Waveform Design for Active Sensing Systems -- a Computational Approach

MISSING IMAGE

Material Information

Title:
Waveform Design for Active Sensing Systems -- a Computational Approach
Physical Description:
1 online resource (161 p.)
Language:
english
Creator:
He,Hao
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Electrical and Computer Engineering
Committee Chair:
Li, Jian
Committee Members:
Zmuda, Henry
Lin, Jenshan
Shabanov, Sergei

Subjects

Subjects / Keywords:
algorithm -- computational -- correlation -- radar -- signal -- sonar -- waveform
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre:
Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Active sensing applications such as radar, sonar and medical imaging, demand proper designs of the probing waveform. A well-synthesized waveform can significantly increase the system performance in terms of signal-to-interference ratio, spectrum containment, beampattern matching, target parameter estimation and so on. The focus of this work is on designing probing waveforms using computational algorithms. We first investigate designing waveforms with good correlation properties, which are widely useful in applications including range compression, channel estimation and spread spectrum. We consider both the design of a single sequence and that of a set of sequences, the former with only auto-correlations and the latter with auto- and cross-correlations. The proposed algorithms leverage FFT (fast Fourier transform) operations and can efficiently generate long sequences that were previously difficult to synthesize. We present a new derivation of the lower bound for sequence correlations that arises from the proposed algorithm framework. We show that such a lower bound can be closely approached by the newly designed sequences. A two-dimensional extension of the time-delay correlation function is the ambiguity function (AF) that involves a Doppler frequency shift. We give an overview of AF properties and discuss how to minimize AF sidelobes in a discrete formation. Besides good correlation properties, we also consider the stopband constraint that is required in the scenario of avoiding reserved frequency bands or strong electronic jammer. We present an algorithm that accounts for both correlation and stopband constraints. We finally consider transmit beampattern synthesis, particularly in the wideband case. We establish the relationship between a desired beampattern and underlying waveforms by using the Fourier transform. We highlight the increased design freedom resulting from the waveform diversity of a MIMO (multi-input multi-output) system.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Hao He.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Li, Jian.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2011
System ID:
UFE0043119:00001


This item is only available as the following downloads:


Full Text

PAGE 1

WAVEFORMDESIGNFORACTIVESENSINGSYSTEMSACOMPUTATIONALAPPROACHByHAOHEADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

PAGE 2

c2011HaoHe 2

PAGE 3

ACKNOWLEDGMENTS Firstandforemost,Ioffersinceregratitudetomyadvisor,Dr.JianLi,whohasguidedmethroughoutmyPhDpursuitwithhergreatknowledgeandpatience.IthasbeenanexceptionalexperiencetoworkwithDr.Liinthepastfouryears.Hermentoringisinspiringandherdedicationtoworkiscontagious.Thedissertationwouldhavebeennexttoimpossiblewithouthervisionandresearchsupport.FurthermoreIamgratefulfortheexcellentcollaborationwithDr.PetreStoicaofUppsalaUniversity,Sweden.IheartilyappreciatehiscontributionofideasthatmotivatedmyresearchandmademyPhDjourneyproductive.IsimplycouldnotwishforabettercollaboratorwhoilluminatesalmostalltopicsIstudied.IacknowledgemycommitteemembersatUniversityofFlorida:Dr.JenshanLin,Dr.SergeiShabanovandDr.HenryZmuda.Iamtrulythankfulforthetimeandeffortsthattheyspentonreviewingandcommentingmyresearchproposalanddissertationdefense.IamfortunatetoworkwithacheerfulgroupoffellowstudentsatDr.Li'slab.IappreciateallthehelpandencouragementthatIreceivedfromlabmembersduringbothmypersonalandprofessionaltime.Researchwouldhavebeenlesscolorfulwithoutthewittyremarksinthelabnowandthen.Finally,Iamdeeplyindebtedtomyparentsandmywife.Theyhaveprovidedmewithimmenseunderstandingandmoralsupportalltheseyears.Ihaveenjoyedeverymomentwespenttogetherwithcareandlove.Thisdissertationisdedicatedtothem. 3

PAGE 4

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 3 LISTOFTABLES ...................................... 6 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 1.1SignalModel .................................. 14 1.2DesignMetrics ................................. 17 1.3ReviewofExistingWaveforms ........................ 19 2SINGLESEQUENCEDESIGN ........................... 29 2.1CyclicAlgorithm-New(CAN) ......................... 31 2.2WeightedCyclicAlgorithm-New(WeCAN) .................. 35 2.3NumericalExamples .............................. 39 2.3.1IntegratedSidelobeLevel(ISL)Design ................ 39 2.3.2WeightedIntegratedSidelobeLevel(WISL)Design ......... 40 2.3.3ChannelEstimation ........................... 40 3SEQUENCESETDESIGN ............................. 48 3.1TheMulti-CANAlgorithm ........................... 49 3.2TheMulti-WeCANAlgorithm ......................... 53 3.3TheMulti-CA-Original(Multi-CAO)Algorithm ................ 58 3.4NumericalExamples .............................. 60 3.4.1Multi-CAN ................................ 60 3.4.2Multi-WeCAN .............................. 62 3.4.3Multi-WeCAN(cont.) .......................... 64 3.4.4QuantizationEffects .......................... 64 3.4.5SyntheticApertureRadar(SAR)Imaging .............. 65 4CORRELATIONLOWERBOUNDS ......................... 80 4.1BoundDerivation ................................ 81 4.2ApproachingtheBound ............................ 82 5AMBIGUITYFUNCTION .............................. 89 5.1AFProperties .................................. 89 5.2Discrete-AF ................................... 91 4

PAGE 5

5.3MinimizingtheDiscrete-AFSidelobes .................... 94 6STOPBANDCONSTRAINT ............................. 102 6.1StopbandCAN(SCAN) ............................ 103 6.2WeightedStopbandCyclicAlgorithm-New(WeSCAN) ........... 106 6.3NumericalExamples .............................. 109 6.3.1SCAN .................................. 109 6.3.2WeSCAN ................................ 111 6.3.3RelaxedAmplitudeConstraint ..................... 111 6.3.4UsingaDifferentFrequencyFormulation ............... 111 7TRANSMITBEAMPATTERNSYNTHESIS ..................... 118 7.1ProblemFormulation .............................. 119 7.2TheProposedDesignMethodology ..................... 123 7.2.1BeampatterntoSpectrum ....................... 123 7.2.2SpectrumtoWaveform ......................... 125 7.3NumericalExamples .............................. 128 7.3.1TheIdealizedTime-DelayedCase .................. 128 7.3.2ANarrowMainbeam .......................... 130 7.3.3TwoMainbeams ............................ 131 7.3.4AWideMainbeam ........................... 131 8CONCLUSIONS ................................... 144 APPENDIX ACONNECTIONSTOAPHASE-RETRIEVALALGORITHM ............ 146 BDERIVATIONOFAUNITARYMATRIXSOLUTION ................ 150 CPROPERTIESOFAMBIGUITYFUNCTION .................... 151 DNARROWBANDTRANSMITBEAMPATTERN ................... 152 ERECEIVEBEAMPATTERN ............................. 153 REFERENCES ....................................... 154 BIOGRAPHICALSKETCH ................................ 161 5

PAGE 6

LISTOFTABLES Table page 1-1Notation ........................................ 23 2-1MMFvalues ...................................... 43 3-1ComparisonbetweenMulti-CAN,Multi-CAOandMulti-WeCANsequencesets 68 3-2ComparisonbetweenMulti-CANandMulti-WeCANsequencesets ....... 68 4-1BISLvs.ISLofMulti-CANsequencesets ...................... 88 7-1Optimizedcriterionvaluesforasynthesizedbeampatternwithonemainbeam 133 7-2Optimizedcriterionvaluesforasynthesizedbeampatternwithtwomainbeams 133 6

PAGE 7

LISTOFFIGURES Figure page 1-1Achirpsignalanditsauto-correlations ....................... 24 1-2Auto-correlationofaP4sequence ......................... 25 1-3AP4waveformanditsauto-correlations ...................... 26 1-4Alinearfeedbackshiftregister ........................... 27 1-5Auto-correlationofanm-sequence ......................... 27 1-6Auto-correlationofapolyphaseBarkersequence ................. 28 1-7Auto-correlationofaBarkersequence ....................... 28 2-1ThemeritfactorsoftheGolomb,Frank,CAN(G)andCAN(F)sequencesoflengthsfrom32upto1002. .............................. 42 2-2CorrelationlevelsoftheGolombandCANsequencesoflengthN=102designedundertheISLmetric ................................. 43 2-3CorrelationlevelsoftheGolombandCANsequencesoflengthN=103designedundertheISLmetric ................................. 44 2-4ThecorrelationlevelofaWeCANsequencedesignedundertheWISLmetric 45 2-5ThesimulatedchannelimpulseresponseandtheprobingWeCANsequence 46 2-6MSEofthechannelestimation ........................... 47 3-1Correlationsofthe40-by-3CEandMulti-CANsequencesets:r11(k)andr12(k) 69 3-2Correlationsofthe40-by-3CEandMulti-CANsequencesets:r13(k)andr22(k) 70 3-3Correlationsofthe40-by-3CEandMulti-CANsequencesets:r23(k)andr33(k) 71 3-4ComparsionbetweenMulti-CANandHadamard+PNsequencesetsintermsoftheauto-correlationsidelobepeakandthecross-correlationpeak ...... 72 3-5ComparsionbetweenMulti-CANandHadamard+PNsequencesetsintermsofthettingerror ................................... 73 3-6CorrelationlevelsofMulti-CAOandMulti-WeCANsequencesetswithN=256,M=4andP=50 ............................... 74 3-7CorrelationlevelsofMulti-CANandMulti-WeCANsequencesetswithN=256,M=4andcertainweightsfngN)]TJ /F6 7.97 Tf 6.58 0 Td[(1n=0 ...................... 75 3-8Quantizationeffect .................................. 76 7

PAGE 8

3-9Thetruetargetimage ................................ 77 3-10TheestimatedtargetimagesusingtheHadamard+PNwaveform ........ 78 3-11TheestimatedtargetimagesusingtheMulti-WeCANwaveform ......... 79 4-1Auto-correlationsoftwoCANsequenceswithdifferentPARs .......... 87 4-2ISLofCANsequencesvs.PAR ........................... 88 5-1AFofachirp ..................................... 98 5-2AFofaGolombsequence-codedwaveform .................... 99 5-3AFofaCANsequence-codedwaveform ...................... 100 5-4Thesynthesizeddiscrete-AF ............................ 101 6-1ThespectralpowerandcorrelationlevelofaSCANsequenceundertheconstraintofonestopband ................................... 112 6-2PstopandPcorrvs.from0:1to1 ........................... 113 6-3ThespectralpowerandcorrelationlevelofaSCANsequenceundertheconstraintofseveralstopbands ................................. 114 6-4ThespectralpowerandcorrelationlevelofaWeSCANsequence ........ 115 6-5ThespectralpowerandcorrelationlevelofaSCANsequencewitharelaxedPARconstraint .................................... 116 6-6ThespectralpowerandcorrelationlevelofaSCANsequencewithacontinuousfrequencyformulation ................................ 117 7-1TheULAarrayconguration ............................. 133 7-2Anidealizedtime-delayedbeampattern ...................... 134 7-3AWB-CAbeampatternunderonlythetotalenergyconstraint .......... 135 7-4AWB-CAbeampatternundertheunit-modulusconstraint ............ 136 7-5Overlaidspectraldensitiesoftheobtainedcontinuouswaveforms ........ 137 7-6TheWB-CAbeampatternoftheobtainedcontinuouswaveforms ........ 138 7-7AWB-CAbeampatternwithtwomainbeamsundertheunit-modulusconstraint 139 7-8AWB-CAbeampatternwithtwomainbeamsunderthePAR2constraint ... 140 7-9AWB-CAbeampatternwithawidemainbeamundertheunit-modulusconstraint 141 7-10AWB-CAbeampatternwithawidemainbeamunderthePAR2constraint 142 8

PAGE 9

7-11AWB-CAbeampatternwithalargebandwidth .................. 143 9

PAGE 10

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyWAVEFORMDESIGNFORACTIVESENSINGSYSTEMSACOMPUTATIONALAPPROACHByHaoHeAugust2011Chair:JianLiMajor:ElectricalandComputerEngineeringActivesensingapplicationssuchasradar,sonarandmedicalimaging,demandproperdesignsoftheprobingwaveform.Awell-synthesizedwaveformcansignicantlyincreasethesystemperformanceintermsofsignal-to-interferenceratio,spectrumcontainment,beampatternmatching,targetparameterestimationandsoon.Thefocusofthisworkisondesigningprobingwaveformsusingcomputationalalgorithms.Werstinvestigatedesigningwaveformswithgoodcorrelationproperties,whicharewidelyusefulinapplicationsincludingrangecompression,channelestimationandspreadspectrum.Weconsiderboththedesignofasinglesequenceandthatofasetofsequences,theformerwithonlyauto-correlationsandthelatterwithauto-andcross-correlations.TheproposedalgorithmsleverageFFT(fastFouriertransform)operationsandcanefcientlygeneratelongsequencesthatwerepreviouslydifculttosynthesize.Wepresentanewderivationofthelowerboundforsequencecorrelationsthatarisesfromtheproposedalgorithmframework.Weshowthatsuchalowerboundcanbecloselyapproachedbythenewlydesignedsequences.Atwo-dimensionalextensionofthetime-delaycorrelationfunctionistheambiguityfunction(AF)thatinvolvesaDopplerfrequencyshift.WegiveanoverviewofAFpropertiesanddiscusshowtominimizeAFsidelobesinadiscreteformation.Besidesgoodcorrelationproperties,wealsoconsiderthestopbandconstraintthatisrequiredinthescenarioofavoidingreservedfrequencybandsorstrongelectronic 10

PAGE 11

jammer.Wepresentanalgorithmthataccountsforbothcorrelationandstopbandconstraints.Wenallyconsidertransmitbeampatternsynthesis,particularlyinthewidebandcase.WeestablishtherelationshipbetweenadesiredbeampatternandunderlyingwaveformsbyusingtheFouriertransform.WehighlighttheincreaseddesignfreedomresultingfromthewaveformdiversityofaMIMO(multi-inputmulti-output)system. 11

PAGE 12

CHAPTER1INTRODUCTIONThegoalofanactivesensingsystem,suchasradarorsonar,istodetermineusefulpropertiesofthetargetsorofthepropagationmediumbytransmittingcertainwaveformstowardanareaofinterestandanalyzingthereceivedsignals.Forexample,aland-basedsurveillanceradarsendselectromagneticwavesinthedirectionofsky,whereobjectssuchasairplanescanreecta(usuallyaverytiny)fractionofthetransmittedsignalbacktotheradar.Bymeasuringtheround-triptimedelay,thedistance(calledrange)betweentheradarandthetargetcanbeestimatedsincethespeedofpropagationforradiowavesisknown(3108m/s).Additionaltargetpropertiescanbeobtainedbyperformingfurtherprocessingatthereceiverside;e.g.,thespeedofatargetcanbeestimatedbymeasuringtheDopplerfrequencyshiftofthereceivedsignal.Animalslikedolphinsandbatshaveusedsoundwavestocommunicateanddetectobjectsformillionsofyears,whilehumansdidnotstarttodosountiltheearlytwentiethcentury.In1904,aGermanengineernamedChristianHulsmeyercarriedouttherstradarexperimentusinghistelemobiloscopetodetectshipsindensefogbymeansofradiowaves.Astosonar,ReginaldFessenden,aCanadianengineer,demonstratedin1914usingasoundechodevice,thoughnotsuccessfully,foricebergdetectionofftheeastcoastofCanada.Itwasamongstseveralotherexperimentsandpatentssaidtobepromptedbythe1912Titanicdisaster.Radarandsonarunderwentgreatdevelopmentduringthetwoworldwarsandlateronspreadintodiverseeldsincludingweathermonitoring,ightcontrolandunderwatersensing.Therearetwofactorsthatarecriticaltothesystemperformance,namelythereceivelterandthetransmitwaveform.Thereceivelterisusedtoextractfromthereceivedsignalstheinformationofinterest,e.g.,thetargetlocationsinradar/sonarapplications[ 1 ]orthechannelconditionsincommunications[ 2 ].Thetransmitwaveform, 12

PAGE 13

notsurprisingly,interplayswiththereceivelter.Agooddesignofthetransmitwaveformlendsitselftoaccurateparameterestimationandareducedcomputationalburdenatthereceiver.Arguablythemostcommonlyusedreceivelteristhematchedlter,whichmaximizesthesignal-to-noiseratio(SNR)inthepresenceofstochasticadditivewhitenoise[ 3 ].Examplesofotherwell-knownreceiveltersincludethemismatchedlterwhichisalsocalledtheinstrumental-variable(IV)method[ 4 6 ],theCaponestimator[ 7 ],theamplitudeandphaseestimation(APES)algorithm[ 8 10 ]andmoreadvanceddata-adaptivetechniquessuchastheiterativeadaptiveapproach(IAA)[ 11 ].Weconcentrateourattentionontransmitwaveformdesigninthiswork.Particularlyweareinterestedinsynthesizingwaveformsthathavegoodcorrelationproperties(Chapters 2 3 and 4 ).Inradarrangecompression,lowauto-correlationsidelobesimprovethedetectionperformanceofweaktargets[ 12 13 ];incodedivisionmultipleaccess(CDMA)systems,lowauto-correlationsidelobesaredesiredforsynchronizationpurposesandlowcross-correlationsreduceinterferencesfromotherusers[ 14 15 ];andthesituationissimilarinmanyotheractivesensingapplicationssuchasultra-sonicimaging[ 16 ].Anemittedprobingwaveformwithlowauto-correlationsidelobesmaximizesthesignal-to-noiseratiowhencomplementedbyamatchedlteratthereceiversidewhilesignicantlyweakeningsignalsfromadjacentrangebins.InChapter 5 ,thecorrelationfunctionisextendedtothetwo-dimensionalambiguityfunction(AF)byincorporatingtheimpactofDopplerdelays.Inadditiontocorrelationproperties,goodspectrumcontainmentisdesiredfortransmittedsignals(Chapter 6 ).Inpractice,manyfrequencybandshavealreadybeenreservedforparticularusessuchasnavigationormilitarycommunications;ortherecouldexiststrongemitters(e.g.,electronicjamming)whoseoperatingfrequenciesshouldbeavoided.Thereforeitisdesiredthatthetransmitwaveformdeliveraslittleenergyaspossibleinthosefrequencybands. 13

PAGE 14

Chapter 7 discussestheproblemoftransmitbeampatternsynthesisinaMIMO(multi-inputmulti-output)system.Aclassicalphased-arraysteersanarrowbeamtowarddifferentanglesbyadjustingonlythewaveformphasesacrossantennaelements.InamodernMIMOsystem,however,waveformscanbechosenfreelyandthiswaveformdiversityallowsformoreexibilityinbeampatternsynthesis.Notation:Wewilluseboldfacelowercaseanduppercaseletterstodenotevectorsandmatrices,respectively.SeeTable 1-1 forothercommonnotationusedthroughoutthiswork. 1.1SignalModelLets(t)denotethetransmittedsignalwithtindicatingtime.Supposethats(t)consistsofNsymbolss(t)=NXn=1x(n)pn(t) (1)wherepn(t)istheshapingpulseandfx(n)gNn=1aretheNsymbols.Theshapingpulsepn(t)(withdurationtp)canbeanidealrectangularpulsepn(t)=1 p tprectt)]TJ /F5 11.955 Tf 11.95 0 Td[((n)]TJ /F5 11.955 Tf 11.95 0 Td[(1)tp tp;n=1;:::;N; (1)whererect(t)=8><>:1;0t1;0;elsewhere; (1)orotherpulses,suchastheraised-cosinepulse[ 2 ].Notethattheactualtransmittedwaveformiscomposedofthein-phaseandquadraturecomponentsofs(t)ej2fctwherefcisthecarrierfrequency.Itisassumedthatthesignaldemodulationhasalreadybeenperformedatthereceiversideandthusthecarriertermej2fctcanbesafelyignoredintheanalysis. 14

PAGE 15

Inpractice,hardwarecomponentssuchasanalog-to-digitalconvertersandpowerampliershaveamaximumsignalamplitudeclip.Inordertomaximizethetransmittedpowerthatisavailableinthesystem,itisdesirablethatthetransmitsequencesareunimodularorhavelowpeak-to-averagepowerratios(PAR).Inourdesign,weimposethefollowingunit-modulusconstraintwheneverfeasible:x(n)=ej(n);n=1;:::;N (1)wheref(n)garephases.Notethat( 1 )combinedwith( 1 )providesaphase-codedsignalrepresentation.Therearemanyothertypesofsignalsthatarewidelyusedorhavebeendiscussedintheliterature,whichincludethewell-knownchirpwaveform(Section 1.3 ),discretefrequency-codedwaveforms[ 17 18 ]andwaveformsconstructedfromaparticularsetoffunctionssuchastheprolatespheroidal[ 19 ]ortheHermitewavefunctions[ 20 ].Inthisworkwechosetofocusspecicallyonthephase-codedsignalmodel,whichservesasapracticalandeffectiveframeworkfordesigningwaveformswithvariousdesirableproperties.Thewaveforms(t)istransmittedinthedirectionofasceneofinterestandisreectedbyvarioustargetsatdifferentrangelocations.Thereectedsignals,whicharetime-shiftedandweightedversionsofs(t),arrivelinearlycombinedatthereceiverside:y(t)=Xkks(t)]TJ /F3 11.955 Tf 11.96 0 Td[(k)+e(t) (1)wherekistheround-triptimedelayforthekthtarget,kisthecoefcientrelatedtothetargetreectionsuchastheradarcrosssection(RCS)ande(t)isthenoise.Supposethatweaimtoestimatethecoefcientk0byapplyingthelterw(t)atthereceiver:^k0=Z1w(t)y(t)dt: (1) 15

PAGE 16

Morepreciselyspeaking,accordingtoaconventionalconvolutiondenition,( 1 )isthereceiveroutputattimeinstant0wheny(t)istheinputandw()]TJ /F3 11.955 Tf 9.3 0 Td[(t)isthelter.However,wesimplyrefertow(t)asthelterinthereceiverprocessingindicatedby( 1 ),withoutintroducinganyambiguityindiscussionsafterwards.Todetermineaproperw(t),wedecomposey(t)intothreeparts:y(t)=k0s(t)]TJ /F3 11.955 Tf 11.95 0 Td[(k0)| {z }signal+Xk6=k0ks(t)]TJ /F3 11.955 Tf 11.96 0 Td[(k)| {z }clutter+e(t)|{z}noise: (1)Ifthereisnoclutterande(t)iszero-meanwhitenoise,thenthematchedlterw(t)=s(t)]TJ /F3 11.955 Tf 11.96 0 Td[(k0)willgivethelargestsignal-to-noiseratio(SNR).Adirectproofgoesasfollows:SNR4=R1w(t)k0s(t)]TJ /F3 11.955 Tf 11.95 0 Td[(k0)dt2 ER1w(t)e(t)dt2=jk0j2R1w(t)s(t)]TJ /F3 11.955 Tf 11.96 0 Td[(k0)dt2 2eR1jw(t)j2dt (1)jk0j2R1js(t)]TJ /F3 11.955 Tf 11.95 0 Td[(k0)j2dt 2e=jk0j22s 2e (1)whereEdenotestheexpectation,and2eand2sarethenoisepowerandsignalpower,respectively.Notethat( 1 )isduetothewhitenoiseassumptionEfe(t1)e(t2)g=2et1)]TJ /F7 7.97 Tf 6.58 0 Td[(t2andthat( 1 )resultsfromtheCauchy-SchwartzinequalityRw(t)s(t)]TJ /F3 11.955 Tf 11.96 0 Td[(k0)dt2Rjw(t)j2dtRjs(t)]TJ /F3 11.955 Tf 12.08 0 Td[(k0)j2dt.ThemaximumvalueofSNRin( 1 )isachievedifandonlyifthelterw(t)isascaledversionofs(t)]TJ /F3 11.955 Tf 11.96 0 Td[(k0),whichconcludestheproof.Forthepurposeofnormalization,thematchedlterw(t)ischosentobes(t)]TJ /F3 11.955 Tf -424.47 -23.91 Td[(k0)=Rjs(t)j2dtandthecorrespondingestimateofk0in( 1 )isgivenby^k0=R1s(t)]TJ /F3 11.955 Tf 11.96 0 Td[(k0)y(t)dt R1js(t)j2dt: (1)Besidesboostingthesignalcomponentandsuppressingthenoise,thematchedltercanalsoeliminatethecluttercomponent(aseasilyseenfrom( 1 )and( 1 ))ifr()=Z1s(t)s(t)]TJ /F3 11.955 Tf 11.95 0 Td[()dt;<<1; (1) 16

PAGE 17

iszeroforall6=0.Ther()asdenedin( 1 )iscalledtheauto-correlationofs(t). 1.2DesignMetricsTheprevioussectionhasoutlinedthebenetofsmallauto-correlationsidelobesr()(for6=0).Formostpracticalcases,weonlyneedtofocusonthedelaythatisanintegermultipleofthesymbollengthtp.Oneofthereasonsisthatinmodernsystemsdigitallteringisusuallyperformedatthereceiverside,thatis,theintegralin( 1 )isimplementedasasummationofsampledsignals.Inaddition,iftherectangularshapingpulse(Eq.( 1 ))isused,thevaluesofr()canbeobtainedexactlybyalinearinterpolationoftwoneighboringauto-correlationsamples[e.g. 13 ,Chapter6]:r()=)]TJ /F3 11.955 Tf 11.96 0 Td[(t1 tpr(t2)+t2)]TJ /F3 11.955 Tf 11.95 0 Td[( tpr(t1); (1)wheret1= tptpandt2=t1+tp:Suchauto-correlationsatintegermultipledelaysfktpgN)]TJ /F6 7.97 Tf 6.59 0 Td[(1k=)]TJ /F7 7.97 Tf 6.59 0 Td[(N+1canbecalculatedfork0asr(ktp)=Z1s(t)s(t)]TJ /F3 11.955 Tf 11.96 0 Td[(ktp)dt=ZNtpktpNXn=k+1x(n)pn(t)x(n)]TJ /F3 11.955 Tf 11.96 0 Td[(k)pn(t)dt=NXn=k+1x(n)x(n)]TJ /F3 11.955 Tf 11.96 0 Td[(k)(N)]TJ /F3 11.955 Tf 11.96 0 Td[(k)Ztp0jpn(t)j2dt=(N)]TJ /F3 11.955 Tf 11.95 0 Td[(k)NXn=k+1x(n)x(n)]TJ /F3 11.955 Tf 11.96 0 Td[(k): (1)Correlationsatnegativedelayscanbeobtainedfromr(ktp)=r()]TJ /F3 11.955 Tf 9.3 0 Td[(ktp).Whenshapingpulsesotherthantherectangularoneareused,itcanstillbeexpectedthatr()iswellcontrolledaslongasr(k)ismadesufcientlysmall.Itfollowsfromtheabovediscussionsthatthecorrelationsofinterestaregivenbyr(k)=NXn=k+1x(n)x(n)]TJ /F3 11.955 Tf 11.96 0 Td[(k)=r()]TJ /F3 11.955 Tf 9.3 0 Td[(k);k=0;:::;N)]TJ /F5 11.955 Tf 11.96 0 Td[(1: (1) 17

PAGE 18

Theabovefr(k)giscalledtheauto-correlationofthediscretesequencefx(n)g.Notethatthenotationrisslightlyabusedin( 1 )and( 1 )todenotebothcontinuous-timeanddiscrete-timeauto-correlations,yetdistinctioncanbemadeeasilybyexaminingthetwodifferenttimevariables.Forfr(k)gN)]TJ /F6 7.97 Tf 6.59 0 Td[(1k=)]TJ /F7 7.97 Tf 6.58 0 Td[(N+1denedabove,r(0)iscalledthein-phasecorrelation,whichisalwaysequaltothesignalenergy.Allotherauto-correlations,i.e.,fr(k);k=)]TJ /F3 11.955 Tf 9.3 0 Td[(N+1;:::;)]TJ /F5 11.955 Tf 9.3 0 Td[(1;1;:::;N)]TJ /F5 11.955 Tf 12.06 0 Td[(1g,arecollectivelycalledtheauto-correlationsidelobes.OneofthemaininterestsinChapters 2 and 3 istodesignphase-codedsequencesfx(n)gwhoseauto-correlationsidelobesareaslowaspossible.Chapter 5 discussestheambiguityfunctionsynthesis,whichcanbeconsideredasatwo-dimensionalextensiontothecorrelationdesign.Morepreciselyspeaking,thefr(k)gdenedin( 1 )istheaperiodicauto-correlation.Theperiodicauto-correlationofthesequencefx(n)gisdenedas~r(k)=NXn=1x(n)x((n)]TJ /F3 11.955 Tf 11.96 0 Td[(k)modN)=~r()]TJ /F3 11.955 Tf 9.3 0 Td[(k) (1)=~r(N)]TJ /F3 11.955 Tf 11.95 0 Td[(k)k=0;:::;N)]TJ /F5 11.955 Tf 11.95 0 Td[(1;wheremodisthemodulooperatorpmodN=8><>:p)-222(bp NcN;p6=0orN;N;p=0orN: (1)Therelationshipbetweentheaperiodiccorrelation( 1 )andtheperiodiccorrelation( 1 )canbeeasilyobtainedasfollows:~r(k)=KXn=1x(n)x(n)]TJ /F3 11.955 Tf 11.96 0 Td[(k+N)+NXn=k+1x(n)x(n)]TJ /F3 11.955 Tf 11.96 0 Td[(k) (1)=NXm=(N)]TJ /F7 7.97 Tf 6.58 0 Td[(k)+1x(m)]TJ /F5 11.955 Tf 11.96 0 Td[((N)]TJ /F3 11.955 Tf 11.95 0 Td[(k))x(m)+NXn=k+1x(n)x(n)]TJ /F3 11.955 Tf 11.95 0 Td[(k)=r(N)]TJ /F3 11.955 Tf 11.95 0 Td[(k)+r(k): 18

