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Growth Curve Models in Signal Processing Applications

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IfeelextremelyluckytohavemetallthepeopleIhaveandtohavereceivedtheirhelp.Foremost,Iwouldliketoexpressmymostsinceregratitudetomyadvisor,Dr.JianLi,forherconstantsupport,encouragement,andguidance.Iamspeciallygratefulfortheopportunityshehasoeredmetopursuemyresearchunderhersupervision.Shegivesmefreedomtotrynewideas,whileprovidingtheneededassistanceattherighttime.Sheisalwayswillingtoshareherknowledgeandcareerexperienceasanadvisor,collaborator,andfriend.Theexperienceworkingwithherhasproventobeinvaluableandmemorable.IwouldalsoliketothankDr.JenshenLin,Dr.ClintSlatton,andDr.RonglingWuinthedepartmentofstatisticsforservingonmysupervisorycommitteeandfortheirvaluableadvice.Theirwonderfulteachinghaswidenedmyhorizontotheirintriguingresearchelds,andwasandwillalwaysbehelpfultomyresearchwork.MyspecialgratitudeisduetoDr.PetreStoicaatUppsalaUniversity,Sweden,forhisguidanceinmanyinterestingtopics.Hiswideknowledge,stronganalyticalskill,andkeeninsighthaveneverceasedtoamazeme.Iwassofortunatetohavetheopportunitytoworkwithhimandtobenetfromhisinsightfulideasandconstructiveadvice.IamgratefultoDr.MingzhouDinginthedepartmentofbiomedicalengineeringforhissupportandguidanceinmyresearchonneuraldataanalysis.IalsowanttothankmygroupmatesinSpectralAnalysisLab:BinGuo,YiJiang,JianhuaLiu,ZhipengLiu,GuoqingLiu,WilliamRoherts,YijunSun,YanweiWang,ZhisongWang,YaoXie,HongXiong,ChangjiangXu,XiayuZheng,XuminZhu,andothers.Theirfriendship,support,andhelphavemademylifeeasierandmoreenjoyable. iv

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v

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page ACKNOWLEDGMENTS ............................. iv LISTOFFIGURES ................................ vii ABSTRACT .................................... viii CHAPTER 1INTRODUCTION .............................. 1 1.1Growth-CurveModelandItsVariations ............... 1 1.2Multiple-InputMultiple-OutputRadar ................ 4 1.3StudyOverview ............................. 6 2GROWTH-CURVEMODEL ........................ 8 2.1IntroductionandProblemFormulation ................ 8 2.2MultivariateParameterEstimation .................. 9 2.2.1MultivariateCaponEstimation ................ 9 2.2.2MultivariateMaximumLikelihoodEstimation ........ 10 2.3PerformanceAnalysis .......................... 12 2.3.1PerformanceAnalysisoftheMultivariateMLEstimator ... 12 2.3.1.1Biasanalysis ..................... 13 2.3.1.2Mean-squared-erroranalysis ............. 13 2.3.2PerformanceAnalysisoftheMultivariateCaponEstimator 19 2.3.2.1Biasanalysis ..................... 19 2.3.2.2Mean-squared-erroranalysis ............. 23 2.4NumericalExamples .......................... 27 2.5Conclusions ............................... 30 3DIAGONALGROWTH-CURVEMODEL ................. 33 3.1IntroductionandProblemFormulation ................ 33 3.2ApproximateMaximumLikelihoodEstimation ............ 34 3.3PerformanceAnalysis .......................... 38 3.3.1BiasAnalysis .......................... 38 3.3.2Mean-Squared-ErrorAnalysis ................. 39 3.4NumericalExamples .......................... 40 3.4.1ExamplesinArraySignalProcessing ............. 40 3.4.2SpectralAnalysisExamples .................. 44 vi

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............................... 50 4BLOCKDIAGONALGROWTH-CURVEMODEL ............ 51 4.1IntroductionandProblemFormulation ................ 51 4.2PreliminaryResults ........................... 52 4.3ApproximateMaximumLikelihoodEstimation ............ 55 4.4PerformanceAnalysis .......................... 58 4.4.1BiasAnalysis .......................... 58 4.4.2Mean-Squared-Error(MSE)Analysis ............. 59 4.5NumericalResults ............................ 60 4.6Conclusions ............................... 63 5ITERATIVEGENERALIZEDLIKELIHOODRATIOTESTFORMIMORADAR .................................... 65 5.1IntroductionandSignalModel ..................... 65 5.2SeveralSpatialSpectralEstimators .................. 67 5.2.1Capon .............................. 68 5.2.2APES .............................. 69 5.3GeneralizedLikelihoodRatioTest ................... 70 5.3.1GeneralizedLikelihoodRatioTest(GLRT) .......... 70 5.3.2ConditionalGeneralizedLikelihoodRatioTest(cGLRT) .. 73 5.3.3IterativeGeneralizedLikelihoodRatioTest(iGLRT) .... 78 5.4NumericalExamples .......................... 79 5.4.1Cramer-RaoBound ....................... 79 5.4.2TargetDetectionandLocalization ............... 81 5.5Conclusions ............................... 88 6CONCLUSIONSANDFUTUREWORK .................. 89 APROOFSFORTHEGROWTH-CURVEMODEL ............ 92 A.1Cramer-RaoBoundfortheGCModel ................ 92 A.2ProofofLemma 2.1 ........................... 94 A.3ProofofLemma 2.2 ........................... 96 A.4ProofofLemma 2.3 ........................... 96 A.5ProofofLemma 2.4 ........................... 98 BPROOFSFORTHEDIAGONALGROWTH-CURVEMODEL ..... 102 B.1ProofofLemma 3.1 ........................... 102 B.2ProofofLemma 3.2 ........................... 102 B.3GeneralizedLeast-Squares(GLS)InterpretationofAMLin( 3{18 ) 102 B.4Cramer-RaoBoundfortheDGCmodel ................ 104 CCRAMER-RAOBOUNDFORTHEMIMORADAR ........... 105 vii

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................................... 108 BIOGRAPHICALSKETCH ............................ 116 viii

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Figure page 2{1BiasversusLwhenSNR=10dB,K=2,N=2. ............. 28 2{2BiasversusKwhenSNR=10dB,L=16,N=2. ............ 29 2{3MSEversusLwhenSNR=10dB,K=2,N=2. ............. 30 2{4MSEversusSNRwhenL=16,K=2,N=2. ............... 31 2{5MSEversusKwhenSNR=10dB,L=16,N=2. ............ 31 2{6MSEversuswhenSNR=10dB,L=16,K=2,N=2. ........ 32 3{1EmpiricalMSE'sandtheCRBversusLwhenSNR=10dB,M=6,andN=3withlinearlyindependentsteeringvectorsandlinearlyindependentwaveforms. .................................. 41 3{2EmpiricalMSE'sandtheCRBversusSNRwhenL=128,M=6,andN=3withlinearlyindependentsteeringvectorsandlinearlyindependentwaveforms. .................................. 42 3{3EmpiricalMSE'sandtheCRBversusLwhenSNR=10dB,M=6,andN=3withidenticalwaveforms. ....................... 43 3{4EmpiricalMSE'sandtheCRBversusSNRwhenL=128,M=6,andN=3withidenticalwaveforms. ....................... 44 3{5EmpiricalMSE'sandtheCRBversusLwhenSNR=10dB,M=6,andN=13withlinearlydependentsteeringvectors .............. 45 3{6EmpiricalMSE'sandtheCRBversusSNRwhenL=128,M=6,andN=13withlinearlydependentsteeringvectors. .............. 46 3{7EmpiricalMSE'sandtheCRBversuslocalSNRwhenL0=32,M=8,andtheobservationnoiseiscolored.(a)For3and(b)for1. ..... 48 3{8EmpiricalMSE'sandtheCRBversusMwhenL0=32,2=0:01,andtheobservationnoiseiscolored.(a)For3and(b)for1. ........ 48 4{1EmpiricalMSE'sandtheCRBversusLwhenSNR=10dB.(a)For1,(b)for2,and(c)for3. .......................... 63 ix

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......................... 64 5{1CumulativedensityfunctionsoftheCramer-Raoboundsfor(a)1and(b)3. .................................... 80 5{2OutageCRBversusSNR.(a)CRB0:01for1,(b)CRB0:01for2,(c)CRB0:1for1,and(d)CRB0:1for2. ................... 82 5{3Spatialspectra,andGLRandcGLRPseudo-Spectra,when1=40,2=20,and3=0.(a)LS,(b)Capon,(c)APES,(d)GLRT,and(e)iGLRT. .................................. 83 5{4Spatialspectra,andGLRandcGLRPseudo-Spectra,when1=40,2=4,and3=0.(a)Capon,(b)APES,(c)GLRT,and(d)iGLRT. 85 5{5GLRandcGLRPseudo-SpectraobtainedinStepsIandIIofiGLRT,when1=40,2=4,and3=0.(a)(),(b)(j^1),(c)(j^1;^2),and(d)(j^1;^2;^3). ...................... 86 5{6CumulativedensityfunctionsoftheCRBsandMSEsfor(a)1and(b)3. 86 5{7OutageCRB0:1andMSE0:1versusSNRfor(a)1and(b)3. ...... 87 5{8OutageCRB0:1andMSE0:1versusSNRfor(a)1and(b)3. ...... 87 x

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Asapowerfulstatisticaltool,thegrowth-curve(GC)modelisattractingincreasingattentionsinvariousareas.Inthisdissertation,westudyseveralvariationsofthegrowth-curvemodel,anddiscusstheirapplicationstotheemergingmultiple-inputmultiple-output(MIMO)radarsystem. WerststudythestatisticalpropertiesoftwoestimatorsfortheregressioncoecientmatrixintheGCmodel,i.e.,themaximumlikelihood(ML)andCaponmethods.Wederivetheclosed-formexpressionoftheCramer-Raobound(CRB)fortheunknownregressioncoecientmatrix,andthenanalyzethebiaspropertiesandmean-squarederrors(MSEs)ofthetwoestimators.WeshowthatthemultivariateMLestimatorisunbiasedwhereasthemultivariateCaponestimatorisbiaseddownwardfornitedatasamples.Bothestimatorsareasymptoticallystatisticallyecientwhenthenumberofdatasamplesislarge. Next,weconsideravariationoftheGCmodel,referredtoasthediagonalgrowth-curve(DGC)model,wheretheregressionmatrixisconstrainedtobediagonal.Aclosed-formapproximatemaximumlikelihood(AML)estimatorforthismodelisderivedbasedonthemaximumlikelihoodprinciple.Weanalyze xi

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Thenweconsiderageneralgrowth-curvemodel,referredtoastheblockdiagonalgrowth-curve(BDGC)model,wheretheunknownregressioncoecientmatrixisconstrainedtobeblock-diagonal,andwhichcanunifytheGCandDGCmodels.Weproposedaclosed-formapproximatemaximumlikelihood(AML)estimatorfortheblock-diagonalconstrainedmatrix,whichisprovedtobeunbiasedandasymptoticallystatisticallyecientforalargedatasamplenumber.Severalapplicationsofthismodelinsignalprocessingarethenpresented. Finally,weconsideramultiple-inputmultiple-output(MIMO)radarsystemwithageneralantennaconguration,i.e.,boththetransmitterandreceiverhavemultiplewell-separatedsubarrayswitheachsubarraycontainingclosely-spacedantennas.Hence,boththecoherentprocessinggainandthespatialdiversitygaincanbeachievedbythesystemsimultaneously.Weintroduceseveralspatialspectralestimators,includingCaponandAPES,fortargetdetectionandparameterestimation.Wealsoprovideageneralizedlikelihoodratiotest(GLRT)andaconditionalgeneralizedlikelihoodratiotest(cGLRT)forthesystem.BasedonGLRTandiGLRT,wethenproposeaniterativeGLRT(iGLRT)procedurefortargetdetectionandparameterestimation.Viaseveralnumericalexamples,weshowthatiGLRTcanprovideexcellentdetectionandestimationperformanceatalowcomputationalcost. xii

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1 ]forinvestigatinggrowthcurveproblemsinstatisticalapplications.Sincethenithasbeenstudiedbymanyauthors,includingRao[ 2 ]-[ 4 ],Khatri[ 5 ],GleserandOlkin[ 6 ],Geisser[ 7 ],vonRosen[ 8 ]-[ 10 ],VerbylaandVenables[ 11 ],andSrivastava[ 12 ].Itisoneofthemaintoolsfordealingwithlongitudinaldata,especiallyforserialcorrelations[ 13 ]aswellasrepeatedmeasurements[ 14 ]-[ 18 ],andisattractingincreasingattentionsinvariousareas,suchaseconomics,biology,medicalresearchandepidemiology.Recently,thismodelwasextendedtothecomplex-valuedeldandwasadoptedinthesignalprocessingliterature[ 19 ]-[ 22 ]. ConsideranobserveddatamatrixX2CML,whichcanbewrittenas In( 1{1 ),A2CMNandS2CKLarebothknownmatrices,B2CNKisanunknownregressionmatrix,andZ2CMListheerrormatrixwhosecolumnsareindependentlyandidenticallydistributed(i.i.d.)zero-meanGaussianrandomvectorswithanunknowncovariancematrix.TheproblemofinterestistoestimateBfromtheobserveddatamatrixX. Aspecialcaseof( 1{1 ),whenN=K=1,hasbeenstudiedwidelyinsignalprocessing,suchashigh-resolutionspectralanalysis[ 23 ][ 24 ]andarraysignalprocessing[ 25 ]-[ 35 ].Inthiscase,theGCmodelreducestoaunivariateGC 1

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(UGC)model withaandsTbeingcolumnandrowvectors,respectively,andbeinganunknownscalarvariable.Theperformanceofthemaximum-likelihood(ML)andCaponestimatorsfortheUGCmodelhasbeenthoroughlystudiedin[ 25 ].ItwasshowntheoreticallythatMLisunbiasedwhereasCaponisbiaseddownward,andbothestimatorsareasymptoticallystatisticallyecientforalargenumberofdatasamples(i.e.,LM).Inthisdissertation,wewillextendthisresulttotheGCmodel. Inmanypracticalapplications,theobservedsignalconsistsofmultiplecomponents.Forthisscenario,theUGCmodelin( 1{2 )canbeextendedas withfakgandfsTkgbeingtheknowncolumnandrowvectors,andfkgbeingunknownscalarvariables.Obviously,themodelin( 1{3 )canberewrittenas wherediag()denotesadiagonalmatrixwithitsdiagonalformedbytheelementsofthevector,A=[a1aK],S=[s1sK]T,and=[1K]T.Wenotethatthemodelin( 1{4 )issimilartotheGCmodelin( 1{1 )exceptthattheunknownregressioncoecientmatrixBisconstrainedtobediagonal.Hence,itisreferredtoasadiagonalgrowth-curve(DGC)model[ 36 ].DespitetheseeminglyminordierencebetweentheGCandDGCmodels,theMLestimator[ 21 ][ 22 ]fortheGCmodelisinvalidfortheDGCmodel.Infact,toourknowledge,noclosed-formMLestimatorforin( 1{4 )existsintheliterature.Inthisdissertation,weproposeanapproximatemaximumlikelihood(AML)estimatorforin( 1{4 ).Wealsoinvestigateitsstatisticalpropertiesviatheoreticalanalysisandnumerical

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simulations,andshowthattheAMLestimatorfortheDGCmodelisunbiasedandasymptoticallystatisticallyecientforalargenumberofdatasamples. AmoregeneralvariationoftheGCmodelwasstudiedbyVerbyla[ 11 ],Rosen[ 8 ]andSrivastava[ 12 ],wheretheauthorsconsideranestimationproblemofunknownregressionmatricesfBkgfromtheobserveddatamatrixXintheequation Again,in( 1{5 ),fAkgandfSkg(k=1;2;;K)areallknownmatrices,andZisdenedasintheGCmodel.Notethat( 1{5 )canberewrittenintheformoftheGCmodelin( 1{1 )byconstrainingtheunknownregressioncoecientmatrixBtobeablock-diagonalmatrix;hence( 1{5 )isreferredtoasablockdiagonalgrowth-curve(BDGC)modelinthisdissertation.AniterativenumericalapproachfortheestimationoftheunknownregressioncoecientmatricesBkin( 1{5 )wasproposedbyVerbyla[ 11 ]byusingthecanonicalreductionmethod.However,thisapproachisbothconceptuallyandpracticallycomplicated.Moreover,beinganiterativemethod,itmaysuerfromconvergenceproblems.TwonestedvariationsoftheBDGCmodelwerestudiedbyRosen[ 8 ]andSrivastava[ 12 ],independently,whereexplicitformsoftheMLestimatorswerepresented.However,someadditionalassumptionshavebeimposedin[ 8 ]and[ 12 ].In[ 8 ],therowsofSk(k=1;2;K)areassumedtobenested,i.e.,R(STK)R(STK1)R(ST1),withR()denotingtherangespace;andin[ 12 ]thecolumnsofAk(k=1;2;K)areassumedtobenested,i.e.,R(AK)R(AK1)R(A1).Neitherofthesetwonestedsubspaceconditionscanbesatisedinsignalprocessingapplications.Inthisdissertation,wewillconsiderageneralBDGCmodelin( 1{5 ),andproposeanapproximatemaximumlikelihood(AML)estimator[ 37 ]fortheunknownregressioncoecientmatricesBk,whichwillbeshownboththeoreticallyandnumericallytobeunbiasedandasymptoticallystatisticallyecient.

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38 ][ 39 ].Ithasbeenshownthatbyexploitingthiswaveformdiversity,MIMOradarcanovercomeperformancedegradationscausedtheradarcross-section(RCS)uctuations[ 40 ]-[ 43 ],achieveexiblespatialtransmitbeampatterndesign[ 44 ][ 45 ],providehigh-resolutionspatialspectralestimates[ 46 ]-[ 57 ],andsignicantlyimprovetheparameteridentiability[ 58 ]. ThestatisticalMIMOradar,studiedin[ 40 ]-[ 43 ],aimsatresistingthe\scintillation"eectencounteredinradarsystems.Itiswell-knownthattheRCSofatarget,whichrepresentstheamountofenergyreectedfromthetargettowardthereceiver,changesrapidlyasafunctionofthetargetaspect[ 59 ]andthelocationsofthetransmittingandreceivingantennas.Thetarget\scintillation"causesseveredegradationsinthetargetdetectionandparameterestimationperformanceoftheradar.Byspacingthetransmitantennas,whichtransmitlinearlyindependentsignals,farawayfromeachother,aspatialdiversitygaincanbeobtainedasintheMIMOwirelesscommunicationstothis\scintillation"eect[ 40 ]-[ 43 ]. Flexibletransmitbeampatterndesignsareinvestigatedin[ 44 ]and[ 45 ].Dierentfromthe\statistical"MIMOradarabove,thetransmittingantennasarecloselyspaced.Theauthorsin[ 44 ]and[ 45 ]showthatthewaveformstransmittedviaitsantennascanbeoptimizedtoobtainseveraltransmitbeampatterndesignswithsuperiorperformance.Forexample,thecovariancematrixofthewaveformscanbeoptimizedtomaximizethepoweraroundthelocationsofinterestandalsotominimizethecross-correlationofthesignalsreectedbacktotheradarbythesetargets,therebysignicantlyimprovingtheperformanceoftheadaptiveMIMO

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radartechniques.DuetothesignicantlylargernumberofdegreesoffreedomofaMIMOsystem,amuchbettertransmitbeampatternwithaMIMOradarcanbeachievedthanwithitsphased-arraycounterpart. In[ 48 ],aMIMOradartechniqueissuggestedtoimprovetheradarresolution.TheideaistotransmitN(N>1)orthogonalcodedwaveformsbyNantennasandtoreceivethereectedsignalsbyM(M>1)antennas.Ateachreceivingantennaoutput,thesignalismatched-lteredusingeachofthetransmittedwaveformstoobtainNMchannels,towhichthedata-adaptiveCaponbeamformer[ 60 ]isapplied.Itisprovedin[ 48 ]thatthebeampatternoftheproposedMIMOradarisobtainedbythemultiplicationofthetransmittingandreceivingbeampatterns,whichgiveshighresolution.However,onlythesingle-targetcaseisconsideredin[ 48 ]. AMIMOradarschemeisconsideredin[ 55 ]-[ 57 ]thatcandealwiththepresenceofmultipletargets.SimilartosomeoftheaforementionedMIMOradarapproaches,linearlyindependentwaveformsaretransmittedsimultaneouslyviamultipleantennas.Duetothedierentphaseshiftsassociatedwithdierentpropagationpathsfromtransmittingantennastotargets,theseindependentwaveformsarelinearlycombinedatthetargetswithdierentphasefactors.Asaresult,thesignalwaveformsreectedfromdierenttargetsarelinearlyindependentofeachother,whichallowsthedirectapplicationofmanyadaptivetechniquestoachievehighresolutionandexcellentinterferencerejectioncapability.Severaladaptivenonparametricalgorithmsinthepresenceorabsenceofsteeringvectorerrorsarepresentedin[ 55 ]-[ 57 ]. NotethattheMIMOradarsdiscussedintheaforementionedliteraturecanbegroupedintotwoclassesaccordingtotheirantennacongurations.Oneistheconventionalradararray,inwhichbothtransmittingandreceivingantennasarecloselyspacedforcoherenttransmissionanddetection[ 44 ]-[ 57 ].Theotheristhe

