 1 n = )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(c n d dt )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j! 0 nt = P 1 n = )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(c n j! 0 ne j! 0 nt .29 Adierentiator attenuates thelowfrequenciesin f t and accentuates thehighfrequencies.Itremoves generaltrendsandaccentuatesareasofsharpvariation. note: Acommonwaytomathematicallymeasurethesmoothnessofafunction f t istoseehow manyderivativesareniteenergy. ThisisdonebylookingattheFouriercoecientsofthesignal,specicallyhowfastthey decay as n !1 If Ff f t g = c n and j c n j hastheform 1 n k ,then F d m dt m f t = jn! 0 m c n andhastheform n m n k .Sofor the m th derivativetohaveniteenergy,weneed X j n m n k j 2 < 1

PAGE 130

122 CHAPTER6.CONTINUOUSTIMEFOURIERSERIES thus n m n k decays faster than 1 n whichimpliesthat 2 k )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 m> 1 or k> 2 m +1 2 ThusthedecayrateoftheFourierseriesdictatessmoothness. 6.5.5IntegrationintheFourierDomain If Ff f t g = c n .30 then F Z t f d = 1 j! 0 n c n .31 note: If c 0 6 =0 ,thisexpressiondoesn'tmakesense. Integrationaccentuateslowfrequenciesandattenuateshighfrequencies.Integratorsbringoutthe generaltrends insignalsandsuppressshorttermvariationwhichisnoiseinmanycases.Integratorsare much nicerthandierentiators. 6.5.6SignalMultiplication Givenasignal f t withFouriercoecients c n andasignal g t withFouriercoecients d n ,wecandene anewsignal, y t ,where y t = f t g t .WendthattheFourierSeriesrepresentationof y t e n ,is suchthat e n = P 1 k = c k d n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k .Thisistosaythatsignalmultiplicationinthetimedomainisequivalent todiscrete-timeconvolutionSection4.2inthefrequencydomain.Theproofofthisisasfollows e n = 1 T R T 0 f t g t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt = 1 T R T 0 )]TJ 4.566 -0.597 Td [(P 1 k = )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(c k e j! 0 kt g t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt = P 1 k = c k 1 T R T 0 g t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k t dt = P 1 k = c k d n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k .32 6.6SymmetryPropertiesoftheFourierSeries 9 6.6.1SymmetryProperties 6.6.1.1RealSignals RealsignalshaveaconjugatesymmetricFourierseries. Theorem6.3: If f t isrealitimpliesthat f t = f t f t isthecomplexconjugateof f t then c n = c )]TJ/F10 6.9738 Tf 6.227 0 Td [(n whichimpliesthat Re c n = Re c )]TJ/F10 6.9738 Tf 6.226 0 Td [(n i.e. therealpartof c n iseven,and Im c n = )]TJ/F8 9.9626 Tf 9.409 0 Td [(Im c )]TJ/F10 6.9738 Tf 6.226 0 Td [(n i.e. theimaginarypartof c n isodd.SeeFigure6.10.Italsoimpliesthat j c n j = j c )]TJ/F10 6.9738 Tf 6.227 0 Td [(n j ,i.e.that magnitudeiseven,andthat c n = )]TJ/F11 9.9626 Tf 9.963 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(n i.e. thephaseisodd. 9 Thiscontentisavailableonlineat.

PAGE 131

123 Proof: c )]TJ/F10 6.9738 Tf 6.226 0 Td [(n = 1 T R T 0 f t e j! 0 nt dt = 1 T R T 0 f t e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 nt dt ;f t = f t = 1 T R T 0 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt = c n .33 a b Figure6.10: Re c n = Re c )]TJ/F59 5.9776 Tf 5.756 0 Td [(n ,and Im c n = )]TJ/F56 8.9664 Tf 8.703 0 Td [(Im c )]TJ/F59 5.9776 Tf 5.756 0 Td [(n .

PAGE 132

124 CHAPTER6.CONTINUOUSTIMEFOURIERSERIES a b Figure6.11: j c n j = j c )]TJ/F59 5.9776 Tf 5.756 0 Td [(n j ,and c n = )]TJ/F58 8.9664 Tf 9.215 0 Td [(c )]TJ/F59 5.9776 Tf 5.756 0 Td [(n 6.6.1.2RealandEvenSignals RealandevensignalshaverealandevenFourierseries. Theorem6.4: If f t = f t and f t = f )]TJ/F11 9.9626 Tf 7.748 0 Td [(t i.e. thesignalisrealandeven, then c n = c )]TJ/F10 6.9738 Tf 6.227 0 Td [(n and c n = c n Proof: c n = 1 T R T 2 )]TJ/F8 9.9626 Tf 6.226 -0.747 Td [( T 2 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt = 1 T R 0 )]TJ/F8 9.9626 Tf 6.226 -0.747 Td [( T 2 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt + 1 T R T 2 0 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt = 1 T R T 2 0 f )]TJ/F11 9.9626 Tf 7.749 0 Td [(t e j! 0 nt dt + 1 T R T 2 0 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt = 2 T R T 2 0 f t cos 0 nt dt .34 f t and cos 0 nt arebothrealwhichimpliesthat c n isreal.Also cos 0 nt =cos )]TJ/F8 9.9626 Tf 9.409 0 Td [( 0 nt so c n = c )]TJ/F10 6.9738 Tf 6.227 0 Td [(n .Itisalsoeasytoshowthat f t =2 P 1 n =0 c n cos 0 nt since f t c n ,and cos 0 nt areallrealandeven. 6.6.1.3RealandOddSignals RealandoddsignalshaveFourierSeriesthatareoddandpurelyimaginary. Theorem6.5: If f t = )]TJ/F8 9.9626 Tf 9.409 0 Td [( f )]TJ/F11 9.9626 Tf 7.749 0 Td [(t and f t = f t i.e. thesignalisrealandodd, then c n = )]TJ/F11 9.9626 Tf 7.749 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(n and c n = )]TJ/F8 9.9626 Tf 9.409 0 Td [( c n ,i.e. c n isoddandpurelyimaginary. Proof: Doitathome.

PAGE 133

125 If f t isodd,thenwecanexpanditintermsof sin 0 nt : f t = 1 X n =1 c n sin 0 nt 6.6.2Summary Insummary,wecannd f e t ,anevenfunction,and f o t ,anoddfunction,suchthat f t = f e t + f o t .35 whichimpliesthat,forany f t ,wecannd f a n g and f b n g suchthat f t = 1 X n =0 a n cos 0 nt + 1 X n =1 b n sin 0 nt .36 Example6.3:TriangleWave Figure6.12: T =1 and 0 =2 f t isrealandodd. c n = 8 > > < > > : 4 A j 2 n 2 if n = f :::; )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; 1 ; 5 ; 9 ;::: g )]TJ/F1 9.9626 Tf 9.409 11.059 Td [( 4 A j 2 n 2 if n = f :::; )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 ; 3 ; 7 ; 11 ;::: g 0 if n = f :::; )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 0 ; 2 ; 4 ;::: g Does c n = )]TJ/F11 9.9626 Tf 7.749 0 Td [(c )]TJ/F10 6.9738 Tf 6.227 0 Td [(n ?

PAGE 134

126 CHAPTER6.CONTINUOUSTIMEFOURIERSERIES Figure6.13: TheFourierseriesofatrianglewave. note: Wecanoftengatherinformationaboutthe smoothness ofasignalbyexaminingits Fouriercoecients. Takealookattheaboveexamples.ThepulseandsawtoothwavesarenotcontinuousandthereFourier series'fallolike 1 n .Thetrianglewaveiscontinuous,butnotdierentiableanditsFourierseriesfallso like 1 n 2 Thenext3propertieswillgiveabetterfeelforthis. 6.7CircularConvolutionPropertyofFourierSeries 10 6.7.1SignalCircularConvolution Givenasignal f t withFouriercoecients c n andasignal g t withFouriercoecients d n ,wecandene anewsignal, v t ,where v t = f t ~ g t WendthattheFourierSeriesSection6.2representation of y t a n ,issuchthat a n = c n d n f t ~ g t isthecircularconvolutionSection4.3oftwoperiodic signalsandisequivalenttotheconvolutionoveroneinterval, i.e. f t ~ g t = R T 0 R T 0 f g t )]TJ/F11 9.9626 Tf 9.962 0 Td [( ddt note: CircularconvolutioninthetimedomainisequivalenttomultiplicationoftheFourier coecients. Thisisprovedasfollows a n = 1 T R T 0 v t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt = 1 T 2 R T 0 R T 0 f g t )]TJ/F11 9.9626 Tf 9.962 0 Td [( de )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt = 1 T R T 0 f 1 T R T 0 g t )]TJ/F11 9.9626 Tf 9.963 0 Td [( e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt d = 1 T R T 0 f 1 T R T )]TJ/F10 6.9738 Tf 6.227 0 Td [( )]TJ/F10 6.9738 Tf 6.226 0 Td [( g e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 + d d; = t )]TJ/F11 9.9626 Tf 9.962 0 Td [( = 1 T R T 0 f 1 T R T )]TJ/F10 6.9738 Tf 6.227 0 Td [( )]TJ/F10 6.9738 Tf 6.226 0 Td [( g e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 n d e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 n d = 1 T R T 0 f d n e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 n d = d n 1 T R T 0 f e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 n d = c n d n .37 10 Thiscontentisavailableonlineat.

PAGE 135

127 Example6.4 Takealookatasquarepulsewithaperiod, T 1 = T 4 : Figure6.14 Forthissignal c n = 8 < : 1 T if n =0 1 2 sin 2 n 2 n otherwise Exercise6.8 Solutiononp.140. WhatsignalhasFouriercoecients a n = c n 2 = 1 4 sin 2 2 n 2 n 2 ? 6.8FourierSeriesandLTISystems 11 6.8.1IntroducingtheFourierSeriestoLTISystems Beforelookingatthismodule,oneshouldbefamiliarwiththeconceptsofeigenfunctionandLTIsystems Section5.5.Recall,for H LTIsystemwegetthefollowingrelationship Figure6.15: InputandoutputsignalstoourLTIsystem. where e st isaneigenfunctionof H .ItscorrespondingeigenvalueSection5.2 H s canbecalculated usingtheimpulseresponseSection1.5 h t H s = Z 1 h e )]TJ/F7 6.9738 Tf 6.226 0 Td [( s d 11 Thiscontentisavailableonlineat.

PAGE 136

128 CHAPTER6.CONTINUOUSTIMEFOURIERSERIES So,usingtheFourierSeriesSection6.2expansionforperiodicSection6.1 f t whereweinput f t = X n )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c n e j! 0 nt intothesystem, Figure6.16: LTIsystem ouroutput y t willbe y t = X n )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(H j! 0 n c n e j! 0 nt Sowecanseethatbyapplyingthefourierseriesexpansionequations,wecangofrom f t to c n andvice versa,andwedothesameforouroutput, y t 6.8.2EectsofFourierSeries WecanthinkofanLTIsystemas shaping thefrequencycontentoftheinput.Keepinmindthebasic LTIsystemwepresentedaboveinFigure6.16.TheLTIsystem, H ,simplymultipliesallofourFourier coecientsandscalesthem. GiventheFouriercoecients f c n g oftheinputandtheeigenvaluesofthesystem f H jw 0 n g ,theFourier seriesoftheoutputis f H jw 0 n c n g simpleterm-by-termmultiplication. note: Theeigenvalues H jw 0 n completelydescribewhataLTIsystemdoestoperiodicsignals withperiod T =2 w 0 Example6.5 Whatdoesthissystemdo? Figure6.17 Example6.6 Whataboutthissystem?

PAGE 137

129 a b Figure6.18 6.8.3Examples Example6.7:RCCircuit h t = 1 RC e )]TJ/F9 4.9813 Tf 5.397 0 Td [(t RC u t WhatdoesthissystemdototheFourierSeriesofaninput f t ? Calculatetheeigenvaluesofthissystem H s = R 1 h e )]TJ/F7 6.9738 Tf 6.227 0 Td [( s d = R 1 0 1 RC e )]TJ/F9 4.9813 Tf 5.396 0 Td [( RC e )]TJ/F7 6.9738 Tf 6.226 0 Td [( s d = 1 RC R 1 0 e )]TJ/F10 6.9738 Tf 6.227 0 Td [( 1 RC + s d = 1 RC 1 1 RC + s e )]TJ/F10 6.9738 Tf 6.227 0 Td [( 1 RC + s j 1 =0 = 1 1+ RCs .38 Now,saywefeedtheRCcircuitaperiodicperiod T =2 w 0 input f t Lookattheeigenvaluesfor s = jw 0 n j H jw 0 n j = 1 j 1+ RCjw 0 n j = 1 p 1+ R 2 C 2 w 0 2 n 2 TheRCcircuitisa lowpass system:itpasseslowfrequencies n around 0 andattenuates highfrequencieslarge n Example6.8:SquarepulsewavethroughRCcircuit InputSignal:Takingthefourierseriesof f t c n = 1 2 sin )]TJ/F10 6.9738 Tf 5.762 -4.148 Td [( 2 n 2 n 1 t at n =0 System:eigenvalues H jw 0 n = 1 1+ jRCw 0 n

PAGE 138

130 CHAPTER6.CONTINUOUSTIMEFOURIERSERIES OutputSignal:Takingthefourierseriesof y t d n = H jw 0 n c n = 1 1+ jRCw 0 n 1 2 sin )]TJ/F10 6.9738 Tf 5.762 -4.147 Td [( 2 n 2 n d n = 1 1+ jRCw 0 n 1 2 sin )]TJ/F10 6.9738 Tf 5.762 -4.147 Td [( 2 n 2 n y t = X )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(d n e jw 0 nt Whatcanweinferabout y t from f d n g ? 1.Is y t real? 2.Is y t evensymmetric?oddsymmetric? 3.Qualitatively,whatdoes y t looklike?Isit"smoother"than f t ?decayrateof d n vs. c n d n = 1 1+ jRCw 0 n 1 2 sin )]TJ/F10 6.9738 Tf 5.762 -4.147 Td [( 2 n 2 n j d n j = 1 q 1+ RCw 0 2 n 2 1 2 sin )]TJ/F10 6.9738 Tf 5.762 -4.147 Td [( 2 n 2 n 6.9ConvergenceofFourierSeries 12 6.9.1Introduction Beforelookingatthismodule,hopefullyyouhavebecomefullyconvincedofthefactthat any periodic function, f t ,canberepresentedasasumofcomplexsinusoidsSection1.4.Ifyouarenot,thentry lookingbackateigen-stuinanutshellSection5.4oreigenfunctionsofLTIsystemsSection5.5.We haveshownthatwecanrepresentasignalasthesumofexponentialsthroughtheFourierSeriesSection6.2 equationsbelow: f t = X n )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(c n e j! 0 nt .39 c n = 1 T Z T 0 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt .40 JosephFourier 13 insistedthattheseequationsweretrue,butcouldnotproveit.Lagrangepubliclyridiculed Fourier,andsaidthatonlycontinuousfunctionscanberepresentedby.39indeedheprovedthat.39 holdsforcontinuous-timefunctions.However,weknownowthattherealtruthliesinbetweenFourierand Lagrange'spositions. 12 Thiscontentisavailableonlineat. 13 http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Fourier.html

PAGE 139

131 6.9.2UnderstandingtheTruth Formulatingourquestionmathematically,let f N 0 t = N X n = )]TJ/F10 6.9738 Tf 6.226 0 Td [(N )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(c n e j! 0 nt where c n equalstheFouriercoecientsof f t see.40. f N 0 t isa"partialreconstruction"of f t usingtherst 2 N +1 Fouriercoecients. f N 0 t approximates f t ,withtheapproximationgettingbetterandbetteras N getslarge.Therefore,wecanthink oftheset f N 0 t ;N = f 0 ; 1 ;::: g asa sequenceoffunctions ,eachoneapproximating f t better thantheonebefore. Thequestionis,doesthissequenceconvergeto f t ?Does f N 0 t f t as N !1 ?Wewilltryto answerthisquestionbythinkingaboutconvergenceintwodierentways: 1.Lookingatthe energy oftheerrorsignal: e N t = f t )]TJ/F11 9.9626 Tf 9.963 0 Td [(f N 0 t 2.Lookingat lim N !1 f N 0 t at eachpoint andcomparingto f t 6.9.2.1Approach#1 Let e N t bethedierence i.e. errorbetweenthesignal f t anditspartialreconstruction f N 0 t e N t = f t )]TJ/F11 9.9626 Tf 9.963 0 Td [(f N 0 t .41 If f t 2 L 2 [0 ;T ] niteenergy,thentheenergyof e N t 0 as N !1 is Z T 0 j e N t j 2 dt = Z T 0 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(f t )]TJ/F11 9.9626 Tf 9.963 0 Td [(f N 0 t 2 dt 0 .42 WecanprovethisequationusingParseval'srelation: lim N !1 Z T 0 )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [(j f t )]TJ/F11 9.9626 Tf 9.962 0 Td [(f N 0 t j 2 dt = lim N !1 1 X N = )]TJ/F14 9.9626 Tf 4.567 -8.07 Td [(jF n f t )-222(F n f N 0 t j 2 = lim N !1 X j n j >N j c n j 2 =0 wherethelastequationbeforezeroisthetailsumoftheFourierSeries,whichapproacheszerobecause f t 2 L 2 [0 ;T ] .Sincephysicalsystemsrespondtoenergy,theFourierSeriesprovidesanadequaterepresentation forall f t 2 L 2 [0 ;T ] equalingniteenergyoveroneperiod. 6.9.2.2Approach#2 Thefactthat e N 0 saysnothingabout f t and lim N !1 f N 0 t being equal atagivenpoint.Takethetwo functionsgraphedbelowforexample:

PAGE 140

132 CHAPTER6.CONTINUOUSTIMEFOURIERSERIES a b Figure6.19 Giventhesetwofunctions, f t and g t ,thenwecanseethatforall t f t 6 = g t ,but Z T 0 j f t )]TJ/F11 9.9626 Tf 9.962 0 Td [(g t j 2 dt =0 Fromthiswecanseethefollowingrelationships: energyconvergence 6 =pointwiseconvergence pointwiseconvergence convergenceinL 2 [0 ;T ] However,thereverseoftheabovestatementdoesnotholdtrue. Itturnsoutthatif f t hasa discontinuity ascanbeseeningureof g t aboveat t 0 ,then f t 0 6 = lim N !1 f N 0 t 0 Butaslongas f t meetssomeotherfairlymildconditions,then f t 0 = lim N !1 f N 0 t 0 if f t is continuous at t = t 0 6.10DirichletConditions 14 NamedaftertheGermanmathematician,PeterDirichlet,the Dirichletconditions arethesucientconditionstoguarantee existence and convergence oftheFourierseriesSection6.2ortheFouriertransform 15 6.10.1TheWeakDirichletConditionfortheFourierSeries Rule6.1: TheWeakDirichletCondition FortheFourierSeriestoexist,theFouriercoecientsmustbenite.The WeakDirichlet Condition guaranteesthisexistence.Itessentiallysaysthattheintegraloftheabsolutevalueof thesignalmustbenite.ThelimitsofintegrationaredierentfortheFourierSeriescasethanfor theFourierTransformcase.Thisisadirectresultofthedieringdenitionsofthetwo. 14 Thiscontentisavailableonlineat. 15 "DerivationoftheFourierTransform"

PAGE 141

133 Proof: TheFourierSeriesexiststhecoecientsareniteif WeakDirichletConditionfortheFourierSeries Z T 0 j f t j dt< 1 .43 ThiscanbeshownfromtheinitialconditionthattheFourierSeriescoecientsbenite. j c n j = j 1 T Z T 0 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt j 1 T Z T 0 j f t jj e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt j dt .44 RememberingourcomplexexponentialsSection1.6,weknowthatintheaboveequation j e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 nt j =1 ,whichgivesus 1 T Z T 0 j f t j dt = 1 T Z T 0 j f t j dt .45 < 1 .46 note: Ifwehavethefunction: f t = 1 t ; 0
PAGE 142

134 CHAPTER6.CONTINUOUSTIMEFOURIERSERIES 6.10.2.1Example Letusassumewehavethefollowingfunctionandequality: f 0 t = lim N !1 f N 0 t .48 If f t meetsallthreeconditionsoftheStrongDirichletConditions,then f = f 0 atevery atwhich f t iscontinuous.Andwhere f t isdiscontinuous, f 0 t isthe average ofthevalues ontherightandleft.SeeFigure6.20asanexample: a b Figure6.20: Discontinuousfunctions, f t note: ThefunctionsthatfailtheDirchletconditionsareprettypathological-asengineers,we arenottoointerestedinthem. 6.11Gibbs'sPhenomena 16 6.11.1Introduction TheFourierSeriesSection6.2istherepresentationofcontinuous-time,periodicsignalsintermsofcomplex exponentials.TheDirichletconditionsSection6.10suggestthatdiscontinuoussignalsmayhaveaFourier Seriesrepresentationsolongasthereareanitenumberofdiscontinuities.Thisseemscounter-intuitive, however,ascomplexexponentialsSection1.6arecontinuousfunctions.Itdoesnotseempossibletoexactly reconstructadiscontinuousfunctionfromasetofcontinuousones.Infact,itisnot.However,itcanbeif werelaxtheconditionof'exactly'andreplaceitwiththeideaof'almosteverywhere'.Thisistosaythatthe reconstructionisexactlythesameastheoriginalsignalexceptatanitenumberofpoints.Thesepoints, notnecessarilysurprisingly,occuratthepointsofdiscontinuities. 6.11.1.1History Inthelate1800s,manymachineswerebuilttocalculateFouriercoecientsandre-synthesize: f N 0 t = N X n = )]TJ/F10 6.9738 Tf 6.226 0 Td [(N )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c n e j! 0 nt .49 16 Thiscontentisavailableonlineat.

PAGE 143

135 AlbertMichelsonanextraordinaryexperimentalphysicistbuiltamachinein1898thatcouldcompute c n upto n = 79 ,andhere-synthesized f 79 0 t = 79 X n = )]TJ/F7 6.9738 Tf 6.226 0 Td [(79 )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(c n e j! 0 nt .50 Themachineperformedverywellonalltestsexceptthoseinvolving discontinuousfunctions .Whena squarewave,likethatshowninFigure6.21Fourierseriesapproximationofasquarewave,wasinputedinto themachine,"wiggles"aroundthediscontinuitiesappeared,andevenasthenumberofFouriercoecients approachedinnity,thewigglesneverdisappeared-thesecanbeseeninthelastplotinFigure6.21Fourier seriesapproximationofasquarewave.J.WillardGibbsrstexplainedthisphenomenonin1899,and thereforethesediscontinuouspointsarereferredtoas GibbsPhenomenon 6.11.2Explanation Webeginthisdiscussionbytakingasignalwithanitenumberofdiscontinuitieslikea squarepulse and ndingitsFourierSeriesrepresentation.WethenattempttoreconstructitfromtheseFouriercoecients. Whatwendisthatthemorecoecientsweuse,themorethesignalbeginstoresembletheoriginal. However,aroundthediscontinuities,weobserveripplingthatdoesnotseemtosubside.Asweconsidereven morecoecients,wenoticethattheripplesnarrow,butdonotshorten.Asweapproachaninnitenumber ofcoecients,thisripplingstilldoesnotgoaway.Thisiswhenweapplytheideaofalmosteverywhere. Whiletheseripplesremainneverdroppingbelow9%ofthepulseheight,theareainsidethemtendstozero, meaningthattheenergyofthisripplegoestozero.Thismeansthattheirwidthisapproachingzeroand wecanassertthatthereconstructionisexactlytheoriginalexceptatthepointsofdiscontinuity.Sincethe Dirichletconditionsassertthattheremayonlybeanitenumberofdiscontinuities,wecanconcludethat theprincipleofalmosteverywhereismet.Thisphenomenonisaspeciccaseof nonuniformconvergence Belowwewillusethesquarewave,alongwithitsFourierSeriesrepresentation,andshowseveralgures thatrevealthisphenomenonmoremathematically. 6.11.2.1SquareWave TheFourierseriesrepresentationofasquaresignalbelowsaysthattheleftandrightsidesare"equal."In ordertounderstandGibbsPhenomenonwewillneedtoredenethewaywelookatequality. s t = a 0 + 1 X k =1 a k cos 2 kt T + 1 X k =1 b k sin 2 kt T .51 6.11.2.2Example Figure6.21FourierseriesapproximationofasquarewaveshowsseveralFourierseriesapproximationof thesquarewave 17 usingavariednumberofterms,denotedby K : 17 "FourierSeriesApproximationofaSquareWave"

PAGE 144

136 CHAPTER6.CONTINUOUSTIMEFOURIERSERIES Fourierseriesapproximationofasquarewave Figure6.21: Fourierseriesapproximationto sq t .ThenumberoftermsintheFouriersumisindicated ineachplot,andthesquarewaveisshownasadashedlineovertwoperiods. WhencomparingthesquarewavetoitsFourierseriesrepresentationinFigure6.21Fourierseriesapproximationofasquarewave,itisnotclearthatthetwoareequal.Thefactthatthesquarewave'sFourier seriesrequiresmoretermsforagivenrepresentationaccuracyisnotimportant.However,closeinspectionof Figure6.21Fourierseriesapproximationofasquarewavedoesrevealapotentialissue:DoestheFourier seriesreallyequalthesquarewaveat all valuesof t ?Inparticular,ateachstep-changeinthesquarewave, theFourierseriesexhibitsapeakfollowedbyrapidoscillations.Asmoretermsareaddedtotheseries, theoscillationsseemtobecomemorerapidandsmaller,butthepeaksarenotdecreasing.Considerthis mathematicalquestionintuitively:Canadiscontinuousfunction,likethesquarewave,beexpressedasa sum,evenaninniteone,ofcontinuousones?Oneshouldatleastbesuspicious,andinfact,itcan'tbe thusexpressed.ThisissuebroughtFourier 18 muchcriticismfromtheFrenchAcademyofScienceLaplace, Legendre,andLagrangecomprisedthereviewcommitteeforseveralyearsafteritspresentationon1807. Itwasnotresolvedforalsoacentury,anditsresolutionisinterestingandimportanttounderstandfroma 18 http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Fourier.html

PAGE 145

137 practicalviewpoint. Theextraneouspeaksinthesquarewave'sFourierseries never disappear;theyaretermed Gibb's phenomenon aftertheAmericanphysicistJosiahWillardGibbs.Theyoccurwheneverthesignalisdiscontinuous,andwillalwaysbepresentwheneverthesignalhasjumps. 6.11.2.3RedeneEquality Let'sreturntothequestionofequality;howcantheequalsigninthedenitionoftheFourierseriesbe justied?Thepartialansweristhatpointwiseeachandeveryvalueof t equalityis not guaranteed.What mathematicianslaterinthenineteenthcenturyshowedwasthatthermserroroftheFourierserieswas alwayszero. lim K !1 rms K =0 .52 WhatthismeansisthatthedierencebetweenanactualsignalanditsFourierseriesrepresentationmay notbezero,butthesquareofthisquantityhas zero integral!Itisthroughtheeyesofthermsvaluethat wedeneequality:Twosignals s 1 t s 2 t aresaidtobeequalinthe meansquare if rms s 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(s 2 =0 Thesesignalsaresaidtobeequal pointwise if s 1 t = s 2 t forallvaluesof t .ForFourierseries,Gibb's phenomenonpeakshaveniteheightandzerowidth:Theerrordiersfromzeroonlyatisolatedpoints whenevertheperiodicsignalcontainsdiscontinuitiesandequalsabout9%ofthesizeofthediscontinuity. Thevalueofafunctionatanitesetofpointsdoesnotaectitsintegral.Thiseectunderliesthereason whydeningthevalueofadiscontinuousfunctionatitsdiscontinuityismeaningless.Whateveryoupick foravaluehasnopracticalrelevanceforeitherthesignal'sspectrumorforhowasystemrespondstothe signal.TheFourierseriesvalue"at"thediscontinuityistheaverageofthevaluesoneithersideofthejump. Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10092/latest/FFTSymbolic.llb 6.12FourierSeriesWrap-Up 19 BelowwewillhighlightsomeofthemostimportantconceptsabouttheFourierSeriesSection6.2and ourunderstandingofitthrougheigenfunctionsandeigenvalues.Hopefullyyouarefamiliarwithallofthis material,sothisdocumentwillsimplyserveasarefresher,butifnot,thenrefertothemanylinksbelowfor moreinformationonthevariousideasandtopics. 1.WecanrepresentaperiodicfunctionSection6.1orafunctiononaninterval f t asacombination ofcomplexexponentialsSection1.6: f t = 1 X n = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c n e j! 0 nt .53 c n = 1 T Z T 0 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt .54 Wherethefouriercoecients, c n ,approximatelyequalhowmuchoffrequency 0 n isinthesignal. 2.Since e j! 0 nt areeigenfunctionsofLTIsystemsSection5.5,wecaninterprettheactionofasystem onasignalintermsofitseigenvaluesSection5.2: H j! 0 n = Z 1 h t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt .55 19 Thiscontentisavailableonlineat.

PAGE 146

138 CHAPTER6.CONTINUOUSTIMEFOURIERSERIES j H j! 0 n j islarge system accentuates frequency 0 n j H j! 0 n j issmall system attenuates frequency 0 n 3.Inaddition,the f c n g ofaperiodicfunction f t cantellusabout: symmetriesin f t smoothnessof f t ,wheresmoothnesscanbeinterpretedasthedecayrateof j c n j 4.Wecan approximate afunctionbyre-synthesizingusingonlysomeoftheFouriercoecientstruncatingtheF.S. f N 0 t = X n j N j )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c n e j! 0 nt .56 Thisapproximationworkswellwhere f t iscontinuous,butnotsowellwhere f t isdiscontinuous. ThisideaisexplainedbyGibb'sPhenomena. Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10749/latest/TransferFunctions.llb

PAGE 147

139 SolutionstoExercisesinChapter6 SolutiontoExercise6.1p.113 Thetrickypartoftheproblemisndingawaytorepresenttheabovefunctionintermsofitsbasis, e j! 0 nt Todothis,wewilluseourknowledgeofEuler'sRelationSection1.6.2:Euler'sRelationtorepresentour cosinefunctionintermsoftheexponential. f t = 1 2 e j! 0 t + e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 t Nowfromthisformofourfunctionandfrom.3,byinspectionwecanseethatourfouriercoecientswill be: c n = 8 < : 1 2 if j n j =1 0 otherwise SolutiontoExercise6.2p.113 Asdoneinthepreviousexample,wewillagainuseEuler'sRelationSection1.6.2:Euler'sRelationto representoursinefunctionintermsofexponentialfunctions. f t = 1 2 j e j! 0 t )]TJ/F11 9.9626 Tf 9.963 0 Td [(e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 t Andsoourfouriercoecientsare c n = 8 > > < > > : )]TJ/F10 6.9738 Tf 6.227 0 Td [(j 2 if n = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 j 2 if n =1 0 otherwise SolutiontoExercise6.3p.113 Onceagainwewillusethesametechniqueaswasusedintheprevioustwoproblems.Thebreakdownof ourfunctionyields f t =3+4 1 2 e j! 0 t + e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 t +2 1 2 e j 2 0 t + e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j 2 0 t Andfromthiswecanndourfouriercoecientstobe: c n = 8 > > > > > < > > > > > : 3 if n =0 2 if j n j =1 1 if j n j =2 0 otherwise SolutiontoExercise6.4p.114 Wewillbeginbypluggingourabovefunction, f t ,into.4.Ourintervalofintegrationwillnowchange tomatchtheintervalspeciedbythefunction. c n = 1 T Z T 1 )]TJ/F10 6.9738 Tf 6.226 0 Td [(T 1 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt Noticethatwemustconsidertwocases: n =0 and n 6 =0 .For n =0 wecantellbyinspectionthatwewill get c n = 2 T 1 T ;n =0

PAGE 148

140 CHAPTER6.CONTINUOUSTIMEFOURIERSERIES For n 6 =0 ,wewillneedtotakeafewmorestepstosolve.Wecanbeginbylookingatthebasicintegralof theexponentialwehave.Rememberingourcalculus,wearereadytointegrate: c n = 1 T 1 j! 0 n e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 nt j T 1 t = )]TJ/F10 6.9738 Tf 6.226 0 Td [(T 1 Letusnowevaluatetheexponentialfunctionsforthegivenlimitsandexpandourequationto: c n = 1 T 1 )]TJ/F8 9.9626 Tf 9.409 0 Td [( j! 0 n e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 nT 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(e j! 0 nT 1 Nowifwemultipletherightsideofourequationby 2 j 2 j anddistributeournegativesignintotheparenthesis, wecanutilizeEuler'sRelationSection1.6.2:Euler'sRelationtogreatlysimplifyourexpressioninto: c n = 1 T 2 j j! 0 n sin 0 nT 1 Now,recallearlierthatwedened 0 = 2 T .Wecansolvethisequationfor T andsubstitutein. c n = 2 j! 0 j! 0 n 2 sin 0 nT 1 Andnally,ifwemakeafewsimplecancellationswewillarriveatournalanswerfortheFouriercoecients of f t : c n = sin 0 nT 1 n ;n 6 =0 SolutiontoExercise6.5p.121 j c n j 2 < 1 for f t tohaveniteenergy. SolutiontoExercise6.6p.121 Yes,because j c n j 2 = 1 n 2 ,whichissummable. SolutiontoExercise6.7p.121 No,because j c n j 2 = 1 n ,whichisnotsummable. SolutiontoExercise6.8p.127 Figure6.22: Atrianglepulsetrainwithaperiodof T 4 .

PAGE 149

Chapter7 DiscreteFourierTransform 7.1FourierAnalysis 1 Fourieranalysisisfundamentaltounderstandingthebehaviorofsignalsandsystems.Thisisaresultof thefactthatsinusoidsareEigenfunctionsSection5.5oflinear,time-invariantLTISection2.1systems. ThisistosaythatifwepassanyparticularsinusoidthroughaLTIsystem,wegetascaledversionof thatsamesinusoidontheoutput.Then,sinceFourieranalysisallowsustoredenethesignalsinterms ofsinusoids,allweneedtodoisdeterminehowanygivensystemeectsallpossiblesinusoidsitstransfer function 2 andwehaveacompleteunderstandingofthesystem.Furthermore,sinceweareabletodene thepassageofsinusoidsthroughasystemasmultiplicationofthatsinusoidbythetransferfunctionatthe samefrequency,wecanconvertthepassageofanysignalthroughasystemfromconvolutionSection3.3 intimetomultiplicationinfrequency.TheseideasarewhatgiveFourieranalysisitspower. Now,afterhopefullyhavingsoldyouonthevalueofthismethodofanalysis,wemustexamineexactly whatwemeanbyFourieranalysis.ThefourFouriertransformsthatcomprisethisanalysisaretheFourier Series 3 ,Continuous-TimeFourierTransformSection11.1,Discrete-TimeFourierTransformSection10.4 andDiscreteFourierTransform 4 .Forthisdocument,wewillviewtheLaplaceTransformSection13.1 andZ-TransformSection14.2assimplyextensionsoftheCTFTandDTFTrespectively.Allofthese transformsactessentiallythesameway,byconvertingasignalintimetoanequivalentsignalinfrequency sinusoids.However,dependingonthenatureofaspecicsignal i.e. whetheritisnite-orinnite-length andwhetheritisdiscrete-orcontinuous-timethereisanappropriatetransformtoconvertthesignalinto thefrequencydomain.BelowisatableofthefourFouriertransformsandwheneachisappropriate.Italso includestherelevantconvolutionforthespeciedspace. TableofFourierRepresentations Transform TimeDomain FrequencyDomain Convolution Continuous-Time FourierSeries L 2 [0 ;T l 2 Z Continuous-TimeCircular continuedonnextpage 1 Thiscontentisavailableonlineat. 2 "TransferFunctions" 3 "Continuous-TimeFourierSeriesCTFS" 4 "DiscreteFourierTransform" 141

PAGE 150

142 CHAPTER7.DISCRETEFOURIERTRANSFORM Continuous-Time FourierTransform L 2 R L 2 R Continuous-TimeLinear Discrete-TimeFourier Transform l 2 Z L 2 [0 ; 2 Discrete-TimeLinear DiscreteFourierTransform l 2 [0 ;N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1] l 2 [0 ;N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1] Discrete-TimeCircular Table7.1 7.2FourierAnalysisinComplexSpaces 5 7.2.1Introduction BynowyoushouldbefamiliarwiththederivationoftheFourierseriesSection6.2forcontinuous-time, periodicSection6.1functions.Thisderivationleadsustothefollowingequationsthatyoushouldbequite familiarwith: f t = X n )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c n e j! 0 nt .1 c n = 1 T R n f t e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 nt dt = 1 T .2 where c n tellsustheamountoffrequency 0 n in f t Inthismodule,wewillderiveasimilarexpansionfordiscrete-time,periodicfunctions.Indoingso,we willderivethe DiscreteTimeFourierSeries DTFS,ortheDiscreteFourierTransformSection10.2 DFT. 7.2.2DerivationofDTFS Muchlikeaperiodic,continuous-timefunctioncanbethoughtofasafunctionontheinterval [0 ;T ] a b Figure7.1: Wewilljustconsideroneintervaloftheperiodicfunctionthroughoutthissection.a PeriodicFunctionbFunctionontheinterval [0 ;T ] 5 Thiscontentisavailableonlineat.