PAGE 19

AswillbeshowninChapter 7 ,waveformcorrelationscanserveasabridgeconnectingtheunderlyingwaveformtothedesiredbeampatternof,e.g.,anantennaarray.Particularly,thewaveformdiversityinaMIMOsystemleadstoaexiblecontrolofwaveformcorrelationswhichfurtherleadtoanagiletransmitbeampatternsynthesis.Inthenextsection,wewillreviewseveralwell-knownwaveformsthathavegoodauto-correlationproperties,especiallythosethatarephase-coded( 1 ). 1.3ReviewofExistingWaveformsWestartwiththewell-knownchirpwaveform.Achirpwaveformisalinearfrequency-modulated(LFM)pulse,whosefrequencyissweptlinearlyoverabandwidthBinatimedurationT.ChirpsignalshavebeenwidelyusedinradarapplicationssinceWorldWarII,astheypossessrelativelylowcorrelationsidelobesandaremostlytoleranttoDopplerfrequencyshifts[ 13 ].Inaddition,thepowerofachirpsignalisdispersedevenlythroughoutthefrequencyspectrum,whichallowsforhighspectralefciency.Achirpsignals(t)canbewrittenass(t)=1 p TejB Tt2;0tT (1)whereB Tisthechirprate.Figure 1-1A showstherealpartofs(t)withparametersT=100secandB=1Hz.Figure 1-1B showsitsauto-correlationfunctionr()(normalizedbyr(0)andusinga20lgscale),wherethepeaksidelobeis)]TJ /F5 11.955 Tf 9.3 0 Td[(13:4dB.Manyphasecodescanbederivedfromthechirpsignal.Wesamples(t)attimeintervalsts=1 B(N=BT)andobtainthefollowingsequence:x(n)4=s(nts)=ejB T(n B)2=ejn2 BT=ejn2 N;n=1;:::;N: (1)Thesequencefx(n)gshownabovehasperfectperiodicauto-correlationsifNiseven,meaningthatallperiodicauto-correlationsidelobesarezero:~r(k)=0fork6=0. 19

PAGE 20

AsequencewithperfectperiodiccorrelationsforanyoddNcanbeconstructedbychangingthesequencephasesin( 1 )tox(n)=ejn(n)]TJ /F14 5.978 Tf 5.75 0 Td[(1) N;n=1;:::;N (1)whichistheGolombsequence[ 21 ].TheChusequence[ 22 ],interestingly,isacombinationoftheabovetwosequences:x(n)=8><>:ejQn2 N;NisevenejQn(n)]TJ /F14 5.978 Tf 5.76 0 Td[(1) N;Nisodd (1)whereQisanyintegerthatisprimetoN.Asexpected,theChusequencehasperfectperiodiccorrelationsforany(positive)integerN.BesidestheGolombandtheChusequences,therearemanyotherphase-codedsequenceswhosephasesarequadraticfunctionsofn,suchasthewell-knownFranksequenceandtheP4sequence.TheFranksequenceisdenedforN=L2as:x((m)]TJ /F5 11.955 Tf 11.95 0 Td[(1)L+p)=ej2(m)]TJ /F14 5.978 Tf 5.75 0 Td[(1)(p)]TJ /F14 5.978 Tf 5.76 0 Td[(1) L;m;p=1;:::;L: (1)TheP4sequenceisdenedforanylengthNasx(n)=ej2 N(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)(n)]TJ /F14 5.978 Tf 5.76 0 Td[(1)]TJ /F12 5.978 Tf 5.76 0 Td[(N 2);n=1;:::;N: (1)BothFrankandP4sequenceshaveperfectperiodiccorrelations.Figure 1-2 showstheauto-correlationr(k)oftheP4sequenceoflengthN=100.Notethatfromthesequencefx(n)g,wecanconstructacontinuous-timewaveforms(t)using( 1 ).Wechooserectangularpulseshapingandtp=1secsothatthesignaldurationis100sandthatthesignalbandwidthisroughly1Hz(i.e.,1=tp),whicharethesameparametersasusedinFigure 1-1 .Therealandimaginarypartsoftheso-constructeds(t)areshownseparatelyinFig. 1-3A .Theauto-correlationr()ofthis 20

PAGE 21

s(t)isshowninFigure 1-3B .Thepeaksidelobeis)]TJ /F5 11.955 Tf 9.29 0 Td[(26:3dB,whichismuchlowerthanthatofthechirpwaveforminFigure 1-1B .Theauto-correlationpropertiesoftheGolomb,ChuorFranksequencesaresimilartothoseoftheP4sequenceandareomittedforbrevity.Anotherwidelyusedsequenceisthemaximumlengthsequence(m-sequence)[e.g. 2 ].Anm-sequenceisatypeofpseudorandombinarysequencethatisgeneratedbyamaximallinearfeedbackshiftregister(LFSR).Fig. 1-4 showsalength-3LFSRwheretheplusoperatorindicates`exclusiveor'.Eachregisterblockcanstore0or1sothreeblocksamounttoeightdifferentstates.Whenfedwithanyinitialbinarysequence(notallzeros),suchashiftregisterwillcyclethroughalleightstatesexceptfortheall-zerostate.Forinstance,startingfrom`001',theregisterinFig. 1-4 willpassrepeatedlythroughthefollowingsevenstates`001',`100',`010',`101',`110',`111',`011'.Bytakingonlytheoutputfromthethirdblockandreplacing`0'with`-1',weobtainalength-7m-sequencef1;)]TJ /F5 11.955 Tf 9.3 0 Td[(1;)]TJ /F5 11.955 Tf 9.3 0 Td[(1;1;)]TJ /F5 11.955 Tf 9.3 0 Td[(1;1;1g.ItsaperiodicaswellasperiodiccorrelationsareshowninFig. 1-5 .Oneoftheprominentfeaturesofanm-sequenceisthatitsperiodiccorrelationsidelobesarealwaysequalto)]TJ /F5 11.955 Tf 9.3 0 Td[(1,ascanbeobservedfromFig. 1-5B .Itsaperiodiccorrelationsidelobes,though,donothavearegularpatternandcanberelativelyhigh.AlsonotethattheLFSRcanbeefcientlyimplementedinhardware,whichlargelyfacilitatestheuseofm-sequencesinpractice.Theaforementionedsequences/waveformsallhaveclosed-formconstructionmethods.Researchershavealsousedgradientdescentorstochasticoptimizationtechniquestondsequenceswithlowerauto-correlations[ 23 25 ].ThesealgorithmsareusuallycomputationallyexpensiveandworkwellonlyforsmallvaluesofN,suchasN102.AnexampleisthesearchforpolyphaseBarkersequenceswhoseauto-correlationsidelobesarealllessthanorequalto1.Alength-45polyphaseBarker 21

PAGE 22

sequenceisgivenby[ 24 ]x(n)=expfj(n)g;n=1;:::;45 (1)f(n)g=2 90f00717671766356738799142553625323585694076572698356572155289481168266263773195812ganditsauto-correlationisshowninFigure 1-6 .Itspeaksidelobeis)]TJ /F5 11.955 Tf 9.29 0 Td[(33:1dB.Notethatforaunit-modulussequence,thelowestpossiblepeaksidelobeis1,sincejr(N)]TJ /F5 11.955 Tf 12.2 0 Td[(1)j=jx(N)x(1)j=1.InthecaseofN=45,1correspondsto20lg(1=N)=)]TJ /F5 11.955 Tf 9.3 0 Td[(33:1dB.ThusapolyphaseBarkercodehasthebestauto-correlationpropertiesintermsofthelowestpeaksidelobe.WementioninpassingonthefactthattheBarkercodewasoriginallydenedasabinarysequencewithcorrelationsidelobesnotlargerthan1[ 26 ].ThelongestknownBarkercodehaslength13andisshownbelowfx(n)g=11111)]TJ /F5 11.955 Tf 9.3 0 Td[(1)]TJ /F5 11.955 Tf 9.29 0 Td[(111)]TJ /F5 11.955 Tf 9.3 0 Td[(11)]TJ /F5 11.955 Tf 9.3 0 Td[(11; (1)whoseauto-correlationisplottedinFigure 1-7 .Wenallypointoutthatthedesignofunit-modulussequenceswithlowaperiodiccorrelationsismuchmoredifcultthanthedesignthatisconcernedwithlowperiodiccorrelations.TheaforementionedsequencessuchasChu,P4andm-sequenceshaveloworzeroperiodiccorrelationsidelobes;andtherearemanyotherswiththesamepropertysuchasGoldsequences[ 27 ]andKasamisequences[ 28 ].Notethesymmetricpropertyofperiodiccorrelationsin( 1 ):~r(k)=~r(N)]TJ /F3 11.955 Tf 12.35 0 Td[(k).Itleadstothefactthatinordertominimizeallperiodiccorrelationsidelobes,weneedtoconsideronly~r(1);:::;~r(N=2)foranevenNandonly~r(1);:::;~r(N)]TJ /F6 7.97 Tf 6.58 0 Td[(1 2)foranoddN.Theaperiodiccorrelations,ontheotherhand,donothavesuchsymmetricproperty.Inaddition,theabsolutevalueofthemaximum-lagaperiodiccorrelationr(N)]TJ /F5 11.955 Tf 12.4 0 Td[(1)isalwaysequalto1(thuscannotbeminimized):jr(N)]TJ /F5 11.955 Tf 12.61 0 Td[(1)j=jx(N)x(1)j=1becauseeachelementof 22

PAGE 23

thesequencehasunit-modulus.Wecanalsoobservefrom( 1 )thatall-zeroperiodiccorrelationsidelobesimplyr(k)=)]TJ /F3 11.955 Tf 9.3 0 Td[(r(N)]TJ /F3 11.955 Tf 12.56 0 Td[(k)(fork=1;:::;N)]TJ /F5 11.955 Tf 12.55 0 Td[(1)andviceversa.Inthefollowingchapterswewillfocusonsequencedesignfortheaperiodiccorrelationcasethatisrelativelylesstouchedintheliterature.Forthesakeofbrevity,thewordcorrelationindicatestheaperiodiccorrelationunlessperiodiccorrelationisexplicitlypointedout. Table1-1.Notation a:complexconjugateofascalaraRefag:realpartofascalaraImfag:imaginarypartofascalarakak:EuclideannormofavectoraA:complexconjugateofamatrixAAT:transposeofamatrixAAH:conjugatetransposeofamatrixAtr(A):traceofamatrixAkAk:FrobeniusnormofamatrixAA0:matrixAispositivesemi-deniteAB:matrixB)]TJ /F15 11.955 Tf 11.96 0 Td[(Aispositivesemi-deniten:Kroneckerdelta:n=1ifn=0andn=0otherwisemn:anextensionofn:mn=1ifm=nandmn=0otherwiseIM:identitymatrixofdimensionMMf(x)g(x):convolutionoftwofunctionsf(x)andg(x)bxc:biggestintegerlessthanorequaltox(real-valued)arg(x):phaseangle(inradians)ofx 23

PAGE 24

A B Figure1-1.Achirpsignals(t)anditsauto-correlations.A)Therealpartofs(t)in( 1 )withT=100sandB=1Hz.B)Theauto-correlationfunctionofs(t). 24

PAGE 25

Figure1-2.Theauto-correlationfunctionr(k)oftheP4sequence,asdenedin( 1 ),oflengthN=100. 25

PAGE 26

A B Figure1-3.AP4waveformanditsauto-correlations.A)Therealandimaginarypartsofs(t)in( 1 )whentheP4sequenceoflength100andrectangularshapingpulsesareused.B)Theauto-correlationfunctionr()ofthiss(t). 26

PAGE 27

Figure1-4.Alinearfeedbackshiftregisteroflength3. A B Figure1-5.Auto-correlationofanm-sequence.A)Theaperiodicauto-correlationandB)theperiodiccorrelationofalength-7m-sequencef1;)]TJ /F5 11.955 Tf 9.3 0 Td[(1;)]TJ /F5 11.955 Tf 9.3 0 Td[(1;1;)]TJ /F5 11.955 Tf 9.3 0 Td[(1;1;1g. 27

PAGE 28

Figure1-6.Theauto-correlationfunctionr(k)ofapolyphaseBarkersequenceoflengthN=45,asdenedin( 1 ). Figure1-7.Theauto-correlationfunctionr(k)ofaBarkersequenceoflengthN=13,asdenedin( 1 ). 28

PAGE 29

CHAPTER2SINGLESEQUENCEDESIGNFromthediscussionsgiveninChapter 1 ,wehavemadeitclearthatthegoalofaperiodicsequencedesignistomakefr(k)gk6=0assmallaspossible.Inthischapter,wefocusontheintegratedsidelobelevel(ISL)metricwhichisdenedas:ISL=N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=)]TJ /F6 7.97 Tf 6.59 0 Td[((N)]TJ /F6 7.97 Tf 6.59 0 Td[(1)k6=0jr(k)j2=2N)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xk=1jr(k)j2: (2)OurgoalistopresentefcientcomputationalgorithmstominimizetheISLmetricorISL-relatedmetricsundertheconstraintofsynthesizingunit-modulussequences.NotethattheminimizationoftheISLmetricisequivalenttothemaximizationofthemeritfactor(MF)denedasfollows:MF=jr(0)j2 N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=)]TJ /F14 5.978 Tf 5.76 0 Td[((N)]TJ /F14 5.978 Tf 5.76 0 Td[(1)k6=0jr(k)j2=N2 ISL: (2)Owingtothesignicantpracticalinterestinthedesignofunit-modulussequenceswithlowISL(orequivalentlylargeMF)values,aspointedoutinChapter 1 ,itcomesasnosurprisethattheliteratureonthistopicisextensive[ 25 29 32 ].AnextensiontotheISLandMFmetricsaretheweightedISL(WISL)andmodiedMF(MMF)metrics:WISL=2N)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xk=1wkjr(k)j2;wk0; (2)MMF=jr(0)j2 WISL (2)wherefwkgN)]TJ /F6 7.97 Tf 6.59 0 Td[(1k=1isanarbitrarysetofweights.SuchweightedISLmetricsareimportantinapplicationswherewewanttoreduce,asmuchaspossible,theinterferenceduetoaknownmultipathoraknownclutterdiscrete.Forexample,therearecasesinwhichthemaximumdifferencebetweenthearrivaltimesofthesequenceofinterestandofthe 29

PAGE 30

interferenceis(much)smallerthanthedurationoftheemittedsequence[ 30 33 35 ].Consequently,insuchcasestheinterestliesinmakingfjr(k)jgP)]TJ /F6 7.97 Tf 6.58 0 Td[(1k=1smallforsomeP
PAGE 31

2.1CyclicAlgorithm-New(CAN)ThederivationofCANinvolvesseveralsteps,therstofwhichconsistsofexpressingtheISLmetricinthefrequencydomain.Itiswell-knownthat,forany!2[0;2],NXn=1x(n)e)]TJ /F7 7.97 Tf 6.58 0 Td[(j!n2=N)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xk=)]TJ /F6 7.97 Tf 6.58 0 Td[((N)]TJ /F6 7.97 Tf 6.58 0 Td[(1)r(k)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j!k,(!) (2)[ 38 ].Furthermore,itcanbeshownthattheISLmetricin( 2 )canbeequivalentlywrittenas:ISL=1 2N2NXp=1[(!p))]TJ /F3 11.955 Tf 11.96 0 Td[(N]2; (2)wheref!pgarethefollowingFourierfrequencies:!p=2 2Np;p=1;:::;2N: (2)(Notethat( 2 )isaParseval-typeequality.)Toprove( 2 ),letkdenotetheKroneckerdeltaandusethecorrelogram-basedexpressionfor(!)in( 2 )toverifythat:2NXp=1[(!p))]TJ /F3 11.955 Tf 11.96 0 Td[(N]2=2NXp=124N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=)]TJ /F6 7.97 Tf 6.59 0 Td[((N)]TJ /F6 7.97 Tf 6.59 0 Td[(1)(r(k))]TJ /F3 11.955 Tf 11.95 0 Td[(Nk)e)]TJ /F7 7.97 Tf 6.58 0 Td[(j!pk352=N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=)]TJ /F6 7.97 Tf 6.58 0 Td[((N)]TJ /F6 7.97 Tf 6.58 0 Td[(1)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1X~k=)]TJ /F6 7.97 Tf 6.59 0 Td[((N)]TJ /F6 7.97 Tf 6.59 0 Td[(1)(r(k))]TJ /F3 11.955 Tf 11.95 0 Td[(Nk)(r(~k))]TJ /F3 11.955 Tf 11.95 0 Td[(N~k)"2NXp=1e)]TJ /F7 7.97 Tf 6.58 0 Td[(j!p(k)]TJ /F6 7.97 Tf 6.78 2.1 Td[(~k)#: (2)Because,forjk)]TJ /F5 11.955 Tf 12.27 3.15 Td[(~kj2N)]TJ /F5 11.955 Tf 11.95 0 Td[(2,2NXp=1e)]TJ /F7 7.97 Tf 6.59 0 Td[(j!p(k)]TJ /F6 7.97 Tf 6.78 2.1 Td[(~k)=e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2 2N(k)]TJ /F6 7.97 Tf 6.78 2.1 Td[(~k)e)]TJ /F7 7.97 Tf 6.58 0 Td[(j2(k)]TJ /F6 7.97 Tf 6.78 2.1 Td[(~k))]TJ /F5 11.955 Tf 11.96 0 Td[(1 e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2 2N(k)]TJ /F6 7.97 Tf 6.78 2.1 Td[(~k))]TJ /F5 11.955 Tf 11.95 0 Td[(1=2N(k)]TJ /F6 7.97 Tf 6.78 2.11 Td[(~k); (2)weobtainfrom( 2 )thefollowingequation:1 2N2NXp=1[(!p))]TJ /F3 11.955 Tf 11.96 0 Td[(N]2=N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=)]TJ /F6 7.97 Tf 6.58 0 Td[((N)]TJ /F6 7.97 Tf 6.58 0 Td[(1)jr(k))]TJ /F3 11.955 Tf 11.95 0 Td[(Nkj2=2N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=1jr(k)j2=ISL; (2) 31

PAGE 32

whichis( 2 ).Usingtheperiodogram-basedexpressionfor(!)(Eq.( 2 ))in( 2 )showsthattheproblemofminimizingtheISLisequivalenttotheminimizationofthefollowingfrequency-domainmetric:2NXp=124NXn=1x(n)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j!pn2)]TJ /F3 11.955 Tf 11.96 0 Td[(N352: (2)Thisequivalenceresulthasanobviousintuitiveinterpretation:minimizingtheISLmakesthesequencebehavelikewhitenoise,andconsequentlyitsperiodogramshouldbenearlyconstantinfrequency.Thenextpointtonoteisthatthecriterionin( 2 )isaquarticfunctionoffx(n)g.However,itcanbeveriedthattheminimizationof( 2 )withrespecttofx(n)gisalmostequivalenttothefollowingsimplerproblem(whosecriterionisaquadraticfunctionoffx(n)g):minfx(n)gNn=1;f pg2Np=12NXp=1NXn=1x(n)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j!pn)]TJ 11.96 10.71 Td[(p Nej p2: (2)Brieyspeaking,ifthecriterionin( 2 )takesonasmallvalue,thensodoes( 2 ),andviceversa.Morespecically,( 2 )isequaltozeroifandonlyifthecriterionin( 2 )isequaltozero.Consequently,bycontinuityarguments,iftheglobalminimumvalueof( 2 )issufcientlysmall,thenthesequencesminimizing( 2 )and,respectively,thecriterionin( 2 )canbeexpectedtobeclosetooneanother.SeeAppendix A foradetaileddiscussiononthisalmostequivalence.LetaHp=e)]TJ /F7 7.97 Tf 6.58 0 Td[(j!pe)]TJ /F7 7.97 Tf 6.59 0 Td[(j2N!p; (2) 32

PAGE 33

letAHbethefollowingunitary2N2NFFTmatrix:AH=1 p 2N266664aH1...aH2N377775; (2)andletzbethesequencefx(n)gNn=1paddedwithNzeros:z=x(1)x(N)00T2N1: (2)Thenthecriterionin( 2 )canberewritteninthefollowingmorecompactform(towithinamultiplicativeconstant):AHz)]TJ /F15 11.955 Tf 11.96 0 Td[(v2; (2)wherev=1 p 2ej 1ej 2NT: (2)Forgivenfx(n)g,theminimizationof( 2 )withrespecttof pgisimmediate:letf=AHz (2)denotetheFFTofz;then p=arg(fp);p=1;:::;2N: (2)Similarly,forgivenv,letg=Av (2)denotetheInverse-FFT(IFFT)ofv.BecausekAHz)]TJ /F15 11.955 Tf 11.33 0 Td[(vk2=kz)]TJ /F15 11.955 Tf 11.34 0 Td[(Avk2,itfollowsthattheminimizingsequencefx(n)gisgivenby:x(n)=ejarg(gn);n=1;:::;N: (2) 33

PAGE 34

TheCANforthecycliclocalminimizationoftheISL-relatedmetricin( 2 )issummarizedbelow.Owingtoitssimple(I)FFToperations,CANcanbeusedforverylargevaluesofN,suchasN106.TheCANalgorithm: Step0:Setfx(n)gNn=1torandominitialvalues(e.g.,fx(n)gcanbesettofej2(n)gwheref(n)gareindependentrandomvariablesuniformlydistributedin[0;2]),orfx(n)gNn=1canbeinitializedbyagoodexistingsequencesuchasaGolombsequence. Step1:Computethef pg2Np=1thatminimizethemetricforfx(n)gNn=1xedattheirmostrecentvalues(Eq.( 2 )). Step2:Computethesequencefx(n)gNn=1thatminimizesthemetric,undertheconstraintjx(n)j=1,forf pg2Np=1xedattheirmostrecentvalues(Eq.( 2 )). Iteration:RepeatSteps1and2untilapre-speciedstopcriterionissatisede.g.kx(i))]TJ /F15 11.955 Tf 11.77 0 Td[(x(i+1)k<,wherex(i)isthesequenceobtainedattheithiteration,andisapredenedthreshold,suchas10)]TJ /F6 7.97 Tf 6.59 0 Td[(3.TheCANalgorithmdescribedaboveconsiderstheconstraintthatthesequencefx(n)gisunit-modulus,i.e.,itspeak-to-averagepowerratio(PAR)isequalto1.AllowingthePARvaluetobelargerthan1leadstobettersuppressionofcorrelationsidelobes.SeeChapter 4 foranextendedCANalgorithmthatdealswitharelaxedPARconstraint.Inthenextsection,wepresentanextendedversionofCANwhichcandealwiththeWISLmetric(witharbitrarilychosenweights)asdenedin( 2 ).TheextendedalgorithmiscalledWeCAN(weightedCAN).ThepricepaidforWeCAN'sabilitytodealwithageneralWISLmetricisanincreasedcomputationalburdencomparedtoCAN.Specically,aswillbeshowninthenextsection,eachiterationofWeCANrequiresNcomputationsof2N-point(I)FFT's;thusthenumberofopsrequiredbyWeCANisroughlyNtimeslargerthanthatofCAN.Nonetheless,WeCANcanstillbeusedforrelativelylargevaluesofN,suchasN104. 34

PAGE 35

2.2WeightedCyclicAlgorithm-New(WeCAN)Similarlytotheproofof( 2 )inSection 2.1 ,wecanderivethefollowingexpressionfortheWISLmetric(kbelowisrelatedtotheweightwkin( 2 )aswk=2k):WISL=2N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=12kjr(k)j2 (2)=1 2N2NXp=1[~(!p))]TJ /F3 11.955 Tf 11.96 0 Td[(0N]2; (2)where~(!p)4=N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=)]TJ /F6 7.97 Tf 6.58 0 Td[((N)]TJ /F6 7.97 Tf 6.58 0 Td[(1)kr(k)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j!pk;!p=2 2Np;p=1;:::;2N; (2)andwherefkgN)]TJ /F6 7.97 Tf 6.59 0 Td[(1k=1arereal-valued(withk=)]TJ /F7 7.97 Tf 6.59 0 Td[(k).NotethatbychoosingfkgN)]TJ /F6 7.97 Tf 6.59 0 Td[(1k=1appropriately,wecanweighthecorrelationlagsin( 2 )inanydesiredway.Regarding0,whichdoesnotenterinto( 2 ),itwillbechosentoensurethatthematrix)]TJ /F5 11.955 Tf 11.4 0 Td[(=1 026666666401N)]TJ /F6 7.97 Tf 6.59 0 Td[(110...............1N)]TJ /F6 7.97 Tf 6.59 0 Td[(110377777775 (2)ispositivesemi-denite,whichwedenoteby)]TJ /F2 11.955 Tf 12.61 0 Td[(0.Thiscanbedoneinthefollowingsimpleway:let~)]TJ /F1 11.955 Tf 11.41 0 Td[(bethematrix0)]TJ /F1 11.955 Tf 11.41 0 Td[(withalldiagonalelementssetto0,andletmindenotetheminimumeigenvalueof~)]TJ /F1 11.955 Tf 8.08 0 Td[(;then)]TJ /F2 11.955 Tf 11.83 0 Td[(0ifandonlyif0+min0,aconditionthatcanalwaysbesatisedbyselecting0.Nextwewillderiveacriterionthatisalmostequivalentto( 2 )andwhichdependsquadraticallyontheunknownsfx(n)gNn=1,similarlytowhatwehavedoneintheprevioussection.Todoso,wemustapparentlyobtainasquarerootof~(!p)in( 2 )thatislinearinfx(n)gNn=1.NotethefollowingDFTpairs:fr(k)g !(!)=jX(!)j2 35

PAGE 36

fkr(k)g !~(!)=\(!)jX(!)j2; (2)whereX(!)=NXn=1x(n)e)]TJ /F7 7.97 Tf 6.59 0 Td[(jn!;\(!)=N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=)]TJ /F6 7.97 Tf 6.59 0 Td[((N)]TJ /F6 7.97 Tf 6.59 0 Td[(1)ke)]TJ /F7 7.97 Tf 6.59 0 Td[(j!k; (2)andwhereistheconvolutionoperator.Thus~(!p)canbeexpressedas~(!p)=1 2Z)]TJ /F7 7.97 Tf 6.59 0 Td[(\(!p)]TJ /F3 11.955 Tf 11.96 0 Td[( )jX( )j2d =1 2Z)]TJ /F7 7.97 Tf 6.59 0 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=)]TJ /F6 7.97 Tf 6.58 0 Td[((N)]TJ /F6 7.97 Tf 6.58 0 Td[(1)ke)]TJ /F7 7.97 Tf 6.59 0 Td[(jk(!p)]TJ /F7 7.97 Tf 6.59 0 Td[( )NXn=1x(n)e)]TJ /F7 7.97 Tf 6.59 0 Td[(jn NX~n=1x(~n)ej~n d =N)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xk=)]TJ /F6 7.97 Tf 6.59 0 Td[((N)]TJ /F6 7.97 Tf 6.59 0 Td[(1)NXn=1NX~n=1kx(n)x(~n)1 2Z)]TJ /F7 7.97 Tf 6.59 0 Td[(ej (k)]TJ /F7 7.97 Tf 6.59 0 Td[(n+~n)d e)]TJ /F7 7.97 Tf 6.59 0 Td[(j!pk: (2)Itiseasytoverifythat1 2Z)]TJ /F7 7.97 Tf 6.58 0 Td[(ej (k)]TJ /F7 7.97 Tf 6.58 0 Td[(n+~n)d =k)]TJ /F6 7.97 Tf 6.58 0 Td[((n)]TJ /F6 7.97 Tf 7.04 0 Td[(~n): (2)Thus~(!p)=NXn=1NX~n=1n)]TJ /F6 7.97 Tf 7.04 0 Td[(~nx(n)x(~n)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j!p(n)]TJ /F6 7.97 Tf 7.04 0 Td[(~n)=~xHp(0)]TJ /F5 11.955 Tf 8.08 0 Td[()~xp; (2)where~xp=x(1)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j!px(2)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2!px(N)e)]TJ /F7 7.97 Tf 6.59 0 Td[(jN!pT (2)and)]TJ /F1 11.955 Tf 11.41 0 Td[(isdenedin( 2 ).ThereforetheWISLmetricin( 2 )canbewrittenasWISL=20 2N2NXp=1~xHp)]TJ /F5 11.955 Tf 8.7 .17 Td[(~xp)]TJ /F3 11.955 Tf 11.96 0 Td[(N2: (2)ThisexpressionsuggeststhatthefollowingproblemcanbeexpectedtobealmostequivalenttotheminimizationoftheWISLmetric:minfx(n)gNn=1;fpg2Np=12NXp=1kC~xp)]TJ /F18 11.955 Tf 11.95 0 Td[(pk2 (2) 36

PAGE 37

s.t.kpk2=N;p=1;:::;2N;jx(n)j=1;n=1;:::;N;wheres.t.standsforsubjecttoandtheNNmatrixCisasquarerootof)]TJ /F1 11.955 Tf 8.08 -.01 Td[(,i.e.,)]TJ /F5 11.955 Tf 11.4 0 Td[(=CTC.Acyclicalgorithmfor( 2 ),whichwewillcallWeCAN,canbederivedasfollows.Forgivenfx(n)gNn=1,( 2 )decouplesinto2Nindependentproblemseachofwhichhasthefollowingform:minpkfp)]TJ /F18 11.955 Tf 11.96 0 Td[(pk2 (2)s.t.kpk2=NwheretheN1vectorfp=C~xpisgiven.Notethatundertheconstraintkpk2=Nwehavekfp)]TJ /F18 11.955 Tf 11.95 0 Td[(pk2=const)]TJ /F5 11.955 Tf 11.95 0 Td[(2ReffHppgconst)]TJ /F5 11.955 Tf 11.96 0 Td[(2kfpkkpk=const)]TJ /F5 11.955 Tf 11.95 0 Td[(2Nkfpk; (2)wheretheequalityisachievedifandonlyifp=p Nfp kfpk: (2)Thisisthereforethesolutiontotheminimizationproblemin( 2 )forgivenfx(n)gNn=1.Notethatthecomputationofffpg2Np=1canbedonebymeansoftheFFT.Indeed,letckndenotethe(k;n)thelementofCanddenezk=ck1x(1)ck2x(2)ckNx(N)00T(2N1) (2)andF=p 2NAHz1z2zN2NN (2) 37