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diverseantennaconguration,wheretheantennasareseparatedfarawayfromeachothertoachievespatialdiversitygain[ 40 ]-[ 43 ].Toreapthebenetsofbothschemes,inthisdissertation,weconsiderageneralantennaconguration,i.e.,boththetransmittingandreceivingantennaarraysconsistofseveralwell-separatedsubarrayswitheachsubarraycontainingclosely-spacedantennas[ 61 ].Byusingsomeresultsofthegrowth-curvemodels,weprovideageneralizedlikelihoodratiotest(GLRT)andaconditionalgeneralizedlikelihoodratiotest(cGLRT)forthesystem.BasedonGLRTandiGLRT,wethenproposeaniterativeGLRT(iGLRT)procedurefortargetdetectionandparameterestimation.Viaseveralnumericalexamples,weshowthatiGLRTcanprovideexcellentdetectionandestimationperformanceatalowcomputationalcost. 2 ,weconsiderestimatingtheunknownregressioncoecientmatrixBintheGCmodelin( 1{1 ).Twomultivariateapproaches,MaximumLikelihood(ML)andCapon,areprovided.Wederivetheclosed-formexpressionoftheCramer-Raobound(CRB)fortheunknowncomplexamplitudes.WealsoanalyzethebiaspropertiesandMeanSquaredErrors(MSE)ofthetwoestimators.AcomparativestudyshowsthatthemultivariateMLestimatorisunbiasedwhereasthemultivariateCaponestimatorisbiaseddownwardfornitedatasamples.Bothestimatorsareasymptoticallystatisticallyecientwhenthenumberofdatasamplesislarge. InChapter 3 ,weconsideravariationoftheGCmodel,referredtoasthediagonalgrowth-curve(DGC)model,wherethematricesAandSin( 1{4 )arebothknownandtheregressioncoecientmatrixBisconstrainedtobediagonal.Aclosed-formapproximatemaximumlikelihood(AML)estimatorforthismodelisderivedbasedonthemaximumlikelihoodprinciple.WeanalyzethestatisticalpropertiesofthismethodtheoreticallyandshowthattheAML

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estimateisunbiasedandasymptoticallystatisticallyecientforalargenumberofdatasamples.Viaseveralnumericalexamplesinarraysignalprocessingandspectralanalysis,wealsoshowthattheproposedAMLestimatorcanachievebetterestimationaccuracyandexhibitgreaterrobustnessthanthebestexistingmethods. InChapter 4 ,weconsiderageneralvariationofthegrowth-curve(GC)model,referredtoastheblockdiagonalgrowth-curve(BDGC)model,whichcanunifytheGCandDGCmodelsinChapters2and3.InBDGC,theunknownregressioncoecientmatrixisconstrainedtobeblock-diagonal.Aclosed-formapproximatemaximumlikelihood(AML)estimatorforthismodelisthenderived,whichisshowntobeunbiasedandasymptoticallystatisticallyecientforalargenumberofdatasamples. InChapter 5 ,weconsideramultiple-inputmultiple-output(MIMO)radarsystemwithageneralantennaconguration,i.e.,boththetransmitterandreceiverhavemultiplewell-separatedsubarrayswitheachsubarraycontainingclosely-spacedantennas.Weintroduceseveralspatialspectralestimators,includingCaponandAPES,fortargetdetectionandparameterestimation.Wealsoprovideageneralizedlikelihoodratiotest(GLRT)andaconditionalgeneralizedlikelihoodratiotest(cGLRT)forthesystem.BasedonGLRTandiGLRT,wethenproposeaniterativeGLRT(iGLRT)procedurefortargetdetectionandparameterestimation. Finally,wesummarizethedissertationandpointoutfutureresearchdirectionsinChapter 6

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In( 2{1 ),X2CMLdenotestheobservedsnapshotswithLbeingthenumberofsnapshots.ThecolumnsinA2CMNaretheknownlinearlyindependentspatialvectors,e.g.,steeringvectors.TherowsinS2CKLaretheknowntemporalvectors,e.g.,waveforms,assumedtobelinearlyindependentofeachotherornotcompletelycorrelatedwitheachother.ThematrixB2CNKcontainsthemultivariateunknowncomplexamplitudes.Throughoutthischapter,weassumethatMNandLK+M.ThecolumnsoftheinterferenceandnoisematrixZ2CMLarestatisticallyindependentcircularlysymmetriccomplexGaussianrandomvectorswithzero-meanandunknowncovariancematrixQ.TheproblemofinterestistoestimatetheunknownmatrixB. Wenotethatthedatamodelof( 2{1 )hasgeneralapplications.Itsreal-valuedcounterpart,calledgrowth-curve(GC)Model,hasbeenstudiedandusedwidelyforinvestigatinggrowthproblemsinthestatisticseld[ 62 ][ 63 ][ 18 ].Thisreal-valuedgrowth-curvemodelwasextendedandintroducedtothesignalprocessingeldin[ 21 ].Usingtheextendedmodel,theauthorsin[ 21 ]uniedmanyexistingalgorithmsproposedforradararrayprocessing[ 64 ][ 65 ],spectralanalysis[ 23 ][ 66 ]andwirelesscommunication[ 67 ]-[ 70 ]applications. 8

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ThefocusofthischapterisontheperformanceanalysisofthemultivariateMaximumLikelihood(ML)andCaponestimatorsforthedatamodelin( 2{1 ).Wederivetheclosed-formexpressionoftheCramer-RaoBound(CRB)oftheunknowncomplexamplitudeparameters.WealsoanalyzethebiaspropertiesandMean-Squared-Errors(MSE)ofthetwoestimators.AcomparativestudyshowsthatthemultivariateMLestimatorisunbiasedwhereasthemultivariateCaponestimatorisbiaseddownwardfornitesnapshots.YetinnitedatasamplesandatlowSNR,CaponcanprovideasmallerMSEthanML.Bothestimatorsareasymptoticallystatisticallyecientwhenthenumberofsnapshotsislarge. Theremainderofthechapterisorganizedasfollows.Section 2.2 providesthemultivariateCaponandMLestimators.Section 2.3 givestheperformanceanalysisofthetwoestimatorsandtheCRBoftheunknowncomplexamplitudes.NumericalexamplesareprovidedinSection 2.4 .Finally,wepresentourconclusionsinSection 2.5 2{1 ),wedescribethemultivariateCaponandMLestimatorsinthissection. 60 ][ 71 ][ 72 ].TheotheristheLeast-Squares(LS)estimation[ 16 ][ 73 ],whichisbasicallythematchedltering. WerstconsidertheCaponbeamforming.Let Then,theCaponbeamformercanbeformulatedas ^W=argminWtr(WHRW)subjecttoWHA=I;(2{3)

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where^Wisamultivariateweightingmatrixfornoiseandinterferencesuppressionwhilekeepingthedesiredsignalsundistorted.Solvingtheaboveoptimizationproblemyields ^W=R1A(AHR1A)1:(2{4) NotethatsinceMNandthecolumnsinAarelinearlyindependentofeachother,AHR1AhasfullrankNwithprobabilityone.Thebeamformingoutput,denotedbyY,is NowweconsidertheLSestimation.Substituting( 2{1 )into( 2{5 )yields EstimatingBfromYbasedon( 2{6 )isastandardMultivariateAnalysisofVariance(MANOVA)problem[ 62 ][ 63 ][ 21 ].Notethatafterspatialbeamforming,thenoisevectorsremaintemporallywhite,andhencetheLSestimatorgivesthebestperformance.UsingtheLSalgorithmyields ^BCapon=YSH(SSH)1:(2{7) Substituting( 2{5 )into( 2{7 ),themultivariateCaponestimatorhastheform ^BCapon=(AHR1A)1AHR1XSH(SSH)1:(2{8) NotethattheCaponestimatorfortheunivariatecasein[ 25 ]isaspecialcaseof( 2{8 ). 21 ].Inthischapter,weassumethatbothAandSareknownandthemultivariateMLestimatorcanbebrieyderivedasfollowstomakethischapterself-contained.

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Basedonthedatamodelin( 2{1 ),thenegativelog-likelihoodfunctionisproportionalto wherejj,tr()and()Hdenotethedeterminant,traceandconjugatetransposeofamatrix,respectively. Minimizingthenegativelog-likelihoodfunctionwithrespecttoQyields ^Q=1 Inserting( 2{10 )into( 2{9 ),theMLestimatorofBcanbeformulatedas ^BML=argminBj(XABS)(XABS)Hj:(2{11) Notethat 2ABXSH(SSH)1(SSH)ABXSH(SSH)1HT1 2=jTjI+(SSH)1 2ABXSH(SSH)1HT1ABXSH(SSH)1(SSH)1 2=jTjI+(SSH)1 2SXHT1T1A(AHT1A)1AHT1XSH(SSH)1 2+(SSH)1 2B(AHT1A)1AHT1XSH(SSH)1H(AHT1A)B(AHT1A)1AHT1XSH(SSH)1(SSH)1 2jTjI+(SSH)1 2SXHT1T1A(AHT1A)1AHT1XSH(SSH)1 2;(2{12) where

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andIisanidentitymatrix.IntheabovederivationwehaveusedthefactthatjI+XYj=jI+YXj[ 74 ].SincetherowsinSarelinearlyindependentofeachotherandLK+M,therankofISH(SSH)1S,whichisLK,isgreaterthanorequaltoM.Hence,TandAHT1Ain( 2{12 )havefullranksMandN,respectively,withprobabilityone. From( 2{12 ),theMLestimatorofBiswrittenas ^BML=(AHT1A)1AHT1XSH(SSH)1:(2{14) NoteagainthattheMLestimatorfortheunivariatecasein[ 25 ]isaspecialcaseof( 2{14 ). TobetterunderstandtheaboveMLestimatorintuitively,weinsert( 2{1 )into( 2{13 )andget Itshowsthat1 2{14 )canbedividedintotwosteps,includingtheMLbeamformingspatiallycorrespondingtotheleft-multiplicationmatrix(AHT1A)1AHT1andtheLSestimationtemporallycorrespondingtotheright-multiplicationmatrixSH(SSH)1. Notethatliketheunivariatecasein[ 25 ],theonlydierencebetweentheCaponandMLestimatorsisthatthematrixRin( 2{8 )isreplacedbyTin( 2{14 ).However,aswewillshowinthefollowinganalysis,thisseeminglyminordierenceinfactleadstosignicantandinterestingperformancedierencesbetweenthetwoestimators. 2.3.1PerformanceAnalysisoftheMultivariateMLEstimator 25 ]canbeextendedtothe

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multivariatecase,i.e.,themultivariateMLestimatorisunbiasedanditisasymptoticallystatisticallyecientwhenthenumberofsnapshotsislarge.Inthefollowingderivations,severaltechniquespresentedin[ 25 ],[ 62 ]and[ 18 ]areemployed. Clearly,wehave NotethatthecolumnsofZareindependentzero-meanGaussianrandomvectors.NotealsothatthecolumnsofSHareorthogonaltothoseofISH(SSH)1S.BythepropertyofjointGaussiandistribution,ZSandZ?SaretwoindependentGaussianrandommatrices[ 75 ].Hence,XSandTarealsoindependentofeachother.Utilizingthisconclusion,wecanreadilyshowthat whereEC[:]denotescalculatingtheexpectationwithrespecttotherandommatrixC. Hence,themultivariateMLestimatoris,likeitsunivariatecounterpartin[ 25 ],unbiased. A.1 showsthattheCRBofBbasedon

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thedatamodelin( 2{1 )hasthefollowingform CRB(B),CRB(vec(B))=(SST)1(AHQ1A)1;(2{20) wherevec()denotesstackingthecolumnsofamatrixontopofeachother,()denotesthecomplexconjugateanddenotestheKroneckermatrixproduct[ 74 ]. BeforecalculatingtheMSEofthemultivariateMLestimator,weintroducethefollowingthreelemmaswhichwillbeusedinourderivation.Thecounterpartsofthethreelemmasforreal-valuedvariableshavebeenprovedandusedinthestatisticsliterature[ 62 ][ 63 ][ 18 ]andpartofLemma1hasbeenprovedin[ 75 ]. (i). (ii). (iii). Theconditionaldistributionof12given22isthematrix-variatecomplexGaussiandistributionCN(1212222;11:2;22)[ 74 ][ 75 ],whoseprobabilitydensityfunction(pdf)isgivenby Proof:Appendix A.2

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E(CH)=tr(C)+HCHH:(2{24) Proof:Appendix A.3 E(1)=1 Proof:Appendix A.4 NowweconsidertheMSEof^BML.TheerrorofthemultivariateMLestimateofBis B,^BMLB=(AHT1A)1AHT1(ABS+Z)SH(SSH)1B=(AHT1A)1AHT1ZS(SSH)1:(2{26) Since[SH(SSH)1 2]H[SH(SSH)1 2]=I,i.e.,SH(SSH)1 2isanLKsemi-unitarymatrix,wecanconstructanLLunitarymatrix 2:(2{27) Thus, SincethecolumnrandomvectorsofZarestatisticallyindependentofeachotherandthecolumnsofU2areorthogonaltoeachother,ZU2CN(0;QMM;ILK)andQ1 2ZU2CN(0;IM;ILK)[ 75 ].Accordingtothedenitionofthecomplex

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Wishartdistribution,wehave ~T,Q1 2TQ1 2=(Q1 2ZU2)(Q1 2ZU2)HCW(LK;M;I):(2{29) Let ~ZS=Q1 2ZS(SSH)1 2;(2{30) whichhastheCN(0;IM;IK)distribution[ 75 ].Denote ~A=Q1 2A(AHQ1A)1 2:(2{31) Then,inserting( 2{29 ),( 2{30 )and( 2{31 )into( 2{26 )gives B=(AHQ1A)1 2(~AH~T1~A)1~AH~T1~ZS(SSH)1 2:(2{32) Since~AH~A=I,wecandecompose~Aas ~A=~UP;(2{33) where~UisanMMunitarymatrixwithitsrstNcolumnsbeing~A;PMN=[IN0]T.Since~Uisunitary,like~T,~UH~T~UremainstobethecomplexWishartdistribution[ 75 ],i.e., Let whichobviouslyhastheCN(0;IM;IK)distribution. WenextpartitionMK,MM,and1MM,respectively,asfollows.

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where1and2areNKand(MN)K,respectively,andboth11and11areNNmatrices. Inserting( 2{33 )to( 2{36 )into( 2{32 )gives B=(AHQ1A)1 2(PH1P)1PH1(SSH)1 2=(AHQ1A)1 2(1121222)(SSH)1 2:(2{37) Toobtain( 2{37 ),wehaveusedtheinversionlemmaofpartitionedmatrices. Hence,usingthelemmathatvec(XYZ)=(ZTX)vec(Y)andthefactthat1,2aretwoindependentrandommatriceswiththedistributionsCN(0;IN;IK)andCN(0;IMN;IK),respectively,aswellas( 2{20 ),wehave MSE(^BML),Efvec(B)vec(B)Hg=[(SST)1 2(AHQ1A)1 2]Efvec(1121222)vec(1121222)Hg[(SST)1 2(AHQ1A)1 2]=[CRB(B)]1 2fI+E[vec(121222)vec(121222)H]g[CRB(B)]1 2:(2{38) Usingthefactsthatvec(XY)=(IX)vec(Y)and2CN(0;I;I)yields whereEAjB()denotestheexpectationwithrespecttorandommatrixAgivenB. ByLemma 2.1 and( 2{34 ),weknowthat12given22hastheCN(0;I;22)distribution.Hence,applyingLemma 2.2 gives

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Furthermore,byLemma 2.1 weknowthat22CW(LK;MN;IMN).Hence,byLemma3,wehave Thus,itfollowsfrom( 2{38 ),( 2{39 ),( 2{40 )and( 2{41 )that MSE(^BML)=LK LKM+NCRB(B):(2{42) From( 2{42 ),wenotethatMSE(^BML)approachesCRB(B)forlargeL,whichmeansthatthemultivariateMLestimatorisasymptoticallystatisticallyecientforlargenumberofsnapshotsL.Hencetheeciencyconditionfortheunivariatecasein[ 25 ]canbeextendedtothemultivariatecaseaswell.WhenL,K,MandNarexed,theMSEofthemultivariateMLestimatorisproportionaltoCRB(B).HenceitisexpectedthattheMSE-versus-SNRlineswillbeparalleltotheCRB-versus-SNRlines.ThistheoreticalresultwillbeveriedvianumericalsimulationsinSection 2.4 Furthermore,theCRBofBdependson(SST)1and(AHQ1A)1.AsweshowinAppendix A.1 ,orthogonalitiesamongtherowsofSandamongthecolumnsofQ1 2Aleadtosmalldiagonalelementsfor(SST)1and(AHQ1A)1,respectively,whichinturnreducetheCRB. WealsonotethatwhenM=N,whichimpliesthatAisasquarematrix,themultivariateMLestimatorisecient.However,weshouldnotthinkofitasasignicantadvantagetomakeNaslargeaspossible.AsweshowinAppendix A.1 ,inthecasethatthecolumnsofQ1 2Aarenotorthogonaltoeachother,whichoftenhappensinpractice,largeNcausesCRBtoincrease. NowwesummarizethestatisticalpropertiesofthemultivariateCaponestimatorbythefollowingtheorem.

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2{1 ),themultivariateMLestimateofB,givenby( 2{14 ),isunbiasedandasymptoticallystatisticallyecientforlargenumberofdatasamples.ItsMSEmatrixcanbeexpressedas MSE(^BML),E[vec(^BML)vec(^BML)H]=LK LKM+NCRB(B)(2{43) CRB(B)=(SST)1(AHQ1A)1;(2{44)vec(),(),()Tanddenotethedirectoperator(stackingthecolumnsofamatrixontopofeachother),complexconjugate,transposeandKroneckerproductofmatrices,respectively. 2.3.1 ,weknowthatthemultivariateMLestimatorisunbiased.WewillinvestigatethebiasofthemultivariateCaponestimatorbystudyingtherelationshipbetweenthetwoestimators. Comparing( 2{2 )and( 2{13 ),wenotethat Applyingthematrixinversionlemmagives

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and (AHR1A)1=[AHT1AAHT1XSH(SXHT1XSH+SSH)1SXHT1A]1=(AHT1A)1(AHT1A)1AHT1XSHSXHT1A(AHT1A)1AHT1XSHSXHT1XSHSSH1SXHT1A(AHT1A)1:(2{47) Substituting( 2{46 )and( 2{47 )into( 2{8 ),andaftersomestraightforwardmanipulations,weget ^BCapon=^BML;(2{48) where with Theninserting( 2{1 )into( 2{50 )gives Notethattherearetworandommatrices,i.e.,TandZS,in( 2{26 )and( 2{51 ).SincethecolumnsofZarestatisticallyindependentzero-meanGuassianrandomvectorswhilethecolumnsofSHareorthogonaltothoseofISH(SSH)S,bythepropertyofjointGaussiandistributionweknowthatZSandZ?S,Z[ISH(SSH)S]aretwoindependentGaussianrandommatrices.Hence,ZSandT=Z?S(Z?S)Harealsoindependentofeachother.ByLemmas1.9and1.11in[ 18 ],whichcanbereadilyextendedtothecomplex-valuedcase,wehaveZSCN(0;Q;SSH)andTCW(LK;M;Q).

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SinceZSandTarestatisticallyindependentofeachotherandby( 2{26 )and( 2{51 ),weknowthat(^BMLB)isanoddfunctionwithrespecttoZS.HencereplacingZSwithZSyields Ontheotherhand,sinceZSisazero-meanGaussianrandommatrix,ZS,asarandommatrixtransformedfromZS,retainsallthestatisticalpropertiesofZS.Hence,replacingZSbyZSwillnotchangetheexpectationof(^BMLB),i.e., Itfollowsfrom( 2{52 )and( 2{53 )that Therefore,by( 2{48 )and( 2{54 )wehave NowwefollowthesametechniqueusedintheprevioussubsectiontosimplifyandVviatransformationofrandommatrices. Followingthedenitionsin( 2{29 ),( 2{30 )and( 2{31 )andinsertingtheminto( 2{51 ),weget 2~ZHS~T1~T1~A(~AH~T1~A)1~AH~T1~ZS(SS)1 2:(2{56) Thenweadoptthedecompositionin( 2{33 ),thedenitionsin( 2{34 )and( 2{35 ),andthepartitionsin( 2{36 ),andinserttheminto( 2{56 ).Bytheinversion

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lemmaofpartitionedmatrices,weobtain 2H11P(PH1P)1PH1(SSH)1 2=(SSH)1 2H8><>:2641112212237526411122121(11)1123759>=>;(SSH)1 2=(SSH)1 2H21222(SSH)1 2:(2{57) From( 2{49 )and( 2{57 )andbythematrixinversionlemma,itfollowsthat 2I+H212221(SSH)1 2=I(SSH)1 2H2(22+2H2)12(SSH)1 2:(2{58) Tocalculatetheexpectationof,weusethefollowinglemma. l+pIp; (l+p)(l+p+1)vec(Ip)vec(Ip)H+Dp; (l+p)2(l+p+1)forlargel+p,andDpisap2p2matrixwithitselementatthe[(c11)p+r1]throwandthe[(c21)p+r2]thcolumn(r1;c1;r2;c2=1;2;:::p)being Proof:Appendix A.5

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Applyingtheabovelemmatoand,whichbyconstructionsatisfytheassumptionsinthelemma,wehaveimmediately LI:(2{63) Inserting( 2{63 )into( 2{55 ),weget LB:(2{64) TheaboveequationshowsthatthemultivariateCaponestimatorsharesthesamepropertiesastheunivariateCaponin[ 25 ].Inotherwords,itisbiaseddownwardfornitesnapshotnumberL.However,forlargeL,itisasymptoticallyunbiased.ItisalsoworthnotingthatthebiasoftheCaponestimatorisnotrelatedtoK,whichmeansthatincreasingthenumberofrowsinthetemporalinformationmatrixSwillnotcausehigherbias. Moreover,wenotethatwhenM=N,themultivariateCaponestimatorbecomesunbiasedasthemultivariateMLestimator.Inthiscase,bothMLandCaponreducetothesameestimatorA1XSH(SSH)1.Hence,forthesamereasonthatwehavestatedinSection 2.3.1 ,thisunbiasednessofthemultivariateCaponestimatorshouldnotbeseenasasignicantadvantage. 2{54 ),wecanprovethatBandB(I)areuncorrelated.Hence,from( 2{26 )and( 2{48 ),wehave MSE(^BCapon),E[vec(^BCaponB)vec(^BCaponB)H]=Efvec[BB(I)]vec[BB(I)]Hg=E[vec(B)vec(B)H]+E[vec(B(I))vec(B(I))H]:(2{65)

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WerstcalculateE[vec(B)vec(B)H]. By( 2{37 )and( 2{58 ),wehave B=(AHQ1A)1 2[1121222][I+H21222]1(SSH)1 2:(2{66) Usingthefactthatvec(XYZ)=(ZTX)vec(Y)aswellas( 2{20 )yields 2F[CRB(B)]1 2;(2{67) where Thenusingthelemmathatvec(XY)=(YTI)vec(X)andthefactthat1and2areindependentstandardmatrix-variateGaussiandistributions,aftersomemanipulations,weget Notethat Togettheaboveequation,wehaveutilizedLemma2in[ 76 ],i.e.,12j22CN(0;I;22).Hence,bythecomplex-valuedcounterpartofLemma1.8in[ 18 ],weknowthatthecovariancematrixofvec(12)given22is22I.