PAGE 151

143 Aperiodic,discrete-timesignalwithperiod N canbethoughtofasa nite setofnumbers.For example,saywehavethefollowingsetofnumbersthatdescribeaperiodic,discrete-timesignal,where N =4 : f :::; 3 ; 2 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 1 ; 3 ;::: g Wecanrepresentthissignalaseitheraperiodicsignalorasjustasingleintervalasfollows: a b Figure7.2: Herewecanlookatjustoneperiodofthesignalthathasavectorlengthoffourandis containedin C 4 .aPeriodicFunctionbFunctionontheinterval [0 ;T ] note: Thesetofdiscretetimesignalswithperiod N equal C N Justlikethecontinuouscase,wearegoingtoformabasisusing harmonicsinusoids .Beforewelookinto this,itwillbeworthourtimetolookatthediscrete-time,complexsinusoidsinalittlemoredetail. 7.2.2.1ComplexSinusoids Ifyouarefamiliarwiththebasicsinusoidsignal 6 andwithcomplexexponentialsSection1.6thenyou shouldnothaveanyproblemunderstandingthissection.Inmosttexts,youwillseethethediscrete-time, complexsinusoidnotedas: e j!n Example7.1 Figure7.3: Complexsinusoidwithfrequency =0 6 "ElementalSignals"

PAGE 152

144 CHAPTER7.DISCRETEFOURIERTRANSFORM Example7.2 Figure7.4: Complexsinusoidwithfrequency = 4 7.2.2.1.1IntheComplexPlane Thecomplexsinusoidcanbedirectlymappedontoourcomplexplane 7 ,whichallowsustoeasilyvisualize changestothecomplexsinusoidandextractcertainproperties.Theabsolutevalueofourcomplexsinusoid hasthefollowingcharacteristic: j e j!n j =1 ;n 2 R .3 whichtellsthatourcomplexsinusoidonlytakesvaluesontheunitcircle.Asfortheangle,thefollowing statementholdstrue: e j!n = wn .4 As n increases,wecanpicture e j!n equalingthevalueswegetmovingcounterclockwisearoundtheunit circle.SeeFigure7.5foranillustration: a b c Figure7.5: Theseimagesshowthatas n increases,thevalueof e j!n movesaroundtheunitcircle counterclockwise.a n =0 b n =1 c n =2 7 "TheComplexPlane"

PAGE 153

145 note: For e j!n tobeperiodicSection6.1,weneed e j!N =1 forsome N Example7.3 Forourrstexampleletuslookataperiodicsignalwhere = 2 7 and N =7 a b Figure7.6: a N =7 bHerewehaveaplotof Re e j 2 7 n Example7.4 Nowletuslookattheresultsofplottinganon-periodicsignalwhere =1 and N =7 a b Figure7.7: a N =7 bHerewehaveaplotof Re )]TJ/F58 8.9664 Tf 4.223 -7.242 Td [(e jn 7.2.2.1.2Aliasing Ourcomplexsinusoidshavethefollowingproperty: e j!n = e j +2 n .5 Giventhisproperty,ifwehaveasinusoidwithfrequency +2 ,thenthissignal"aliases"toasinusoid withfrequency note: Each e j!n isuniquefor 2 [0 ; 2

PAGE 154

146 CHAPTER7.DISCRETEFOURIERTRANSFORM 7.2.2.1.3"Negative"Frequencies Ifwearegivenasignalwithfrequency
PAGE 155

147 Inanswertotheabovequestion,letustrythe"harmonic"sinusoidswithafundamentalfrequency 0 = 2 N : HarmonicSinusoid e j 2 N kn .7 a b c Figure7.10: ExamplesofourHarmonicSinusoidsaHarmonicsinusoidwith k =0 bImaginary partofsinusoid, Im e j 2 N 1 n ,with k =1 cImaginarypartofsinusoid, Im e j 2 N 2 n ,with k =2 e j 2 N kn isperiodicwithperiod N andhas k "cycles"between n =0 and n = N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Theorem7.1: Ifwelet b k [ n ]= 1 p N e j 2 N kn ;n = f 0 ;:::;N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 g wheretheexponentialtermisavectorin C N ,then f b k gj k = f 0 ;:::;N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 g isan orthonormalbasis Section15.7.3:OrthonormalBasis for C N Proof: Firstofall,wemustshow f b k g isorthonormal, i.e. = kl = N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X n =0 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(b k [ n ] b l [ n ] = 1 N N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X n =0 e j 2 N kn e )]TJ/F8 9.9626 Tf 6.227 -0.748 Td [( j 2 N ln = 1 N N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X n =0 e j 2 N l )]TJ/F10 6.9738 Tf 6.226 0 Td [(k n .8 If l = k ,then = 1 N P N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 n =0 =1 .9 If l 6 = k ,thenwemustusethe"partialsummationformula"shownbelow: N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X n =0 n = 1 X n =0 n )]TJ/F13 6.9738 Tf 15.041 12.454 Td [(1 X n = N n = 1 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [( )]TJ/F11 9.9626 Tf 15.959 6.74 Td [( N 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( = 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( N 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [( = 1 N N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X n =0 e j 2 N l )]TJ/F10 6.9738 Tf 6.226 0 Td [(k n

PAGE 156

148 CHAPTER7.DISCRETEFOURIERTRANSFORM whereintheaboveequationwecansaythat = e j 2 N l )]TJ/F10 6.9738 Tf 6.227 0 Td [(k ,andthuswecanseehowthisisinthe formneededtoutilizeourpartialsummationformula. = 1 N 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(e j 2 N l )]TJ/F10 6.9738 Tf 6.227 0 Td [(k N 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(e j 2 N l )]TJ/F10 6.9738 Tf 6.226 0 Td [(k = 1 N 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(e j 2 N l )]TJ/F10 6.9738 Tf 6.226 0 Td [(k =0 So, = 8 < : 1 if k = l 0 if k 6 = l .10 Therefore: f b k g isanorthonormalset. f b k g isalsoabasisSection5.1.3:Basis,sincethereare N vectorswhicharelinearlyindependentSection5.1.1:LinearIndependenceorthogonalityimplies linearindependence. Andnally,wehaveshownthattheharmonicsinusoids n 1 p N e j 2 N kn o formanorthonormalbasis for C n 7.2.2.2Discrete-TimeFourierSeriesDTFS UsingthestepsshownaboveinthederivationandourpreviousunderstandingofHilbertSpacesSection15.3 andOrthogonalExpansionsSection15.8,therestofthederivationisautomatic.Givenadiscrete-time, periodicsignalvectorin C n f [ n ] ,wecanwrite: f [ n ]= 1 p N N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 c k e j 2 N kn .11 c k = 1 p N N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X n =0 f [ n ] e )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( j 2 N kn .12 Note:Mostpeoplecollectboththe 1 p N termsintotheexpressionfor c k note: HereisthecommonformoftheDTFSwiththeabovenotetakenintoaccount: f [ n ]= N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 c k e j 2 N kn c k = 1 N N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X n =0 f [ n ] e )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( j 2 N kn Thiswhatthe fft commandinMATLABdoes. 7.3MatrixEquationfortheDTFS 8 TheDTFSSection7.2.2.2:Discrete-TimeFourierSeriesDTFSisjustachangeofbasisSection5.1.3: Basisin C N .Tostart,wehave f [ n ] intermsofthe standardbasis f [ n ]= f [0] e 0 + f [1] e 1 + + f [ N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1] e N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = P n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 k =0 f [ k ] [ k )]TJ/F11 9.9626 Tf 9.963 0 Td [(n ] .13 8 Thiscontentisavailableonlineat.

PAGE 157

149 0 B B B B B B B B @ f [0] f [1] f [2] . f [ N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1] 1 C C C C C C C C A = 0 B B B B B B B B @ f [0] 0 0 . 0 1 C C C C C C C C A + 0 B B B B B B B B @ 0 f [1] 0 . 0 1 C C C C C C C C A + 0 B B B B B B B B @ 0 0 f [2] . 0 1 C C C C C C C C A + + 0 B B B B B B B B @ 0 0 0 . f [ N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1] 1 C C C C C C C C A .14 TakingtheDTFS,wecanwrite f [ n ] intermsofthesinusoidalFourierbasis f [ n ]= N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 c k e j 2 N kn .15 0 B B B B B B B B @ f [0] f [1] f [2] . f [ N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1] 1 C C C C C C C C A = c 0 0 B B B B B B B B @ 1 1 1 . 1 1 C C C C C C C C A + c 1 0 B B B B B B B B @ 1 e j 2 N e j 4 N . e j 2 N N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 C C C C C C C C A + c 2 0 B B B B B B B B @ 1 e j 4 N e j 8 N . e j 4 N N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 C C C C C C C C A + ::: .16 Wecanformthebasismatrixwe'llcallit W hereinsteadof B bystackingthebasisvectorsinascolumns W = b 0 [ n ] b 1 [ n ] :::b N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 [ n ] = 0 B B B B B B B B @ 111 ::: 1 1 e j 2 N e j 4 N :::e j 2 N N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 1 e j 4 N e j 8 N :::e j 2 N 2 N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 . . . . . . . 1 e j 2 N N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 e j 2 N 2 N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 :::e j 2 N N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 C C C C C C C C A .17 with b k [ n ]= e j 2 N kn note: theentryinthek-throwandn-thcolumnis W j;k = e j 2 N kn = W n;k So,herewehaveanadditionalsymmetry W = W T W T = W = 1 N W )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 since f b k [ n ] g areorthogonal WecannowrewritetheDTFSequationsinmatrixformwherewehave: f =signalvectorin C N c =DTFScoes.vectorin C N "synthesis" f = W c f [ n ]= < c ;b n > "analysis" c = W T f = W f c [ k ]= < f ;b k > Table7.2 FindingandinvertingtheDTFSisjust matrixmultiplication Everythingin C N is clean :nolimits,noconvergencequestions,justgoodolematrixarithmetic.

PAGE 158

150 CHAPTER7.DISCRETEFOURIERTRANSFORM 7.4PeriodicExtensiontoDTFS 9 7.4.1Introduction Nowthatwehaveanunderstandingofthediscrete-timeFourierseriesDTFSSection7.2.2.2:DiscreteTimeFourierSeriesDTFS,wecanconsiderthe periodicextension of c [ k ] theDiscrete-timeFourier coecients.Figure7.11showsasimpleillustrationofhowwecanrepresentasequenceasaperiodicsignal mappedoveraninnitenumberofintervals. a b Figure7.11: avectorsbperiodicsequences Exercise7.1 Solutiononp.162. WhydoesaperiodicSection6.1extensiontotheDTFScoecients c [ k ] makesense? 7.4.2Examples Example7.6:Discretetimesquarewave 9 Thiscontentisavailableonlineat.

PAGE 159

151 Figure7.12 CalculatetheDTFS c [ k ] using: c [ k ]= 1 N N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X n =0 f [ n ] e )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( j 2 N kn .18 JustlikecontinuoustimeFourierseries,wecantakethesummationoveranyinterval,sowehave c k = 1 N N 1 X n = )]TJ/F10 6.9738 Tf 6.226 0 Td [(N 1 e )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( j 2 N kn .19 Let m = n + N 1 sowecangetageometricseriesstartingat0 c k = 1 N P 2 N 1 m =0 e )]TJ/F8 9.9626 Tf 6.227 -0.748 Td [( j 2 N m )]TJ/F10 6.9738 Tf 6.227 0 Td [(N 1 k = 1 N e j 2 N k P 2 N 1 m =0 e )]TJ/F8 9.9626 Tf 6.226 -0.747 Td [( j 2 N mk .20 Now,usingthe"partialsummationformula" M X n =0 a n = 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(a M +1 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a .21 c k = 1 N e j 2 N N 1 k P 2 N 1 m =0 e )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( j 2 N k m = 1 N e j 2 N N 1 k 1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(e )]TJ/F8 9.9626 Tf 5.396 -1.245 Td [( j 2 N N 1 +1 1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(e )]TJ/F8 9.9626 Tf 5.396 -1.245 Td [( jk 2 N .22 Manipulatetomakethislooklikeasincfunctiondistribute: c k = 1 N e )]TJ/F8 9.9626 Tf 5.397 -1.246 Td [( jk 2 2 N e jk 2 N N 1 + 1 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(e )]TJ/F8 9.9626 Tf 5.397 -1.246 Td [( jk 2 N N 1 + 1 2 e )]TJ/F8 9.9626 Tf 5.396 -1.246 Td [( jk 2 2 N e jk 2 N 1 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(e )]TJ/F8 9.9626 Tf 5.397 -1.246 Td [( jk 2 N 1 2 = 1 N sin 2 k N 1 + 1 2 N sin k N = digitalsinc .23

PAGE 160

152 CHAPTER7.DISCRETEFOURIERTRANSFORM note: It'speriodic!Figure7.13,Figure7.14,andFigure7.15showourabovefunctionand coecientsforvariousvaluesof N 1 a b Figure7.13: N 1 =1 aPlotof f [ n ] .bPlotof c [ k ] a b Figure7.14: N 1 =3 aPlotof f [ n ] .bPlotof c [ k ] a b Figure7.15: N 1 =7 aPlotof f [ n ] .bPlotof c [ k ] .

PAGE 161

153 7.5CircularShifts 10 ThemanypropertiesoftheDFTSection7.2.2.2:Discrete-TimeFourierSeriesDTFSbecomereally straightforward very similartotheFourierSeriesSection6.2oncewehaveonceconceptdown: Circular Shifts 7.5.1Circularshifts WecanpictureperiodicSection6.1sequencesashavingdiscretepointsonacircleasthedomain Figure7.16 Shiftingby m f n + m ,correspondstorotatingthecylinder m notchesACWcounterclockwise.For m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ,wegetashiftequaltothatinthefollowingillustration: Figure7.17: for m = )]TJ/F56 8.9664 Tf 7.167 0 Td [(2 10 Thiscontentisavailableonlineat.

PAGE 162

154 CHAPTER7.DISCRETEFOURIERTRANSFORM Figure7.18 Tocyclicshiftwefollowthesesteps: 1Write f n onacylinder,ACW Figure7.19: N =8

PAGE 163

155 2Tocyclicshiftby m ,spincylindermspotsACW f [ n ] f [ n + m N ] Figure7.20: m = )]TJ/F56 8.9664 Tf 7.168 0 Td [(3 Example7.7 If f n =[0 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7] ,then f n )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 N =[3 ; 4 ; 5 ; 6 ; 7 ; 0 ; 1 ; 2] It'scalled circularshifting ,sincewe'removingaroundthecircle.Theusualshiftingiscalled "linearshifting"shiftingalongaline. 7.5.1.1Notesoncircularshifting f [ n + N N ]= f [ n ] Spinning N spotsisthesameasspinningallthewayaround,ornotspinningatall. f [ n + N N ]= f [ n )]TJ/F8 9.9626 Tf 9.962 0 Td [( N )]TJ/F11 9.9626 Tf 9.962 0 Td [(m N ] ShiftingACW m isequivalenttoshiftingCW N )]TJ/F11 9.9626 Tf 9.962 0 Td [(m

PAGE 164

156 CHAPTER7.DISCRETEFOURIERTRANSFORM Figure7.21 f [ )]TJ/F11 9.9626 Tf 7.748 0 Td [(n N ] Theaboveexpression,simplywritesthevaluesof f [ n ] clockwise.

PAGE 165

157 a b Figure7.22: a f [ n ] b f )]TJ/F58 8.9664 Tf 7.168 0 Td [(n N 7.5.2CircularshiftsandtheDFT Theorem7.2: CircularShiftsandDFT If f [ n ] DFT $ F [ k ] then f [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m N ] DFT $ e )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( j 2 N km F [ k ] i.e. circularshiftintimedomain=phaseshiftinDFT Proof: f [ n ]= 1 N N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k =0 F [ k ] e j 2 N kn .24 sophaseshiftingtheDFT f [ n ]= 1 N P N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k =0 F [ k ] e )]TJ/F8 9.9626 Tf 6.226 -0.747 Td [( j 2 N kn e j 2 N kn = 1 N P N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k =0 F [ k ] e j 2 N k n )]TJ/F10 6.9738 Tf 6.227 0 Td [(m = f [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(m N ] .25 7.6CircularConvolutionandtheDFT 11 7.6.1Introduction YoushouldbefamiliarwithDiscrete-TimeConvolutionSection4.2,whichtellsusthatgiventwodiscretetimesignals x [ n ] ,thesystem'sinput,and h [ n ] ,thesystem'sresponse,wedenetheoutputofthesystem as y [ n ]= x [ n ] h [ n ] = P 1 k = x [ k ] h [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ] .26 11 Thiscontentisavailableonlineat.

PAGE 166

158 CHAPTER7.DISCRETEFOURIERTRANSFORM WhenwearegiventwoDFTsnite-lengthsequencesusuallyoflength N ,wecannotjustmultiplythem togetheraswedointheaboveconvolutionformula,oftenreferredtoas linearconvolution .Becausethe DFTsareperiodic,theyhavenonzerovaluesfor n N andthusthemultiplicationofthesetwoDFTswillbe nonzerofor n N .Weneedtodeneanewtypeofconvolutionoperationthatwillresultinourconvolved signalbeingzerooutsideoftherange n = f 0 ; 1 ;:::;N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 g .Thisidealedtothedevelopmentof circular convolution ,alsocalledcyclicorperiodicconvolution. 7.6.2CircularConvolutionFormula WhathappenswhenwemultiplytwoDFT'stogether,where Y [ k ] istheDFTof y [ n ] ? Y [ k ]= F [ k ] H [ k ] .27 when 0 k N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 UsingtheDFTsynthesisformulafor y [ n ] y [ n ]= 1 N N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 F [ k ] H [ k ] e j 2 N kn .28 Andthenapplyingtheanalysisformula F [ k ]= P N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m =0 f [ m ] e )]TJ/F10 6.9738 Tf 6.226 0 Td [(j 2 N kn y [ n ]= 1 N P N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k =0 P N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m =0 f [ m ] e )]TJ/F10 6.9738 Tf 6.227 0 Td [(j 2 N kn H [ k ] e j 2 N kn = P N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 m =0 f [ m ] 1 N P N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k =0 H [ k ] e j 2 N k n )]TJ/F10 6.9738 Tf 6.227 0 Td [(m .29 wherewecanreducethesecondsummationfoundintheaboveequationinto h [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m N ]= 1 N P N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k =0 H [ k ] e j 2 N k n )]TJ/F10 6.9738 Tf 6.226 0 Td [(m y [ n ]= N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X m =0 f [ m ] h [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m N ] whichequalscircularconvolution!Whenwehave 0 n N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 intheabove,thenweget: y [ n ] f [ n ] ~ h [ n ] .30 note: Thenotation ~ representscyclicconvolution"modN". 7.6.2.1StepsforCyclicConvolution Stepsforcyclicconvolutionarethesameastheusualconvo,exceptallindexcalculationsaredone"modN" ="onthewheel" StepsforCyclicConvolution Step1:"Plot" f [ m ] and h [ )]TJ/F11 9.9626 Tf 7.748 0 Td [(m N ]

PAGE 167

159 a b Figure7.23: Step1 Step2:"Spin" h [ )]TJ/F11 9.9626 Tf 7.749 0 Td [(m N ] n notchesACWcounter-clockwisetoget h [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m N ] i.e.Simply rotatethesequence, h [ n ] ,clockwiseby n steps. Figure7.24: Step2 Step3:Pointwisemultiplythe f [ m ] wheelandthe h [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(m N ] wheel. sum= y [ n ] Step4:Repeatforall 0 n N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 Example7.8:Convolven=4 a b Figure7.25: Twodiscrete-timesignalstobeconvolved.

PAGE 168

160 CHAPTER7.DISCRETEFOURIERTRANSFORM h [ )]TJ/F11 9.9626 Tf 7.748 0 Td [(m N ] Figure7.26 Multiply f [ m ] and sum toyield: y [0]=3 h [ )]TJ/F11 9.9626 Tf 9.962 0 Td [(m N ] Figure7.27 Multiply f [ m ] and sum toyield: y [1]=5 h [ )]TJ/F11 9.9626 Tf 9.962 0 Td [(m N ] Figure7.28 Multiply f [ m ] and sum toyield: y [2]=3 h [ )]TJ/F11 9.9626 Tf 9.962 0 Td [(m N ]

PAGE 169

161 Figure7.29 Multiply f [ m ] and sum toyield: y [3]=1 Example7.9 Thefollowingdemonstrationallowsyoutoexplorethisalgorithmforcircularconvolution.See here 12 forinstructionsonhowtousethedemo. Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10786/latest/DTCircularConvolution.llb 7.6.2.2AlternativeAlgorithm AlternativeCircularConvolutionAlgorithm Step1:CalculatetheDFTof f [ n ] whichyields F [ k ] andcalculatetheDFTof h [ n ] whichyields H [ k ] Step2:Pointwisemultiply Y [ k ]= F [ k ] H [ k ] Step3:InverseDFT Y [ k ] whichyields y [ n ] Seemslikearoundaboutwayofdoingthings, but itturnsoutthatthereare extremely fastwaysto calculatetheDFTofasequence. Tocircularilyconvolve 2 N -pointsequences: y [ n ]= N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X m =0 f [ m ] h [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m N ] Foreach n : N multiples, N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 additions N pointsimplies N 2 multiplications, N N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 additionsimplies O )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(N 2 complexity. 12 "HowtousetheLabVIEWdemos"

PAGE 170

162 CHAPTER7.DISCRETEFOURIERTRANSFORM SolutionstoExercisesinChapter7 SolutiontoExercise7.1p.150 Aliasing: b k = e j 2 N kn b k + N = e j 2 N k + N n = e j 2 N kn e j 2 n = e j 2 N n = b k .31 DTFScoecientsarealsoperiodicwithperiod N .

PAGE 171

Chapter8 FastFourierTransformFFT 8.1DFT:FastFourierTransform 1 Wenowhaveawayofcomputingthespectrumforanarbitrarysignal:TheDiscreteFourierTransform DFT 2 computesthespectrumat N equallyspacedfrequenciesfromalengthN sequence.Anissuethat neverarisesinanalog"computation,"likethatperformedbyacircuit,ishowmuchworkittakestoperform thesignalprocessingoperationsuchasltering.Incomputation,thisconsiderationtranslatestothenumber ofbasiccomputationalstepsrequiredtoperformtheneededprocessing.Thenumberofsteps,knownas the complexity ,becomesequivalenttohowlongthecomputationtakeshowlongmustwewaitforan answer.Complexityisnotsomuchtiedtospeciccomputersorprogramminglanguagesbuttohowmany stepsarerequiredonanycomputer.Thus,aprocedure'sstatedcomplexitysaysthatthetimetakenwillbe proportional tosomefunctionoftheamountofdatausedinthecomputationandtheamountdemanded. Forexample,considertheformulaforthediscreteFouriertransform.Foreachfrequencywechose,we mustmultiplyeachsignalvaluebyacomplexnumberandaddtogethertheresults.Forareal-valuedsignal, eachreal-times-complexmultiplicationrequirestworealmultiplications,meaningwehave 2 N multiplications toperform.Toaddtheresultstogether,wemustkeeptherealandimaginarypartsseparate.Adding N numbersrequires N )]TJ/F8 9.9626 Tf 10.272 0 Td [(1 additions.Consequently,eachfrequencyrequires 2 N +2 N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1=4 N )]TJ/F8 9.9626 Tf 10.272 0 Td [(2 basic computationalsteps.Aswehave N frequencies,thetotalnumberofcomputationsis N N )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 Incomplexitycalculations,weonlyworryaboutwhathappensasthedatalengthsincrease,andtakethe dominanttermherethe 4 N 2 termasreectinghowmuchworkisinvolvedinmakingthecomputation. Asmultiplicativeconstantsdon'tmattersincewearemakinga"proportionalto"evaluation,wendthe DFTisan O )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(N 2 computationalprocedure.Thisnotationisread"order N -squared".Thus,ifwedouble thelengthofthedata,wewouldexpectthatthecomputationtimetoapproximatelyquadruple. Exercise8.1 Solutiononp.168. InmakingthecomplexityevaluationfortheDFT,weassumedthedatatobereal.Threequestionsemerge.Firstofall,thespectraofsuchsignalshaveconjugatesymmetry,meaningthat negativefrequencycomponents k = N 2 +1 ;:::;N +1 intheDFT 3 canbecomputedfromthe correspondingpositivefrequencycomponents.DoesthissymmetrychangetheDFT'scomplexity? Secondly,supposethedataarecomplex-valued;whatistheDFT'scomplexitynow? Finally,alessimportantbutinterestingquestionissupposewewant K frequencyvaluesinstead of N ;nowwhatisthecomplexity? 1 Thiscontentisavailableonlineat. 2 "DiscreteFourierTransform",:DiscreteFouriertransform 3 "DiscreteFourierTransform",:DiscreteFouriertransform 163

PAGE 172

164 CHAPTER8.FASTFOURIERTRANSFORMFFT 8.2TheFastFourierTransformFFT 4 8.2.1Introduction TheFastFourierTransformFFTisanecientONlogNalgorithmforcalculatingDFTs originallydiscoveredbyGaussintheearly1800's rediscoveredbyCooleyandTukeyatIBMinthe1960's C.S.Burrus,RiceUniversity'sveryownDeanofEngineering,literally"wrotethebook"onfastDFT algorithms. TheFFT 5 exploitssymmetriesinthe W matrixtotakea"divideandconquer"approach.Wewon'ttalk abouttheactualFFTalgorithmhere,seethesenotes 6 ifyouareinterestedinreadingalittlemoreonthe ideabehindFFT. 8.2.2SpeedComparison HowmuchbetterisONlogNthanO N 2 ? Figure8.1: ThisgureshowshowmuchslowerthecomputationtimeofanONlogNprocessgrows. N 10 100 1000 10 6 10 9 N 2 100 10 4 10 6 10 12 10 18 N log N 1 200 3000 6 10 6 9 10 9 Table8.1 Sayyouhavea1MFLOPmachineamillion"oatingpoint"operationspersecond.Let N =1million= 10 6 AnO N 2 algorithmtakes 10 12 ors 10 6 seconds 11.5days. AnO N log N algorithmtakes 6 10 6 Flors 6seconds. note: N =1million is not unreasonable. Example8.1 3megapixeldigitalcameraspitsout 3 10 6 numbersforeachpicture.Sofortwo N pointsequences f [ n ] and h [ n ] .Ifcomputing f [ n ] ~ h [ n ] directly:O N 2 operations. 4 Thiscontentisavailableonlineat. 5 "FastFourierTransformFFT" 6 "FastFourierTransformFFT"

PAGE 173

165 takingFFTsONlogN multiplyingFFTsON inverseFFTsONlogN. thetotalcomplexityisONlogN. note: FFT+digitalcomputerwerealmostcompletelyresponsibleforthe"explosion"ofDSPin the60's. note: RicewasandstillisoneoftheplacestodoresearchinDSP. 8.3DerivingtheFastFourierTransform 7 ToderivetheFFT,weassumethatthesignal'sdurationisapoweroftwo: N =2 l .Considerwhathappens totheeven-numberedandodd-numberedelementsofthesequenceintheDFTcalculation. S k = s + s e )]TJ/F73 7.9701 Tf 6.587 0 Td [(j 2 2 k N + + s N )]TJ/F93 11.9552 Tf 11.955 0 Td [(2 e )]TJ/F73 7.9701 Tf 6.587 0 Td [(j 2 N )]TJ/F57 5.9776 Tf 5.756 0 Td [(2 k N + s e )]TJ/F73 7.9701 Tf 6.587 0 Td [(j 2 k N + s e )]TJ/F73 7.9701 Tf 6.586 0 Td [(j 2 +1 k N + + s N )]TJ/F93 11.9552 Tf 11.955 0 Td [(1 e )]TJ/F73 7.9701 Tf 6.586 0 Td [(j 2 N )]TJ/F57 5.9776 Tf 5.756 0 Td [(2+1 k N = s + s e )]TJ/F73 7.9701 Tf 6.587 0 Td [(j 2 k N 2 + + s N )]TJ/F93 11.9552 Tf 11.955 0 Td [(2 e )]TJ/F73 7.9701 Tf 6.586 0 Td [(j 2 N 2 )]TJ/F57 5.9776 Tf 5.756 0 Td [(1 k N 2 + 0 @ s + s e )]TJ/F73 7.9701 Tf 6.587 0 Td [(j 2 k N 2 + + s N )]TJ/F93 11.9552 Tf 11.956 0 Td [(1 e )]TJ/F73 7.9701 Tf 6.587 0 Td [(j 2 N 2 )]TJ/F57 5.9776 Tf 5.756 0 Td [(1 k N 2 1 A e )]TJ/F57 5.9776 Tf 5.756 0 Td [( j 2 k N .1 Eachterminsquarebracketshasthe form ofa N 2 -lengthDFT.TherstoneisaDFToftheevennumberedelements,andthesecondoftheodd-numberedelements.TherstDFTiscombinedwiththe secondmultipliedbythecomplexexponential e )]TJ/F6 4.9813 Tf 5.397 0 Td [( j 2 k N .Thehalf-lengthtransformsareeachevaluatedat frequencyindices k 2f 0 ;:::;N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 g .Normally,thenumberoffrequencyindicesinaDFTcalculationrange betweenzeroandthetransformlengthminusone.The computationaladvantage oftheFFTcomesfrom recognizingtheperiodicnatureofthediscreteFouriertransform.TheFFTsimplyreusesthecomputations madeinthehalf-lengthtransformsandcombinesthemthroughadditionsandthemultiplicationby e )]TJ/F6 4.9813 Tf 5.397 0 Td [( j 2 k N ,whichisnotperiodicover N 2 ,torewritethelength-NDFT.Figure8.2Length-8DFTdecomposition illustratesthisdecomposition.Asitstands,wenowcomputetwolengthN 2 transformscomplexity 2 O N 2 4 ,multiplyoneofthembythecomplexexponentialcomplexity O N ,andaddtheresultscomplexity O N .Atthispoint,thetotalcomplexityisstilldominatedbythehalf-lengthDFTcalculations,butthe proportionalitycoecienthasbeenreduced. Nowforthefun.Because N =2 l ,eachofthehalf-lengthtransformscanbereducedtotwoquarter-length transforms,eachofthesetotwoeighth-lengthones,etc.Thisdecompositioncontinuesuntilweareleftwith length-2transforms.Thistransformisquitesimple,involvingonlyadditions.Thus,therststageofthe FFThas N 2 length-2transformsseethebottompartofFigure8.2Length-8DFTdecomposition.Pairs ofthesetransformsarecombinedbyaddingonetotheothermultipliedbyacomplexexponential.Eachpair requires4additionsand4multiplications,givingatotalnumberofcomputationsequaling 8 N 4 = N 2 .This numberofcomputationsdoesnotchangefromstagetostage.Becausethenumberofstages,thenumberof timesthelengthcanbedividedbytwo,equals log 2 N ,thecomplexityoftheFFTis O N log N 7 Thiscontentisavailableonlineat.

PAGE 174

166 CHAPTER8.FASTFOURIERTRANSFORMFFT Length-8DFTdecomposition a b Figure8.2: Theinitialdecompositionofalength-8DFTintothetermsusingeven-andodd-indexed inputsmarkstherstphaseofdevelopingtheFFTalgorithm.Whenthesehalf-lengthtransformsare successivelydecomposed,weareleftwiththediagramshowninthebottompanelthatdepictsthe length-8FFTcomputation. Doinganexamplewillmakecomputationalsavingsmoreobvious.Let'slookatthedetailsofalength-8 DFT.AsshownonFigure8.2Length-8DFTdecomposition,werstdecomposetheDFTintotwolength4DFTs,withtheoutputsaddedandsubtractedtogetherinpairs.ConsideringFigure8.2Length-8DFT decompositionasthefrequencyindexgoesfrom0through7,werecyclevaluesfromthelength-4DFTs intothenalcalculationbecauseoftheperiodicityoftheDFToutput.Examininghowpairsofoutputsare collectedtogether,wecreatethebasiccomputationalelementknownasa buttery Figure8.3Buttery.

PAGE 175

167 Buttery Figure8.3: ThebasiccomputationalelementofthefastFouriertransformisthebuttery.Ittakes twocomplexnumbers,representedby a and b ,andformsthequantitiesshown.Eachbutteryrequires onecomplexmultiplicationandtwocomplexadditions. Byconsideringtogetherthecomputationsinvolvingcommonoutputfrequenciesfromthetwohalf-length DFTs,weseethatthetwocomplexmultipliesarerelatedtoeachother,andwecanreduceourcomputational workevenfurther.Byfurtherdecomposingthelength-4DFTsintotwolength-2DFTsandcombiningtheir outputs,wearriveatthediagramsummarizingthelength-8fastFouriertransformFigure8.2Length-8 DFTdecomposition.Althoughmostofthecomplexmultipliesarequitesimplemultiplyingby e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j meansnegatingrealandimaginaryparts,let'scountthoseforpurposesofevaluatingthecomplexityasfull complexmultiplies.Wehave N 2 =4 complexmultipliesand 2 N =16 additionsforeachstageand log 2 N =3 stages,makingthenumberofbasiccomputations 3 N 2 log 2 N aspredicted. Exercise8.2 Solutiononp.168. NotethattheorderingoftheinputsequenceinthetwopartsofFigure8.2Length-8DFT decompositionaren'tquitethesame.Whynot?Howistheorderingdetermined? Other"fast"algorithmswerediscovered,allofwhichmakeuseofhowmanycommonfactorsthetransform lengthNhas.Innumbertheory,thenumberofprimefactorsagivenintegerhasmeasureshow composite itis.Thenumbers16and81arehighlycompositeequaling 2 4 and 3 4 respectively,thenumber18islessso 2 1 3 2 ,and17notatallit'sprime.InoverthirtyyearsofFouriertransformalgorithmdevelopment,the originalCooley-Tukeyalgorithmisfarandawaythemostfrequentlyused.Itissocomputationallyecient thatpower-of-twotransformlengthsarefrequentlyusedregardlessofwhattheactuallengthofthedata.

PAGE 176

168 CHAPTER8.FASTFOURIERTRANSFORMFFT SolutionstoExercisesinChapter8 SolutiontoExercise8.1p.163 Whenthesignalisreal-valued,wemayonlyneedhalfthespectralvalues,butthecomplexityremains unchanged.Ifthedataarecomplex-valued,whichdemandsretainingallfrequencyvalues,thecomplexityis againthesame.Whenonly K frequenciesareneeded,thecomplexityis O KN SolutiontoExercise8.2p.167 TheupperpanelhasnotusedtheFFTalgorithmtocomputethelength-4DFTswhiletheloweronehas. Theorderingisdeterminedbythealgorithm.

PAGE 177

Chapter9 Convergence 9.1ConvergenceofSequences 1 9.1.1WhatisaSequence? Denition9.1:sequence Asequenceisafunction g n denedonthepositiveintegers' n '.Weoftendenoteasequenceby f g n gj 1 n =1 Example Arealnumbersequence: g n = 1 n Example Avectorsequence: g n = 0 @ sin )]TJ/F10 6.9738 Tf 5.761 -4.147 Td [(n 2 cos )]TJ/F10 6.9738 Tf 5.762 -4.148 Td [(n 2 1 A Example Afunctionsequence: g n t = 8 < : 1 if 0 t< 1 n 0 otherwise note: Afunctioncanbethoughtofasaninnitedimensionalvectorwhereforeachvalueof' t wehaveonedimension 9.1.2ConvergenceofRealSequences Denition9.2:limit Asequence f g n gj 1 n =1 convergestoalimit g 2 R iffor every > 0 thereisaninteger N suchthat j g i )]TJ/F11 9.9626 Tf 9.963 0 Td [(g j <;i N 1 Thiscontentisavailableonlineat. 169

PAGE 178

170 CHAPTER9.CONVERGENCE Weusuallydenotealimitbywriting lim i !1 g i = g or g i g Theabovedenitionmeansthatnomatterhowsmallwemake ,exceptforanitenumberof g i 's,all pointsofthesequencearewithindistance of g Example9.1 Wearegiventhefollowingconvergentsequence: g n = 1 n .1 Intuitivelywecanassumethefollowinglimit: lim n !1 g n =0 Letusprovethisrigorously.Saythatwearegivenarealnumber > 0 .Letuschoose N = d 1 e where d x e denotesthesmallestintegerlargerthan x .Thenfor n N wehave j g n )]TJ/F8 9.9626 Tf 9.963 0 Td [(0 j = 1 n 1 N < Thus, lim n !1 g n =0 Example9.2 Nowletuslookatthefollowingnon-convergentsequence g n = 8 < : 1 if n =even )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 if n =odd Thissequenceoscillatesbetween1and-1,soitwillthereforeneverconverge. 9.1.2.1Problems Forpractice,saywhichofthefollowingsequencesconvergeandgivetheirlimitsiftheyexist. 1. g n = n 2. g n = 8 < : 1 n if n =even )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 n if n =odd 3. g n = 8 < : 1 n if n 6 =powerof10 1 otherwise 4. g n = 8 < : n if n< 10 5 1 n if n 10 5 5. g n =sin )]TJ/F10 6.9738 Tf 5.77 -4.148 Td [( n 6. g n = j n

PAGE 179

171 9.2ConvergenceofVectors 2 9.2.1ConvergenceofVectors Wenowdiscusspointwiseandnormconvergenceofvectors.Othertypesofconvergencealsoexist,andone inparticular,uniformconvergenceSection9.3,canalsobestudied.Forthisdiscussion,wewillassume thatthevectorsbelongtoanormedvectorspaceSection15.2. 9.2.1.1PointwiseConvergence AsequenceSection9.1 f g n gj 1 n =1 converges pointwise tothelimit g ifeachelementof g n convergesto thecorrespondingelementin g .Belowarefewexamplestotryandhelpillustratethisidea. Example9.3 g n = 0 @ g n [1] g n [2] 1 A = 0 @ 1+ 1 n 2 )]TJ/F7 6.9738 Tf 11.635 3.922 Td [(1 n 1 A Firstwendthefollowinglimitsforourtwo g n 's: lim n !1 g n [1]=1 lim n !1 g n [2]=2 Thereforewehavethefollowing, lim n !1 g n = g pointwise,where g = 0 @ 1 2 1 A Example9.4 g n t = t n ;t 2 R Asdoneabove,werstwanttoexaminethelimit lim n !1 g n t 0 = lim n !1 t 0 n =0 where t 0 2 R .Thus lim n !1 g n = g pointwisewhere g t =0 forall t 2 R 9.2.1.2NormConvergence ThesequenceSection9.1 f g n gj 1 n =1 convergesto g innormif lim n !1 k g n )]TJ/F11 9.9626 Tf 10.125 0 Td [(g k =0 .Here kk isthenorm Section15.2ofthecorrespondingvectorspaceof g n 's.Intuitivelythismeansthedistancebetweenvectors g n and g decreasesto 0 Example9.5 g n = 0 @ 1+ 1 n 2 )]TJ/F7 6.9738 Tf 11.635 3.922 Td [(1 n 1 A 2 Thiscontentisavailableonlineat.