PAGE 38

wheretheunitary2N2NFFTmatrixAHhasbeendenedin( 2 ).ThenitisnotdifculttoseethatthetransposeofthevectorfpisgivenbythepthrowofF.Nextweshowthat,forgivenfpg2Np=1,theminimizationproblemin( 2 )withrespecttofx(n)gNn=1alsohasaclosed-formsolution.LetpkdenotethekthelementofpandletaHpbegivenby( 2 ).Usingthisnotation,thecriterionin( 2 )canbewrittenas2NXp=1kC~xp)]TJ /F18 11.955 Tf 11.96 0 Td[(pk2=NXk=12NXp=1aHpzk)]TJ /F3 11.955 Tf 11.95 0 Td[(pk2=NXk=1AHzk)]TJ /F18 11.955 Tf 11.96 0 Td[(k2=NXk=1kzk)]TJ /F15 11.955 Tf 11.95 0 Td[(Akk2; (2)wherek=1 p 2N1k2k2N;kT;k=1;:::;N: (2)Foragenericelementoffx(n)gNn=1,denotedasx,( 2 )becomesNXk=1jkx)]TJ /F3 11.955 Tf 11.95 0 Td[(kj2=const)]TJ /F5 11.955 Tf 11.96 0 Td[(2Re" NXk=1kk!x#; (2)wherekandkaregivenbythecorrespondingelementsinzkandAk,respectively.Undertheunimodularconstraint,theminimizerxofthecriterionin( 2 )isgivenbyx=ej;=arg NXk=1kk!: (2)ThisobservationconcludesthederivationofthemainstepsoftheWeCANalgorithm,whosesummaryisasfollows.TheWeCANalgorithm: Step0:Setthefx(n)gNn=1tosomeinitialvaluesandselectthedesiredweightsfkgN)]TJ /F6 7.97 Tf 6.59 0 Td[(1k=1;alsochoose0suchthatthematrix)]TJ /F1 11.955 Tf 11.41 0 Td[(in( 2 )ispositivesemidenite. Step1:Computethefpg2Np=1thatminimizethecriterionin( 2 )forfx(n)gNn=1xedattheirmostrecentvalues(Eq.( 2 )). 38

PAGE 39

Step2:Computethesequencefx(n)gNn=1thatminimizesthecriterionin( 2 )forfpg2Np=1xedattheirmostrecentvalues(Eq.( 2 )). Iteration:RepeatSteps1and2untilapre-speciedstopcriterionissatised. 2.3NumericalExamples 2.3.1IntegratedSidelobeLevel(ISL)DesignWecomparethemeritfactorsoftheGolombsequence[ 39 ],oftheFranksequence[ 40 ],andoftheCANsequenceinitializedbyoneofthesetwotypesofsequences(denotedasCAN(G)andCAN(F),respectively).ThedenitionsoftheGolombandFranksequencesaregivenin( 1 )and( 1 ),respectively.(NotethattheGolomborFranksequencescanbeeasilycomputedforanyvalueofNofpossiblepracticalinterest,withtheonlyrestrictionthatNmustbeaperfectsquarefortheFranksequence.)Wecomputethemeritfactorsoftheabovefourtypesofsequences(Golomb,Frank,CAN(G)andCAN(F))forthefollowinglengths:N=32;52;102;152;202;302;702and1002.TheresultsareshowninFigure 2-1 usingalog-logscale.Forallsequencelengthsweconsider,theCAN(G)andCAN(F)sequencesgivenearlythesamemeritfactors;botharemuchlargerthanthemeritfactorsgivenbytheGolomborFranksequence.WhenN=104,theCAN(G)sequenceprovidesthelargestmeritfactorof1839:76,whichismorethantentimeslargerthanthatgivenbytheGolombsequence(whichis157:10).WealsoshowthecorrelationlevelsoftheGolombandCAN(G)sequencesoflengthsN=102and103inFigures 2-2 and 2-3 ,respectively.Thecorrelationlevelisdenedascorrelationlevel=20log10r(k) r(0);k=1;:::;N)]TJ /F5 11.955 Tf 11.95 0 Td[(1: (2)WenotethatthecorrelationsidelobesoftheGolombsequencearecomparativelylargeforkcloseto0andN)]TJ /F5 11.955 Tf 12.27 0 Td[(1(thesameistruefortheFranksequence),whiletheCAN(G)sequencehasrelativelymoreuniformcorrelationsidelobesaskincreasesfrom0toN)]TJ /F5 11.955 Tf 11.95 0 Td[(1. 39

PAGE 40

2.3.2WeightedIntegratedSidelobeLevel(WISL)DesignConsiderthedesignofasequenceoflengthN=100.Supposethatweareinterestedinsuppressingthecorrelationsr1;:::;r25andr70;:::;r79.WeapplytheWeCANalgorithmwiththefollowingweightsusedinthematrix)]TJ /F1 11.955 Tf 11.4 0 Td[(in( 2 ):k=8><>:1;k2[1;25][[70;79]0;k2[26;69][[80;99]: (2)(0ischosentoensurethepositivesemi-denitenessof)]TJ /F1 11.955 Tf 8.09 0 Td[(;moreexactlywechoose0=12:05followingthediscussionrightafterEq.( 2 ).)Inthisscenario,theMMFisasdenedinEq.( 2 )withwk=2k=8><>:1;k2[1;25][[70;79]0;k2[26;69][[80;99]: (2)ArandomlygeneratedsequenceisusedtoinitializetheWeCANalgorithm.ThecorrelationlevelofthedesignedsequenceisshowninFigure 2-4 .TheWeCANsequencehascorrelationsidelobesthatarepracticallyzeroattherequiredlags,andwhicharemuchsmallerthanthesidelobesoftheGolomborCAN(G)sequenceinthelastsubsection(Figure 2-2A and 2-2B ).Table 2-1 presentsthecorrespondingMMFvalues.TheMMFoftheWeCANsequence(whichispracticallyinnite)issignicantlylargerthantheotherMMFvaluesinthetable. 2.3.3ChannelEstimationConsideranFIR(niteimpulseresponse)channelimpulseresponsefhpgP)]TJ /F6 7.97 Tf 6.59 0 Td[(1p=0whoseestimationisourmaingoal(thenumberofchanneltapsPisassumedtobeknown).Supposethatwetransmitaprobingsequencefx(n)gNn=1andobtainthereceivedsignalyn=P)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xp=0hpx(n)]TJ /F3 11.955 Tf 11.95 0 Td[(p)+en;n=1;:::;N+P)]TJ /F5 11.955 Tf 11.95 0 Td[(1; (2) 40

PAGE 41

wherefengN+P)]TJ /F6 7.97 Tf 6.58 0 Td[(1n=1isani.i.d.complexGaussianwhitenoisesequencewithzeromeanandvariance2.Eq.( 2 )canbewritteninthefollowingmorecompactform:y=Xh+e (2)whereXisdenedasX=2666666666666664x(1)0.........x(1)x(N).........0x(N)3777777777777775(N+P)]TJ /F6 7.97 Tf 6.59 0 Td[(1)P; (2)andy=y1yN+P)]TJ /F6 7.97 Tf 6.59 0 Td[(1T;h=h0hP)]TJ /F6 7.97 Tf 6.58 0 Td[(1T;e=e1eN+P)]TJ /F6 7.97 Tf 6.59 0 Td[(1T: (2)LetxpdenotethepthcolumnofthematrixX.Weusexpasamatchedltertodeterminehpfromy,whichleadstothefollowingestimateofhp:^hp=1 Nxpy: (2)LetthenumberofchanneltapsbeP=40.Figure 2-5A showsthemagnitudeofthesimulatedchannelimpulseresponsefjhpjgP)]TJ /F6 7.97 Tf 6.58 0 Td[(1p=0.WeperformtwoexperimentstocomparetheGolombsequenceandtheWeCANsequence.Thelatterisgeneratedwiththefollowingweightsk=8><>:1;k2[1;39]0;k2[40;N)]TJ /F5 11.955 Tf 11.95 0 Td[(1]; 41

PAGE 42

wk=2k;k=1;:::;N)]TJ /F5 11.955 Tf 11.96 0 Td[(1; (2)(andasusual,0isselectedsuchthat)]TJ /F2 11.955 Tf 13.15 0 Td[(0)anditsauto-correlationlevelisshowninFig. 2-5B forthecaseofN=100.Inoneexperimentthenoisepower2isxedat10)]TJ /F6 7.97 Tf 6.58 0 Td[(4andthesequencelengthNisvariedfrom100to500;IntheotherexperimentNisxedat200and2isvariedfrom10)]TJ /F6 7.97 Tf 6.59 0 Td[(6to1.Foreachpair(N;2),500Monte-Carlotrialsarerun(inwhichthenoisesequenceeisvaried)andthemean-squarederror(MSE)of^hisrecorded.Figure 2-6 showstheMSEof^hinthetwosituations.Duetobetterautocorrelationproperties,theWeCANsequencegeneratesconsistentlysmallerMSEthantheGolombsequence.Inparticular,itisinterestingtoobservefromFigure 2-6B thatas2decreases,theMSEof^hcorrespondingtotheWeCANsequenceisdecreasinglinearly(anditbecomes0as2goesto0),whiletheperformanceoftheGolombsequenceislimitedtoacertainlevelbecauseofitsnon-zerocorrelationsidelobes,whichinduceanestimationbias. Figure2-1.ThemeritfactorsoftheGolomb,Frank,CAN(G)andCAN(F)sequencesoflengthsfrom32upto1002. 42

PAGE 43

A B Figure2-2.CorrelationlevelsoftheGolombandCANsequencesoflengthN=102designedundertheISLmetric.A)TheGolombsequence,N=102andB)theCAN(G)sequence,N=102. Table2-1.MMFvaluesfortheweightsin( 2 )andN=100 GolombCAN(G)WeCAN MMF32:55142:641:061021 43

PAGE 44

A B Figure2-3.CorrelationlevelsoftheGolombandCANsequencesoflengthN=103designedundertheISLmetric.A)theGolombsequence,N=103andB)theCAN(G)sequence,N=103. 44

PAGE 45

Figure2-4.ThecorrelationlevelofaWeCANsequencedesignedundertheWISLmetricwithweightsin( 2 ). 45

PAGE 46

A B Figure2-5.ThesimulatedchannelimpulseresponseandtheprobingWeCANsequence.A)Themagnitudeofthesimulatedchannelimpulseresponseh.B)ThecorrelationleveloftheWeCANsequencedesignedundertheWISLmetricwithweightsin( 2 ). 46

PAGE 47

A B Figure2-6.TheMSEoftheestimated^husingtwotrainingsequences:theGolombsequenceandtheWeCANsequence.A)Thenoisepower2isxedat10)]TJ /F6 7.97 Tf 6.59 0 Td[(4andthesequencelengthNisvariedfrom100to500.B)Nisxedat200and2isvariedfrom10)]TJ /F6 7.97 Tf 6.59 0 Td[(6to1. 47

PAGE 48

CHAPTER3SEQUENCESETDESIGNChapter 2 dealswithdesigningasinglesequencewithgoodauto-correlationproperties.Inasimilarway,inmanyapplicationsasetofsequencesthathavegoodcorrelationpropertiescanbedesired,suchasinMIMO(multi-inputandmulti-output)radarandCDMA(codedevisionmultipleaccess)systems.Forexample,whentransmittingorthogonalwaveforms,aMIMOradarsystemcanachieveagreatlyincreasedvirtualaperturecomparedtoitsphased-arraycounterpart.ThisincreasedvirtualapertureenablesmanyoftheMIMOradaradvantages,suchasbetterdetectionperformance[ 41 ],improvedparameteridentiability[ 42 ],enhancedresolution[ 43 ]anddirectapplicabilityofadaptivearraytechniques[ 44 ].Inthecaseofwaveformsetdesign,whichisalsoreferredtoasmulti-waveformdesign,bothauto-andcross-correlationsareinvolved.Goodauto-correlationmeansthatatransmittedwaveformisnearlyuncorrelatedwithitsowntime-shiftedversions,whilegoodcross-correlationindicatesthatanyoneofthetransmittedwaveformisnearlyuncorrelatedwithothertime-shiftedtransmittedwaveforms.Goodcorrelationpropertiesintheabovesensereducetheriskthatthereceivedsignalofinterestisdrawnincorrelatedmultipathorclutterinterferences.Thereisanextensiveliteratureaboutmulti-waveformdesign.In[ 18 ]and[ 30 ],orthogonalwaveformsaredesignedwithgoodauto-andcross-correlationproperties,atopicthatisdirectlytiedtothisChapter.MorerelatedtoChapter 7 ,[ 45 ]and[ 46 ]focusonoptimizingthecovariancematrixofthetransmittedwaveformstoachieveagiventransmitbeampatternandin[ 32 ]thewaveformsaredesignedtoapproximateagivencovariancematrix.Otherworksinclude[ 47 ][ 48 ]and[ 49 ],wheresomepriorinformationisassumed(e.g.,thetargetimpulseresponse)andthewaveformsaredesignedtooptimizeastatisticalcriterion(e.g.,themutualinformationbetweenthetargetimpulseresponseandthereectedsignals).Wealsonotethatintheareaofmultipleaccess 48

PAGE 49

wirelesscommunications,thespreadingsequencedesignbasicallyaddressesthesameproblemofsynthesizingwaveformswithgoodauto-andcross-correlationproperties[ 50 ].Letfxm(n)g(m=1;:::;Mandn=1;:::;N)denoteasetofMsequences,eachofwhichisoflengthN.The(aperiodic)cross-correlationoffxm1(k)gNk=1andfxm2(k)gNk=1atlagnisdenedasrm1m2(n)=NXk=n+1xm1(k)xm2(k)]TJ /F3 11.955 Tf 11.96 0 Td[(n)=rm2m1()]TJ /F3 11.955 Tf 9.3 0 Td[(n); (3)m1;m2=1;:::;M;n=0;:::;N)]TJ /F5 11.955 Tf 11.95 0 Td[(1:Whenm1=m2,( 3 )becomestheauto-correlationoffxm1(k)gNk=1.ExtendingtheapproachesinChapter 2 ,wepresentinthischaptercyclicalgorithmsformulti-sequencedesign.TherstalgorithmiscalledMulti-CAN,whichaimstominimizeallcorrelationsidelobes.ThesecondalgorithmiscalledMulti-WeCAN,whichfocusesonminimizingthecorrelationsidelobesinacertaintimelaginterval. 3.1TheMulti-CANAlgorithmTheMulti-CANalgorithmaimstominimizethefollowingmetricE=MXm=1N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xn=)]TJ /F7 7.97 Tf 6.58 0 Td[(N+1;n6=0jrmm(n)j2+MXm1=1MXm2=1;m26=m1N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xn=)]TJ /F7 7.97 Tf 6.59 0 Td[(N+1jrm1m2(n)j2: (3)Tofacilitatethediscussion,denotethematrixofthetransmittedwaveformsbyX=x1x2xMNM (3)wherexm=xm(1)xm(2)xm(N)T (3) 49

PAGE 50

isthemthwaveform.ThewaveformcovariancematricesfordifferenttimelagsaregivenbyRn=266666664r11(n)r12(n)r1M(n)r21(n)r22(n)r2M(n).........rM1(n)rMM(n)377777775;n=)]TJ /F3 11.955 Tf 9.3 0 Td[(N+1;:::;0;:::;N)]TJ /F5 11.955 Tf 11.95 0 Td[(1: (3)ByusingthefollowingshiftingmatrixJn=2666666666664n+1z }| {10...103777777777775NN=JT)]TJ /F7 7.97 Tf 6.59 0 Td[(n;n=0;:::;N)]TJ /F5 11.955 Tf 11.96 0 Td[(1; (3)theRnin( 3 )canberewrittenas:Rn=(XHJnX)T=RH)]TJ /F7 7.97 Tf 6.58 0 Td[(n;n=0;:::;N)]TJ /F5 11.955 Tf 11.95 0 Td[(1: (3)Withtheabovenotation,thecriterionin( 3 )canbewrittenmorecompactlyasE=kR0)]TJ /F3 11.955 Tf 11.95 0 Td[(NIMk2+2N)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xn=1kRnk2=N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xn=)]TJ /F6 7.97 Tf 6.58 0 Td[((N)]TJ /F6 7.97 Tf 6.58 0 Td[(1)kRn)]TJ /F3 11.955 Tf 11.95 0 Td[(NIMnk2: (3)ThefollowingParseval-typeequalityholdstrue(theproofissimilartothatforthecaseofM=1inChapter 2 ):N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xn=)]TJ /F6 7.97 Tf 6.59 0 Td[((N)]TJ /F6 7.97 Tf 6.59 0 Td[(1)kRn)]TJ /F3 11.955 Tf 11.95 0 Td[(NIMnk2=1 2N2NXp=1k(!p))]TJ /F3 11.955 Tf 11.96 0 Td[(NIMk2; (3)where(!)4=N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xn=)]TJ /F7 7.97 Tf 6.59 0 Td[(N+1Rne)]TJ /F7 7.97 Tf 6.58 0 Td[(j!n (3) 50

PAGE 51

isthespectraldensitymatrixofthevectorsequencex1(n)xM(n)Tand!p=2 2Np;p=1;:::;2N: (3)The(!)denedin( 3 )canbewritteninthefollowingperiodogram-likeform[ 38 ]:(!)=~y(!)~yH(!) (3)where~y(!)=NXn=1y(n)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j!n;y(n)=x1(n)x2(n)xM(n)T: (3)Itfollowsfrom( 3 )and( 3 )that( 3 )canberewrittenasE=1 2N2NXp=1~yp~yHp)]TJ /F3 11.955 Tf 11.95 0 Td[(NIM2:~yp4=~y(!p) (3)Remark:TheEin( 3 )cannotbemadeverysmall,evenwithouttheunit-modulusconstraintontheelementsofX,becausetherank1matrix~yp~yHpcannotapproximatewellafullrankmatrixNI.Eq.( 3 )isaquartic(i.e.,fourth-order)functionoftheunknownsfxm(n)gM;Nm=1;n=1.Togetasimplerquadraticcriterionfunctionoffxm(n)g,notethatE=1 2N2NXp=1~yp~yHp)]TJ /F3 11.955 Tf 11.95 0 Td[(NI2=1 2N2NXp=1tr(~yp~yHp)]TJ /F3 11.955 Tf 11.95 0 Td[(NI)(~yp~yHp)]TJ /F3 11.955 Tf 11.95 0 Td[(NI)H=1 2N2NXp=1(k~ypk4)]TJ /F5 11.955 Tf 11.96 0 Td[(2Nk~ypk2+N2M)=2N2NXp=1 ~yp p 2N2)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2!2+N2(M)]TJ /F5 11.955 Tf 11.96 0 Td[(1): (3)Insteadofminimizing( 3 )withrespecttoX,weconsiderthefollowingminimizationproblem:minX;fpg2Np=12NXp=11 p 2N~yp)]TJ /F18 11.955 Tf 11.96 0 Td[(p2 (3) 51

PAGE 52

s.t.jxm(n)j=1;m=1;:::;Mandn=1;:::;Nkpk2=1 2;p=1;:::;2N(pisM1)wheres.t.standsforsubjectto,andfpgareauxiliaryvariables.Evidently,if( 3 )(withouttheconstanttermN2(M)]TJ /F5 11.955 Tf 13.13 0 Td[(1))canbemadeequaltozero(orsmall)bychoosingX,socan( 3 ),andviceversa.Tosolvetheminimizationproblemin( 3 ),deneaHp=e)]TJ /F7 7.97 Tf 6.58 0 Td[(j!pe)]TJ /F7 7.97 Tf 6.59 0 Td[(j2N!p;A=1 p 2Na1a2N; (3)~X=264X03752NM;V=12NT:Thenitisnotdifculttoobservethat2NXp=11 p 2N~yp)]TJ /F18 11.955 Tf 11.95 0 Td[(p2=kAH~X)]TJ /F15 11.955 Tf 11.96 0 Td[(Vk2=k~X)]TJ /F15 11.955 Tf 11.95 0 Td[(AVk2: (3)(Thesecondequalityin( 3 )followsfromthefactthatAisunitary.)Thecriterionin( 3 )canbeminimizedbymeansoftwoiterative(cyclic)steps.Forgiven~X(i.e.,Xisgiven),theminimizerfpg2Np=1of( 3 )isgivenbyp=1 p 2cp kcpk;p=1;:::;2N (3)wherecTp=thepthrowof(AH~X): (3)ForgivenV(i.e.,fpg2Np=1aregiven),theminimizerfxm(n)gof( 3 )isgivenbyxm(n)=exp(jarg(dnm));m=1;:::;Mandn=1;:::;N (3) 52

PAGE 53

wherednm=the(n;m)thelementof(AV): (3)TheMulti-CANalgorithmthusobtainedissummarizedinbelow: Step0:InitializeXbyarandomlygeneratedNMmatrixorbysomegoodexistingsequences. Step1:Fix~XandcomputeVaccordingto( 3 ). Step2:FixVandcompute~Xaccordingto( 3 ). Iteration:RepeatSteps1and2untilapre-speciedstopcriterionissatised,e.g.,kX(i))]TJ /F15 11.955 Tf 12.26 0 Td[(X(i+1)k<10)]TJ /F6 7.97 Tf 6.59 0 Td[(3,whereX(i)isthewaveformmatrixobtainedattheithiteration.NotethattheAH~Xin( 3 )istheFFTofeachcolumnof~XandthattheAVin( 3 )istheIFFTofeachcolumnofV.Becauseofthese(I)FFT-basedcomputations,theMulti-CANalgorithmisquitefast.Indeed,itcanbeusedtodesignlongsequencesuptoN103andM10,whichcanhardlybehandledbyotheralgorithmssuggestedinthepreviousliterature. 3.2TheMulti-WeCANAlgorithmInsomeradarapplicationslikesyntheticapertureradar(SAR)imaging,thetransmittedpulseisrelativelylong(i.e.,Nislarge)andthesignalsbackscatteredfromobjectsinthenearandfarrangebinsoverlapsignicantly[ 37 ].Inthiscase,onlythewaveformcorrelationpropertiesinacertainlagintervalaroundn=0arerelevanttorangeresolutionandaminimizationcriteriondifferentfrom( 3 )isgivenby~E=kR0)]TJ /F3 11.955 Tf 11.95 0 Td[(NIMk2+2P)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xn=1kRnk2; (3)whereP)]TJ /F5 11.955 Tf 11.96 0 Td[(1isthemaximumlagthatweareinterestedin.Morespecically,(P)]TJ /F5 11.955 Tf 11.96 0 Td[(1)tp(tpisasmentionedin( 1 ))shouldbechosenlargerthanthemaximumdifferenceofroundtripdelaysofthesignalsbackscatteredfromnearandfarrangebins. 53

PAGE 54

Remark:RecallthatitisnotpossibletomakethecriterionEin( 3 )verysmall.Onewaytounderstandthisproblemistoexaminethecriterion~Edenedin( 3 )whereonlyR0;R1;:::;RP)]TJ /F6 7.97 Tf 6.59 0 Td[(1(whicharecomplex-valuedMMmatrices)areconsidered.R0isHermitianwithalldiagonalelementsequaltoN,sosettingR0=NIleadstoM(M)]TJ /F5 11.955 Tf 12.99 0 Td[(1)(real-valued)equations.R1;:::;RP)]TJ /F6 7.97 Tf 6.58 0 Td[(1donothaveanyspecialstructureandsettingthemtozeroadds2M2equationsforeachofthem.ThusthetotalnumberofequationsisK=M(M)]TJ /F5 11.955 Tf 12.12 0 Td[(1)+(P)]TJ /F5 11.955 Tf 12.12 0 Td[(1)2M2.Comparedtothis,thenumberofvariablesthatwecanmanipulateisM(N)]TJ /F5 11.955 Tf 12.38 0 Td[(1)(foreachoftheMwaveformsthereareN)]TJ /F5 11.955 Tf 12.09 0 Td[(1freephases,astheinitialphasedoesnotmatter).Therefore,abasicrequirementforgoodperformanceisthatKM(N)]TJ /F5 11.955 Tf 12.32 0 Td[(1),whichcanbesimpliedto:PN+M 2M.Putdifferently,onlywhenPN+M 2MisitpossibleinprincipletodesignunimodularwaveformsXthatmake~Ezero;inothercases~EorEcannotbemadeequaltozero.TheMulti-WeCAN(multi-sequenceweighted-CAN)algorithmaimsatminimizingthefollowingcriterion:^E=20kR0)]TJ /F3 11.955 Tf 11.95 0 Td[(NIMk2+2N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xn=12nkRnk2; (3)wherefngN)]TJ /F6 7.97 Tf 6.59 0 Td[(1n=0arereal-valuedweights.Forinstance,ifwechoosen=1forn=0;:::;P)]TJ /F5 11.955 Tf 11.96 0 Td[(1andn=0otherwise,^Ebecomesthe~Edenedin( 3 ).Similarlyto( 3 ),wecanshowthat^E=1 2N2NXp=1~(!p))]TJ /F3 11.955 Tf 11.96 0 Td[(0NIM2; (3)wheref!pg2Np=1isgivenby( 3 )and~(!)4=N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xn=)]TJ /F6 7.97 Tf 6.59 0 Td[((N)]TJ /F6 7.97 Tf 6.59 0 Td[(1)nRne)]TJ /F7 7.97 Tf 6.58 0 Td[(j!n; (3) 54

PAGE 55

andwheren=)]TJ /F7 7.97 Tf 6.59 0 Td[(nforn=1;:::;N)]TJ /F5 11.955 Tf 12.29 0 Td[(1.Tofacilitatelaterdevelopments,0ischosensuchthatthematrix)]TJ /F5 11.955 Tf 11.4 0 Td[(=26666666401N)]TJ /F6 7.97 Tf 6.59 0 Td[(110...............1N)]TJ /F6 7.97 Tf 6.59 0 Td[(110377777775 (3)ispositivesemi-denite(denotedas)]TJ /F2 11.955 Tf 13.11 0 Td[(0).Seethediscussionsfollowing( 2 )inChapter 2 forawaytodetermine0.Thecondition)]TJ /F2 11.955 Tf 13.23 0 Td[(0isnecessarybecausethematrixsquarerootof)]TJ /F1 11.955 Tf 11.4 0 Td[(isneededlateron(Eq.( 3 )).Similarlyto( 3 ),itcanbeshownthat:~(!)=ZT(!)Z(!); (3)whereZT(!)=y(1)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j!y(2)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j!2y(N)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j!NMN: (3)Bycombining( 3 )and( 3 ),thecriterionbecomes^E=1 2N2NXp=1ZHpZp)]TJ /F3 11.955 Tf 11.96 0 Td[(0NIM2(Zp4=Z(!p)): (3)Insteadofminimizing( 3 )withrespecttoX,weconsiderthefollowingminimizationproblem(thediscussionfollowing( 3 )):minX;U2NXp=1CZp)]TJ /F11 11.955 Tf 11.95 10.74 Td[(p 0NUp2; (3)s.t.jxm(n)j=1;m=1;:::;Mandn=1;:::;N;UHpUp=I;p=1;:::;2N;(UpisNM)wheretheNNmatrixCisasquarerootof)]TJ /F1 11.955 Tf 11.4 0 Td[((i.e.,CHC=)]TJ /F1 11.955 Tf 8.08 0 Td[(). 55

PAGE 56

Theminimizationproblemin( 3 )canbesolvedinacyclicwayasfollows.ForgivenfZpg2Np=1(i.e.,Xisgiven),( 3 )decouplesinto2Nindependentproblems,eachofwhichcanbewrittenasCZp)]TJ /F11 11.955 Tf 11.96 10.74 Td[(p 0NUp2=const)]TJ /F5 11.955 Tf 11.95 0 Td[(2Rentrhp 0NUpZHpCHio; (3)p=1;:::;2N;whereconstdenotesatermthatisindependentofthevariableUp.LetZHpCH=U1UH2 (3)denotetheeconomicSVD(singularvaluedecomposition)ofZHpCH,whereU1isMM,isMMandU2isNM.ThentheminimizerUpof( 3 ),forxedZp,isgivenby(Eq.( 3 )( 3 )forthesameoptimizationproblemandAppendix B fortheproof):Up=U2UH1: (3)NotethatthecomputationoffCZpg2Np=1canbedonebymeansoftheFFT.Toseethis,let~Xm=CTxmxmxmNN;m=1;:::;M; (3)andF=p 2NAH~F;~F=264~X1~X2~XM0NN0NN0NN3752NNM; (3)whereAhasbeendenedin( 3 ).LetfTpdenotethepthrowofF.WedividetheNM1vectorfpequallyintoMpieces,whichwecanobservethatcorrespondtotheMcolumns(fromlefttoright)ofCZp.ThusthematrixCZpcanbeobtainedfromF,whichiscalculatedfromp 2NAH~F,anFFToperation. 56

PAGE 57

ForgivenfUpg2Np=1,theminimizationproblemin( 3 )alsohasaclosed-formsolutionwithrespecttoX.LetG2NNM=g1g2g2NT; (3)wheregpdenotestheNM1vectorgivenbythecolumnsofp 0NUpstackedontopofeachother.Thenthecriterionin( 3 )canbewrittenas2NXp=1CZp)]TJ /F11 11.955 Tf 11.96 10.74 Td[(p 0NUp2=p 2NAH~F)]TJ /F15 11.955 Tf 11.96 0 Td[(G2 (3)=2N~F)]TJ /F5 11.955 Tf 23.44 8.09 Td[(1 p 2NAG2:Theabovefunctioncanbeminimizedwithrespecttoeachelementoffxm(n)gM;Nm=1;n=1separately.Letxdenoteagenericelementoffxm(n)g.Thenthecorrespondingproblemistominimizethefollowingcriterionwithrespecttox:NXk=1jkx)]TJ /F3 11.955 Tf 11.95 0 Td[(kj2=const)]TJ /F5 11.955 Tf 11.95 0 Td[(2Re" NXk=1kk!x#; (3)wherefkgNk=1aregivenbytheelementsof~Fwhichcontainx,andfkgNk=1aregivenbytheelementsof1 p 2NAGwhosepositionsarethesameasthoseoffkgNk=1in~F.(Morespecically,fork=1;:::;N,kisgivenbythe(k;n)thelementofCandkisgivenbythe(n;(m)]TJ /F5 11.955 Tf 12.58 0 Td[(1)N+k)thelementof1 p 2NAG.)Undertheunit-modulusconstraint,theminimizerxofthecriterionin( 3 )isgivenbyx=ej;=arg NXk=1kk!: (3)TheMulti-WeCANalgorithmfollowsfromtheabovediscussionsanditissummarizedbelow: Step0:InitializeXandselectthedesiredweightsfngN)]TJ /F6 7.97 Tf 6.59 0 Td[(1n=0suchthatthematrix)]TJ /F1 11.955 Tf -419.52 -14.44 Td[(in( 3 )ispositivesemi-denite. Step1:FixfZpg2Np=1(i.e.,Xisgiven)andcomputefUpg2Np=1accordingto( 3 ). 57