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Inserting( 2{70 )into( 2{69 )andrecalling( 2{58 )and( 2{63 )yield 2E()(SSH)1 2]I=1MN LI:(2{71) From( 2{67 )and( 2{71 ),theequation LCRB(B)(2{72) followsdirectly. Nowweconsiderthesecondtermin( 2{65 ).By( 2{58 ),weknowthat 2H2(22+2H2)12(SSH)1 2:(2{73) Weknowthat2and22areindependentofeachotherwithCN(0;IMN;IK)andCW(LK;MN;IMN)distributions,respectively.Thenusingthefactthatvec(XYZ)=(ZTX)vec(Y),thefollowingequationisobtainedfollowingLemma 2.2 2[B(SSH)1 2]Evec[H2(22+2H2)12]vec[H2(22+2H2)12]H(SST)1 2[B(SSH)1 2]H=(MN)(MN+1) 2[B(SSH)1 2]DK(SST)1 2[(SSH)1 2BH];(2{74) whereDKisaK2K2matrixdenedas( 2{62 ),andisascalarandapproximatelyequalto(MN)(LM+N)

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By( 2{65 ),( 2{72 )and( 2{74 ),wegettheMSEofthemultivariateCaponestimator MSE(^BCapon)=LM+N LCRB(B)+(MN)(MN+1) 2[B(SSH)1 2]oDKn(SST)1 2[(SSH)1 2BH]o:(2{75) Equation( 2{75 )givesanapproximateclosed-formexpressionoftheMSEofthemultivariateCaponestimator.Inthisequation,wenotethattheMSEconsistsofthreeterms.ThersttermisproportionaltoCRB(B).Thesecondtermisproportionaltotheouter-productofvec(B)andisnotrelatedtotheparameterKandthetemporalinformationmatrixS.Inthethirdterm,althoughthereisnoexplicitdependenceoftheparameterK,thenumberofnon-zeroelementsinDKisdependentofK.Hence,thethirdtermwillincreaseasKincreases.Moreover,thethirdtermisafunctionof(SST),whichdependsonthethecorrelationamongtherowsofS.Aswewillseeinthefollowingnumericalsimulations,forSwithcorrelatedrows,theMSEofanelementofBincreasesasKincreasesand/orastheotherelementsinBincrease.Onthecontrary,whenK=1,thethirdtermiszerobecausethematrixDbecomesascalar0accordingtoitsdenition.IfwefurthersetN=1,then( 2{75 )reducestotheconclusionintheunivariatecasein[ 25 ]. WealsonotethatwhenthenumberofsnapshotsLislarge,thelasttwotermsapproachzerowhilethersttermapproachesCRB(B).Hence,themultivariateCaponestimatorisalsoasymptoticallystatisticallyecientforlargeL. Furthermore,wenotethatwhenM=N,theMSEofthemultivariateCaponestimatorissimpliedtoCRB(B)likethemultivariateMLestimator.ThisisconsistentwithourconclusionintheabovesubsectionthatthetwomultivariatemethodsreducetothesameestimatorwhenM=N.Forthesamereasonthatwe

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statedinSection 2.3.1 ,thiseciencyofthemultivariateCaponestimatorshouldnotbeseenasasignicantadvantage. NowwesummarizethestatisticalpropertiesofthemultivariateCaponestimatorbythefollowingtheorem. 2{1 ),themultivariateCaponestimateofBin( 2{8 )isbiaseddownward.However,forlargenumberofdatasamples,itisasymptoticallyunbiasedandstatisticallyecient.ItsbiasandMSEmatricesaregivenby( 2{64 )and( 2{75 ),respectively. 2{6 ,theelementsinBareallsettobe1.Theinterferenceandnoiseterminourdatamodelin( 2{1 )istemporallywhitebutspatiallycoloredzero-meancircularlysymmetriccomplexGaussianwiththespatialcovariancematrixQgivenby [Q]ij=(0:9)jijj;(2{76) where=1=SNRand[]ijdenotestheithrowandjthcolumnelementofamatrix.Thegurebelowareallfor[B]11.TheguresforotherelementsofBaresimilar.WeobtaintheempiricalresultsinFig. 2{2 using10000MonteCarlotrialswhiletheothers1000trails. Werstinvestigatethebiasperformance.Fig. 2{1 showsthebiaspropertiesofthetwomultivariateestimators(denotedby\MV-ML"and\MV-Capon")from

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Figure2{1: BiasversusLwhenSNR=10dB,K=2,N=2. boththeoreticalpredictions(denotedby\Theo.")andMonteCarlotrials(denotedby\Empi.").Asexpected,themultivariateMLisunbiasedwhereasCaponisbiaseddownwardfornitesnapshots.However,whenthenumberofsnapshotsLislarge,thebiasofthemultivariateCaponapproacheszero,aspredictedbyourtheoreticalanalysis. Fig. 2{2 illustratestherelationshipbetweenthebiasandthenumberofrowsofthetemporalinformationmatrixS,i.e.,K,whenthefrequencydierenceofthecomplexsinusoidsinSis0:04Hz.Aspredictedbyourtheoreticalanalysis,thebiasofthemultivariateCaponestimatorisindependentofK. Fig. 2{3 illustratestheMSEsofthemultivariateestimatorsaswellastheCRBasafunctionofL.Asillustrated,thetheoreticalandempiricalMSEsareconsistent.TheperformanceofthemultivariateMLestimatorisbetterthanthemultivariateCaponandveryclosetothecorrespondingCRB.AswehavepredictedinSection 2.3 thatbothmultivariateestimatorsareasymptoticallystatistically

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Figure2{2: BiasversusKwhenSNR=10dB,L=16,N=2. ecientforlargenumberofsnapshots,andtheperformancecurvesofthetwoestimatorsapproachtheCRBasLincreases. Fig. 2{4 showstherelationshipbetweentheMSEandSNR.NotethattheerrorooroccursathighSNRforthemultivariateCaponestimatorduetoitsbias.Asshowninourtheoreticalanalyses,foraxedM,L,NandK,theMSEofMLisproportionaltoCRB(B),andhenceno\thresholdeect"occurs.Notealsothat,likeintheunivariatecase,theCaponestimatecanprovideasmallerMSEthanMLatlowSNR.AtsuchalowSNR,though,bothMLandCaponperformpoorly. Fig. 2{5 givestheMSEsofthemultivariateCaponandMLestimatorsaswellasthecorrespondingCRBasafunctionofKwhenthefrequencydierenceofthecomplexsinusoidsinSis0:04Hz.Aswecansee,boththeCRBandtheMSEsofthetwomultivariateestimatorsincreaseasKincreases.However,duetothecontributionofthethirdtermin( 2{75 ),theMSEofCaponincreasesmorequicklythantheCRBandtheMSEofML.

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Figure2{3: MSEversusLwhenSNR=10dB,K=2,N=2. InFig. 2{6 ,weconsiderthecasewhereBhasunequalelements.WesetN=K=2,[B]11=[B]21=1and[B]12=[B]22=1 2,whereisthepowerratiobetweenthetwocomplexsinusoidsinS.Fig. 2{6 givestheCRBandMSEsof[B]11asvaries.Asillustrated,theMSEofthemultivariateMLestimatorisalmostconstantwithrespectto,whiletheMSEofthemultivariateCaponestimatorincreasesrapidlywhenisdecreasedtobelowerthan0dBduetoitsbiasednature.

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Figure2{4: MSEversusSNRwhenL=16,K=2,N=2. Figure2{5: MSEversusKwhenSNR=10dB,L=16,N=2.

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Figure2{6: MSEversuswhenSNR=10dB,L=16,K=2,N=2.

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wherediag()denotesadiagonalmatrixwithitsdiagonalformedbytheelementsofthevector.In( 3{1 ),X2CMLdenotestheobservedsnapshotswithLbeingthesnapshotnumberandMthesnapshotdimension.ThecolumnsinA2CMNaretheknownspatialinformationvectors,referredtoasthesteeringvectors.TherowsinS2CNLaretheknowntemporalinformationvectors,referredtoaswaveforms.Theelementsin2CN1aretheunknowncomplexamplitudes.ThecolumnsoftheinterferenceandnoisematrixZ2CMLareindependentlyandidenticallydistributed(i.i.d.)circularlysymmetriccomplexGaussianrandomvectorswithzero-meanandunknowncovariancematrixQ.Throughoutthischapter,weassumethatM+rLwithrbeingtherowrankofS.Theproblemofinterestistoestimatetheunknowncomplexamplitudes.NotethatintheDGCmodelwedonotneedtoassumethelinearindependenceofthesteeringvectorsorwaveforms,unlikeintheGCmodel. TheonlydierencebetweenDGCandGCisthatthecomplexamplitudematrixBinDGCisconstrainedtobediagonalwhileitisarbitraryinGC.However,thisseeminglyminordierencemakesthederivationsofthemultivariateMLestimatorin[ 21 ]and[ 22 ]invalid.Infact,toourknowledge,noclosed-formMLestimatorforin( 3{1 )existsintheliterature.Inthischapter,wepropose 33

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anapproximatemaximumlikelihood(AML)estimatorforinthismodel.Wealsoinvestigateitsstatisticalpropertiesviatheoreticalanalysesandnumericalsimulations. Weremarkthatalthoughinthischapterwefocusonthecomplexamplitudeestimationofsignalswithknownsteeringvectorsandwaveforms,theproposedDGCmodelandAMLestimatorcanalsobeusedinthecasewhereboththesteeringvectorsandwaveformsareparameterizedbyunknownparameters.UsingtheproposedAMLmethod,wecanconstructaconcentrated(approximate)likelihoodfunctionoftheunknownparameters[ 27 ][ 30 ][ 31 ].Theunknownparametersinthesteeringvectorsandwaveformscanthenbeestimatedviamaximizingtheconcentratedlikelihoodfunction. Theremainderofthechapterisorganizedasfollows.InSection 3.2 ,weintroducetheAMLestimatorforbasedontheDGCmodelin( 3{1 ).Section 3.3 presentstheperformanceanalysisoftheAMLestimatorandtheCRBfortheunknowncomplexamplitudes.NumericalexamplesarepresentedinSection 3.4 .Finally,Section 3.5 containsourconclusions. 3{1 ).ThisapproachisbasedontheMLprinciple,butanapproximationismadetogetaclosed-formsolution. Itfollowsfrom( 3{1 )thatthenegativelog-likelihoodfunctionoftheobserveddatasamples(towithinanadditiveconstant)is wheretr(),jjand()Hdenotethetrace,thedeterminantandtheconjugatetransposeofamatrix,respectively.Minimizingthenegativelog-likelihoodfunction

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withrespecttoQyields ^Q=1 Notethat^Qisnonsingular,i.e.,j^Qj6=0,withprobabilityonewhenML.Substituting( 3{3 )into( 3{2 ),theMLestimateofcanbeformulatedas ^ML=argminlnj(XABS)(XABS)HjwithB=diag():(3{4) Ingeneral,theoptimizationproblemin( 3{4 )doesnotappeartoadmitaclosed-formsolution.Here,weusethetechniquein[ 25 ]and[ 27 ]tosolvethisproblemapproximately.LetSand?Sdenotetheorthogonalprojectionmatricesgivenby and where()denotesthegeneralizedinverseofamatrix[ 74 ].Notethatwhenthewaveforms,i.e.,therowsofS,arenotlinearlyindependentofeachother,thematrix(SSH)issingularandhenceitsgeneralizedinverseisnotunique.However,Sand?Sareunique[ 74 ]. Using( 3{5 )and( 3{6 ),itfollowsthat wherethematrixTisdenedby

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Notethattherankof?SisLr.Hence,ThasfullrankMwithprobabilityonewhenM+rL.NotealsothatTisanestimateofthecovariancematrixQ(towithinamultiplicativeconstant)obtainedbyprojectingoutthedesiredsignalcomponentsfromX. Using( 3{7 )andLemma4in[ 25 ](orTheorem1in[ 27 ]),thesolutionoftheoptimizationproblemin( 3{4 )canbeapproximatedforalargesnapshotnumberasfollows. ^AML=argmintr[(XSABS)HT1(XSABS)]=argminkvec(T1 2XS)vec(T1 2ABS)k2(3{9) whereT1 2istheHermitiansquarerootofT1,kkdenotestheEuclideanvectornormandvec()denotesthevectorizationoperator(stackingthecolumnsofamatrixontopofeachother). Tosolvetheaboveoptimizationproblem,werstintroducethefollowinglemmas. vec(UGVT)=(VU)vecd(G);(3{10) 77 ][ 78 ]. B.1 forthecompletenessofthischapter. vecd(UGV)=(VUT)Tvec(G):(3{11)

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vecd(UGV)=(UVT)vecd(G);(3{12) B.2 Proof:ThislemmaisastraightforwardextensionofLemmaA2in[ 78 ]. ByLemma 3.1 ,itfollowsfrom( 3{9 )that ^AML=argminkvec(T1 2XS)k2(3{14) with,ST(T1 2A).Notethat( 3{14 )isinaquadraticfunctionof.Minimizing( 3{14 )withrespecttoyields ^AML=(H)1Hvec(T1 2XS):(3{15) In( 3{15 ),wehaveassumedthatthecolumnsofarelinearlyindependenceofeachotherandhenceHisinvertible.NotethatthisassumptionismuchweakerthanthatintheGCmodel. Ontheotherhand,usingLemmas 3.3 and 3.2 ,respectively,wehavethat and 2XS)=vecd(AHT1XSH):(3{17)

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Substituting( 3{16 )and( 3{17 )into( 3{14 )yieldsthefollowingexpressionfortheAMLestimateof Notethat( 3{18 )doesnotrequiretheexistenceoftheinverseofAHT1A.Moreover,( 3{18 )doesnotrequireSSHtobeinvertible.Therefore,unliketheestimatorsintheGCmodel[ 22 ],theAMLestimatorinDGCdoesnotrequirethelinearindependenceofthesteeringvectorsorofthewaveforms. Appendix B.3 givesanotherinterpretationofAMLfromageneralizedleast-squares(GLS)pointofview. 3{1 )into( 3{18 )gives ^AML=[(AHT1A)(SSH)T]1vecd(AHT1ABSSH)+[(AHT1A)(SSH)T]1vecd(AHT1ZSH):(3{19) ByLemma 3.2 weget vecd[(ATT1A)B(SSH)]=[(AHT1A)(SSH)T]:(3{20) Hence,from( 3{19 )weobtaintheestimationerror ^AML=[(AHT1A)(SSH)T]1vecd(AHT1ZSH):(3{21) Ontheotherhand,substituting( 3{1 )into( 3{8 )yields

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From( 3{22 ),weseethatTisanevenfunctionofZ,andhence^AMLisanoddfunctionofZ.Moreover,wehaveassumedthatZisazero-meanGaussianrandommatrix.Usingthestatisticalpropertiesofthezero-meanGaussiandistribution[ 75 ],wecanreadilyshowthattheexpectationof( 3{21 )iszero,i.e.,^AMLisunbiased. Fromthetheoryonthelinearstatisticalmodel[ 16 ],weknowthatT 3{21 )canbeapproximatedby ^AMLCRB()vecd(AHQ1ZSH);(3{23) where CRB()=[(AHQ1A)(SSH)T]1(3{24) istheCramer-RaoboundforthatisthelowestpossibleMSEofanyunbiasedestimatorof(seeAppendix B.3 forthederivationofCRB). From( 3{23 )andLemma 3.2 ,itfollowsthat var(^AML),E[(^AML)(^AML)H]CRB()E[vecd(AHQ1ZSH)vecd(AHQ1ZSH)H]CRB()=CRB()[SH(AHQ1)T]TE[vec(Z)vec(Z)H][SH(AHQ1)T]CRB()=CRB()[ST(Q1A)]H(IQ)[ST(Q1A)]CRB()(3{25) where()anddenotethecomplexconjugateandtheKroneckermatrixproduct,respectively.From( 3{25 ),similarlytoEquation( B{7 )inAppendix B.3 ,wegetthe

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MSEof^AMLas var(^AML)=[(AHQ1A)(SSH)T]1=CRB():(3{26) WeseethattheAMLestimateofisasymptoticallystatisticallyecientforalargesnapshotnumber.ThistheoreticalresultisveriedbythenumericalexamplesinSection 3.4 [Q]m;n=0:99jmnjej(mn) where=1=SNRand[]m;ndenotesthe(m;n)thelementofamatrix. Inthefollowingexamples,wepresenttheMSEoftheAMLestimatorin( 3{18 )aswellasthecorrespondingCRBin( 3{24 ).Forcomparison,wealsoshowtheMSEoftheMLestimatorin[ 21 ]and[ 22 ]basedontheGCmodel,whichignoresthediagonalconstraintonB,andoftheleastsquares(LS)estimatorwhichassumesthattheinterferenceandnoisevectorsinZareuncorrelatedbothspatiallyandtemporally.TheLSestimatorcanbeobtainedimmediatelyviareplacingTin

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Figure3{1: EmpiricalMSE'sandtheCRBversusLwhenSNR=10dB,M=6,andN=3withlinearlyindependentsteeringvectorsandlinearlyindependentwaveforms. ( 3{18 )byanidentitymatrix,i.e., ^LS=[(AHA)(SSH)T]1vecd(AHXSH):(3{28) Theempiricalestimationperformancesareallobtainedby1000Monte-Carlosimulations. Werstconsiderthecasewherethesteeringvectorsarelinearlyindependentofeachotherandsoarethewaveforms.SupposethatN=3signalswithknownwaveformsarriveatthesensorarrayfrom1=10,2=5and3=10,respectively.AssumethattheDOAsofthesignalshavebeenestimatedaccuratelyandthereforetheycanbeconsideredtobeknown.Thesignalwaveformsaregeneratedindependentlyandhenceareuncorrelatedwitheachother.Duetospacelimitations,weonlyshowbelowtheMSEofthecomplexamplitudeestimateofthesignalarrivingfrom2=5.Theperformancesforothersignalsaresimilar.