PAGE 180

172 CHAPTER9.CONVERGENCE Let g = 0 @ 1 2 1 A k g n )]TJ/F83 9.9626 Tf 9.963 0 Td [(g k = q )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ 1 n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 2 + )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 )]TJ/F7 6.9738 Tf 11.635 3.923 Td [(1 n 2 = q 1 n 2 + 1 n 2 = p 2 n .2 Thus lim n !1 k g n )]TJ/F83 9.9626 Tf 9.963 0 Td [(g k =0 Therefore, g n g innorm. Example9.6 g n t = 8 < : t n if 0 t 1 0 otherwise Let g t =0 forall t k g n t )]TJ/F11 9.9626 Tf 9.963 0 Td [(g t k = R 1 0 t 2 n 2 dt = t 3 3 n 2 j 1 n =0 = 1 3 n 2 .3 Thus lim n !1 k g n t )]TJ/F11 9.9626 Tf 9.963 0 Td [(g t k =0 Therefore, g n t g t innorm. 9.2.2Pointwisevs.NormConvergence Theorem9.1: For R m ,pointwiseandnormconvergenceareequivalent. Proof: Pointwise Norm g n [ i ] g [ i ] Assumingtheabove,then k g n )]TJ/F83 9.9626 Tf 9.962 0 Td [(g k 2 = m X i =1 g n [ i ] )]TJ/F11 9.9626 Tf 9.963 0 Td [(g [ i ] 2 Thus, lim n !1 k g n )]TJ/F83 9.9626 Tf 9.962 0 Td [(g k 2 = lim n !1 P m i =1 2 = P m i =1 lim n !1 2 =0 .4 Proof: Norm Pointwise k g n )]TJ/F83 9.9626 Tf 9.962 0 Td [(g k! 0 lim n !1 P m i =1 2= P m i =1 lim n !1 2 =0 .5 Sinceeachtermisgreaterthanorequalzero,all' m 'termsmustbezero.Thus, lim n !1 2=0

PAGE 181

173 forall i .Therefore, g n g pointwise note: Ininnitedimensionalspacestheabovetheoremisnolongertrue.Weprovethiswith counterexamplesshownbelow. 9.2.2.1CounterExamples Example9.7:Pointwise [U+21CF] Norm Wearegiventhefollowingfunction: g n t = 8 < : n if 0
PAGE 182

174 CHAPTER9.CONVERGENCE 9.2.2.2Problems Proveifthefollowingsequencesarepointwiseconvergent,normconvergent,orbothandthenstatetheir limits. 1. g n t = 8 < : 1 nt if 0 0 thereis aninteger N suchthat n N implies j g n t )]TJ/F11 9.9626 Tf 9.962 0 Td [(g t j .7 for all t 2 R ObviouslyeveryuniformlyconvergentsequenceispointwiseSection9.2convergent.Thedierence betweenpointwiseanduniformconvergenceisthis:If f g n g convergespointwiseto g ,thenforevery > 0 andforevery t 2 R thereisaninteger N dependingon and t suchthat.7holdsif n N .If f g n g convergesuniformlyto g ,itispossibleforeach > 0 tond one integer N thatwilldoforall t 2 R Example9.9 g n t = 1 n ;t 2 R Let > 0 begiven.Thenchoose N = d 1 e .Obviously, j g n t )]TJ/F8 9.9626 Tf 9.963 0 Td [(0 j ;n N forall t .Thus, g n t convergesuniformlyto 0 Example9.10 g n t = t n ;t 2 R Obviouslyforany > 0 wecannotndasinglefunction g n t forwhich.7holdswith g t =0 forall t .Thus g n isnotuniformlyconvergent.Howeverwedohave: g n t g t pointwise note: Uniformconvergencealwaysimpliespointwiseconvergence,butpointwiseconvergencedoes notguaranteeuniformconvergence. 3 Thiscontentisavailableonlineat.

PAGE 183

175 9.3.1.1Problems Rigorouslyproveifthefollowingfunctionsconvergepointwise,uniformly,orboth. 1. g n t = sin t n 2. g n t = e t n 3. g n t = 8 < : 1 nt if t> 0 0 if t 0

PAGE 184

176 CHAPTER9.CONVERGENCE

PAGE 185

Chapter10 DiscreteTimeFourierTransform DTFT 10.1DiscreteFourierTransformation 1 10.1.1N-pointDiscreteFourierTransformDFT X [ k ]= N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X n =0 x [ n ] e )]TJ/F10 6.9738 Tf 6.227 0 Td [(j 2 n kn ;k = f 0 ;:::;N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 g .1 x [ n ]= 1 N N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 X [ k ] e j 2 n kn ;n = f 0 ;:::;N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 g .2 Notethat: X [ k ] istheDTFTevaluatedat = 2 N k;k = f 0 ;:::;N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 g Zero-padding x [ n ] to M samplespriortotheDFTyieldsan M -pointuniformsampledversionofthe DTFT: X e j 2 M k = N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X n =0 x [ n ] e )]TJ/F10 6.9738 Tf 6.227 0 Td [(j 2 M k .3 X e j 2 M k = N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X n =0 x zp [ n ] e )]TJ/F10 6.9738 Tf 6.226 0 Td [(j 2 M k X e j 2 M k = X zp [ k ] ;k = f 0 ;:::;M )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 g The N -ptDFTissucienttoreconstructtheentireDTFTofan N -ptsequence: X )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j! = N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X n =0 x [ n ] e )]TJ/F10 6.9738 Tf 6.227 0 Td [(j !n .4 X )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j! = N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X n =0 1 N N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 X [ k ] e j 2 N kn e )]TJ/F10 6.9738 Tf 6.227 0 Td [(j !n 1 Thiscontentisavailableonlineat. 177

PAGE 186

178 CHAPTER10.DISCRETETIMEFOURIERTRANSFORMDTFT X )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j! = N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 X [ k ] 1 N N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 e )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F6 4.9813 Tf 7.422 2.677 Td [(2 N k n X )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j! = N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k =0 X [ k ] 1 N sin )]TJ/F10 6.9738 Tf 5.762 -4.147 Td [(!N )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 k 2 sin )]TJ/F10 6.9738 Tf 5.762 -4.147 Td [(!N )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 k 2 N e )]TJ/F10 6.9738 Tf 6.226 0 Td [(j )]TJ/F6 4.9813 Tf 7.422 2.678 Td [(2 N k N )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 2 Figure10.1: Dirichletsinc, 1 N sin !N 2 sin 2 TheDFThasaconvenientmatrixrepresentation.Dening W N = e )]TJ/F10 6.9738 Tf 6.226 0 Td [(j 2 N 0 B B B B B @ X [0] X [1] . X [ N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1] 1 C C C C C A = 0 B B B B B @ W 0 N W 0 N W 0 N W 0 N ::: W 0 N W 1 N W 2 N W 3 N ::: W 0 N W 2 N W 4 N W 6 N ::: . . . . . . . 1 C C C C C A 0 B B B B B @ x [0] x [1] . x [ N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1] 1 C C C C C A .5 where X = W x respectively. W hasthefollowingproperties: W isVandermonde:the n thcolumnof W isapolynomialin W n N W issymmetric: W = W T 1 p N W isunitary: 1 p N W 1 p N W H = 1 p N W H 1 p N W = I 1 N W = W )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ,theIDFTmatrix. For N apowerof2,theFFTcanbeusedtocomputetheDFTusingabout N 2 log 2 N ratherthan N 2 operations. N N 2 log 2 N N 2 16 32 256 64 192 4096 256 1024 65536 1024 5120 1048576

PAGE 187

179 Table10.1 10.2DiscreteFourierTransformDFT 2 Thediscrete-timeFouriertransformandthecontinuous-timetransformaswellcanbeevaluatedwhenwe haveananalyticexpressionforthesignal.Supposewejusthaveasignal,suchasthespeechsignalused inthepreviouschapter,forwhichthereisnoformula.Howthenwouldyoucomputethespectrum?For example,howdidwecomputeaspectrogramsuchastheoneshowninthespeechsignalexample 3 ?The DiscreteFourierTransformDFTallowsthecomputationofspectrafromdiscrete-timedata.Whilein discrete-timewecan exactly calculatespectra,foranalogsignalsnosimilarexactspectrumcomputation exists.Foranalog-signalspectra,usemustbuildspecialdevices,whichturnoutinmostcasestoconsistof A/Dconvertersanddiscrete-timecomputations.Certainlydiscrete-timespectralanalysisismoreexible thancontinuous-timespectralanalysis. TheformulafortheDTFT 4 isasum,whichconceptuallycanbeeasilycomputedsavefortwoissues. Signalduration .Thesumextendsoverthesignal'sduration,whichmustbenitetocomputethe signal'sspectrum.Itisexceedinglydiculttostoreaninnite-lengthsignalinanycase,sowe'll assumethatthesignalextendsover [0 ;N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1] Continuousfrequency .Subtlerthanthesignaldurationissueisthefactthatthefrequencyvariable iscontinuous:Itmayonlyneedtospanoneperiod,like )]TJ/F1 9.9626 Tf 9.409 8.069 Td [()]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(1 2 ; 1 2 or [0 ; 1] ,buttheDTFTformulaas itstandsrequiresevaluatingthespectraat all frequencieswithinaperiod.Let'scomputethespectrum atafewfrequencies;themostobviousonesaretheequallyspacedones f = k K k 2f 0 ;:::;K )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 g Wethusdenethe discreteFouriertransform DFTtobe S k = N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X n =0 s n e )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( j 2 nk K ;k 2f 0 ;:::;K )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 g .6 Here, S k isshorthandfor S e j 2 k K Wecancomputethespectrumatasmanyequallyspacedfrequenciesaswelike.Notethatyoucanthink aboutthiscomputationallymotivatedchoiceas sampling thespectrum;moreaboutthisinterpretationlater. Theissuenowishowmanyfrequenciesareenoughtocapturehowthespectrumchangeswithfrequency. OnewayofansweringthisquestionisdetermininganinversediscreteFouriertransformformula:given S k k = f 0 ;:::;K )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 g howdowend s n n = f 0 ;:::;N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 g ?Presumably,theformulawillbeoftheform s n = P K )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 k =0 S k e j 2 nk K .SubstitutingtheDFTformulainthisprototypeinversetransformyields s n = K )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X k =0 N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X m =0 s m e )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( j 2 mk K e j 2 nk K .7 Notethattheorthogonalityrelationweusesooftenhasadierentcharacternow. K )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X k =0 e )]TJ/F8 9.9626 Tf 6.226 -0.748 Td [( j 2 km K e j 2 kn K = 8 < : K if m = f n; n K ; n 2 K ;::: g 0 otherwise .8 2 Thiscontentisavailableonlineat. 3 "ModelingtheSpeechSignal",Figure5:spectrogram 4 "Discrete-TimeFourierTransformDTFT",

PAGE 188

180 CHAPTER10.DISCRETETIMEFOURIERTRANSFORMDTFT Weobtainnonzerovaluewheneverthetwoindicesdierbymultiplesof K .Wecanexpressthisresultas K P l m )]TJ/F11 9.9626 Tf 9.963 0 Td [(n )]TJ/F11 9.9626 Tf 9.962 0 Td [(lK .Thus,ourformulabecomes s n = N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X m =0 s m K 1 X l = m )]TJ/F11 9.9626 Tf 9.963 0 Td [(n )]TJ/F11 9.9626 Tf 9.962 0 Td [(lK .9 Theintegers n and m bothrangeover f 0 ;:::;N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 g .Tohaveaninversetransform,weneedthesum tobea single unitsamplefor m n inthisrange.Ifitdidnot,then s n wouldequalasumofvalues, andwewouldnothaveavalidtransform:Oncegoingintothefrequencydomain,wecouldnotgetback unambiguously!Clearly,theterm l =0 alwaysprovidesaunitsamplewe'lltakecareofthefactorof K soon.Ifweevaluatethespectrumat fewer frequenciesthanthesignal'sduration,thetermcorresponding to m = n + K willalsoappearforsomevaluesof m n = f 0 ;:::;N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 g .Thissituationmeansthatour prototypetransformequals s n + s n + K forsomevaluesof n .Theonlywaytoeliminatethisproblem istorequire K N :We must haveatleastasmanyfrequencysamplesasthesignal'sduration.Inthis way,wecanreturnfromthefrequencydomainweenteredviatheDFT. Exercise10.1 Solutiononp.188. Whenwehavefewerfrequencysamplesthanthesignal'sduration,somediscrete-timesignalvalues equalthesumoftheoriginalsignalvalues.Giventhesamplinginterpretationofthespectrum, characterizethiseectadierentway. Anotherwaytounderstandthisrequirementistousethetheoryoflinearequations.Ifwewriteoutthe expressionfortheDFTasasetoflinearequations, s + s + + s N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1= S .10 s + s e )]TJ/F10 6.9738 Tf 6.227 0 Td [(j 2 K + + s N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 e )]TJ/F10 6.9738 Tf 6.227 0 Td [(j 2 N )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 K = S . s + s e )]TJ/F10 6.9738 Tf 6.227 0 Td [(j 2 K )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 K + + s N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 e )]TJ/F10 6.9738 Tf 6.227 0 Td [(j 2 N )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 K )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 K = S K )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 wehave K equationsin N unknownsifwewanttondthesignalfromitssampledspectrum.Thisrequirementisimpossibletofulllif K
PAGE 189

181 Example10.2 UsethisdemonstrationtosynthesizeasignalfromaDFTsequence. Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10249/latest/DFT_Component_Manipulation.llb 10.3TableofCommonFourierTransforms 5 TimeDomainSignal FrequencyDomainSignal Condition e )]TJ/F7 6.9738 Tf 6.227 0 Td [( at u t 1 a + j! a> 0 e at u )]TJ/F11 9.9626 Tf 7.749 0 Td [(t 1 a )]TJ/F10 6.9738 Tf 6.227 0 Td [(j! a> 0 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( a j t j 2 a a 2 + 2 a> 0 te )]TJ/F7 6.9738 Tf 6.226 0 Td [( at u t 1 a + j! 2 a> 0 t n e )]TJ/F7 6.9738 Tf 6.226 0 Td [( at u t n a + j! n +1 a> 0 t 1 1 2 e j! 0 t 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(! 0 cos 0 t )]TJ/F11 9.9626 Tf 9.962 0 Td [(! 0 + + 0 sin 0 t j + 0 )]TJ/F11 9.9626 Tf 9.963 0 Td [( )]TJ/F11 9.9626 Tf 9.963 0 Td [(! 0 u t + 1 j! sgn t 2 j! cos 0 t u t 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(! 0 + + 0 + j! 0 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(! 2 sin 0 t u t 2 j )]TJ/F11 9.9626 Tf 9.963 0 Td [(! 0 )]TJ/F11 9.9626 Tf 9.962 0 Td [( + 0 + 0 0 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(! 2 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( at sin 0 t u t 0 a + j! 2 + 0 2 a> 0 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( at cos 0 t u t a + j! a + j! 2 + 0 2 a> 0 u t + )]TJ/F11 9.9626 Tf 9.963 0 Td [(u t )]TJ/F11 9.9626 Tf 9.963 0 Td [( 2 sin ! =2 sinc !t continuedonnextpage 5 Thiscontentisavailableonlineat.

PAGE 190

182 CHAPTER10.DISCRETETIMEFOURIERTRANSFORMDTFT 0 sin 0 t 0 t = 0 sinc 0 u + 0 )]TJ/F11 9.9626 Tf 9.963 0 Td [(u )]TJ/F11 9.9626 Tf 9.963 0 Td [(! 0 )]TJ/F10 6.9738 Tf 6.497 -4.148 Td [(t +1 )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(u )]TJ/F10 6.9738 Tf 6.497 -4.148 Td [(t +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(u )]TJ/F10 6.9738 Tf 6.497 -4.148 Td [(t + )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F10 6.9738 Tf 6.497 -4.147 Td [(t +1 )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(u )]TJ/F10 6.9738 Tf 6.497 -4.147 Td [(t )]TJ/F11 9.9626 Tf 9.962 0 Td [(u )]TJ/F10 6.9738 Tf 6.497 -4.147 Td [(t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = triag )]TJ/F10 6.9738 Tf 8.483 -4.147 Td [(t 2 sinc 2 )]TJ/F10 6.9738 Tf 5.761 -4.148 Td [(! 2 0 2 sinc 2 )]TJ/F10 6.9738 Tf 5.762 -3.985 Td [(! 0 t 2 ! 0 +1 u ! 0 +1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(u ! 0 + )]TJ/F1 9.9626 Tf 9.409 11.059 Td [( ! 0 +1 u ! 0 )]TJ/F11 9.9626 Tf 9.963 0 Td [(u ! 0 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = triag 2 0 P 1 n = t )]TJ/F11 9.9626 Tf 9.963 0 Td [(nT 0 P 1 n = )]TJ/F11 9.9626 Tf 9.962 0 Td [(n! 0 0 = 2 T e )]TJ/F27 6.9738 Tf 6.227 7.681 Td [( t 2 2 2 p 2 e )]TJ/F27 6.9738 Tf 6.227 7.681 Td [( 2 2 2 Table10.2 10.4Discrete-TimeFourierTransformDTFT 6 Discrete-TimeFourierTransform X = 1 X n = x n e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j!n .12 InverseDiscrete-TimeFourierTransform x n = 1 2 Z 2 0 X e j!n d! .13 10.4.1RelevantSpaces TheDiscrete-TimeFourierTransform 7 mapsinnite-length,discrete-timesignalsin l 2 tonite-lengthor 2 -periodic,continuous-frequencysignalsin L 2 Figure10.2: Mapping l 2 Z inthetimedomainto L 2 [0 ; 2 inthefrequencydomain. 6 Thiscontentisavailableonlineat. 7 "Discrete-TimeFourierTransformDTFT"

PAGE 191

183 10.5Discrete-TimeFourierTransformProperties 8 Discrete-TimeFourierTransformProperties SequenceDomain FrequencyDomain Linearity a 1 s 1 n + a 2 s 2 n a 1 S 1 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 f + a 2 S 2 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(e j 2 f ConjugateSymmetry s n real S )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 f = S )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j 2 f EvenSymmetry s n = s )]TJ/F11 9.9626 Tf 7.748 0 Td [(n S )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(e j 2 f = S )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j 2 f OddSymmetry s n = )]TJ/F8 9.9626 Tf 9.409 0 Td [( s )]TJ/F11 9.9626 Tf 7.749 0 Td [(n S )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(e j 2 f = )]TJ/F1 9.9626 Tf 9.409 8.069 Td [()]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(S )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j 2 f TimeDelay s n )]TJ/F11 9.9626 Tf 9.963 0 Td [(n 0 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j 2 fn 0 S )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 f ComplexModulation e j 2 f 0 n s n S )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 f )]TJ/F10 6.9738 Tf 6.227 0 Td [(f 0 AmplitudeModulation s n cos f 0 n S e j 2 f )]TJ/F9 4.9813 Tf 5.396 0 Td [(f 0 + S e j 2 f + f 0 2 s n sin f 0 n S e j 2 f )]TJ/F9 4.9813 Tf 5.396 0 Td [(f 0 )]TJ/F10 6.9738 Tf 6.226 0 Td [(S e j 2 f + f 0 2 j Multiplicationbyn ns n 1 )]TJ/F7 6.9738 Tf 6.226 0 Td [( j d df )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(S )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 f Sum P 1 n = s n S )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 0 ValueatOrigin s R 1 2 )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( 1 2 S )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 f df Parseval'sTheorem P 1 n = j s n j 2 R 1 2 )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( 1 2 )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [(j S )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 f j 2 df Figure10.3: Discrete-timeFouriertransformpropertiesandrelations. 10.6Discrete-TimeFourierTransformPair 9 Whenweobtainthediscrete-timesignalviasamplingananalogsignal,theNyquistfrequencycorresponds tothediscrete-timefrequency 1 2 .Toshowthis,notethatasinusoidattheNyquistfrequency 1 2 T s hasa sampledwaveformthatequals SinusoidatNyquistFrequency1/2T cos 2 1 2 T s nT s =cos n = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 n .14 TheexponentialintheDTFTatfrequency 1 2 equals e )]TJ/F6 4.9813 Tf 5.397 0 Td [( j 2 n 2 = e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jn = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 n ,meaningthatthe correspondencebetweenanaloganddiscrete-timefrequencyisestablished: Analog,Discrete-TimeFrequencyRelationship f D = f A T s .15 8 Thiscontentisavailableonlineat. 9 Thiscontentisavailableonlineat.

PAGE 192

184 CHAPTER10.DISCRETETIMEFOURIERTRANSFORMDTFT where f D and f A representdiscrete-timeandanalogfrequencyvariables,respectively.Thealiasing gure 10 providesanotherwayofderivingthisresult.Asthedurationofeachpulseintheperiodicsampling signal p T s t narrows,theamplitudesofthesignal'sspectralrepetitions,whicharegovernedbytheFourier seriescoecientsof p T s t ,becomeincreasinglyequal. 11 Thus,thesampledsignal'sspectrumbecomes periodicwithperiod 1 T s .Thus,theNyquistfrequency 1 2 T s correspondstothefrequency 1 2 Theinversediscrete-timeFouriertransformiseasilyderivedfromthefollowingrelationship: Z 1 2 )]TJ/F8 9.9626 Tf 6.226 -0.747 Td [( 1 2 e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j 2 fm e + jfn df = 8 < : 1 if m = n 0 if m 6 = n .16 Therefore,wendthat R 1 2 )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( 1 2 S )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 f e + j 2 fn df = R 1 2 )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( 1 2 P m )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(s m e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j 2 fm e + j 2 fn df = P m s m R 1 2 )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( 1 2 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j 2 f m )]TJ/F10 6.9738 Tf 6.226 0 Td [(n df = s n .17 TheFouriertransformpairsindiscrete-timeare FourierTransformPairsinDiscreteTime S )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 f = X n s n e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j 2 fn .18 FourierTransformPairsinDiscreteTime s n = Z 1 2 )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( 1 2 S )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 f e + j 2 fn df .19 10.7DTFTExamples 12 Example10.3 Let'scomputethediscrete-timeFouriertransformoftheexponentiallydecayingsequence s n = a n u n ,where u n istheunit-stepsequence.Simplypluggingthesignal'sexpressionintothe Fouriertransformformula, FourierTransformFormula S )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(e j 2 f = P 1 n = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a n u n e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j 2 fn = P 1 n =0 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(ae )]TJ/F7 6.9738 Tf 6.227 0 Td [( j 2 f n .20 Thissumisaspecialcaseofthe geometricseries 10 "TheSamplingTheorem",Figure2:aliasing 11 Examinationoftheperiodicpulsesignalrevealsthatas decreases,thevalueof c 0 ,thelargestFouriercoecient,decreases tozero: j c 0 j = A T .Thus,tomaintainamathematicallyviableSamplingTheorem,theamplitude A mustincreaseas 1 becominginnitelylargeasthepulsedurationdecreases.Practicalsystemsuseasmallvalueof ,say 0 : 1 T s anduseampliers torescalethesignal. 12 Thiscontentisavailableonlineat.

PAGE 193

185 GeometricSeries 1 X n =0 n = 1 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( ; j j < 1 .21 Thus,aslongas j a j < 1 ,wehaveourFouriertransform. S )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(e j 2 f = 1 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(ae )]TJ/F7 6.9738 Tf 6.227 0 Td [( j 2 f .22 UsingEuler'srelation,wecanexpressthemagnitudeandphaseofthisspectrum. j S )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 f j = 1 q )]TJ/F11 9.9626 Tf 9.963 0 Td [(a cos f 2 + a 2 sin 2 f .23 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(S )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 f = )]TJ/F1 9.9626 Tf 9.409 14.048 Td [( arctan a sin f 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(a cos f .24 Nomatterwhatvalueof a wechoose,theaboveformulaeclearlydemonstratetheperiodic natureofthespectraofdiscrete-timesignals.Figure10.4showsindeedthatthespectrumisa periodicfunction.Weneedonlyconsiderthespectrumbetween )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(1 2 and 1 2 tounambiguously deneit.When a> 0 ,wehavealowpassspectrumthespectrumdiminishesasfrequency increasesfrom 0 to 1 2 withincreasing a leadingtoagreaterlowfrequencycontent;for a< 0 wehaveahighpassspectrumFigure10.5. Figure10.4: Thespectrumoftheexponentialsignal a =0 : 5 isshownoverthefrequencyrange [ )]TJ/F56 8.9664 Tf 7.168 0 Td [(2 ; 2] ,clearlydemonstratingtheperiodicityofalldiscrete-timespectra.Theanglehasunitsofdegrees.

PAGE 194

186 CHAPTER10.DISCRETETIMEFOURIERTRANSFORMDTFT Figure10.5: Thespectraofseveralexponentialsignalsareshown.Whatistheapparentrelationship betweenthespectrafor a =0 : 5 and a = )]TJ/F56 8.9664 Tf 7.167 0 Td [(0 : 5 ? Example10.4 Analogoustotheanalogpulsesignal,let'sndthespectrumofthelengthN pulsesequence. s n = 8 < : 1 if 0 n N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 0 otherwise .25 TheFouriertransformofthissequencehastheformofatruncatedgeometricseries. S )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 f = N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X n =0 e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j 2 fn .26 Fortheso-callednitegeometricseries,weknowthat FiniteGeometricSeries N + n 0 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X n = n 0 n = n 0 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( N 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( .27 for all valuesof Exercise10.2 Solutiononp.188. Derivethisformulaforthenitegeometricseriessum.The"trick"istoconsiderthedierence betweentheseries';sumandthesumoftheseriesmultipliedby ApplyingthisresultyieldsFigure10.6. S )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(e j 2 f = 1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(e )]TJ/F6 4.9813 Tf 5.396 0 Td [( j 2 fN 1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(e )]TJ/F6 4.9813 Tf 5.396 0 Td [( j 2 f = e )]TJ/F7 6.9738 Tf 6.226 0 Td [( jf N )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 sin fN sin f .28

PAGE 195

187 Theratioofsinefunctionshasthegenericformof sin Nx sin x ,whichisknownasthe discrete-timesinc function dsinc x .Thus,ourtransformcanbeconciselyexpressedas S )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(e j 2 f = e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jf N )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 dsinc f .Thediscrete-timepulse'sspectrumcontainsmanyripples,thenumberofwhichincreasewith N ,thepulse's duration. Figure10.6: Thespectrumofalength-tenpulseisshown.Canyouexplaintherathercomplicated appearanceofthephase?

PAGE 196

188 CHAPTER10.DISCRETETIMEFOURIERTRANSFORMDTFT SolutionstoExercisesinChapter10 SolutiontoExercise10.1p.180 Thissituationamountstoaliasinginthetime-domain. SolutiontoExercise10.2p.186 N + n 0 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X n = n 0 n )]TJ/F10 6.9738 Tf 9.962 12.564 Td [(N + n 0 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X n = n 0 n = N + n 0 )]TJ/F11 9.9626 Tf 9.963 0 Td [( n 0 .29 which,aftermanipulation,yieldsthegeometricsumformula.

PAGE 197

Chapter11 ContinuousTimeFourierTransform CTFT 11.1Continuous-TimeFourierTransformCTFT 1 11.1.1Introduction Duetothelargenumberofcontinuous-timesignalsthatarepresent,theFourierseries 2 providedusthe rstglimpseofhowmewemayrepresentsomeofthesesignalsinageneralmanner:asasuperpositionofa numberofsinusoids.Now,wecanlookatawaytorepresentcontinuous-timenonperiodicsignalsusingthe sameideaofsuperposition.Belowwewillpresentthe Continuous-TimeFourierTransform CTFT, alsoreferredtoasjusttheFourierTransformFT.BecausetheCTFTnowdealswithnonperiodicsignals, wemustnowndawaytoinclude all frequenciesinthegeneralequations. 11.1.1.1Equations Continuous-TimeFourierTransform F = Z 1 f t e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j t dt .1 InverseCTFT f t = 1 2 Z 1 F e j t d .2 warning: Donotbeconfusedbynotation-itisnotuncommontoseetheaboveformulawritten slightlydierent.Oneofthemostcommondierencesamongmanyprofessorsisthewaythatthe exponentialiswritten.Aboveweusedtheradialfrequencyvariable intheexponential,where =2 f ,butonewilloftenseeprofessorsincludethemoreexplicitexpression, j 2 ft ,inthe exponential.Clickhere 3 foranoverviewofthenotationusedinConnexion'sDSPmodules. TheaboveequationsfortheCTFTanditsinversecomedirectlyfromtheFourierseriesandourunderstandingofitscoecients.FortheCTFTwesimplyutilizeintegrationratherthansummationtobeable toexpresstheaperiodicsignals.ThisshouldmakesensesincefortheCTFTwearesimplyextendingthe 1 Thiscontentisavailableonlineat. 2 "ClassicFourierSeries" 3 "DSPnotation" 189

PAGE 198

190 CHAPTER11.CONTINUOUSTIMEFOURIERTRANSFORMCTFT ideasoftheFourierseriestoincludenonperiodicsignals,andthustheentirefrequencyspectrum.Lookat theDerivationoftheFourierTransform 4 foramoreindepthlookatthis. 11.1.2RelevantSpaces TheContinuous-TimeFourierTransformmapsinnite-length,continuous-timesignalsin L 2 toinnitelength,continuous-frequencysignalsin L 2 .ReviewtheFourierAnalysisSection7.1foranoverviewofall thespacesusedinFourieranalysis. Figure11.1: Mapping L 2 R inthetimedomainto L 2 R inthefrequencydomain. FormoreinformationonthecharacteristicsoftheCTFT,pleaselookatthemoduleonPropertiesofthe FourierTransformSection11.2. 11.1.3ExampleProblems Exercise11.1 Solutiononp.194. FindtheFourierTransformCTFTofthefunction f t = 8 < : e )]TJ/F7 6.9738 Tf 6.227 0 Td [( t if t 0 0 otherwise .3 Exercise11.2 Solutiononp.194. FindtheinverseFouriertransformofthesquarewavedenedas X = 8 < : 1 if j j M 0 otherwise .4 11.2PropertiesoftheContinuous-TimeFourierTransform 5 ThismodulewilllookatsomeofthebasicpropertiesoftheContinuous-TimeFourierTransformSection11.1CTFT.Therstsectioncontainsatablethatillustratestheproperties,andthesectionsfollowing itdiscussafewofthemoreinterestingpropertiesinmoredepth.Inthetable,clickontheoperationname tobetakentothepropertiesexplanationfoundlateronthispage.LookatthismoduleSection5.6foran expandedtableofmoreFouriertransformproperties. 4 "DerivationoftheFourierTransform" 5 Thiscontentisavailableonlineat.

PAGE 199

191 note: Wewillbediscussingthesepropertiesforaperiodic,continuous-timesignalsbutunderstand thatverysimilarpropertiesholdfordiscrete-timesignalsandperiodicsignalsaswell. 11.2.1TableofCTFTProperties OperationName Signal f t Transform F AdditionSection11.2.2.1:Linearity f 1 t + f 2 t F 1 + F 2 ScalarMultiplicationSection11.2.2.1:Linearity f t F t SymmetrySection11.2.2.2: Symmetry F t 2 f )]TJ/F11 9.9626 Tf 7.749 0 Td [(! TimeScalingSection11.2.2.3: TimeScaling f t 1 j j F )]TJ/F10 6.9738 Tf 5.761 -4.147 Td [(! TimeShiftSection11.2.2.4: TimeShifting f t )]TJ/F11 9.9626 Tf 9.963 0 Td [( F e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! ModulationFrequencyShift Section11.2.2.5:Modulation FrequencyShift f t e jt F )]TJ/F11 9.9626 Tf 9.963 0 Td [( ConvolutioninTimeSection11.2.2.6:Convolution f 1 t ;f 2 t F 1 t F 2 t ConvolutioninFrequencySection11.2.2.6:Convolution f 1 t f 2 t 1 2 F 1 t ;F 2 t DierentiationSection11.2.2.7: TimeDierentiation d n dt n f t j! n F Table11.1 11.2.2DiscussionofFourierTransformProperties AfterglancingattheabovetableandgettingafeelforthepropertiesoftheCTFT,wewillnowtakealittle moretimetodiscusssomeofthemoreinteresting,andmoreuseful,properties. 11.2.2.1Linearity Thecombinedadditionandscalarmultiplicationpropertiesinthetableabovedemonstratethebasicproperty oflinearity.WhatyoushouldseeisthatifonetakestheFouriertransformofalinearcombinationofsignals thenitwillbethesameasthelinearcombinationoftheFouriertransformsofeachoftheindividualsignals. ThisiscrucialwhenusingatableSection10.3oftransformstondthetransformofamorecomplicated signal. Example11.1 Wewillbeginwiththefollowingsignal: z t = f 1 t + f 2 t .5

PAGE 200

192 CHAPTER11.CONTINUOUSTIMEFOURIERTRANSFORMCTFT Now,afterwetaketheFouriertransform,shownintheequationbelow,noticethatthelinear combinationofthetermsisunaectedbythetransform. Z = F 1 + F 2 .6 11.2.2.2Symmetry SymmetryisapropertythatcanmakelifequiteeasywhensolvingproblemsinvolvingFouriertransforms. Basicallywhatthispropertysaysisthatsincearectangularfunctionintimeisasincfunctioninfrequency, thenasincfunctionintimewillbearectangularfunctioninfrequency.Thisisadirectresultofthesimilarity betweentheforwardCTFTandtheinverseCTFT.Theonlydierenceisthescalingby 2 andafrequency reversal. 11.2.2.3TimeScaling Thispropertydealswiththeeectonthefrequency-domainrepresentationofasignalifthetimevariable isaltered.Themostimportantconcepttounderstandforthetimescalingpropertyisthatsignalsthatare narrowintimewillbebroadinfrequencyand viceversa .Thesimplestexampleofthisisadeltafunction, aunitpulse 6 witha very smallduration,intimethatbecomesaninnite-lengthconstantfunctionin frequency. Thetableaboveshowsthisideaforthegeneraltransformationfromthetime-domaintothefrequencydomainofasignal.Youshouldbeabletoeasilynoticethattheseequationsshowtherelationshipmentioned previously:ifthetimevariableisincreasedthenthefrequencyrangewillbedecreased. 11.2.2.4TimeShifting Timeshiftingshowsthatashiftintimeisequivalenttoalinearphaseshiftinfrequency.Sincethefrequency contentdependsonlyontheshapeofasignal,whichisunchangedinatimeshift,thenonlythephase spectrumwillbealtered.ThispropertycanbeeasilyprovedusingtheFourierTransform,sowewillshow thebasicstepsbelow: Example11.2 Wewillbeginbyletting z t = f t )]TJ/F11 9.9626 Tf 9.963 0 Td [( .NowletustaketheFouriertransformwiththeprevious expressionsubstitutedinfor z t Z = Z 1 f t )]TJ/F11 9.9626 Tf 9.962 0 Td [( e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j!t dt .7 Nowletusmakeasimplechangeofvariables,where = t )]TJ/F11 9.9626 Tf 10.103 0 Td [( .Throughthecalculationsbelow, youcanseethatonlythevariableintheexponentialarealteredthusonlychangingthephasein thefrequencydomain. Z = R 1 f e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! + t d = e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! R 1 f e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! d = e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! F .8 6 "ElementalSignals":SectionPulse

PAGE 201

193 11.2.2.5ModulationFrequencyShift Modulationisabsolutelyimperativetocommunicationsapplications.Beingabletoshiftasignaltoadierent frequency,allowsustotakeadvantageofdierentpartsoftheelectromagneticspectrumiswhatallowsus totransmittelevision,radioandotherapplicationsthroughthesamespacewithoutsignicantinterference. TheproofofthefrequencyshiftpropertyisverysimilartothatofthetimeshiftSection11.2.2.4:Time Shifting;however,herewewouldusetheinverseFouriertransforminplaceoftheFouriertransform.Since wewentthroughthestepsintheprevious,time-shiftproof,belowwewilljustshowtheinitialandnalstep tothisproof: z t = 1 2 Z 1 F )]TJ/F11 9.9626 Tf 9.962 0 Td [( e j!t d! .9 Nowwewouldsimplyreducethisequationthroughanotherchangeofvariablesandsimplifytheterms. Thenwewillprovethepropertyexpressedinthetableabove: z t = f t e jt .10 11.2.2.6Convolution Convolutionisoneofthebigreasonsforconvertingsignalstothefrequencydomain,sinceconvolutionin timebecomesmultiplicationinfrequency.Thispropertyisalsoanotherexcellentexampleofsymmetry betweentimeandfrequency.Italsoshowsthattheremaybelittletogainbychangingtothefrequency domainwhenmultiplicationintimeisinvolved. Wewillintroducetheconvolutionintegralhere,butifyouhavenotseenthisbeforeorneedtorefreshyour memory,thenlookatthecontinuous-timeconvolutionSection3.2moduleforamoreindepthexplanation andderivation. y t = f 1 t ;f 2 t = R 1 f 1 f 2 t )]TJ/F11 9.9626 Tf 9.963 0 Td [( d .11 11.2.2.7TimeDierentiation SinceLTISection2.1systemscanberepresentedintermsofdierentialequations,itisapparentwith thispropertythatconvertingtothefrequencydomainmayallowustoconvertthesecomplicateddierential equationstosimplerequationsinvolvingmultiplicationandaddition.Thisisoftenlookedatinmoredetail duringthestudyoftheLaplaceTransformSection13.1.

PAGE 202

194 CHAPTER11.CONTINUOUSTIMEFOURIERTRANSFORMCTFT SolutionstoExercisesinChapter11 SolutiontoExercise11.1p.190 InordertocalculatetheFouriertransform,allweneedtouseis.1Continuous-TimeFourierTransform, complexexponentialsSection1.6,andbasiccalculus. F = R 1 f t e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j t dt = R 1 0 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j t dt = R 1 0 e )]TJ/F10 6.9738 Tf 6.227 0 Td [(t + j dt =0 )]TJ/F13 6.9738 Tf 16.4 3.923 Td [()]TJ/F7 6.9738 Tf 6.226 0 Td [(1 + j .12 F = 1 + j .13 SolutiontoExercise11.2p.190 Herewewilluse.2InverseCTFTtondtheinverseFTgiventhat t 6 =0 x t = 1 2 R M )]TJ/F10 6.9738 Tf 6.226 0 Td [(M e j t d = 1 2 e j t j ; = e jw = 1 t sin Mt .14 x t = M sinc Mt .15

PAGE 203

Chapter12 SamplingTheorem 12.1Sampling 1 12.1.1Introduction Thedigitalcomputercanprocess discretetimesignals usingextremelyexibleandpowerfulalgorithms. However,mostsignalsofinterestare continuoustime ,whichishowthealmostalwaysappearinnature. Thismoduleintroducestheideaoftranslatingcontinuoustimeproblemsintodiscretetime,andyoucan readontolearnmoreofthedetailsandimportanceof sampling KeyQuestions Howdoweturnacontinuoustimesignalintoadiscretetimesignalsampling,A/D? WhencanwereconstructSection12.2aCTsignalexactlyfromitssamplesreconstruction,D/A? ManipulatingtheDTsignaldoeswhattothereconstructedsignal? 12.1.2Sampling Samplingandreconstructionarebestunderstoodinthefrequencydomain.We'llstartbylookingatsome examples Exercise12.1 Solutiononp.214. WhatCTsignal f t hastheCTFTSection11.1shownbelow? f t = 1 2 Z 1 F jw e jwt dw Figure12.1: TheCTFTof f t 1 Thiscontentisavailableonlineat. 195

PAGE 204

196 CHAPTER12.SAMPLINGTHEOREM Hint: F jw = F 1 jw F 2 jw wherethetwopartsof F jw are: a b Figure12.2 Exercise12.2 Solutiononp.214. WhatDTsignal f s [ n ] hastheDTFTSection10.4shownbelow? f s [ n ]= 1 2 Z )]TJ/F10 6.9738 Tf 6.227 0 Td [( f s )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e jw e jwn dw Figure12.3: DTFTthatisaperiodicwith period=2 versionof F jw inFigure12.1.