PAGE 58

Step2:FixfUpg2Np=1andcomputeXaccordingto( 3 ). Iteration:RepeatSteps1and2untilapre-speciedstopcriterionissatised,e.g.,kX(i))]TJ /F15 11.955 Tf 11.96 0 Td[(X(i+1)k<,whereX(i)isthewaveformmatrixobtainedattheithiteration.Multi-WeCANisnotsocomputationallyefcientasMulti-CAN,butitcanstillbeusedforrelativelylargevaluesofNandM,uptoN103andM10. 3.3TheMulti-CA-Original(Multi-CAO)AlgorithmTheoriginalcyclicalgorithm(CA)forwaveformdesignproposedin[ 32 37 ]aimsatminimizingaparticularformofthecriterion^Ein( 3 ):^ECAO=PkR0)]TJ /F3 11.955 Tf 11.96 0 Td[(NIMk2+2P)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xn=1(P)]TJ /F3 11.955 Tf 11.96 0 Td[(n)kRnk2; (3)whichcanbeobtainedfrom( 3 )bychoosingtheweights2n=P)]TJ /F3 11.955 Tf 13.12 0 Td[(nforn=0;:::;P)]TJ /F5 11.955 Tf 12.33 0 Td[(1and2n=0otherwise.WerefertothisoriginalCAalgorithmasMulti-CAO(multi-sequenceCAoriginal)tomakethenamingconsistentwiththatofMulti-CANandMulti-WeCAN.TheabovechoiceoffngN)]TJ /F6 7.97 Tf 6.58 0 Td[(1n=0resultsfromthefollowingproblemformulationthatissimpleanddirect.ConsiderthematrixX=X1XM(N+P)]TJ /F6 7.97 Tf 6.59 0 Td[(1)MP (3)whereXm=2666666666666664xm(1).........xm(1)xm(N).........xm(N)3777777777777775(N+P)]TJ /F6 7.97 Tf 6.58 0 Td[(1)P;m=1;:::;M: (3) 58

PAGE 59

Thenitiseasytoobservethatthe^ECAOdenedin( 3 )canbeexpressedas^ECAO=XHX)]TJ /F3 11.955 Tf 11.95 0 Td[(NIMP2: (3)Theminimizationof( 3 )canbetackledbysolvingthefollowingproblemminX;UkX)]TJ 11.96 10.7 Td[(p NUk2; (3)s.t.jxm(n)j=1;m=1;:::;Mandn=1;:::;N;UHU=I;(Uis(N+P)]TJ /F5 11.955 Tf 11.95 0 Td[(1)MP):Notethattheproblemofminimizing( 3 )andthatof( 3 )arealmostequivalent,andsoistherelationshipbetween( 3 )and( 3 ),aswellas( 3 )and( 3 ).Seethediscussionfollowing( 2 )inChapter 2 aswellasAppendix A formoreinformation.Regarding( 3 ),wenotethefollowingfacts.ForgivenX,letXH=U1SUH2 (3)denotetheeconomicSVD(singularvaluedecomposition)ofX.HereU1isanMPMPunitarymatrix,U2isan(N+P)]TJ /F5 11.955 Tf 12.26 0 Td[(1)MPsemi-unitarymatrixandSisanMPMPdiagonalmatrix.ThenthesolutionUof( 3 ),forxedX,isgivenby(Appendix B )U=U2UH1: (3)NextnotethatforxedU,theminimizationofthecriterionin( 3 )alsohasasimpleclosed-formsolution.Toseethis,letxdenoteanarbitraryelementfromfxm(n)g.Thenagenericformoftheminimizationproblemin( 3 )withrespecttoxisgivenbyminxPXk=1jx)]TJ /F3 11.955 Tf 11.95 0 Td[(kj2; (3)wherefkgPk=1aretheelementsofthematrixp NUwhosepositionsarethesameasthepositionsofxinX.Moreprecisely,forx=xm(n)thecorrespondingsequence 59

PAGE 60

fkgPk=1isgivenbythe[n)]TJ /F5 11.955 Tf 12.44 0 Td[(1+r;(m)]TJ /F5 11.955 Tf 12.44 0 Td[(1)P+r]thelementsofp NU,forr=1;:::;P.Becausejxj=1,thecriterionin( 3 )canberewrittenasPXk=1jx)]TJ /F3 11.955 Tf 11.96 0 Td[(kj2=const)]TJ /F5 11.955 Tf 11.96 0 Td[(2Re"xPXk=1k# (3)=const)]TJ /F5 11.955 Tf 11.95 0 Td[(2PXk=1kcos"arg(x))]TJ /F5 11.955 Tf 11.96 0 Td[(arg PXk=1k!#whereconstdenotesthetermthatdoesnotdependonx.Hencetheminimizerxofthecriterionin( 3 )isgivenbyx=exp(jarg PXk=1k!): (3)TheMulti-CAOalgorithmfor( 3 )followsfromtheabovediscussionandissummarizedbelow: Step0:SetthematrixXtoaninitialvalue. Step1:Fixfxm(n)g(i.e.,Xisgiven)andcomputeUaccordingto( 3 ). Step2:FixUandcomputefxm(n)gaccordingto( 3 ). Iteration:RepeatSteps1and2untilapre-speciedstopcriterionissatised.ThecriterionminimizedbyMulti-CAOisaspecialcaseofthatminimizedbyMulti-WeCAN.Multi-CAOandMulti-WeCANprovidesimilarperformancesintermsofcorrelationsidelobesuppression,thoughthederivationanditerationstepsofMulti-CAOarerelativelylessinvolved. 3.4NumericalExamples 3.4.1Multi-CANConsiderminimizingthecriterionEin( 3 ),i.e.,minimizingallcorrelationsidelobes:rmm(n)forallmandn6=0,andrm1m2(n)forallm16=m2andn.SupposethatthenumberoftransmitsequencesisM=3andthelengthofeachsequenceisN=40.WecomparetheMulti-CANsequencesetwiththeCE(crossentropy)sequencesetin[ 30 ].WeuserandomlygeneratedsequencestoinitializeMulti-CAN.100Monte-Carlo 60

PAGE 61

trialsarerun(i.e.,100randominitializations)andthesequencesetwiththelowestcorrelationsidelobepeakiskept.The40-by-3CEsequencesetisgiveninTable1of[ 30 ].Figures 3-1 3-2 and 3-3 showthecorrelations(r11;r12;:::;r33,normalizedbyN)oftheMulti-CANsequencesetandCEsequenceset.TheCEsequencesetisslightlybetterthantheMulti-CANsequencesetintermsofcorrelationsidelobepeaks.However,ourgoalistominimizeEorequivalentlythefollowingnormalizedttingerror:Enorm=E MN2= kR0)]TJ /F3 11.955 Tf 11.95 0 Td[(NIk2+2N)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xn=1kRnk2!=(MN2): (3)TheMulti-CANsequencesetgivesattingerrorof2:00,whereastheCEsequencesethasabiggerttingerrorequalto2:23.NotethatalthoughtheMulti-CANandCEsequencesetsshowcomparableperformances(alsocomparabletotheperformanceofothersequencesetsliketheonesin[ 18 ]),theMulti-CANalgorithmworksmuchfasterthanotherexistingalgorithms,becauseMulti-CANisbasedonFFTcomputations.Fortheaboveparameterset(N=40andM=3),theMulti-CANalgorithmconsumeslessthanonesecondonanordinaryPCtocompleteoneMonte-Carlotrial.TheoverallcomputationtimeisstillshortifwerunplentyofMonte-Carlotrialsandpickupthebestsequenceset.Moreover,thecomputationtimeofMulti-CANgrowsroughlyasO(MNlogN)sothatMulti-CANcanhandleverylargevaluesofN,uptoN105.Incontrast,theCrossEntropy[ 30 ]orSimulatedAnnealingbasedmethods[ 18 ]arerelativelyinvolvedandbecomeimpracticalforlargevaluesofN.Infact,wewereunabletondintheliteratureanycodesetthatisdesignedforgood(aperiodic)correlationsandatthesametimeissufcientlylongtobecomparablewithwhatcanbeobtainedusingtheMulti-CANalgorithm.ForrelativelylargevaluesofN,wedecidedtoemploytheHadamardsequencesetforcomparison[ 15 ],whichiseasytogenerate(forvirtuallyanylengththatisapowerof2)andisfrequentlyusedinwirelesscommunications.WescrambledtheHadamard 61

PAGE 62

sequencesetwithaPN(pseudo-noise)sequencesettoloweritscorrelationsidelobes.WecomparetheMulti-CANsequenceset(100Monte-CarlotrialsarerunforeachNandtheresultwiththelowestcorrelationsidelobepeakisshown)andtheQPSK(quadraturephase-shiftkeying)Hadamard+PNsequencesetforM=3andN=27;:::;213.Figures 3-4 and 3-5 comparethesequencesetsintermsofthreecriteria:theauto-correlationsidelobepeak,thecross-correlationpeakandthenormalizedttingerror(denedin( 3 )).TheMulti-CANsequencesetoutperformstheHadamard+PNsequencesetwithrespecttoeachcriterion.Infact,theadvantageoftheMulti-CANalgorithmliesnotonlyinthesignicantlengthandthelowcorrelationsidelobesofthedesignedsequencesets,butalsointheeasygeneration(usingdifferentinitialconditions)ofmanysequencesetswhichareofthesameN-by-Mdimensionandallhavereasonablylowcorrelationsidelobes.Theserandomlydistributedwaveformsetsareusefulinsomeapplicationareas,likeforcounteringthecoherentrepeaterjamminginradarsystems[ 1 18 ]. 3.4.2Multi-WeCANConsiderminimizingthecriterion~Ein( 3 ),i.e.,minimizingthecorrelationsidelobesforlagsnotlargerthanP)]TJ /F5 11.955 Tf 10.94 0 Td[(1:rmm(n)forallmand1nP)]TJ /F5 11.955 Tf 10.94 0 Td[(1,andrm1m2(n)forallm16=m2and0nP)]TJ /F5 11.955 Tf 12.48 0 Td[(1.SupposethatthenumberoftransmitsequencesisM=4,thelengthofeachsequenceisN=256andthenumberofcorrelationlagswewanttoconsiderisP=50.Similarlyto( 3 ),thenormalizedttingerrorforthisscenarioisdenedas~Enorm=~E MN2= kR0)]TJ /F3 11.955 Tf 11.96 0 Td[(NIk2+2P)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xn=1kRnk2!=(MN2): (3)Wealsodenethecorrelationlevelascorrelationlevel=20lgkRn)]TJ /F3 11.955 Tf 11.95 0 Td[(NInk p MN2;n=)]TJ /F3 11.955 Tf 9.3 0 Td[(N+1;:::;0;:::;N)]TJ /F5 11.955 Tf 11.95 0 Td[(1 (3)whichmeasuresthecorrelationsidelobesforeachtimelag. 62

PAGE 63

WecomparetheMulti-WeCANalgorithmandtheMulti-CAOalgorithm.WeusearandomlygeneratedunimodularsequencesettoinitializebothMulti-WeCANandMulti-CAO.ForMulti-WeCAN,wechoose2n=8><>:1;n2[1;P)]TJ /F5 11.955 Tf 11.96 0 Td[(1]0;n2[P;N)]TJ /F5 11.955 Tf 11.95 0 Td[(1] (3)and0ischosentoensurethat)]TJ /F3 11.955 Tf 11.4 0 Td[(>0(moreexactlywechoose0=25:5).Table 3-1 comparestheMulti-CAOsequencesetandtheMulti-WeCANsequencesetintermsoftheauto-correlationsidelobepeak(intheconsideredlaginterval),thecross-correlationpeak(intheconsideredlaginterval)andthe~Enormdenedin( 3 ).The2564Multi-CANsequencesetisalsoaddedinTable 3-1 forcomparison.TheMulti-WeCANsequencesetgivesthelowestcorrelationsidelobepeakandttingerror.Figure 3-6 showsthecorrelationleveloftheMulti-CAOandMulti-WeCANsequencesets.WeobservefromFigure 3-6 thattheMulti-WeCANsequencesetprovidesauniformlylowcorrelationlevelintherequiredlaginterval[1;P)]TJ /F5 11.955 Tf 11.82 0 Td[(1],whilethecorrelationleveloftheMulti-CAOsequencesetincreasesasthelagincreasesfrom1toP)]TJ /F5 11.955 Tf 12.71 0 Td[(1.ThisbehaviorisattributedtothefactthatMulti-WeCANmakesuseofuniformweightsfn=1gP)]TJ /F6 7.97 Tf 6.59 0 Td[(1n=1in( 3 )whereasMulti-CAOimplicitlyassumesunevenweightsfn=P)]TJ /F3 11.955 Tf 12.16 0 Td[(ngP)]TJ /F6 7.97 Tf 6.59 0 Td[(1n=1(Eq.( 3 )),sothebiggerthelag,thesmallertheweight.Wealsonotethatthecorrelationlevelatn=0fortheMulti-WeCANsequencesetisverylow(around)]TJ /F5 11.955 Tf 9.3 0 Td[(85dB).Thereasonisthatwechose0=25:5,whichismuchlargerthantheotherweights(Eq.( 3 ))andthusthe-lagcorrelationttingerrorkR0)]TJ /F3 11.955 Tf 12.41 0 Td[(NIkisemphasizedthemostinthecriterion^Ein( 3 ). 63

PAGE 64

3.4.3Multi-WeCAN(cont.)ConsiderusingtheMulti-WeCANalgorithmtominimizethecriterion^Ein( 3 )withN=256;M=4andthefollowingweights:2n=8><>:1;n2[1;19][[236;255]0;n2[20;235] (3)(asbefore,0ischosentoensurethepositivesemi-denitenessof)]TJ /F1 11.955 Tf 10.64 0 Td[(in( 3 )).WestillusearandomlygeneratedsequencetoinitializeMulti-WeCAN.Inthisscenario,thenormalizedttingerrorisdenedas^Enorm=^E=(MN2).Table 3-2 comparestheMulti-WeCANsequenceandthe2564Multi-CANsequence.TheMulti-WeCANsequenceprovidesmuchlowercorrelationsidelobepeaksandmuchsmallerttingerror.Figure 3-7 showsthecorrespondingcorrelationlevelsoftheMulti-CANandMulti-WeCANsequences,fromwhichweseethatMulti-WeCANsucceedsmuchbetterinsuppressingthecorrelationsattherequiredlags.Notethatbecausejrm1m2(N)]TJ /F5 11.955 Tf 12.15 0 Td[(1)j=1forallm1andm2,thecorrelationlevelcorrespondingtothemaximumlagN)]TJ /F5 11.955 Tf 12.04 0 Td[(1isalwaysequalto20log10(p M2=p MN2),whichis)]TJ /F5 11.955 Tf 9.3 0 Td[(42:14dBinthiscase(theendpointsinbothFigures 3-7A and 3-7B ). 3.4.4QuantizationEffectsWehaveassumedthatthephasesofthedesignedsequencescantakeonanyvaluesfrom0to2.Inpracticeitmightberequiredthatthephasesaredrawnfromadiscreteconstellation.Thuswebrieyexamineheretheperformanceofthedesignedsequencesunderquantization.Letfxm(n)gM;Nm=1;n=1denotethesequencesetthatisobtainedfromoneofthealgorithmsdiscussedinthispaper.Supposethatthequantizationlevelis2qwhereq1isaninteger.Thenthequantizedsequencecanbeexpressedas^xm(n)=expjargfxm(n)g 2=2q2 2q;m=1;:::;Mandn=1;:::;N: (3) 64

PAGE 65

WequantizetheMulti-CANsequenceusedinFigure 3-4 into32levels(i.e.q=5)anddothesamecomparisonswiththeHadamard+PNsequence.TheresultsareshowninFigure 3-8 ,fromwhichweseethatthecurvesrepresentingtheCANsequencemoveupalittlebuttheyarestillbelowthecorrespondingcurvesoftheHadamard+PNsequence(exceptforthepointofN=4096inFigure 3-8B ).WedonotplotthettingerrorhereaswasdoneinFigure 3-5 ,becausethettingerroroftheCANsequencealmostdoesnotchangeafterthis32-levelquantization.Similarsituationsoccurifwequantizesequencesgeneratedfromtheotheralgorithmsdiscussedinthischapter.Inourtest,theperformancedegradation(i.e.thecorrelationsidelobeincrease)wasquitelimitedprovidedthatthequantizationlevelwasnotverysmall(e.g.q6). 3.4.5SyntheticApertureRadar(SAR)ImagingConsideraMIMOradarangle-rangeimagingexample(intra-pulseDopplereffectsareassumedtobenegligible)usinguniformlineararrayswithcolocatedM=4transmitandMr=4receiveantennas.Theinter-elementspacingofthetransmitandreceiveantennasisequalto2and0:5wavelengths,respectively.SupposethatallpossibletargetsareinafareldconsistingofP=30rangebins(whichmeansthatthemaximumroundtripdelaydifferencewithintheilluminatedsceneisnotlargerthan29subpulses)andascanningangleareaof()]TJ /F5 11.955 Tf 9.3 0 Td[(30;30)degrees.ThelengthoftheprobingwaveformforeachtransmitantennaisN=256.LetXdenotetheNMtransmittedprobingwaveformmatrix(Eq.( 3 )),andlet~X=264X0375(N+P)]TJ /F6 7.97 Tf 6.58 0 Td[(1)M (3) 65

PAGE 66

where0isa(P)]TJ /F5 11.955 Tf 10.92 0 Td[(1)Mmatrixofzeros.ThentheMr(N+P)]TJ /F5 11.955 Tf 10.92 0 Td[(1)receiveddatamatrixcanbewrittenas[ 44 ]DH=P)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xp=0KXk=1pkakbTk~XHJp+EH; (3)whereJpisan(N+P)]TJ /F5 11.955 Tf 12.8 0 Td[(1)(N+P)]TJ /F5 11.955 Tf 12.8 0 Td[(1)shiftingmatrixasdenedin( 3 )(withthesamestructurebutdifferentdimension),EHisthenoisematrixwhosecolumnsareindependentandidenticallydistributed(i.i.d.)randomvectorswithmeanzeroandcovariancematrixQ,fpkgP)]TJ /F6 7.97 Tf 6.58 0 Td[(1;Kp=0;k=1arecomplexamplitudeswhichareproportionaltotheradar-cross-sections(RCS)ofthescatters,andakandbkarethereceiveandtransmitsteeringvectors,respectively,whicharegivenbyak=1e)]TJ /F7 7.97 Tf 6.58 0 Td[(jsin(k)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j(Mr)]TJ /F6 7.97 Tf 6.58 0 Td[(1)sin(k)T; (3)andbk=1e)]TJ /F7 7.97 Tf 6.58 0 Td[(jMrsin(k)e)]TJ /F7 7.97 Tf 6.58 0 Td[(j(M)]TJ /F6 7.97 Tf 6.59 0 Td[(1)Mrsin(k)T; (3)wherefkgKk=1arethescanningangles.OurgoalistoestimatefpkgP)]TJ /F6 7.97 Tf 6.59 0 Td[(1;Kp=0;k=1fromthecollecteddataDH.FirstweapplythefollowingmatchedltertothedatamatrixDH:~XMFp4=JHp~X(~XH~X))]TJ /F6 7.97 Tf 6.59 0 Td[(1;(N+P)]TJ /F5 11.955 Tf 11.96 0 Td[(1)M; (3)(notethatweassumeN+P)]TJ /F5 11.955 Tf 12.76 0 Td[(1Mandthus~XHJp~XMFp=IM)toperformrangecompressionforthepthrangebin,i.e.,~DHp4=DH~XMFp= P)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xq=0KXk=1qkakbTk~XHJq!~XMFp+EH~XMFp (3)=KXk=1pkakbTk~XHJp~XMFp+Zp=KXk=1pkak~bHk+Zp;(~bHk4=bTk); 66

PAGE 67

whereZp=0BB@P)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xq=0q6=pKXk=1qkakbTk~XHJq1CCA~XMFp+EH~XMFp.Theltereddatain( 3 )leadsnaturallytothefollowingleastsquares(LS)estimateofpk:^LSpk=aHk~DHp~bk kakk2k~bkk2;k=1;:::;Kandp=0;:::;P)]TJ /F5 11.955 Tf 11.96 0 Td[(1; (3)aswellastothefollowingCaponestimate:^Caponpk=aHk^R)]TJ /F6 7.97 Tf 6.59 0 Td[(1p~DHp~bk aHk^R)]TJ /F6 7.97 Tf 6.59 0 Td[(1pakk~bkk2;k=1;:::;Kandp=0;:::;P)]TJ /F5 11.955 Tf 11.95 0 Td[(1; (3)where^Rp=~DHp~Dpdenotesthecovariancematrixofthecompressedreceiveddata([ 44 ]formoredetailsabouttheseestimatesofpk).Toobtainalargersyntheticaperture,weusetheSARprincipleandthusrepeattheprocessofsendingaprobingwaveformandcollectingdataat~N=20differentpositions;thecollecteddatamatricesaredenotedasDH1;DH2:::;DH~Nrespectively.SupposethattwoadjacentpositionsarespacedMMr 2wavelengthsapart,whichinducesaphaseshiftof k=)]TJ /F5 11.955 Tf 9.3 0 Td[(2MMr 2sin(k)forboththetransmitandreceivesteeringvectorscorrespondingtothetwoadjacentpositions.(Aslongasthetargetsinthefar-eldassumptionholds,thedistancebetweentwoadjacentpositionscanbechosenatwillandcanbedifferentfordifferentadjacentpositions;weonlyneedtochangethephaseshift kaccordingly.)Inthiscase,welet~DHp=DH1~XMFpDH2~XMFpDH~N~XMFpMr~NM; (3)and~bHk=bTkbTkej2 kbTkej2(~N)]TJ /F6 7.97 Tf 6.59 0 Td[(1) k1~NM: (3)Usingthisnotation,theexpressionsfortheestimatesofpkin( 3 )and( 3 )canbeusedmutatismutandis. 67

PAGE 68

Table3-1.ComparisonbetweenMulti-CAN,Multi-CAOandMulti-WeCANunder~E(N=256;M=4;P=50) Auto-corrsidelobepeak(dB)Cross-corrpeak(dB)~Enorm Multi-CAN)]TJ /F5 11.955 Tf 9.3 0 Td[(20:54)]TJ /F5 11.955 Tf 9.3 0 Td[(18:190.91Multi-CAO)]TJ /F5 11.955 Tf 9.3 0 Td[(21:08)]TJ /F5 11.955 Tf 9.3 0 Td[(20:770.088Multi-WeCAN)]TJ /F5 11.955 Tf 9.3 0 Td[(31:10)]TJ /F5 11.955 Tf 9.3 0 Td[(29:090.072 Table3-2.ComparisonbetweenMulti-CANandMulti-WeCANunder^E(N=256;M=4) Auto-corrsidelobepeak(dB)Cross-corrpeak(dB)^Enorm Multi-CAN)]TJ /F5 11.955 Tf 9.3 0 Td[(20:53)]TJ /F5 11.955 Tf 9.3 0 Td[(17:680:40Multi-WeCAN)]TJ /F5 11.955 Tf 9.3 0 Td[(45:17)]TJ /F5 11.955 Tf 9.3 0 Td[(45:819:5410)]TJ /F6 7.97 Tf 6.59 0 Td[(4 Inthenumericalsimulation,thenoisecovariancematrixQischosenas2IMr,where2=0:001.ThetargetsarechosentoformaUFshape(Fig. 3-9 )andtheRCS-relatedparametersfpkgP)]TJ /F6 7.97 Tf 6.58 0 Td[(1;Kp=0;k=1aresimulatedasi.i.d.complexsymmetricGaussianrandomvariableswithmean0andvariance1atthetargetlocationsandzeroelsewhere.Theaverage(transmitted)signal-to-noiseratio(SNR)isgivenbySNR=tr(XHX)=N tr(Q)=M Mr2=30dB: (3)Weusetwodifferentprobingsequencesets:aQPSKHadamard+PNsequencesetandaMulti-WeCANsequencesetwithN=256;M=4andP=30.Thetransmittedwaveformisphase-modulatedbytheprobingsequenceset(onesequenceelementcorrespondstoonesubpulse)andweassumepropersamplingsothattheconsidereddiscretemodelsareappropriate.TheestimatedfpkgP)]TJ /F6 7.97 Tf 6.59 0 Td[(1;Kp=0;k=1usingthesetwowaveformsareshowninFigures 3-10 and 3-11 .TheMulti-WeCANwaveformgivesmuchclearerangle-rangeimagesthantheHadamard+PNwaveform.NotefromFigure 3-11B thattheMulti-WeCANwaveformleadstoanalmostperfectrangecompressionviathematchedlter(thefalsescatterersareduetothepresenceofnoise)andthattheCaponestimatorprovidesaradarimagewithahighangleresolution. 68

PAGE 69

A B Figure3-1.Thecorrelationsofthe40-by-3CEandMulti-CANsequencesets.A)r11(k)andB)r12(k). 69

PAGE 70

A B Figure3-2.Thecorrelationsofthe40-by-3CEandMulti-CANsequencesets.A)r13(k)andB)r22(k). 70

PAGE 71

A B Figure3-3.Thecorrelationsofthe40-by-3CEandMulti-CANsequencesets.A)r23(k)andB)r33(k). 71

PAGE 72

A B Figure3-4.ComparisonbetweentheMulti-CANsequencesetandtheHadamard+PNsequencesetwithM=3andN=27;:::;213intermsofA)theauto-correlationsidelobepeakandB)thecross-correlationpeak. 72

PAGE 73

Figure3-5.ComparisonbetweentheMulti-CANsequencesetandtheHadamard+PNsequencesetwithM=3andN=27;:::;213intermsofthenormalizedttingerrorasdenedin( 3 ). 73

PAGE 74

A B Figure3-6.CorrelationlevelsoftheMulti-CAOsequencesetandtheMulti-WeCANsequencesetwithN=256,M=4andP=50.(Thedottedverticallinessignifytheboundaryofthetimelagintervalunderconsideration.)A)TheMulti-CAOsequencesetandB)theMulti-WeCANsequenceset. 74

PAGE 75

A B Figure3-7.CorrelationlevelsoftheMulti-CANsequenceandtheMulti-WeCANsequencewithN=256,M=4andweightsfngN)]TJ /F6 7.97 Tf 6.58 0 Td[(1n=0asspeciedin( 3 ).(Thedottedverticallinessignifytheboundariesofthetimelagintervalsunderconsideration.)A)TheMulti-CANsequenceandB)theMulti-WeCANsequence. 75

PAGE 76

A B Figure3-8.ThesamecomparisonsasshowninFigures 3-4 ,exceptthatthephasesoftheCANsequenceusedherearequantizedinto32levels. 76

PAGE 77

Figure3-9.Thetruetargetimage(theabsolutevaluesoffpkgP)]TJ /F6 7.97 Tf 6.59 0 Td[(1;Kp=0;k=1areshown). 77

PAGE 78

A B Figure3-10.TheestimatedtargetimagesintermsoftheRCS-relatedparametersfjpkjgP)]TJ /F6 7.97 Tf 6.59 0 Td[(1;Kp=0;k=1usingtheHadamard+PNwaveform.A)TheLSestimateandB)theCaponestimate. 78

PAGE 79

A B Figure3-11.TheestimatedtargetimagesintermsoftheRCS-relatedparametersfjpkjgP)]TJ /F6 7.97 Tf 6.59 0 Td[(1;Kp=0;k=1usingtheMulti-WeCANwaveform.A)TheLSestimateandB)theCaponestimate. 79

PAGE 80

CHAPTER4CORRELATIONLOWERBOUNDSThepreviouschaptershaveshowntheimportanceofdesigningsequencesorsequencesetswithlowcorrelationsidelobes.Letfxk(n)g(k=1;:::;Mandn=1;:::;N)denoteasetofMsequencesoflengthN,whicharerestrictedtohavethesameenergy:NXn=1jxk(n)j2=N;k=1;:::;M: (4)Excludingthein-phase(i.e.,thezero-delay)auto-correlations,allothercorrelationsarecategorizedtobecorrelationsidelobesandcorrespondinglythepeaksidelobelevel(PSL)metricisdenedasPSL=maxfjrks(l)jg; (4)k;s=1;:::;Mandl=0;:::;N)]TJ /F5 11.955 Tf 11.95 0 Td[(1(l6=0ifk=s):ThefollowingPSLlowerboundisduetoWelch[ 51 ]:PSLNr M)]TJ /F5 11.955 Tf 11.96 0 Td[(1 2NM)]TJ /F3 11.955 Tf 11.95 0 Td[(M)]TJ /F5 11.955 Tf 11.96 0 Td[(14=BPSL: (4)TheISLmetric,usedpreviouslyinthesinglesequencecase(Eq.( 2 )inChapter 2 ),isdenedinthepresentmulti-sequencecaseas(also( 3 )inChapter 3 ):ISL=MXk=1N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xp=)]TJ /F7 7.97 Tf 6.59 0 Td[(N+1p6=0jrkk(p)j2+MXk=1MXs=1s6=kN)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xp=)]TJ /F7 7.97 Tf 6.59 0 Td[(N+1jrks(p)j2: (4)AlowerboundonISLwasimplicitlyderivedin[ 52 ].InthenextsectionwederiveboththeISLandPSLlowerboundsusingtheMulti-CANframework. 80

PAGE 81

4.1BoundDerivationItisshowninChapter 3 thattheISLmetricin( 4 )canbetransformedtothefrequencydomainas:ISL=1 2N2NXp=1)]TJ /F2 11.955 Tf 5.47 -9.68 Td[(kypk2)]TJ /F3 11.955 Tf 11.95 0 Td[(N2+(M)]TJ /F5 11.955 Tf 11.95 0 Td[(1)N2; (4)whereyp=266664y1(p)...yM(p)377775;yk(p)=NXn=1xk(n)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2 2N(p)]TJ /F6 7.97 Tf 6.58 0 Td[(1)(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1);k=1;:::;M: (4)Notethatfyk(p)g2Np=1istheDFT(discreteFouriertransform)ofthesequencefxk(n)gNn=1paddedwithNzerosinthetail.Inthissectionwestartfromthisfrequency-domainexpressionofISLtoderivealowerboundonit.Letzkp=jyk(p)j2.Thentheenergyconstraintin( 4 )isrelatedtofzkpgviatheParsevalequality:2NXp=1zkp=2NNXn=1jxk(n)j2=2N2;k=1;:::;M: (4)Expanding( 4 )andpluggingin( 4 ),weobtainISL=1 2N2NXp=1 MXk=1zkp!2)]TJ /F3 11.955 Tf 11.95 0 Td[(MN2: (4)MakinguseoftheCauchy-Schwartzinequalityleadstothefollowingresult:ISL=1 (2N)2"2NXp=112#242NXp=1 MXk=1zkp!235)]TJ /F3 11.955 Tf 11.95 0 Td[(MN21 4N2"2NXp=11 MXk=1zkp!#2)]TJ /F3 11.955 Tf 11.95 0 Td[(MN2 (4)=M2N2)]TJ /F3 11.955 Tf 11.95 0 Td[(MN2; (4) 81