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Figure3{2: EmpiricalMSE'sandtheCRBversusSNRwhenL=128,M=6,andN=3withlinearlyindependentsteeringvectorsandlinearlyindependentwaveforms. Fig. 3{1 andFig. 3{2 showtheestimationperformanceasafunctionofthesnapshotnumberLandtheSNR,respectively.TheSNRisxedat10dBinFig. 3{1 whilethesnapshotnumberisxedat128inFig. 3{2 .NotethattheMLestimatorbasedontheGCmodelgivesrelativelypoorestimationperformance.ThismethodignorestheaprioriinformationaboutthediagonalstructureofthecomplexamplitudematrixB,whichincreasesthenumberofunknownsfromNtoN2andhenceresultsinmuchlargerestimationvariances.InFig. 3{1 ,weseethattheAMLestimatoroutperformsLSsignicantly,andapproachestheCRBrapidly,aspredictedtheoretically.InFig. 3{2 ,weseethattheAMLestimatoroutperformstheothermethodsandisveryclosetotheCRBinthewholerangeofSNRconsidered. Fig. 3{3 andFig. 3{4 showtheestimationperformancewhenthewaveformsarelinearlydependentofeachother.Weretainthesamesimulationparametersas

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Figure3{3: EmpiricalMSE'sandtheCRBversusLwhenSNR=10dB,M=6,andN=3withidenticalwaveforms. inFig. 3{1 andFig. 3{2 exceptthatthewaveformsofthethreeincidentsignalsareidenticalinthiscase.SincetheMLestimatorbasedontheGCmodelrequiresthatthewaveformsarelinearlyindependentofeachother[ 22 ],itfailstoworkproperlyinthiscase.Ontheotherhand,whiletheMSE'sofAMLaresomewhatlargerthanthoseforlinearlyindependentwaveforms,theAMLestimatorremainsasymptoticallystatisticallyecientandprovideshigherestimationaccuracythanLS. NextweconsideranexamplewhereN=13incidentsignalsimpingeonthesensorarrayfrom=30;25;30.TheothersimulationparametersarethesameasthoseforFig. 3{1 andFig. 3{2 .NotethatinthiscaseN>M,whichinparticularmeansthatthesteeringvectorsarelinearlydependentandhenceitdoesnotsatisfytheassumptionsrequiredbytheGCmodel;consequentlyagaintheMLestimatorbasedontheGCmodelcannotbeused.FromFig. 3{5 andFig. 3{6 ,we

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Figure3{4: EmpiricalMSE'sandtheCRBversusSNRwhenL=128,M=6,andN=3withidenticalwaveforms. seethattheAMLestimatorhasabetterestimationaccuracythanLSandremainsasymptoticallystatisticallyecient. 24 ].WeapplytheAMLestimatortothis1-Dspectralestimationproblem.Asweshowbelow,theproposedAMLestimatorcanachievebetterestimationaccuracyandexhibitgreaterrobustnessthanthemethodsin[ 24 ]. Considerthenoise-corruptedobservationsofNcomplex-valuedsinusoids wherenisthecomplexamplitudeofthenthsinusoidwithfrequency!n;L0isthenumberofdatasamples;z(l)istheobservationnoise,whichiscomplexvaluedand

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Figure3{5: EmpiricalMSE'sandtheCRBversusLwhenSNR=10dB,M=6,andN=13withlinearlydependentsteeringvectors assumedtobestationaryandpossiblycoloredwithzero-meanandunknownnitepowerspectraldensity(PSD).Weassumethatf!ngNn=1areknown,with!n6=!kforn6=k.TheproblemofinterestistoestimatefngNn=1fromtheobservationsfx(l)gL01l=0. Tosolvethisproblem,wedividethedatasequenceintooverlappingsub-sequenceswithshorterlengths[ 60 ][ 23 ].Thenwereformulatethedataequationin( 3{29 )intotheformofaDGCmodel,andusetheproposedAMLapproachtoestimatethecomplexamplitudesofthesinusoids. Let

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Figure3{6: EmpiricalMSE'sandtheCRBversusSNRwhenL=128,M=6,andN=13withlinearlydependentsteeringvectors. and Then,( 3{29 )canbereadilywrittenintheDGCformin( 3{1 )with=[1;2;N]T,L=L0M+1andthenoisematrixZdenedsimilarlytoXin( 3{30 ).Hence,theAMLestimatorin( 3{18 )canbeapplieddirectly.IndoingsoweignorethefactthatthecolumnsoftheinterferenceandnoisematrixZ

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arecorrelatedduetotheoverlappingofdatasamples,whichiscommonlydoneintheliteraturetoretainthesimplicityoftheparameterestimationalgorithm. Weconsideranumericalexampleusedin[ 24 ].TheobserveddatawithL0=32consistsofN=3complexsinusoidswithfrequenciesf1=0:10,f2=0:11,f3=0:30(fk=!k=2)andcomplexamplitudes1=ej wheree(l)isacomplexwhiteGaussiannoisewithzero-meanandvariance2.ThePSDofthetestdataisshowninFig.1of[ 24 ]when2=0:01. Forcomparison,weprovidetheestimationperformanceoftheproposedAMLmethodandofthematched-lterbank(MAFI)approach[ 24 ],whichisthemostcompetitiveoneamongthealgorithmspresentedin[ 24 ],aswellasthecorrespondingCRB.NotethatinthisapplicationthecolumnsoftheinterferenceandnoisematrixZarenotstatisticallyindependent,andhencetheCRBin( 3{24 )isnotapplicable.Instead,weutilizetheCRBformulapresentedinEquation(9)of[ 24 ].TheMSEvaluespresentedinthefollowingexamplesareallobtainedvia1000Monte-Carlosimulations.SincethePSDoftheARnoisevariesinthefrequencydomain,weutilizethelocalSNRasameasureofthesignalqualityforaparticularsinusoid[ 24 ][ 79 ]. Fig. 3{7 showstheMSE'softhetwoamplitudeestimatorsfor3and1alongwiththecorrespondingCRBasthecorrespondinglocalSNRvaries,whenM=L0=4=8.(Theresultsfor2areomittedbecausetheyresemblethosefor1.)InFig. 3{7 (a),MAFIandAMLbothprovidehighestimationaccuracyandareveryclosetotheCRB.However,theestimationperformanceinFig. 3{7 (b)issignicantlypoorerthanthatinFig. 3{7 (a)forbothestimatorsduetotheinterferencefromthesecondsinusoidatf2.(Notethatf2f1=0:01,whichis

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(a)(b) Figure3{7: EmpiricalMSE'sandtheCRBversuslocalSNRwhenL0=32,M=8,andtheobservationnoiseiscolored.(a)For3and(b)for1. (a)(b) Figure3{8: EmpiricalMSE'sandtheCRBversusMwhenL0=32,2=0:01,andtheobservationnoiseiscolored.(a)For3and(b)for1.

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smallerthantheFourierresolutionlimit,i.e.,1=L00:03.)Aswecansee,MAFIdeviatesawayfromtheCRBathighSNRbecauseoftheinterferenceatf2,whichintroducesabiasintotheestimateof1thatdominatestheMSEathighSNR.TheAMLestimatorprovidesbetterestimationaccuracythanMAFI,especiallyathighSNR.Notethatinthepresentedspectralanalysisapplication,thecolumnsoftheinterferenceandnoisematrixZarenoti.i.d.andhencethetheoreticalanalysisinSection 3.3 showingthatAMLisasymptoticallystatisticallyecientisnolongervalid.However,aswecanseefromFig. 3{7 (a)andFig. 3{7 (b),theMSEofAMLremainsclosetotheCRBwhichshowsthehighinterference-resistantcapabilityofAML. Intuitively,wecanexpectthatasMincreases,theAMLestimatoraswellasMAFI(andalsoothermethodspresentedin[ 24 ])candealbetterwiththecaseofcloselyspacedsinusoids,buttheirstatisticalaccuracy,ingeneral,decreases[ 71 ].Hence,thereisatradeowhenchoosingM.ThefollowingexampleexaminestheeectofMontheperformanceoftheseestimators.ThescenarioissimilartotheexampleinFig. 3{7 ,exceptthatwex2=0:01,whichcorrespondstoalocalSNRof30:8dBfortherstsinusoid(atf1=0:1)and39:2dBforthethirdsinusoid(atf3=0:3).ThesubvectorlengthMisvariedfrom1to16forAMLandfrom3to16forMAFI(MAFIrequiresthatMN=3).TheMSE'softheamplitudeestimatesof3and1andthecorrespondingCRB'sareshowninFig. 3{8 (a)andFig. 3{8 (b).Aswecansee,whennosinusoidsareclosetotheonebeingestimated,suchasthethirdsinusoidinthisexample,bothAMLandMAFIperformquitewellforawiderangeofMvalues.ForthemoredicultcaseshowninFig. 3{8 (b),thechoiceofMbecomescritical.Aswecansee,theAMLestimatoroutperformsMAFIoverawiderangeofMvaluesanddemonstratesitsrobustnesstothechoiceofM.

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where and In( 4{1 ),X2CMLcontainstheobserveddatasampleswithMbeingthesnapshotdimensionandLbeingthesnapshotnumber.ThecolumnsinAj2CMNjandtherowsinSj2CKjLareknownandassumedtobelinearlyindependentofeachother.ThematricesBj2CNjKjcontaintheunknownregressioncoecients.Throughoutthischapter,weassumethatMNjandM+rLwithrbeingtherowrankofS.ThecolumnsoftheerrormatrixZareassumedtobei.i.d.circularlysymmetriccomplexGaussianrandomvectorswithzero-meanandunknowncovariancematrixQ.Theproblemofinteresthereinistoestimatetheunknownblock-diagonalmatrixB. NotethatintheBDGCmodelthelinearindependenceamongthecolumnsofAjandamongtherowsofSjisonlyrequiredwithinthesubmatrices.Inotherwords,thecolumnsfromdierentsubmatricesofAandtherowsfromdierentsubmatricesofScanbelinearlydependentoneachother.Notealsothat 51

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theBDGCmodeluniestheGCmodel[ 18 ]-[ 21 ],andtheDGCmodelin[ 36 ].Weremarkthatinthisdissertationwefocusonthecomplex-valuedparameterestimationproblem;howevertheproposedAMLestimatorcanbeuseddirectlyforreal-valuedparameterestimationasrequiredinsomestatisticalapplications[ 11 ]. Theremainderofthischapterisorganizedasfollows.WerstgivesomepreliminarymatrixresultsinSection 4.2 ,andthenderivetheAMLestimatorinSection 4.3 .ThetheoreticalperformanceanalysisandnumericalexamplesareprovidedinSections 4.4 and 4.5 ,respectively.Finally,Section 4.6 containstheconclusions. vecb(G),[vec(G1;1)Tvec(G2;2)Tvec(GJ;J)T]T;(4{4) 80 ]).

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Notethattheblock-diagonalvectorizationvecb()andthegeneralizedKhatri-Raoproduct~aredenedbasedonaparticularmatrixpartitioning,i.e.,dierentmatrixpartitioningswillleadtodierentresults.Notealsothatthestandardvectorizationvec()[ 74 ]andthediagonalvectorizationvecd()[ 36 ]arebothspecialcasesoftheblock-diagonalvectorizationvecb(),whiletheKroneckerproduct,theHadamardproductandtheKhatri-Raoproduct[ 77 ][ 78 ]areallspecialcasesofthegeneralizedKhatri-Raoproductbasedondierentmatrixpartitionings.Itisalsoworthpointingoutthatmatrixpartitioningmaybe\inherited"throughmatrixoperations.Forexample,forthepartitionedmatrixAgivenby( 4{2 ),AHAisapartitionedmatrixwiththe(i;j)th(i;j=1;2;;J)submatrixbeingAHiAj;hereafter,thesuperscriptHdenotestheconjugatetransposeofamatrix. Next,wegivetwolemmasonpartitionedmatrixoperations. vecb(GT)=(~)vecb(G):(4{6) Proof:WenotethatGTisapartitionedmatrixwiththe(k;k)th(k=1;2;;K)submatrixbeing [GT]k;k=JXj=1[]k;j[G]j;j[]Tk;j:(4{7) Hence, vec([GT]k;k)=JXj=1vec([]k;j[G]j;j[]Tk;j)=JXj=1([]k;j[]k;j)vec([G]j;j)=([]k;1[]k;1)([]k;J[]k;J)vecb(G);(4{8)

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wherewehaveusedthefactthatvec(ABC)=(CTA)vec(B)[ 74 ].Arrangingthevectorsvec([GT]k;k)(k=1;2;J)in( 4{8 )intoacolumnvectoryields( 4{6 ). Proof:NotethatU~Visapartitionedmatrixwith1blockrowandJblockcolumns,andwiththe(1;j)thsubmatrixbeing [U~V]1;j=[U]1;j[V]1;j:(4{10) Similarly,H~Fisapartitionedmatrixwith1blockrowandKblockcolumns,andwiththe(1;k)thsubmatrixbeing [H~F]1;k=[H]1;k[F]1;k:(4{11) Hence,(U~V)H(H~F)isapartitionedmatrixwithJblockrowsandKblockcolumns,andwiththe(j;k)thsubmatrixbeing [(U~V)H(H~F)]j;k=([U]1;j[V]1;j)H([H]1;k[F]1;k)=([U]H1;j[H]1;k)([V]H1;j[F]1;k);(4{12) wherewehaveusedthefactthat(AB)H(CD)=(AHC)(BHD)[ 74 ]. Ontheotherhand,UHHandVHFaretwopartitionedmatriceswithJblockrowsandKblockcolumns,andwiththe(j;k)thsubmatricesbeing [UHH]j;k=[U]H1;j[H]1;k(4{13) [VHF]j;k=[V]H1;j[F]1;k:(4{14)

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From( 4{12 ),( 4{13 )and( 4{14 ),weobtain [(U~V)(H~F)]j;k=[UHH]j;k[VHF]j;k;(4{15) whichyields( 4{9 )immediately. WeremarkthatLemmas 3.1 and 3.2 inChapter 3 arespecialcasesofLemma 4.1 inthischapter,whileLemmasA1andA2in[ 78 ]andLemma 3.3 in[ 36 ]arespecialcasesofLemma 4.2 inthischapter. 4{1 ).OurapproachisbasedontheMLprinciple,butanapproximationismadetogetaclosed-formsolution.Inthefollowingderivation,weutilizethepartitionedmatrixoperationsandlemmasofSection 4.2 .NotethatthepartitionedmatrixoperationsusedinthederivationsarebasedonthematrixpartitioningsofA,SandBin( 4{2 ),( 4{3 )and( 4{1 ),respectively.ThematricesXandZarebothtreatedasnon-partitionedmatrices. Forconvenience,wearrangetheunknownsinBintoacolumnvector,i.e.,=vecb(B).Itfollowsfrom( 4{1 )thatthenegativelog-likelihoodfunctionoftheobserveddatasamplesis(towithinanadditiveconstant) wheretr()andjjdenotethetraceandthedeterminantofamatrix,respectively. Minimizingthenegativelog-likelihoodfunctionwithrespecttoQyields ^Q=1 ForML,^Qisnonsingularwithprobabilityone.

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Substituting( 4{17 )into( 4{16 )yieldstheMLestimateof Ingeneral,theoptimizationproblemin( 4{18 )doesnotappeartoadmitaclosed-formsolution.Here,weusethetechniquein[ 25 ]and[ 27 ]tosolvethisproblemapproximately.LetSand?S,respectively,denotetheorthogonalprojectionmatricesgivenby and where()denotesthepseudoorgeneralizedinverseofamatrix[ 74 ].NotethatwhentherowsofSarenotlinearlyindependentofeachother,thematrix(SSH)issingularanditsgeneralizedinverseisnotunique.However,Sand?Sareunique[ 74 ]. Using( 4{19 )and( 4{20 ),weget wherethematrixTisdenedby Notethattherankof?SisLr.Hence,ThasfullrankMwithprobabilityonewhenM+rL.

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Using( 4{21 )andLemma4in[ 25 ](orTheorem1in[ 27 ]),thesolutionoftheoptimizationproblemin( 4{18 )canbeapproximated,foralargesnapshotnumberL,asfollows. ^AML=argmintr[(XSABS)HT1(XSABS)]=argminkvec(T1 2XS)vec(T1 2ABS)k2;(4{23) whereT1 2istheHermitiansquarerootofT1andkkdenotestheEuclideanvectornorm. ByusingLemma 4.1 (withK=1),wehave vec(T1 2ABS)=vecb(T1 2ABS)=[ST~(T1 2A)]:(4{24) Substituting( 4{24 )into( 4{23 )yieldsaquadraticfunctionof,whoseminimizerisgivenby ^AML=(H)1Hvec(T1 2XS)(4{25) where 2A):(4{26) ToguaranteethatthematrixHisinvertible,weassumethatthecolumnsofarelinearlyindependentofeachother,whichrequiresthelinearindependenceamongthecolumnswithineachsubmatrixofAandamongtherowswithineachsubmatrixofS.However,thecolumnsfromdierentsubmatricesofAandtherowsfromdierentsubmatricesofScanbelinearlydependentoneachother. ByLemmas 4.2 and 4.1 ,respectively,wehavethat and 2XS)=Hvecb(T1 2XS)=vecb(AHT1XSH):(4{28)

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Substituting( 4{27 )and( 4{28 )into( 4{25 )yieldstheAMLestimateof Wenotethatin( 4{29 )whenJ=1thegeneralizedKhatri-RaoproductreducestotheKroneckerproductandvecb()reducestovec().Then,usingthefactthatvec(ABC)=(CTA)vec(B),itcanbereadilyshownthattheAMLestimatorin( 4{29 )reducesto ^BGC-ML=(AHT1A)1AHT1XSH(SSH)1;(4{30) whichistheexactMLestimatorbasedontheGCmodel[ 17 ][ 18 ][ 21 ][ 22 ].Ontheotherhand,whenNj=Kj=1forj=1;2;J,thegeneralizedKhatri-RaoproductreducestotheHadamardproductandvecb()reducestovecd().Hence,( 4{29 )reducestotheAMLestimatorfortheDGCmodelin[ 36 ],namely ^DGC-AML=[(AHT1A)(SSH)T]1vecd(AHT1XSH);(4{31) wheredenotestheHadamardproduct(elementwisemultiplication)betweentwomatrices,andvecb()denotesacolumnvectorformedbythediagonalelementsofamatrix. 4{1 )into( 4{29 )andusingLemma 4.1 ,wehavethat ^AML=[(SSH)T~(AHT1A)]1vecb(AHT1ZSH):(4{32) Ontheotherhand,substituting( 4{1 )into( 4{22 )yields

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From( 4{33 ),weseethatTisanevenfunctionofZ,andhence^AMLisanoddfunctionofZ.Moreover,wehaveassumedthatZisazero-meanGaussianrandommatrix.Usingthestatisticalpropertiesofthezero-meanGaussiandistribution,wecanreadilyshowthattheexpectationof( 4{32 )iszero,i.e.,theAMLestimatorbasedontheBDGCmodelisunbiased. 4.1 ,( 4{1 )canberewrittenas vec(X)=(ST~A)+vec(Z):(4{34) Notethat( 4{34 )isalinearstatisticalmodelwithunknownnoisecovariancematrixIQ.ItcanbeeasilyveriedthattheFisherinformationmatrixforthismodelisablock-diagonalmatrixwithrespecttoandQ[ 73 ].Hence,theunknownsinQdonotaecttheCRBfor.ItfollowsthattheCRBforcanbereadilywritten[ 73 ]as CRB()=[(ST~A)H(IQ)1(ST~A)]1:(4{35) Then,byusingLemma 4.2 ,( 4{35 )canbesimpliedas CRB()=[(SSH)T~(AHQ1A)]1:(4{36) Next,weturntotheMSEanalysisof^AML.TheexactMSEof^AMLisdiculttodetermine,ifnotimpossible.Herein,weprovideanapproximateexpressionfortheMSEof^AML,whichholdsforalargesnapshotnumberL. Fromthetheoryofthelinearstatisticalmodel[ 16 ],weknowthatT 4{32 )canbeapproximated

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by ^AMLCRB()vecd(AHQ1ZSH):(4{37) UsingLemmas 4.1 and 4.2 (andviewingasaspecialcaseof~),itfollowsfrom( 4{37 )that MSE(^AML),E[(^AML)(^AML)H]CRB()E[vecb(AHQ1ZSH)vecb(AHQ1ZSH)H]CRB()=CRB()[S~(AHQ1)]E[vec(Z)vec(Z)H][ST~(AHQ1)H]CRB()=CRB()[S~(AHQ1)](IQ)[ST~(Q1A)]CRB()=CRB();(4{38) where()denotesthecomplexconjugate. Weseefrom( 4{38 )thattheAMLestimateofisasymptoticallystatisticallyecientforalargesnapshotnumberL.ThistheoreticalresultisillustratednumericallyinSection 4.5 .ItcanbeeasilyveriedthattheCRBfortheGCmodelinEquation(13)of[ 22 ]andtheCRBfortheDGCmodelinEquation(24)of[ 36 ])arespecialcasesof( 4{36 ). 4.4 WeconsideraDS-CDMAreceiverwithauniformlineararrayconsistingofM=6antennaswithhalf-wavelengthspacingbetweenadjacentantennas.Thearrayelementsareassumedtobeomni-directional.ConsiderJ=2transmitters,whosesignalsaremodulatedbytwodierentpseudo-noise(PN)sequences[ 81 ],respectively.ThePNsequencesareknownaprioribythereceiver.Supposethatthesignalfromthe1sttransmitterarrivesatthearraythroughN1=2paths

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withdirectionsofarrival(DOAs)of1=10and2=5,whilethesignalfromthe2ndtransmitterarrivesatthearraythroughN2=1pathwithDOAof3=10.WeassumethattheDOAshavebeenestimatedaccuratelyusing,e.g.,themethodofdirectionestimation(MODE)algorithm[ 82 ][ 83 ],andthereforetheycanbeconsideredtobeknown.Theproblemofinterestistoestimatetheunknowncomplexamplitudes1and2forthetwopathsofthe1sttransmitter,and3forthe2ndtransmitter,whichcontainthetransmittedinformation.Theestimatesoffjg3j=1canbeusedforsymboldetection.TheerrormatrixZcontainsthenoiseandinterference,whosecolumnsareassumedtobei.i.d.circularlysymmetriccomplexGaussianrandomvectorswithzero-meanandanunknowncovariancematrixQgivenby where=1=SNRwithSNRbeingthesignal-to-noiseratioandQm;ndenotesthe(m;n)thelementofQ. Let wherea()=[1ejsin()ej2sin()ej(M1)sin()]TisthesteeringvectorofthesignalwithDOAof.Let LetS12C1LandS22C1LbetheknownPNsequencesforthetwotransmitters,respectively,withLbeingthelengthofeachPNsequence.UnderthepreviousassumptionswecandescribethereceivedsignalusingaBDGCmodelwithM=6,J=2,N1=2,N2=1andK1=K2=1.Hence,theAMLestimatorin( 4{29 )canbeapplieddirectly. WepresenttheMSEoftheAMLestimatorin( 4{29 )aswellasthecorrespondingCRBin( 4{36 ).Forcomparisonpurposes,wealsoshowtheperformancesoftheGC

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method,theleastsquares(LS)andtheexactMLestimators.IntheGCmethod,weestimatethefullmatrixofBusing( 4{30 )[ 17 ][ 18 ][ 21 ][ 22 ],andthenpickupthecorrespondingblock-diagonalsubmatricesof^BGC-MLastheestimateofBj.TheLSestimatorassumesthattheinterferenceandnoisevectorsinZareuncorrelatedbothspatiallyandtemporally,andcanbeobtainedimmediatelyfrom( 4{29 )byreplacingTtherebyanidentitymatrix,i.e., ^LS=[(SSH)T~(AHA)]1vecb(AHXSH):(4{42) TheexactMLestimatesareobtainedbyapplyingthecyclicmaximization(CM)technique[ 84 ]tothecostfunctionin( 4{18 )withrespecttovariousBj.Ineachstepofthisiterativealgorithm,weassumethatallregressioncoecientsubmatrices,exceptforBk,areknown,whichmeansthatBkcanbereadilyestimatedbyusing( 4{30 ).WeusetheAMLestimatesastheinitialvaluesoftheregressioncoecientmatricesintheexactMLestimator.Theempiricalestimationperformancesareallobtainedfrom500Monte-Carlosimulations. Figs. 4{1 and 4{1 showtheestimationperformanceasfunctionsofthelengthofthePNsequencesLandtheSNR,respectively.TheSNRisxedat10dBinFig. 4{1 whileLisxedtobe128inFig. 4{2 .NotethattheMLestimatorbasedontheGCmodelhasarelativelypoorestimationperformance.Thismethodignorestheaprioriinformationabouttheblock-diagonalstructureofthecomplexamplitudematrixB,whichdoublesthenumberofunknownsandhenceresultsinmuchlargerestimationvariances.Notealsothatduetothe(quasi-)orthogonalpropertyofthePNsequenceswehaveRSS=LI(approximately),andhencetheCRBin( 4{36 )reducesto1 4{1 (a)4{1 (c),theCRBdecreaseslinearlyasthesnapshotnumberLincreases,whenbothCRBandLarepresentedinlog-scales.FromFigs. 4{1 (a)4{1 (c),wecanalsoseethattheAMLestimatorachievesaverysimilarperformanceasthe

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(a)(b) (c) Figure4{1: EmpiricalMSE'sandtheCRBversusLwhenSNR=10dB.(a)For1,(b)for2,and(c)for3. exactML;anditoutperformsLSandGCsignicantly.Aspredictedtheoretically,boththeexactMLandAMLareasymptoticallystatisticallyecientforalargesnapshotnumber,andtheyapproachthecorrespondingCRBrapidly.FromFig. 4{2 ,wenotethatAMLandtheexactMLprovidealmostidenticalperformances,andtheirestimatesareveryclosetotheCRBfortheentirerangeofSNRconsidered.Again,AMLoutperformstheLSandGCestimatorssignicantly.