PAGE 205

197 Figure12.4: f t isthecontinuous-timesignalaboveand f s [ n ] isthediscrete-time,sampledversion of f t 12.1.2.1Generalization Ofcourse,theresultsfromtheaboveexamplescanbegeneralizedto any f t with F jw =0 j w j > where f t is bandlimited to [ )]TJ/F11 9.9626 Tf 7.748 0 Td [(; ] a b Figure12.5: F jw istheCTFTof f t

PAGE 206

198 CHAPTER12.SAMPLINGTHEOREM a b Figure12.6: F s )]TJ/F58 8.9664 Tf 4.224 -7.243 Td [(e jw istheDTFTof f s [ n ] F s )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e jw isaperiodicSection6.1withperiod 2 versionof F jw F s )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e jw istheDTFTofsignal sampledattheintegers. F jw istheCTFTofsignal. note: If f t isbandlimitedto [ )]TJ/F11 9.9626 Tf 7.749 0 Td [(; ] thentheDTFTofthesampledversion f s [ n ]= f n isjustaperiodicwithperiod 2 versionof F jw 12.1.3TurningaDiscreteSignalintoaContinuousSignal Now,let'slookatturningaDTsignalintoacontinuoustimesignal.Let f s [ n ] beaDTsignalwithDTFT F s )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e jw a b Figure12.7: F s )]TJ/F58 8.9664 Tf 4.224 -7.243 Td [(e jw istheDTFTof f s [ n ] Now,set f imp t = 1 X n = f s [ n ] t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n TheCTsignal, f imp t ,isnon-zeroonlyontheintegerswherethereareimpulsesofheight f s [ n ] .

PAGE 207

199 Figure12.8 Exercise12.3 Solutiononp.214. WhatistheCTFTof f imp t ? Now,giventhesamples f s [ n ] ofabandlimitedto [ )]TJ/F11 9.9626 Tf 7.749 0 Td [(; ] signal,ournextstepwillbetoseehowwecan reconstructSection12.2 f t Figure12.9: Blockdiagramshowingtheverybasicstepsusedtoreconstruct f t .Canwemakeour resultsequal f t exactly? 12.2Reconstruction 2 12.2.1Introduction Thereconstructionprocessbeginsbytakingasampledsignal,whichwillbeindiscretetime,andperforming afewoperationsinordertoconvertthemintocontinuous-timeand,withanyluck,intoanexactcopyof theoriginalsignal.Abasicmethodusedtoreconstructa [ )]TJ/F11 9.9626 Tf 7.748 0 Td [(; ] bandlimitedsignalfromitssamplesonthe integeristodothefollowingsteps: turnthesamplesequence f s [ n ] intoanimpulsetrain f imp t lowpasslter f imp t togetthereconstruction f t cutofreq.= 2 Thiscontentisavailableonlineat.

PAGE 208

200 CHAPTER12.SAMPLINGTHEOREM Figure12.10: ReconstructionblockdiagramwithlowpasslterLPF. Thelowpasslter'simpulseresponseis g t .Thefollowingequationsallowustoreconstructoursignal Figure12.11, f t f t = g t f imp t = g t P 1 n = f s [ n ] t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n = f t = P 1 n = f s [ n ] g t t )]TJ/F11 9.9626 Tf 9.962 0 Td [(n = P 1 n = f s [ n ] g t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n .1 Figure12.11 12.2.1.1ExamplesofFiltersg Example12.1:ZeroOrderHold Thistype"lter"isoneofthemostbasictypesofreconstructionlters.Itsimplyholdsthevalue thatisin f s [ n ] for seconds.Thiscreatesablockorsteplikefunctionwhereeachvalueofthe pulsein f s [ n ] issimplydraggedovertothenextpulse.Theequationsandillustrationsbelow Figure12.12depicthowthisreconstructionlterworkswiththefollowing g : g t = 8 < : 1 if 0
PAGE 209

201 a b Figure12.12: ZeroOrderHold note: Howdoes f t reconstructedwithazeroorderholdcomparetotheoriginal f t inthe frequencydomain? Example12.2:NthOrderHold HerewewilllookatafewquickexamplesofvariancestotheZeroOrderHoldlterdiscussedin thepreviousexample. a b c Figure12.13: NthOrderHoldExamplesnthorderholdisequaltoannthorderB-splineaFirst OrderHoldbSecondOrderHoldc 1 OrderHold

PAGE 210

202 CHAPTER12.SAMPLINGTHEOREM 12.2.2UltimateReconstructionFilter note: Whatistheultimatereconstructionlter? RecallthatseeFigure12.14 Figure12.14: Ourcurrentreconstructionblockdiagram.Notethateachofthesesignalshasitsown correspondingCTFTorDTFT. If G j! hasthefollowingshapeFigure12.15: Figure12.15: Ideallowpasslter then f t = f t Therefore,anideallowpasslterwillgiveusperfectreconstruction! Inthetimedomain,impulseresponse g t = sin t t .3 f t = P 1 n = f s [ n ] g t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n = P 1 n = f s [ n ] sin t )]TJ/F10 6.9738 Tf 6.226 0 Td [(n t )]TJ/F10 6.9738 Tf 6.226 0 Td [(n = f t .4

PAGE 211

203 12.2.3AmazingConclusions If f t isbandlimitedto [ )]TJ/F11 9.9626 Tf 7.749 0 Td [(; ] ,itcanbereconstructedperfectlyfromitssamplesontheintegers f s [ n ]= f t j t = n f t = 1 X n = f s [ n ] sin t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n t )]TJ/F11 9.9626 Tf 9.962 0 Td [(n .5 TheaboveequationforperfectreconstructiondeservesacloserlookSection12.3,whichyoushould continuetoreadinthefollowingsectiontogetabetterunderstandingofreconstruction.Hereareafew thingstothinkaboutfornow: Whatdoes sin t )]TJ/F10 6.9738 Tf 6.227 0 Td [(n t )]TJ/F10 6.9738 Tf 6.227 0 Td [(n equalatintegersotherthann? Whatisthesupportof sin t )]TJ/F10 6.9738 Tf 6.227 0 Td [(n t )]TJ/F10 6.9738 Tf 6.226 0 Td [(n ? 12.3MoreonReconstruction 3 12.3.1Introduction InthepreviousmoduleonreconstructionSection12.2,wegaveanintroductionintohowreconstruction worksandbrieyderivedanequationusedtoperformperfectreconstruction.Letusnowtakeacloserlook attheperfectreconstructionformula: f t = 1 X n = f s sin t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n .6 Wearewriting f t intermsofshiftedandscaledsincfunctions. sin t )]TJ/F11 9.9626 Tf 9.962 0 Td [(n t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n n 2 Z isa basisSection5.1.3:Basis forthespaceof [ )]TJ/F11 9.9626 Tf 7.749 0 Td [(; ] bandlimitedsignals.Butwait.... 12.3.1.1DeriveReconstructionFormulas Whatis < sin t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n ; sin t )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t )]TJ/F11 9.9626 Tf 9.962 0 Td [(k > =? .7 ThisinnerproductSection15.3canbehardtocalculateinthetimedomain,solet'susePlancharel TheoremSection15.12 < ; > = 1 2 Z )]TJ/F10 6.9738 Tf 6.227 0 Td [( e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j!n e j!k d! .8 3 Thiscontentisavailableonlineat.

PAGE 212

204 CHAPTER12.SAMPLINGTHEOREM a b Figure12.16 if n = k < sinc n ; sinc k > = 1 2 R )]TJ/F10 6.9738 Tf 6.227 0 Td [( e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j!n e j!k d! =1 .9 if n 6 = k < sinc n ; sinc k > = 1 2 R )]TJ/F10 6.9738 Tf 6.226 0 Td [( e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j!n e j!n d! = 1 2 R )]TJ/F10 6.9738 Tf 6.226 0 Td [( e j! k )]TJ/F10 6.9738 Tf 6.226 0 Td [(n d! = 1 2 sin k )]TJ/F10 6.9738 Tf 6.226 0 Td [(n j k )]TJ/F10 6.9738 Tf 6.227 0 Td [(n =0 .10 note: In.10weusedthefactthattheintegralofsinusoidoveracompleteintervalis0to simplifyourequation. So, < sin t )]TJ/F11 9.9626 Tf 9.962 0 Td [(n t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n ; sin t )]TJ/F11 9.9626 Tf 9.963 0 Td [(k t )]TJ/F11 9.9626 Tf 9.963 0 Td [(k > = 8 < : 1 if n = k 0 if n 6 = k .11 Therefore sin t )]TJ/F11 9.9626 Tf 9.962 0 Td [(n t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n n 2 Z isanorthonormalbasisSection15.7.3:OrthonormalBasisONBforthespaceof [ )]TJ/F11 9.9626 Tf 7.749 0 Td [(; ] bandlimited functions. note: SamplingisthesameascalculatingONBcoecients,whichisinnerproductswithsincs 12.3.1.2Summary Onelasttimefor f t [ )]TJ/F11 9.9626 Tf 7.749 0 Td [(; ] bandlimited Synthesis f t = 1 X n = f s [ n ] sin t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n t )]TJ/F11 9.9626 Tf 9.962 0 Td [(n .12

PAGE 213

205 Analysis f s [ n ]= f t j t = n .13 Inordertounderstandalittlemoreabouthowwecanreconstructasignalexactly,itwillbeusefulto examinetherelationships 4 betweenthefouriertransformsCTFTandDTFTinmoredepth. 12.4NyquistTheorem 5 12.4.1Introduction EarlieryoushouldhavebeenexposedtotheconceptsbehindsamplingSection12.1andthesampling theorem.Whilelearningabouttheseideas,youshouldhavebeguntonoticethatifwesampleattoolowof arate,thereisachancethatouroriginalsignalwillnotbeuniquelydenedbyoursampledsignal.Ifthis happens,thenthereisnoguaranteethatwecancorrectlyreconstructSection12.2thesignal.Asaresult ofthis,the NyquistTheorem wascreated.Below,wewilldiscussjustwhatexactlythistheoremtellsus. 12.4.2NyquistTheorem Wewilllet T equaloursamplingperioddistancebetweensamples.Thenlet s = 2 T samplingfrequency inradians/sec.Wehaveseenthatif f t isbandlimitedto [ )]TJ/F8 9.9626 Tf 7.749 0 Td [( B ; B ] andwesamplewithperiod T< b 2 s < B s > 2 B thenwecanreconstruct f t fromitssamples. Theorem12.1: NyquistTheorem"FundamentalTheoremofDSP" If f t isbandlimitedto [ )]TJ/F8 9.9626 Tf 7.749 0 Td [( B ; B ] ,wecanreconstructit perfectly fromitssamples f s [ n ]= f nT for s = 2 T > 2 B N =2 B iscalledthe" Nyquistfrequency "for f t .Forperfectreconstructiontobepossible s 2 B where s isthesamplingfrequencyand B isthehighestfrequencyinthesignal. Figure12.17: IllustrationofNyquistFrequency Example12.3:Examples: Humanearhearsfrequenciesupto20kHz CDsamplerateis44.1kHz. Phonelinepassesfrequenciesupto4kHz phonecompanysamplesat8kHz. 4 "ExamingReconstructionRelations" 5 Thiscontentisavailableonlineat.

PAGE 214

206 CHAPTER12.SAMPLINGTHEOREM 12.4.2.1Reconstruction Thereconstructionformulainthetimedomainlookslike f t = 1 X n = f s [ n ] sin )]TJ/F10 6.9738 Tf 6.196 -4.147 Td [( T t )]TJ/F11 9.9626 Tf 9.962 0 Td [(nT T t )]TJ/F11 9.9626 Tf 9.963 0 Td [(nT Wecanconclude,justasbefore,that sin )]TJ/F10 6.9738 Tf 6.196 -4.148 Td [( T t )]TJ/F11 9.9626 Tf 9.963 0 Td [(nT T t )]TJ/F11 9.9626 Tf 9.963 0 Td [(nT ;n 2 Z isabasisSection5.1forthespaceof [ )]TJ/F8 9.9626 Tf 7.748 0 Td [( B ; B ] bandlimitedfunctions, B = T .Theexpansioncoecient forthisbasisarecalculatedbysampling f t atrate 2 T =2 B note: Thebasisisalsoorthogonal.TomakeitorthonormalSection15.8,weneedanormalization factorof p T 12.4.2.2TheBigQuestion Exercise12.4 Solutiononp.214. Whatif s < 2 B ?WhathappenswhenwesamplebelowtheNyquistrate? Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10791/latest/NyquistPlot.llb 12.5Aliasing 6 12.5.1Introduction WhenconsideringthereconstructionSection12.2ofasignal,youshouldalreadybefamiliarwiththeidea oftheNyquistrateSection12.4.Thisconceptallowsustondthesamplingratethatwillprovidefor perfectreconstructionofoursignal.IfwesampleattoolowofaratebelowtheNyquistrate,thenproblems willarisethatwillmakeperfectreconstructionimpossible-thisproblemisknownas aliasing .Aliasing occurswhenthereisanoverlapintheshifted,perioidiccopiesofouroriginalsignal'sFT, i.e. spectrum. Inthefrequencydomain,onewillnoticethatpartofthesignalwilloverlapwiththeperiodicsignalsnext toit.Inthisoverlapthevaluesofthefrequencywillbeaddedtogetherandtheshapeofthesignalsspectrum willbeunwantinglyaltered.Thisoverlapping,oraliasing,makesitimpossibletocorrectlydeterminethe correctstrengthofthatfrequency.Figure12.18providesavisualexampleofthisphenomenon: 6 Thiscontentisavailableonlineat.

PAGE 215

207 Figure12.18: ThespectrumofsomebandlimitedtoWHzsignalisshowninthetopplot.Ifthe samplinginterval T s ischosentoolargerelativetothebandwidth W ,aliasingwilloccur.Inthebottom plot,thesamplingintervalischosensucientlysmalltoavoidaliasing.Notethatifthesignalwerenot bandlimited,thecomponentspectrawouldalwaysoverlap. 12.5.2AliasingandSampling Ifwesampletooslowly, i.e. s < 2 B ;T> B Wecannotrecoverthesignalfromitssamplesduetoaliasing. Example12.4 Let f 1 t haveCTFT. Figure12.19: Inthisgure,notethefollowingequation: B )]TJ/F57 5.9776 Tf 10.411 3.806 Td [( s 2 = a Let f 2 t haveCTFT.

PAGE 216

208 CHAPTER12.SAMPLINGTHEOREM Figure12.20: Thehorizontalportionsofthesignalresultfromoverlapwithshiftedreplicas-showing visualproofofaliasing. Trytosketchandanswerthefollowingquestionsonyourown: WhatdoestheDTFTof f 1 ;s [ n ]= f 1 nT looklike? WhatdoestheDTFTof f 2 ;s [ n ]= f 2 nT looklike? DoanyothersignalshavethesameDTFTas f 1 ;s [ n ] and f 2 ;s [ n ] ? CONCLUSION:IfwesamplebelowtheNyquistfrequency,therearemanysignalsthatcouldhaveproduced thatgivensamplesequence. Figure12.21: Theseareallequal! Whytheterm"aliasing"?BecausethesamesamplesequencecanrepresentdierentCTsignalsas opposedtowhenwesampleabovetheNyquistfrequency,thenthesamplesequencerepresentsauniqueCT signal.

PAGE 217

209 Figure12.22: Thesetwosignalscontainthesamefoursamples,yetareverydierentsignals. Example12.5 f t =cos t Figure12.23: Thecosinefunction, f t =cos t ,anditsCTFT. Case1:Sample s = rad sec T = 1 4 sec note: s > 2 B Case2:Sample w s = )]TJ/F7 6.9738 Tf 5.761 -4.148 Td [(8 3 rad sec T = 3 4 sec note: s < 2 B WhenweruntheDTFTfromCase#2throughthereconstructionsteps,werealizethatweend upwiththefollowingcosine: f t =cos 2 t Thisisa"stretched"outversionofouroriginal.Clearly,oursamplingratewasnothighenough toensurecorrectreconstructionfromthesamples. Youmayhaveseensomeeectsofaliasingsuchasawagonwheelturningbackwardsinawesternmovie. Aliasinginimages 7 canresultinMoirePatterns.HereisanexampleofanimagethathasMoireartifacts 8 asaresultofscanningattoolowafrequency. 7 http://ptolemy.eecs.berkeley.edu/eecs20/week13/moire.html 8 http://www.dvp.co.il/lter/moire.html

PAGE 218

210 CHAPTER12.SAMPLINGTHEOREM Thisisanunsupportedmediatype.Toview,pleaseseehttp://cnx.org/content/m10793/latest/alias.llb 12.6Anti-AliasingFilters 9 12.6.1Introduction TheideaofaliasingSection12.5hasbeendescribedastheproblemthatoccursifasignalisnotsampled Section12.1atahighenoughrateforexample,belowtheNyquistFrequencySection12.4.Butexactly whatkindofdistortiondoesaliasingproduce? a b Figure12.24 Highfrequenciesintheoriginalsignal"foldback"intolowerfrequencies. Highfrequenciesmasqueradingaslowerfrequenciesproduces highlyundesirable artifactsinthereconstructedsignal. warning: Wemustavoidaliasinganywaywecan. 12.6.2AvoidingAliasing Whatifitisimpractical/impossibletosampleat s > 2 B ? Filteroutthefrequenciesabove s 2 before yousample.Thebestwaytovisualizedoingthisistoimagine thefollowingsimplesteps: 1.TaketheCTFTofthesignal, f t 9 Thiscontentisavailableonlineat.

PAGE 219

211 2.Sendthissignalthroughalowpasslterwiththefollowingspecication, c = s 2 3.Wenowhaveagraphofoursignalinthefrequencydomainwithallvaluesof j j > s 2 equaltozero. Now,wetaketheinverseCTFTtogetbackourcontinuoustimesignal, f a t 4.Andnallywearereadytosampleoursignal! Example12.6 Sampleratefor CD=44 : 1KHz Manymusicalinstruments e.g. highhatcontainfrequenciesabove 22KHz eventhoughwe cannothearthem. Becauseofthis,wecanltertheoutputsignalfromtheinstrumentbeforewesampleitusing thefollowinglter: Figure12.25: Thislterwillcutothehigher,unnecessaryfrequencies,where j c j > 2 22kHz Nowthesignalisreadytobesampled! Example12.7:AnotherExample Speechbandwidthis > kHz ,butitisperfectlyintelligiblewhenlowpasslteredtoa kHz range.Becauseofthis,wecantakeanormalspeechsignalandpassitthroughalterliketheone showninFigure12.25,wherewenowset j c j > 2 4kHz .Thesignalwereceivefromthislteronly containsvalueswhere j j > 8 k Nowwecansampleat 16 k =8kHz standardtelephonyrate. 12.7DiscreteTimeProcessingofContinuousTimeSignals 10 Figure12.26: DSPSystem 10 Thiscontentisavailableonlineat.

PAGE 220

212 CHAPTER12.SAMPLINGTHEOREM HowistheCTFTofytrelatedtotheCTFTofftFigure1? Let G j! =reconstructionlterfreq.response Y j! = G j! Y imp j! where Y imp j! isimpulsesequencecreatedfrom y s [ n ] .So, Y j! = G j! Y s )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(e j!T = G j! H )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(e j!T F s )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(e j!T Y j! = G j! H )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j!T 1 T 1 X r = F j !F 2 r T Y j! = 1 T G j! H )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j!T 1 X r = F j !F 2 r T Now,letsassumethatftisbandlimitedto )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F10 6.9738 Tf 6.196 -4.147 Td [( T ; T = )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F7 6.9738 Tf 5.762 -3.985 Td [( s 2 ; s 2 and G j! isaperfectreconstruction lter.Then Y j! = 8 < : F j! H )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j!T if j j T 0 otherwise note: Y j! hasthesame"bandlimit"as F j! So,forbandlimitedsignals,andwithahighenoughsamplingrateandaperfectreconstructionlterFigure 2 Figure12.27: FT'soforiginalanalogsignalftandsampledversionofftrespectively. isequivalenttousingananalogLTIlterFigure3

PAGE 221

213 Figure12.28: ImplementingadiscretetimelterHinanalog where H a j! = 8 < : H )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j!T if j j T 0 otherwise So,bybeingcarefulwecanimplementLTIsystemsforbandlimitedsignals onourcomputer!!! Importantnote: H a j! =lterinducedbyoursystem. H a j! isLTIonlyif h ,theDTsystem,isLTI F j! ,theinput,isbandlimitedandthesamplerateishighenough.

PAGE 222

214 CHAPTER12.SAMPLINGTHEOREM SolutionstoExercisesinChapter12 SolutiontoExercise12.1p.195 f t = 1 2 Z 1 F jw e jwt dw SolutiontoExercise12.2p.196 Since F jw =0 outsideof [ )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 ; 2] f t = 1 2 Z 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 F jw e jwt dw Also,sinceweonlyuseoneintervaltoreconstruct f s [ n ] fromitsDTFT,wehave f s [ n ]= 1 2 Z 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 f s )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e jw e jwn dw Since F jw = F s )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e jw on [ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 2] f s [ n ]= f t j t = n i.e. f s [ n ] isa sampled versionof f t SolutiontoExercise12.3p.199 f imp t = 1 X n = f s [ n ] t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n F imp jw = R 1 f imp t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jwt dt = R 1 )]TJ 4.566 -0.597 Td [(P 1 n = f s [ n ] t )]TJ/F11 9.9626 Tf 9.963 0 Td [(n e )]TJ/F7 6.9738 Tf 6.226 0 Td [( jwt dt = )]TJ 4.566 -0.598 Td [(P 1 n = f s [ n ] R 1 t )]TJ/F11 9.9626 Tf 9.962 0 Td [(n e )]TJ/F7 6.9738 Tf 6.226 0 Td [( jwt dt = )]TJ 4.566 -0.598 Td [(P 1 n = f s [ n ] e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jwn = F s )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e jw .14 So,theCTFTof f imp t is equal totheDTFTof f s [ n ] note: Weusedthesiftingpropertytoshow R 1 t )]TJ/F11 9.9626 Tf 9.962 0 Td [(n e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jwt dt = e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jwn SolutiontoExercise12.4p.206 Gothroughthesteps:seeFigure12.29

PAGE 223

215 Figure12.29 Finally,whatwillhappento F s )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j! now?Toanswerthisnalquestion,wewillnowneedtolookinto theconceptofaliasingSection12.5.

PAGE 224

216 CHAPTER12.SAMPLINGTHEOREM

PAGE 225

Chapter13 LaplaceTransformandSystemDesign 13.1TheLaplaceTransforms 1 TheLaplacetransformisageneralizationoftheContinuous-TimeFourierTransformSection11.1.However,insteadofusingcomplexsinusoidsSection7.2oftheform e j!t ,astheCTFTdoes,theLaplace transformusesthemoregeneral, e st ,where s = + j! AlthoughLaplacetransformsarerarelysolvedusingintegrationtablesSection13.3andcomputers e.g. Matlabaremuchmorecommon,wewillprovidethe bilateralLaplacetransformpair here.Thesedene theforwardandinverseLaplacetransformations.Noticethesimilaritiesbetweentheforwardandinverse transforms.ThiswillgiverisetomanyofthesamesymmetriesfoundinFourieranalysisSection7.1. LaplaceTransform F s = Z 1 f t e )]TJ/F7 6.9738 Tf 6.226 0 Td [( st dt .1 InverseLaplaceTransform f t = 1 2 j Z c + j 1 c )]TJ/F10 6.9738 Tf 6.227 0 Td [(j 1 F s e st ds .2 13.1.1FindingtheLaplaceandInverseLaplaceTransforms 13.1.1.1SolvingtheIntegral ProbablythemostdicultandleastusedmethodforndingtheLaplacetransformofasignalissolving theintegral.Althoughitistechnicallypossible,itisextremelytimeconsuming.Givenhoweasythenext twomethodsareforndingit,wewillnotprovideanymorethanthis.Theintegralsareprimarilytherein ordertounderstandwherethefollowingmethodsoriginatefrom. 13.1.1.2UsingaComputer UsingacomputertondLaplacetransformsisrelativelypainless.Matlabhastwofunctions, laplace and ilaplace ,thatarebothpartofthesymbolictoolbox,andwillndtheLaplaceandinverseLaplace transformsrespectively.Thismethodisgenerallypreferredformorecomplicatedfunctions.Simplerand morecontrivedfunctionsareusuallyfoundeasilyenoughbyusingtablesSection13.1.1.3:UsingTables. 1 Thiscontentisavailableonlineat. 217

PAGE 226

218 CHAPTER13.LAPLACETRANSFORMANDSYSTEMDESIGN 13.1.1.3UsingTables WhenrstlearningabouttheLaplacetransform,tablesarethemostcommonmeansforndingit.With enoughpractice,thetablesthemselvesmaybecomeunnecessary,asthecommontransformscanbecome secondnature.Forthepurposeofthissection,wewillfocusontheinverseLaplacetransform,sincemost designapplicationswillbeginintheLaplacedomainandgiverisetoaresultinthetimedomain.The methodisasfollows: 1.Writethefunctionyouwishtotransform, H s ,asasumofotherfunctions, H s = P m i =1 H i s whereeachofthe H i isknownfromatableSection13.3. 2.Inverteach H i s togetits h i t 3.Sumupthe h i t toget h t = P m i =1 h i t Example13.1 Compute h t for H s = 1 s +5 ;Re s > )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 ThiscanbesolveddirectlyfromthetableSection13.3tobe h t = e )]TJ/F7 6.9738 Tf 6.227 0 Td [( t Example13.2 Findthetimedomainrepresentation, h t ,of H s = 25 s +10 ;Re s > )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 Tosolvethis,werstnoticethat H s canalsobewrittenas 25 1 s +10 .Wecanthengotothe tableSection13.3tond h t =25 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( t Example13.3 Wecannowextendthetwopreviousexamplesbynding h t for H s = 1 s +5 + 25 s +10 ;Re s > )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Todothis,wetakeadvantageoftheadditivepropertyoflinearityandthethree-stepmethod describedabovetoyieldtheresult h t = e )]TJ/F7 6.9738 Tf 6.227 0 Td [( t +25 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( t Formorecomplicatedexamples,itmaybemorediculttobreakupthetransferfunctionintopartsthat existinatable.Inthiscase,itisoftennecessarytousepartialfractionexpansion 2 togetthetransfer functionintoamoreusableform. 13.1.2VisualizingtheLaplaceTransform WiththeFouriertransform,wehada complex-valuedfunction ofa purelyimaginaryvariable F j! Thiswassomethingwecouldenvisionwithtwo2-dimensionalplotsrealandimaginarypartsormagnitude andphase.However,withLaplace,wehavea complex-valuedfunction ofa complexvariable .In ordertoexaminethemagnitudeandphaseorrealandimaginarypartsofthisfunction,wemustexamine 3-dimensionalsurfaceplotsofeachcomponent. 2 "PartialFractionExpansion"

PAGE 227

219 realandimaginarysampleplots a b Figure13.1: Realandimaginarypartsof H s arenoweach3-dimensionalsurfaces.aTheReal partof H s bTheImaginarypartof H s magnitudeandphasesampleplots a b Figure13.2: Magnitudeandphaseof H s arealsoeach3-dimensionalsurfaces.Thisrepresentation ismorecommonthanrealandimaginaryparts.aTheMagnitudeof H s bThePhaseof H s WhilethesearelegitimatewaysoflookingatasignalintheLaplacedomain,itisquitediculttodraw and/oranalyze.Forthisreason,asimplermethodhasbeendeveloped.Althoughitwillnotbediscussed indetailhere,themethodofPolesandZerosSection13.6ismucheasiertounderstandandistheway boththeLaplacetransformanditsdiscrete-timecounterparttheZ-transformSection14.1arerepresented graphically. 13.2PropertiesoftheLaplaceTransform 3 3 Thiscontentisavailableonlineat.

PAGE 228

220 CHAPTER13.LAPLACETRANSFORMANDSYSTEMDESIGN Property Signal LaplaceTransform Regionof Convergence Linearity x 1 t + x 2 t X 1 s + X 2 s Atleast ROC 1 T ROC 2 TimeShifting x t )]TJ/F11 9.9626 Tf 9.962 0 Td [( e )]TJ/F7 6.9738 Tf 6.226 0 Td [( s X s ROC FrequencyShifting modulation e t x t X s )]TJ/F11 9.9626 Tf 9.962 0 Td [( Shifted ROC s )]TJ/F11 9.9626 Tf 9.963 0 Td [( mustbeintheregionof convergence TimeScaling x t )-222(j j X s )]TJ/F11 9.9626 Tf 9.962 0 Td [( Scaled ROC s )]TJ/F11 9.9626 Tf 9.963 0 Td [( mustbeintheregionof convergence Conjugation x t X s ROC Convolution x 1 t x 2 t X 1 t X 2 t Atleast ROC 1 T ROC 2 TimeDierentiation d dt x t sX s Atleast ROC Frequency Dierentiation )]TJ/F11 9.9626 Tf 7.748 0 Td [(t x t d ds X s ROC IntegrationinTime R t x d )]TJ/F11 9.9626 Tf 9.963 0 Td [(s X s Atleast ROC T Re s > 0 Table13.1

PAGE 229

221 13.3TableofCommonLaplaceTransforms 4 Signal LaplaceTransform RegionofConvergence t 1 All s t )]TJ/F11 9.9626 Tf 9.963 0 Td [(T e )]TJ/F7 6.9738 Tf 6.227 0 Td [( sT All s u t 1 s Re s > 0 )]TJ/F8 9.9626 Tf 9.409 0 Td [( u )]TJ/F11 9.9626 Tf 7.749 0 Td [(t 1 s Re s < 0 tu t 1 s 2 Re s > 0 t n u t n s n +1 Re s > 0 )]TJ/F8 9.9626 Tf 9.409 0 Td [( t n u )]TJ/F11 9.9626 Tf 7.749 0 Td [(t n s n +1 Re s < 0 e )]TJ/F7 6.9738 Tf 6.226 0 Td [( t u t 1 s + Re s > )]TJ/F11 9.9626 Tf 7.748 0 Td [( )]TJ/F14 9.9626 Tf 4.566 -8.069 Td [()]TJ/F1 9.9626 Tf 9.409 8.069 Td [()]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(e )]TJ/F7 6.9738 Tf 6.227 0 Td [( t u )]TJ/F11 9.9626 Tf 7.748 0 Td [(t 1 s + Re s < )]TJ/F11 9.9626 Tf 7.748 0 Td [( te )]TJ/F7 6.9738 Tf 6.227 0 Td [( t u t 1 s )]TJ/F10 6.9738 Tf 6.227 0 Td [( 2 Re s > )]TJ/F11 9.9626 Tf 7.748 0 Td [( t n e )]TJ/F7 6.9738 Tf 6.227 0 Td [( t u t n s + n +1 Re s > )]TJ/F11 9.9626 Tf 7.748 0 Td [( )]TJ/F1 9.9626 Tf 9.409 8.069 Td [()]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(t n e )]TJ/F7 6.9738 Tf 6.226 0 Td [( t u )]TJ/F11 9.9626 Tf 7.749 0 Td [(t n s + n +1 Re s < )]TJ/F11 9.9626 Tf 7.748 0 Td [( cos bt u t s s 2 + b 2 Re s > 0 sin bt u t b s 2 + b 2 Re s > 0 e )]TJ/F7 6.9738 Tf 6.226 0 Td [( at cos bt u t s + a s + a 2 + b 2 Re s > )]TJ/F11 9.9626 Tf 7.748 0 Td [(a e )]TJ/F7 6.9738 Tf 6.226 0 Td [( at sin bt u t b s + a 2 + b 2 Re s > )]TJ/F11 9.9626 Tf 7.748 0 Td [(a d n dt n t s n All s Table13.2 13.4RegionofConvergencefortheLaplaceTransform 5 WiththeLaplacetransformSection13.1,thes-planerepresentsasetofsignalscomplexexponentials Section1.6.ForanygivenLTISection2.1system,someofthesesignalsmaycausetheoutputofthe systemtoconverge,whileotherscausetheoutputtodiverge"blowup".Thesetofsignalsthatcausethe system'soutputtoconvergelieinthe regionofconvergenceROC .Thismodulewilldiscusshowto ndthisregionofconvergenceforanycontinuous-time,LTIsystem. RecallthedenitionoftheLaplacetransform, LaplaceTransform H s = Z 1 h t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( st dt .3 IfweconsideracausalSection1.1,complexexponential, h t = e )]TJ/F7 6.9738 Tf 6.227 0 Td [( at u t ,wegettheequation, Z 1 0 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( at e )]TJ/F7 6.9738 Tf 6.226 0 Td [( st dt = Z 1 0 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( a + s t dt .4 4 Thiscontentisavailableonlineat. 5 Thiscontentisavailableonlineat.

PAGE 230

222 CHAPTER13.LAPLACETRANSFORMANDSYSTEMDESIGN Evaluatingthis,weget )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 s + a lim t !1 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( s + a t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 .5 Noticethatthisequationwilltendtoinnitywhen lim t !1 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( s + a t tendstoinnity.Tounderstandwhen thishappens,wetakeonemorestepbyusing s = + j! torealizethisequationas lim t !1 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j!t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( + a t .6 Recognizingthat e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j!t issinusoidal,itbecomesapparentthat e )]TJ/F7 6.9738 Tf 6.227 0 Td [( a t isgoingtodeterminewhether thisblowsupornot.Whatwendisthatif + a ispositive,theexponentialwillbetoanegativepower, whichwillcauseittogotozeroas t tendstoinnity.Ontheotherhand,if + a isnegativeorzero,the exponentialwillnotbetoanegativepower,whichwillpreventitfromtendingtozeroandthesystemwill notconverge.Whatallofthistellsusisthatforacausalsignal,wehaveconvergencewhen ConditionforConvergence Re s > )]TJ/F11 9.9626 Tf 7.749 0 Td [(a .7 Althoughwewillnotgothroughtheprocessagainforanticausalsignals,wecould.Indoingso,wewould ndthatthenecessaryconditionforconvergenceiswhen NecessaryConditionforAnti-CausalConvergence Re s < )]TJ/F11 9.9626 Tf 7.749 0 Td [(a .8 13.4.1GraphicalUnderstandingofROC Perhapsthebestwaytolookattheregionofconvergenceistoviewitinthes-plane.Whatweobserveis thatforasinglepole,theregionofconvergenceliestotherightofitforcausalsignalsandtotheleftfor anti-causalsignals. a b Figure13.3: aTheRegionofConvergenceforacausalsignal.bTheRegionofConvergencefor ananti-causalsignal.

PAGE 231

223 Oncewehaverecognizedthis,thenaturalquestionbecomes:Whatdowedowhenwehavemultiple poles?Thesimpleansweristhatwetaketheintersectionofalloftheregionsofconvergenceoftherespective poles. Example13.4 Find H s andstatetheregionofconvergencefor h t = e )]TJ/F7 6.9738 Tf 6.227 0 Td [( at u t + e )]TJ/F7 6.9738 Tf 6.227 0 Td [( bt u )]TJ/F11 9.9626 Tf 7.748 0 Td [(t Breakingthisupintoitstwoterms,wegettransferfunctionsandrespectiveregionsofconvergenceof H 1 s = 1 s + a ;Re s > )]TJ/F11 9.9626 Tf 7.749 0 Td [(a .9 and H 2 s = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 s + b ;Re s < )]TJ/F11 9.9626 Tf 7.748 0 Td [(b .10 Combiningthese,wegetaregionofconvergenceof )]TJ/F11 9.9626 Tf 7.749 0 Td [(b>Re s > )]TJ/F11 9.9626 Tf 7.749 0 Td [(a .If a>b ,wecanrepresent thisgraphically.Otherwise,therewillbenoregionofconvergence. Figure13.4: TheRegionofConvergenceof h t if a>b 13.5TheInverseLaplaceTransform 6 13.5.1ToCome InTheTransferFunction 7 weshallestablishthattheinverseLaplacetransformofafunction h is )]TJ/F14 9.9626 Tf 4.567 -8.07 Td [(L )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 h t = 1 2 Z 1 e c + yj t h c + yj t dy .11 6 Thiscontentisavailableonlineat. 7 "EigenvalueProblem:TheTransferFunction"

PAGE 232

224 CHAPTER13.LAPLACETRANSFORMANDSYSTEMDESIGN where j p )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 andtherealnumber c ischosensothatallofthe singularities of h lietotheleftofthe lineofintegration. 13.5.2ProceedingwiththeInverseLaplaceTransform WiththeinverseLaplacetransformonemayexpressthesolutionof x 0 = B x + g ,as x t = L )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 sI )]TJ/F11 9.9626 Tf 9.962 0 Td [(B )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Lf g g + x .12 Asanexample,letustaketherstcomponentof Lf x g ,namely L x 1 s = 0 : 19 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(s 2 +1 : 5 s +0 : 27 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(s + 1 6 4 s 3 +1 : 655 s 2 +0 : 4078 s +0 : 0039 Wedene: Denition13.1:poles Alsocalledsingularities,thesearethepoints s atwhich L x 1 s blowsup. Theseareclearlytherootsofitsdenominator,namely )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 100 )]TJ/F8 9.9626 Tf 7.749 0 Td [(329 = 400 p 73 16 ,and )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = 6 ..13 Allfourbeingnegative,itsucestotake c =0 andsotheintegrationin.11proceedsuptheimaginary axis.Wedon'tsupposethereadertohavealreadyencounteredintegrationinthecomplexplanebuthope thatthisexamplemightprovidethemotivationnecessaryforabriefoverviewofsuch.Beforethathowever wenotethatMATLABhasdigestedthecalculuswewishtodevelop.Referringagaintob3.m 8 fordetails wenotethatthe ilaplace commandproduces x 1 t =211 : 35 e )]TJ/F59 5.9776 Tf 5.756 0 Td [(t 100 )]TJ/F93 11.9552 Tf 22.196 0 Td [( : 0554 t 3 +4 : 5464 t 2 +1 : 085 t +474 : 19 e )]TJ/F59 5.9776 Tf 5.756 0 Td [(t 6 + e )]TJ/F57 5.9776 Tf 5.756 0 Td [( t 400 262 : 842cosh p 73 t 16 +262 : 836sinh p 73 t 16 8 http://www.caam.rice.edu/ caam335/cox/lectures/b3.m

PAGE 233

225 Figure13.5: The3potentialsassociatedwiththeRCcircuitmodelgure 9 Theotherpotentials,seethegureabove,possesssimilarexpressions.Pleasenotethateachofthepoles of Lf x 1 g appearasexponentsin x 1 andthatthecoecientsoftheexponentialsarepolynomialswhose degreesisdeterminedbythe order oftherespectivepole. 13.6PolesandZeros 10 13.6.1Introduction ItisquitediculttoqualitativelyanalyzetheLaplacetransformSection13.1andZ-transformSection14.1,sincemappingsoftheirmagnitudeandphaseorrealpartandimaginarypartresultinmultiple mappingsof2-dimensionalsurfacesin3-dimensionalspace.Forthisreason,itisverycommontoexaminea plotofatransferfunction's 11 polesandzerostotrytogainaqualitativeideaofwhatasystemdoes. Givenacontinuous-timetransferfunctionintheLaplacedomain, H s ,oradiscrete-timeoneinthe Z-domain, H z ,azeroisanyvalueof s or z suchthatthetransferfunctioniszero,andapoleisanyvalue of s or z suchthatthetransferfunctionisinnite.Todenethemprecisely: Denition13.2:zeros 1.Thevaluesfor z wherethe numerator ofthetransferfunctionequalszero 2.Thecomplexfrequenciesthatmaketheoverallgainoftheltertransferfunctionzero. Denition13.3:poles 1.Thevaluesfor z wherethe denominator ofthetransferfunctionequalszero 2.Thecomplexfrequenciesthatmaketheoverallgainoftheltertransferfunctioninnite. 9 "NerveFibersandtheDynamicStrangQuartet",Figure1:AnRCmodelofanerveber 10 Thiscontentisavailableonlineat. 11 "TransferFunctions"

PAGE 234

226 CHAPTER13.LAPLACETRANSFORMANDSYSTEMDESIGN 13.6.2Pole/ZeroPlots Whenweplottheseintheappropriates-orz-plane,werepresentzeroswith"o"andpoleswith"x".Refer tothismoduleSection14.7foradetailedlookingatplottingthepolesandzerosofaz-transformonthe Z-plane. Example13.5 Findthepolesandzerosforthetransferfunction H s = s 2 +6 s +8 s 2 +2 andplottheresultsinthe s-plane. Therstthingwerecognizeisthatthistransferfunctionwillequalzerowheneverthetop, s 2 +6 s +8 ,equalszero.Tondwherethisequalszero,wefactorthistoget, s +2 s +4 .This yieldszerosat s = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 and s = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 .Hadthisfunctionbeenmorecomplicated,itmighthavebeen necessarytousethequadraticformula. Forpoles,wemustrecognizethatthetransferfunctionwillbeinnitewheneverthebottom partiszero.Thatiswhen s 2 +2 iszero.Tondthis,weagainlooktofactortheequation.This yields )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(s + j p 2 )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(s )]TJ/F11 9.9626 Tf 9.963 0 Td [(j p 2 .Thisyieldspurelyimaginaryrootsof + j p 2 and )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(j p 2 PlottingthisgivesFigure13.6PoleandZeroPlot PoleandZeroPlot Figure13.6: Samplepole-zeroplot Nowthatwehavefoundandplottedthepolesandzeros,wemustaskwhatitisthatthisplotgivesus. Basicallywhatwecangatherfromthisisthatthemagnitudeofthetransferfunctionwillbelargerwhen itisclosertothepolesandsmallerwhenitisclosertothezeros.Thisprovidesuswithaqualitative understandingofwhatthesystemdoesatvariousfrequenciesandiscrucialtothediscussionofstability Section3.4. 13.6.3RepeatedPolesandZeros Itispossibletohavemorethanonepoleorzeroatanygivenpoint.Forinstance,thediscrete-timetransfer function H z = z 2 willhavetwozerosattheoriginandthecontinuous-timefunction H s = 1 s 25 willhave

PAGE 235

227 25polesattheorigin. 13.6.4Pole-ZeroCancellation Aneasymistaketomakewithregardstopolesandzerosistothinkthatafunctionlike s +3 s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 isthe sameas s +3 .Intheorytheyareequivalent,asthepoleandzeroat s =1 canceleachotheroutinwhatis knownas pole-zerocancellation .However,thinkaboutwhatmayhappenifthiswereatransferfunction ofasystemthatwascreatedwithphysicalcircuits.Inthiscase,itisveryunlikelythatthepoleandzero wouldremaininexactlythesameplace.Aminortemperaturechange,forinstance,couldcauseoneofthem tomovejustslightly.Ifthisweretooccuratremendousamountofvolatilityiscreatedinthatarea,since thereisachangefrominnityatthepoletozeroatthezeroinaverysmallrangeofsignals.Thisisgenerally averybadwaytotrytoeliminateapole.Amuchbetterwayistouse controltheory tomovethepole toabetterplace.