PAGE 82

where( 4 )wasusedtoget( 4 )from( 4 ).TheaboveresultontheISLlowerboundissummarizedas:ISLN2M(M)]TJ /F5 11.955 Tf 11.95 0 Td[(1)4=BISL: (4)ThePSLlowerboundin( 4 )canbeeasilyobtainedfromBISLasacorollary.ItfollowsfromthedenitionofISLin( 4 )thatISL2M(N)]TJ /F5 11.955 Tf 11.95 0 Td[(1)PSL2+M(M)]TJ /F5 11.955 Tf 11.96 0 Td[(1)(2N)]TJ /F5 11.955 Tf 11.95 0 Td[(1)PSL2: (4)Substituting( 4 )in( 4 ),weobtainBPSLin( 4 ).Notethattheequalityin( 4 )holdsifandonlyifPMk=1zkp=cforallp=1;:::;2Nwherecisaconstant.Becauseoftheenergyconstraintin( 4 ),itiseasytoseethatc=NM.Inotherwords,asetofenergy-constrainedsequencesfxk(n)gmeettheISLlowerboundifandonlyiftheir2N-pointDFTvaluessatisfykypk2=NMforallp=1;:::;2N(Eq.( 4 )forthedenitionofkypk).Anexampleofsuchasequencesetisgivenin( 4 )below. 4.2ApproachingtheBoundAnaturalquestionarisesastowhetherwecangeneratesequencesetsthatachievethecorrelationlowerboundBISLorBPSL.HerewefocusontryingtomeetBISL.AtrivialsolutiontomeetingBISListhefollowingsequenceset(recallthattheenergyconstraintin( 4 )isalwaysimposed):xk(n)=8><>:p N;n=1;0;n=2;:::;N;k=1;:::;M; (4)whosecorrelationsidelobesareallzeroexceptforthezero-lagcross-correlationwhichisN.AsetofMsequencesleadstoM(M)]TJ /F5 11.955 Tf 12.71 0 Td[(1)pairsofcross-correlationsandthustheISLfortheabovesequencesetisexactlyequaltothelowerboundN2M(M)]TJ /F5 11.955 Tf 12.51 0 Td[(1).However,thesequencesetin( 4 )isnotpracticallyusefulbecauseitsPSLisashigh 82

PAGE 83

asthein-phaseauto-correlation.Moreover,transmitingonlyatonetimeinstantwhilekeepingsilentatallothertimes,asevidencedbythezerosforn=2;:::;Nin( 4 ),resultsinahigh(infact,themaximumpossible)peak-to-averagepowerratio(PAR),whichisonceagainundesirableinpractice.TheMulti-CANalgorithmintroducedinChapter 3 aimstondunimodularsequencesetswithlowISL.Theunimodularconstraintreferstothefactthateverysequenceelementhasunitmodulus,i.e.jxk(n)j=1.Inthiscasetheenergyconstraintin( 4 )isautomaticallysatised.Notethatunimodularsequencesareoftenpreferredinpracticeduetohardwarerestrictions,suchasusinganeconomicalnon-linearamplierwhichisessentiallyworkingwellonlywhenthePARis1orcloseto1.Althoughtheunimodularconstraintiscertainlymorestringentthantheenergyconstraint,theunimodularsequencesetsgeneratedbyMulti-CANhaveanISLthatisfairlyclosetoBISL,providedthatthereareatleasttwosequencesintheset(theM=1situationturnsouttobespecialandistakencareoflateron).Toillustrate,weshowtheISLofsequencesetsgeneratedbyMulti-CANandthecorrespondingBISLinTable 4-1 ,forvariouscombinationsofMandN.NotethattheMulti-CANalgorithmisrunfromarandominitialization,andthatdifferentrandominitializationsleadtodifferentsequencesetsbutwithsimilarlylowcorrelations.RegardingTable 4-1 ,onlyonesuchrealizationispresentedforeachpairof(M;N).ThegoodperformanceofMulti-CANsynthesizedunimodularsequencesets,comparedtoBISL,cannolongerbeguaranteedwhenM=1inwhichcaseBISL=0.NotethattheMulti-CANalgorithmbecomestheCANalgorithmwhenM=1.Hereafterinthissection,onlytheauto-correlationofasinglesequenceisconsidered.Forasequencefx(n)gNn=1withjx(n)j=1foralln,itholdsthatjr(N)]TJ /F5 11.955 Tf 12.45 0 Td[(1)j=jx(N)x(1)j=1andthusISL1.Hence,obviouslyBISLcannotbereachedbyusingunimodularsequences.ActuallytheISLofasinglesequencegeneratedbyCANismuchlargerthan1(e.g.,ontheorderof103whenN=200),althoughaCANsequencecanpossessmuchlower 83

PAGE 84

correlationsidelobesthanmanywell-knownunimodularsequencesintheliterature,suchastheGolomborFranksequence(Chapter 2 ).WeconsiderbelowrelaxingtheunimodularconstraintintheCANalgorithmsoastoobtainlowercorrelations.Moreprecisely,denethePARofthesequencex=x(1)x(N)Tas:PAR(x)=maxnjx(n)j2 1 NPNn=1jx(n)j2=maxnjx(n)j2; (4)wherethesecondequalityisduetotheenergyconstraint.TheCANalgorithmgeneratessequenceswithPAR=1.HereweextendittothemoregeneralcaseofPARwherecanbeanynumberbetween1andN.FollowingthediscussionsinChapter 2 ,theISLmetricin( 4 ),forthecaseofM=1,canbemadesmallbysolvingthefollowingminimizationproblem:minfx(n)gNn=1;f (p)g2Np=1f=kAHz)]TJ /F15 11.955 Tf 11.95 0 Td[(vk2=kz)]TJ /F15 11.955 Tf 11.96 0 Td[(Avk2 (4)s.t.kxk2=N;PAR(x);wherez=x(1)x(N)00T2N1; (4)v=1 p 2ej (1)ej (2N)T2N1;f (p)gareauxiliaryvariablesandAHisaunitary2N2NDFTmatrix(i.e.,AHxgivesthe2N-pointDFTofanyvectorxoflength2N).Notethat( 4 )wouldreducetotheproblemdiscussedinChapter 2 (Section 2.1 )ifitssecondconstraintwerereplacedbyPAR(x)=1. 84

PAGE 85

Theproblemin( 4 )canbesolvedinacyclicway.Werstxzandcomputethevthatminimizesf: (p)=argfthepthelementofAHzg;p=1;:::;2N: (4)Nextwexvandnotethattheminimizationproblemcanbecastasminxkx)]TJ /F15 11.955 Tf 11.95 0 Td[(sk2 (4)s.t.kxk2=N;PAR(x);wheresisanN1vectormadefromtherstNelementsofAv.Thenearest-vectorproblemin( 4 )hasbeenconsideredin[ 53 ];hereinwebrieyoutlineitssolution.Tobeginwith,notethatthesolutionto( 4 )withoutthePARconstraintisgivenby^x=p Ns=ksk.ThennotethatthePARconstraintisequivalenttomaxnjx(n)jp .Henceifthemagnitudesofallelementsin^xarebelowp ,then^xisasolution;ifnot,weresorttoarecursiveprocedureasfollows.Theelementinxcorrespondingtothelargestelement(intermsofmagnitude)ins,says,isgivenbyp expfjarg(s)g.TheotherN)]TJ /F5 11.955 Tf 12.74 0 Td[(1elementsinxareobtainedbysolvingthesameproblemasin( 4 ),exceptthatnowxandsare(N)]TJ /F5 11.955 Tf 12.86 0 Td[(1)1andthattheenergyconstraintiskxk2=N)]TJ /F3 11.955 Tf 12.46 0 Td[(.Sincethescalarcaseof( 4 )istrivial,sucharecursiveprocedureisguaranteedtoyieldasolution.Wereferthereadersto[ 53 ]formoredetails.Tosummarize,weiteratebetween( 4 )and( 4 )untilconvergence(forinstance,untilthenormofthedifferencebetweenthex'sobtainedintwoconsecutiveiterationsislessthanapredenedthreshold,e.g.10)]TJ /F6 7.97 Tf 6.58 0 Td[(3).Thecriterionin( 4 )isdecreasedineveryiterationstepsolocalconvergenceisguaranteed(i.e.theso-obtainedxisatleastalocalminimumsolutionto( 4 )).Theiterativeprocesscanbestartedfromarandomphaseinitializationofx,e.g.fx(n)=ej(n)gNn=1,whereeach(n)isdrawnindependentlyfromauniformdistributionover[0;2];suchan 85

PAGE 86

initializationisusedwheneverweconsiderrandominitializationbelow.Alternativelyxcanbeinitializedbyanygoodexistingsequence(goodmeaningthatthesequenceitselfalreadyhasrelativelylowcorrelations),e.g.,theP4sequence.TheresultingalgorithmisstillnamedCANinviewofthefactthattheCANalgorithmproposedinChapter 2 isjustaspecialcaseof( 4 )(correspondingtoPAR=1)andsurelyanimportantone;noambiguitywillbeintroducedbyusingthisnamesincehereafterinthisChapterwewillspecifythePARvaluewheneverweapplyCAN.ConsidernextusingCANtogenerateasequenceoflengthN=512withenergyN.Figure 4-1A showstheauto-correlations(normalizedbyNandindB)oftwoCANsequences,onewithPAR=1andtheotherwithPAR=4,bothinitializedbyarandomlygeneratedsequence.Figure 4-1B isforthesamesettingas 4-1A exceptthattheP4sequencewasusedtoinitializetheCANalgorithm.Clearlyplaysanimportantrole:alargerleadstosignicantlylowercorrelationsidelobelevels.(Notethatwedonotplot,forcomparison,thecorrelationsoftheP4orotherwell-knownsequencessuchasGolomborFrank,becausetheyhavehighercorrelationsidelobesthantheCANsequencewithPAR=1;seeChapter 2 forexamples.)Figure 4-2 illustratestheISLofaCANsequencewithlengthN=512andrangingfrom1to10.Asbefore,weuseeitherarandomlygeneratedsequenceortheP4sequencetoinitializeCAN.TheP4initializationgiveslowerISLthantherandominitialization.Interestingly,whenisrelativelysmall,thedecreaseofISLcausedbyevenasmallincreaseofissignicant.NotethatinthecaseofP4initialization,theISLcanbedecreasedbymorethan2ordersofmagnitudeifisincreasedjustfrom1to1:2.However,afterreachingacertainpoint,theincreaseofdoesnotpushISLanylower.TheISLoftheCANsequenceinitializedbyP4when=4is5:38,avaluerelativelyclosetotheISLlowerboundofBISL=0.AfullexplanationisstilllackingastowhytheISLoftheCANsequencedoesnotgotozerowhenissufcientlylarge,thoughthepossibletrappingofthealgorithminlocalminimaisalikelyreason. 86

PAGE 87

A B Figure4-1.Auto-correlationsoftwoCANsequenceswithdifferentPARs.A)Theauto-correlations(normalizedbyNandshownindB)oftwoCANsequencesoflengthN=512,onewithPAR=1andtheotherwithPAR=4,bothinitializedbyarandomlygeneratedsequence.B)ThesameasAexceptthattheP4sequenceisusedtoinitializetheCANalgorithm. 87

PAGE 88

Table4-1.BISLvs.ISLofMulti-CANsequencesets ISLBISL M=2;N=20080013.780000M=2;N=512524385.8524288M=4;N=5123145752.23145728M=4;N=100012000044.812000000 Figure4-2.TheISLofCANsequences(withlengthN=512andinitializedeitherrandomlyorbyP4)vs.. 88

PAGE 89

CHAPTER5AMBIGUITYFUNCTIONTheambiguityfunction(AF)showstheresponseofamatchedltertoasignalwithvarioustimedelaysandDopplerfrequencyshifts.Letu(t)denoteaprobingsignalwithtimesupport[0;T](i.e.,u(t)isassumedtobezerooutside[0;T]).The(continuous-time)AFofu(t)isdenedas(;f)=Z1u(t)u(t)]TJ /F3 11.955 Tf 11.96 0 Td[()e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2f(t)]TJ /F7 7.97 Tf 6.59 0 Td[()dt (5)whereisthetimedelayandistheDopplerfrequencyshift.Thereexistsanextensiveliteratureonradarambiguityfunctions,suchas[ 13 17 20 54 59 ]. 5.1AFPropertiesFigure 5-1 showstheambiguityfunctionofachirpsignal(Eq.( 1 )inChapter 1 )withparametersT=10sandB=5Hz.Inthegurenotethattheabsolutevalueof(;f)isnormalizedsothatthepeakvalueattheoriginis1,thatthedelayisnormalizedbyTandthattheDopplershiftfisnormalizedby1=T.SuchnormalizationswillalsobeusedinmostotherAFplotstoprovideaconsistentscaling.TwooftheambiguityfunctionfeaturescanbeeasilyobservedfromFigure 5-1 .Therstfeatureisthatthemaximumvalueofj(;f)jisachievedbyj(0;0)j,whichinfactequalstheenergyofu(t).Theotheroneisthesymmetrywithrespecttotheorigin,i.e.,j(;f)j=j()]TJ /F3 11.955 Tf 9.3 0 Td[(;)]TJ /F3 11.955 Tf 9.3 0 Td[(f)j,sothatitsufcestoshowtheambiguityfunctiononlyforhalfofthe(;f)plane(aswasdoneinFigure 5-1A ).AnotherprominentfeatureofAF,whichislessobviousthanthetwomentionedabove,istheconstantvolumeproperty:Z1Z1j(;f)j2ddf=E2 (5)whereE=Z1ju(t)j2dt (5) 89

PAGE 90

istheenergyofu(t).SeeAppendix C forproofsofthethreepropertiesmentionedabove.TheAFofthechirpsignalillustratedinFig. 5-1 hastheDoppler-toleranceproperty,inthesensethatamismatchintheDopplerfrequencycanstillleadtoapeakinthematched-lteringprocessalthoughatthecostofatime-delayestimationerror.ThereforeevenifabankofltersmatchedtodifferentDopplerfrequenciesisnotavailableatthereceiverend,targetswithunknownDopplershiftsmaystillbedetectedduetothisdelay-Dopplercouplingproperty.AsdiscussedinSection 1.3 ofChapter 1 ,manysequencessuchastheGolombandFranksequencescanbederivedfromthechirpsignal.Notsurprisingly,theirambiguityfunctionsinherittheDoppler-toleranceproperty.Figure 5-2 showstheAFofthewaveformin( 1 )whenfx(n)gisalength-50Golombsequenceandpn(t)istherectangularshapingpulse.(LateronwhenwerefertotheAFofasequence,weimplicitlymeantheAFoftheunderlyingcodedwaveform.)ComparingFig. 5-1 andFig. 5-2 ,itisinterestingtoobservehowpartoftheAFvolumeinthecentralridgeismovedtotheedgeinFig. 5-2 (thetotalvolumebeingconstant,see( 5 )).AsdiscussedinChapter 2 ,theCANalgorithmcanbeusedtodesignsequenceswithlowcorrelationsidelobes.Theauto-correlationfunctionisnothingbutthezero-DopplercutofAF.TheAFoftwolength-50CANsequencesareshowninFig. 5-3 .ForFig. 5-3A arandomlygeneratedsequenceisusedtoinitializetheCANalgorithmwhileforFig. 5-3B theGolombsequenceofthesamelengthisusedforinitialization.Thewhitestripeatzero-Dopplerfrequencyindicatesthelowcorrelationsidelobes.WeobservethattheAFinFig. 5-3A isthumbtack-shapedwhichleadstohighresolutionforbothdelayandDopplerestimation.WealsoobservethattheAFinFig. 5-3B exhibitsDopplertoleranceduetotheinitializationbytheGolombsequence.Afactworthnotingisthatthezero-DelaycutofAFistheFouriertransformofu(t)u(t).Forthewaveformsdiscussedabovethathaveunitmodulus,thezero-Delaycut 90

PAGE 91

ofAFcanbeeasilycalculatedas(0;f)=Z1u(t)u(t)e)]TJ /F7 7.97 Tf 6.58 0 Td[(j2ftdt=ZT0e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2ftdt (5)=1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F7 7.97 Tf 6.58 0 Td[(j2fT j2f=e)]TJ /F7 7.97 Tf 6.59 0 Td[(jfTTsinc(fT)wheresinc(x)=sin(x)=x.Thereforej(0;f)j=jTsinc(fT)jregardlessofu(t)(providedthatu(t)isunimodular).Sincesinc(fT)diesoutquicklyasfincreases,thezero-DelaycutappearsasaverticalstripeofsmallvaluesinFig. 5-3 ,aswellasinFigs. 5-1 and 5-2 (thoughsomewhatlessobvious).Wehaveillustratedtheambiguityfunctionsofseveralwaveforms.Howtorealizeadesiredambiguityfunctionusingpracticalsignalshasbeenaclassicalprobleminthewaveformdesignareaandthereexistsaconsiderableliteratureonthistopic[ 13 17 20 55 56 58 60 61 ].Despitethisextensiveliterature,apparentlythereisnouniversalmethodthatcansynthesizeanarbitraryambiguityfunction.Infact,matchingonlythezero-Dopplercutofanambiguityfunctionorminimizingthesidelobesoftheauto-correlationfunctionisadifcultprobleminitself,asdiscussedinChapters 2 and 3 .Inwhatfollows,weintroducethediscrete-AFconceptandshowthatthesidelobesofadiscrete-AFcanbesuppressedinaregionclosetotheorigin. 5.2Discrete-AFWerestraintheattentiontobasebandwaveformsmodulatedinthefollowingway(e.g.,( 1 )inChapter 1 ):u(t)=NXn=1x(n)pn(t);0tT (5)wherefx(n)gNn=1isthemodulatingcodesequencethatistobedesigned(assumedtobezerowhenn=2[1;N])andpn(t)=8><>:1 p tp;(n)]TJ /F5 11.955 Tf 11.96 0 Td[(1)tptntp0;elsewhere; (5) 91

PAGE 92

isanidealrectangularshapingpulseoftimelengthtp(andthusT=Ntp).Theenergyoffx(n)gNn=1isconstrainedtobeN,i.e.,NXn=1jx(n)j2=N: (5)Aspointedoutseveraltimesbefore,itisusuallydesiredtohaveaunimodularsequence:x(n)=ejn;n=1;:::;N; (5)wherefngarethephases.Forthewaveformdenedin( 5 ),theambiguityfunction( 5 )canbesimpliedasfollows.Substitute( 5 )into( 5 )toobtain(;f)=ZT0 NXn=1x(n)pn(t)! NXm=1x(m)pm(t)]TJ /F3 11.955 Tf 11.96 0 Td[()!e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2f(t)]TJ /F7 7.97 Tf 6.59 0 Td[()dt (5)=NXm=1NXn=1x(m)ZT0pn(t)pm(t)]TJ /F3 11.955 Tf 11.96 0 Td[()e)]TJ /F7 7.97 Tf 6.58 0 Td[(j2f(t)]TJ /F7 7.97 Tf 6.58 0 Td[()dtx(n):Considerthetimegridft=ktpg(k=)]TJ /F3 11.955 Tf 9.3 0 Td[(N+1;:::;0;:::;N)]TJ /F5 11.955 Tf 12.03 0 Td[(1)whosepointsareintegermultiplesofthesubpulselengthtp.Itisnotdifculttocalculate(;f)at=ktp:(ktp;f)=NXm=1NXn=1x(m) Zntp(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)tpjpn(t)j2e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2f(t)]TJ /F7 7.97 Tf 6.58 0 Td[(ktp)dt!m+k;nx(n) (5)=ejftpsin(ftp) ftpNXn=1x(n)x(n)]TJ /F3 11.955 Tf 11.95 0 Td[(k)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2ftp(n)]TJ /F7 7.97 Tf 6.58 0 Td[(k): (5)Notethat(ktp;f)equalszeroifkliesoutside[)]TJ /F3 11.955 Tf 9.3 0 Td[(N+1;N)]TJ /F5 11.955 Tf 11.93 0 Td[(1].Similarlytothetimegrid,weconsiderff=p Ntpgforanintegerp.Thenweobtain(ktp;p Ntp)=ejp Nsinc(p N)r(k;p) (5)wheresinc(x)=sin(x)=xandr(k;p)iswhatwecallthediscrete-AF:r(k;p)=NXn=1x(n)x(n)]TJ /F3 11.955 Tf 11.96 0 Td[(k)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2(n)]TJ /F12 5.978 Tf 5.76 0 Td[(k)p N; (5) 92

PAGE 93

k=)]TJ /F3 11.955 Tf 11.95 0 Td[(N+1;:::;N)]TJ /F5 11.955 Tf 11.95 0 Td[(1;p=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(N 2;:::;N 2)]TJ /F5 11.955 Tf 11.95 0 Td[(1:Therangeofpischosenas)]TJ /F7 7.97 Tf 10.49 4.71 Td[(N 2;:::;N 2)]TJ /F5 11.955 Tf 10.76 0 Td[(1becauseitcorrespondstothelargestDopplerfrequencyrangethatcanbeunambiguouslyidentied(notethatthebandwidthofu(t)isapproximatelyequalto1=tp.).Alsonotethatwithoutlossofgenerality,wehaveimplicitlyassumedandwillassumehereafteranevenNinthischapter.AnoddNleadstotherangep=)]TJ /F7 7.97 Tf 10.5 4.71 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1 2;:::;N)]TJ /F6 7.97 Tf 6.59 0 Td[(1 2,whichwillbringlittledifferenceinthediscussionsbelow.Inpractice,theDopplerfrequencyfisusuallymuchsmallerthanthebandwidthoftheprobingwaveform.Forexample,assuminganX-bandradaroperatingatthewavelength=3cm,aghterjetmovingatthespeedofMach3(v=1020m/s)canonlyinduceaDopplerfrequencyoff=2v =21020m/s 0:03m=68kHz; (5)whichismuchsmallerthanthebandwidththatisontheorderofmanyMHz.Asanotherexample,forasonaroperatingatthefrequencyf=20kHz(correspondingtoawavelengthof1500=20=75mm),afast-movingsubmarineatthespeedof25knots(v=13m/s)inducesaDopplerfrequencyoff=2v =213m/s 0:075m=346Hz; (5)whichcanalsobeconsideredtobeverysmallcomparedwiththeseveral-kHzbandwidththatisnormallyused.Therefore,itissafetoconneourattentiontovaluesofjpjN,inwhichcasesinc(p N)1andthusj(ktp;p Ntp)jjr(k;p)j;k=)]TJ /F3 11.955 Tf 9.29 0 Td[(N+1;:::;N)]TJ /F5 11.955 Tf 11.95 0 Td[(1;jpjN: (5)SincealmostalwaysonlythemagnitudeofAFmattersintargetdetectionapplications,( 5 )showsthatitsufcestoconsiderthediscrete-AFasdenedin( 5 ). 93

PAGE 94

Thewell-knownAFpropertiesofsymmetryandconstantvolumealsoholdforthediscretizedversionin( 5 ):jr(k;p)j=jr()]TJ /F3 11.955 Tf 9.3 0 Td[(k;)]TJ /F3 11.955 Tf 9.3 0 Td[(p)j; (5)N+1Xk=)]TJ /F7 7.97 Tf 6.59 0 Td[(N+1N=2)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xp=)]TJ /F7 7.97 Tf 6.59 0 Td[(N=2jr(k;p)j2=N3: (5)Inaddition,r(0;0)isalwaysequaltoNbecauseoftheenergyconstraintin( 5 );werefertor(k;p)fork6=0orp6=0asthesidelobes.Notethattheabovepropertiesfollowdirectlyfromthediscrete-AFdenitionin( 5 )inspiteoftheapproximaterelationshipwiththeoriginalAFasshownin( 5 ).Inthenextsectionwewillbemainlyconcernedwithdesigningthesequencefx(n)gNn=1soastominimizethesidelobesofthediscrete-AFinacertainregion:minfx(n)gC1=Xk2KXp2Pjr(k;p)j2 (5)whereKandParetheindexsetsthatspecifytheregionofinterest.Becausethetotalvolumeofr(k;p)isxed,( 5 )ismeaningfulonlywhenKandParestrictlysubsetsoff)]TJ /F3 11.955 Tf 15.28 0 Td[(N+1;:::;N)]TJ /F5 11.955 Tf 11.95 0 Td[(1gandf)]TJ /F3 11.955 Tf 15.28 0 Td[(N=2;:::;N=2)]TJ /F5 11.955 Tf 11.96 0 Td[(1grespectively. 5.3MinimizingtheDiscrete-AFSidelobesFollowingthenotationspeciedin( 5 ),assumethatthetimedelaysetofinterestisgivenbyK=f0;1;:::;(K)]TJ /F5 11.955 Tf 12.32 0 Td[(1)gandthattheDopplerfrequencysetofinterestisgivenbyP=f0;1;:::;(P)]TJ /F5 11.955 Tf 11.95 0 Td[(1)g.DeneasetofPsequencesfxm(n)gPm=1asfollows:fx1(n)=x(n)gNn=1 (5)fx2(n)=x(n)ej2n NgNn=1...fxP(n)=x(n)ej2n(P)]TJ /F14 5.978 Tf 5.76 0 Td[(1) NgNn=1: 94

PAGE 95

Notethatfxm(n)gPm=1arezerowhenn=2[1;N].Letfrml(k)gdenotethecorrelationbetweenfxm(n)gandfxl(n)g:rml(k)=NXn=1xm(n)xl(n)]TJ /F3 11.955 Tf 11.95 0 Td[(k) (5)=ej2(m)]TJ /F14 5.978 Tf 5.76 0 Td[(1)k NNXn=1x(n)x(n)]TJ /F3 11.955 Tf 11.95 0 Td[(k)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2(n)]TJ /F12 5.978 Tf 5.76 0 Td[(k)(l)]TJ /F12 5.978 Tf 5.75 0 Td[(m) N;k2Km;l=1;:::;P:Itisstraightforwardtoverifythatallvaluesoffjr(k;p)jg(k2Kandp2P)arecontainedinthesetfjrml(k)jg(k2Kandm;l=1;:::;P).Interestingly,mandldonotneedtoincreasestepwisefrom1toP.Forexample,fjrml(k)jg(m;l=1;2;5;7)alreadycoversallvaluesoffjr(k;p)jg(p=0;:::;6)(afactwhichbearsresemblancetotheminimumredundancylineararray[ 62 ]).Thisobservationsavescomputationbutdoesnotimprovethealgorithmperformance,so( 5 )willbeusedform;l=1;:::;Ptokeepthenotationsimple.Theforegoingdiscussionimpliesthatbyminimizingthecorrelationsofthesequencesetin( 5 ),weequivalentlyminimizethediscrete-AFsidelobes,i.e.,thecriterionC1in( 5 ).TheMulti-CAOalgorithmdiscussedinChapter 3 whichwasusedtodesignsetsofwaveformswithgoodcorrelations,canbeadaptedtotheproblemofminimizingC1asexplainedinwhatfollows.DeneX=X1XP(N+K)]TJ /F6 7.97 Tf 6.59 0 Td[(1)KP (5) 95

PAGE 96

whereXm=2666666666666664xm(1)0.........xm(1)xm(N).........0xm(N)3777777777777775(N+K)]TJ /F6 7.97 Tf 6.59 0 Td[(1)K;m=1;:::;P (5)andfxm(n)garedenedin( 5 ).Itisnotdifculttoseethatallfrml(k)g(k2Kandm;l=1;:::;P)appearinthematrixXHX.AlsonotethatthediagonalelementsofXHXareequaltoNbecauseoftheenergyconstraintin( 5 ).Therefore,thecorrelationsofthesequencesetin( 5 )canbemadesmallthroughminimizingthefollowingcriterion:^C1=XHX)]TJ /F3 11.955 Tf 11.96 0 Td[(NIKP2: (5)Notethatthecriterion^C1equalszeroifthematrixXisasemi-unitarymatrixscaledbyp N,anobservationwhichleadstothefollowingminimizationproblemthathasasimplerformthan( 5 ):minX;UkX)]TJ 11.95 10.7 Td[(p NUk2; (5)s.t.jx(n)j=1;n=1;:::;Nxm(n)=x(n)ej2n(m)]TJ /F14 5.978 Tf 5.75 0 Td[(1) N;m=1;:::;P;n=1;:::;NUHU=I(Uis(N+K)]TJ /F5 11.955 Tf 11.96 0 Td[(1)KP):Theminimizationproblemin( 5 )canbesolvedbythecyclicalgorithmdescribedbelow: Step0:Randomlyinitializethesequencefx(n)g. Step1:ForxedX,theminimizerUisgivenby(Section 3.3 ofChapter 3 )U=U2UH1 (5) 96

PAGE 97

wherethematricesU1(KPKP)andU2((N+K)]TJ /F5 11.955 Tf 12.43 0 Td[(1)KP)comefromtheeconomicSVDofXH:XH=U1UH2. Step2:ForxedU,thecriterionin( 5 )canbewrittenas(notetheunit-modulusconstraintin( 5 ))kX)]TJ 11.95 10.71 Td[(p NUk2=NXn=1KPXl=1jnlx(n))]TJ /F3 11.955 Tf 11.95 0 Td[(fnlj2=const)]TJ /F5 11.955 Tf 11.95 0 Td[(2NXn=1Re" KPXl=1nlfnl!x(n)# (5)whereconstisaconstantthatdoesnotdependonfx(n)g,fnlgaregivenbytheelementsofXthatcontainx(n):[n1n;KP]="11| {z }Kej2n Nej2n N| {z }Kej2n(P)]TJ /F14 5.978 Tf 5.76 0 Td[(1) Nej2n(P)]TJ /F14 5.978 Tf 5.76 0 Td[(1) N| {z }K#1KP (5)andffnlgaregivenbytheelementsofp NUwhosepositionsarethesameasthoseoffnlginX.Theminimizerx(n)(moreexactlyitsphase)isobtainedimmediately:n=arg KPXl=1nlfnl!;n=1;:::;N: (5) Iteration:RepeatSteps1and2untilconvergence.Asanexample,considerascenariowithN=100;K=10andP=3.Weusethecyclicalgorithmoutlinedabovetodesignaunimodularsequencefx(n)gNn=1.Theso-obtaineddiscrete-AF,jr(k;p)j,isshowninFig. 5-4 .Thewhiteareainthecenterindicatesthatthesidelobesneartheoriginweresuccessfullysuppressed. 97

PAGE 98

A B Figure5-1.TheAFofachirpsignalwithT=10sandB=5Hz.A)3DplotofthepositiveDopplerplaneandB)2Dplotofthewholeplane. 98

PAGE 99

A B Figure5-2.TheAFofalength-50Golombsequence.A)3DplotofthepositiveDopplerplaneandB)2Dplotofthewholeplane. 99

PAGE 100

A B Figure5-3.TheAFofalength-50CANsequence.A)ArandomlygeneratedsequencewasusedtoinitializeCANandB)theGolombsequencewasusedtoinitializeCAN. 100