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(a)(b) (c) Figure4{2: EmpiricalMSE'sandtheCRBversusSNR(dB)whenL=128.(a)For1,(b)for2,and(c)for3. matrixisconstrainedtobeblock-diagonal.Viaatheoreticalanalysis,wehaveshownthattheAMLestimatorisunbiasedandasymptoticallystatisticallyecientforalargesnapshotnumber.WehaveappliedtheAMLestimatortoacomplexamplitudeestimationprobleminwirelesscommunications.ThenumericalexamplesprovidecompellingevidencethattheAMLmethodcanachieveexcellentestimationaccuracy.

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1 ,wehavediscussedseveralmultiple-inputmultiple-output(MIMO)radarsystems.Accordingtotheirantennacongurations,theMIMOradarsdiscussedcanbegroupedintotwoclasses.Oneistheconventionalradararray,inwhichbothtransmittingandreceivingantennasarecloselyspacedforcoherenttransmissionanddetection[ 44 ]-[ 57 ].Theotheristhediverseantennaconguration,wheretheantennasareseparatedfarawayfromeachothertoachievespatialdiversitygain[ 40 ]-[ 43 ].Toreapthebenetsofbothschemes,inthischapter,weconsiderageneralantennaconguration,i.e.,boththetransmittingandreceivingantennaarraysconsistofseveralwell-separatedsubarrayswitheachsubarraycontainingclosely-spacedantennas.Weestablishthegrowth-curvemodelsinChapters 2 4 anddeviseseveralestimatorsfortheproposedMIMOradarsystem. Consideranarrow-bandMIMOradarsystemwith~Nand~Msubarraysfortransmittingandreceiving,respectively.Thenthtransmitandmthreceivesubarrayshave,respectively,NnandMmclosely-spacedantennas,n=1;2;;~N,m=1;2;;~M.Weassumethatthesubarraysaresucientlyseparated,andhenceforeachtargetitsradarcross-sections(RCS)fordierenttransmitandreceivesubarraypairsarestatisticallyindependentofeachother.Letvn()andam()bethesteeringvectorsofthenthtransmittingsubarrayandthemthreceivingsubarray,respectively,wheredenotesthetargetlocationparameter,forexampleitsangularlocation.Lettherowsofnbethewaveformstransmittedfromtheantennasofthenthtransmitsubarray.Weassumethatthearrivaltimeis 65

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known.Then,thesignalreceivedbythemthsubarrayduetothereectionofthetargetatcanbewrittenas wheremn;isthecomplexamplitudeproportionaltotheRCSforthe(m;n)threceiveandtransmitsubarraypairandforthetargetatthelocation,andthematrixZmdenotestheresidualtermcontainingtheunmodellednoise,interferencesfromtargetsotherthanandatotherrangebins,andintentionalorunintentionaljamming.Fornotationalsimplicity,wewillnotshowexplicitlythedependenceofZmon. Let and whereM=M1++M~MandN=N1++N~Narethetotalnumbersofreceiveandtransmitantennas,respectively,Listhenumberofdatasamplesofthetransmittedwaveforms,()Tdenotesthetransposeoperator,andDiag(a1;;a~M)isablock-diagonalmatrixwitha1;;a~Mbeingitsdiagonalsubmatrices.Then,( 5{1 )canbereadilyrewritteninthegrowth-curve(GC)modelinChapter 2 ,i.e., wherethe(m;n)thelementofthe~M~NmatrixBismn;,ZisdenedsimilarlytoXin( 5{2 ),andtherowsofS()arethereectedwaveformsbythetargetat

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location,i.e., Notethatwhen~N=~M=1,thesignalmodelin( 5{6 )reducestotheMIMOradarmodelin[ 55 ]-[ 57 ],whereaswhenN=~NandM=~Mitreducestothediversitydatamodelin[ 40 ]-[ 42 ].Basedonthisdatamodel,Webelowproposetwoclassesofnonparametricmethods,i.e.,spatialspectralestimationandgeneralizedlikelihoodratiotest(GLRT),fortargetdetectionandlocalization. TheremainderofthisChapterisorganizedasfollows.InSection 5.2 ,weintroduceseveraladaptivespatialspectralestimatorsincludingCapon[ 60 ]andAPES[ 23 ].InSection 5.3 ,wedescribeageneralizedlikelihoodratiotest(GLRT)andaconditionalgeneralizedlikelihoodratiotest(cGLRT),andwethenproposeaniterativeGLRT(iGLRT)procedurefortargetdetectionandparameterestimation.NumericalexamplesareprovidedinSection 5.4 .WerstcomparetheCramer-Raobounds(CRBs)forMIMOradarswithdierentantennacongurations,andthenpresentthedetectionandlocalizationperformanceoftheproposedmethods.Finally,Section 5.5 containstheconclusions. AsimplewaytoestimateBin( 5{6 )isviatheLeast-Squares(LS)method,i.e., ^BLS;=[AH()A()]1A()XSH()[S()SH()]1;(5{8)

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where()Hdenotestheconjugatetranspose.However,asanyotherdata-independentbeamforming-typemethod,theLSmethodsuersfromhigh-sidelobesandlowresolution.Inthepresenceofstronginterferenceandjamming,themethodcompletelyfailstowork.Weintroducebelowtwoadaptivespatialspectralestimationapproachesthatoermuchhigherresolutionandinterferencesuppressioncapabilities. 5{6 )consistsoftwomainsteps[ 60 ][ 85 ][ 22 ].TherstisageneralizedCaponbeamformingstep.ThesecondisaLSestimationstep,whichinvolvesbasicallyamatchedlteringtotheknownwaveformS(). ThegeneralizedCaponbeamformercanbeformulatedas minWtr(WHRW)subjecttoWHA()=I;(5{9) whereW2CM~Mistheweightingmatrixusedtoachievenoise,interferenceandjammingsuppressionwhilekeepingthedesiredsignalundistorted,tr()denotesthetraceofamatrix,and ^R=1 isthesamplecovariancematrixwithLbeingthenumberofdatasamples. Solvingtheoptimizationproblemin( 5{9 ),wecanreadilyhave ^WCapon=^R1A()[AH()^R1A()]1:(5{11) Byusing( 5{11 )and( 5{6 ),theoutputoftheCaponbeamformercanbewrittenas [AH()^R1A()]1AH()^R1X=BS()+[AH()^R1A()]1AH()^R1Z:(5{12)

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Byapplyingtheleast-squares(LS)methodto( 5{12 ),theCaponestimateofBfollowsreadily,i.e., ^BCapon;=[AH()^R1A()]1AH()^R1XSH()[S()SH()]1:(5{13) 23 ][ 24 ],whichcanbeformulatedas minW;BkWHXBS()k2subjecttoWHA()=I;(5{14) wherekkdenotestheFrobeniusnorm,andWistheweightingmatrix.Minimizingthecostfunctionin( 5{14 )withrespecttoByields ^BAPES;=WHXSH()[S()SH()]1:(5{15) Thentheoptimizationproblemreducesto mintr(WH^QW)subjecttoWHA()=I;(5{16) with ^Q=^R1 Fornotionalsimplicity,wehaveomittedthedependenceof^Qon. Solvingtheoptimizationproblemof( 5{16 )givesthegeneralizedAPESbeamformerweightingmatrix ^WAPES;=[AH()^QA()]1^Q1A():(5{18) Inserting( 5{18 )in( 5{15 ),wereadilygettheAPESestimateofBas ^BAPES;=[AH()^Q1A()]1AH()^Q1XSH()[S()SH()]1:(5{19)

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Interestingly,wenotethat( 5{19 )hasthesameformastheMLestimateinChapter 2 .However,theAPESestimateisderivedbasedonthebeamformingmethod,and,unliketheMLinChapter 2 ,itdoesnotneedprobabilitydensityfunction(pdf)ofZ. 5{6 )areindependentlyandidenticallydistributed(i.i.d.)circularlysymmetriccomplexGaussianrandomvectorswithmeanzeroandanunknowncovariancematrixQ. Considerthefollowinghypothesistestproblem i.e.,wewanttotestifthereexistsatargetatlocationornot.Similarlyto[ 65 ]and[ 86 ],wedeneageneralizedlikelihoodratio(GLR) maxB;Qf(XjH1)1 wheref(XjHi)(i=0;1)isthepdfofXunderthehypothesisHi.From( 5{21 ),wenotethatthevalueoftheGLR,(),liesbetween0and1.Ifthereisatargetatalocationofinterest,wehavemaxB;Qf(XjH1)maxQf(XjH0),i.e.,1;otherwise0.

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UnderHypothesisH0,wehave wherejjdenotesthedeterminantofamatrix.Maximizing( 5{22 )withrespecttoQyields maxQf(XjH0)=(e)LMj^RjL;(5{23) where^Risdenedin( 5{10 ). Similarly,underHypothesisH1,wehave Maximizing( 5{24 )withrespecttoQyields maxQf(XjH1)=(e)LM1 Hence,theoptimizationprobleminthedenominatorof( 5{21 )reducesto minB1

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Following[ 21 ]and[ 22 ]anddroppingthedependenceofA,SandBonfornotionalconvenience,wehave 2ABXSH(SSH)1(SSH)ABXSH(SSH)1H^Q1 2=j^QjI+1 2ABXSH(SSH)1H^Q1ABXSH(SSH)1(SSH)1 2 =j^QjI+1 2SXH^Q1^Q1A(AH^Q1A)1AH^Q1XSH(SSH)1 2+1 2B(AH^Q1A)1AH^Q1XSH(SSH)1H(AH^Q1A)B(AH^Q1A)1AH^Q1XSH(SSH)1(SSH)1 2j^QjI+1 2SXH[^Q1^Q1A(AH^Q1A)1AH^Q1]XSH(SSH)1 2 where^Qisdenedin( 5{17 ).Toget( 5{27 ),wehaveusedthefactthatjI+XYj=jI+YXj[ 74 ],andtheequalityin( 5{28 )holdswhenBequatestotheAPESestimatein( 5{19 ).Notethat 2SXH[^Q1^Q1A(AH^Q1A)1AH^Q1]XSH(SSH)1 2=j^QjjI+[^Q1^Q1A(AH^Q1A)1AH^Q1](^R^Q)j=j^RA(AH^Q1A)1AH^Q1(^R^Q)j=j^RjjIAH(AH^Q1A)1AH(^Q1^R1)j=j^Rjj(AH^Q1A)1(AH^R1A)j; From( 5{25 ),( 5{28 )and( 5{29 ),itfollowsthat maxB;Qf(XjH1)=(e)LMj^RjLjAH()^Q1A()jLjAH()^R1A()jL:(5{30)

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Substituting( 5{23 )and( 5{30 )into( 5{21 )yields jAH()^Q1A()j)L:(5{31) Weremarkthatwhentherearemultipletargets,andthenumberoftargets(sayK)areknownapriori,theGLRTin( 5{31 )canbeextendedtoamultivariatecounterpartbyconsideringthefollowinghypothesistestingproblem Asaparametricmethod,thismultivariateGLRTcanprovidebettertargetdetectionandparameterestimationperformancethanitsunivariatecounterpart.However,themultivariateGLRTiscomputationallyintensiveduetothefactthatitneedstosearchintheKdimensionalparameterspacefkgKk=1.Moreover,thenumberoftargetsishardlyknownaprioriinpractice. WeproposebelowaniterativeGLRT(iGLRT),whichrequireonlyone-dimensionalsearch(liketheunivariateGLRT),butprovidesatargetdetectionandparameterestimationperformanceclosetothemultivariateGLRT.

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Notethatboththeequationsin( 5{33 )areintheformoftheblockdiagonalgrowthcurve(BDGC)modelstudiedinChapter 4 .Forconvenience,werewrite( 5{33 )as where ~Bp=Diag(B^1;;B^p) (5{35) ~Ap=[A(^1)A(^p)]; ~Sp=[ST(^1)ST(^p)]T; ~Bp+1=Diag(B;B^1;;B^p) (5{38) ~Ap+1=[A()A(^1)A(^p)]; ~Sp+1=[ST()ST(^1)ST(^p)]T; andDiag(1;;K)denotesablockdiagonalmatrixformedfrom1;;K. Similarly,wedeneaconditionalgeneralizedlikelihoodratio(cGLR) max~Bp+1;Qf(XjHp+1)#1 wheref(XjHi)isthepdfofXundertheHihypothesis,andQisthecovariancematrixofthecolumnsofZ. Werstconsidertheoptimizationproblemofthenumeratorin( 5{41 ).Maximizingf(XjHp)withrespecttoQyields maxQf(XjHp)=(e)ML1 Hence,theoptimizationproblemreducesto min~Bp1

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Theoptimizationproblemof( 5{43 )doesnotappeartoadmitaclosed-formsolution,duetotheconstraintthat~Bpisablock-diagonalmatrix.Herein,weadoptatechniqueusedinChapters 3 and 4 togetapproximateclosed-formsolution. Notethat where and ~Qp=1 with()denotingthegeneralizedmatrixinverse. Considertheidempotentmatrices~Spand?~Sp.Assumethatthenumberofdatasamplesislargeenough,i.e.,L~Np.Notethat~Spisan~NpLmatrix.Hence,wehave rank(~Sp)~Npandrank(?~Sp)L~Np;(5{47) withrank()denotingtherankofamatrix.Then,wehave ~Qp=O(1);(5{48)

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1 Therefore,weget 1 LetfigMi=1betheeigenvaluesofthematrixin( 5{50 ),whichobviouslysatisfythat0i1.Byusingsomematrixmanipulations,weobtain 2pX~Sp)vec(~Q1 2p~Ap~Bp~Sp)k2; wherekkandvec()denotetheEuclideannormandvectorizationoperator(stackingthecolumnsofamatrixontopofeachother),respectively,and~Q1 2pistheHermitiansquarerootof~Q1p.In( 5{51 ),wehaveomittedthehigh-ordertermsoffigfortheapproximation. Hence,foralargenumberofdatasamples,theoptimizationproblemin( 5{43 )canbeapproximatedas min~Bpkvec(~Q1 2pX~Sp)vec(~Q1 2p~Ap~Bp~Sp)k2with~Bp=Diag(B^1;;B^p): Tosolvetheaboveoptimizationproblem,weneedtheblock-matrixoperationsandlemmas 4.1 and 4.2 inChapter 4 .Throughoutthischapter,thepartitionedmatrixoperationareallbasedonthepartitioningsin( 5{35 )-( 5{40 ).

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Now,let ~p=vecb(~Bp);(5{53) wherevecb()denotestheblock-diagonalvectorizationoperator(Denition 4.1 ),i.e., ~p=hvecT(B^1)vecT(B^p)iT:(5{54) ByusingLemma 4.1 (withK=1andJ=p)ofChapter 4 ,weobtain vec(~Q1 2p~Ap~Bp~Sp)=[~STp~(~Q1 2p~A)]~p,~p~p;(5{55) where~denotesthegeneralizedKhatri-Raoproduct(Denition 4.2 ).Hence, 2pX~Sp)vec(~Q1 2p~Ap~Bp~Sp)k2=kvec(~Q1 2pX~Sp)p~pk2kvec(~Q1 2pX~Sp)k2vecH(~Q1 2pX~Sp)~p(~Hp~p)1~Hpvec(~Q1 2pX~Sp)=Ltr(~Q1p^R)LMvecbH(~AHp~Q1pX~SHp)h(~Sp~SHp)T~(~AHp~Q1p~Ap)i1vecb(~AHp~Q1pX~SHp); wherewehaveusedLemmas and 4.2 ,andtheequalityholdswhen ~=(~Hp~p)1~Hpvec(~Q1 2pX~Sp):(5{57) Byusing( 5{42 ),( 5{44 ),( 5{51 )and( 5{56 ),itfollowsthat (e)Mg(^1;;^p)j~Qpj;(5{58) where

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Similarly,wehave (e)Mg(;^1;;^p)j~Qp+1j;(5{60) where~Qp+1andg(;^1;;^p)aredenedsimilarlyto~Qpin( 5{46 )andg(^1;;^p)in( 5{59 ),respectively. Substituting( 5{58 )and( 5{60 )into( 5{41 )yieldstheconditionalGLR Calculate()in( 5{31 )foreach. Compare()toathreshold(say0).If()<0forall,thenStop;otherwise,^1=argmax(),gotoStepII. Calculatejf^igki=1in( 5{61 )foreach. Ifjf^igki=1<0forall,thengotoStepIII;otherwise,^k+1=argmaxjf^igki=1.

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Calculatejf^igi6=kforeach. Update^kbyargmaxjf^igi6=k. 37 ],i.e., ^^K=(~AH^K~Q1^K~A^K)~(~SH^K~S^K)T1vecb(~AH^K~Q1X~SH^K);(5{62) where~A^K,~S^Kand~Q^Karedenedsimilarlyto~Ap,~Apand~Qpin( 5{37 ),( 5{38 )and( 5{46 ),respectively. WenotethatStepIIIoftheaboveiGLRTalgorithmactuallyminimizethefunctiong(1;;^K)withrespecttofkg^Kk=1byusingthecyclicminimization(CM)technique[ 84 ].Underamildcondition,i.e.,L~N^K,wehaveg(1;;^K)0.Furthermore,weknowthattheCMalgorithmmonotonicallydecreasesthecostfunction.HencetheiGLRTalgorithmisconvergent.When^Kisthetruenumberoftargets,iGLRTreducestoanapproximate(parametric)maximumlikelihoodestimator.Aswewillshowvianumericalexamples,themean-squared-error(MSE)oftheestimateofiGLRTapproachesthecorrespondingCramer-Raobound(CRB)foralargenumberofdatasamples.Ontheotherhand,wenotethatiGLRTneedsonlyone-dimensionalsearchandhenceiscomputationallyecient. 5.4.1Cramer-RaoBound

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groupedintomultiplesubarrays(eachbeingauniformlineararraywithhalf-wavelengthspacingbetweenadjacentelements).Weconsiderthefollowingantennacongurationschemes. Weassumethatthetransmittedwaveformsarelinearlyorthogonaltoeachotherandthetotaltransmittedpowerisxedtobe1,i.e.,R=1 WeconsiderascenarioinwhichK=3targetsarelocatedat1=40,2=4and3=0,andtheelementsoffBkg3k=1areindependentlyandidenticallydistributed(i.i.d.)circularlysymmetriccomplexGaussianrandomvariableswithzeromeanandunitvariance.Thereisastrongjammerat10withamplitude100,i.e.,40dBabovethereectedsignals.ThereceivedsignalhasL=128snapshotsandiscorruptedbyazero-meanspatiallycoloredGaussiannoisewithanunknowncovariancematrix.The(p;q)thelementoftheunknownnoisecovariancematrixis1 5{1 (a)and 5{1 (b)showthecumulativedensityfunctions(CDFs)oftheCRBsforMIMOradarwithvariousantennacongurationswhenSNR=20dB.(TheCRBof2issimilartothatof3andhenceisnotshown.)TheCDFsareobtainedby2000Monte-Carlotrials.Ineachtrial,wegeneratetheelementsoffBkg3k=1randomly,andthencalculatethecorrespondingCRBsusing( C{18 )givenbyintheAppendix C .Forcomparisonpurposes,wealsoprovidetheCDFofthephased-array(single-inputmultiple-output)counterpart,i.e.,thespecialcaseoftheaboveMIMOradarwhenN=1,

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(a)(b) Figure5{1: CumulativedensityfunctionsoftheCramer-Raoboundsfor(a)1and(b)3. withthesametotaltransmissionpower.Asexpected,theMIMOradarprovidesmuchbetterperformancethanthephased-arraycounterpart.DuetothefadingeectoftheelementsoffBkg3k=1,theCRBofMIMORadarAvarieswithinalargerange.Withina95%condenceinterval(i.e.,whenCDFvariesfrom2.5%to97.5%),itsCRBfor1variesapproximatelyfrom5107to5105.TheCRBsforMIMOradarCvarieswithinasmallrange. ToevaluatetheCRBperformance,wedeneanoutageCRB[ 43 ]foragivenprobabilityp,denotedbyCRBp,as Figs. 5{2 (a)5{2 (d)showtheoutageCRB0:01andCRB0:1of1and3,asfunctionsofSNR.Asexpected,theSNRgainsdependontheprobabilityp.Aswecansee,whenp=0:01,MIMOradarCoutperformstheotherradarcongurations,andprovidesaround20dBand12dBimprovementsintermsofSNRcomparedtothephased-arrayandMIMOradarA,respectively.Ontheotherhand,Fig. 5{2 (d)showsthatMIMOradarsAandBoutperformotherswhenp=0:1.

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(a)(b) (c)(d) Figure5{2: OutageCRBversusSNR.(a)CRB0:01for1,(b)CRB0:01for2,(c)CRB0:1for1,and(d)CRB0:1for2.