PAGE 236

228 CHAPTER13.LAPLACETRANSFORMANDSYSTEMDESIGN

PAGE 237

Chapter14 Z-TransformandDigitalFiltering 14.1TheZTransform:Denition 1 14.1.1BasicDenitionoftheZ-Transform The z-transform ofasequenceisdenedas X z = 1 X n = )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x [ n ] z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n .1 Sometimesthisequationisreferredtoasthe bilateralz-transform .Attimesthez-transformisdened as X z = 1 X n =0 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(x [ n ] z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n .2 whichisknownasthe unilateralz-transform Thereisacloserelationshipbetweenthez-transformandthe Fouriertransform ofadiscretetime signal,whichisdenedas X )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j! = 1 X n = x [ n ] e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j!n .3 Noticethatthatwhenthe z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n isreplacedwith e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j!n thez-transformreducestotheFourierTransform. WhentheFourierTransformexists, z = e j! ,whichistohavethemagnitudeof z equaltounity. 14.1.2TheComplexPlane InordertogetfurtherinsightintotherelationshipbetweentheFourierTransformandtheZ-Transformit isusefultolookatthecomplexplaneor z-plane .Takealookatthecomplexplane: 1 Thiscontentisavailableonlineat. 229

PAGE 238

230 CHAPTER14.Z-TRANSFORMANDDIGITALFILTERING Z-Plane Figure14.1 TheZ-planeisacomplexplanewithanimaginaryandrealaxisreferringtothecomplex-valuedvariable z .Thepositiononthecomplexplaneisgivenby re j! ,andtheanglefromthepositive,realaxisaround theplaneisdenotedby X z isdenedeverywhereonthisplane. X )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j! ontheotherhandisdened onlywhere j z j =1 ,whichisreferredtoastheunitcircle.Soforexample, =1 at z =1 and = at z = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 .Thisisusefulbecause,byrepresentingtheFouriertransformasthez-transformontheunitcircle, theperiodicityofFouriertransformiseasilyseen. 14.1.3RegionofConvergence Theregionofconvergence,knownasthe ROC ,isimportanttounderstandbecauseitdenestheregion wherethez-transformexists.TheROCforagiven x [ n ] ,isdenedastherangeof z forwhichthez-transform converges.Sincethez-transformisa powerseries ,itconvergeswhen x [ n ] z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n isabsolutelysummable. Stateddierently, 1 X n = )]TJ/F14 9.9626 Tf 4.567 -8.07 Td [(j x [ n ] z )]TJ/F10 6.9738 Tf 6.226 0 Td [(n j < 1 .4 mustbesatisedforconvergence.ThisisbestillustratedbylookingatthedierentROC'softheztransformsof n u [ n ] and n u [ n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1] Example14.1 For x [ n ]= n u [ n ] .5

PAGE 239

231 Figure14.2: x [ n ]= n u [ n ] where =0 : 5 X z = P 1 n = x [ n ] z )]TJ/F10 6.9738 Tf 6.226 0 Td [(n = P 1 n = n u [ n ] z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n = P 1 n =0 n z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n = P 1 n =0 \000 z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 n .6 Thissequenceisanexampleofaright-sidedexponentialsequencebecauseitisnonzerofor n 0 Itonlyconvergeswhen j z )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 j < 1 .Whenitconverges, X z = 1 1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(z )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 = z z )]TJ/F10 6.9738 Tf 6.227 0 Td [( .7 If j z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 j 1 ,thentheseries, P 1 n =0 \000 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 n doesnotconverge.ThustheROCistherangeof valueswhere j z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 j < 1 .8 or,equivalently, j z j > j j .9

PAGE 240

232 CHAPTER14.Z-TRANSFORMANDDIGITALFILTERING Figure14.3: ROCfor x [ n ]= n u [ n ] where =0 : 5 Example14.2 For x [ n ]= )]TJ/F8 9.9626 Tf 9.409 0 Td [( n u [ )]TJ/F11 9.9626 Tf 7.749 0 Td [(n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1] .10

PAGE 241

233 Figure14.4: x [ n ]= )]TJ/F56 8.9664 Tf 8.703 0 Td [( n u [ )]TJ/F58 8.9664 Tf 7.168 0 Td [(n )]TJ/F56 8.9664 Tf 9.216 0 Td [(1] where =0 : 5 X z = P 1 n = x [ n ] z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n = P 1 n = )]TJ/F8 9.9626 Tf 9.409 0 Td [( n u [ )]TJ/F8 9.9626 Tf 7.749 0 Td [(n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1] z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n = )]TJ/F1 9.9626 Tf 9.409 11.058 Td [( P )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 n = n z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n = )]TJ/F1 9.9626 Tf 9.409 11.059 Td [( P )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 n = )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [( )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n = )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ 4.566 -0.598 Td [(P 1 n =1 \000 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 z n =1 )]TJ/F1 9.9626 Tf 9.963 7.472 Td [(P 1 n =0 \000 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 z n .11 TheROCinthiscaseistherangeofvalueswhere j )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 z j < 1 .12 or,equivalently, j z j < j j .13 IftheROCissatised,then X z =1 )]TJ/F7 6.9738 Tf 23.549 3.923 Td [(1 1 )]TJ/F10 6.9738 Tf 6.227 0 Td [( )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 z = z z )]TJ/F10 6.9738 Tf 6.227 0 Td [( .14

PAGE 242

234 CHAPTER14.Z-TRANSFORMANDDIGITALFILTERING Figure14.5: ROCfor x [ n ]= )]TJ/F56 8.9664 Tf 8.703 0 Td [( n u [ )]TJ/F58 8.9664 Tf 7.167 0 Td [(n )]TJ/F56 8.9664 Tf 9.216 0 Td [(1] 14.2TableofCommonz-Transforms 2 Thetablebelowprovidesanumberofunilateralandbilateral z-transformsSection14.1 .Thetable alsospeciestheregionofconvergenceSection14.3. note: Thenotationfor z foundinthetablebelowmaydierfromthatfoundinothertables.For example,thebasicz-transformof u [ n ] canbewrittenaseitherofthefollowingtwoexpressions, whichareequivalent: z z )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 = 1 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 .15 2 Thiscontentisavailableonlineat.

PAGE 243

235 Signal Z-Transform ROC [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k ] z )]TJ/F10 6.9738 Tf 6.227 0 Td [(k All z u [ n ] z z )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 j z j > 1 )]TJ/F8 9.9626 Tf 9.409 0 Td [( u [ )]TJ/F11 9.9626 Tf 7.748 0 Td [(n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1] z z )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 j z j < 1 nu [ n ] z z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 2 j z j > 1 n 2 u [ n ] z z +1 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 3 j z j > 1 n 3 u [ n ] z z 2 +4 z +1 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 4 j z j > 1 )]TJ/F8 9.9626 Tf 9.41 0 Td [( n u [ )]TJ/F11 9.9626 Tf 7.749 0 Td [(n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1] z z )]TJ/F10 6.9738 Tf 6.226 0 Td [( j z j < j j n u [ n ] z z )]TJ/F10 6.9738 Tf 6.226 0 Td [( j z j > j j n n u [ n ] z z )]TJ/F10 6.9738 Tf 6.227 0 Td [( 2 j z j > j j n 2 n u [ n ] z z + z )]TJ/F10 6.9738 Tf 6.227 0 Td [( 3 j z j > j j Q m k =1 n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k +1 m m n u [ n ] z z )]TJ/F10 6.9738 Tf 6.227 0 Td [( m +1 n cos n u [ n ] z z )]TJ/F10 6.9738 Tf 6.227 0 Td [( cos z 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [( cos z + 2 j z j > j j n sin n u [ n ] z sin z 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [( cos z + 2 j z j > j j Table14.1 14.3RegionofConvergencefortheZ-transform 3 14.3.1TheRegionofConvergence Theregionofconvergence,knownasthe ROC ,isimportanttounderstandbecauseitdenestheregion wherethez-transformSection14.1exists.The z-transform ofasequenceisdenedas X z = 1 X n = )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(x [ n ] z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n .16 TheROCforagiven x [ n ] ,isdenedastherangeof z forwhichthez-transformconverges.Sincethe z-transformisa powerseries ,itconvergeswhen x [ n ] z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n isabsolutelysummable.Stateddierently, 1 X n = )]TJ/F14 9.9626 Tf 4.567 -8.07 Td [(j x [ n ] z )]TJ/F10 6.9738 Tf 6.226 0 Td [(n j < 1 .17 mustbesatisedforconvergence. 14.3.2PropertiesoftheRegionofConvergencec TheRegionofConvergencehasanumberofpropertiesthataredependentonthecharacteristicsofthe signal, x [ n ] TheROCcannotcontainanypoles. Bydenitionapoleisawhere X z isinnite.Since X z mustbeniteforall z forconvergence,therecannotbeapoleintheROC. 3 Thiscontentisavailableonlineat.

PAGE 244

236 CHAPTER14.Z-TRANSFORMANDDIGITALFILTERING If x [ n ] isanite-durationsequence,thentheROCistheentirez-plane,exceptpossibly z =0 or j z j = 1 A nite-durationsequence isasequencethatisnonzeroinaniteinterval n 1 n n 2 .Aslongaseachvalueof x [ n ] isnitethenthesequencewillbeabsolutelysummable. When n 2 > 0 therewillbea z )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 termandthustheROCwillnotinclude z =0 .When n 1 < 0 thenthe sumwillbeinniteandthustheROCwillnotinclude j z j = 1 .Ontheotherhand,when n 2 0 then theROCwillinclude z =0 ,andwhen n 1 0 theROCwillinclude j z j = 1 .Withtheseconstraints, theonlysignal,then,whoseROCistheentirez-planeis x [ n ]= c [ n ] Figure14.6: Anexampleofanitedurationsequence. Thenextpropertiesapplytoinnitedurationsequences.Asnotedabove,thez-transformconverges when j X z j < 1 .Sowecanwrite j X z j = j 1 X n = )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(x [ n ] z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n j 1 X n = )]TJ/F14 9.9626 Tf 4.566 -8.069 Td [(j x [ n ] z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n j = 1 X n = j x [ n ] j j z j )]TJ/F10 6.9738 Tf 6.227 0 Td [(n .18 Wecanthensplittheinnitesumintopositive-timeandnegative-timeportions.So j X z j N z + P z .19 where N z = )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X n = j x [ n ] j j z j )]TJ/F10 6.9738 Tf 6.227 0 Td [(n .20 and P z = 1 X n =0 j x [ n ] j j z j )]TJ/F10 6.9738 Tf 6.226 0 Td [(n .21 Inorderfor j X z j tobenite, j x [ n ] j mustbebounded.Letusthenset j x n j C 1 r 1 n .22

PAGE 245

237 for n< 0 and j x n j C 2 r 2 n .23 for n 0 Fromthissomefurtherpropertiescanbederived: If x [ n ] isaright-sidedsequence,thentheROCextendsoutwardfromtheoutermostpole in X z A right-sidedsequence isasequencewhere x [ n ]=0 for nr 2 ,andthereforetheROCofaright-sidedsequenceis oftheform j z j >r 2 Figure14.7: Aright-sidedsequence.

PAGE 246

238 CHAPTER14.Z-TRANSFORMANDDIGITALFILTERING Figure14.8: TheROCofaright-sidedsequence. If x [ n ] isaleft-sidedsequence,thentheROCextendsinwardfromtheinnermostpole in X z A right-sidedsequence isasequencewhere x [ n ]=0 for n>n 2 > .Lookingatthe negative-timeportionfromtheabovederivation,itfollowsthat N z C 1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X n = r 1 n j z j )]TJ/F10 6.9738 Tf 6.227 0 Td [(n = C 1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 X n = r 1 j z j n = C 1 1 X k =1 j z j r 1 k .25 Thusinorderforthissumtoconverge, j z j
PAGE 247

239 Figure14.10: TheROCofaleft-sidedsequence. If x [ n ] isatwo-sidedsequence,theROCwillbearinginthez-planethatisboundedon theinteriorandexteriorbyapole. A two-sidedsequence isansequencewithinniteduration inthepositiveandnegativedirections.Fromthederivationoftheabovetwoproperties,itfollowsthat if r 2 < j z j
PAGE 248

240 CHAPTER14.Z-TRANSFORMANDDIGITALFILTERING Figure14.12: TheROCofatwo-sidedsequence. 14.3.3Examples Togainfurtherinsightitisgoodtolookatacoupleofexamples. Example14.3 Letstake x 1 [ n ]= 1 2 n u [ n ]+ 1 4 n u [ n ] .26 Thez-transformof )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(1 2 n u [ n ] is z z )]TJ/F6 4.9813 Tf 7.422 2.677 Td [(1 2 withanROCat j z j > 1 2 Figure14.13: TheROCof )]TJ/F57 5.9776 Tf 5.419 -3.569 Td [(1 2 n u [ n ]

PAGE 249

241 Thez-transformof )]TJ/F13 6.9738 Tf 5.762 -4.147 Td [()]TJ/F7 6.9738 Tf 6.226 0 Td [(1 4 n u [ n ] is z z + 1 4 withanROCat j z j > )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 4 Figure14.14: TheROCof )]TJ/F61 5.9776 Tf 5.419 -3.569 Td [()]TJ/F57 5.9776 Tf 5.756 0 Td [(1 4 n u [ n ] Duetolinearity, X 1 [ z ]= z z )]TJ/F6 4.9813 Tf 7.422 2.678 Td [(1 2 + z z + 1 4 = 2 z z )]TJ/F6 4.9813 Tf 7.422 2.677 Td [(1 8 z )]TJ/F6 4.9813 Tf 7.422 2.677 Td [(1 2 z + 1 4 .27 Byobservationitisclearthattherearetwozeros,at 0 and 1 8 ,andtwopoles,at 1 2 ,and )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 4 Followingtheoboveproperties,theROCis j z j > 1 2 .

PAGE 250

242 CHAPTER14.Z-TRANSFORMANDDIGITALFILTERING Figure14.15: TheROCof x 1 [ n ]= )]TJ/F57 5.9776 Tf 5.419 -3.57 Td [(1 2 n u [ n ]+ )]TJ/F61 5.9776 Tf 5.419 -3.57 Td [()]TJ/F57 5.9776 Tf 5.756 0 Td [(1 4 n u [ n ] Example14.4 Nowtake x 2 [ n ]= )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 4 n u [ n ] )]TJ/F1 9.9626 Tf 9.962 14.047 Td [( 1 2 n u [ )]TJ/F11 9.9626 Tf 7.749 0 Td [(n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1] .28 Thez-transformandROCof )]TJ/F13 6.9738 Tf 5.761 -4.147 Td [()]TJ/F7 6.9738 Tf 6.227 0 Td [(1 4 n u [ n ] wasshownintheexampleaboveExample14.3.The z-transormof )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F1 9.9626 Tf 9.409 8.07 Td [(\000 1 2 n u [ )]TJ/F11 9.9626 Tf 7.749 0 Td [(n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1] is z z )]TJ/F6 4.9813 Tf 7.423 2.678 Td [(1 2 withanROCat j z j > 1 2 .

PAGE 251

243 Figure14.16: TheROCof )]TJ/F60 8.9664 Tf 4.223 -7.243 Td [()]TJ/F62 8.9664 Tf 8.704 7.243 Td [(\000 1 2 n u [ )]TJ/F58 8.9664 Tf 7.167 0 Td [(n )]TJ/F56 8.9664 Tf 9.215 0 Td [(1] Onceagain,bylinearity, X 2 [ z ]= z z + 1 4 + z z )]TJ/F6 4.9813 Tf 7.422 2.678 Td [(1 2 = z 2 z )]TJ/F6 4.9813 Tf 7.422 2.677 Td [(1 8 z + 1 4 z )]TJ/F6 4.9813 Tf 7.422 2.677 Td [(1 2 .29 Byobservationitisagainclearthattherearetwozeros,at 0 and 1 16 ,andtwopoles,at 1 2 ,and )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 4 inthscasethough,theROCis j z j < 1 2 .

PAGE 252

244 CHAPTER14.Z-TRANSFORMANDDIGITALFILTERING Figure14.17: TheROCof x 2 [ n ]= )]TJ/F61 5.9776 Tf 5.419 -3.57 Td [()]TJ/F57 5.9776 Tf 5.756 0 Td [(1 4 n u [ n ] )]TJ/F62 8.9664 Tf 9.215 7.243 Td [()]TJ/F57 5.9776 Tf 5.419 -3.57 Td [(1 2 n u [ )]TJ/F58 8.9664 Tf 7.167 0 Td [(n )]TJ/F56 8.9664 Tf 9.216 0 Td [(1] 14.4InverseZ-Transform 4 Whenusingthez-transformSection14.1 X z = 1 X n = )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x [ n ] z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n .30 itisoftenusefultobeabletond x [ n ] given X z .Thereareatleast4dierentmethodstodothis: 1.InspectionSection14.4.1:InspectionMethod 2.Partial-FractionExpansionSection14.4.2:Partial-FractionExpansionMethod 3.PowerSeriesExpansionSection14.4.3:PowerSeriesExpansionMethod 4.ContourIntegrationSection14.4.4:ContourIntegrationMethod 14.4.1InspectionMethod This"method"istobasicallybecomefamiliarwiththez-transformpairtablesSection14.2andthen "reverseengineer". Example14.5 Whengiven X z = z z )]TJ/F11 9.9626 Tf 9.963 0 Td [( withanROCSection14.3of j z j > 4 Thiscontentisavailableonlineat.

PAGE 253

245 wecoulddetermine"byinspection"that x [ n ]= n u [ n ] 14.4.2Partial-FractionExpansionMethod Whendealingwith lineartime-invariantsystems thez-transformoftenintheform X z = B z A z = P M k =0 b k z )]TJ/F9 4.9813 Tf 5.396 0 Td [(k P N k =0 a k z )]TJ/F9 4.9813 Tf 5.396 0 Td [(k .31 Thiscanalsoexpressedas X z = a 0 b 0 Q M k =1 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(c k z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Q N k =1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(d k z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 .32 where c k representsthenonzerozerosof X z and d k representsthenonzeropoles. If M 2 .Inthiscase M = N =2 ,sowehavetouselongdivisiontoget X z = 1 2 + 1 2 + 7 2 z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 +2 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Nextfactorthedenominator. X z =2+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+5 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Nowdopartial-fractionexpansion. X z = 1 2 + A 1 1 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 + A 2 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 1 2 + 9 2 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Noweachtermcanbeinvertedusingtheinspectionmethodandthez-transformtable.Thus,since theROCis j z j > 2 x [ n ]= 1 2 [ n ]+ 9 2 2 n u [ n ]+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 u [ n ] 5 "PartialFractionExpansion"

PAGE 254

246 CHAPTER14.Z-TRANSFORMANDDIGITALFILTERING 14.4.3PowerSeriesExpansionMethod Whenthez-transformisdenedasapowerseriesintheform X z = 1 X n = )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(x [ n ] z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n .35 theneachtermofthesequence x [ n ] canbedeterminedbylookingatthecoecientsoftherespectivepower of z )]TJ/F10 6.9738 Tf 6.226 0 Td [(n Example14.7 Nowlookatthez-transformofa nite-lengthsequence X z = z 2 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(1+2 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )]TJ/F8 9.9626 Tf 10.792 -8.069 Td [(1 )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(1 2 z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F8 9.9626 Tf 10.793 -8.069 Td [(1+ z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = z 2 + 5 2 z + 1 2 + )]TJ/F14 9.9626 Tf 4.566 -8.069 Td [()]TJ/F1 9.9626 Tf 9.409 8.069 Td [()]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(z )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 .36 Inthiscase,sincetherewerenopoles,wemultipliedthefactorsof X z .Now,byinspection,it isclearthat x [ n ]= [ n +2]+ 5 2 [ n +1]+ 1 2 [ n ]+ )]TJ/F8 9.9626 Tf 9.409 0 Td [( [ n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1] Oneoftheadvantagesofthepowerseriesexpansionmethodisthatmanyfunctionsencounteredinengineeringproblemshavetheirpowerseries'tabulated.Thusfunctionssuchaslog,sin,exponent,sinh,etc, canbeeasilyinverted. Example14.8 Suppose X z =log n )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Notingthat log n + x = 1 X n =1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 n +1 x n n Then X z = 1 X n =1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 n +1 n z )]TJ/F10 6.9738 Tf 6.227 0 Td [(n n Therefore X z = 8 < : )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 n +1 n n if n 1 0 if n 0 14.4.4ContourIntegrationMethod Withoutgoingintomuchdetail x [ n ]= 1 2 j I r X z z n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 dz .37 where r isacounter-clockwisecontourintheROCof X z encirclingtheoriginofthez-plane.Tofurther expandonthismethodofndingtheinverserequirestheknowledgeofcomplexvariabletheoryandthus willnotbeaddressedinthismodule.

PAGE 255

247 14.5RationalFunctions 6 14.5.1Introduction Whendealingwithoperationsonpolynomials,theterm rationalfunction isasimplewaytodescribea particularrelationshipbetweentwopolynomials. Denition14.1:rationalfunction Foranytwopolynomials,AandB,theirquotientiscalledarationalfunction. Example Belowisasimpleexampleofabasicrationalfunction, f x .Notethatthenumeratorand denominatorcanbepolynomialsofanyorder,buttherationalfunctionisundenedwhenthe denominatorequalszero. f x = x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 2 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 .38 IfyouhavebeguntostudytheZ-transformSection14.1,youshouldhavenoticedbynowtheyareall rationalfunctions.Belowwewilllookatsomeofthepropertiesofrationalfunctionsandhowtheycanbe usedtorevealimportantcharacteristicsaboutaz-transform,andthusasignalorLTIsystem. 14.5.2PropertiesofRationalFunctions Inordertoseewhatmakesrationalfunctionsspecial,letuslookatsomeoftheirbasicpropertiesand characteristics.Ifyouarefamiliarwithrationalfunctionsandbasicalgebraicproperties,skiptothenext sectionSection14.5.3:RationalFunctionsandtheZ-Transformtoseehowrationalfunctionsareuseful whendealingwiththez-transform. 14.5.2.1Roots Tounderstandmanyofthefollowingcharacteristicsofarationalfunction,onemustbeginbyndingthe rootsoftherationalfunction.Inordertodothis,letusfactorbothofthepolynomialssothattheroots canbeeasilydetermined.Likeallpolynomials,therootswillprovideuswithinformationonmanykey properties.Thefunctionbelowshowstheresultsoffactoringtheaboverationalfunction,.38. f x = x +2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x +3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 .39 Thus,therootsoftherationalfunctionareasfollows: Rootsofthenumeratorare: f)]TJ/F8 9.9626 Tf 12.73 0 Td [(2 ; 2 g Rootsofthedenominatorare: f)]TJ/F8 9.9626 Tf 12.731 0 Td [(3 ; 1 g note: Inordertounderstandrationalfunctions,itisessentialtoknowandunderstandtheroots thatmakeuptherationalfunction. 14.5.2.2Discontinuities Becausewearedealingwithdivisionoftwopolynomials,wemustbeawareofthevaluesofthevariablethat willcausethedenominatorofourfractiontobezero.Whenthishappens,therationalfunctionbecomes undened, i.e. wehaveadiscontinuityinthefunction.Becausewehavealreadysolvedforourroots,it isveryeasytoseewhenthisoccurs.Whenthevariableinthedenominatorequalsanyoftherootsofthe denominator,thefunctionbecomesundened. 6 Thiscontentisavailableonlineat.

PAGE 256

248 CHAPTER14.Z-TRANSFORMANDDIGITALFILTERING Example14.9 Continuingtolookatourrationalfunctionabove,.38,wecanseethatthefunctionwillhave discontinuitiesatthefollowingpoints: x = f)]TJ/F8 9.9626 Tf 12.73 0 Td [(3 ; 1 g InrespecttotheCartesianplane,wesaythatthediscontinuitiesarethevaluesalongthex-axiswherethe functioninundened.Thesediscontinuitiesoftenappearas verticalasymptotes onthegraphtorepresent thevalueswherethefunctionisundened. 14.5.2.3Domain Usingtherootsthatwefoundabove,the domain oftherationalfunctioncanbeeasilydened. Denition14.2:domain Thegroup,orset,ofvaluesthataredenedbyagivenfunction. Example Usingtherationalfunctionabove,.38,thedomaincanbedenedasanyrealnumber x where x doesnotequal1ornegative3.Writtenoutmathematical,wegetthefollowing: f x 2 R j x 6 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 andx 6 =1 g .40 14.5.2.4Intercepts The x-intercept isdenedasthepointswhere f x i.e. theoutputoftherationalfunctions,equalszero. Becausewehavealreadyfoundtherootsoftheequationthisprocessisverysimple.Fromalgebra,weknow thattheoutputwillbezerowheneverthenumeratoroftherationalfunctionisequaltozero.Therefore,the functionwillhaveanx-interceptwherever x equalsoneoftherootsofthenumerator. The y-intercept occurswhenever x equalszero.Thiscanbefoundbysettingallthevaluesof x equal tozeroandsolvingtherationalfunction. 14.5.3RationalFunctionsandtheZ-Transform Aswehavestatedabove,allz-transformscanbewrittenasrationalfunctions,whichhavebecomethemost commonwayofrepresentingthez-transform.Becauseofthis,wecanusethepropertiesabove,especially thoseoftheroots,inordertorevealcertaincharacteristicsaboutthesignalorLTIsystemdescribedbythe z-transform. Belowisthegeneralformofthez-transformwrittenasarationalfunction: X z = b 0 + b 1 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 + + b M z )]TJ/F10 6.9738 Tf 6.226 0 Td [(M a 0 + a 1 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 + + a N z )]TJ/F10 6.9738 Tf 6.227 0 Td [(N .41 IfyouhavealreadylookedatthemoduleaboutUnderstandingPole/ZeroPlotsandtheZ-transformSection14.7,youshouldseehowtherootsoftherationalfunctionplayanimportantroleinunderstandingthe z-transform.Theequationabove,.41,canbeexpressedinfactoredformjustaswasdoneforthesimple rationalfunctionabove,see.39.Thus,wecaneasilyndtherootsofthenumeratoranddenominator ofthez-transform.Thefollowingtworelationshipsbecomeapparent: RelationshipofRootstoPolesandZeros Therootsofthenumeratorintherationalfunctionwillbethe zeros ofthez-transform Therootsofthedenominatorintherationalfunctionwillbethe poles ofthez-transform

PAGE 257

249 14.5.4Conclusion Oncewehaveusedourknowledgeofrationalfunctionstonditsroots,wecanmanipulateaz-transformin anumberofusefulways.WecanapplythisknowledgetorepresentinganLTIsystemgraphicallythrough aPole/ZeroPlotSection14.7,ortoanalyzeanddesignadigitallterthroughFilterDesignfromthe Z-TransformSection14.8. 14.6DierenceEquation 7 14.6.1Introduction OneofthemostimportantconceptsofDSPistobeabletoproperlyrepresenttheinput/outputrelationshiptoagivenLTIsystem.Alinearconstant-coecient dierenceequation LCCDEservesasaway toexpressjustthisrelationshipinadiscrete-timesystem.Writingthesequenceofinputsandoutputs, whichrepresentthecharacteristicsoftheLTIsystem,asadierenceequationhelpinunderstandingand manipulatingasystem. Denition14.3:dierenceequation Anequationthatshowstherelationshipbetweenconsecutivevaluesofasequenceandthedierencesamongthem.Theyareoftenrearrangedasarecursiveformulasothatasystemsoutputcan becomputedfromtheinputsignalandpastoutputs. Example y [ n ]+7 y [ n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1]+2 y [ n )]TJ/F8 9.9626 Tf 9.963 0 Td [(2]= x [ n ] )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x [ n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1] .42 14.6.2GeneralFormulasfromtheDierenceEquation Asstatedbrieyinthedenitionabove,adierenceequationisaveryusefultoolindescribingandcalculating theoutputofthesystemdescribedbytheformulaforagivensample n .Thekeypropertyofthedierence equationisitsabilitytohelpeasilyndthetransform, H z ,ofasystem.Inthefollowingtwosubsections, wewilllookatthegeneralformofthedierenceequationandthegeneralconversiontoaz-transformdirectly fromthedierenceequation. 14.6.2.1DierenceEquation Thegeneralformofalinear,constant-coecientdierenceequationLCCDE,isshownbelow: N X k =0 a k y [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ]= M X k =0 b k x [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k ] .43 Wecanalsowritethegeneralformtoeasilyexpressarecursiveoutput,whichlookslikethis: y [ n ]= )]TJ/F1 9.9626 Tf 9.409 17.036 Td [( N X k =1 a k y [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ] + M X k =0 b k x [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ] .44 Fromthisequation,notethat y [ n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k ] representstheoutputsand x [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ] representstheinputs.Thevalue of N representsthe order ofthedierenceequationandcorrespondstothememoryofthesystembeing represented.Becausethisequationreliesonpastvaluesoftheoutput,inordertocomputeanumerical solution,certainpastoutputs,referredtoasthe initialconditions ,mustbeknown. 7 Thiscontentisavailableonlineat.

PAGE 258

250 CHAPTER14.Z-TRANSFORMANDDIGITALFILTERING 14.6.2.2ConversiontoZ-Transform Usingtheaboveformula,.43,wecaneasilygeneralizethe transferfunction H z ,foranydierence equation.Belowarethestepstakentoconvertanydierenceequationintoitstransferfunction, i.e. ztransform.TherststepinvolvestakingtheFourierTransform 8 ofallthetermsin.43.Thenweuse thelinearitypropertytopullthetransforminsidethesummationandthetime-shiftingpropertyofthe z-transformtochangethetime-shiftingtermstoexponentials.Oncethisisdone,wearriveatthefollowing equation: a 0 =1 Y z = )]TJ/F1 9.9626 Tf 9.409 17.037 Td [( N X k =1 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a k Y z z )]TJ/F10 6.9738 Tf 6.226 0 Td [(k + M X k =0 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(b k X z z )]TJ/F10 6.9738 Tf 6.226 0 Td [(k .45 H z = Y z X z = P M k =0 b k z )]TJ/F9 4.9813 Tf 5.396 0 Td [(k 1+ P N k =1 a k z )]TJ/F9 4.9813 Tf 5.397 0 Td [(k .46 14.6.2.3ConversiontoFrequencyResponse Oncethez-transformhasbeencalculatedfromthedierenceequation,wecangoonestepfurthertodene thefrequencyresponseofthesystem,orlter,thatisbeingrepresentedbythedierenceequation. note: Rememberthatthereasonwearedealingwiththeseformulasistobeabletoaidusin lterdesign.ALCCDEisoneoftheeasiestwaystorepresentFIRlters.Bybeingabletond thefrequencyresponse,wewillbeabletolookatthebasicpropertiesofanylterrepresentedby asimpleLCCDE. Belowisthegeneralformulaforthefrequencyresponseofaz-transform.Theconversionissimpleamatter oftakingthez-transformformula, H z ,andreplacingeveryinstanceof z with e jw H w = H z j z;z = e jw = P M k =0 b k e )]TJ/F6 4.9813 Tf 5.396 0 Td [( jwk P N k =0 a k e )]TJ/F6 4.9813 Tf 5.396 0 Td [( jwk .47 Onceyouunderstandthederivationofthisformula,lookatthemoduleconcerningFilterDesignfromthe Z-TransformSection14.8foralookintohowalloftheseideasoftheZ-transformSection14.1,Dierence Equation,andPole/ZeroPlotsSection14.7playaroleinlterdesign. 14.6.3Example Example14.10:FindingDierenceEquation Belowisabasicexampleshowingtheoppositeofthestepsabove:givenatransferfunctionone caneasilycalculatethesystemsdierenceequation. H z = z +1 2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(z )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(1 2 )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(z + 3 4 .48 Giventhistransferfunctionofatime-domainlter,wewanttondthedierenceequation.To beginwith,expandbothpolynomialsanddividethembythehighestorder z H z = z +1 z +1 z )]TJ/F6 4.9813 Tf 7.422 2.678 Td [(1 2 z + 3 4 = z 2 +2 z +1 z 2 +2 z +1 )]TJ/F6 4.9813 Tf 7.422 2.677 Td [(3 8 = 1+2 z )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 + z )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 1+ 1 4 z )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 )]TJ/F6 4.9813 Tf 7.422 2.677 Td [(3 8 z )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 .49 8 "DerivationoftheFourierTransform"

PAGE 259

251 Fromthistransferfunction,thecoecientsofthetwopolynomialswillbeour a k and b k values foundinthegeneraldierenceequationformula,.43.Usingthesecoecientsandtheabove formofthetransferfunction,wecaneasilywritethedierenceequation: x [ n ]+2 x [ n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1]+ x [ n )]TJ/F8 9.9626 Tf 9.963 0 Td [(2]= y [ n ]+ 1 4 y [ n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1] )]TJ/F8 9.9626 Tf 11.158 6.74 Td [(3 8 y [ n )]TJ/F8 9.9626 Tf 9.963 0 Td [(2] .50 Inournalstep,wecanrewritethedierenceequationinitsmorecommonformshowingthe recursivenatureofthesystem. y [ n ]= x [ n ]+2 x [ n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1]+ x [ n )]TJ/F8 9.9626 Tf 9.963 0 Td [(2]+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 4 y [ n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1]+ 3 8 y [ n )]TJ/F8 9.9626 Tf 9.962 0 Td [(2] .51 14.6.4SolvingaLCCDE Inorderforalinearconstant-coecientdierenceequationtobeusefulinanalyzingaLTIsystem,wemust beabletondthesystemsoutputbaseduponaknowninput, x n ,andasetofinitialconditions.Two commonmethodsexistforsolvingaLCCDE:the directmethod andthe indirectmethod ,thelater beingbasedonthez-transform.BelowwewillbrieydiscusstheformulasforsolvingaLCCDEusingeach ofthesemethods. 14.6.4.1DirectMethod Thenalsolutiontotheoutputbasedonthedirectmethodisthesumoftwoparts,expressedinthefollowing equation: y n = y h n + y p n .52 Therstpart, y h n ,isreferredtoasthe homogeneoussolution andthesecondpart, y h n ,isreferred toas particularsolution .Thefollowingmethodisverysimilartothatusedtosolvemanydierential equations,soifyouhavetakenadierentialcalculuscourseoruseddierentialequationsbeforethenthis shouldseemveryfamiliar. 14.6.4.1.1HomogeneousSolution Webeginbyassumingthattheinputiszero, x n =0 .Nowwesimplyneedtosolvethehomogeneous dierenceequation: N X k =0 a k y [ n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ]=0 .53 Inordertosolvethis,wewillmaketheassumptionthatthesolutionisintheformofanexponential.We willuselambda, ,torepresentourexponentialterms.Wenowhavetosolvethefollowingequation: N X k =0 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(a k n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k =0 .54 Wecanexpandthisequationoutandfactoroutallofthelambdaterms.Thiswillgiveusalargepolynomial inparenthesis,whichisreferredtoasthe characteristicpolynomial .Therootsofthispolynomialwill bethekeytosolvingthehomogeneousequation.Iftherearealldistinctroots,thenthegeneralsolutionto theequationwillbeasfollows: y h n = C 1 1 n + C 2 2 n + + C N N n .55

PAGE 260

252 CHAPTER14.Z-TRANSFORMANDDIGITALFILTERING However,ifthecharacteristicequationcontainsmultiplerootsthentheabovegeneralsolutionwillbeslightly dierent.Belowwehavethemodiedversionforanequationwhere 1 has K multipleroots: y h n = C 1 1 n + C 1 n 1 n + C 1 n 2 1 n + + C 1 n K )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 n + C 2 2 n + + C N N n .56 14.6.4.1.2ParticularSolution Theparticularsolution, y p n ,willbeanysolutionthatwillsolvethegeneraldierenceequation: N X k =0 a k y p n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k = M X k =0 b k x n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k .57 Inordertosolve,ourguessforthesolutionto y p n willtakeontheformoftheinput, x n .Afterguessing atasolutiontotheaboveequationinvolvingtheparticularsolution,oneonlyneedstoplugthesolutioninto thedierenceequationandsolveitout. 14.6.4.2IndirectMethod Theindirectmethodutilizestherelationshipbetweenthedierenceequationandz-transform,discussed earlierSection14.6.2:GeneralFormulasfromtheDierenceEquation,tondasolution.Thebasicidea istoconvertthedierenceequationintoaz-transform,asdescribedaboveSection14.6.2.2:Conversionto Z-Transform,togettheresultingoutput, Y z .Thenbyinversetransformingthisandusingpartial-fraction expansion,wecanarriveatthesolution. 14.7UnderstandingPole/ZeroPlotsontheZ-Plane 9 14.7.1IntroductiontoPolesandZerosoftheZ-Transform OncetheZ-transformofasystemhasbeendetermined,onecanusetheinformationcontainedinfunction's polynomialstographicallyrepresentthefunctionandeasilyobservemanydeningcharacteristics.The Z-transformwillhavethebelowstructure,basedonRationalFunctionsSection14.5: X z = P z Q z .58 Thetwopolynomials, P z and Q z ,allowustondthepolesandzerosSection13.6oftheZTransform. Denition14.4:zeros 1.Thevaluesfor z where P z =0 2.Thecomplexfrequenciesthatmaketheoverallgainoftheltertransferfunctionzero. Denition14.5:poles 1.Thevaluesfor z where Q z =0 2.Thecomplexfrequenciesthatmaketheoverallgainoftheltertransferfunctioninnite. Example14.11 Belowisasimpletransferfunctionwiththepolesandzerosshownbelowit. H z = z +1 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(z )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(1 2 )]TJ/F11 9.9626 Tf 10.793 -8.069 Td [(z + 3 4 9 Thiscontentisavailableonlineat.