PAGE 101

Figure5-4.Thesynthesizeddiscrete-AF:jr(k;p)j. 101

PAGE 102

CHAPTER6STOPBANDCONSTRAINTAmongthetasksassociatedwithcognitiveradar[ 63 ],animportantoneistoadaptthespectrumoftransmittedwaveformstothechangingenvironment.Inparticular,thetransmittedsignalshouldnotusecertainfrequencybandsthathavealreadybeenreserved,suchasthebandsfornavigationandmilitarycommunications;ortherecouldexiststrongemitterswhoseoperatingfrequenciesshouldbeavoided.Thereforeitisrequiredthatthespectralpoweroftransmittedwaveformsbesmallforcertainfrequencybands[ 64 67 ].Themainfocusinthischapterisondesigningadiscretesequencewhosespectralpowerissmallincertainspeciedfrequencybands.Thedesignedsequencecanbeusedinactivesensingsystemslikeradar/sonarasaprobingsequence.ItcanalsobeusedasaspreadingsequenceinspreadspectrumapplicationssuchasaCDMA(codedivisionmultipleaccess)system.Besidesfrequencynotching,wealsotakeintoaccountthecorrelationpropertiesofthedesignedsequence.Aspointedoutseveraltimesinpreviouschapters(e.g.,inChapter 1 ),inradar/sonarapplicationslowauto-correlationoftheprobingsequenceimprovestargetdetectionwhenrangecompressionisappliedinthereceiver.Furthermore,practicalhardwarecomponentssuchasanalog-to-digitalconvertersandpowerampliershavemaximumsignalamplitudeclip.Inordertomaximizethetransmittedpowerthatisavailableinthesystem,unimodularsequencesaredesired.InthischapterweproposeanalgorithmnamedSCAN(stopbandCAN)forunimodulartransmitsequencedesign.SCANisanextensionoftheCANalgorithmintroducedinChapter 2 .CANaimsatgeneratingunimodularsequenceswithlowcorrelationsidelobes.SCANextendsCANinsuchawaythatbothfrequencystopbandsandcorrelationsidelobesareconsidered.TheSCANalgorithmiscomputationallyefcient(asisbasedonFFToperations)andthusitfacilitateslongsequencegeneration 102

PAGE 103

andpossiblyreal-timewaveformupdate.AnotheradvantageofSCANisthatthealgorithmcanstartfromrandominitializationsandthatdifferentinitializationsleadtodifferentsequences,butallwithsimilarlygoodproperties.TheproblemformulationandtheSCANalgorithmarepresentedinSection 6.1 .AvariationoftheSCANalgorithmnamedWeSCAN,whichhasmoreexibilityincontrollingthecorrelationlevels(butatthecostofincreasedcomputation),isdiscussedinSection 6.2 .SeveralsimulationresultsareshowninSection 6.3 6.1StopbandCAN(SCAN)Werstformulatethetwodesigncriteriathatarerelated,respectively,tospectralbandsuppressionandcorrelationsidelobesuppression.Withoutlossofgenerality,onlynormalizedfrequencies(from0to1Hz)areconsideredfornotationalsimplicity.Supposethatthesetoffrequencieswhichfx(n)gNn=1shouldavoidcanbeexpressedas=Ns[k=1(fk1;fk2) (6)where(fk1;fk2)identiesonestopbandandNsisthenumberofstopbands.Correspondingto,wechooseanumber~NthatislargeenoughsothatpointsoftheDFT(discreteFouriertransform)frequencygridfp=~Ng~N)]TJ /F6 7.97 Tf 6.59 0 Td[(1p=0coverdensely.LetF~Ndenotethe~N~NDFTmatrixwhose(k;l)thelementisgivenby[F~N]kl=1 p ~Nexpj2kl ~N;k;l=0;:::;~N)]TJ /F5 11.955 Tf 11.96 0 Td[(1 (6)wherethecoefcient1=p ~NmakesF~Nunitary.WeformamatrixSfromthecolumnsofF~Ncorrespondingtothefrequenciesin.Forexample,if=[0:2;0:3]Hzandwechoose~N=100,Swillbethe10011submatrixofF~Ncomprisingits20thto30thcolumns(indexedfrom0).AfterconstructingS,letGdenotethematrixcomprisingtheremainingcolumnsinF~N. 103

PAGE 104

Itfollowsfromtheabovediscussionthatwecansuppressthespectralpoweroffx(n)ginbyminimizingthefollowingcriterion:kSH~xk2 (6)where~x="x(1)x(N)00| {z }~N)]TJ /F7 7.97 Tf 6.59 0 Td[(N#T: (6)Observethat( 6 )wouldbecomezeroif~xliedinthenullspaceofSH.SincethenullspaceofSHisspannedbythecolumnsofG,theproblemofminimizing( 6 )canbeequivalentlyformulatedasminx;J1(x;)=k~x)]TJ /F15 11.955 Tf 11.96 0 Td[(Gk2 (6)s.t.jx(n)j=1;n=1;:::;Nwherex=x(1)x(N)T,isanauxiliaryvariableands.t.isshortforsubjectto.TheproblemofsuppressingthecorrelationsidelobescanbedealtwithusingtheCANalgorithmformulationdiscussedinChapter 2 .AsshowninSection 2.1 ofChapter 2 ,thecorrelationsidelobescanbesuppressedbysolvingthefollowingproblem:minx;vJ2(x;v)=FH2N264x0N1375)]TJ /F15 11.955 Tf 11.96 0 Td[(v2 (6)s.t.jx(n)j=1;n=1;:::;Njvnj=1 p 2;n=1;:::;2NwhereF2Nisthe2N2NFFTmatrix,x=[x(1):::x(N)]andvisanauxiliaryvariable. 104

PAGE 105

Combining( 6 )and( 6 ),weobtainthefollowingminimizationproblemthatincorporatesbothfrequencystopbandandcorrelationsidelobeconstraints:minx;;vJ(x;;v)=k~x)]TJ /F15 11.955 Tf 11.96 0 Td[(Gk2+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()FH2N264x0N1375)]TJ /F15 11.955 Tf 11.96 0 Td[(v2 (6)s.t.jx(n)j=1;n=1;:::;Njvnj=1 p 2;n=1;:::;2Nwhere01controlstherelativeweightonthetwopenaltyfunctionsJ1andJ2.NotethattherearethreevariablesinthecriterionJ(x;;v).WeminimizeJ(x;;v)withrespecttoonlyonevariableatatimeandtheniterate.Theiterationstepsaresummarizedbelow: Step0:Initializefx(n)gNn=1witharandomlygeneratedunimodularsequence. Step1:Forxedxandv,J(x;;v)isaconvexquadraticfunctionof.Bysetting@J @=0andusingGHG=I,wegettheminimizer:=GH~x: (6)v=1 p 2exp)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(jargFH2N[xT01N]T: (6) Step3:Forxedandv,JcanbewrittenasJ=const.)]TJ /F5 11.955 Tf 11.95 0 Td[(2Re[xH(c1+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()c2)] (6)wherec1=therstNelementsofG (6)andc2=therstNelementsofF2Nv: (6)Thentheminimizerxisgivenbyx=exp(jargfc1+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()c2g): (6) 105

PAGE 106

Iteration:Repeat( 6 ),( 6 )and( 6 )untilconvergence(forinstance,untilthenormofthedifferencebetweenthex'sobtainedintwoconsecutiveiterationsislessthanapredenedthreshold,e.g.10)]TJ /F6 7.97 Tf 6.59 0 Td[(3).TheresultingalgorithmisnamedSCAN(stopbandCAN),asitisanextensionoftheCANalgorithmdiscussedinChapter 2 .Notethatallmatrixoperationsinvolvedintheupdatingformulaeof( 6 ),( 6 )and( 6 )canbedoneviaFFT.ThereforetheSCANalgorithmiscomputationallyefcient.Indeed,itcanbeusedtogeneratesequencesoflengthuptoN106.Beforeproceedingtothenextsection,wepointoutthatamoregeneralconstraintthanunimodularityistoconstrainthepeak-to-averagepowerratio(PAR)ofthetransmittedsequence(Eq.( 4 )inChapter 4 ).IfaPARlargerthan1isallowed,theSCANalgorithmcanbekeptasiswiththeexceptionthattheminimizerxin( 6 )isnowgivenbythesolutiontothefollowingproblem:minxkx)]TJ /F5 11.955 Tf 11.96 0 Td[([c1+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()c2]k2 (6)s.t.PAR(x)where1NistheprescribedlargestallowablePAR.Thenearest-vectorproblemin( 6 )hasalreadybeendiscussedinSection 4.2 ofChapter 4 towhichwereferfordetails.IntheexamplesofSection 6.3 ,however,wewillusemostlytheunimodularconstraintunlessotherwisestated. 6.2WeightedStopbandCyclicAlgorithm-New(WeSCAN)InthissectionwepresenttheWeSCAN(weightedSCAN)algorithmwhichcanbeviewedasanextensionofbothSCANandWeCAN.ForSCAN,minimizingthefunctionJ2in( 6 )isawayofminimizingtheintegratedsidelobelevel(ISL)metric(Eq.( 2 )inChapter 2 ):ISL=2N)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xk=1jr(k)j2: (6) 106

PAGE 107

ThefollowingmoregeneralWISL(weightedISL)metricassociatesaweight2kwitheachcorrelationtermr(k)(Eq.( 2 )inChapter 2 ):WISL=2N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=12kjr(k)j2: (6)Theweightsfkgcanbechosentosatisfyourneeds.Forexample,wecanset1=0;2=0andk=1fork=3;:::;N)]TJ /F5 11.955 Tf 12.65 0 Td[(1totradethecorrelationmainlobewidthforsidelobesuppression.Denetheweightingmatrix)]TJ /F5 11.955 Tf 11.4 0 Td[(=1 026666666401N)]TJ /F6 7.97 Tf 6.59 0 Td[(11...N)]TJ /F6 7.97 Tf 6.59 0 Td[(2.........N)]TJ /F6 7.97 Tf 6.59 0 Td[(10377777775 (6)where0>0islargeenoughtomake)]TJ /F1 11.955 Tf 11.41 0 Td[(positivesemi-denite.LetDbeasquarerootof)]TJ /F1 11.955 Tf 11.4 0 Td[(andletdkldenotethe(k;l)thelementofD.ThenitisshowninSection 2.2 ofChapter 2 thattheWISLmetriccanbeminimizedbysolvingthefollowingproblem:minx;fpg2Np=1~J2(x;V)=kFH2NZ)]TJ /F15 11.955 Tf 11.96 0 Td[(Vk2 (6)s.t.jx(n)j=1;n=1;:::;Nkpk2=1;p=1;:::;2NwhereZ=z1z2zN2NN; (6)zk=dk1x(1)dkNx(N)00T2N1;k=1;:::;NandV=1 p 2122NT(2NN): (6) 107

PAGE 108

WereplacethepenaltyfunctionJ2in( 6 )bythe~J2in( 6 ),followalldiscussionsafter( 6 )inSection 6.1 andmakethenecessarychangesthatarestraightforward.TheresultingalgorithmisnamedWeSCAN(weightedSCAN).WewillshowinSection 6.3 thatcomparedtoSCAN,WeSCANisabletogeneratesequenceswithmuchbetterfrequencystopbandsuppression,forinstanceatthecostofanincreasedcorrelationmainlobewidth.Toconcludethissection,weremarkonthefactthatthefrequencystopbandpenaltyfunction(e.g.,in( 6 ))canbeformulatedusingcontinuousfrequencies[ 64 ].LetX(f)=NXn=1x(n)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2f(n)]TJ /F6 7.97 Tf 6.58 0 Td[(1);f2[0;1] (6)bethediscrete-timeFouriertransformoffx(n)gNn=1.Thefrequencystopbandisstillgivenby( 6 ).ThespectralpowerinthekthbandiscalculatedasZfk2fk1jX(f)j2df=Zfk2fk1NXn=1x(n)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2f(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)2df (6)=NXn=1NXm=1x(m)Zfk2fk1ej2f(m)]TJ /F7 7.97 Tf 6.59 0 Td[(n)dfx(n);andthestopbandcriterion~J1isdenedtobethesummationof( 6 )overthestopbands:~J1=NsXk=1Zfk2fk1jX(f)j2df (6)=NXn=1NXm=1x(m)"NsXk=1Zfk2fk1ej2f(m)]TJ /F7 7.97 Tf 6.59 0 Td[(n)df#x(n):LettingRbeanNNmatrixwhose(m;n)thelementisgivenbyRmn=NsXk=1Zfk2fk1ej2f(m)]TJ /F7 7.97 Tf 6.59 0 Td[(n)df (6)=NsXk=18><>:ej2fk2(m)]TJ /F7 7.97 Tf 6.59 0 Td[(n))]TJ /F3 11.955 Tf 11.95 0 Td[(ej2fk1(m)]TJ /F7 7.97 Tf 6.59 0 Td[(n) j2(m)]TJ /F3 11.955 Tf 11.96 0 Td[(n);m6=nfk2)]TJ /F3 11.955 Tf 11.95 0 Td[(fk1;m=n 108

PAGE 109

leadstothefollowingexpressionfor~J1:~J1=xHRx: (6)IngeneralRwillberankdecient[ 38 ].Supposethatrank(R)=^N
PAGE 110

calculatedasPstop=10log10maxkjy(k)j2fork)]TJ /F5 11.955 Tf 11.96 0 Td[(1 ~N2: (6)Forthisexample,therangeofkin( 6 )isfrom201to300(notethatkcorrespondstothefrequency(k)]TJ /F5 11.955 Tf 11.95 0 Td[(1)=~N).Thespectralpower(i.e.thenormalizedfjy(k)j2g)andcorrelationlevel(fjr(k)j=Ng)ofthegeneratedsequenceareshowninFigs. 6-1A and 6-1B ,respectively.Correspondingtothetwogures,wehavePstop=)]TJ /F5 11.955 Tf 9.3 0 Td[(8:3dBandPcorr=)]TJ /F5 11.955 Tf 9.3 0 Td[(19:2dB.Toillustratehowaffectstheperformance,weincreasefrom0:1to1andplotthevaluesofPstopandPcorrinFig. 6-2 (otherparametersarekeptthesameasbefore).Wedonotshowtheresultsforlessthan0:1becausesuchasmallleadstoverylittlestopbandsuppression.FromFig. 6-2 itiseasytoseethatalargergivesmoreweighttothestopbandpenaltyfunctionandthusresultsinasmallerPstopatthecostofanincreasedPcorr.Remark:ThecurveofPcorrinFig. 6-2 doesnotincreasestrictlymonotonicallyasincreases,nordoesPstopdecreasemonotonically.ThereasonisthattheSCANalgorithmisinitializedbyarandomsequence.DifferentinitializationsleadtodifferentnalsequenceswhosePstoporPcorrvaries.Whilegenerallyalargerfavorsthestopbandsuppression,itisnotguaranteedthat,e.g.,=0:75givesasmallerPstopandalargerPcorrthan=0:7.Nextweconsiderasituationwheretheallowedbandishighlysegmentedbystopbands:=[0;0:11)[[0:13;0:19)[[0:25;0:36) (6)[[0:40;0:65)[[0:8;0:87)[[0:94;1)Hz:ThesequencelengthisN=104andwechoose~N=NsinceNisalreadylargeenoughtoensureadenseDFT-frequencygrid.Weset=0:9toemphasizethe 110

PAGE 111

stopbandsuppression.Fig. 6-3 showsthespectralpowerandcorrelationlevelofthegeneratedSCANsequence.HerePstop=)]TJ /F5 11.955 Tf 9.3 0 Td[(15:1dBandPcorr=)]TJ /F5 11.955 Tf 9.3 0 Td[(7:3dB.Notethatthepeakcorrelationsidelobeoccursclosetotheorigin(i.e.thein-phasecorrelationpoint)andthatcorrelationsidelobesfarawayfromtheoriginaremuchlowerthanPcorr. 6.3.2WeSCANInthissubsectionweshowthatbyusingtheWeSCANalgorithm,wecantrade-offcorrelationpropertiesforanimprovementofPstop.WeusethesamesettingasinFig. 6-1 exceptthatweapplytheWeSCANalgorithm.Assumingthatarelativelywidecorrelationmainlobeisacceptable,wechoosethecorrelationweights(Eq.( 6 ))as1=0,2=0andk=1fork=3;:::;N)]TJ /F5 11.955 Tf 11.9 0 Td[(1.Weshowthespectralpowerandcorrelationleveloftheso-generatedWeSCANsequenceinFig. 6-4 .ComparedtoFig. 6-1 ,thestopbandpower(Pstop=)]TJ /F5 11.955 Tf 9.3 0 Td[(34:9dB)issmallerbymorethan20dB,atthecostofanenlargedcorrelationmainlobe. 6.3.3RelaxedAmplitudeConstraintInthisexampleweillustratetheeffectofarelaxedpeak-to-averageratio(i.e.PAR>1).ThesamesettingasinFig. 6-1 isusedexceptthat( 6 )replaces( 6 )intheSCANalgorithm.Wechoose=2whichconstrainsthePARofthedesignedsequencetobelessthanorequalto2.Thespectralpowerandcorrelationleveloftheso-generatedSCANsequenceareshowninFig. 6-5 ,wherePstop=)]TJ /F5 11.955 Tf 9.3 0 Td[(9:0dBandPcorr=)]TJ /F5 11.955 Tf 9.3 0 Td[(19:3dB.IntermsofPstopandPcorr,thisrelaxedPARdoesnotleadtomuchbetterperformances.However,comparedtoFig. 6-1 ,thespectralpowerinFig. 6-5A jitterslessandthecorrelationlevelinFig. 6-5B issmallerforlargetimelags.ThussuchadesignisworthconsideringifPAR>1isallowedintherealsystem. 6.3.4UsingaDifferentFrequencyFormulationFinallyweshowanexampleinwhichthediscrete-frequencyformulationin( 6 )isreplacedbythecontinuous-frequencyonein( 6 ).WestillusetheparametersettinginFig. 6-1 .InthiscasetherankofRequals22.Thespectralpowerandcorrelationlevel 111

PAGE 112

ofthegeneratedSCANsequenceareshowninFig. 6-6 ,wherePstop=)]TJ /F5 11.955 Tf 9.3 0 Td[(9:0dBandPcorr=)]TJ /F5 11.955 Tf 9.3 0 Td[(18:5dB.Wecanobservethatthereisnoperformancegainbyusingarelativelymoreinvolvedcontinuous-frequencystopbandformulation. A B Figure6-1.ThespectralpowerandcorrelationlevelofaSCANsequencegeneratedwithparametersN=100,~N=1000,=0:7and=[0:2;0:3)Hz.A)Thespectralpower(thedashedverticallinessignifythestopbandborders)andB)thecorrelationlevel. 112

PAGE 113

Figure6-2.PstopandPcorrvs.(othersettingsarethesameasthoseinFig. 6-1 ). 113

PAGE 114

A B Figure6-3.ThespectralpowerandcorrelationlevelofaSCANsequencegeneratedwithparametersN=105,~N=105,=0:9and=[0;0:11)S[0:13;0:19)S[0:25;0:36)S[0:40;0:65)S[0:8;0:87)S[0:94;1)Hz.A)ThespectralpowerandB)thecorrelationlevel. 114

PAGE 115

A B Figure6-4.ThespectralpowerandcorrelationlevelofaWeSCANsequencegeneratedwiththesameparametersasinFig. 6-1 .Thecorrelationweightsarechosenas1=0,2=0andk=1fork=3;:::;N)]TJ /F5 11.955 Tf 11.96 0 Td[(1.A)ThespectralpowerandB)thecorrelationlevel. 115

PAGE 116

A B Figure6-5.ThespectralpowerandcorrelationlevelofaSCANsequencegeneratedwiththesameparametersasinFig. 6-1 ,exceptthattheconstraintPAR2isimposedinsteadofunimodularity.A)ThespectralpowerandB)thecorrelationlevel. 116

PAGE 117

A B Figure6-6.ThespectralpowerandcorrelationlevelofaSCANsequencegeneratedwiththesameparametersasinFig. 6-1 ,exceptthat( 6 )isusedinlieuof( 6 )intheSCANalgorithm.A)ThespectralpowerandB)thecorrelationlevel. 117

PAGE 118

CHAPTER7TRANSMITBEAMPATTERNSYNTHESISAntennaarraybeampatterndesignhasbeenawell-studiedtopicandthereisaconsiderableliteratureavailablefromclassicanalyticaldesign[ 62 68 71 ]tomorerecentworksthatresorttonumericaloptimization[ 72 76 ].Thepredominantproblemconsideredintheliteraturereferstothereceivebeampatterndesign,whichisconcernedwithdesigningweightsforthereceivedsignalsothatthesignalcomponentimpingingfromaparticulardirectionisreinforcedwhilethosefromotherdirectionsareattenuated,awayinwhichcertainsignalproperties(e.g.,thesignalpowerordirection-of-arrival)canbeestimated.SuchaproblemusuallyboilsdowntothedesignofanFIR(nite-impulse-response)lterinthenarrowbandcaseorasetofFIRltersinthewidebandcase.Thetransmitbeampatterndesign,ontheotherhand,referstodesigningtheprobingsignalstoapproximateadesiredtransmitbeampattern(i.e.,anenergydistributioninspaceandfrequency).Ithasbeenoftenstatedthatthereceiveandtransmitbeampatterndesignsareessentiallyequivalent,whichispartlytrueinthesensethatthetwoscenariosbearsimilarproblemformulationsandthattheFIRltertapsobtainedviareceivepatterndesigncanbeusedtheoreticallyastheprobingsignaltoachieveanidenticaltransmitpattern.Inpractice,however,thetransmitbeampatterndesignproblemappearstobemuchharderbecauseoftheenergyandpeak-to-averagepowerratio(PAR)constraintsonthetransmitwaveforms.Inparticular,adigital-to-analogconverterscalesthesignalbythemaximumallowableamplitudeandasaturatedpoweramplierworkswellonlywhenthesignalisconstant-modulus[ 1 77 ].Ifthetransmittedsignalshavewildlyvaryingmagnitudes,weriskenergy-lossorevensignaldistortion.Asaresult,thetransmitbeampatterndesignmustbesubjecttotheconstraintthatthetransmitwaveformshaveunit-modulusorlowPARs.Onthecontraryinthereceivebeampatterndesign,theFIRtapscantakeonanyvalues,although 118

PAGE 119

certaineasy-to-meetconstraints(e.g.,thesymmetryoftheltercoefcients)areusuallyimposed.Therefore,exceptinafewsimplesituationssuchasthephasedarraycase,thetransmitbeampatterndesignshouldbetreateddifferentlyfromthemoreprevalentreceivebeampatterndesign.ThenarrowbandtransmitbeampatterndesignproblemhasbeenaddressedintheMIMOradararea[ 32 45 46 78 ],andinbiomedicalimaging[ 79 ].Mostoftheproposedmethodsrstrelatethedesiredbeampatterntothecross-correlationsbetweenthetransmitsignals,thenaimtodesignthesignalcovariancematrixandnallysynthesizetheactualsignals[ 32 ].Inthewidebandcase,similarapproacheshavebeenproposedtodesignthepowerspectraldensitymatrix[ 75 ],butnosignalshavebeensynthesizedduetothedifcultyofimposingtheunit-modulusorPARconstraints.InthischapterweproposeanalgorithmnamedWB-CA(widebandbeampatternCA)todesignunimodularorlow-PARsequencesfortransmitbeampatternsynthesisinwidebandactivesensingsystems.Wedonotformulatetheproblemintermsofthetransmitspectraldensitymatrix(aswasdonein[ 75 ]),butinsteaddirectlylinkthebeampatterntothesignalsthroughtheirFouriertransform.ThedesigncriterionisformulatedinSection 7.1 ,whichisfollowedbythealgorithmdescriptioninSection 7.2 .SimulationexamplesareshowninSection 7.3 7.1ProblemFormulationWefocusonfar-eldbeampatternsynthesisforuniformlineararrays(ULA)asillustratedinFig. 7-1 .Notethattheproposedmethodcanbeeasilyextendedtothenon-ULAcasebyusingamoregeneralsteeringvectorthantheonein( 7 );see,e.g.,thesteeringvectorsusedin[ 79 ].SupposethatthereareMlinearlyspacedisotropicarrayelementsandthattheinter-elementspacingisd.Thesignaltransmittedbythemthelementisdenotedassm(t).Considerthebeampatterninthefareldatangle(0180)measuredwithrespectto(w.r.t.)thearrayline.Itiseasytoseethatthetimedelaybetween 119

PAGE 120

twoneighboringelementsisdcos c,wherecisthespeedofwavepropagation.Weletsm(t)=xm(t)ej2fctwherefcisthecarrierfrequencyandxm(t)isthebasebandsignalwhosespectralsupportisassumedtobeincludedintheinterval[)]TJ /F7 7.97 Tf 10.49 4.71 Td[(B 2;B 2].Byusingtheabovenotation,theresultingfar-eldsignalatanglecanbewrittenasz(t)=MXm=1smt)]TJ /F5 11.955 Tf 13.15 8.09 Td[((m)]TJ /F5 11.955 Tf 11.96 0 Td[(1)dcos c (7)=MXm=1xmt)]TJ /F5 11.955 Tf 13.15 8.08 Td[((m)]TJ /F5 11.955 Tf 11.96 0 Td[(1)dcos cej2fc(t)]TJ /F14 5.978 Tf 7.79 4.03 Td[((m)]TJ /F14 5.978 Tf 5.76 0 Td[(1)dcos c):Supposethatthetimesupportofxm(t)is[0;].ThentheFouriertransformofxm(t)isgivenbyym(f)=Z0xm(t)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2ftdt;f2[)]TJ /F3 11.955 Tf 10.5 8.09 Td[(B 2;B 2] (7)andtheinverseFouriertransformisaccordinglyxm(t)=ZB=2)]TJ /F7 7.97 Tf 6.59 0 Td[(B=2ym(f)ej2ftdf: (7)Substituting( 7 )into( 7 )yields:z(t)=ZB=2)]TJ /F7 7.97 Tf 6.59 0 Td[(B=2Y(;f)ej2(f+fc)tdf (7)whereY(;f)=MXm=1ym(f)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2(f+fc)(m)]TJ /F14 5.978 Tf 5.76 0 Td[(1)dcos c: (7)Itfollowsfrom( 7 )thatthebeampatternatspatialangleandfrequencyf+fccanbedenedasP(;f+fc)=jY(;f)j2=jaH(;f)y(f)j2;f2[)]TJ /F3 11.955 Tf 10.5 8.09 Td[(B 2;B 2] (7) 120

PAGE 121

wherea(;f)=1ej2(f+fc)dcos cej2(f+fc)(M)]TJ /F14 5.978 Tf 5.76 0 Td[(1)dcos cT (7)andy(f)=y1(f)y2(f)yM(f)T: (7)Ourproblemistodesignthesignalsfxm(t)gMm=1(band-limitedto[)]TJ /F7 7.97 Tf 10.49 4.71 Td[(B 2;B 2])sothatthebeampatternP(;f+fc)in( 7 )matchesadesiredone.Inthesequel,thebasebandfrequencyrange[)]TJ /F7 7.97 Tf 10.5 4.7 Td[(B 2;B 2]isexplicitlyindicatedwhennecessary.Digitalsignalprocessingtechniquesdealwiththesampledsignal:xm(n)4=xm(t=nTs);n=1;:::;N (7)whereTsisthesymbolperiodthatsatisesTs=1 BandN=b=Tsc.Then( 7 )becomes(byaslightabuseofnotation)ym(fTs)=TsNXn=1xm(nTs)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2nfTs;f2[)]TJ /F3 11.955 Tf 10.49 8.09 Td[(B 2;B 2]: (7)SincetheintervalforfTsis[)]TJ /F5 11.955 Tf 9.29 0 Td[(0:5;0:5],itisenoughtoconsidertheDFT(discreteFouriertransform)offxm(n)g:ym(p)=NXn=1xm(n)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2(n)]TJ /F14 5.978 Tf 5.76 0 Td[(1)p N;p=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(N 2;:::;0;:::;N 2)]TJ /F5 11.955 Tf 11.96 0 Td[(1 (7)whereNwasassumedtobeeven(pwillrunfrom)]TJ /F5 11.955 Tf 9.3 0 Td[((N)]TJ /F5 11.955 Tf 12.13 0 Td[(1)=2to(N)]TJ /F5 11.955 Tf 12.13 0 Td[(1)=2ifNisodd).Notethatin( 7 )wehavedroppedthemultiplicativeconstantTsfrom( 7 )sincethescalingoffym(p)gdoesnotaffecttheproposedapproach(thediscussionsfollowing( 7 )inSection 7.2 ). 121

PAGE 122

Similarlytothefrequencygridconsideredabove,wealsouseagridwithpointsdenotedasfkgKk=1forthespatialangleinterval[0;180].Fornotationalsimplicity,letakp=a(k;p NTs) (7)andyp=y1(p)y2(p)yM(p)T: (7)Thenitfollowsfrom( 7 )thatthebeampatterncanbeexpressedonthediscreteangle-frequencygridasPkp=jaHkpypj2: (7)Lettingdkpdenotethedesiredbeampattern,weseektosolvethefollowingbeampatternmatchingproblem:minfxm(n)gKXk=1N=2)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xp=)]TJ /F7 7.97 Tf 6.59 0 Td[(N=2dkp)-222(jaHkpypj2 (7)subjecttoPAR(xm);m=1;:::;Mwhere1isapre-denedthresholdandPAR(xm)denotesthePAR(Eq.( 4 )inChapter 4 )ofthemthsequence.Asusual,weimposetheenergyconstraintonthedesignedsequence:PNn=1jxm(n)j2=Nform=1;:::;M.Theoptimizationproblemin( 7 )isnon-convex(andthusdifcultingeneral)becauseofthePARconstraint.Thisnon-convexitycanbeeasilyseeninthecaseof=1:theneachxm(n)canonlytakevaluesfromtheunit-circle,whichisnotaconvexset.Globaloptimizationalgorithms,suchasthesimulatedannealingmethod,areexpectedtobecomputationallytooexpensivefortheproblemin( 7 )duetoitshighdimensionality.InSection 7.2 ,anefcientcyclicalgorithmwillbeproposedtosearchforthelocalminimumof( 7 ). 122