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Werstconsiderascenarioinwhich3targetsarelocatedat1=40,2=20and3=0withthecorrespondingelementsinB1,B2andB3beingxedto2,2and1,respectively.TheothersimulationparametersarethesameasforFig. 5{1 .TheFrobeniusnormofthespatialspectralestimatesofBversus,obtainedbyusingLS,CaponandAPESaregivenbyFigs. 5{3 (a)5{3 (c).Forcomparisonpurposes,weshowthetruespatialspectrumviadashedlinesinthesegures.AsseenfromFig. 5{3 ,theLSmethodsuersfromhigh-sidelobesandpoorresolutionproblems.Duetothepresenceofthestrongjammingsignal,theLSestimatorfailstoworkproperly.CaponandAPESpossessexcellentinterferenceandjammingsuppressioncapabilities.TheCaponmethodgivesverynarrowpeaksaroundthetargetlocations.However,theCaponestimatesofB1,B2andB3arebiaseddownward.TheAPESmethodgivesmoreaccurateestimatesaroundthetargetlocationsbutitsresolutionisworsethanthatofCapon.NotethatinFigs. 5{3 (a), 5{3 (b)and 5{3 (c),afalsepeakoccursat=10duetothepresenceofthestrongjammer.DespitethefactthatthejammerwaveformisstatisticallyindependentofthewaveformstransmittedbytheMIMOradar,afalsepeakstillexistssincethejammeris40dBstrongerthantheweakesttargetandthenumberofdatasamplesisnite.Figs. 5{3 (d)and 5{3 (e)givetheGLRT,andtheiGLRTresults,asfunctionsofthetargetlocationparameter.Forconvenience,inFig. 5{3 (e),wehaveincludedallcGLRfunctionsobtainedbyiGLRT,eachindicatingonetarget.Asexpected,wegethighGLRs(cGLRs)atthetargetlocationsandlowGLRs(cGLRs)atotherlocationsincludingthejammerlocation.BycomparingtheGLRwithathreshold,thefalsepeakduetothestrongjammercanbereadily

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(a)(b) (c)(d) (e) Figure5{3: Spatialspectra,andGLRandcGLRPseudo-Spectra,when1=40,2=20,and3=0.(a)LS,(b)Capon,(c)APES,(d)GLRT,and(e)iGLRT.

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detectedandrejected,andacorrectestimateofthenumberofthetargetscanbeobtainedbybothmethods. Nextweconsideramorechallengingexamplewhere2is4whilealltheothersimulationparametersarethesameasbefore.AsshowninFig. 5{4 (c),inthisexample,theAPES,CaponandGLRTmethodsfailtoresolvethetwocloselyspacedtargetsat2=4and3=0.Ontheotherhand,iGLRTgiveswell-resolvedpeaksaroundthetruetargetlocations.ToillustratetheprocedureoftheiGLRTalgorithm,wegivetheGLR,andcGLRsobtainedinStepsIandIIofiGLRTinFigs. 5{5 (a)5{5 (d).Figs. 5{5 (a)and 5{5 (b)showtheGLR()andthecGLR(j^1),respectively,where^1istheestimatedlocationoftarget1from().Aswecansee,thereisnopeakataround3=0inbothgures.Yetaclearpeakisshownin(j^1;^2)inFig. 5{5 (c),whichindicatestheexistenceandlocationoftarget3.ThecGLR(j^1;^2;^3)inFig. 5{5 (d)showsthatnoadditionaltargetexistsotherthanthetargetsat^1,^2and^3.Inotherwords,theiGLRTmethodcorrectlyestimatesthenumberoftargetstobe3. NowweconsidertheelementsinB1,B2andB3asi.i.dcomplexGaussianrandomvariableswithmeanzeroandunitvariance.TheotherparametersarethesameasthoseinFig. 5{6 .TheFigs. 5{6 (a)and 5{6 (b)presenttheCDFsoftheMSEsof1and3aswellastheCRBs,whenSNR=20dBandL=128.Aswecansee,theMSEsoftheiGLRTareveryclosetothecorrespondingCRBs.Figs. 5{7 (a)and 5{7 (b)showtheoutageMSE0:1andCRB0:1whenp=0:1asfunctionsofSNRwhenL=128.Again,theMSEsareveryclosetothecorrespondingtheCRBs,anddecreasesalmostlinearlyasSNRincreases.Fig. 5{8 givsetheoutageMSE0:1andCRB0:1asfunctionsofLwhenSNR=20dB.Asexpected,theoutageMSE0:1approachesthecorrespondingCRB0:1asLincreases.

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(a)(b) (c)(d) Figure5{4: Spatialspectra,andGLRandcGLRPseudo-Spectra,when1=40,2=4,and3=0.(a)Capon,(b)APES,(c)GLRT,and(d)iGLRT.

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(a)(b) (c)(d) Figure5{5: GLRandcGLRPseudo-SpectraobtainedinStepsIandIIofiGLRT,when1=40,2=4,and3=0.(a)(),(b)(j^1),(c)(j^1;^2),and(d)(j^1;^2;^3). (a)(b) Figure5{6: CumulativedensityfunctionsoftheCRBsandMSEsfor(a)1and(b)3.

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(a)(b) Figure5{7: OutageCRB0:1andMSE0:1versusSNRfor(a)1and(b)3. (a)(b) Figure5{8: OutageCRB0:1andMSE0:1versusSNRfor(a)1and(b)3.

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Inthisdissertation,wehavestudiedseveralvariationsofthegrowth-curvemodel,includinggrowth-curve(GC),diagonalgrowth-curve(DGC)andblockdiagonalgrowth-curve(BDGC).Thesemodelsareinthesameformbutwithdierentconstraintsontheunknownregressioncoecientmatrix.Specically,theregressioncoecientmatrixinGCisunconstrained,whereasitisconstrainedtobediagonalinDGCandblockdiagonalinBDGC.Wehaveprovidedamaximumlikelihood(ML)estimatorfortheregressioncoecientmatrixfortheGCmodel,andproposedtwoapproximatemaximumlikelihood(AML)estimatorsforDGCandBDGC.WehavealsoanalyzedthestatisticalpropertiesoftheMLandAMLestimatorstheoretically,andhaveshownthatallthreeestimatorsareunbiasedandasymptoticallystatisticallyecientforalargedatasamplenumber.Severalapplicationsofthegrowthcurvemodelstosignalprocessing,includingspectralanalysis,arraysignalprocessing,wirelesscommunicationsandmultiple-inputmultiple-output(MIMO)radar,havealsobeeninvestigated. MotivatedbythesuccessoftheMIMOwirelesscommunications,wehaveinvestigatedaMIMOradartopicindetail,whichisattractingincreasingattentionsinbothacademicandindustry.WehaveconsideredaMIMOradarsystemwithageneralantennaconguration,i.e.,boththetransmitterandreceiverhavemultiplewell-separatedsubarrayswitheachsubarraycontainingclosely-spacedantennas.Hence,boththecoherentprocessinggainandthespatialdiversitygaincanbeachievedbythesystemsimultaneously.Byusingourresultsonthegrowth-curvemodel,wehaveprovidedageneralizedlikelihoodratiotest(GLRT)andaconditionalgeneralizedlikelihoodratiotest(cGLRT)forthesystem,and 90

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thenproposedaniterativeGLRT(iGLRT)procedurefortargetdetectionandparameterestimation.Viaseveralnumericalexamples,wehaveshownthatiGLRTcanprovideexcellentdetectionandestimationperformanceatalowcomputationalcost. SinceMIMOradarisanemergingtechnology,manytopicsandissuesofMIMOradarareyettobeaddressed.ThefollowinglistprovidesoftheinterestingtopicsonMIMOradar.FundamentalTradeosinMIMORadar 42 ],multipleantennasareusedtoachievediversity.Ithasbeenshown[ 42 ]thatforaMIMOradarwithMtransmitandNreceiveantennas,themaximaldiversitygainisMN.In[ 58 ],weusetheMIMOradartechnologytoimprovetheparameteridentiability.Wehaveshown[ 58 ]thatthemaximumnumberoftargetsthatcanbeuniquelyidentied(withprobability1)bytheMIMOradarisuptob2MN1 3c,i.e.,approximatelyMtimesthatofitsphased-arraycounterpart.Obviously,byusingthegeneralantennacongurationinChapter 5 ,weneedtomakeatrade-obetweenthespatialdiversitygainandtheparameteridentiability.OneinterestingproblemisweatherthereisafundamentaltradeotheoryforMIMOradar,whichisanaloguetotheoneforwirelesscommunications[ 87 ].TargetDetectionandParameterEstimation

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radarinmorechallengingenvironmentsandforvariousapplicationsneedstobestudied.Forexamples,weneedtodevisealgorithmsfor 88 ]-[ 90 ]MIMOradar, 44 ][ 45 ].Thesemethodsmainlyfocusontheoptimizationofthespatialfeaturesofthetransmittedwaveforms,andhencetheyactuallyarespatialbeampatterndesigns.Ontheotherhand,intheliterature[ 91 ]-[ 94 ]manytemporaldesignshavebeenproposedfortheconventionalradar.In[ 94 ],wehaveproposedaSignalWaveform'sOptimal-under-RestrictionDesign(SWORD)foractivesensing,whichcansignicantlyincreasetheSINRwhilekeepingthedesiredfeatures,forexample,constantmodulusaswellasreasonablerangeresolutionandpeaksidelobelevel.Inthelightofthesetwodierentdesignphilosophies,itisinterestingtodevelopaspace-timedesignofprobingsignals,whichcanachieveoptimalitybothspatiallyandtemporally.

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2{1 ),bothQandBareunknown.Letdenotethevectorcontainingthereal-valuedunknownsinQandB.Then,the(i;k)-thelementofthecorrespondingFisherinformationmatrix(FIM)[ 71 ][ 72 ]is FIM(i;k)=LtrQ1@Q wheretr()denotesthetraceofamatrix,Re()andIm()denotetherealandimaginarypartsofacomplexnumber(ormatrix),respectively,andidenotestheithelementof.BecauseQandBdependonthedierentvariablesin,FIMwillbeablockdiagonalmatrixwithrespecttotheunknownsinQandB.Hence,wecancalculatetheCRBsofBandQseparately.InthisChapter,weareonlyinterestedintheCRBofB. Letbik;Randbik;Idenotetherealandimaginarypartsofthe(i;k)-thelementinB,respectively.ThecorrespondingelementsinFIMwithrespecttoanytworeal-valuedunknownsinBareFIM(bik;R;bmn;R)=2Re(sksTn)(aHiQ1am); wheresk(k=1;2;:::K)andai(i=1;2;:::N)arethekthrowvectorinSandithcolumnvectorinA,respectively. 93

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Arrangingthecomplex-valuedmatrixBtoformareal-valuedvector,i.e., andarranging( A{2 )to( A{5 )toamatrixaccordingtob,wegetthecorrespondingFIM FIM(b)=2264Re()Im()Im()Re()375;(A{7) where Usingthematrixinversionlemmaandtheinversionlemmaofpartitionedmatrices[ 74 ],wegettheCRBinreal-valuedform CRB(b)=[FIM(b)]1=1 2264Re(1)Im(1)Im(1)Re(1)375:(A{9) Transformingtheabovereal-valuedCRBintothecomplex-valuedformyields CRB(B),CRB(vec(B))=(SST)1(AHQ1A)1:(A{10) Clearly,thediagonalelementsofCRB(B)aredeterminedbythediagonalelementsof(SST)1and(AHQ1A)1.TostudytheinuencesofthetemporalinformationmatrixSandthespatialinformationmatrixAonCRB(B),wedenoteRS=(SST)andRA=(AHQ1A).Withoutlossofgenerality,weconsidertheCRBofb11,i.e.,theelementontherstcolumnandrstrowofB.PartitionRS,R1S,RAandR1Aasfollows.

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Obviously,CRB(b11)=(rS)11(rA)11.Bytheinversionlemmaofpartitionedmatrices[ 74 ],wehave (rS)11=1 (rS)11(rS)12(RS)122(rS)21;(A{13) (rA)11=1 (rA)11(rA)12(RA)122(rA)21:(A{14) In( A{13 ),(rS)11=ks1k2istheEuclideannormsquareoftherstrowofS.Itiseasilyveriedthat(RS)22isapositivedenitematrixwhile(rS)12=(rS)H21.Therefore,(rS)11isminimizedwhen(rS)12=0,i.e.,s1sTk=0(k=2;3;:::;K).Hence,tominimizeCRB(B),therowvectorsinSshouldbeorthogonaltoeachother. Similarly,(rA)11isminimizedwhen(rA)12=0,i.e.,(Q1 2a1)H(Q1 2ai)=0(i=2;3;:::N).Notethatwhenthisconditionisnotsatised,largeNcauses(rA)11toincrease.Furthermore,Wenotethat(rA)11=aH1Qa1.Therefore,whena1isproportionaltotheeigenvectorofQcorrespondingtoitssmallesteigenvalue,(rA)11isminimized.Therefore,tominimizeCRB(B),thecolumnsofQ1 2AshouldbeorthogonaltoeachotherandthecolumnsofAshouldcorrespondtothesubspacespannedbytheeigenvectorsofQcorrespondingtoitssmallestNeigenvalues.SinceQisunknownandusuallyAisgivenandcannotbechangedinpractice,wecanonlyhopefortheseconditionsofA. 2.1 75 ]and[ 95 ],thepdfofthecomplexWishartdistributionis

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whereI()isaconstantscalardependenton,i.e., 2m(m1)(l)(l1):::(lm+1)jjm;(A{16) with()beingtheGammafunction. Partition1asin( 2{21 ) Utilizingtheformulathatjj=j22jj11:2j[ 74 ]andthefactthatandarebothHermitiansymmetric,( A{15 )canbewritten Makingatransformationfrom(11,12,22)to(11:2,12,22)whoseJocobiandeterminantiseasilycalculatedtobe1,andaftersomestraightforwardmanipulationsusingtheinversionlemmaofpartitionedmatrices,weget wheref1(11:2)=K1j11:2jlmexptr[111:211:2]; withK1,K2,K3beingconstantscalars. BasedonthepdfsofthecomplexWishartandmatrix-variatecomplexGaussiandistributionsin[ 75 ]and[ 95 ],theconclusionsinLemma 2.1 isimmediatelyreachedfrom( A{19 )to( A{22 ).

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2.2 75 ].Hence, SinceisHermitiansymmetric[ 75 ],jjisareal-valuedscalar,i.e.,jj=jj.Usingtheformulasthattr(XTYZWT)=vec(X)T(WY)vec(Z)andjXlljmjYmmjl=jXYj[ 74 ],wecanwritethepdfof~as Equation( A{24 )isastandardvectorcomplexGaussianpdfwithzero-meanandcovariancematrix.Hence, i.e., where~idenotestheithcolumnvectorof~andikisthe(i;k)-thelementin. Letcikbethe(i;k)-thelementinC.Thenwehave By( A{23 )and( A{27 ),Lemma 2.2 isimmediatelyproved. 2.3

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Let~g=1 2gand~=1 21 2.Obviously,wehave[ 75 ] ~CW(l;m;I);(A{28) and Decomposethevector~gas ~g=Up;(A{30) whereUisaunitarymatrixwithitsrstcolumnbeing~g A{29 )weget Thenpartitionand1asfollows. =26411122122375;1=26411122122375;(A{32) where11and11arebothscalars. Let11:2=111212221:Then, (gH1g)1=(pH1p)1=(gH1g)1(11)1=(gH1g)111:2:(A{33) ByLemma 2.1 ,weknow11:2CW(lm+1;1;1).AccordingtothedenitionofthecomplexWishartdistribution,11:2canbeexpressedasthesumofthenormsquaresoflm+1independentstandardcomplexGaussianrandomvariables.Hence,211:2canbeexpressedasthesumofthesquaresof2(lm+1)independentstandardreal-valuedGaussianrandomvariables,i.e.,211:2hasthecentral2distributionwith2(lm+1)degreesoffreedom[ 96 ].Accordingtothe

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proprietiesofthe2distribution,weknowthat 2(lm):(A{34) Thus,by( A{33 )and( A{34 ) Hence,gHE(1)g=gH(1 2.4 Decomposegas whereUisappunitarymatrixwithitsrstcolumnbeingg Let~=U;obviously~CN(0;In;Ip).Let~i(i=1;2;:::p)betheithcolumnvectorof~.Let~=+Ppi=2~i~Hi;obviously~CW(l+p1;n;I)[ 75 ].Then 1+~H1~1~1:(A{37) Ithasbeenshownintheappendixof[ 64 ]that 1 1+~H1~1~1beta(ln+p;n):(A{38)

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Hence[ 96 ], 1+~H1~1~1=ln+p l+p:(A{39) From( A{37 )and( A{39 ),wehave l+pIpg:(A{40) Theaboveequationshouldbesatisedforanynon-zerovectorg,whichmeansE()=n l+pI. TocalculateE(vec()vec()H),weneedtocalculate wherer1c1denotesthe(r1;c1)-thelementin,kdenotesthekthcolumnvectorinandr1;c1;r2;c2=1;2;3;:::;p. Notethat( A{41 )isafunctionofk(k=1;2;:::;p).Adoptingthesametechniqueusedtoobtain( 2{54 ),wecaneasilyshowthattheexpectationiszerowhen( A{41 )containsoddnumbersofk,e.g.,E(1234)=E(1123)=E(1223)=E(1232)=0. Whenr1=c2,c1=r2butr16=c1, Replacingr1byjr1,wherejistheunitoftheimaginarynumber,yields Ontheotherhand,sincer1isazero-meancircularlysymmetriccomplexGaussianrandomvector,jr1,asarandomvectortransformedfromr1,hasthesamestatisticalpropertyasr1.Hence

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Hence,by( A{43 )and( A{44 )wehaveE(r1c1c1r1)=0. Besidestheabovecases,therearethreecasesleftinwhichE(r1c1r2c2)isnon-zero,i.e., (i). (ii). (iii). Weknowthattheexpectationineachcaseisequal.Forconvenience,wedenotetheexpectationsinthethreecasesasE(j11j2),E(1122),andE(j12j2),respectively. First,wecalculateE(j11j2).Let=+Ppk=2kHk,whichobviouslyhastheCW(l+p1;n;I)distribution.Then, 1+H11121A=12E1 1+H111!+E241 1+H111!235:(A{45) Again,usingtheconclusionintheappendixof[ 64 ],weknow 1 1+H111beta(ln+p;n):(A{46) Calculatingtheexpectationandthe2nd-ordermomentoftheabovebetadistributionandsubstitutingtheminto( A{45 )yields (l+p)(l+p+1):(A{47) Second,weconsiderE(1122).Itisdiculttocalculatethisexpectationdirectly.However,notethat+Hcanbeexpressedasthesumoftheouter-productsofl+pcomplexGaussianrandomvectorswithzeromeanandcovarianceIn.BytheLawofLargeNumbers[ 96 ],forlargel+p,itapproaches(l+p)Iin

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probability.Hence, (l+p)2E(H11)(H22)=n2 Third,inordertocalculateE(j12j2),weusethefactthat Thisequationcanbeprovedasfollows.Letusmakeatransformationfrom(1;2)to 2(1+2);p 2(12).Obviously,p 2(1+2)andp 2(12)aretwoindependentstandardcomplexGaussianrandomvectorsandretainthesamestatisticalpropertiesof(1;2).Hence,replacing(1;2)bythenewrandomvectorsinE(j11j2),theexpectationwillnotchange,i.e., 4Ej(H1+H2)(+H)1(1+2)j2=1 2E(j11j2)+1 2E(j12j2)+1 2E(1122):(A{50) From( A{50 ),Equation( A{49 )isproved. Usingthefactsof( A{47 ),( A{48 )and( A{49 )andarrangingE(r1c1r2c2)intoE(vec()vec()H),( 2{61 )followsimmediately.

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3.1 vec(UGVT)=vecpXi=1(giuivTi)=pXi=1vec(uigivTi)=pXi=1(viui)gi=(VU)g;(B{1) wherewehaveusedthefactthatvec(ABC)=(CTA)vec(B)[ 74 ]. 3.2 [UGV]i;i=uTiGvi=(vTiuTi)vec(G);(B{2) wherewehaveusedthefactthatvec(ABC)=(CTA)vec(B)[ 74 ].Arranging( B{2 )intoacolumnvectoryieldsEquation( 3{11 ). Toprove( 3{12 ),letui;jandvi;jbethe(i;j)thelementsofUandV,respectively.LetgjbethejthdiagonalelementinG.Then [UGV]i;i=uTiGvi=pXj=1(ui;jvj;i)gj=(uTivTi)vecd(G):(B{3) Arranging( B{3 )intoacolumnvectoryieldsEquation( 3{12 ). 3{18 ) 16 ]. 103

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ByLemma1,weknowthat vec(ABS)=(STA):(B{4) Then,theDGCmodelin( 3{1 )canberewrittenas vec(X)=(STA)+vec(Z);(B{5) wherevec(Z)isaGaussianrandomvectorwithzero-meanandcovariancematrix(IQ).From( B{5 ),theGLSestimatorcanbereadilyobtainedas ^GLS=[(STA)H(IQ)1(STA)]1(STA)H(IQ)1vec(X):(B{6) Furthermore,wehave (STA)H(IQ)1(STA)=(STA)H(IQ1 2)(IQ1 2)(STA)=[ST(Q1 2A)]H[ST(Q1 2A)]=(SSH)T(AHQ1A);(B{7) wherewehaveusedthefactthat(UV)(GH)=(UG)(VH)(see,LemmaA1in[ 78 ])andLemma3inSectionII.Also,byLemmaA1in[ 78 ]andLemma2wehave (STA)H(IQ)1vec(X)=[ST(Q1A)]Hvec(X)=vecd(AHQ1XSH):(B{8) Substituting( B{7 )and( B{8 )into( B{6 )yields ^GLS=[(AHQ1A)(SSH)T]1vecd(AHQ1XSH):(B{9)

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Notethatin( B{9 )thecovariancematrixQisunknown.However,wehaveknownthatTisaconsistentestimateofQ(towithinamultiplicativeconstant).ReplacingQbyTin( B{9 )yieldstheAMLestimatorin( 3{18 ). B{5 )isalinearstatisticalmodelwithunknownnoisecovariancematrixIQ.ItcanbeeasilyveriedthattheFisherinformationmatrixforthismodelisablockdiagonalmatrixwithrespecttoandQ.Hence,theunknownsinQdonotaecttheCRBof.Therefore,theCRBofcanbereadilyobtainedas[ 73 ] CRB()=(STA)H(IQ1)(STA):(B{10) Thenby( B{7 )theCRBin( 3{24 )followsimmediately.