PAGE 261

253 Thezerosare: f)]TJ/F8 9.9626 Tf 12.73 0 Td [(1 g Thepolesare: 1 2 ; )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F7 6.9738 Tf 5.761 -4.147 Td [(3 4 14.7.2TheZ-Plane OncethepolesandzeroshavebeenfoundforagivenZ-Transform,theycanbeplottedontotheZ-Plane. TheZ-planeisacomplexplanewithanimaginaryandrealaxisreferringtothecomplex-valuedvariable z Thepositiononthecomplexplaneisgivenby re j andtheanglefromthepositive,realaxisaroundthe planeisdenotedby .Whenmappingpolesandzerosontotheplane,polesaredenotedbyan"x"andzeros byan"o".ThebelowgureshowstheZ-Plane,andexamplesofplottingzerosandpolesontotheplanecan befoundinthefollowingsection. Z-Plane Figure14.18 14.7.3ExamplesofPole/ZeroPlots Thissectionlistsseveralexamplesofndingthepolesandzerosofatransferfunctionandthenplotting themontotheZ-Plane. Example14.12:SimplePole/ZeroPlot H z = z )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(z )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(1 2 )]TJ/F11 9.9626 Tf 10.793 -8.069 Td [(z + 3 4 Thezerosare: f 0 g Thepolesare: 1 2 ; )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F7 6.9738 Tf 5.761 -4.148 Td [(3 4

PAGE 262

254 CHAPTER14.Z-TRANSFORMANDDIGITALFILTERING Pole/ZeroPlot Figure14.19: Usingthezerosandpolesfoundfromthetransferfunction,theonezeroismappedto zeroandthetwopolesareplacedat 1 2 and )]TJ/F62 8.9664 Tf 8.703 7.243 Td [()]TJ/F57 5.9776 Tf 5.419 -3.57 Td [(3 4 Example14.13:ComplexPole/ZeroPlot H z = z )]TJ/F11 9.9626 Tf 9.963 0 Td [(j z + j )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(z )]TJ/F1 9.9626 Tf 9.963 8.07 Td [()]TJ/F7 6.9738 Tf 5.761 -4.147 Td [(1 2 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(1 2 j )]TJ/F11 9.9626 Tf 15.359 -8.07 Td [(z )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(1 2 + 1 2 j Thezerosare: f j; )]TJ/F11 9.9626 Tf 7.748 0 Td [(j g Thepolesare: )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; 1 2 + 1 2 j; 1 2 )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(1 2 j Pole/ZeroPlot Figure14.20: Usingthezerosandpolesfoundfromthetransferfunction,thezerosaremappedto j ,andthepolesareplacedat )]TJ/F56 8.9664 Tf 7.167 0 Td [(1 1 2 + 1 2 j and 1 2 )]TJ/F57 5.9776 Tf 10.411 3.673 Td [(1 2 j

PAGE 263

255 MATLAB -IfaccesstoMATLABisreadilyavailable,thenyoucanuseitsfunctionstoeasilycreate pole/zeroplots.Belowisashortprogramthatplotsthepolesandzerosfromtheaboveexampleontothe Z-Plane. %Setupvectorforzeros z=[j;-j]; %Setupvectorforpoles p=[-1;.5+.5j;.5-.5j]; figure; zplanez,p; title'Pole/ZeroPlotforComplexPole/ZeroPlotExample'; 14.7.4Pole/ZeroPlotandRegionofConvergence TheregionofconvergenceROCfor X z inthecomplexZ-planecanbedeterminedfromthepole/zero plot.Althoughseveralregionsofconvergencemaybepossible,whereeachonecorrespondstoadierent impulseresponse,therearesomechoicesthataremorepractical.AROCcanbechosentomakethetransfer functioncausaland/orstabledependingonthepole/zeroplot. FilterPropertiesfromROC IftheROCextendsoutwardfromtheoutermostpole,thenthesystemis causal IftheROCincludestheunitcircle,thenthesystemis stable Belowisapole/zeroplotwithapossibleROCoftheZ-transformintheSimplePole/ZeroPlotExample14.12:SimplePole/ZeroPlotdiscussedearlier.TheshadedregionindicatestheROCchosenforthe lter.Fromthisgure,wecanseethatthelterwillbebothcausalandstablesincetheabovelisted conditionsarebothmet. Example14.14 H z = z )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(z )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(1 2 )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(z + 3 4

PAGE 264

256 CHAPTER14.Z-TRANSFORMANDDIGITALFILTERING RegionofConvergenceforthePole/ZeroPlot Figure14.21: TheshadedarearepresentsthechosenROCforthetransferfunction. 14.7.5FrequencyResponseandtheZ-Plane Thereasonitishelpfultounderstandandcreatethesepole/zeroplotsisduetotheirabilitytohelpuseasily designalter.Basedonthelocationofthepolesandzeros,themagnituderesponseoftheltercanbe quicklyunderstood.Also,bystartingwiththepole/zeroplot,onecandesignalterandobtainitstransfer functionveryeasily.RefertothismoduleSection14.8forinformationontherelationshipbetweenthe pole/zeroplotandthefrequencyresponse. 14.8FilterDesignusingthePole/ZeroPlotofaZ-Transform 10 14.8.1EstimatingFrequencyResponsefromZ-Plane Oneofthemotivatingfactorsforanalyzingthepole/zeroplotsisduetotheirrelationshiptothefrequency responseofthesystem.Basedonthepositionofthepolesandzeros,onecanquicklydeterminethefrequency response.Thisisaresultofthecorrespondencebetweenthefrequencyresponseandthetransferfunction evaluatedontheunitcircleinthepole/zeroplots.Thefrequencyresponse,orDTFT,ofthesystemis denedas: H w = H z j z;z = e jw = P M k =0 b k e )]TJ/F6 4.9813 Tf 5.396 0 Td [( jwk P N k =0 a k e )]TJ/F6 4.9813 Tf 5.396 0 Td [( jwk .59 Next,byfactoringthetransferfunctionintopolesandzerosandmultiplyingthenumeratoranddenominator by e jw wearriveatthefollowingequations: H w = j b 0 a 0 j Q M k =1 )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [(j e jw )]TJ/F11 9.9626 Tf 9.962 0 Td [(c k j Q N k =1 j e jw )]TJ/F11 9.9626 Tf 9.962 0 Td [(d k j .60 10 Thiscontentisavailableonlineat.

PAGE 265

257 From.60wehavethefrequencyresponseinaformthatcanbeusedtointerpretphysicalcharacteristics aboutthelter'sfrequencyresponse.Thenumeratoranddenominatorcontainaproductoftermsofthe form j e jw )]TJ/F11 9.9626 Tf 9.537 0 Td [(h j ,where h iseitherazero,denotedby c k orapole,denotedby d k .Vectorsarecommonlyused torepresentthetermanditspartsonthecomplexplane.Thepoleorzero, h ,isavectorfromtheoriginto itslocationanywhereonthecomplexplaneand e jw isavectorfromtheorigintoitslocationontheunit circle.Thevectorconnectingthesetwopoints, j e jw )]TJ/F11 9.9626 Tf 10.285 0 Td [(h j ,connectsthepoleorzerolocationtoaplaceon theunitcircledependentonthevalueof w .Fromthis,wecanbegintounderstandhowthemagnitudeof thefrequencyresponseisaratioofthedistancestothepolesandzeropresentinthez-planeas w goesfrom zerotopi.Thesecharacteristicsallowustointerpret j H w j asfollows: j H w j = j b 0 a 0 j Q "distancesfromzeros" Q "distancesfrompoles" .61 Inconclusion,usingthedistancesfromtheunitcircletothepolesandzeros,wecanplotthefrequency responseofthesystem.As w goesfrom 0 to 2 ,thefollowingtwoproperties,takenfromtheaboveequations, specifyhowoneshoulddraw j H w j Whilemovingaroundtheunitcircle... 1.ifclosetoazero,thenthemagnitudeissmall.Ifazeroisontheunitcircle,thenthefrequency responseiszeroatthatpoint. 2.ifclosetoapole,thenthemagnitudeislarge.Ifapoleisontheunitcircle,thenthefrequencyresponse goestoinnityatthatpoint. 14.8.2DrawingFrequencyResponsefromPole/ZeroPlot Letusnowlookatseveralexamplesofdeterminingthemagnitudeofthefrequencyresponsefromthe pole/zeroplotofaz-transform.Ifyouhaveforgottenorareunfamiliarwithpole/zeroplots,pleaserefer backtothePole/ZeroPlotsSection14.7module. Example14.15 Inthisrstexamplewewilltakealookattheverysimplez-transformshownbelow: H z = z +1=1+ z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 H w =1+ e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jw Forthisexample,someofthevectorsrepresentedby j e jw )]TJ/F11 9.9626 Tf 8.717 0 Td [(h j ,forrandomvaluesof w ,areexplicitly drawnontothecomplexplaneshowninthegurebelow.Thesevectorsshowhowtheamplitude ofthefrequencyresponsechangesas w goesfrom 0 to 2 ,andalsoshowthephysicalmeaning ofthetermsin.60above.Onecanseethatwhen w =0 ,thevectoristhelongestandthus thefrequencyresponsewillhaveitslargestamplitudehere.As w approaches ,thelengthofthe vectorsdecreaseasdoestheamplitudeof j H w j .Sincetherearenopolesinthetransform,there isonlythisonevectortermratherthanaratioasseenin.60.

PAGE 266

258 CHAPTER14.Z-TRANSFORMANDDIGITALFILTERING aPole/ZeroPlot bFrequencyResponse:|Hw| Figure14.22: Therstgurerepresentsthepole/zeroplotwithafewrepresentativevectorsgraphed whilethesecondshowsthefrequencyresponsewithapeakat+2andgraphedbetweenplusandminus Example14.16 Forthisexample,amorecomplextransferfunctionisanalyzedinordertorepresentthesystem's frequencyresponse. H z = z z )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(1 2 = 1 1 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(1 2 z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 H w = 1 1 )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(1 2 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jw Belowwecanseethetwoguresdescribedbytheaboveequations.TheFigure14.23a Pole/ZeroPlotrepresentsthebasicpole/zeroplotofthez-transform, H w .Figure14.23b FrequencyResponse:|Hw|showsthemagnitudeofthefrequencyresponse.Fromtheformulas andstatementsintheprevioussection,wecanseethatwhen w =0 thefrequencywillpeaksince itisatthisvalueof w thatthepoleisclosesttotheunitcircle.Theratiofrom.60helpsussee themathematicsbehindthisconclusionandtherelationshipbetweenthedistancesfromtheunit circleandthepolesandzeros.As w movesfrom 0 to ,weseehowthezerobeginstomaskthe eectsofthepoleandthusforcethefrequencyresponsecloserto 0 .

PAGE 267

259 aPole/ZeroPlot bFrequencyResponse:|Hw| Figure14.23: Therstgurerepresentsthepole/zeroplotwhilethesecondshowsthefrequency responsewithapeakat+2andgraphedbetweenplusandminus .

PAGE 268

260 CHAPTER14.Z-TRANSFORMANDDIGITALFILTERING

PAGE 269

Chapter15 Appendix:HilbertSpacesand OrthogonalExpansions 15.1VectorSpaces 1 15.1.1 Introduction Denition15.1:Vectorspace Alinearvectorspace S isacollectionof"vectors"suchthatif f 1 2 S f 1 2 S forallscalars where 2 R or 2 C andif f 1 2 S f 2 2 S ,then f 1 + f 2 2 S Todeneanabstractlinearvectorspace,weneed Asetofthingscalled"vectors" X Asetofthingscalled"scalars" A Avectoradditionoperator + Ascalarmultiplicationoperator Theoperatorsneedtohaveallthepropertiesofgivenbelow.Closureisusuallythemostimportanttoshow. 15.1.2VectorSpaces Ifthescalars arereal, S iscalleda realvectorspace Ifthescalars arecomplex, S iscalleda complexvectorspace Ifthe"vectors"in S arefunctionsofacontinuousvariable,wesometimescall S a linearfunctionspace 15.1.2.1Properties Wedeneaset V tobeavectorspaceif 1. x + y = y + x foreach x and y in V 2. x + y + z = x + y + z foreach x y ,and z in V 3.Thereisaunique"zerovector"suchthat x +0= x foreach x in V 4.Foreach x in V thereisauniquevector )]TJ/F83 9.9626 Tf 7.749 0 Td [(x suchthat x + )]TJ/F83 9.9626 Tf 7.749 0 Td [(x =0 5. 1 x = x 6. c 1 c 2 x = c 1 c 2 x foreach x in V and c 1 and c 2 in C 1 Thiscontentisavailableonlineat. 261

PAGE 270

262 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS 7. c x + y = c x + c y foreach x and y in V and c in C 8. c 1 + c 2 x = c 1 x + c 2 x foreach x in V and c 1 and c 2 in C 15.1.2.2Examples R n =realvectorspace C n =complexvectorspace L 1 R = n f t j R 1 j f t j dt< 1 o isavectorspace L 1 R = f f t j f t isbounded g isavectorspace L 2 R = n f t j R 1 j f t j 2 dt< 1 o =niteenergysignals isavectorspace L 2 [0 ;T ]=niteenergyfunctionsoninterval[0 ; T] ` 1 Z ` 2 Z ` 1 Z arevectorspaces Thecollectionoffunctionspiecewiseconstantbetweentheintegersisavectorspace Figure15.1 R + 2 = 8 < : 0 @ x 0 x 1 1 A j x 0 > 0 andx 1 > 0 9 = ; is not avectorspace. 0 @ 1 1 1 A 2 R + 2 ,but 0 @ 1 1 1 A = 2 R + 2 ;< 0 D = f z 2 C ; j z j 1 g is not avectorspace. z 1 =1 2 D z 2 = j 2 D ,but z 1 + z 2 = 2 D j z 1 + z 2 j = p 2 > 1 note: Vectorspacescanbecollectionsoffunctions,collectionsofsequences,aswellascollections oftraditionalvectors i.e. nitelistsofnumbers

PAGE 271

263 15.2Norms 2 15.2.1Introduction Muchofthelanguageinthissectionwillbefamiliartoyou-youshouldhavepreviouslybeenexposedto theconceptsof innerproductsSection15.3 orthogonality basisexpansionsSection15.8 inthecontextof R n .We'regoingtotakewhatweknowaboutvectorsandapplyittofunctionscontinuous timesignals. 15.2.2Norms The norm ofavectorisarealnumberthatrepresentsthe"size"ofthevector. Example15.1 In R 2 ,wecandeneanormtobeavectorsgeometriclength. Figure15.2 x = x 0 ;x 1 T ,norm k x k = p x 0 2 + x 1 2 Mathematically,anorm kk isjustafunctiontakingavectorandreturningarealnumber thatsatisesthreerules. Tobeanorm, kk mustsatisfy: 1.thenormofeveryvectorispositive k x k > 0 ; x 2 S 2.scalingavectorscalesthenormbythesameamount k x k = j jk x k forallvectors x andscalars 3.TriangleProperty: k x + y kk x k + k y k forallvectors x y ."The"size"ofthesumoftwovectors islessthanorequaltothesumoftheirsizes" AvectorspaceSection15.1withawelldenednormiscalleda normedvectorspace or normed linearspace 2 Thiscontentisavailableonlineat.

PAGE 272

264 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS 15.2.2.1Examples Example15.2 R n or C n x = 0 B B B B B @ x 0 x 1 ::: x n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 C C C C C A k x k 1 = P n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 i =0 j x i j R n withthisnormiscalled ` 1 [0 ;n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1] Figure15.3: Collectionofall x 2 R 2 with k x k 1 =1 Example15.3 R n or C n ,withnorm k x k 2 = P n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 i =0 j x i j 2 1 2 R n iscalled ` 2 [0 ;n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1] theusual"Euclidean"norm. Figure15.4: Collectionofall x 2 R 2 with k x k 2 =1

PAGE 273

265 Example15.4 R n or C n ,withnorm k x k 1 =max i fj x i jg iscalled ` 1 [0 ;n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1] Figure15.5: x 2 R 2 with k x k 1 =1 15.2.2.2SpacesofSequencesandFunctions Wecandenesimilarnormsforspacesofsequencesandfunctions. Discretetimesignals=sequencesofnumbers x [ n ]= f :::;x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 ;x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ;x 0 ;x 1 ;x 2 ;::: g k x n k 1 = P 1 i = j x [ i ] j x [ n ] 2 ` 1 Z k x k 1 < 1 k x n k 2 = P 1 i = j x [ i ] j 2 1 2 x [ n ] 2 ` 2 Z k x k 2 < 1 k x n k p = )]TJ 4.567 -0.598 Td [(P 1 i = j x [ i ] j p 1 p x [ n ] 2 ` p Z k x k p < 1 k x n k 1 =sup i j x [ i ] j x [ n ] 2 ` 1 Z k x k 1 < 1 Forcontinuoustimefunctions: k f t k p = R 1 j f t j p dt 1 p f t 2 L p R k f t k p < 1 Ontheinterval k f t k p = R T 0 j f t j p dt 1 p f t 2 L p [0 ;T ] k f t k p < 1 Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10768/latest/NormCalc.llb

PAGE 274

266 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS 15.3InnerProducts 3 15.3.1Denition:InnerProduct Youmayhaverunacross innerproducts ,alsocalled dotproducts ,on R n beforeinsomeofyourmath orsciencecourses.Ifnot,wedenetheinnerproductasfollows,givenwehavesome x 2 R n and y 2 R n Denition15.2:innerproduct Theinnerproductisdenedmathematicallyas: < x ; y > = y T x = y 0 y 1 :::y n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 0 B B B B B @ x 0 x 1 . x n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 C C C C C A = P n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 i =0 x i y i .1 15.3.1.1InnerProductin2-D Ifwehave x 2 R 2 and y 2 R 2 ,thenwecanwritetheinnerproductas < x ; y > = k x kk y k cos .2 where istheanglebetween x and y Figure15.6: Generalplotofvectorsandanglereferredtoinaboveequations. Geometrically,theinnerproducttellsusaboutthe strength of x inthedirectionof y Example15.5 Forexample,if k x k =1 ,then < x ; y > = k y k cos 3 Thiscontentisavailableonlineat.

PAGE 275

267 Figure15.7: Plotoftwovectorsfromaboveexample. Thefollowingcharacteristicsarerevealedbytheinnerproduct: < x ; y > measuresthelengthofthe projection of y onto x < x ; y > is maximum forgiven k x k k y k when x and y areinthesamedirection =0 cos =1 < x ; y > iszerowhen cos =0 =90 i.e. x and y are orthogonal 15.3.1.2InnerProductRules Ingeneral,aninnerproductonacomplexvectorspaceisjustafunctiontakingtwovectorsandreturning acomplexnumberthatsatisescertainrules: ConjugateSymmetry: < x ; y > = < x ; y > Scaling: < x ; y > = < x ; y > Additivity: < x + y ; z > = < x ; z > + < y ; z > "Positivity": < x ; x >> 0 ; x 6 =0 Denition15.3:orthogonal Wesaythat x and y areorthogonalif: < x ; y > =0 Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10755/latest/InnerProductCalc.llb

PAGE 276

268 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS 15.4HilbertSpaces 4 15.4.1HilbertSpaces Avectorspace S withavalidinnerproductSection15.3denedonitiscalledan innerproductspace whichisalsoa normedlinearspace .A Hilbertspace isaninnerproductspacethatiscompletewith respecttothenormdenedusingtheinnerproduct.HilbertspacesarenamedafterDavidHilbert 5 ,who developedthisideathroughhisstudiesofintegralequations.Wedeneourvalidnormusingtheinner productas: k x k = p < x ; x > .3 HilbertspacesareusefulinstudyingandgeneralizingtheconceptsofFourierexpansion,Fouriertransforms, andareveryimportanttothestudyofquantummechanics.Hilbertspacesarestudiedunderthefunctional analysisbranchofmathematics. 15.4.1.1ExamplesofHilbertSpaces BelowwewilllistafewexamplesofHilbertspaces 6 .Youcanverifythatthesearevalidinnerproductsat home. For C n < x ; y > = y T x = y 0 y 1 :::y n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 0 B B B B B @ x 0 x 1 . x n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 C C C C C A = n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 X i =0 x i y i Spaceofniteenergycomplexfunctions: L 2 R < f ; g > = Z 1 f t g t dt Spaceofsquare-summablesequences: ` 2 Z < x ; y > = 1 X i = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x [ i ] y [ i ] 15.5Cauchy-SchwarzInequality 7 15.5.1Introduction Recallin R 2 < x ; y > = k x kk y k cos j < x ; y > jk x kk y k .4 ThesamerelationholdsforinnerproductspacesSection15.3ingeneral... 4 Thiscontentisavailableonlineat. 5 http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hilbert.html 6 "HilbertSpaces" 7 Thiscontentisavailableonlineat.

PAGE 277

269 15.5.1.1Cauchy-SchwarzInequality Denition15.4:Cauchy-SchwarzInequality For x y inaninnerproductspace j < x ; y > jk x kk y k withequalityholding ifandonlyif x and y arelinearlydependentSection5.1.1:LinearIndependence, i.e. x = y forsomescalar 15.5.2MatchedFilterDetector AlsoreferredtoasCauchy-Schwarz's"KillerApp." 15.5.2.1ConceptbehindMatchedFilter Ifwearegiventwovectors, f and g ,thentheCauchy-SchwarzInequalityCSIis maximized when f = g Thistellsus: f isinthesame"direction"as g if f and g arefunctions, f = g means f and g havethesameshape. Forexample,sayweareinasituationwherewehaveasetofsignals,denedas f g 1 t ;g 2 t ;:::;g k t g andwewanttobeabletotellwhich,ifany,ofthesesignalsresembleanothergivensignal f t note: Inordertondthesignalsthatresembles f t wewilltaketheinnerproducts.If g i t resembles f t ,thentheabsolutevalueoftheinnerproduct, j j ,willbe large Thisideaofbeingabletomeasureandrankthe"likeness"oftwosignalsleadsustothe MatchedFilter Detector 15.5.2.2ComparingSignals ThesimplestuseoftheMatchedFilterwouldbetotakeasetof"candidate"signals,sayoursetof f g 1 t ;g 2 t ;:::;g k t g ,andtrytomatchittoa"template"signal, f t .Forexamplesaywearegiventhe belowtemplateFigure15.8TemplateSignalandcandidatesignalsFigure15.9CandidateSignals: TemplateSignal Figure15.8: Oursignalwewishtondmatchof.

PAGE 278

270 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS CandidateSignals a b Figure15.9: Clearlybylookingatthesewecanseewhichsignalwillprovidethebettermatchtoour templatesignal. Nowifouronlyquestionwaswhichfunctionwasaclosermatchto f t thenwecaneasilycomeup withtheanswerbasedoninspectiong 2 t .However,thiswillnotalwaysbethecase.Also,wemaywant todevelopamethod,oralgorithm,thatcouldautomatethesecomparisons.Orperhapswewishtohavea quantitativevalueexpressingjusthowsimilarthesignalsare.Toaddresstheseissues,wewilllayoutamore formalapproachtocomparingthesignals,whichwill,asmentionedabove,bebasedontheinnerproduct. Inordertoseewhichofourcandidatesignals, g 1 t or g 2 t ,bestresembles f t weneedtoperform thefollowingsteps: Normalizethe g i t Taketheinnerproductwith f t Findthebiggest! Or,puttingitmathematically: Bestcandidate= argmax i j < f ;g i > j k g i k .5 15.5.2.3FindingaPattern ExtendingthesethoughtsofusingtheMatchedFiltertondsimilaritiesamongsignals,wecanusethesame ideatosearchforapatterninalongsignal.Theideaissimplytorepeatedlyperformthesamecalculation aswedidpreviously;however,nowinsteadofcalculatingondierentsignalswewillsimplyperformthe innerproductwithdierentshiftedversionsofour"pattern"signal.Forexample,saywehavethefollowing twosignals-apatternsignalFigure15.10PatternSignalandlongsignalFigure15.11LongSignal.

PAGE 279

271 PatternSignal Figure15.10: Thepatternwearelookingforinaourlongsignalhavingalength T LongSignal Figure15.11: Hereisthelongsignalthatcontainsapiecethatresemblesourpatternsignal. Herewewilllookattwoshiftsofourpatternsignal,shiftingthesignalby s 1 and s 2 .Thesetwopossibilities yieldthefollowingcalculationsandresults: Shiftof s 1 : R s 1 + T s 1 g t f t )]TJ/F11 9.9626 Tf 9.963 0 Td [(s 1 dt q R s 1 + T s 1 j g t j 2 dt ="large" .6 Shiftof s 2 : R s 2 + T s 2 g t f t )]TJ/F11 9.9626 Tf 9.963 0 Td [(s 2 dt q R s 2 + T s 2 j g t j 2 dt ="small" .7 Therefore,wecandeneageneralizedequationforourmatchedlter: m s =matchedlter .8 m s = R s + T s g t f t )]TJ/F11 9.9626 Tf 9.963 0 Td [(s dt )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [(k g t kj L 2 [ s;s + T ] .9 wherethenumeratorin.9istheconvolutionof g t f )]TJ/F11 9.9626 Tf 7.748 0 Td [(t .Nowinordertodecidewhetherornot theresultfromourmatchedlterdetectorishighenoughtoindicateanacceptablematchbetweenthetwo signals,wedenesome threshold .If m s 0 threshold thenwehaveamatchatlocation s 0 .

PAGE 280

272 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS 15.5.2.4PracticalExamples 15.5.2.4.1ImageDetection In2-D,thisconceptisusedtomatchimagestogether,suchasverifyingngerprintsforsecurityortomatch photosofsomeone.Forexample,thisideacouldbeusedfortheever-popular"Where'sWaldo?"books!If wearegiventhebelowtemplateFigure15.12aandpieceofa"Where'sWaldo?"bookFigure15.12b, a b Figure15.12: Exampleof"Where'sWaldo?"picture.OurMatchedFilterDetectorcanbeimplementedtondapossiblematchforWaldo. thenwecouldeasilydevelopaprogramtondtheclosestresemblancetotheimageofWaldo'sheadin thelargerpicture.Wewouldsimplyimplementoursamematchlteralgorithm:taketheinnerproductsat eachshiftandseehowlargeourresultinganswersare.Thisideawasimplementedonthissamepicturefor aSignalsandSystemsProject 8 atRiceUniversityclickthelinktolearnmore. 15.5.2.4.2CommunicationsSystems MatchedlterdetectorarealsocommonlyusedinCommunicationsSystems 9 .Infact,theyarethe optimal detectorsinGaussiannoise.Signalsinthereal-worldareoftendistortedbytheenvironmentaroundthem, sothereisaconstantstruggletodevelopwaystobeabletoreceiveadistortedsignalandthenbeableto lteritinsomewaytodeterminewhattheoriginalsignalwas.Matchedltersprovideonewaytocomparea receivedsignalwithtwopossibleoriginal"template"signalsanddeterminewhichoneistheclosestmatch tothereceivedsignal. Forexample,belowwehaveasimpliedexampleofFrequencyShiftKeying 10 FSKwherewehaving thefollowingcodingfor'1'and'0': 8 http://www.owlnet.rice.edu/ elec301/Projects99/waldo/process.html 9 "StructureofCommunicationSystems" 10 "FrequencyShiftKeying"

PAGE 281

273 Figure15.13: FrequencyShiftKeyingfor'1'and'0'. Basedontheabovecoding,wecancreatedigitalsignalsbasedon0'sand1'sbyputtingtogetherthe abovetwo"codes"inaninnitenumberofways.Forthisexamplewewilltransmitabasic3-bitnumber, 101,whichisdisplayedinFigure15.14: Figure15.14: Thebitstream"101"codedwiththeaboveFSK.

PAGE 282

274 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS Now,thesignalpictureaboverepresentsouroriginalsignalthatwillbetransmittedoversomecommunicationsystem,whichwillinevitablypassthroughthe"communicationschannel,"thepartofthesystem thatwilldistortandalteroursignal.Aslongasthenoiseisnottoogreat,ourmatchedltershouldkeepus fromhavingtoworryaboutthesechangestoourtransmittedsignal.Oncethissignalhasbeenreceived,we willpassthenoisysignalthroughasimplesystem,similartothesimpliedversionshowninFigure15.15: Figure15.15: Blockdiagramofmatchedlterdetector. Figure15.15basicallyshowsthatournoisysignalwillbepassedinwewillassumethatitpassesinone "bit"atatimeandthissignalwillbesplitandpassedtotwodierentmatchedlterdetectors.Eachone willcomparethenoisy,receivedsignaltooneofthetwocodeswedenedfor'1'and'0.'Thenthisvaluewill bepassedonandwhichevervalueishigher i.e. whicheverFSKcodesignalthenoisysignalmostresembles willbethevaluethatthereceivertakes.Forexample,therstbitthatwillbesentthroughwillbea'1'so theupperleveloftheblockdiagramwillhaveahighervalue,thusdenotingthata'1'wassentbythesignal, eventhoughthesignalmayappearverynoisyanddistorted. 15.5.3ProofofCSI HerewilllookattheproofofourCauchy-SchwarzInequalityCSIfora realvectorspace Theorem15.1: CSIforRealVectorSpace For f 2 HilbertSpaceS and g 2 HilbertSpaceS ,show: j jk f kk g k .10 withequalityifandonlyif g = f Proof: If g = f ,show j j = k f kk g k j j = j j = j jj j = j j k f k 2 j j = k f k j jk f k = k f kk g k ThisveriesourabovestatementoftheCSI! If g 6 = f ,show j j < k f kk g k wherewehave f + g 6 =0 ; 2 R 0 < k f + g k 2 = = 2 +2 + = 2 k f k 2 +2 + k g k 2

PAGE 283

275 Andwegetaquadraticin .Visually,thequadraticpolynomialin > 0 forall .Also,note thatthispolynomialhasnorealrootsandthediscriminantislessthan0.-BLAHBLAH BLAH a 2 + b + c hasdiscriminant 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 ac wherewehave: a = k f k 2 b =2 c = k g k 2 Therefore,wecanplugthisvaluesintotheabovepolynomialsdiscriminanttoget: 4 j j 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 k f k 2 k g k 2 < 0 .11 j j < k f kk g k .12 AndnallywehaveproventheCauchy-SchwarzInequalityformulaforrealvectorsspaces. note: Whatchangesdowehavetomaketotheproofforacomplexvectorspace?tryto gurethisoutathome 15.6CommonHilbertSpaces 11 15.6.1CommonHilbertSpaces BelowwewilllookatthefourmostcommonHilbertspacesSection15.3thatyouwillhavetodealwith whendiscussingandmanipulatingsignalsandsystems. 15.6.1.1 R n realsscalarsand C n complexscalars,alsocalled ` 2 [0 ;n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1] x = 0 B B B B B @ x 0 x 1 ::: x n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 1 C C C C C A isalistofnumbersnitesequence.TheinnerproductSection15.3forourtwospaces areasfollows: Innerproduct R n : < x ; y > = y T x = P n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 i =0 x i y i .13 Innerproduct C n : < x ; y > = y T x = P n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 i =0 x i y i .14 11 Thiscontentisavailableonlineat.

PAGE 284

276 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS Modelfor:Discretetimesignalsontheinterval [0 ;n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1] or periodicwithperiod n discretetimesignals. 0 B B B B B @ x 0 x 1 ::: x n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 C C C C C A Figure15.16 15.6.1.2 f 2 L 2 [ a;b ] isa niteenergy functionon [ a;b ] InnerProduct = Z b a f t g t dt .15 Modelfor:continuoustimesignalsontheinterval [ a;b ] or periodicwithperiod T = b )]TJ/F11 9.9626 Tf 9.183 0 Td [(a continuoustime signals 15.6.1.3 x 2 ` 2 Z isan innitesequenceofnumbers that'ssquare-summable Innerproduct = 1 X i = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x [ i ] y [ i ] .16 Modelfor:discretetime,non-periodicsignals 15.6.1.4 f 2 L 2 R isa niteenergyfunction onallof R Innerproduct = Z 1 f t g t dt .17 Modelfor:continuoustime,non-periodicsignals

PAGE 285

277 15.6.2AssociatedFourierAnalysis Eachofthese4HilbertspaceshasatypeofFourieranalysisassociatedwithit. L 2 [ a;b ] Fourierseries ` 2 [0 ;n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1] DiscreteFourierTransform L 2 R FourierTransform ` 2 Z DiscreteTimeFourierTransform Butall4ofthesearebasedonthesameprinciplesHilbertspace. note: NotallnormedspacesareHilbertspaces Forexample: L 1 R k f k 1 = R j f t j dt .Tryasyoumight,youcan'tndaninnerproductthatinduces thisnorm, i.e. a < ; > suchthat = R j f t j 2 dt 2 = k f k 1 2 .18 Infact,ofallthe L p R spaces, L 2 R isthe onlyone thatisaHilbertspace. Figure15.17

PAGE 286

278 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS Hilbertspacesarebyfarthenicest.IfyouuseorstudyorthonormalbasisexpansionSection15.8then youwillstarttoseewhythisistrue. 15.7TypesofBasis 12 15.7.1NormalizedBasis Denition15.5:NormalizedBasis abasisSection5.1.3:Basis f b i g whereeach b i hasunitnorm k b i k =1 ;i 2 Z .19 note: TheconceptofbasisappliestoallvectorspacesSection15.1.Theconceptof normalized basis appliesonlytonormedspacesSection15.2. Youcanalwaysnormalizeabasis:justmultiplyeachbasisvectorbyaconstant,suchas 1 k b i k Example15.6 Wearegiventhefollowingbasis: f b 0 ;b 1 g = 8 < : 0 @ 1 1 1 A ; 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 A 9 = ; Normalizedwith ` 2 norm: b 0 = 1 p 2 0 @ 1 1 1 A b 1 = 1 p 2 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 A Normalizedwith ` 1 norm: b 0 = 1 2 0 @ 1 1 1 A b 1 = 1 2 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 A 15.7.2OrthogonalBasis Denition15.6:OrthogonalBasis abasis f b i g inwhichtheelementsare mutuallyorthogonal =0 ;i 6 = j 12 Thiscontentisavailableonlineat.

PAGE 287

279 note: TheconceptoforthogonalbasisappliesonlytoHilbertSpaces. Example15.7 Standardbasisfor R 2 ,alsoreferredtoas ` 2 [0 ; 1] : b 0 = 0 @ 1 0 1 A b 1 = 0 @ 0 1 1 A = 1 X i =0 b 0 [ i ] b 1 [ i ]=1 0+0 1=0 Example15.8 Nowwehavethefollowingbasisandrelationship: 8 < : 0 @ 1 1 1 A ; 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 A 9 = ; = f h 0 ;h 1 g =1 1+1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1=0 15.7.3OrthonormalBasis Pullingtheprevioustwosectionsdenitionstogether,wearriveatthemostimportantandusefulbasis type: Denition15.7:OrthonormalBasis abasisthatisboth normalized and orthogonal k b i k =1 ;i 2 Z ;i 6 = j note: Wecanshortenthesetwostatementsintoone: = ij where ij = 8 < : 1 if i = j 0 if i 6 = j Where ij isreferredtoastheKroneckerdeltafunctionSection1.5andisalsooftenwrittenas [ i )]TJ/F11 9.9626 Tf 9.963 0 Td [(j ] .

PAGE 288

280 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS Example15.9:OrthonormalBasisExample#1 f b 0 ;b 2 g = 8 < : 0 @ 1 0 1 A ; 0 @ 0 1 1 A 9 = ; Example15.10:OrthonormalBasisExample#2 f b 0 ;b 2 g = 8 < : 0 @ 1 1 1 A ; 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 A 9 = ; Example15.11:OrthonormalBasisExample#3 f b 0 ;b 2 g = 8 < : 1 p 2 0 @ 1 1 1 A ; 1 p 2 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 A 9 = ; 15.7.3.1BeautyofOrthonormalBases Orthonormalbasesare very easytodealwith!If f b i g isanorthonormalbasis,wecanwriteforany x x = X i i b i .20 Itiseasytondthe i : = < P k k b k ;b i > = P k k .21 whereintheaboveequationwecanuseourknowledgeofthedeltafunctiontoreducethisequation: = ik = 8 < : 1 if i = k 0 if i 6 = k = i .22 Therefore,wecanconcludethefollowingimportantequationfor x : x = X i b i .23 The i 'sareeasytocomputenointeractionbetweenthe b i 's Example15.12 Giventhefollowingbasis: f b 0 ;b 1 g = 8 < : 1 p 2 0 @ 1 1 1 A ; 1 p 2 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 A 9 = ; represent x = 0 @ 3 2 1 A

PAGE 289

281 Example15.13:SlightlyModiedFourierSeries Wearegiventhebasis 1 p T e j! 0 nt j 1 n = on L 2 [0 ;T ] where T = 2 0 f t = 1 X n = e j! 0 nt 1 p T Wherewecancalculatetheaboveinnerproductin L 2 as = 1 p T Z T 0 f t e j! 0 nt dt = 1 p T Z T 0 f t e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j! 0 nt dt 15.7.3.2OrthonormalBasisExpansionsinaHilbertSpace Let f b i g beanorthonormalbasisforaHilbertspace H .Then,forany x 2 H wecanwrite x = X i i b i .24 where i = "Analysis":decomposing x intermofthe b i i = .25 "Synthesis":building x upoutofaweightedcombinationofthe b i x = X i i b i .26 Thisisanunsupportedmediatype.Toview,pleaseseehttp://cnx.org/content/m10772/latest/ONB.llb 15.8OrthonormalBasisExpansions 13 15.8.1MainIdea Whenworkingwithsignalsmanytimesitishelpfultobreakupasignalintosmaller,moremanageable parts.HopefullybynowyouhavebeenexposedtotheconceptofeigenvectorsSection5.2andthereuse indecomposingasignalintooneofitspossiblebasis.Bydoingthisweareabletosimplifyourcalculations ofsignalsandsystemsthrougheigenfunctionsofLTIsystemsSection5.5. Nowwewouldliketolookatanalternativewaytorepresentsignals,throughtheuseof orthonormal basis .Wecanthinkoforthonormalbasisasasetofbuildingblocksweusetoconstructfunctions.Wewill buildupthesignal/vectorasaweightedsumofbasiselements. 13 Thiscontentisavailableonlineat.