PAGE 123

Remark:Notethatfxm(n)gistypicallyrelatedtofxm(t)gthroughpulseshaping:xm(t)=NXn=1xm(n)p(t)]TJ /F5 11.955 Tf 11.95 0 Td[((n)]TJ /F5 11.955 Tf 11.95 0 Td[(1)Ts);m=1;:::;M (7)wherep(t)isthepulse(Eq.( 1 )inChapter 1 ).Thespectrumofthebasebandsignalxm(t)wouldbeconnedto[)]TJ /F3 11.955 Tf 9.3 0 Td[(B=2;B=2]onlyifp(t)wereaperfectNyquistshapingpulse(i.e.,asincfunctionwhichiscenteredat0andhastherstzero-crossingatTs).Theuseofanypracticalshapingpulsesuchasa(truncated)raisedcosine[ 2 ]willresultinaleakageofthespectrumoutsidethedesiredrange[)]TJ /F3 11.955 Tf 9.3 0 Td[(B=2;B=2];thesefactsmake( 7 )and( 7 )onlyapproximatelyequivalent.WewillexaminetheeffectthisapproximationhasonthedesignviaanexampleinSection 7.3 .Inaddition,notethat:i)thenarrowbandtransmitbeampatterndesignisjustaspecialcaseofthewidebandproblemconsideredhereandthatii)thereceivebeampatterndesigncanbegivenasimilarformulationwhich,however,differsinimportantwaysfromthetransmitproblem;seeAppendix D andAppendix E formoredetailsontheseaspects. 7.2TheProposedDesignMethodologyMinimizationofthecriterionin( 7 )directlyw.r.t.fxm(n)gappearstobeadifculttask(unlessthematrixa1paKpturnsouttobesemi-unitaryforeachvalueofp,whichishardlytrueingeneral).Forthisreasonweadoptatwo-stagedesignapproach:Stage1(beampatterntospectrum):Firstwesolve( 7 )w.r.t.fypgconsideredtobegeneralvectorsinCM1.Stage2(spectrumtowaveform):ThenwettheDFToffx(n)gtotheso-obtainedfypg,subjectto(s.t.)theenforcedPARconstraintonfx(n)g. 7.2.1BeampatterntoSpectrumForagenericterm[d)-221(jaHyj]2of( 7 )itholdsthat(d0)minjdej)]TJ /F15 11.955 Tf 11.95 0 Td[(aHyj2 (7) 123

PAGE 124

=mind2+jaHyj2)]TJ /F5 11.955 Tf 11.96 0 Td[(2RedjaHyjcos()]TJ /F5 11.955 Tf 11.96 0 Td[(arg(aHy))=d)-222(jaHyj2(for=argfaHyg):Consequentlywecangettheminimizerof( 7 )fromthefypgthatminimize,alongwiththeauxiliaryvariablesfkpg,thefollowingcriterion:XkXpdkpejkp)]TJ /F15 11.955 Tf 11.96 0 Td[(aHkpyp2: (7)Theabovecriterioncanbeconvenientlyminimized(w.r.tfypgandfkpg)bythecyclicalgorithmoutlinedbelow.Stage1ofWB-CA(BeampatterntoSpectrum): Step0:Initializefkpg,suchasfkpg=0orfkpg=randomlygeneratedvariablesuniformlydistributedin[0;2]. Step1:Forfkpgsetattheirlatestvalues(denotedasf^kpg),letAp=264aH1p...aHKp375;bp=264d1pej^1p...dKpej^Kp375 (7)Then( 7 )canbewrittenasPpkbp)]TJ /F15 11.955 Tf 12.16 0 Td[(Apypk2.Thustheminimizerfypgisgivenbytheleast-squaresestimate:^yp=(AHpAp))]TJ /F6 7.97 Tf 6.59 0 Td[(1AHpbp;p=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(N 2;:::;0;:::;N 2)]TJ /F5 11.955 Tf 11.95 0 Td[(1: (7) Step2:Forfypgsetattheirlatestvalues,theminimizerfkpgisgivenby(Eq.( 7 ))^kp=arg(aHkp^yp): (7) Iteration:RepeatSteps1and2untilconvergence,e.g.,untilthechangeoffkpgintwoconsecutiveiterationsislessthanapredenedthreshold.Theabovealgorithmmonotonicallydecreasesthecriterion( 7 )ateachiteration,andhenceitmonotonicallydecreasestheoriginalcriterionin( 7 )aswell.Thusitisboundtoconvergetoatleastalocalminimumvalueof( 7 ).Thebasicprincipleofthe 124

PAGE 125

algorithmisrelatedtotheoperationoftheGerchberg-SaxtonAlgorithmasdescribedinAppendix A .Remark:ItfollowsfromtheParsevalequalitythattheenergy-constraintonfxm(n)gimposesthefollowingconstraintonfypg:N=2)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xp=)]TJ /F7 7.97 Tf 6.59 0 Td[(N=2jym(p)j2=NNXn=1jxm(n)j2=N2;m=1;:::;M (7)whereym(p)isthemthelementofyp(Eq.( 7 )).NotethatthestepsinStage1ofWB-CAomittheconstraintin( 7 )forsimplicity(observethatthisconstraintyieldsacouplingoffypg,whichthereforecouldnolongerbedeterminedseparatelyasin( 7 )).Nonetheless,theproposedalgorithmperformsreasonablywelllikelybecausetheenergyconstraintonfxm(n)gisconsideredinStage2anyway. 7.2.2SpectrumtoWaveformInStage2weaimtosynthesizethewaveformfx(n)gNn=1,underthePARconstraint,sothatitsDFTapproximatesthef^ypgN=2)]TJ /F6 7.97 Tf 6.59 0 Td[(1p=)]TJ /F7 7.97 Tf 6.59 0 Td[(N=2fromStage1ascloselyaspossible.Wenoterstthatthef^ypgN=2)]TJ /F6 7.97 Tf 6.59 0 Td[(1p=)]TJ /F7 7.97 Tf 6.59 0 Td[(N=2obtainedinStage1haveaphaseambiguity,whichcanbeobservedfromtheminimizationcriterionin( 7 ):if(fypg;fkpg)areminimizersof( 7 ),then(fypej pg;fkp+ pg)arealsominimizersof( 7 )forany p.Thisphaseambiguityresultsinfactfromtheoriginalmatchingproblemin( 7 ):ypandypej pleadtothesamevalueof( 7 ).Toexploitthisphaseexibilityassociatedwithf^ypg,weintroduceauxiliaryvariablesf pgN=2)]TJ /F6 7.97 Tf 6.59 0 Td[(1p=)]TJ /F7 7.97 Tf 6.58 0 Td[(N=2andminimizethefollowingttingcriterionw.r.t.bothfxm(n)gandf pg:N=2)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xp=)]TJ /F7 7.97 Tf 6.58 0 Td[(N=2^yTpej p)]TJ /F11 11.955 Tf 11.95 16.86 Td[(1e)]TJ /F7 7.97 Tf 6.58 0 Td[(j2p Ne)]TJ /F7 7.97 Tf 6.59 0 Td[(j2(N)]TJ /F14 5.978 Tf 5.75 0 Td[(1)p NX2 (7)whereX=x1x2xM (7) 125

PAGE 126

=266664x1(1)x2(1)xM(1).........x1(N)x2(N)xM(N)377775:WefurtherdeneeHp=1e)]TJ /F7 7.97 Tf 6.58 0 Td[(j2p Ne)]TJ /F7 7.97 Tf 6.58 0 Td[(j2(N)]TJ /F14 5.978 Tf 5.75 0 Td[(1)p N;p=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(N 2;:::;N 2)]TJ /F5 11.955 Tf 11.95 0 Td[(1; (7)FH=266664eH)]TJ /F7 7.97 Tf 6.59 0 Td[(N=2...eHN=2)]TJ /F6 7.97 Tf 6.58 0 Td[(1377775NN;ST=266664^yT)]TJ /F7 7.97 Tf 6.58 0 Td[(N=2ej )]TJ /F12 5.978 Tf 5.75 0 Td[(N=2...^yTN=2)]TJ /F6 7.97 Tf 6.58 0 Td[(1ej N=2)]TJ /F14 5.978 Tf 5.75 0 Td[(1377775NM:Then( 7 )canbewrittenaskST)]TJ /F15 11.955 Tf 11.95 0 Td[(FHXk2=Nk1 NFST)]TJ /F15 11.955 Tf 11.96 0 Td[(Xk2 (7)wheretheequalitycomesfromthefactthat1 p NFisaunitarymatrix.Onceagainweuseacyclicalgorithmtominimize( 7 )(w.r.t.fxm(n)gandf pg);seethestepsbelow.NotethattherequiredmatrixcalculationsFHXandFSTthereincanbedonebytheFFT,whichreducesthecomputationtime.Stage2ofWB-CA(SpectrumtoWaveform): Step0:Initializef pg,forinstanceasf pg=0. Step1:Forf pgxedattheirlatestvalues,theminimizationof( 7 )w.r.t.fxm(n)gdependsontheconsideredPARconstraint.Undertheunit-modulusconstraint(i.e.,jxm(n)j=1),theminimizationof( 7 )isimmediate:^xm(n)=exp(jargfthe(n;m)thelementofFSTg); (7)m=1;:::;M;n=1;:::;N:IfthePAR(>1)constraintisimposed,weneedtosolveMseparateminimizationproblems:minxmkum)]TJ /F15 11.955 Tf 11.95 0 Td[(xmk2 (7)s.t.PAR(xm) 126

PAGE 127

(form=1;:::;M)whereumisthemthcolumnof1 NFST.Thisproblemcanbesolvedefcientlybythenearest-vectormethodoutlinedinSection 4.2 ofChapter 4 Step2:Forfxm(n)gxedattheirmostrecentvalues,theminimizerf pgisgivenby(thederivationof( 7 )issimilarto( 7 )):^ p=argf^yHpvpg;p=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(N 2;:::;N 2)]TJ /F5 11.955 Tf 11.96 0 Td[(1 (7)wherevTpisthe(p+N=2)throwofFHX. Iteration:RepeatSteps1and2untilconvergence.ItiseasytoseethatapossiblescalingofShasnoeffecton( 7 ).Thesameistruefor( 7 )(whichfollowsfromtheoperationofthemethodin[ 53 ]usedtosolve( 7 )).Therefore,wecanchoosethedesiredbeampatternfdkpgin( 7 )withoutanyconcernforapossiblenormalization,asfypgwillautomaticallyscale(Eq.( 7 ))totthechosenfdkpgandthescalingoffypgdoesnotaffectthesynthesisoffxm(n)g.Tosummarize,theproposedtwo-stagedesignmethodology,rstdeterminingfypgandthenfxm(n)g,basicallyreducestheproblemin( 7 )tothedesignofNbeamformingvectorsfypg,oneforeachfrequencybinandthentomatchingthembytheselectionoffxm(n)g.Notethatthereare2MNreal-valuedelementsinfypg,andMNfreevariablesinfxm(n)gundertheunit-modulusconstraint,andmorethanMNdegreesoffreedomifthePARisallowedtobelargerthan1.Inaddition,thef pgprovideNdegreesoffreedom.Hencewecanexpectareasonableperformanceforthematchingstepoftheproposedapproach.AlthoughWB-CAreliesonaniterativeprocess,theupdatingformulasarerelativelysimpleandtheiterationturnsouttoconvergeveryfast.Forthenumericalexamplespresentedinthenextsection,theexecutionoftheWB-CAalgorithmcodedinMATLABtakesonlyafewsecondsonanordinaryPC. 127

PAGE 128

7.3NumericalExamplesUnlessstatedotherwise,thefollowingsettingisusedinthissection:aULAwithM=10elements,thecarrierfrequencyofthetransmittedsignalisfc=1GHz,thebandwidthisB=200MHzandthenumberofsymbolsisN=64.ThesymbolperiodisTs=1=B.Theinter-elementspacingisgivenbyd=c 2(fc+B=2),thatis,halfwavelengthofthehighestin-bandfrequencytoavoidgratinglobes.ThespatialangleisdividedintoK=180gridpoints(i.e.,onedegreepergridstep).Remark:Inpracticalapplicationstheantennaelementsofanarrayaretypicallymutuallycoupled.Theinter-elementspacingdchosenaboveresultsinover-sampling(i.e.samplingintervallessthanhalfwavelength)forlowerin-bandfrequenciesandmayrenderthemutualcouplingeffectsnon-negligible,whichcouldleadtoenergybeingcoupledintotransmitters.However,thisissueliesoutsidethescopeofthischapter(asitdependsonthespecichardwareimplementationsuchasthesystemtoleranceandantennatypes),andwerefertheinterestedreaderto[ 80 82 ]fordiscussionsondecoupling. 7.3.1TheIdealizedTime-DelayedCaseItfollowsfrom( 7 )thatwecansteerthetransmitbeamtowardstheangle0bychoosingthefollowingsignalspectrum:y(f)=p Na(0;f);f2[)]TJ /F3 11.955 Tf 10.5 8.09 Td[(B 2;B 2] (7)wherep Nisduetotheenergyconstraint.Eq.( 7 )leadsto(Eq.( 7 ))Pkp=NMXm=1ej2(p NTs+fc)(m)]TJ /F14 5.978 Tf 5.75 0 Td[(1)d(cos0)]TJ /F14 5.978 Tf 5.75 0 Td[(cosk) c2; (7)whereforaxedvalueoff(i.e.,p),thebeamissteeredinthedirectionof0asinthecaseofanarrowbandphasedarray.Theunderlyingsignals,i.e.,theinverseFourier 128

PAGE 129

transformof( 7 ),aregiven(uptoamultiplicativeconstant)byxm(t)=sinc Tst)]TJ /F5 11.955 Tf 13.15 8.09 Td[((m)]TJ /F5 11.955 Tf 11.96 0 Td[(1)dcos0 c;m=1;:::;M (7)wheresinc(t)=sin(x)=x.Notethatsuchanidealizedfxm(t)ghasaveryhighPARwhichisundesirable.Moreover,becaused=c=1=(2fc+B)Ts,therequiredtimedelayofmdcos0 ccanbetoosmalltobereadilyimplementedinpractice,especiallywhen0isclosebutnotequalto90.Weshowthebeampattern10lg(Pkp=N)in( 7 )for0=120asa2DplotinFigure 7-2A ,aswellasa3DplotinFigure 7-2B .Thebeampatternexhibitsacleanmainlobeat0acrosstheentirefrequencyrange.Remark:Inthenarrowbandcase,foragivenULAaperture,thetransmitbeampatterngeneratedbyaphasedarrayhasthesmallestmainlobewidth.Intheaboveexample,weusedtheimpracticaltime-delayedfxm(t)gin( 7 )togetthephasedarray-likebeampatternin( 7 ),whichthushasthenarrowestmainlobeforeveryxedfrequency.Thereforewecallittheidealizedtime-delayedcase.IthasbeenassumedthatforeacharrayelementanenergyequaltoNisemittedinalldirections;andtheenergyconstraintPpjym(p)j2=N2in( 7 )indicatesthatonaveragejym(p)j2equalsN.Therefore,werethereonlyonearrayelement,PkpwouldequalNateverygridpointintheangle-frequencyplane(Eq.( 7 )).Thisisthereasonwhythenormalization10lg(Pkp=N)isusedinallplots.NowthatthereareMtransmitwaveforms,thecoherentsumgivesmaxPkp=maxjaHkpypj2=jMp Nj2=M2N,whichleadsto10lg(M2N=N)=20dBintheplot.Infact,intheaboveidealizedtime-delayedcase,allMwaveformsaddcoherentlyat=0andtheenergyisevenlydistributedat=0forallfrequencies,whichproducesaconstant20dBmainlobeheight(Figs. 7-2 and( 7 )).Inotherexamples,however,themainlobeheightisnotnecessarily20dBandtheupperlimitofthecolorbaralwayscorrespondstothelargestvalueintheplot. 129

PAGE 130

7.3.2ANarrowMainbeamWeusetheproposedWB-CAalgorithmtosynthesizethefollowingdesiredtransmitbeampattern:d(;f+fc)=8><>:1;=1200;elsewhereforallf2[)]TJ /F3 11.955 Tf 10.5 8.09 Td[(B 2;B 2]; (7)thatis,abeampatternwiththemainlobe(asnarrowaspossible)locatedat120acrossthefrequencysupport.Stage1ofWB-CAgeneratestheDFTvectorsf^ypgN=2)]TJ /F6 7.97 Tf 6.59 0 Td[(1p=)]TJ /F7 7.97 Tf 6.59 0 Td[(N=2,whicharefurthernormalizedtopreservethetotalenergy(i.e.,normalizedsuchthatPpkypk2=MN2).Fig. 7-3 showsthebeampatternPkpthatiscalculateddirectlyfromthesef^ypg.Theso-obtainedbeampatternisquitesimilartotheidealizedoneinFig. 7-2 .However,theunderlyingwaveformscorrespondingtoFig. 7-3 ,givenbytheinverseDFT(IDFT)off^ypgN=2)]TJ /F6 7.97 Tf 6.58 0 Td[(1p=)]TJ /F7 7.97 Tf 6.59 0 Td[(N=2,donotsatisfytheenergyandPARrequirement.Indeed,theMsequencesobtainedfromtheIDFToff^ypghaveenergiesvaryingfrom55:4to71:2andPARsvaryingfrom1:3to1:8.Notethatsuchtransmitsequencesneedtobescaledinpracticesothatthemaximumenergydoesnotexceedthesystemspecications,whichwillinevitablyresultinanenergyloss.WethenproceedtoStage2ofWB-CAandsynthesizethesequencesf^xm(n)gundertheunit-modulusconstraint.Afterthat,wecomputetheDFToff^xm(n)gandobtainthebeampatternusing( 7 );seeFig. 7-4 .Itisclearthatthestrictunit-modulusconstraintdegradesthebeampatternmatching.Table 7-1 showstheminimumvalueofthettingcriterion(Eq.( 7 ))associatedwithFigs. 7-3 and 7-4 ,respectively.Nextweexaminethebeampatternofthecontinuous-timewaveformscorrespondingtotheso-obtainedf^xm(n)g(theRemarkattheendofSection 7.1 ).Morespecically,wepasseachf^xm(n)gNn=1(m=1;:::;M)throughanFIRraised-cosinelter(withtheroll-offfactorequalto0:5)togetthecontinuous-timesignal^xm(t).Thespectraldensity 130

PAGE 131

functionsoff^xm(t)gareshowninFig. 7-5 inanoverlappingmanner,fromwhichweobservethatthespectrumiswellcontainedwithin[fc)]TJ /F3 11.955 Tf 12.8 0 Td[(B=2;fc+B=2]despiteofacertainleakageoutsidethefrequencyrangeofinterest.Thebeampatternoff^xm(t)g,asdenedin( 7 ),isshowninFig. 7-6 .ComparedtoFig. 7-4 ,thebeampatterninFig. 7-6 isapoorerapproximationofthedesiredone.AsdiscussedintheRemarkattheendofSection 7.1 ,apracticalpulseshapingrenders( 7 )and( 7 )notexactlyequivalent.SincetheWB-CAalgorithmaimstomatch( 7 )tothedesiredbeampattern,thenonequivalencebetween( 7 )and( 7 )explainsthedegradationfromFig. 7-4 toFig. 7-6 7.3.3TwoMainbeamsInthisexampleweconsiderthefollowingdesiredbeampattern:d(;f+fc)=8>>>><>>>>:1;fc)]TJ /F3 11.955 Tf 11.96 0 Td[(B=2ffcand=1201;fcffc+B=2and=600;elsewhere: (7)TheWB-CAbeampatternundertheunit-modulusconstraintisshowninFig. 7-7 andthatunderthePAR2constraintisshowninFig. 7-8 .WhileFig. 7-7 alreadyprovidesareasonablygoodbeampatternmatching,relaxingthePARfrom1to2leadstothevisiblybetterresultinFig. 7-8 duetomoredegreesoffreedominthewaveformdesign.Table 7-2 showsthisperformanceimprovementintermsofthecorrespondingttingcriterionvalues. 7.3.4AWideMainbeamInbothexamplesabove,wefocusedonachievingabeampatternwheremainlobe(s)wereasnarrowaspossible.Specicallytheidealizedphasedarray-likebeampatterninFig. 7-2 ,whichhasthenarrowestpossiblemainlobe,waswellapproximatedbyusingpracticalwaveformsinSection 7.3.2 .Ifwewanttoobtainanarrowermainlobe,wehavetousealargervalueofM,i.e.moretransmitantennaelements. 131

PAGE 132

Hereweconsiderinsteadthefollowingbeampatternwithawidermainlobe:d(;f+fc)=8><>:1;1001400;elsewhereforallf2[)]TJ /F3 11.955 Tf 10.5 8.09 Td[(B 2;B 2]: (7)ThecorrespondingWB-CAbeampatternundertheunit-modulusconstraintisshowninFig. 7-9 andthatunderthePAR2constraintinFig. 7-10 .WeobservethatthemainlobeinFig. 7-9 or 7-10 hasanalmostconstantwidthfordifferentfrequencies,unlikethemainlobeinFig. 7-2 whosewidthtendsto(slightly)shrinkasthefrequencyincreases(thewell-knownbeamsquintphenomenon).AlsonotethemainlobesplittinginFigs. 7-9 or 7-10 .Hadwesynthesizedanevenwidermainlobethan( 7 ),thesplittingwouldhavebeenmoresevere(e.g.,themainlobecansplittwicesothattherearethreelocalmaximainthemainlobearea).InallaboveexamplesthebandwidthBwassetto200MHz.Alargerbandwidthmeansmoreconstraintsandthusthebeampatternmatchingisexpectedtobecomemoredifcult.Toillustratethisfact,werepeattheexamplecorrespondingtoFig. 7-4 exceptthatthebandwidthisnowequalto350MHz.TheresultisshowninFig. 7-11 ,wherethebeampatternismoreirregularthaninFig. 7-4 .RegardingchoosingN(thenumberoftransmittedsymbols),wenotethatincreasingNdoesnotnecessarilyleadtoabetterbeampatternmatching.ThereasonisthatwhilealargerNincreasesthenumberofdegreesoffreedomofthewaveformfxm(n)g,italsoincreasesproportionallythenumberofelementsinfypgthataretobematchedinStage2ofWB-CA(thediscussionattheendofSection 7.2 ).Atthesametime,Ncannotbechosentoosmallbecausethefrequencygridshouldbedenseenoughtocoverthefrequencysupportnely.WenallypointoutthattheinitializationofWB-CA(i.e.Step0inbothStages)doesnotplayanimportantroleinthealgorithmperformance.Inallnumericalexamplespresented,randomlygeneratedphaseswereusedforinitialization;differentinitializations 132

PAGE 133

ledtodifferentwaveformsbutallofthesewaveformshadsimilarbeampatterns.Thisalsosigniesthefactthatthebeampatternmatchingproblemishighlymulti-modal. Figure7-1.TheULAarrayconguration. Table7-1.OptimizedcriterionvaluesforFig. 7-3 and 7-4 Fig. 7-3 Fig. 7-4 Criterionin( 7 )59879146048430 Table7-2.OptimizedcriterionvaluesforFig. 7-7 and 7-8 Fig. 7-7 Fig. 7-8 Criterionin( 7 )56988085619700 133

PAGE 134

A B Figure7-2.Theidealizedtime-delayedbeampatternin( 7 ).A)The2DplotandB)the3Dplot. 134

PAGE 135

A B Figure7-3.TheWB-CAbeampatternunderonlythetotalenergyconstraint.Thedesiredbeampatternisgivenin( 7 ).A)The2DplotandB)the3Dplot. 135

PAGE 136

A B Figure7-4.TheWB-CAbeampatternundertheunit-modulusconstraint.Thedesiredbeampatternisgivenin( 7 ).A)The2DplotandB)the3Dplot. 136

PAGE 137

Figure7-5.TheoverlaidspectraldensitiesofthecontinuouswaveformscorrespondingtothesequencesusedinFig. 7-4 .Thetwoverticaldashedlinesrepresenttheboundariesofthefrequencyrangeofinterest. 137

PAGE 138

A B Figure7-6.TheWB-CAbeampatternofthecontinuouswaveformscorrespondingtothesequencesusedinFig. 7-4 .Thedesiredbeampatternisgivenin( 7 ).A)The2DplotandB)the3Dplot. 138

PAGE 139

A B Figure7-7.TheWB-CAbeampatternundertheunit-modulusconstraint.Thedesiredbeampatternisgivenin( 7 ).A)The2DplotandB)the3Dplot. 139

PAGE 140

A B Figure7-8.TheWB-CAbeampatternunderthePAR2constraint.Thedesiredbeampatternisgivenin( 7 ).A)The2DplotandB)the3Dplot. 140

PAGE 141

A B Figure7-9.TheWB-CAbeampatternundertheunit-modulusconstraint.Thedesiredbeampatternisgivenin( 7 ).A)The2DplotandB)the3Dplot. 141

PAGE 142

A B Figure7-10.TheWB-CAbeampatternunderthePAR2constraint.Thedesiredbeampatternisgivenin( 7 ).A)The2DplotandB)the3Dplot. 142

PAGE 143

A B Figure7-11.TheWB-CAbeampatternforthesamesettingsasinFig. 7-4 ,exceptingthatthebandwidthBischangedfrom200to350MHz.A)The2DplotandB)the3Dplot. 143

PAGE 144

CHAPTER8CONCLUSIONSSeveralcomputationalalgorithmshavebeenpresentedfortransmitwaveformdesigninactivesensingapplications.Waveformsaredesignedwiththegoalofachievingcertainpropertiesthatincludelowcorrelationsidelobes,stopbandconstraintandbeampatternmatching.Discussionsinthisworkleantowardformulatingpracticalproblemsmathematicallyandsolvingthemathematicalproblemsusingoptimizationtechniques.Eachalgorithmistestedviaextensivenumericalexamplestoassessitseffectiveness.Particularattentionispaidtomakingthedevelopedalgorithmscomputationallyefcient.Besidesnewlydevelopedalgorithms,aconsiderableportionofthisworkpresentstutorial-likematerials,e.g.,reviewingexistingwaveforms,analyzingpropertiesofambiguityfunctions,describingapplicationscenarios,etc.Itisworthpointingoutthatnoneofthediscussedwaveformdesignrequirements(suchasminimizationofcorrelationsidelobes)hasaclosed-formsolution.Theyyieldnon-convexoptimizationproblemsthatcannotbesolvedglobally.Theproposediterativealgorithmsstartfromaninitialization,minimizetheobjectivefunctioninacyclicway(i.e.,withrespecttovariablesalternately)andareguaranteedforalocalconvergence.Whiledependingontheinitializationisadownsideofthesealgorithms,itisatthesametimeanadvantagebecausedifferentinitializationsleadtodifferentnalwaveformswhichusuallyallhavedesiredproperties.Wehavefocusedmainlyonphase-codedwaveformsinthiswork.Asmentionedinthechapterofintroduction,therearemanyothertypesofsignalsthatarewidelyusedorhavebeendiscussedintheliteraturesuchasfrequency-modulatedwaveforms.Differenttypesofwaveformsleadtodifferentproblemformulations,whichcansignicantlycomplicatethediscussionaswehavemanydifferentrequirementsofwaveformproperties.Themodelofphase-codedwaveformmeetspracticalconstraintssuchastheunit-modulusconstraint,iscomfortabletohandlemathematicallyandmost 144

PAGE 145

importantlyturnsouttohavegoodperformances.Byfocusingononewaveformmodel,wewereabletotellaconsistentstorythroughoutthediscussions.Amongthediscussedtopics,minimizingthesidelobesofanambiguityfunction(AF)isthemostchallengingone.Theattainedperformanceisratherlimitedduetothelargenumberofconstraintsinatwo-dimensionalspace.Asawidelyusedtoolinradarsignalanalysis(aswellasinothereldsofactivesensing),AFdeservesmoreattentioninourfutureendeavors. 145

PAGE 146

APPENDIXACONNECTIONSTOAPHASE-RETRIEVALALGORITHMTheCANalgorithmintroducedinthischapterfordesigningcodesequenceswithimpulse-likecorrelationsisrelatedtotheGerchberg-SaxtonAlgorithm(GSA)introducedforphaseretrievalintheopticsliterature[ 83 ]some40yearsago.ThetechniqueusedinGSAappearedinfactearlierin[ 55 ],soGSAmightbebetternamedtheSussman-Gerchberg-SaxtonAlgorithm.However,tobeconsistentwithmostotherliterature,wewillstillusethenameGSA.Inthisappendix,weattempttodescribeandclarifytherelationshipbetweenCANandGSA.WealsopresentsomefactsonGSAthatappeartobeusefulintheirownright.LetxbeanN1vectorandconsidertheproblemofminimizingw.r.t.(withrespectto)xthecriterion:C(x)=KXk=1jaHkxj)]TJ /F3 11.955 Tf 17.93 0 Td[(dk2 (A)wheredk2R+andak2CN1aregivenandKisanintegerthattypicallysatisesKN.Insomeapplications,thevectorxisfreetovaryinCN1[ 84 ].InotherapplicationsxisconstrainedtoacertainsubsetofCN1,suchastothesetofvectorswithunimodularelements(i.e.jxkj=1).Totakethisfactintoaccount,weletx2SCN1.TheGSAwasintroducedin[ 55 83 ]fortacklingrecoveryproblemstypicallyinvolvingasequenceanditsFouriertransform.Whenusedforproblemsthatcanbeformulatedasin( A ),GSAhastheformdescribedbelow.TheGerchberg-Saxtonalgorithm: Step0:Giveninitialvaluesf0kgKk=1(fkgareauxiliaryvariables;seebelowfordetails),iterateSteps1and2below,fori=0;1;:::untilconvergence. Step1:xi=argminx2SPKk=1jaHkx)]TJ /F3 11.955 Tf 11.96 0 Td[(dkejikj2. Step2:i+1k=arg(aHkxi)andi i+1. 146

PAGE 147

Notethat[ 83 ]proposedtheabovealgorithmonheuristicgrounds,withoutanyreferencetotheminimizationofC(x)in( A ).Howeveritwasrealizedlateronin[ 85 ]thatGSAisaminimizationalgorithmfor( A )whichhastheappealingpropertyofmonotonicallydecreasingthecriterionastheiterationproceeds.Asimpleproofofthisfactisasfollows:C(xi)=KXk=1[jaHkxij)]TJ /F3 11.955 Tf 17.93 0 Td[(dk]2=KXk=1jaHkxi)]TJ /F3 11.955 Tf 11.95 0 Td[(dkeji+1kj2 (A)KXk=1jaHkxi+1)]TJ /F3 11.955 Tf 11.95 0 Td[(dkeji+1kj2KXk=1jaHkxi+1)]TJ /F3 11.955 Tf 11.95 0 Td[(dkeji+2kj2=C(xi+1)wheretherstinequalityisduetoStep1andthesecondinequalityisduetoStep2(theseinequalitiesarestrictifthesolutionscomputedinSteps1and2areunique,whichisusuallythecaseinapplications).Thecalculationin( A )providesawaytomotivateGSAasaminimizationalgorithmforC(x).InthefollowingweoutlineawaytoderiveGSAasaminimizingprocedureforC(x).LetdenoteaK1vectorofauxiliaryvariablesandletD(x;)beafunctionwhichhasthepropertythat:minD(x;)=C(x): (A)Then,underrathergeneralconditions,thexthatminimizesC(x)isthesameasthexobtainedfromtheminimizationofD(x;)w.r.t.bothxand.Evidently,forthisapproachtobeusefultheminimizationofD(x;)shouldbeeasiertohandlethanthatofC(x).Tousetheaboveideainthepresentcaseof( A ),weletD(x;)=KXk=1jaHkx)]TJ /F3 11.955 Tf 11.95 0 Td[(dkejkj2 (A) 147