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ConsideraMIMOradarsystemwithKtargets.Thenthereceivedsignalcanbewrittenas Let and whereRe()andIm()denotetherealandimaginaryparts,respectively.AssumethatthecolumnsofZarei.i.d.circularlysymmetriccomplexGaussianrandomvectorswithzero-meanandanunknowncovariancematrixQ. UsingthesameargumentasinAppendix A ,weknowthattheunknownsinQwillnotaecttheCRBsofand.Hence,weneedonlytocalculatethefollowingFisherinformationmatrixwithrespectto,RandI,i.e., FIM=266664F(;)F(;R)F(;I)F(R;)F(R;R)F(R;I)F(I;)F(I;R)F(I;I)377775;(C{5) whereF(;)denotestheFisherinformationmatrixwithrespecttoand. 106

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Notethat and where _A(k)=@A(k) Inserting( C{7 )into( C{6 )andaftersomematrixmanipulations,weobtain whereR=1 whereFisaKKmatrixwithits(i;j)elementbeing [F]ij=Ltrn[_A(i)HQ1_A(j)][BjVT(j)RV(i)BHi]o+Ltrn[_AH(i)Q1A(j)][Bj_VT(j)RVT(i)BHi]o+Ltrn[AH(i)Q1_A(j)][BjVT(j)R_V(i)BHi]o+LRen[AH(i)Q1A(j)][Bj_VT(j)R_V(i)BHi]o:(C{11)

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Similarly,wehave and whereFandFarebothpartitionedmatricesformedbythefollowingsubmatrices,respectively, [F]ij=Lvecn[AH(i)Q1_A(j)][BjVT(j)RV(i)]o+Lvecn[AH(i)Q1A(j)][Bj_VT(j)RV(i)]o(C{16) and [F]ij=L[VT(i)RV(j)][AH(i)Q1A(j)];(C{17) withi;j=1;2;;KanddenotingtheKroneckerproduct. SubstitutingEquations( C{9 )-( C{15 )into( C{6 ),andaftersomematrixmanipulations,weget CRB()=1 2nReFFF1FHo1;(C{18) and CRB()=F1+F1FHCRB()FF1:(C{19)

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[92] M.R.Bell,\Informationtheoryandradarwaveformdesign,"IEEETransactionsonInformationTheory,vol.39,pp.1578{1597,September1993. [93] D.A.Garren,M.K.Osborn,A.C.Odom,J.S.Goldstein,S.U.Pillai,andJ.Guerci,\Enhancedtargetdetectionandidenticationviaoptimisedradartransmissionpulseshape,"IEEProceedings-Radar,Sonar,andNavigation,vol.148,pp.130{138,June2001. [94] J.Li,J.R.Guerci,andL.Xu,\Signalwaveform'soptimal-under-restrictiondesignforactivesensing,"toappearinIEEESignalProcessingLetters,2006. [95] N.R.Goodman,\Statisticalanalysisbasedonacertainmulti-variatecomplexGaussiandistribution,"Ann.Math.Stat.,vol.34,pp.152{177,March,1963. [96] G.CasellaandR.L.Berger,StatisticalInference(2ndEdition).ThomsonLearning,Inc.,2002.

PAGE 129

LuzhouXureceivedtheB.Eng.andM.S.degreesinelectricalengineeringfromZhejiangUniversity,Hangzhou,China,in1996and1999,respectively,andheisexpectedtoreceivethePh.DdegreeinelectricalengineeringfromUniversityofFlorida,Gainesville,in2006.From1999to2001,hewaswithZhongxingResearchandDevelopmentInstitute,Shanghai,wherehewasinvolvedinthesystemandalgorithmdesignofmobilecommunications.From2001to2003,hewaswithPhilipsResearch.Hisresearchinterestsincludestatisticalandarraysignalprocessingandtheirapplications. 117


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GROWTH CURVE MODELS
IN SIGNAL PROCESSING APPLICATIONS
















By

LUZHOU XU


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006


































Copyright 2006

by

Luzhou Xu

















To my wife, my son, and my parents















ACKNOWLEDGMENTS

I feel extremely lucky to have met all the people I have and to have received their

help. Foremost, I would like to express my most sincere gratitude to my advisor, Dr.

Jian Li, for her constant support, encouragement, and guidance. I am specially

grateful for the opportunity she has offered me to pursue my research under her

supervision. She gives me freedom to try new ideas, while providing the needed

assistance at the right time. She is ah--,i-b willing to share her knowledge and career

experience as an advisor, collaborator, and friend. The experience working with her

has proven to be invaluable and memorable.

I would also like to thank Dr. Jenshen Lin, Dr. Clint Slatton, and Dr. Rongling

Wu in the department of statistics for serving on my supervisory committee and for

their valuable advice. Their wonderful teaching has widened my horizon to their

intriguing research fields, and was and will ah--,i-b be helpful to my research work.

My special gratitude is due to Dr. Petre Stoica at Uppsala University, Sweden,

for his guidance in many interesting topics. His wide knowledge, strong analytical

skill, and keen insight have never ceased to amaze me. I was so fortunate to have the

opportunity to work with him and to benefit from his insightful ideas and constructive

advice. I am grateful to Dr. Mingzhou Ding in the department of biomedical

engineering for his support and guidance in my research on neural data analysis.

I also want to thank my groupmates in Spectral Analysis Lab: Bin Guo, Yi

Jiang, Jianhua Liu, Zhipeng Liu, Guoqing Liu, William Roherts, Yijun Sun, Yanwei

Wi'_.- Zhisong Wang, Yao Xie, Hong Xiong, C'! Ih gjiang Xu, Xiayu Z!. i.- Xumin

Zhu, and others. Their friendship, support, and help have made my life easier and

more enjoi-- 1-,









Last but not least, I wish to thank my wife and my son for all the happiness

they have brought to my life. I am also deeply thankful to my grandparents, parents

and sisters for their constant love and support.















TABLE OF CONTENTS


page


ACKNOW LEDGMENTS .............................

LIST OF FIGURES ................................

A B ST R A CT . . . . . . . . .

CHAPTER

1 INTRODUCTION ..............................

1.1 Growth-Curve Model and Its Variations ..............
1.2 Multiple-Input Multiple-Output Radar ................
1.3 Study Overview . . . . . . . .

2 GROWTH-CURVE MODEL ............


2.1 Introduction and Problem Formulation ...
2.2 Multivariate Parameter Estimation ......
2.2.1 Multivariate Capon Estimation ...
2.2.2 Multivariate Maximum Likelihood Estir
2.3 Performance Analysis .. ............


nation


2.3.1 Performance Analysis of the Multivariate ML Estimator .
2.3.1.1 Bias analysis . . . . . .
2.3.1.2 Mean-squared-error analysis .. ...........
2.3.2 Performance Analysis of the Multivariate Capon Estimator .
2.3.2.1 Bias analysis . . . . . .
2.3.2.2 Mean-squared-error analysis .. ...........
2.4 Numerical Examples .. .....................
2.5 C conclusions . . . . . . . .

3 DIAGONAL GROWTH-CURVE MODEL .. ..............

3.1 Introduction and Problem Formulation .. .............
3.2 Approximate Maximum Likelihood Estimation .. ..........
3.3 Performance Analysis .. ......................
3.3.1 Bias A analysis . . . . . . .
3.3.2 Mean-Squared-Error Analysis .. ..............
3.4 Numerical Examples .. .....................
3.4.1 Examples in Array Signal Processing .. ..........
3.4.2 Spectral Analysis Examples .. ...............


. . .









3.5 Conclusions ..................... . ...... 50

4 BLOCK DIAGONAL GROWTH-CURVE MODEL ........... .51

4.1 Introduction and Problem Formulation ............ 51
4.2 Preliminary Results .......... . . ... 52
4.3 Approximate Maximum Likelihood Estimation . . ... 55
4.4 Performance Analysis .... ........... ..... .. 58
4.4.1 Bias Analysis ........ . . .... 58
4.4.2 Mean-Squared-Error (\!'Sl) Analysis . . 59
4.5 Numerical Results ............... ....... .. 60
4.6 Conclusions ............... ..... 63

5 ITERATIVE GENERALIZED LIKELIHOOD RATIO TEST FOR MIMO
RADAR ..... .............. ................ .. 65

5.1 Introduction and Signal Model ................ .. .. 65
5.2 Several Spatial Spectral Estimators ................. .. 67
5.2.1 Capon .................. ........... .. 68
5.2.2 APES ........ ....... ..... .... .. 69
5.3 Generalized Likelihood Ratio Test ..... . . ..... 70
5.3.1 Generalized Likelihood Ratio Test (GLRT) . ... 70
5.3.2 Conditional Generalized Likelihood Ratio Test (cGLRT) 73
5.3.3 Iterative Generalized Likelihood Ratio Test (iGLRT) . 78
5.4 Numerical Examples .................. ....... .. 79
5.4.1 Cramir-Rao Bound .................. .. 79
5.4.2 Target Detection and Localization . . ..... 81
5.5 Conclusions .................. .......... .. .. 88

6 CONCLUSIONS AND FUTURE WORK ............. .. 89

A PROOFS FOR THE GROWTH-CURVE MODEL . . 92

A.1 Cramir-Rao Bound for the GC Model ............ .. 92
A.2 Proof of Lemma 2.1 .................. ........ .. 94
A.3 Proof of Lemma 2.2 .................. ........ .. 96
A.4 Proof of Lemma 2.3 .................. ........ .. 96
A.5 Proof of Lemma 2.4 .................. ........ .. 98

B PROOFS FOR THE DIAGONAL GROWTH-CURVE MODEL ..... 102

B.1 Proof of Lemma 3.1 .................. ........ 102
B.2 Proof of Lemma 3.2 .... . . ......... 102
B.3 Generalized Least-Squares (GLS) Interpretation of AML in (3-18) 102
B.4 Cramir-Rao Bound for the DGC model . . ..... 104

C CRAMER-RAO BOUND FOR THE MIMO RADAR . . 105









REFERENCES ...................... ............ 108

BIOGRAPHICAL SKETCH ................... ...... 116















LIST OF FIGURES
Figure page

2-1 Bias versus L when SNR = 10 dB, K = 2, N = 2. ............ ..28

2-2 Bias versus K when SNR = 10 dB, L = 16, N = 2. .......... 29

2-3 MSE versus L when SNR 10 dB, K = 2, N = 2 ............. ..30

2-4 MSE versus SNR when L = 16, K = 2, N = 2 ................ .31

2-5 MSE versus K when SNR = 10 dB, L = 16, N = 2. ............. .31

2-6 MSE versus a when SNR 10 dB, L = 16, K = 2, N = 2. ........ 32

3-1 Empirical MSE's and the CRB versus L when SNR = 10 dB, M = 6, and
N = 3 with linearly independent steering vectors and linearly independent
waveform s. . . . . . .. . . 41

3-2 Empirical MSE's and the CRB versus SNR when L = 128, M = 6, and
N = 3 with linearly independent steering vectors and linearly independent
waveform s. . . . . . .. . . 42

3-3 Empirical MSE's and the CRB versus L when SNR = 10 dB, M = 6, and
N = 3 with identical waveforms. .................. .... 43

3-4 Empirical MSE's and the CRB versus SNR when L = 128, M = 6, and
N = 3 with identical waveforms. .................. .... 44

3-5 Empirical MSE's and the CRB versus L when SNR = 10 dB, M = 6, and
N = 13 with linearly dependent steering vectors ............. ..45

3-6 Empirical MSE's and the CRB versus SNR when L = 128, M = 6, and
N = 13 with linearly dependent steering vectors. ............. .46

3-7 Empirical MSE's and the CRB versus local SNR when Lo = 32, M = 8,
and the observation noise is colored. (a) For /3 and (b) for 31. . .48

3-8 Empirical MSE's and the CRB versus M when Lo = 32, a2 = 0.01, and
the observation noise is colored. (a) For /3 and (b) for 3. . . 48

4-1 Empirical MSE's and the CRB versus L when SNR = 10 dB. (a) For /31,
(b) for 32, and (c) for 3. . . . .... . ...... 63









4-2 Empirical MSE's and the CRB versus SNR (dB) when L = 128. (a) For
Pi, (b) for 32, and (c) for /3 .. .. .. .. .... . . .. 64
5-1 Cumulative density functions of the Cramir-Rao bounds for (a) 01 and
(b ) 03 .. . . . . . . . ... 8 0
5-2 Outage CRB versus SNR. (a) CRBo.o0 for 01, (b) CRBo.o0 for 02, (c)
CRBo.1 for 01, and (d) CRBo.1 for 02. ................. 82
5-3 Spatial spectra, and GLR and cGLR Pseudo-Spectra ,when 01 = -40,
02 = -200, and 03 = 0. (a) LS, (b) Capon, (c) APES, (d) GLRT, and
(e) iGLRT. ........... ..... ........... ...... 83

5-4 Spatial spectra, and GLR and cGLR Pseudo-Spectra, when 01 = -40,
02 -40, and 03 = 00. (a) Capon, (b) APES, (c) GLRT, and (d) iGLRT. 85
5-5 GLR and cGLR Pseudo-Spectra obtained in Steps I and II of iGLRT,
when 01 -400, 02 -4, and 03 0. (a) p(O), (b) p(0|11), (c)
p(0|11,02), and (d) p(0801, 82, 3). ................ . 86
5-6 Cumulative density functions of the CRBs and MSEs for (a) 01 and (b) 03. 86

5-7 Outage CRBo.1 and MSEo.1 versus SNR for (a) 01 and (b) 03. ..... 87

5-8 Outage CRBo.1 and MSEo.1 versus SNR for (a) 01 and (b) 03. ..... 87















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

GROWTH CURVE MODELS
IN SIGNAL PROCESSING APPLICATIONS

By

Luzhou Xu

August 2006

C'! ir: Jian Li
Major Department: Electrical and Computer Engineering

As a powerful statistical tool, the growth-curve (GC) model is attracting

increasing attentions in various areas. In this dissertation, we study several

variations of the growth-curve model, and discuss their applications to the

emerging multiple-input multiple-output (\MI M\ 0) radar system.

We first study the statistical properties of two estimators for the regression

coefficient matrix in the GC model, i.e., the maximum likelihood ( llI) and

Capon methods. We derive the closed-form expression of the Cram6r-Rao

bound (CRB) for the unknown regression coefficient matrix, and then analyze

the bias properties and mean-squared errors (\!iSl- ) of the two estimators. We

show that the multivariate ML estimator is unbiased whereas the multivariate

Capon estimator is biased downward for finite data samples. Both estimators are

.,-vmptotically statistically efficient when the number of data samples is large.

Next, we consider a variation of the GC model, referred to as the diagonal

growth-curve (DGC) model, where the regression matrix is constrained to be

diagonal. A closed-form approximate maximum likelihood (AML) estimator for

this model is derived based on the maximum likelihood principle. We analyze









the statistical properties of this method theoretically and show that the AML

estimate is unbiased and .i-i, !!,I .ll ically statistically efficient for a large number

of data samples. Via several numerical examples in array signal processing and

spectral analysis, we also show that the proposed AML estimator can achieve

better estimation accuracy and exhibit greater robustness than the best existing

methods.

Then we consider a general growth-curve model, referred to as the block

diagonal growth-curve (BDGC) model, where the unknown regression coefficient

matrix is constrained to be block-diagonal, and which can unify the GC and DGC

models. We proposed a closed-form approximate maximum likelihood (AML)

estimator for the block-diagonal constrained matrix, which is proved to be unbiased

and .,-mptotically statistically efficient for a large data sample number. Several

applications of this model in signal processing are then presented.

Finally, we consider a multiple-input multiple-output (MI MO\ 0) radar system

with a general antenna configuration, i.e., both the transmitter and receiver have

multiple well-separated subarrays with each subarray containing closely-spaced

antennas. Hence, both the coherent processing gain and the spatial diversity

gain can be achieved by the system simultaneously. We introduce several spatial

spectral estimators, including Capon and APES, for target detection and parameter

estimation. We also provide a generalized likelihood ratio test (GLRT) and a

conditional generalized likelihood ratio test (cGLRT) for the system. Based on

GLRT and iGLRT, we then propose an iterative GLRT (iGLRT) procedure for

target detection and parameter estimation. Via several numerical examples, we

show that iGLRT can provide excellent detection and estimation performance at a

low computational cost.














CHAPTER 1
INTRODUCTION

1.1 Growth-Curve Model and Its Variations

The growth-curve (GC) model is a generalized multivariate analysis of variance

(GMANOVA) model, which was first formulated by Potthoff and Roy in 1964 [1]

for investigating growth curve problems in statistical applications. Since then it has

been studied by many authors, including Rao [2] [4], Khatri [5], Gleser and Olkin

[6], Geisser [7], von Rosen [8] [10], Verbyla and Venables [11], and Srivastava [12].

It is one of the main tools for dealing with longitudinal data, especially for serial

correlations [13] as well as repeated measurements [14] [18], and is attracting

increasing attentions in various areas, such as economics, biology, medical research

and epidemiology. Recently, this model was extended to the complex-valued field

and was adopted in the signal processing literature [19] [22].

Consider an observed data matrix X E CMXL, which can be written as


X ABS + Z. (1-1)


In (1-1), A E CMxN and S E CKxL are both known matrices, B E CNxK is an

unknown regression matrix, and Z e CMXL is the error matrix whose columns

are independently and identically distributed (i.i.d.) zero-mean Gaussian random

vectors with an unknown covariance matrix. The problem of interest is to estimate

B from the observed data matrix X.

A special case of (1-1), when N = K = 1, has been studied widely in signal

processing, such as high-resolution spectral analysis [23] [24] and array signal

processing [25] [35]. In this case, the GC model reduces to a univariate GC









(UGC) model

X a/sT + Z, (1-2)

with a and sT being column and row vectors, respectively, and 3 being an unknown

scalar variable. The performance of the maximum-likelihood (MiI ) and Capon

estimators for the UGC model has been thoroughly studied in [25]. It was shown

theoretically that ML is unbiased whereas Capon is biased downward, and both

estimators are .,-i ,il1 l ically statistically efficient for a large number of data

samples (i.e., L > M). In this dissertation, we will extend this result to the GC

model.

In many practical applications, the observed signal consists of multiple

components. For this scenario, the UGC model in (1-2) can be extended as
K
X = ak/ks + Z, (1-3)
k=1

with {ak} and {sT} being the known column and row vectors, and {/Ik} being

unknown scalar variables. Obviously, the model in (1-3) can be rewritten as


X ABS + Z with B diag(3), (1-4)


where diag(3) denotes a diagonal matrix with its diagonal formed by the elements

of the vector /, A = [ai .. aK], S = [si .. sK]T, and 3 = [/31 /3K]T. We

note that the model in (1-4) is similar to the GC model in (1-1) except that the

unknown regression coefficient matrix B is constrained to be diagonal. Hence, it is

referred to as a diagonal growth-curve (DGC) model [36]. Despite the seemingly

minor difference between the GC and DGC models, the ML estimator [21] [22] for

the GC model is invalid for the DGC model. In fact, to our knowledge, no closed-

form ML estimator for f in (1-4) exists in the literature. In this dissertation, we

propose an approximate maximum likelihood (AML) estimator for f in (1-4).

We also investigate its statistical properties via theoretical analysis and numerical









simulations, and show that the AML estimator for the DGC model is unbiased and

.,-vmptotically statistically efficient for a large number of data samples.

A more general variation of the GC model was studied by V1 i li-1 i [11],

Rosen [8] and Srivastava [12], where the authors consider an estimation problem

of unknown regression matrices {Bk} from the observed data matrix X in the

equation
K
X AkBkSk + Z. (1-5)
k= 1
Again, in (1-5), {Ak} and {Sk} (k = 1, 2, ,.. K) are all known matrices, and Z

is defined as in the GC model. Note that (1-5) can be rewritten in the form of the

GC model in (1-1) by constraining the unknown regression coefficient matrix B to

be a block-diagonal matrix; hence (1-5) is referred to as a block diagonal growth-

curve (BDGC) model in this dissertation. An iterative numerical approach for the

estimation of the unknown regression coefficient matrices Bk in (1-5) was proposed

by V, i 1-1 [11] by using the canonical reduction method. However, this approach

is both conceptually and practically complicated. Moreover, being an iterative

method, it may suffer from convergence problems. Two nested variations of the

BDGC model were studied by Rosen [8] and Srivastava [12], independently, where

explicit forms of the ML estimators were presented. However, some additional

assumptions have be imposed in [8] and [12]. In [8], the rows of Sk (k = 1, 2, .. K)

are assumed to be nested, i.e., i(ST) C R4(ST_ ) C ... C R(ST), with 7(.)

denoting the range space; and in [12] the columns of Ak (k = 1, 2, .. K) are

assumed to be nested, i.e., R(AK) C R(AK-1) C ... C R (A1). Neither of these

two nested subspace conditions can be satisfied in signal processing applications. In

this dissertation, we will consider a general BDGC model in (1-5), and propose an

approximate maximum likelihood (AML) estimator [37] for the unknown regression

coefficient matrices Bk, which will be shown both theoretically and numerically to

be unbiased and .,-i-','.1i, l ically statistically efficient.









1.2 Multiple-Input Multiple-Output Radar

A multiple-input multiple-output (\I \ lO) radar uses multiple antennas to

simultaneously transmit several (possibly linearly independent) waveforms and it

also uses multiple antennas to receive the reflected signals [38] [39]. It has been

shown that by exploiting this waveform diversity, MIMO radar can overcome

performance degradations caused the radar cross-section (RCS) fluctuations [40]

- [43], achieve flexible spatial transmit beampattern design [44] [45], provide

high-resolution spatial spectral estimates [46] [57], and significantly improve the

parameter identifiability [58].