PAGE 290

282 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS Example15.14 Thecomplexsinusoids 1 p T e j! 0 nt forall
PAGE 291

283 15.8.3FindingtheAlphas Nowletusaddressthequestionposedaboveaboutnding i 'singeneralfor R 2 .Westartbyrewriting .28sothatwecanstackour b i 'sascolumnsina2 2matrix. x = 0 b 0 + 1 b 1 .29 x = 0 B B B @ . . . b 0 b 1 . . . 1 C C C A 0 @ 0 1 1 A .30 Example15.16 Hereisasimpleexample,whichshowsalittlemoredetailabouttheaboveequations. 0 @ x [0] x [1] 1 A = 0 0 @ b 0 [0] b 0 [1] 1 A + 1 0 @ b 1 [0] b 1 [1] 1 A = 0 @ 0 b 0 [0]+ 1 b 1 [0] 0 b 0 [1]+ 1 b 1 [1] 1 A .31 0 @ x [0] x [1] 1 A = 0 @ b 0 [0] b 1 [0] b 0 [1] b 1 [1] 1 A 0 @ 0 1 1 A .32 15.8.3.1SimplifyingourEquation Tomakenotationsimpler,wedenethefollowingtwoitemsfromtheaboveequations: BasisMatrix : B = 0 B B B @ . . . b 0 b 1 . . . 1 C C C A CoecientVector : = 0 @ 0 1 1 A Thisgivesusthefollowing,conciseequation: x = B .33 whichisequivalentto x = P 1 i =0 i b i Example15.17 Givenastandardbasis, 8 < : 0 @ 1 0 1 A ; 0 @ 0 1 1 A 9 = ; ,thenwehavethefollowingbasismatrix: B = 0 @ 01 10 1 A

PAGE 292

284 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS Togetthe i 's,wesolveforthecoecientvectorin.33 = B )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x .34 Where B )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 istheinversematrix 14 of B 15.8.3.2Examples Example15.18 Letuslookatthestandardbasisrstandtrytocalculate fromit. B = 0 @ 10 01 1 A = I Where I isthe identitymatrix .Inordertosolvefor letusndtheinverseof B rstwhichis obviouslyverytrivialinthiscase: B )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 0 @ 10 01 1 A Thereforeweget, = B )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x = x Example15.19 Letuslookataever-so-slightlymorecomplicatedbasisof 8 < : 0 @ 1 1 1 A ; 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 A 9 = ; = f h 0 ;h 1 g Then ourbasismatrixandinversebasismatrixbecomes: B = 0 @ 11 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 A B )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 0 @ 1 2 1 2 1 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 2 1 A andforthisexampleitisgiventhat x = 0 @ 3 2 1 A Nowwesolvefor = B )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x = 0 @ 1 2 1 2 1 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 2 1 A 0 @ 3 2 1 A = 0 @ 2 : 5 0 : 5 1 A andweget x =2 : 5 h 0 +0 : 5 h 1 Exercise15.1 Solutiononp.299. Nowwearegiventhefollowingbasismatrixand x : f b 0 ;b 1 g = 8 < : 0 @ 1 2 1 A ; 0 @ 3 0 1 A 9 = ; 14 "MatrixInversion"

PAGE 293

285 x = 0 @ 3 2 1 A Forthisproblem,makeasketchofthebasesandthenrepresent x intermsof b 0 and b 1 note: Achangeofbasissimplylooksat x froma"dierentperspective." B )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 transforms x from thestandardbasistoournewbasis, f b 0 ;b 1 g .Noticethatthisisatotallymechanicalprocedure. 15.8.4ExtendingtheDimensionandSpace Wecanalsoextendalltheseideaspastjust R 2 andlookatthemin R n and C n .Thisprocedureextendsnaturallytohigher > 2dimensions.Givenabasis f b 0 ;b 1 ;:::;b n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 g for R n ,wewanttond f 0 ; 1 ;:::; n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 g suchthat x = 0 b 0 + 1 b 1 + + n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 b n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 .35 Again,wewillsetupabasismatrix B = b 0 b 1 b 2 :::b n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 wherethecolumnsequalthebasisvectorsanditwillalwaysbeann nmatrixalthoughtheabovematrix doesnotappeartobesquaresincewelefttermsinvectornotation.Wecanthenproceedtorewrite.33 x = b 0 b 1 :::b n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 0 B B B @ 0 . n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 C C C A = B and = B )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x 15.9FunctionSpace 15 WecanalsondbasisvectorsSection15.8forvectorspacesSection15.1otherthan R n Let P n bethevectorspaceofn-thorderpolynomialson-1,1withrealcoecientsverify P 2 isa v.s. athome. Example15.20 P 2 ={allquadraticpolynomials}.Let b 0 t =1 b 1 t = t b 2 t = t 2 f b 0 t ;b 1 t ;b 2 t g span P 2 i.e. youcanwriteany f t 2 P 2 as f t = 0 b 0 t + 1 b 1 t + 2 b 2 t forsome i 2 R note: P 2 is3dimensional. 15 Thiscontentisavailableonlineat.

PAGE 294

286 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS f t = t 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 t )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Alternatebasis f b 0 t ;b 1 t ;b 2 t g = 1 ;t; 1 2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 t 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 write f t intermsofthisnewbasis d 0 t = b 0 t d 1 t = b 1 t d 2 t = 3 2 b 2 t )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(1 2 b 0 t f t = t 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 t )]TJ/F8 9.9626 Tf 9.963 0 Td [(4=4 b 0 t )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 b 1 t + b 2 t f t = 0 d 0 t + 1 d 1 t + 2 d 2 t = 0 b 0 t + 1 b 1 t + 2 3 2 b 2 t )]TJ/F8 9.9626 Tf 11.158 6.74 Td [(1 2 b 0 t f t = 0 )]TJ/F8 9.9626 Tf 11.158 6.74 Td [(1 2 b 0 t + 1 b 1 t + 3 2 2 b 2 t so 0 )]TJ/F8 9.9626 Tf 11.158 6.74 Td [(1 2 =4 1 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 3 2 2 =1 thenweget f t =4 : 5 d 0 t )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 d 1 t + 2 3 d 2 t Example15.21 e j! 0 nt j 1 n = isabasisfor L 2 [0 ;T ] T = 2 0 f t = P n )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(C n e j! 0 nt Wecalculatetheexpansioncoecientswith "changeofbasis"formula C n = 1 T Z T 0 f t e )]TJ/F7 6.9738 Tf 6.226 0 Td [( j! 0 nt dt .36 note: Thereareaninnitenumberofelementsinthebasisset,thatmeans L 2 [0 ;T ] isinnite dimensionalscary!. Innite-dimensionalspacesarehardtovisualize.Wecangetahandleontheintuitionbyrecognizing theysharemanyofthesamemathematicalpropertieswithnitedimensionalspaces.Manyconcepts applytobothlike"basisexpansion".Somedon'tchangeofbasisisn'tanicematrixformula. 15.10HaarWaveletBasis 16 15.10.1Introduction FourierseriesSection6.2isausefulorthonormalrepresentationSection15.8on L 2 [0 ;T ] especialllyfor inputsintoLTIsystems.However,itisillsuitedforsomeapplications,i.e.imageprocessingrecallGibb's phenomenaSection6.11. Wavelets ,discoveredinthelast15years,areanotherkindofbasisfor L 2 [0 ;T ] andhavemanynice properties. 16 Thiscontentisavailableonlineat.

PAGE 295

287 15.10.2BasisComparisons Fourierseriesc n givefrequencyinformation.Basisfunctionslasttheentireinterval. Figure15.18: Fourierbasisfunctions Wavelets-basisfunctionsgivefrequencyinfobutare local intime.

PAGE 296

288 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS Figure15.19: Waveletbasisfunctions InFourierbasis,thebasisfunctionsare harmonicmultiples of e j! 0 t Figure15.20: basis= n 1 p T e j! 0 nt o InHaarwaveletbasis 17 ,thebasisfunctionsare scaledandtranslated versionsofa"motherwavelet" t 17 "TheHaarSystemasanExampleofDWT"

PAGE 297

289 Figure15.21 Basisfunctions f j;k t g areindexedbya scale janda shift k. Let t =1 ; 0 t
PAGE 298

290 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS Figure15.22 t = 8 < : 1 if 0 t< T 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 if 0 T 2
PAGE 299

291 Figure15.23 Let j;k t =2 j 2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 j t )]TJ/F11 9.9626 Tf 9.962 0 Td [(k Figure15.24 Larger j "skinnier"basisfunction, j = f 0 ; 1 ; 2 ;::: g 2 j shiftsateachscale: k =0 ; 1 ;:::; 2 j )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 Check:each j;k t hasunitenergy

PAGE 300

292 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS Figure15.25 Z j;k 2 t dt =1 k j;k t k 2 =1 .38 Anytwobasisfunctionsareorthogonal. a b Figure15.26: Integralofproduct=0aSamescalebDierentscale

PAGE 301

293 Also, f j;k ; g span L 2 [0 ;T ] 15.10.3HaarWaveletTransform UsingwhatweknowaboutHilbertspacesSection15.3:Forany f t 2 L 2 [0 ;T ] ,wecanwrite Synthesis f t = X j X k w j;k j;k t + c 0 t .39 Analysis w j;k = Z T 0 f t j;k t dt .40 c 0 = Z T 0 f t t dt .41 note: the w j;k are real TheHaartransformis superuseful especiallyin imagecompression Example15.22 ThisdemonstrationletsyoucreateasignalbycombiningHaarbasisfunctions,illustratingthe synthesisequationoftheHaarWaveletTransform.Seehere 18 forinstructionsonhowtousethe demo. Thisisanunsupportedmediatype.Toview,pleaseseehttp://cnx.org/content/m10764/latest/HaarSyn.llb 15.11OrthonormalBasesinRealandComplexSpaces 19 15.11.1Notation Transposeoperator A T ipsthematrixacrossit'sdiagonal. A = 0 @ a 11 a 12 a 21 a 22 1 A A T = 0 @ a 11 a 21 a 12 a 22 1 A Column i of A is row i of A T 18 "HowtousetheLabVIEWdemos" 19 Thiscontentisavailableonlineat.

PAGE 302

294 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS Recall,innerproduct 20 x = 0 B B B B B @ x 0 x 1 . x n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 C C C C C A y = 0 B B B B B @ y 0 y 1 . y n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 C C C C C A x T y = x 0 x 1 :::x n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 0 B B B B B @ y 0 y 1 . y n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 C C C C C A = X x i y i = < y ; x > on R n Hermitiantranspose A H ,transposeandconjugate A H = A T < y ; x > = x H y = X x i y i on C n Now,let f b 0 ;b 1 ;:::;b n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 g beanorthonormalbasisSection15.7.3:OrthonormalBasisfor C n i = f 0 ; 1 ;:::;n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 g =1 ; i 6 = j = b j H b i =0 Basismatrix: B = 0 B B B @ . . . . b 0 b 1 :::b n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 . . . . 1 C C C A Now, B H B = 0 B B B B B @ :::b 0 H ::: :::b 1 H ::: . :::b n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 H ::: 1 C C C C C A 0 B B B @ . . . . b 0 b 1 :::b n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 . . . . 1 C C C A = 0 B B B B B @ b 0 H b 0 b 0 H b 1 :::b 0 H b n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 b 1 H b 0 b 1 H b 1 :::b 1 H b n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 . b n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 H b 0 b n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 H b 1 :::b n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 H b n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 1 C C C C C A 20 "Conclusion"

PAGE 303

295 Fororthonormalbasiswithbasismatrix B B H = B )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 B T = B )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 in R n B H iseasytocalculatewhile B )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 ishardtocalculate. So,tond f 0 ; 1 ;:::; n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 g suchthat x = X i b i Calculate = B )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x = B H x Usinganorthonormalbasisweridourselvesoftheinverseoperation. 15.12PlancharelandParseval'sTheorems 21 15.12.1PlancharelTheorem Theorem15.2: PlancharelTheorem Theinnerproductoftwovectors/signalsisthesameasthe ` 2 innerproductoftheirexpansion coecients. Let f b i g beanorthonormalbasisforaHilbertSpace H x 2 H y 2 H x = X i b i y = X i b i then H = X i i Example ApplyingtheFourierSeries,wecangofrom f t to f c n g and g t to f d n g Z T 0 f t g t dt = 1 X n = c n d n innerproductintime-domain=innerproductofFouriercoecients. Proof: x = X i b i y = X j b j H = < X i b i ; X j b j > = X i = X i X j = X i i byusinginnerproductrulesp.267 note: =0 when i 6 = j and =1 when i = j IfHilbertspaceHhasaONB,theninnerproductsareequivalenttoinnerproductsin ` 2 AllHwithONBaresomehowequivalentto ` 2 note: square-summablesequencesareimportant 21 Thiscontentisavailableonlineat.

PAGE 304

296 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS 15.12.2Parseval'sTheorem Theorem15.3: Parseval'sTheorem Energyofasignal=sumofsquaresofit'sexpansioncoecients Let x 2 H f b i g ONB x = X i b i Then k x k H 2 = X j i j 2 Proof: DirectlyfromPlancharel k x k H 2 = H = X i i = X j i j 2 Example15.23 FourierSeries 1 p T e jw 0 nt f t = 1 p T X c n 1 p T e jw 0 nt Z T 0 j f t j 2 dt = 1 X n = j c n j 2 Thisisanunsupportedmediatype.Toview,pleaseseehttp://cnx.org/content/m10769/latest/Parsevals Theorem.llb 15.13ApproximationandProjectionsinHilbertSpace 22 15.13.1Introduction Givenaline'l'andapoint'p'intheplane,what'stheclosestpoint'm'to'p'on'l'? Figure15.27: Figureofpoint'p'andline'l'mentionedabove. 22 Thiscontentisavailableonlineat.

PAGE 305

297 Sameproblem:Let x and v bevectorsin R 2 .Say k v k =1 .Forwhatvalueof is k x )]TJ/F11 9.9626 Tf 10.687 0 Td [(v k 2 minimized?whatpointinspan{v} bestapproximates x ? Figure15.28 Theconditionisthat x )]TJ/F15 9.9626 Tf 10.696 6.628 Td [(^ v and v are orthogonal 15.13.2Calculating Howtocalculate ^ ? Weknowthat x )]TJ/F15 9.9626 Tf 10.696 6.628 Td [(^ v isperpendiculartoeveryvectorinspan{v},so =0 ; 8 )]TJ/F15 9.9626 Tf 10.696 6.627 Td [(^ =0 because =1 ,so )]TJ/F15 9.9626 Tf 10.696 6.628 Td [(^ =0 ^ = Closestvectorinspan{v}= v ,where v istheprojectionof x onto v Wecandothesamethinginhigherdimensions. Exercise15.2 Solutiononp.299. Let V H beasubspaceofaHilbertspaceSection15.3H.Let x 2 H begiven.Findthe y 2 V that bestapproximates x .i.e., k x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y k isminimized. Example15.24 x 2 R 3 V =span 0 B B @ 8 > > < > > : 0 B B @ 1 0 0 1 C C A ; 0 B B @ 0 1 0 1 C C A 9 > > = > > ; 1 C C A x = 0 B B @ a b c 1 C C A .So, y = 2 X i =1 b i = a 0 B B @ 1 0 0 1 C C A + b 0 B B @ 0 1 0 1 C C A = 0 B B @ a b 0 1 C C A Example15.25 V={spaceofperiodicsignalswithfrequencynogreaterthan 3 w 0 }.Givenperiodicft,whatis thesignalinVthatbestapproximatesf?

PAGE 306

298 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS 1.{ 1 p T e jw 0 kt ,k=-3,-2,...,2,3}isanONBforV 2. g t = 1 T P 3 k = )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e jw 0 kt istheclosestsignalinVtoft reconstructft usingonly7termsofitsFourierseriesSection6.2. Example15.26 LetV={functionspiecewiseconstantbetweentheintegers} 1.ONBforV. b i = 8 < : 1 if i )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 tb i = Z 1 f t b i t dt = Z i i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 f t dt Example15.27 ThisdemonstrationexploresapproximationusingaFourierbasisandaHaarWaveletbasis.See here 23 forinstructionsonhowtousethedemo. Thisisanunsupportedmediatype.Toview,pleasesee http://cnx.org/content/m10766/latest/Approximation.llb 23 "HowtousetheLabVIEWdemos"

PAGE 307

299 SolutionstoExercisesinChapter15 SolutiontoExercise15.1p.284 Inordertorepresent x intermsof b 0 and b 1 wewillfollowthesamestepsweusedintheaboveexample. B = 0 @ 12 30 1 A B )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 0 @ 0 1 2 1 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 6 1 A = B )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x = 0 @ 1 2 3 1 A Andnowwecanwrite x intermsof b 0 and b 1 x = b 0 + 2 3 b 1 Andwecaneasilysubstituteinourknownvaluesof b 0 and b 1 toverifyourresults. SolutiontoExercise15.2p.297 1.FindanorthonormalbasisSection15.7.3:OrthonormalBasis f b 1 ;:::;b k g for V 2.Project x onto V using y = k X i =1 b i then y istheclosestpointinVtoxandx-y ? V =0 ; 8 v 2 V

PAGE 308

300 CHAPTER15.APPENDIX:HILBERTSPACESANDORTHOGONAL EXPANSIONS

PAGE 309

Chapter16 HomeworkSets 16.1Homework#1 1 note: Noon,Thursday,September5,2002 16.1.1Assignment1 Homework,tests,andsolutionsfrompreviousoeringsofthiscourseareolimits,underthehonorcode. 16.1.1.1Problem1 Formastudygroupof3-4members.Withyourgroup,discussandsynthesizethemajor themes ofthis weekoflectures.Turninaonepagesummaryofyourdiscussion.Youneedturninonlyonesummaryper group,butincludethenamesofallgroupmembers.Pleasedonotwriteupjusta"tableofcontents." 16.1.1.2Problem2 ConstructaWWWpagewithyour picture andemailMikeWakinwakin@rice.eduyournameasyou wantittoappearontheclasswebpageandtheURL.Ifyouneedassistancesettingupyourpageor taking/scanningapicturebothareeasy!,askyourclassmates. 16.1.1.3Problem3:LearningStyles Followthislearningstyleslink 2 alsofoundontheElec301webpage 3 andlearnaboutthebasicsof learningstyles.Writeashortsummaryofwhatyoulearned.Also,completethe"Indexoflearningstyles" self-scoringtestonthewebandbringyourresultstoclass. 16.1.1.4Problem4 Makesureyouknowthematerialin Lathi ,ChapterB,Sections1-4,6.1,6.2,7.Specically,besuretoreview topicssuchas: complexarithmeticadding,multiplying,powers ndingcomplexrootsofpolynomials complexplane 4 andplottingroots 1 Thiscontentisavailableonlineat. 2 http://www2.ncsu.edu/unity/lockers/users/f/felder/public/Learning_Styles.html 3 http://www-dsp.rice.edu/courses/elec301/ 4 "TheComplexPlane" 301

PAGE 310

302 CHAPTER16.HOMEWORKSETS vectorsadding,innerproducts 16.1.1.5Problem5:ComplexNumberApplet Reacquaintyourselfwithcomplexnumbers 5 bygoingtothecourseappletswebpage 6 andclickingonthe ComplexNumbersappletmaytakeafewsecondstoload. aChangethedefaultaddfunctiontoexponentialexp.Clickonthecomplexplanetogetabluearrow, whichisyourcomplexnumber z .Clickagainanywhereonthecomplexplanetogetayellowarrow,which isequalto e z .Nowdragthetipofthebluearrowalongtheunitcircleonwith j z j =1 smallercircle.For whichvaluesof z ontheunitcircledoes e z alsolieontheunitcircle?Why? bExperimentwiththefunctionsabsoluteabs,realpartre,andimaginarypartimandreport yourndings. 16.1.1.6Problem6:ComplexArithmetic Reducethefollowingtothe Cartesian form, a + jb .Do not useyourcalculator! a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(j p 2 20 b 1+2 j 3+4 j c 1+ p 3 j p 3 )]TJ/F10 6.9738 Tf 6.226 0 Td [(j d p j e j j 16.1.1.7Problem7:RootsofPolynomials Findtherootsofeachofthefollowingpolynomialsshowyourwork.UseMATLABtocheckyouranswer withthe roots commandandtoplottherootsinthecomplexplane.Marktherootlocationswithan'o'. Putalloftherootsonthesameplotandidentifythecorrespondingpolynomial a b etc. ... a z 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 z b z 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 z +4 c z 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 z +8 d z 2 +8 e z 2 +4 z +8 f 2 z 2 +4 z +8 16.1.1.8Problem8:NthRootsofUnity e j 2 N iscalledan NthRootofUnity aWhy? bLet z = e j 2 7 .Draw z;z 2 ;:::;z 7 inthecomplexplane. cLet z = e j 4 7 .Draw z;z 2 ;:::;z 7 inthecomplexplane. 16.1.1.9Problem9:WritingVectorsinTermsofOtherVectors Apairofvectors u 2 C 2 and v 2 C 2 arecalled linearlyindependent if u + v =0 ifandonlyif = =0 Itisafactthatwecanwriteanyvectorin C 2 asa weightedsum or linearcombination ofanytwo linearlyindependentvectors,wheretheweights and arecomplex-valued. 5 "ComplexNumbers" 6 http://www.dsp.rice.edu/courses/elec301/applets.shtml

PAGE 311

303 aWrite 0 @ 3+4 j 6+2 j 1 A asalinearcombinationof 0 @ 1 2 1 A and 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 3 1 A .Thatis,nd and suchthat 0 @ 3+4 j 6+2 j 1 A = 0 @ 1 2 1 A + 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 3 1 A bMoregenerally,write x = 0 @ x 1 x 2 1 A asalinearcombinationof 0 @ 1 2 1 A and 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 3 1 A .Wewilldenote theanswerforagiven x as x and x cWritetheanswertoainmatrixform, i.e. nda2 2matrix A suchthat A 0 @ x 1 x 2 1 A = 0 @ x x 1 A dRepeatbandcforageneralsetoflinearlyindependentvectors u and v 16.1.1.10Problem10:FunwithFractals AJuliaset J isobtainedbycharacterizingpointsinthecomplexplane.Specically,let f x = x 2 + with complex,anddene g 0 x = x g 1 x = f g 0 x = f x g 2 x = f g 1 x = f f x . g n x = f g n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x Thenforeach x inthecomplexplane,wesay x 2 J ifthesequence fj g 0 x j ; j g 1 x j ; j g 2 x j ;::: g does not tendtoinnity.Noticethatif x 2 J ,theneachelementofthesequence f g 0 x ;g 1 x ;g 2 x ;::: g alsobelongsto J Formostvaluesof ,theboundaryofaJuliasetisafractalcurve-itcontains"jagged"detailno matterhowfaryouzoominonit.Thewell-knownMandelbrotsetcontainsallvaluesof forwhichthe correspondingJuliasetisconnected. aLet = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 .Is x =1 in J ? bLet =0 .Whatconditionson x ensurethat x belongsto J ? cCreateanapproximatepictureofaJuliasetinMATLAB.Theeasiestwayistocreateamatrixof complexnumbers,decideforeachnumberwhetheritbelongsto J ,andplottheresultsusingthe imagesc command.Todeterminewhetheranumberbelongsto J ,itishelpfultodenealimit N onthenumberof iterationsof g .Foragiven x ,ifthemagnitude j g n x j remainsbelowsomethreshold M forall 0 n N wesaythat x belongsto J .Thecodebelowwillhelpyougetstarted:

PAGE 312

304 CHAPTER16.HOMEWORKSETS N=100;%Max#ofiterations M=2;%Magnitudethreshold mu=-0.75;%Juliaparameter realVals=[-1.6:0.01:1.6]; imagVals=[-1.2:0.01:1.2]; xVals=oneslengthimagVals,1*realVals+... j*imagVals'*ones,lengthrealVals; Jmap=onessizexVals; g=xVals;%Startwithg0 %InsertcodeheretofillinelementsofJmap.Leavea'1' %inlocationswherexbelongstoJ,insert'0'inthe %locationsotherwise.Itisnotnecessarytostoreall100 %iterationsofg! imagescrealVals,imagVals,Jmap; colormapgray; xlabel'Rex'; ylabel'Imagx'; Thiscreatesthefollowingpicturefor = )]TJ/F8 9.9626 Tf 7.749 0 Td [(0 : 75 N =100 ,and M =2 .

PAGE 313

305 Figure16.1: Exampleimagewherethex-axisis Re x andthey-axisis Im x Usingthe same valuesfor N M ,and x ,createapictureoftheJuliasetfor = )]TJ/F8 9.9626 Tf 7.749 0 Td [(0 : 391 )]TJ/F8 9.9626 Tf 9.709 0 Td [(0 : 587 j .Print outthispictureandhanditinwithyourMATLABcode. note: TryassigningdierentcolorvaluestoJmap.Forexample,letJmapindicatetherst iterationwhenthemagnitudeexceeds M .Tip:try imagesclogJmap and colormapjet fora neatpicture. 16.2Homework#1Solutions 7 16.2.1Problem#1 Nosolutionsprovided. 16.2.2Problem#2 Nosolutionsprovided. 16.2.3Problem#3 Nosolutionsprovided. 16.2.4Problem#4 Nosolutionsprovided. 7 Thiscontentisavailableonlineat.

PAGE 314

306 CHAPTER16.HOMEWORKSETS 16.2.5Problem#5 16.2.5.1Parta e z liesontheunitcirclefor z = j .When z = j e z = e j =cos 1+ j sin 1 e j = )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(cos 2 1+sin 2 1 1 2 =1 .1 whichgivesustheunitcircle! Thinkofitthisway:for z = + j ,youwanta =0 sothat e + j reducesas e + j = e e j = e 0 e j = e j WeknowbyEuler'sformulaSection1.6.2:Euler'sRelationthat e j =cos + j sin Themagnitudeofthisisgivenby sin 2 +cos 2 ,whichis1whichimpliesthat e j isontheunitcircle. So,weknowwewanttopicka z = Aj thatisontheunitcirclefromtheproblemstatement,sowe havetochoose A = 1 togetunitmagnitude. 16.2.5.2Partb jj gives magnitude ofcomplexnumber Re gives realpart ofcomplexnumber Im gives imaginarypart ofcomplexnumber 16.2.6Problem#6 16.2.6.1Parta )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(j p 2 20 = p 2 e 5 4 p 2 20 = e 5 4 20 = e j 25 = e j = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 16.2.6.2Partb 1+2 j 3+4 j = 1+2 j 3+4 j 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 j 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 j = 3+6 j )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 j +8 9+16 = 11+2 j 25 = 11 25 + 2 25 j 16.2.6.3Partc 1+ p 3 j p 3 )]TJ/F11 9.9626 Tf 9.963 0 Td [(j = 2 e j 3 2 e j )]TJ/F9 4.9813 Tf 5.397 0 Td [( 6 = e j 2 = j 16.2.6.4Partd p j = )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(e j 2 1 2 = e j 4 =cos 4 + j sin 4 = p 2 2 + p 2 2 j

PAGE 315

307 16.2.6.5Parte j j = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 j = e j 2 2 = e )]TJ/F9 4.9813 Tf 5.397 0 Td [( 2 16.2.7Problem#7 16.2.7.1Parta z 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 z = z z )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Rootsofz= f 0 ; 4 g 16.2.7.2Partb z 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 z +4= z )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 2 Rootsofz= f 2 ; 2 g 16.2.7.3Partc z 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 z +8 Rootsofz= 4 p 16 )]TJ/F8 9.9626 Tf 9.963 0 Td [(32 2 =2 2 j 16.2.7.4Partd z 2 +8 Rootsofz= p )]TJ/F8 9.9626 Tf 7.749 0 Td [(32 2 = 2 p 2 j 16.2.7.5Parte z 2 +4 z +8 Rootsofz= )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 p 16 )]TJ/F8 9.9626 Tf 9.963 0 Td [(32 2 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 2 j 16.2.7.6Partf 2 z 2 +4 z +8 Rootsofz= )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 p 16 )]TJ/F8 9.9626 Tf 9.963 0 Td [(64 4 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 p 3 j 16.2.7.7MatlabCodeandPlot %%%%%%%%%%%%%%% %%%%%PROBLEM7 %%%%%%%%%%%%%%% rootsA=roots[1-40] rootsB=roots[1-44] rootsC=roots[1-48]

PAGE 316

308 CHAPTER16.HOMEWORKSETS rootsD=roots[108] rootsE=roots[148] rootsF=roots[248] zplane[rootsA;rootsB;rootsC;rootsD;rootsE;rootsF]; gtext'a' gtext'a' gtext'b' gtext'b' gtext'c' gtext'c' gtext'd' gtext'd' gtext'e' gtext'e' gtext'f' gtext'f' Figure16.2: Plotofalltheroots.

PAGE 317

309 16.2.8Problem#8 16.2.8.1Parta Raise e j 2 N tothe N thpower. e j 2 N N = e j 2 =1 note: Similarly, 1 N = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(e j 2 1 N = e j 2 N 16.2.8.2Partb For z = e j 2 7 z k = e j 2 7 k = e j 2 k 7 Wewillhavepointsontheunitcirclewithangleof 2 7 ; 2 2 7 ;:::; 2 7 7 .Thecodeusedtoplotthesein MATLABcanbefoundbelow,followedbytheplot. %%%%%%%%%%%%%%% %%%%%PROBLEM8 %%%%%%%%%%%%%%% %%%Partb figure; clf; holdon; th=[0:0.01:2*pi]; unitCirc=expj*th; plotunitCirc,'--'; fork=1:7 z=expj*2*pi*k/7; plotz,'o'; text.2*realz,1.2*imagz,strcat'z^',num2strk; end xlabel'realpart'; ylabel'imagpart'; title'Powersofexpj2 n pi/7ontheunitcircle'; axis[-1.51.5-1.51.5]; axissquare;

PAGE 318

310 CHAPTER16.HOMEWORKSETS Figure16.3: MATLABplotofpartb. 16.2.8.3Partc For z = e j 4 7 z k = e j 4 7 k = e j 2 2 k 7 Wherewehave z;z 2 ;:::;z 7 = n e j 2 2 7 ;e j 2 4 7 ;e j 2 6 7 ;e j 2 1 7 ;e j 2 3 7 ;e j 2 5 7 ; 1 o ThecodeusedtoplottheseinMATLABcanbefoundbelow,followedbytheplot. %%%Partc figure; clf; holdon; th=[0:0.01:2*pi]; unitCirc=expj*th; plotunitCirc,'--';

PAGE 319

311 fork=1:7 z=expj*4*pi*k/7; plotz,'o'; text.2*realz,1.2*imagz,strcat'z^',num2strk; end xlabel'realpart'; ylabel'imagpart'; title'Powersofexpj4 n pi/7ontheunitcircle'; axis[-1.51.5-1.51.5]; axissquare; Figure16.4: MATLABplotofpartc.

PAGE 320

312 CHAPTER16.HOMEWORKSETS 16.2.9Problem#9 16.2.9.1Parta 0 @ 3+4 j 6+2 j 1 A = 0 @ 1 2 1 A + 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 3 1 A Tosolvefor wemustsolvethefollowingsystemofequations: )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 =3+4 j 2 +3 =6+2 j Ifwemultiplythetopequationby )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 wewillgetthefollowing,whichallowsustocanceloutthealpha terms: )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 +10 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 j 2 +3 =6+2 j Andnowwehave, 13 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 j = )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 13 j Andtosolvefor wehavethefollowingequation: =3+4 j +5 =3+4 j +5 )]TJ/F13 6.9738 Tf 5.762 -4.147 Td [()]TJ/F7 6.9738 Tf 6.227 0 Td [(6 13 j =3+ 22 13 j .2 16.2.9.2Partb 0 @ x 1 x 2 1 A = 0 @ 1 2 1 A + 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 3 1 A x 1 = )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x 2 =2 +3 Solvingfor and weget: x = 3 x 1 +5 x 2 13 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x 1 + x 2 13 16.2.9.3Partc 0 @ x x 1 A = 0 @ 3 13 5 13 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 13 1 13 1 A 0 @ x 1 x 2 1 A

PAGE 321

313 16.2.9.4Partd Write u = 0 @ u 1 u 2 1 A and v = 0 @ v 1 v 2 1 A .Thensolve 0 @ x 1 x 2 1 A = 0 @ u 1 u 2 1 A + 0 @ v 1 v 2 1 A whichcorrespondstothesystemofequations x 1 = u 1 + v 1 x 2 = u 2 + v 2 Solvingfor and weget x = v 2 x 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(v 1 x 2 u 1 v 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(u 2 v 1 x = u 2 x 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(u 1 x 2 v 1 u 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(u 1 v 2 Forthematrix A weget A = 1 u 1 v 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(u 2 v 1 0 @ v 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(v 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(u 2 u 1 1 A 16.2.10Problem#10 16.2.10.1Parta If u = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ,then f x = x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 .Examinethesequence f g 0 x ;g 1 x ;::: g : g 0 x =1 g 1 x =1 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1=0 g 2 x =0 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1= )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 g 3 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1=0 g 4 x =0 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1= )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 . Themagnitudesequenceremainsboundedso x =1 belongsto J .

PAGE 322

314 CHAPTER16.HOMEWORKSETS 16.2.10.2Partb If u =0 ,then f x = x 2 .Sowehave g 0 x = x g 1 x = x 2 g 2 x = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 2 = x 4 . g n x = )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 2 n = x 2 n Writing x = re j ,wehave g n x = x 2 n = r 2 n e j 2 n ,andsowehave j g n x j = r 2 n Themagnitudesequenceblowsupifandonlyif r> 1 .Thus x belongsto J ifandonlyif j x j 1 .So, J correspondstotheunitdisk. 16.2.10.3Partc %%%%%%%%%%%%%%%% %%%%%PROBLEM10 %%%%%%%%%%%%%%%% %%%Partc-solutioncode N=100;%Max#ofiterations M=2;%Magnitudethreshold mu=-0.391-0.587*j;%Juliaparameter realVals=[-1.6:0.01:1.6]; imagVals=[-1.2:0.01:1.2]; xVals=oneslengthimagVals,1*realVals+... j*imagVals'*ones,lengthrealVals; Jmap=onessizexVals; g=xVals;%Startwithg0 forn=1:N g=g.^2+mu; big=absg > M; Jmap=Jmap.*-big; end imagescrealVals,imagVals,Jmap;colormapgray; xlabel'Rex';ylabel'Imagx';

PAGE 323

315 Figure16.5: MATLABplotofpartc. 16.2.10.4JustforFunSolution %%%Justforfuncode N=100;%Max#ofiterations M=2;%Magnitudethreshold mu=-0.391-0.587*j;%Juliaparameter realVals=[-1.6:0.005:1.6]; imagVals=[-1.2:0.005:1.2]; xVals=oneslengthimagVals,1*realVals+... j*imagVals'*ones,lengthrealVals; Jmap=zerossizexVals; %Now,weputzerosinthe'middle',fora %cooleffect. g=xVals;%Startwithg0 forn=1:N g=g.^2+mu; big=absg > M; notAlreadyBig=Jmap==0;

PAGE 324

316 CHAPTER16.HOMEWORKSETS Jmap=Jmap+n*big.*notAlreadyBig; end imagescrealVals,imagVals,logJmap;colormapjet; xlabel'Rex';ylabel'Imagx'; Figure16.6: MATLABplot.