PAGE 148

(whereisthevectormadefromfkgKk=1)andnotethattheabovefunctionhastherequiredproperty:minD(x;) (A)=minKXk=1jaHkxj2+d2k)]TJ /F5 11.955 Tf 11.95 0 Td[(2jaHkxjdkcos(arg(aHkx))]TJ /F3 11.955 Tf 11.96 0 Td[(k)=KXk=1jaHkxj)]TJ /F3 11.955 Tf 17.93 0 Td[(dk2=C(x):ClearlytheminimizationofD(x;)w.r.t.x(unconstrainedasin[ 84 ]orconstrainedasinthischapter)forxedand,respectively,w.r.t.forxedxhassimpleclosed-formsolutions.ConsequentlyD(x;),andhenceC(x),canbeminimizedconvenientlyviaacyclicalgorithminwhichisxedtoitsmostrecentvalueandD(x;)isminimizedw.r.t.x,andviceversa.Theso-obtainedalgorithmisnothingbutGSAanditspropertyin( A )followsimmediatelyfrom( A )andthefactthatthecyclicminimizationofD(x;)yieldsamonotonicallydecreasingsequenceofcriterionvalues:C(xi)=D(xi;i+1)D(xi+1;i+2)=C(xi+1).ThecentralproblemdealtwithinChapter 2 wasthedesignofacodesequencewithimpulse-likeaperiodicand,respectively,periodiccorrelations.Amainresultprovedisthefactthatthesaidproblemcanbereducedtothatofminimizingacriterionoftheform:~C(x)=KXk=1[jaHkxj2)]TJ /F3 11.955 Tf 11.95 0 Td[(d2k]2 (A)foracertainKandcertainfakgandfdkg(whoseexactdenitionsarenotofimportancetothepresentdiscussion).Thecriterionin( A )mightseemrathersimilartotheC(x)in( A ),butinfactthereareimportantdifferencesbetweenthesetwocriteria.Arstdifferenceisthat( A )( A )obviouslydonotholdfor~C(x).ConsequentlyonecannotderiveaGS-typealgorithmfor( A )byfollowingtheapproachbasedon( A )and( A ).Ofcourse,we 148

PAGE 149

couldusea~D(x;)denedas~D(x;)=KXk=1j(aHkx)2)]TJ /F3 11.955 Tf 11.95 0 Td[(d2kejkj2 (A)forwhichitholdsthatmin~D(x;)=~C(x),asrequired.However,theminimizationof~D(x;)isnoteasierthanthatof~C(x).Togetaroundtheaboveproblem,aprincipalobservationmadeinChapter 2 wasthat,undercertainconditions,theminimizationof( A )isalmostequivalenttothatofD(x;)in( A ).Usingthisobservationandtheminimizationapproachoutlinedintheparagraphfollowing( A ),thecyclicalgorithmtermedCANwasintroducedforminimizingD(x;).CANhasthesameformasGSA.However,notethatnowtheminimizationofD(x;)doesnotnecessarilyprovideasolutiontotheproblemofminimizing~C(x).Inparticular,aseconddifferencebetweenthecriteriaC(x)and~C(x)isthattheproposedalgorithmsdonotguaranteethatthecriterion~C(x)monotonicallydecreasesastheiterationproceeds(onlyD(x;)ismonotonicallydecreasedbyeachiteration).Finally,weremarkonthefactthattheWeCAN(weightedCAN)andmulti-variateCANalgorithms(Chapter 3 ),althoughrelatedtoGSAintheirbasicprinciples,haveaweakerconnectiontoGSAthanCANandPeCAN.Thesealgorithms,whichhavebeenobtainedbymeansofthealmostequivalentminimizationapproachmentionedinthepreviousparagraph,canbeviewedasextensionsofGSAtoproblemsthathavemoreinvolvedformsthan( A ). 149

PAGE 150

APPENDIXBDERIVATIONOFAUNITARYMATRIXSOLUTIONWeproveinthefollowingthatthesolutionof( 3 )forxedXisgivenby( 3 ).Thecriterionin( 3 )canbewrittenaskX)]TJ 11.95 10.71 Td[(p NUk2=trn(XH)]TJ 11.96 10.71 Td[(p NUH)(X)]TJ 11.95 10.71 Td[(p NU)o (B)=const)]TJ /F5 11.955 Tf 11.96 0 Td[(2p NRetr(XHU)whereconstdenotesthetermthatdoesnotdependonU(notethatXisassumedknownandthatUHU=I).Using( 3 )weobtainRetr(XHU)=Retr(U1SUH2U)=Retr(UH2UU1S)=MPXk=1Re[UH2UU1]kkSkk (B)where[]kkdenotesthe(k;k)thelementofamatrix.Fornotationalsimplicity,letB=UH2UU1andthenwehavejRefBkkgj2jBkkj2[BBH]kk=[UH2UU1UH1UHU2]kk=[UH2UUHU2]kk: (B)NotethatUisatallsemi-unitarymatrix,whichleadstothefactthatUUHIandthusjRefBkkgj2[UH2U2]kk=1: (B)Itfollowsfrom( B )( B )thatkX)]TJ 11.96 10.71 Td[(p NUk2=const)]TJ /F5 11.955 Tf 11.95 0 Td[(2p NMPXk=1Re[UH2UU1]kkSkkconst)]TJ /F5 11.955 Tf 11.96 0 Td[(2p NMPXk=1Skk (B)whichisanotherconstantindependentofU.Itisnotdifculttoseethattheequalityin( B )holdsifandonlyifU=U2UH1,whichconcludestheproof. 150

PAGE 151

APPENDIXCPROPERTIESOFAMBIGUITYFUNCTIONHereweprovethethreeAFpropertiesmentionedinSection 5.1 Maximumvalueattheorigin:ByusingtheCauchy-Schwartzinequalitywecangetj(;f)j2Z1ju(t)j2dtZ1ju(t)]TJ /F3 11.955 Tf 11.96 0 Td[()e)]TJ /F7 7.97 Tf 6.58 0 Td[(j2f(t)]TJ /F7 7.97 Tf 6.58 0 Td[()j2dt=E2 (C)whereEdenotestheenergyofu(t)(Eq.( 5 )).Sincej(0;0)j=E,itfollowsthatthemaximumvalueofj(;f)jisachievedattheorigin. SymmetryAsimplevariablechange(t t+)showsthat()]TJ /F3 11.955 Tf 9.3 0 Td[(;)]TJ /F3 11.955 Tf 9.3 0 Td[(f)=Z1u(t)u(t+)ej2f(t+)dt=Z1u(t)]TJ /F3 11.955 Tf 11.96 0 Td[()u(t)ej2ftdt: (C)whichimpliesthesymmetryproperty:j(;f)j=j()]TJ /F3 11.955 Tf 9.3 0 Td[(;)]TJ /F3 11.955 Tf 9.3 0 Td[(f)j. ConstantvolumeThevolumeofj(;f)j2isgivenbyV=Z1Z1j(;f)j2ddf (C)=Z1Z1Z1u(t)u(t)]TJ /F3 11.955 Tf 11.95 0 Td[()e)]TJ /F7 7.97 Tf 6.58 0 Td[(j2ftdt2ddf:LetW(f)denotetheFouriertransformofu(t)u(t)]TJ /F3 11.955 Tf 12.7 0 Td[().ByusingtheParsevalequalitywegetZ1jW(f)j2df=Z1Z1u(t)u(t)]TJ /F3 11.955 Tf 11.96 0 Td[()e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2ftdt2df (C)=Z1ju(t)u(t)]TJ /F3 11.955 Tf 11.95 0 Td[()j2dt:ThereforeV=Z1Z1ju(t)u(t)]TJ /F3 11.955 Tf 11.95 0 Td[()j2dtd=Z1Z1ju(x)u(y)j2dxdy=Z1ju(x)j2dxZ1ju(y)j2dy=E2 (C)wherethechangeofvariablesfx=t;y=t)]TJ /F3 11.955 Tf 11.95 0 Td[(gareused. 151

PAGE 152

APPENDIXDNARROWBANDTRANSMITBEAMPATTERNInthenarrowbandcase,Bfcandthereforethedistributionofenergyversusfrequencyfisoflessinterest.Instead,thetotalenergyoverfisthequalityofinterest,whichisgivenby(Eq.( 7 ))P(k)=N=2)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xp=)]TJ /F7 7.97 Tf 6.58 0 Td[(N=2Pkp=N=2)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xp=)]TJ /F7 7.97 Tf 6.59 0 Td[(N=2jaHkpypj2: (D)Sincetheinter-elementspacingdisontheorderofthecarrierwavelength,thenarrowbandassumptionBfcimpliesthatfdcos c0()]TJ /F3 11.955 Tf 9.3 0 Td[(B=2fB=2),whichmeansthatthesteeringvectorakpisindependentoffrequency.Thusitssubscriptpcanbedroppedand( D )becomesP(k)=aHk24N=2)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xp=)]TJ /F7 7.97 Tf 6.59 0 Td[(N=2ypyHp35ak=aHk24NXu=1NXv=1x(u)xH(v)0@N=2)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xp=)]TJ /F7 7.97 Tf 6.58 0 Td[(N=2e)]TJ /F7 7.97 Tf 6.58 0 Td[(j2(u)]TJ /F12 5.978 Tf 5.76 0 Td[(v)p N1A35ak=aHk"NNXn=1x(n)xH(n)#ak (D)wherex(n)isdenedasx(n)=x1(n)x2(n)xM(n)T;n=1;:::;N: (D)Theresultin( D ),uptoamultiplicativeconstant,coincideswiththenarrowbandbeampatternexpressionusedin[ 46 ]. 152

PAGE 153

APPENDIXERECEIVEBEAMPATTERNParallelingthediscussioninSection 7.1 ,webrieyformulateherethereceivebeampatternsynthesisproblemforwidebandsignals.Supposethatawidebandsignalg(t)ej2fctwithfrequencyband[fc)]TJ /F3 11.955 Tf 10.36 0 Td[(B=2;fc+B=2]isimpingingfromangle(0180)onaULA.LetG(f)denotetheFouriertransformofg(t).Thesignalreceivedatthemtharrayelementcanbewrittenasrm(t)=gt)]TJ /F5 11.955 Tf 13.15 8.08 Td[((m)]TJ /F5 11.955 Tf 11.95 0 Td[(1)dcos cej2fc(t)]TJ /F14 5.978 Tf 7.78 4.03 Td[((m)]TJ /F14 5.978 Tf 5.75 0 Td[(1)dcos c) (E)=ZB=2)]TJ /F7 7.97 Tf 6.59 0 Td[(B=2G(f)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2(f+fc)(m)]TJ /F14 5.978 Tf 5.76 0 Td[(1)dcos cej2(f+fc)tdf:LetHm(f)denotethefrequencyresponseoftheFIRlterusedtoprocessthedemodulatedsignalrm(t)e)]TJ /F7 7.97 Tf 6.58 0 Td[(j2fct.ThenthereceivebeampatterncanbeexpressedinthefrequencydomainasA(;f+fc)=MXm=1Hm(f)e)]TJ /F7 7.97 Tf 6.59 0 Td[(j2(f+fc)(m)]TJ /F14 5.978 Tf 5.75 0 Td[(1)dcos c2; (E)f2[)]TJ /F3 11.955 Tf 10.5 8.09 Td[(B 2;B 2]whereG(f)isomittedbecauseitisthesameforallarrayelements.ThereceivebeampatternsynthesisproblemcanbestatedasdesigningasetofMltersfhm(t)gMm=1(theFouriertransformofhm(t)isHm(f))suchthatA(;f+fc)matchesadesiredpattern.AspointedoutinthebeginningofChapter 7 ,thereisnoessentialconstraintonfhm(t)gMm=1,thedesignofwhichcanthereforebedonebyahostofapproaches,suchasclassiclterdesignmethods[ 71 ]orconvexoptimization[ 72 ].Ontheotherhand,thetransmitbeampatterndesign,whichhasbeenthetopicofthischapter,ismuchharderbecauseofthePARconstraint,despitethefactthat( 7 )and( E )havethesameform. 153

PAGE 154

REFERENCES [1] M.I.Skolnik,RadarHandbook,SecondEdition.NewYork,NY:McGraw-Hill,1990. [2] J.G.Proakis,DigitalCommunications,Fourthedition.McGraw-HillInc.,2001. [3] G.L.Turin,Anintroductiontomatchedlters,IRETransactionsonInformationTheory,vol.6,pp.311,June1960. [4] M.H.AckroydandF.Ghani,Optimummismatchedltersforsidelobesuppression,IEEETransactionsonAerospaceandElectronicSystems,vol.9,no.2,pp.214,March1973. [5] S.Zoraster,Minimumpeakrangesidelobeltersforbinaryphase-codedwaveforms,IEEETransactionsonAerospaceandElectronicSystems,vol.16,no.1,pp.112,January1980. [6] P.Stoica,J.Li,andM.Xue,Transmitcodesandreceiveltersforradar,IEEESignalProcessingMagazine,vol.25,no.6,pp.94,November2008. [7] J.Capon,Highresolutionfrequency-wavenumberspectrumanalysis,ProceedingsoftheIEEE,vol.57,pp.1408,August1969. [8] J.LiandP.Stoica,AnadaptivelteringapproachtospectralestimationandSARimaging,IEEETransactionsonSignalProcessing,vol.44,no.6,pp.1469,June1996. [9] P.Stoica,A.Jakobsson,andJ.Li,Capon,APESandmatched-lterbankspectralestimation,SignalProcessing,vol.66,no.1,pp.45,April1998. [10] P.Stoica,H.Li,andJ.Li,AnewderivationoftheAPESlter,IEEESignalPro-cessingLetter,vol.6,no.8,pp.205,August1999. [11] T.Yardibi,J.Li,P.Stoica,M.Xue,andA.B.Baggeroer,Sourcelocalizationandsensing:Anonparametriciterativeadaptiveapproachbasedonweightedleastsquares,IEEETransactionsonAerospaceandElectronicSystems,vol.46,no.1,pp.425443,Jan.2010. [12] G.W.Stimson,IntroductiontoAirborneRadar.Mendham,NJ:SciTechPublishing,Inc.,1998. [13] N.LevanonandE.Mozeson,RadarSignals.NY:Wiley,2004. [14] N.Suehiro,Asignaldesignwithoutco-channelinterferenceforapproximatelysynchronizedCDMAsystems,IEEEJournalonSelectedAreasinCommunica-tions,vol.12,no.5,pp.837,Jun1994. [15] D.TseandP.Viswanath,FundamentalsofWirelessCommunication.NewYork,NY:CambridgeUniversityPress,2005. 154

PAGE 155

[16] V.Diaz,J.Urena,M.Mazo,J.Garcia,E.Bueno,andA.Hernandez,UsingGolaycomplementarysequencesformulti-modeultrasonicoperation,IEEE7thInt.Conf.onEmergingTechnologiesandFactoryAutomation,UPCBarcelona,Catalonia,Spain,October1999. [17] J.P.Costas,Astudyofaclassofdetectionwaveformshavingnearlyidealrange-dopplerambiguityproperties,ProceedingsoftheIEEE,vol.72,no.8,pp.996,August1984. [18] H.Deng,Polyphasecodedesignfororthogonalnettedradarsystems,IEEETransactionsonSignalProcessing,vol.52,no.11,pp.3126,November2004. [19] I.C.MooreandM.Cada,Prolatespheroidalwavefunctions,anintroductiontotheslepianseriesanditsproperties,AppliedandComputationalHarmonicAnalysis,vol.16,no.3,pp.208,May2004. [20] I.GladkovaandD.Chebanov,Onthesynthesisproblemforawaveformhavinganearlyidealambiguityfunctions,InternationalConferenceonRadarSystems,Toulouse,France,October2004. [21] S.Golomb,ShiftRegisterSequences.SanFrancisco,CA:Holden-Day,Inc.,1967. [22] D.C.Chu,Polyphasecodeswithgoodperiodiccorrelationproperties(correspondence),IEEETransactionsonInformationTheory,vol.18,no.4,pp.531,July1972. [23] M.FrieseandH.Zottmann,PolyphaseBarkersequencesuptolength31,ElectronicsLetters,vol.30,no.23,pp.1930,November1994. [24] A.R.Brenner,PolyphaseBarkersequencesuptolength45withsmallalphabets,ElectronicsLetters,vol.34,no.16,pp.1576,August1998. [25] P.BorweinandR.Ferguson,Polyphasesequenceswithlowautocorrelation,IEEETransactionsonInformationTheory,vol.51,no.4,pp.1564,April2005. [26] R.H.Barker,Groupsynchronizingofbinarydigitalsystems.InCommunicationTheory,W.Jackson(ed.).London:Butterworths,1953. [27] R.Gold,Optimalbinarysequencesforspreadspectrummultiplexing,IEEETransactionsonInformationTheory,vol.IT-13,pp.619,1967. [28] T.Kasami,Weightdistributionformulaforsomeclassofcycliccodes,Report0475236atCoordinatedScienceLab,UniversityofIllinoisatUrbana,April1966. [29] J.Jedwab,Asurveyofthemeritfactorproblemforbinarysequences.SequencesandTheirApplicationsSETA2004,T.Helleseth,D.Sarwate,H.Y.Song,andK.Yang,Eds.Springer-Verlag,Heidelberg,2005,vol.3486,LectureNotesinComputerScience,pp.30. 155

PAGE 156

[30] H.A.Khan,Y.Zhang,C.Ji,C.J.Stevens,D.J.Edwards,andD.O'Brien,Optimizingpolyphasesequencesfororthogonalnettedradar,IEEESignalProcessingLetters,vol.13,no.10,pp.589,October2006. [31] T.Hholdt,Themeritfactorproblemforbinarysequences.AppliedAlgebra,AlgebraicAlgorithmsandError-CorrectingCodes,M.Fossorier,H.Imai,S.Lin,andA.Poli,Eds.Springer-Verlag,Heidelberg,2006,vol.3857,LectureNotesinComputerScience,pp.51. [32] P.Stoica,J.Li,andX.Zhu,Waveformsynthesisfordiversity-basedtransmitbeampatterndesign,IEEETransactionsonSignalProcessing,vol.56,no.6,pp.2593,June2008. [33] F.F.KretschmerJr.andK.Gerlach,Lowsideloberadarwaveformsderivedfromorthogonalmatrices,IEEETransactionsonAerospaceandElectronicSystems,vol.27,no.1,pp.92,January1991. [34] C.V.Jakowatz,Jr.,D.E.Wahl,P.H.Eichel,D.C.Ghiglia,andP.A.Thompson,Spotlight-ModeSyntheticApertureRadar:ASignalProcessingApproach.Norwell,MA:KluwerAcademicPublishers,1996. [35] J.Ling,T.Yardibi,X.Su,H.He,andJ.Li,Enhancedchannelestimationandsymboldetectionforhighspeedmulti-inputmulti-outputunderwateracousticcommunications,JournaloftheAcousticalSocietyofAmerica,vol.125,no.5,pp.3067,May2009. [36] H.D.SchottenandH.D.Luke,Onthesearchforlowcorrelatedbinarysequences,InternationalJournalofElectronicsandCommunications,vol.59,no.2,pp.67,2005. [37] J.Li,P.Stoica,andX.Zheng,SignalsynthesisandreceiverdesignforMIMOradarimaging,IEEETransactionsonSignalProcessing,vol.56,no.8,pp.3959,August2008. [38] P.StoicaandR.L.Moses,SpectralAnalysisofSignals.UpperSaddleRiver,NJ:Prentice-Hall,2005. [39] N.ZhangandS.W.Golomb,Polyphasesequencewithlowautocorrelations,IEEETransactionsonInformationTheory,vol.39,no.3,pp.1085,May1993. [40] R.Frank,Polyphasecodeswithgoodnonperiodiccorrelationproperties,IEEETransactionsonInformationTheory,vol.9,no.1,pp.43,January1963. [41] E.Fishler,A.Haimovich,R.Blum,L.Cimini,D.Chizhik,andR.Valenzuela,Spatialdiversityinradars-modelsanddetectionperformance,IEEETransactionsonSignalProcessing,vol.54,no.3,pp.823,March2006. [42] J.Li,P.Stoica,L.Xu,andW.Roberts,OnparameteridentiabilityofMIMOradar,IEEESignalProcessingLetters,vol.14,no.12,pp.968,December2007. 156

PAGE 157

[43] D.W.BlissandK.W.Forsythe,Multiple-inputmultiple-output(MIMO)radarandimaging:degreesoffreedomandresolution,37thAsilomarConferenceonSignals,SystemsandComputers,PacicGrove,CA,vol.1,pp.54,November2003. [44] L.Xu,J.Li,andP.Stoica,TargetdetectionandparameterestimationforMIMOradarsystems,IEEETransactionsonAerospaceandElectronicSystems,vol.44,no.3,pp.927,July2008. [45] D.R.FuhrmannandG.SanAntonio,TransmitbeamformingforMIMOradarsystemsusingsignalcross-correlation,IEEETransactionsonAerospaceandElectronicSystems,vol.44,no.1,pp.1,January2008. [46] P.Stoica,J.Li,andY.Xie,OnprobingsignaldesignforMIMOradar,IEEETransactionsonSignalProcessing,vol.55,no.8,pp.4151,August2007. [47] Y.YangandR.S.Blum,MIMOradarwaveformdesignbasedonmutualinformationandminimummean-squareerrorestimation,IEEETransactionsonAerospaceandElectronicSystems,vol.43,no.1,pp.330,January2007. [48] B.Friedlander,WaveformdesignforMIMOradars,IEEETransactionsonAerospaceandElectronicSystems,vol.43,no.3,pp.1227,July2007. [49] Y.YangandR.S.Blum,MinimaxrobustMIMOradarwaveformdesign,IEEEJournalofSelectedTopicsinSignalProcessing,vol.1,no.1,pp.147,June2007. [50] J.OppermannandB.S.Vucetic,Complexspreadingsequenceswithawiderangeofcorrelationproperties,IEEETransactionsonCommunications,vol.45,no.3,pp.365,March1997. [51] L.R.Welch,Lowerboundsonthemaximumcorrelationofsignals,IEEETransac-tionsonInformationTheory,vol.IT-20,no.3,pp.397,May1974. [52] D.V.Sarwate,MeetingtheWelchboundwithequality,inSequencesandTheirApplications(ProceedingsofSETA'98),pp.79,Springer,London,UK,1999. [53] J.A.Tropp,I.S.Dhillon,R.W.Heath,andT.Strohmer,Designingstructuredtightframesviaanalternatingprojectionmethod,IEEETransactionsonInformationTheory,vol.51,no.1,pp.188,January2005. [54] P.M.Woodward,ProbabilityandInformationTheorywithApplicationstoRadar.NewYork,NY:Pergamon,1957. [55] S.Sussman,Least-squaresynthesisofradarambiguityfunctions,IEEETransac-tionsonInformationTheory,vol.8,no.3,pp.246,1962. [56] J.D.Wolf,G.M.Lee,andC.E.Suyo,Radarwaveformsynthesisbymean-squareoptimizationtechniques,IEEETransactionsonAerospaceandElectronicSystems,vol.5,no.4,pp.611,1968. 157

PAGE 158

[57] J.C.GueyandM.R.Bell,Diversitywaveformsetsfordelay-dopplerimaging,IEEETransactionsonInformationTheory,vol.44,no.4,July1998. [58] A.Bonami,G.Garrigos,andP.Jaming,Discreteradarambiguityproblems,AppliedandComputationalHarmonicAnalysis,vol.23,pp.388,November2007. [59] Y.I.AbramovichandG.J.Frazer,BoundsonthevolumeandheightdistributionsfortheMIMOradarambiguityfunction,IEEESignalProcessingLetters,vol.15,pp.505,2008. [60] S.Stein,Algorithmsforambiguityfunctionprocessing,IEEETransactionsonAcoustics,SpeechandSignalProcessing,vol.29,no.3,June1981. [61] R.Sharma,AnalysisofMIMOradarambiguityfunctionsandimplicationsonclearregion,IEEEInternationalRadarConference,WashingtonDC,USA,May2010. [62] H.L.VanTrees,OptimumArrayProcessing:PartIVofDetection,Estimation,andModulationTheory.NewYork,NY:JohnWiley&Sons,2002. [63] S.Haykin,Cognitiveradar:awayofthefuture,IEEESignalProcessingMagazine,vol.23,no.1,pp.30,January2006. [64] M.J.Lindenfeld,Sparsefrequencytransmitandreceivewaveformdesign,IEEETransactionsonAerospaceandElectronicSystems,vol.40,no.3,July2004. [65] J.Salzman,D.Akamine,andR.Lefevre,OptimalwaveformsandprocessingforsparsefrequencyUWBoperation,ProceedingsofIEEERadarConference,pp.105,2001. [66] G.WangandY.Lu,Designingsingle/multiplesparsefrequencytrainsmitwaveformswithsidelobeconstraint,submittedtoIETRadarSonar&Naviga-tion,October2009. [67] J.M.HeadrickandM.I.Skolnik,Over-the-horizonradarintheHFband,Proceed-ingsoftheIEEE,vol.62,no.6,pp.664,June1974. [68] R.J.Mailloux,Phasedarraytheoryandtechnology,ProceedingsoftheIEEE,vol.70,no.3,pp.246,March1982. [69] C.L.Dolph,Acurrentdistributionforbroadsidearrayswhichoptimizestherelationshipbetweenbeamwidthandside-lobelevel,ProceedingsoftheIRE,vol.34,no.6,pp.335,June1946. [70] R.Elliott,Designoflinesourceantennasfornarrowbeamwidthandasymmetriclowsidelobes,IEEETransactionsonAntennasandPropagation,vol.23,no.1,pp.100,January1975. 158

PAGE 159

[71] D.B.Ward,R.A.Kennedy,andR.C.Williamson,FIRlterdesignforfrequencyinvariantbeamformers,IEEESignalProcessingLetters,vol.3,no.3,pp.69,March1996. [72] H.LebretandS.Boyd,Antennaarraypatternsynthesisviaconvexoptimization,IEEETransactionsonSignalProcessing,vol.45,no.3,pp.526,March1997. [73] D.R.ScholnikandJ.O.Coleman,Formulatingwidebandarray-patternoptimizations,ProceedingofIEEEInternationalConferenceonPhasedArraySystemsandTechnology,pp.489,May2000. [74] G.Cardone,G.Cincotti,andM.Pappalardo,Designofwide-bandarraysforlowside-lobelevelbeampatternsbysimulatedannealing,IEEETransactionsonUltrasonics,FerroelectricsandFrequencyControl,vol.49,no.8,August2002. [75] G.SanAntonioandD.R.Fuhrmann,BeampatternsynthesisforwidebandMIMOradarsystems,TheFirstIEEEInternationalWorkshoponComputationalAdvancesinMulti-SensorAdaptiveProcessing,PuertoVallarta,Mexico,pp.105,December13-152005. [76] J.Li,Y.Xie,P.Stoica,X.Zheng,andJ.Ward,Beampatternsynthesisviaamatrixapproachforsignalpowerestimation,IEEETransactionsonSignalProcessing,vol.55,no.12,pp.5643,December2007. [77] L.K.PattonandB.D.Rigling,Modulusconstraintsinadaptiveradarwaveformdesign,IEEERadarConference,Rome,Italy,May2008. [78] K.W.ForsytheandD.W.Bliss,WaveformcorrelationandoptimizationissuesforMIMOradar,39thAsilomarConferenceonSignals,SystemsandComputers,PacicGrove,CA,pp.1306,November2005. [79] B.GuoandJ.Li,Waveformdiversitybasedultrasoundsystemforhyperthermiatreatmentofbreastcancer,IEEETransactionsonBiomedicalEngineering,vol.55,no.2,pp.822,February2008. [80] H.T.Hui,Decouplingmethodsforthemutualcouplingeffectinantennaarrays:Areview,RecentPatentsonEngineering,vol.1,no.2,pp.187,2007. [81] G.Frazer,Y.Abramovich,andB.Johnson,Spatiallywaveformdiverseradar:PerspectivesforhighfrequencyOTHR,IEEERadarConference,Boston,MA,USA,pp.385,June2007. [82] T.Svantesson,Modelingandestimationofmutualcouplinginauniformlineararrayofdipoles,The1999IEEEInternationalConferenceonAcoustics,Speech,andSignalProcessing,Phoenix,Arizona,USA,March1999. [83] R.W.GerchbergandW.O.Saxton,Apracticalalgorithmforthedeterminationofthephasefromimageanddiffractionplanepictures,Optik(Stuttgart),vol.35,pp.237,1972. 159

PAGE 160

[84] A.J.WeissandJ.Picard,Maximum-likelihoodpositionestimationofnetworknodesusingrangemeasurements,IETSignalProcessing,vol.2,no.4,pp.394,2008. [85] J.R.Fienup,Phaseretrievalalgorithms:acomparison,AppliedOptics,vol.21,no.15,pp.2758,February1982. 160

PAGE 161

BIOGRAPHICALSKETCH HaoHereceivedthedegreeofBachelorofSciencefromtheUniversityofScienceandTechnologyofChina,Hefei,China,in2007,andthedegreeofMasterofSciencefromtheUniversityofFlorida,Gainesville,FL,in2009,bothinelectricalengineering.HegraduatedwiththedegreeofDoctorofPhilosophyfromtheDepartmentofElectricalandComputerEngineeringatUniversityofFloridainthesummerof2011.Hisresearchinterestsareintheareasofradar/sonarwaveformdesignandspectralestimation.Hewonthebeststudentpaperaward(thirdplace)attheIEEE13thDSPWorkshop&5thSPEWorkshop,MarcoIsland,FL,USA,January,2009,andthebeststudentpaperaward(top10)atthe2ndInternationalWorkshoponCognitiveInformationProcessing,ElbaIsland,Italy,June,2010.HeworkedasasoftwaredevelopmentengineerduringasummerinternshipatAmazonCorporateLLC,Seattle,WA,in2010. 161