PAGE 325

Chapter17 ViewingEmbeddedLabVIEWContent 1 InordertoviewLabVIEWcontentembeddedinConnexionsmodules,youmustinstalltheLabVIEWRuntimeEngineonyourcomputer.Thefollowingaresetsofinstructionsforinstallingthesoftwareondierent platforms. note: EmbeddedLabVIEWcontentiscurrentlysupportedonlyunderWindows2000/XP.Also, youmusthaveversion8.0.1oftheLabViewRun-timeEnginetorunmuchoftheembeddedcontent inConnexions. 17.1InstallingtheLabVIEWRun-timeEngineonMicrosoftWindows2000/XP 1.PointyourwebbrowsertotheLabVIEWRun-timeEnginedownloadpageat: http://digital.ni.com/softlib.nsf/websearch/077b51e8d15604bd8625711c006240e7 2 2.Ifyou'renotloggedintoNI,clickthelinktocontinuethedownloadprocessatthebottomofthepage. 3.LoginorcreateaprolewithNItocontinue. 4.Onceloggedin,clickthe LabVIEW_8.0.1_Runtime_Engine.exe linkandsavetheletodisk. 5.Oncethelehasdownloaded,doubleclickitandfollowthestepstoinstalltherun-timeengine. 6.DownloadtheLabVIEWBrowserPlug-inat:http://zone.ni.com/devzone/conceptd.nsf/webmain/7DBFD404C6AD0B24862570BB0072F83B/$FILE/LVBrowserPlugin.ini 3 7.Putthe LVBrowserPlugin.ini leinthe MyDocuments n LabVIEWData folder.Youmayhaveto createthisfolderifitdoesn'talreadyexist. 8.Restartyourwebbrowsertocompletetheinstallationoftheplug-in. 1 Thiscontentisavailableonlineat. 2 http://digital.ni.com/softlib.nsf/websearch/077b51e8d15604bd8625711c006240e7 3 http://zone.ni.com/devzone/conceptd.nsf/webmain/7DBFD404C6AD0B24862570BB0072F83B/$FILE/LVBrowserPlugin.ini 317

PAGE 326

318 GLOSSARY Glossary B Basis Abasisfor C n isasetofvectorsthat:spans C n and islinearlyindependent. C Cauchy-SchwarzInequality For x y inaninnerproductspace j < x ; y > jk x kk y k withequalityholding ifandonlyif x and y arelinearlydependentSection5.1.1:Linear Independence, i.e. x = y forsomescalar D dierenceequation Anequationthatshowstherelationshipbetweenconsecutivevaluesofasequenceandthe dierencesamongthem.Theyareoftenrearrangedasarecursiveformulasothatasystems outputcanbecomputedfromtheinputsignalandpastoutputs. Example: y [ n ]+7 y [ n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1]+2 y [ n )]TJ/F8 9.9626 Tf 9.962 0 Td [(2]= x [ n ] )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x [ n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1] .42 domain Thegroup,orset,ofvaluesthataredenedbyagivenfunction. Example: Usingtherationalfunctionabove,.38,thedomaincanbedenedasanyreal number x where x doesnotequal1ornegative3.Writtenoutmathematical,wegetthe following: f x 2 R j x 6 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 andx 6 =1 g .40 E eigenvector Aneigenvectorof A isavector v 2 C n suchthat A v = v .2 where iscalledthecorresponding eigenvalue A onlychangesthe length of v ,notits direction. I innerproduct Theinnerproductisdenedmathematicallyas: < x ; y > = y T x = y 0 y 1 :::y n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 0 B B B B B @ x 0 x 1 . x n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 C C C C C A = P n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 i =0 x i y i .1

PAGE 327

GLOSSARY 319 L l p [0 ;N )]TJ/F8 9.9626 Tf 9.962 0 Td [(1]= n f 2 C N ; k f k p < 1 o butfrompreviousdiscussion l p [0 ;N )]TJ/F8 9.9626 Tf 9.963 0 Td [(1]= C N L p R = n f; k f k p < 1 o k f k p = Z 1 j f [ n ] j p dt 1 p where 1 p< 1 k f k 1 = esssup j f t j where 0 thereisaninteger N suchthat j g i )]TJ/F11 9.9626 Tf 9.962 0 Td [(g j <;i N Weusuallydenotealimitbywriting lim i !1 g i = g or g i g LinearlyIndependent

PAGE 328

320 GLOSSARY Foragivensetofvectors, f x 1 ;x 2 ;:::;x n g ,theyarelinearlyindependentif c 1 x 1 + c 2 x 2 + + c n x n =0 onlywhen c 1 = c 2 = = c n =0 Example: Wearegiventhefollowingtwovectors: x 1 = 0 @ 3 2 1 A x 2 = 0 @ )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 1 A Theseare notlinearlyindependent asprovenbythefollowingstatement,which,by inspection,canbeseentonotadheretothedenitionoflinearindependencestatedabove. x 2 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 x 1 2 x 1 + x 2 =0 Anotherapproachtorevealavectorsindependenceisbygraphingthevectors.Lookingatthese twovectorsgeometricallyasinFigure5.1,onecanagainprovethatthesevectorsare not linearlyindependent. N NormalizedBasis abasisSection5.1.3:Basis f b i g whereeach b i hasunitnorm k b i k =1 ;i 2 Z .19 O OrthogonalBasis abasis f b i g inwhichtheelementsare mutuallyorthogonal =0 ;i 6 = j orthogonal Wesaythat x and y areorthogonalif: < x ; y > =0 OrthonormalBasis abasisthatisboth normalized and orthogonal k b i k =1 ;i 2 Z ;i 6 = j P poles Alsocalledsingularities,thesearethepoints s atwhich L x 1 s blowsup. poles 1.Thevaluesfor z where Q z =0 .

PAGE 329

GLOSSARY 321 2.Thecomplexfrequenciesthatmaketheoverallgainoftheltertransferfunctioninnite. poles 1.Thevaluesfor z wherethe denominator ofthetransferfunctionequalszero 2.Thecomplexfrequenciesthatmaketheoverallgainoftheltertransferfunctioninnite. R rationalfunction Foranytwopolynomials,AandB,theirquotientiscalledarationalfunction. Example: Belowisasimpleexampleofabasicrationalfunction, f x .Notethatthe numeratoranddenominatorcanbepolynomialsofanyorder,buttherationalfunctionis undenedwhenthedenominatorequalszero. f x = x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 2 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 .38 S sequence Asequenceisafunction g n denedonthepositiveintegers' n '.Weoftendenoteasequenceby f g n gj 1 n =1 Example: Arealnumbersequence: g n = 1 n Example: Avectorsequence: g n = 0 @ sin )]TJ/F10 6.9738 Tf 5.762 -4.148 Td [(n 2 cos )]TJ/F10 6.9738 Tf 5.762 -4.147 Td [(n 2 1 A Example: Afunctionsequence: g n t = 8 < : 1 if 0 t< 1 n 0 otherwise note: Afunctioncanbethoughtofasaninnitedimensionalvectorwhereforeach valueof' t 'wehaveonedimension Span Thespan 4 ofasetofvectors f x 1 ;x 2 ;:::;x k g isthesetofvectorsthatcanbewrittenasalinear combinationof f x 1 ;x 2 ;:::;x k g span f x 1 ;:::;x k g = f 1 x 1 + 2 x 2 + + k x k ; i 2 C n g Example: Giventhevector x 1 = 0 @ 3 2 1 A thespanof x 1 isa line 4 "Subspaces",Denition2:"Span"

PAGE 330

322 GLOSSARY Example: Giventhevectors x 1 = 0 @ 3 2 1 A x 2 = 0 @ 1 2 1 A thespanofthesevectorsis C 2 U UniformConvergence ThesequenceSection9.1 f g n gj 1 n =1 convergesuniformlytofunction g ifforevery > 0 thereis aninteger N suchthat n N implies j g n t )]TJ/F11 9.9626 Tf 9.962 0 Td [(g t j .7 for all t 2 R V Vectorspace Alinearvectorspace S isacollectionof"vectors"suchthatif f 1 2 S f 1 2 S forall scalars where 2 R or 2 C andif f 1 2 S f 2 2 S ,then f 1 + f 2 2 S Z zeros 1.Thevaluesfor z where P z =0 2.Thecomplexfrequenciesthatmaketheoverallgainoftheltertransferfunctionzero. zeros 1.Thevaluesfor z wherethe numerator ofthetransferfunctionequalszero 2.Thecomplexfrequenciesthatmaketheoverallgainoftheltertransferfunctionzero.

PAGE 331

INDEX 323 IndexofKeywordsandTerms Keywords arelistedbythesectionwiththatkeywordpagenumbersareinparentheses.Keywords donotnecessarilyappearinthetextofthepage.Theyaremerelyassociatedwiththatsection. Ex. apples,1.1 Terms arereferencedbythepagetheyappearon. Ex. apples,1 A alias,12.5,12.6 aliasing,7.2,12.5,206, 12.6 almosteverywhere,6.11 alphabet,1.7,34 analog,1.1,2,31,10.6,10.7 analysis,114 Anti-Aliasing,12.6 anticausal,1.1,3 aperiodic,1.1,112 approximation,15.13 B bandlimited,197 baraniuk,16.2 bases,5.1 basis,5.1,94,94,94,7.2,146, 7.3,203,15.7278,15.8,282, 15.9,15.10,15.11 basismatrix,15.8,283 bestapproximates,297 BIBO,3.4 bilateralLaplacetransformpair,217 bilateralz-transform,229 boundedinputboundedoutput,3.4 boundedinput-boundedoutputBIBO,40 boxcarlter,72 buttery,8.3,166 C Cartesian,302 cascade,2.2 cauch-schwarz,15.5 cauchy,15.5 cauchy-schwarzinequality,15.5,269 causal,1.1,3,2.1,39,2.2,255 characteristicequation,52 characteristicpolynomial,251 circular,4.3,6.7,4.3 circularconvolution,4.3,78,6.7, 4.3,158 circularshift,7.5 circularshifting,155 circularshifts,7.5,153 coecient,6.2 coecientvector,15.8,283 commutative,53,60,73 complex,1.7,7.2,14.7 complexamplitude,29 complexcontinuous-timeexponentialsignal, 29 complexexponential,23,1.6,28 complexexponentialsequence,32 complexplane,1.6 complexsinusoids,7.2 complexvectorspace,261 complexvectorspaces,15.1 complex-valued,1.7 complex-valuedfunction,218,218 complexity,163,8.3 composite,167 computationaladvantage,165 conjugates,146 Constant-Coecient,4.4 continuous,1,132 continuousfrequency,10.4,11.1 continuoussystem,37 continuoustime,1.1,1.4,3.1, 3.2,3.4,7.1,10.3, 11.1,11.2,12.1,195, 12.7,13.1,13.2, 13.3,13.4,13.6 continuous-time,3.1 Continuous-TimeFourierTransform,189 controltheory,227 converge,6.9,9.3 convergence,6.9,132,9.1, 9.2,9.3 converges,9.3 convolution,3.2,3.3,4.2,73, 4.3,6.7,4.3,11.2 convolutionintegral,53 convolutionsum,73 convolutions,4.3,4.3 convolve,4.3,4.3

PAGE 332

324 INDEX Cooley-Tukey,8.1,8.3 csi,15.5 CTFT,11.1,12.1 cuachy,15.5 D de,3.1 DecayingExponential,23 decompose,1.7,15.8,282 deltafunction,15.7 determinant,5.2 deterministicsignal,7 dft,4.3,7.5,4.3, 8.2,10.1,10.2 dierenceequation,4.1,69,82,249,249 DierenceEquations,4.4,4.5 dierential,3.1 dierentialequations,3.1 digital,1.1,2,10.6,10.7 digitalsignalprocessing,4.1,10.7 Diracdelta,23 diracdeltafunction,1.4,1.5,25 directmethod,251 dirichletconditions,6.10,132 Dirichletsinc,10.1177 discontinuity,6.9,132 discontinuousfunctions,135 discrete,1,7.2,12.7 discretefouriertransform,4.3, 4.3,10.2,179 discretesystem,37 discretetime,1.1,3.4,4.2, 7.1,7.5,10.4,12.1, 13.6,14.2 discretetimefourierseries,7.2,142, 7.3,7.4 discretetimeprocessing,12.7 discretetimesignals,195 discrete-time,1.7,7.3, 10.5,10.6,10.7 discrete-timeconvolution,72 discrete-timeexponentialsignal,29 discrete-timeFouriertransform,10.7 Discrete-TimeFourierTransformproperties, 10.5 discrete-timesincfunction,187 discrete-timesystems,4.1 domain,248,248 dotproduct,15.3 dotproducts,266 DSP,4.1,10.7,12.4, 14.2 DT,4.2 dtfs,7.2,7.3,7.4 DTFT,10.1,10.4,12.1 dynamiccontent,17 E edgy,118 eigen,5.4 eigenfunction,5.2,5.4,5.5, 106,6.4 eigenfunctions,5.4,6.2, 6.4 eigensignal,106 eigenvalue,5.2,318,97,5.4, 5.5 eigenvalues,5.2,5.4 eigenvector,5.2,96,5.4, 5.5 eigenvectors,5.4 Elec301,1.2,4.4,4.5, 16.1,16.2 elec301,16.1,16.2 embedded,17 energy,131 Euclideannorm,15.2 Euler'sIdentity,29 Euler'sRelation,30 evensignal,1.1,4,6.6 example,10.7 examples,10.7 existence,132 expansion,15.9,15.10 exponential,1.4,1.731,6.3 exponentialfunction,28 F fastFouriertransform,8.1,8.2, 8.3 FFT,8.1,8.2,8.3, 10.1 lter,12.6,14.8 nite-durationsequence,236 nite-lengthsequence,246 nite-lengthsignal,8 FIR,72 form,165 fourier,5.2,6.2,6.3, 6.4,6.5,6.6,6.7, 6.8,6.9,6.10,6.12, 7.2,7.3,7.4,7.5, 8.2,15.10 fourieranalysis,7.2 fouriercoecient,6.9 fouriercoecients,6.2,113,6.3 fourierseries,5.296,6.2,6.3,

PAGE 333

INDEX 325 6.4,6.6,6.7,6.9, 6.10,6.11,6.12, 7.1,7.2,7.3,7.4, 15.7,15.10 fouriertransform,4.3,6.10, 7.1,7.5,4.3,8.2, 10.2,10.3,10.4, 10.5,10.6,10.7, 11.1,11.2,14.1,229 fouriertransformpairs,5.6 frequency,10.6 FrequencyDomain,10.5 frequencyshiftkeying,15.5 fsk,15.5 function,14.5 functionsequences,9.3 functionspace,15.9 functionspaces,95,15.9 fundamentalperiod,2 G geometricseries,184 Gibb'sphenomenon,137 gibbsphenomenon,6.11,135 graphicalmethod,56 GrowingExponential,23 H haar,15.10 haarwavelet,15.10 harmonic,7.2 harmonicsinusoids,7.2,143 hermitian,15.11 hilbert,15.4,15.5,15.6275, 15.8,15.10 Hilbertspace,268,15.6,15.7278, 15.13 hilbertspaces,15.4,15.6, 15.8,15.10 homework1,16.1 homeworkone,16.1 homogeneoussolution,251 I identitymatrix,284 IIR,71 imaginarypart,306 Importantnote:,213 impulse,1.4,1.5 impulseresponse,28,4.2 independence,5.1 indirectmethod,251 innite-lengthsignal,8 initialconditions,70,249 inner,15.4 innerproduct,15.3,266,15.4, 15.5,15.11 innerproductspace,268 innerproducts,15.3,266,15.5 interpolation,12.2 InverseLaplaceTransform,13.5 inversetransform,6.2,114 K kronecker,15.7 L LabVIEW,17 laplacetransform,3.4,7.1, 13.1,13.2,13.3, 13.4,13.6 left-handed,7 limit,169 linear,2.1,37,2.2,4.4 linearalgebra,5.1 linearandtime-invariant,72 linearcombination,302 linearconvolution,78,158 linearfunctionspace,261 linearfunctionspaces,15.1 linearindependence,5.1 linearsystem,3.147,5.2 lineartimeinvariant,3.253,5.5 lineartime-invariantsystems,245 lineartransformation,6.5,120 linearity,11.2 linearlyindependent,91,91,302 LTI,3.2,72,5.4,5.5, 6.2,6.8 LTIsystem,6.2,6.4 M magnitude,306 matchedlter,15.5 matchedlterdetector,15.5,269 matchedlters,15.5 matrixequation,7.3 maxima,133 maximum,267 meansquare,137 minima,133 modulation,11.2 mutuallyorthogonal,320 N noisysignals,118 nonanticipative,39 noncausal,1.1,3,2.1,39 nonlinear,2.1,37 nonuniformconvergence,6.11,135 Norm,1.2,9.1,9.2, 15.2,263,15.3266

PAGE 334

326 INDEX normconvergence,9.1,9.2 normalization,15.2 normalized,15.7,320 normalizedbasis,15.7,278,278 normedlinearspace,15.2263,263,268 normedspace,15.6 normedvectorspace,15.2,263 norms,15.2 not,164 NthRootofUnity,302 Nyquist,10.6,12.4 Nyquistfrequency,10.7,12.4, 205 Nyquisttheorem,12.4,205 O oddsignal,1.1,4,6.6 onourcomputer!!!,213 order,8.3,13.5,225,249 orthogonal,15.3,267,267,15.7, 320,297 orthogonalbasis,15.7,278 orthonormal,7.2,15.7278, 15.8 orthonormalbasis,7.22,147, 15.7,279,15.8,281 P parallel,2.2 Parseval,15.12 Parseval'sTheorem,10.7 particularsolution,251 perfect,12.3 period,2,6.1,111,7.4 periodic,1.1,6.1,7.4 periodicfunction,6.1,111 periodicity,6.1 phasor,29 Plancharel,15.12 pointwise,9.1,9.2 pointwise,6.9,137,9.1, 9.2,171 pointwiseconvergence,6.9,9.2 pole,3.4,13.4,13.6225, 14.7,14.8 pole-zerocancellation,227 poles,13.5,224,225,248,252 polynomial,14.5 powerseries,230,235 processing,12.7 projection,15.3,267,15.13 properties,6.6 property,3.3 proportional,163 R randomsignal,7 rational,14.5 rationalfunction,14.5,247,247 rationalfunctions,14.5 realpart,306 realvectorspace,261,274 realvectorspaces,15.1 real-valued,1.7 reconstruct,12.3 reconstruction,12.2,12.3203, 12.4,12.5 regionofconvergence,13.4 regionofconvergenceROC,221 rest,87 right-handed,7 right-sidedsequence,237,238 ROC,13.4,14.1,230,235 rootmeansquared,17 S s-plane,31 sample,12.4,12.5 sampling,179,195,12.3,12.4, 12.5,12.6 schwarz,15.5 sequence,169 sequenceoffunctions,131 Sequence-Domain,10.5 sequences,1.7,9.1,9.3 shift-invariant,69 shift-invariantsystems,4.1 siftingproperty,1.4,1.5,27 signal,6.1,6.12137 signals,1.3,1.4,1.5, 1.6,1.7,2.1,3.2, 3.3,3.4,4.2,6.2, 6.3,7.1,10.3,13.4, 13.6 signalsandsystems,1.1,4.2 sinc,12.3 sine,1.7 singularities,13.5,224 sinusoid,1.7,6.3 smoothsignals,118 span,5.1,93,5.4,285 squarepulse,135 stability,3.4 stable,40,255 standardbasis,94,7.3,15.8 strongdirichletcondition,6.10 StrongDirichletConditions,133 superposition,2.2,4.1 symbolic-valuedsignals,1.7

PAGE 335

INDEX 327 symmetry,6.5,6.6,11.2 symmetryproperties,6.6 symmetryproperty,6.6 synthesis,114 system,4.3,5.4,6.4, 6.12,4.3 systems,1.7,3.2,3.3, 3.4,7.1,10.3,13.4, 13.6 T t-periodic,6.1 threshold,271 timedierentiation,11.2 timedomain,4.1 timeinvariant,2.1,38 timereversal,1.3 timescaling,1.3,11.2 timeshifting,1.3,11.2 timevariant,39 timevarying,2.1 time-invariant,2.2 transferfunction,250 transform,6.2,114 transformpairs,14.2 transforms,102,285 transpose,15.11 two-sidedsequence,239 U uniform,9.3 uniformconvergence,9.3,174 unilateral,14.2 unilateralz-transform,229 unique,282 unitimpulse,23 unitsample,1.7,33 unitstep,1.4 unit-stepfunction,24 unity,25 unstable,40 V vector,9.21,15.1 vectorspace,15.1,261,15.6, 15.7,15.9 vectorspaces,15.1,15.6, 15.7 vectors,9.2 verticalasymptotes,248 VI,17 virtualinstrument,17 W wavelet,15.10 wavelets,15.10,286 weakdirichletcondition,6.10,132 weightedsum,302 X x-intercept,248 Y y-intercept,248 Z ztransform,3.4,7.1,13.6, 14.2 z-plane,229,14.7 z-transform,14.1,229,14.2,235, 14.5 z-transforms,234 zero,3.4,13.4221,13.6, 14.7,14.8 zero-inputresponse,47 zero-stateresponse,47 zeros,225,248,252

PAGE 336

328 ATTRIBUTIONS Attributions Collection: SignalsandSystems Editedby:RichardBaraniuk URL:http://cnx.org/content/col10064/1.11/ License:http://creativecommons.org/licenses/by/1.0 Module:"SignalClassicationsandProperties" By:MelissaSelik,RichardBaraniuk,MichaelHaag URL:http://cnx.org/content/m10057/2.17/ Pages:1-9 Copyright:MelissaSelik,RichardBaraniuk,MichaelHaag License:http://creativecommons.org/licenses/by/1.0 Module:"SizeofASignal:Norms" By:RichardBaraniuk URL:http://cnx.org/content/m12363/1.2/ Pages:9-19 Copyright:RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"SignalOperations" By:RichardBaraniuk URL:http://cnx.org/content/m10125/2.8/ Pages:19-22 Copyright:RichardBaraniuk,AdamBlair License:http://creativecommons.org/licenses/by/1.0 Module:"UsefulSignals" By:MelissaSelik,RichardBaraniuk URL:http://cnx.org/content/m10058/2.13/ Pages:22-25 Copyright:MelissaSelik,RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"TheImpulseFunction" By:MelissaSelik,RichardBaraniuk URL:http://cnx.org/content/m10059/2.19/ Pages:25-28 Copyright:MelissaSelik,RichardBaraniuk,AdamBlair License:http://creativecommons.org/licenses/by/1.0 Module:"TheComplexExponential" By:RichardBaraniuk URL:http://cnx.org/content/m10060/2.20/ Pages:28-31 Copyright:RichardBaraniuk,AdamBlair License:http://creativecommons.org/licenses/by/1.0

PAGE 337

ATTRIBUTIONS 329 Module:"Discrete-TimeSignals" By:DonJohnson URL:http://cnx.org/content/m0009/2.24/ Pages:31-34 Copyright:DonJohnson License:http://creativecommons.org/licenses/by/1.0 Module:"SystemClassicationsandProperties" By:MelissaSelik,RichardBaraniuk URL:http://cnx.org/content/m10084/2.19/ Pages:37-40 Copyright:MelissaSelik,RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"PropertiesofSystems" By:ThanosAntoulas,JPSlavinsky URL:http://cnx.org/content/m2102/2.16/ Pages:41-46 Copyright:ThanosAntoulas,JPSlavinsky License:http://creativecommons.org/licenses/by/1.0 Module:"CTLinearSystemsandDierentialEquations" By:MichaelHaag,RichardBaraniuk URL:http://cnx.org/content/m10855/2.6/ Pages:47-52 Copyright:MichaelHaag,RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"Continuous-TimeConvolution" By:MelissaSelik,RichardBaraniuk URL:http://cnx.org/content/m10085/2.27/ Pages:53-59 Copyright:MelissaSelik,RichardBaraniuk,AdamBlair License:http://creativecommons.org/licenses/by/1.0 Module:"PropertiesofConvolution" By:MelissaSelik,RichardBaraniuk URL:http://cnx.org/content/m10088/2.14/ Pages:59-65 Copyright:MelissaSelik,RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"BIBOStability" By:RichardBaraniuk URL:http://cnx.org/content/m10113/2.9/ Pages:65-67 Copyright:RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"Discrete-TimeSystemsintheTime-Domain" By:DonJohnson URL:http://cnx.org/content/m10251/2.23/ Pages:69-72 Copyright:DonJohnson License:http://creativecommons.org/licenses/by/1.0

PAGE 338

330 ATTRIBUTIONS Module:"Discrete-TimeConvolution" By:RicardoRadaelli-Sanchez,RichardBaraniuk URL:http://cnx.org/content/m10087/2.18/ Pages:72-78 Copyright:RicardoRadaelli-Sanchez,RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"CircularConvolutionandtheDFT" By:JustinRomberg URL:http://cnx.org/content/m10786/2.8/ Pages:157-161 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"LinearConstant-CoecientDierenceEquations" By:RichardBaraniuk URL:http://cnx.org/content/m12325/1.3/ Pages:82-83 Copyright:RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"SolvingLinearConstant-CoecientDierenceEquations" By:RichardBaraniuk URL:http://cnx.org/content/m12326/1.3/ Pages:83-88 Copyright:RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"LinearAlgebra:TheBasics" By:MichaelHaag,JustinRomberg URL:http://cnx.org/content/m10734/2.4/ Pages:91-95 Copyright:MichaelHaag,JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"EigenvectorsandEigenvalues" By:MichaelHaag,JustinRomberg URL:http://cnx.org/content/m10736/2.7/ Pages:96-100 Copyright:MichaelHaag,JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"MatrixDiagonalization" By:MichaelHaag URL:http://cnx.org/content/m10738/2.5/ Pages:101-103 Copyright:MichaelHaag License:http://creativecommons.org/licenses/by/1.0 Module:"Eigen-stuinaNutshell" By:MichaelHaag,JustinRomberg URL:http://cnx.org/content/m10742/2.4/ Page:104 Copyright:MichaelHaag,JustinRomberg License:http://creativecommons.org/licenses/by/1.0

PAGE 339

ATTRIBUTIONS 331 Module:"EigenfunctionsofLTISystems" By:JustinRomberg URL:http://cnx.org/content/m10500/2.7/ Pages:105-107 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"FourierTransformProperties" By:DonJohnson URL:http://cnx.org/content/m0045/2.8/ Page:108 Copyright:DonJohnson License:http://creativecommons.org/licenses/by/1.0 Module:"PeriodicSignals" By:MichaelHaag,JustinRomberg URL:http://cnx.org/content/m10744/2.6/ Pages:111-112 Copyright:MichaelHaag,JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"FourierSeries:EigenfunctionApproach" By:JustinRomberg URL:http://cnx.org/content/m10496/2.21/ Pages:112-115 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"DerivationofFourierCoecientsEquation" By:MichaelHaag URL:http://cnx.org/content/m10733/2.6/ Pages:115-116 Copyright:MichaelHaag License:http://creativecommons.org/licenses/by/1.0 Module:"FourierSeriesinaNutshell" By:MichaelHaag,JustinRomberg URL:http://cnx.org/content/m10751/2.3/ Pages:116-119 Copyright:MichaelHaag,JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"FourierSeriesProperties" By:JustinRomberg,BenjaminFite URL:http://cnx.org/content/m10740/2.7/ Pages:119-122 Copyright:JustinRomberg,BenjaminFite License:http://creativecommons.org/licenses/by/1.0 Module:"SymmetryPropertiesoftheFourierSeries" By:JustinRomberg URL:http://cnx.org/content/m10838/2.4/ Pages:122-126 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0

PAGE 340

332 ATTRIBUTIONS Module:"CircularConvolutionPropertyofFourierSeries" By:JustinRomberg URL:http://cnx.org/content/m10839/2.4/ Pages:126-127 Copyright:RichardBaraniuk,JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"FourierSeriesandLTISystems" By:JustinRomberg URL:http://cnx.org/content/m10752/2.7/ Pages:127-130 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"ConvergenceofFourierSeries" By:MichaelHaag,JustinRomberg URL:http://cnx.org/content/m10745/2.3/ Pages:130-132 Copyright:MichaelHaag,JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"DirichletConditions" By:RicardoRadaelli-Sanchez URL:http://cnx.org/content/m10089/2.9/ Pages:132-134 Copyright:RicardoRadaelli-Sanchez License:http://creativecommons.org/licenses/by/1.0 Module:"Gibbs'sPhenomena" By:RicardoRadaelli-Sanchez,RichardBaraniuk URL:http://cnx.org/content/m10092/2.9/ Pages:134-137 Copyright:RicardoRadaelli-Sanchez,RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"FourierSeriesWrap-Up" By:MichaelHaag,JustinRomberg URL:http://cnx.org/content/m10749/2.4/ Pages:137-138 Copyright:MichaelHaag,JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"FourierAnalysis" By:RichardBaraniuk URL:http://cnx.org/content/m10096/2.10/ Pages:141-142 Copyright:RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"FourierAnalysisinComplexSpaces" By:MichaelHaag,JustinRomberg URL:http://cnx.org/content/m10784/2.7/ Pages:142-148 Copyright:MichaelHaag,JustinRomberg License:http://creativecommons.org/licenses/by/1.0

PAGE 341

ATTRIBUTIONS 333 Module:"MatrixEquationfortheDTFS" By:RoyHa URL:http://cnx.org/content/m10771/2.6/ Pages:148-149 Copyright:RoyHa License:http://creativecommons.org/licenses/by/1.0 Module:"PeriodicExtensiontoDTFS" By:RoyHa URL:http://cnx.org/content/m10778/2.8/ Pages:150-153 Copyright:RoyHa License:http://creativecommons.org/licenses/by/1.0 Module:"CircularShifts" By:JustinRomberg URL:http://cnx.org/content/m10780/2.6/ Pages:153-157 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"CircularConvolutionandtheDFT" By:JustinRomberg URL:http://cnx.org/content/m10786/2.8/ Pages:157-161 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"DFT:FastFourierTransform" By:DonJohnson URL:http://cnx.org/content/m0504/2.8/ Page:163 Copyright:DonJohnson License:http://creativecommons.org/licenses/by/1.0 Module:"TheFastFourierTransformFFT" By:JustinRomberg URL:http://cnx.org/content/m10783/2.5/ Pages:164-165 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"DerivingtheFastFourierTransform" By:DonJohnson URL:http://cnx.org/content/m0528/2.7/ Pages:165-167 Copyright:DonJohnson License:http://creativecommons.org/licenses/by/1.0

PAGE 342

334 ATTRIBUTIONS Module:"ConvergenceofSequences" By:RichardBaraniuk URL:http://cnx.org/content/m10883/2.4/ Pages:169-170 Copyright:RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"ConvergenceofVectors" By:MichaelHaag URL:http://cnx.org/content/m10894/2.2/ Pages:171-174 Copyright:MichaelHaag License:http://creativecommons.org/licenses/by/1.0 Module:"UniformConvergenceofFunctionSequences" By:MichaelHaag,RichardBaraniuk URL:http://cnx.org/content/m10895/2.5/ Pages:174-175 Copyright:MichaelHaag License:http://creativecommons.org/licenses/by/1.0 Module:"DiscreteFourierTransformation" By:PhilSchniter URL:http://cnx.org/content/m10421/2.11/ Pages:177-179 Copyright:PhilSchniter License:http://creativecommons.org/licenses/by/1.0 Module:"DiscreteFourierTransformDFT" By:DonJohnson URL:http://cnx.org/content/m10249/2.27/ Pages:179-181 Copyright:DonJohnson License:http://creativecommons.org/licenses/by/1.0 Module:"TableofCommonFourierTransforms" By:MelissaSelik,RichardBaraniuk URL:http://cnx.org/content/m10099/2.9/ Pages:181-182 Copyright:MelissaSelik,RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"Discrete-TimeFourierTransformDTFT" By:RichardBaraniuk URL:http://cnx.org/content/m10108/2.11/ Page:182 Copyright:RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0

PAGE 343

ATTRIBUTIONS 335 Module:"Discrete-TimeFourierTransformProperties" By:DonJohnson URL:http://cnx.org/content/m0506/2.6/ Page:183 Copyright:DonJohnson License:http://creativecommons.org/licenses/by/1.0 Module:"Discrete-TimeFourierTransformPair" By:DonJohnson URL:http://cnx.org/content/m0525/2.6/ Pages:183-184 Copyright:DonJohnson License:http://creativecommons.org/licenses/by/1.0 Module:"DTFTExamples" By:DonJohnson URL:http://cnx.org/content/m0524/2.11/ Pages:184-187 Copyright:DonJohnson License:http://creativecommons.org/licenses/by/1.0 Module:"Continuous-TimeFourierTransformCTFT" By:RichardBaraniuk,MelissaSelik URL:http://cnx.org/content/m10098/2.9/ Pages:189-190 Copyright:RichardBaraniuk,MelissaSelik License:http://creativecommons.org/licenses/by/1.0 Module:"PropertiesoftheContinuous-TimeFourierTransform" By:MelissaSelik,RichardBaraniuk URL:http://cnx.org/content/m10100/2.13/ Pages:190-193 Copyright:MelissaSelik,RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"Sampling" By:JustinRomberg URL:http://cnx.org/content/m10798/2.6/ Pages:195-199 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"Reconstruction" By:JustinRomberg URL:http://cnx.org/content/m10788/2.5/ Pages:199-203 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0

PAGE 344

336 ATTRIBUTIONS Module:"MoreonPerfectReconstruction" Usedhereas:"MoreonReconstruction" By:RoyHa,JustinRomberg URL:http://cnx.org/content/m10790/2.4/ Pages:203-205 Copyright:RoyHa,JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"NyquistTheorem" By:JustinRomberg URL:http://cnx.org/content/m10791/2.5/ Pages:205-206 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"Aliasing" By:JustinRomberg,DonJohnson URL:http://cnx.org/content/m10793/2.6/ Pages:206-210 Copyright:JustinRomberg,DonJohnson License:http://creativecommons.org/licenses/by/1.0 Module:"Anti-AliasingFilters" By:JustinRomberg URL:http://cnx.org/content/m10794/2.4/ Pages:210-211 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"DiscreteTimeProcessingofContinuousTimeSignals" By:JustinRomberg URL:http://cnx.org/content/m10797/2.9/ Pages:211-213 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"TheLaplaceTransforms" By:RichardBaraniuk URL:http://cnx.org/content/m10110/2.12/ Pages:217-219 Copyright:RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"PropertiesoftheLaplaceTransform" By:MelissaSelik,RichardBaraniuk URL:http://cnx.org/content/m10117/2.9/ Pages:219-220 Copyright:MelissaSelik,RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0

PAGE 345

ATTRIBUTIONS 337 Module:"TableofCommonLaplaceTransforms" By:MelissaSelik,RichardBaraniuk URL:http://cnx.org/content/m10111/2.10/ Page:221 Copyright:MelissaSelik,RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"RegionofConvergencefortheLaplaceTransform" By:RichardBaraniuk URL:http://cnx.org/content/m10114/2.8/ Pages:221-223 Copyright:RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"TheInverseLaplaceTransform" By:StevenCox URL:http://cnx.org/content/m10170/2.8/ Pages:223-225 Copyright:StevenCox License:http://creativecommons.org/licenses/by/1.0 Module:"PolesandZeros" By:RichardBaraniuk URL:http://cnx.org/content/m10112/2.11/ Pages:225-227 Copyright:RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"TheZTransform:Denition" By:BenjaminFite URL:http://cnx.org/content/m10549/2.9/ Pages:229-234 Copyright:BenjaminFite License:http://creativecommons.org/licenses/by/1.0 Module:"TableofCommonz-Transforms" By:MelissaSelik,RichardBaraniuk URL:http://cnx.org/content/m10119/2.13/ Pages:234-235 Copyright:MelissaSelik,RichardBaraniuk License:http://creativecommons.org/licenses/by/1.0 Module:"RegionofConvergencefortheZ-transform" By:BenjaminFite URL:http://cnx.org/content/m10622/2.5/ Pages:235-244 Copyright:BenjaminFite License:http://creativecommons.org/licenses/by/1.0 Module:"InverseZ-Transform" By:BenjaminFite URL:http://cnx.org/content/m10651/2.4/ Pages:244-246 Copyright:BenjaminFite License:http://creativecommons.org/licenses/by/1.0

PAGE 346

338 ATTRIBUTIONS Module:"RationalFunctions" By:MichaelHaag URL:http://cnx.org/content/m10593/2.7/ Pages:247-249 Copyright:MichaelHaag License:http://creativecommons.org/licenses/by/1.0 Module:"DierenceEquation" By:MichaelHaag URL:http://cnx.org/content/m10595/2.5/ Pages:249-252 Copyright:MichaelHaag License:http://creativecommons.org/licenses/by/1.0 Module:"UnderstandingPole/ZeroPlotsontheZ-Plane" By:MichaelHaag URL:http://cnx.org/content/m10556/2.8/ Pages:252-256 Copyright:MichaelHaag License:http://creativecommons.org/licenses/by/1.0 Module:"FilterDesignusingthePole/ZeroPlotofaZ-Transform" By:MichaelHaag URL:http://cnx.org/content/m10548/2.9/ Pages:256-259 Copyright:MichaelHaag License:http://creativecommons.org/licenses/by/1.0 Module:"VectorSpaces" By:MichaelHaag,StevenCox,JustinRomberg URL:http://cnx.org/content/m10767/2.4/ Pages:261-262 Copyright:MichaelHaag,StevenCox,JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"Norms" By:MichaelHaag,JustinRomberg URL:http://cnx.org/content/m10768/2.4/ Pages:263-265 Copyright:MichaelHaag,JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"InnerProducts" By:MichaelHaag,JustinRomberg URL:http://cnx.org/content/m10755/2.6/ Pages:266-267 Copyright:MichaelHaag,JustinRomberg License:http://creativecommons.org/licenses/by/1.0

PAGE 347

ATTRIBUTIONS 339 Module:"HilbertSpaces" By:JustinRomberg URL:http://cnx.org/content/m10840/2.4/ Page:268 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"Cauchy-SchwarzInequality" By:MichaelHaag,JustinRomberg URL:http://cnx.org/content/m10757/2.5/ Pages:268-275 Copyright:MichaelHaag,JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"CommonHilbertSpaces" By:RoyHa,JustinRomberg URL:http://cnx.org/content/m10759/2.5/ Pages:275-278 Copyright:RoyHa,JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"TypesofBasis" By:MichaelHaag,JustinRomberg URL:http://cnx.org/content/m10772/2.5/ Pages:278-281 Copyright:MichaelHaag,JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"OrthonormalBasisExpansions" By:MichaelHaag,JustinRomberg URL:http://cnx.org/content/m10760/2.4/ Pages:281-285 Copyright:MichaelHaag,JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"FunctionSpace" By:JustinRomberg URL:http://cnx.org/content/m10770/2.4/ Pages:285-286 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"HaarWaveletBasis" By:RoyHa,JustinRomberg URL:http://cnx.org/content/m10764/2.7/ Pages:286-293 Copyright:RoyHa,JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"OrthonormalBasesinRealandComplexSpaces" By:JustinRomberg URL:http://cnx.org/content/m10765/2.7/ Pages:293-295 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0

PAGE 348

340 ATTRIBUTIONS Module:"PlancharelandParseval'sTheorems" By:JustinRomberg URL:http://cnx.org/content/m10769/2.5/ Pages:295-296 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"ApproximationandProjectionsinHilbertSpace" By:JustinRomberg URL:http://cnx.org/content/m10766/2.7/ Pages:296-298 Copyright:JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"Homework1" By:RichardBaraniuk,JustinRomberg URL:http://cnx.org/content/m10826/2.9/ Pages:301-305 Copyright:RichardBaraniuk,JustinRomberg License:http://creativecommons.org/licenses/by/1.0 Module:"Homework1Solutions" By:JustinRomberg,RichardBaraniuk,MichaelHaag URL:http://cnx.org/content/m10830/2.4/ Pages:305-316 Copyright:JustinRomberg,RichardBaraniuk,MichaelHaag License:http://creativecommons.org/licenses/by/1.0 Module:"ViewingEmbeddedLabVIEWContent" By:MatthewHutchinson URL:http://cnx.org/content/m13753/1.3/ Page:317 Copyright:MatthewHutchinson License:http://creativecommons.org/licenses/by/2.0/

PAGE 349

SignalsandSystems Thiscoursedealswithsignals,systems,andtransforms,fromtheirtheoreticalmathematicalfoundationsto practicalimplementationincircuitsandcomputeralgorithms.AttheconclusionofELEC301,youshould haveadeepunderstandingofthemathematicsandpracticalissuesofsignalsincontinuousanddiscretetime, lineartimeinvariantsystems,convolution,andFouriertransforms. AboutConnexions Since1999,Connexionshasbeenpioneeringaglobalsystemwhereanyonecancreatecoursematerialsand makethemfullyaccessibleandeasilyreusablefreeofcharge.WeareaWeb-basedauthoring,teachingand learningenvironmentopentoanyoneinterestedineducation,includingstudents,teachers,professorsand lifelonglearners.Weconnectideasandfacilitateeducationalcommunities. Connexions'smodular,interactivecoursesareinuseworldwidebyuniversities,communitycolleges,K-12 schools,distancelearners,andlifelonglearners.Connexionsmaterialsareinmanylanguages,including English,Spanish,Chinese,Japanese,Italian,Vietnamese,French,Portuguese,andThai.Connexionsispart ofanexcitingnewinformationdistributionsystemthatallowsfor PrintonDemandBooks .Connexions haspartneredwithinnovativeon-demandpublisherQOOPtoacceleratethedeliveryofprintedcourse materialsandtextbooksintoclassroomsworldwideatlowerpricesthantraditionalacademicpublishers.