The statistical MIMO radar, studied in [40] [43], aims at resisting the

"scintillation" effect encountered in radar systems. It is well-known that the

RCS of a target, which represents the amount of energy reflected from the target

toward the receiver, changes rapidly as a function of the target aspect [59] and the

locations of the transmitting and receiving antennas. The target "scintillation"

causes severe degradations in the target detection and parameter estimation

performance of the radar. By spacing the transmit antennas, which transmit

linearly independent signals, far away from each other, a spatial diversity gain can

be obtained as in the MIMO wireless communications to this "scintillation" effect

[40] [43].

Flexible transmit beampattern designs are investigated in [44] and [45].

Different from the -i iI i-I -i !" MIMO radar above, the transmitting antennas are

closely spaced. The authors in [44] and [45] show that the waveforms transmitted

via its antennas can be optimized to obtain several transmit beampattern designs

with superior performance. For example, the covariance matrix of the waveforms

can be optimized to maximize the power around the locations of interest and also

to minimize the cross-correlation of the signals reflected back to the radar by these

targets, thereby significantly improving the performance of the adaptive MIMO






5


radar techniques. Due to the significantly larger number of degrees of freedom of

a MIMO system, a much better transmit beampattern with a MIMO radar can be

achieved than with its phased-array counterpart.

In [48], a MIMO radar technique is -i -.-- -I. -I to improve the radar resolution.

The idea is to transmit N (N > 1) orthogonal coded waveforms by N antennas and

to receive the reflected signals by M (M > 1) antennas. At each receiving antenna

output, the signal is matched-filtered using each of the transmitted waveforms

to obtain NM channels, to which the data-adaptive Capon beamformer [60] is

applied. It is proved in [48] that the beampattern of the proposed MIMO radar

is obtained by the multiplication of the transmitting and receiving beampatterns,

which gives high resolution. However, only the single-target case is considered in

[48].

A MIMO radar scheme is considered in [55] [57] that can deal with the

presence of multiple targets. Similar to some of the aforementioned MIMO radar

approaches, linearly independent waveforms are transmitted simultaneously via

multiple antennas. Due to the different phase shifts associated with different

propagation paths from transmitting antennas to targets, these independent

waveforms are linearly combined at the targets with different phase factors. As a

result, the signal waveforms reflected from different targets are linearly independent

of each other, which allows the direct application of many adaptive techniques

to achieve high resolution and excellent interference rejection capability. Several

adaptive nonparametric algorithms in the presence or absence of steering vector

errors are presented in [55] [57].

Note that the MIMO radars discussed in the aforementioned literature can

be grouped into two classes according to their antenna configurations. One is the

conventional radar array, in which both transmitting and receiving antennas are

closely spaced for coherent transmission and detection [44] [57]. The other is the









diverse antenna configuration, where the antennas are separated far away from

each other to achieve spatial diversity gain [40] [43]. To reap the benefits of both

schemes, in this dissertation, we consider a general antenna configuration, i.e., both

the transmitting and receiving antenna arrays consist of several well-separated

subarrays with each subarray containing closely-spaced antennas [61]. By using

some results of the growth-curve models, we provide a generalized likelihood ratio

test (GLRT) and a conditional generalized likelihood ratio test (cGLRT) for the

system. Based on GLRT and iGLRT, we then propose an iterative GLRT (iGLRT)

procedure for target detection and parameter estimation. Via several numerical

examples, we show that iGLRT can provide excellent detection and estimation

performance at a low computational cost.

1.3 Study Overview

In C'! lpter 2, we consider estimating the unknown regression coefficient

matrix B in the GC model in (1-1). Two multivariate approaches, Maximum

Likelihood (ill') and Capon, are provided. We derive the closed-form expression

of the Cramir-Rao bound (CRB) for the unknown complex amplitudes. We also

analyze the bias properties and Mean Squared Errors (S\! 1 ) of the two estimators.

A comparative study shows that the multivariate ML estimator is unbiased whereas

the multivariate Capon estimator is biased downward for finite data samples.

Both estimators are .,-imptotically statistically efficient when the number of data

samples is large.

In C!I Ipter 3, we consider a variation of the GC model, referred to as the

diagonal growth-curve (DGC) model, where the matrices A and S in (1-4)

are both known and the regression coefficient matrix B is constrained to be

diagonal. A closed-form approximate maximum likelihood (AML) estimator for

this model is derived based on the maximum likelihood principle. We analyze

the statistical properties of this method theoretically and show that the AML









estimate is unbiased and .,-i-! ,1,' i ically statistically efficient for a large number

of data samples. Via several numerical examples in array signal processing and

spectral analysis, we also show that the proposed AML estimator can achieve

better estimation accuracy and exhibit greater robustness than the best existing

methods.

In C'!i lpter 4, we consider a general variation of the growth-curve (GC) model,

referred to as the block diagonal growth-curve (BDGC) model, which can unify

the GC and DGC models in C'!i lpters 2 and 3. In BDGC, the unknown regression

coefficient matrix is constrained to be block-diagonal. A closed-form approximate

maximum likelihood (AML) estimator for this model is then derived, which is

shown to be unbiased and .,-imptotically statistically efficient for a large number of

data samples.

In C'!i lpter 5, we consider a multiple-input multiple-output (! liO) radar

system with a general antenna configuration, i.e., both the transmitter and receiver

have multiple well-separated subarrays with each subarray containing closely-

spaced antennas. We introduce several spatial spectral estimators, including

Capon and APES, for target detection and parameter estimation. We also

provide a generalized likelihood ratio test (GLRT) and a conditional generalized

likelihood ratio test (cGLRT) for the system. Based on GLRT and iGLRT, we then

propose an iterative GLRT (iGLRT) procedure for target detection and parameter

estimation.

Finally, we summarize the dissertation and point out future research directions

in C'! i pter 6.















CHAPTER 2
GROWTH-CURVE MODEL

2.1 Introduction and Problem Formulation

In this chapter, we consider the following multivariate complex amplitude

estimation problem

X = ABS + Z. (2-1)

In (2-1), X E CMXL denotes the observed snapshots with L being the number of

snapshots. The columns in A E CMXN are the known linearly independent spatial

vectors, e.g., steering vectors. The rows in S E CKxL are the known temporal

vectors, e.g., waveforms, assumed to be linearly independent of each other or

not completely correlated with each other. The matrix B E CNxK contains the

multivariate unknown complex amplitudes. Throughout this chapter, we assume

that M > N and L > K + M. The columns of the interference and noise matrix

Z e CMxL are statistically independent circularly symmetric complex Gaussian

random vectors with zero-mean and unknown covariance matrix Q. The problem of

interest is to estimate the unknown matrix B.

We note that the data model of (2-1) has general applications. Its real-valued

counterpart, called growth-curve (GC) Model, has been studied and used widely

for investigating growth problems in the statistics field [62] [63] [18]. This real-

valued growth-curve model was extended and introduced to the signal processing

field in [21]. Using the extended model, the authors in [21] unified many existing

algorithms proposed for radar array processing [64] [65], spectral analysis [23] [66]

and wireless communication [67] [70] applications.









The focus of this chapter is on the performance analysis of the multivariate

Maximum Likelihood (il I) and Capon estimators for the data model in (2-1). We

derive the closed-form expression of the Cram6r-Rao Bound (CRB) of the unknown

complex amplitude parameters. We also analyze the bias properties and Mean-

Squared-Errors (\ !Sl) of the two estimators. A comparative study shows that the

multivariate ML estimator is unbiased whereas the multivariate Capon estimator is

biased downward for finite snapshots. Yet in finite data samples and at low SNR,

Capon can provide a smaller MSE than ML. Both estimators are .i-ii11 .' ically

statistically efficient when the number of snapshots is large.

The remainder of the chapter is organized as follows. Section 2.2 provides the

multivariate Capon and ML estimators. Section 2.3 gives the performance analysis

of the two estimators and the CRB of the unknown complex amplitudes. Numerical

examples are provided in Section 2.4. Finally, we present our conclusions in Section

2.5.

2.2 Multivariate Parameter Estimation

Based on the data model in (2-1), we describe the multivariate Capon and ML

estimators in this section.

2.2.1 Multivariate Capon Estimation

The multivariate Capon estimator consists of two main steps. The first is the

Capon beamforming [60] [71] [72]. The other is the Least-Squares (LS) estimation

[16] [73], which is basically the matched filtering.

We first consider the Capon beamforming. Let


R XXH. (2-2)


Then, the Capon beamformer can be formulated as


W argmintr(WHRW) subject to WHA I,
w


(2-3)









where W is a multivariate weighting matrix for noise and interference suppression

while keeping the desired signals undistorted. Solving the above optimization

problem yields

W R-A(AHR-1A)-1. (2-4)

Note that since M > N and the columns in A are linearly independent of each

other, AHR-1A has full rank N with probability one. The beamforming output,

denoted by Y, is

Y WHX = (AHR-1A)-IAHR-1X. (2-5)

Now we consider the LS estimation. Substituting (2-1) into (2-5) yields

Y = BS + (AHR-1A)-IAHR-1Z. (2-6)

Estimating B from Y based on (2-6) is a standard Multivariate Analysis of

Variance (\ilANOVA) problem [62] [63] [21]. Note that after spatial beamforming,

the noise vectors remain temporally white, and hence the LS estimator gives the

best performance. Using the LS algorithm yields


Bepon YSH(SSH)-1. (2-7)

Substituting (2-5) into (2-7), the multivariate Capon estimator has the form


Bcapon (AHR-1A)-IAHR-1XSH(SSH)-1. (2-8)

Note that the Capon estimator for the univariate case in [25] is a special case of

(2-8).

2.2.2 Multivariate Maximum Likelihood Estimation

A general derivation of the multivariate ML estimator has been given in [21].

In this chapter, we assume that both A and S are known and the multivariate ML

estimator can be briefly derived as follows to make this chapter self-contained.








Based on the data model in (2-1), the negative log-likelihood function is
proportional to

(B, Q) = LlnQ| + tr[Q-1(X ABS)(X ABS)H], (2-9)

where I |, tr(.) and (.)H denote the determinant, trace and conjugate transpose of
a matrix, respectively.
Minimizing the negative log-likelihood function with respect to Q yields

Q (X- ABS)(X- ABS)H. (2-10)
L

Inserting (2-10) into (2-9), the ML estimator of B can be formulated as

BML = arg min (X ABS)(X ABS)HI. (2-11)
B

Note that

I(X- ABS)(X ABS)H

L [AB- XSH(SSH)-] (SSH)[AB- XSH(SSH)-H + T

= |T I+ T- [AB XSH(SSH)] (SSH) [AB- XSH(SSH)-1] HT

= |T I+ (SSH) [AB- XSH(SSH)-1]HT-1[AB XSH(SSH)- I](SSH'

|T I + (SSH) -SXH T-1 T-1A(AHT-lA)-'AHT-1]XSH(SSH) +

(SSH) [B (AHT-lA)- AHT-1XSH(SSH)-1] H(AHT-lA)

[B- (AHT- A)- AHT -XSH(SSH)-1](SSH)

> |T I+ (SSH) -SXH[T-1 T-1A(AH '-A)-AH T-1]XSH(SSH)-
(2-12)

where


T -X[I SH(SSH)-1S]XH,


(2-13)









and I is an identity matrix. In the above derivation we have used the fact that

1I + XYI =I + YX| [74]. Since the rows in S are linearly independent of each

other and L > K + M, the rank of I SH(SSH)-IS, which is L K, is greater

than or equal to M. Hence, T and AHT-1A in (2-12) have full ranks M and N,

respectively, with probability one.

From (2-12), the ML estimator of B is written as


BML = (AHT-1A)-IAHT-1XSH(SSH)-1. (2-14)


Note again that the ML estimator for the univariate case in [25] is a special case of

(2-14).

To better understand the above ML estimator intuitively, we insert (2-1) into

(2-13) and get

T Z[I SH(SSH)-S]ZH. (2-15)

It shows that IT is an estimate of the unknown noise covariance Q. We also

note that the estimator in (2-14) can be divided into two steps, including the

ML beamforming spatially corresponding to the left-multiplication matrix

(AHT-1A)-IAHT-1 and the LS estimation temporally corresponding to the

right-multiplication matrix SH(SSH)- 1

Note that like the univariate case in [25], the only difference between the

Capon and ML estimators is that the matrix R in (2-8) is replaced by T in (2-14).

However, as we will show in the following analysis, this seemingly minor difference

in fact leads to significant and interesting performance differences between the two

estimators.

2.3 Performance Analysis

2.3.1 Performance Analysis of the Multivariate ML Estimator

We consider below the statistical performance analysis of the multivariate

ML estimator. We show that the conclusions in [25] can be extended to the









multivariate case, i.e., the multivariate ML estimator is unbiased and it is

.,-vmptotically statistically efficient when the number of snapshots is large. In

the following derivations, several techniques presented in [25], [62] and [18] are

employ, ,1

2.3.1.1 Bias analysis

For convenience, we denote

Xs XSH, X X[I- SH(SSH)-lS], (2-16)


Zs ZSH, Z- Z[I- SH(SSH)-S]. (2-17)

Clearly, we have


Xs ABSSH + Zs, T X(X )H Z#(Z#)H. (2-18)


Note that the columns of Z are independent zero-mean Gaussian random

vectors. Note also that the columns of SH are orthogonal to those of I -

SH(SSH)-'S. By the property of joint Gaussian distribution, Zs and Z1 are

two independent Gaussian random matrices [75]. Hence, Xs and T are also

independent of each other. Utilizing this conclusion, we can readily show that


E(BML) ET[(AHT-1A)-1AHT-1Ezs(ABSSH + Zs)(SSH)-1] B, (2-19)

where Ec[.] denotes calculating the expectation with respect to the random matrix

C.

Hence, the multivariate ML estimator is, like its univariate counterpart in [25],

unbiased.

2.3.1.2 Mean-squared-error analysis

First, we consider the best possible performance bound for any unbiased

estimator of B, i.e., the CRB. Appendix A.1 shows that the CRB of B based on









the data model in (2-1) has the following form

CRB(B) A CRB(vec(B)) (S*ST)-1 (AHQ-1A)-1, (220)

where vec(.) denotes stacking the columns of a matrix on top of each other, (.)*
denotes the complex conjugate and 0 denotes the Kronecker matrix product [74].
Before calculating the MSE of the multivariate ML estimator, we introduce
the following three lemmas which will be used in our derivation. The counterparts
of the three lemmas for real-valued variables have been proved and used in the
statistics literature [62] [63] [18] and part of Lemma 1 has been proved in [75].
Lemma 2.1. Suppose T is an m x m random matrix with the complex Wishart
distribution with covariance matrix Emxm and I (1 > m) degrees of freedom,
denoted by T ~ CW(1, m; E). Let Y and E be partitioned as

11 T12 11 12
Y = E 1 El 1 (2-21)
S T21 Y22J (21 22)

where YTl and El are mi x ml matrices and Y22 and E22 are m2 x 2 matrices
with mi + m2 = m. Let T1n.2 = n T12 2T21 and 11.2 E12ECi1E21

Then the following properties hold.

(i). 11.2 is independent of T22 and Y12 and T11.2 CW(l m2, Mi1; 11.2);
(ii). Y22 CW(l,M; X22);
(iii). The conditional distribution of T12 given T22 is the matrix-variate complex
Gaussian distribution CN(CE12E IT2; 11.2, T22) [74 [75/, whose ji,,.,-.l.:l:/;
1. ,..:/1 function (I.Ilf) is given by

fT12T22 (2T)-nl.2 )11.2 22

exp tr[El2(T 12- 12222)22(12- 1222122)H }.

(2-22)


Proof: Appendix A.2.









Lemma 2.2. Let C be a pxp constant matrix. Suppose Txp ~ CN(Hxp; E,., xpxp),

i.e.,


f(T) (27)- 1PXl-P f-"exp tr[E- (T- H)-I(TY


H)H]}


E(TCTH) = tr(C)E + HCHH.


(2-23)


(2-24)


Proof: Appendix A.3.

Lemma 2.3. If r ~ CW(1, m; E), then


E(T-') =
(1 m)


(2-25)


Proof: Appendix A.4.

Now we consider the MSE of BML. The error of the multivariate ML estimate

of B is


AB A BML B

S(AHT-1A)- AHT

S(AHT-1A)- AHT

Since [SH(SSH) ~-H[SH(SS )-]

unitary matrix, we can construct an L x


U [UI U21]


l(ABS + Z)SH(SSH)- B

1Zs(SSH)-1


I, i.e., SH(SSH)- is an L x K semi-

L unitary matrix


U1 SH(SSH)
(ss)-


Thus,


Since the column random vectors of Z are statistically independent of each other

and the columns of U2 are orthogonal to each other, ZU2 ~ CN(0; QMXM, IL-K)

and Q- ZU2 ~ CN(0; IM, L-K) [75]. According to the definition of the complex


then


(2-26)


(2-27)


T = Z[UUH UUH]ZH ZU2(ZU2)H.


(2-28)









Wishart distribution, we have


A Q-2TQ-2


(Q-ZU2)(Q-~ZU2)H ~ CW(L


Zs Q-Zs(SSH)-,

which has the CN(0; IM, IK) distribution [75]. Denote

A Q-aA(AHQ-A)- .

Then, inserting (2-29), (2-30) and (2-31) into (2-26) gives

AB = (AHQ-1A)- (AH l )H -lH A-1 s(SSH) .


Since AHA


I, we can decompose A as


A UP,


where U is an M x M unitary matrix with its first N columns being A; PMxN =

[IN 0]'. Since U is unitary, like T, UHTU remains to be the complex Wishart
distribution [75], i.e.,

r UHTU CW(L K, M; I). (2

Let

E = UH (2

which obviously has the CN(0; IM, IK) distribution.

We next partition EMXK, FMxM, and IF-xM, respectively, as follows.


F11 F12
F F
F21 F22


plF pl2
and r-1 -
p21 p22


K,M; I).


(2-29)


(2-30)


(2-31)


(2-32)


(2-33)


34)


35)


(2-36)


32








where '1 and E2 are N x K and (M N) x K, respectively, and both Fll and F11
are N x N matrices.
Inserting (2-33) to (2-36) into (2-32) gives

AB (AHQ-1A)- (PH-1p)-1pHF-lE(SSH)-
(2-37)
S(AHQ-1A)- (El F12r2 2)(ssH)-

To obtain (2-37), we have used the inversion lemma of partitioned matrices.
Hence, using the lemma that vec(XYZ) = (ZT 0 X)vec(Y) and the fact that
51, 2 are two independent random matrices with the distributions CN(0; IN, IK)
and CN(0; IM-N, IK), respectively, as well as (2-20), we have

MSE(BML) ^E{vec(AB) vec(AB)H}

[(S*ST)- 0 (AHQ -A)-]E{vec(Ei i122E2)

vec(El F12F2E2)H}[(S*S )- 0 (AHQ-1A)-]

=[CRB(B)] {I + E[vec(l12rF22E2)vec(l12r-212)H]} [CRB(B)]I.
(2-38)

Using the facts that vec(XY) (I 0 X)vec(Y) and E2 ~ CN(0; I, I) yields

E[veco(F12F12) vvec(F12F2 )HI

E{ [I (D(12F2'2)]E2 r2,r22[vec(E2) vec(2)H] [I0 ( F12I2 } (2-39)

I 0 E(F12F22F 2 1F2),

where EAIB(.) denotes the expectation with respect to random matrix A given B.
By Lemma 2.1 and (2-34), we know that F12 given F22 has the CN(0; I, F22)
distribution. Hence, applying Lemma 2.2 gives

E(r12rrF22 ) = Er, [Er, r22 (12r2 2r)]

{Er22[tr(-21)]} I (2-40)

= {tr[Er22(F22)]} I.









Furthermore, by Lemma 2.1 we know that F22 ~ CW(L K, M N; IM-N). Hence,

by Lemma 3, we have

1
Er(22(F) L K M NIM-N (2 41)
22 L-K-M+N (2-4t

Thus, it follows from (2-38), (2-39), (2-40) and (2-41) that


MSE(BML) K CRB(B). (242)
L-K- M + N

From (2-42), we note that MSE(BML) approaches CRB(B) for large L, which

means that the multivariate ML estimator is .,-vmptotically statistically efficient

for large number of snapshots L. Hence the efficiency condition for the univariate

case in [25] can be extended to the multivariate case as well. When L, K, M and

N are fixed, the MSE of the multivariate ML estimator is proportional to CRB(B).

Hence it is expected that the MSE-versus-SNR lines will be parallel to the CRB-

versus-SNR lines. This theoretical result will be verified via numerical simulations

in Section 2.4.

Furthermore, the CRB of B depends on (S*ST)-1 and (AHQ-1A)-1. As

we show in Appendix A.1, orthogonalities among the rows of S and among the

columns of Q-}A lead to small diagonal elements for (S*ST)-1 and (AHQ-1A)-1,

respectively, which in turn reduce the CRB.

We also note that when M = N, which implies that A is a square matrix,

the multivariate ML estimator is efficient. However, we should not think of it as

a significant advantage to make N as large as possible. As we show in Appendix

A.1, in the case that the columns of Q-2A are not orthogonal to each other, which

often happens in practice, large N causes CRB to increase.

Now we summarize the statistical properties of the multivariate Capon

estimator by the following theorem.









Theorem 2.1. For the data model in (2-1), the multivariate ML estimate of

B, given by (2-14), is unbiased and '- /l,, '''I. ,'//l/ i -l.:'l.: ll,. efficient for '.i,,

number of data samples. Its MSE matrix can be expressed as

A L-K
MSE(BM,) E[vec(Bm,) vec(BL)H] = -N CRB(B) (2-43)


where

CRB(B) (S*ST)-1 0 (AHQ-1A)-1, (2-44)

vec(-), (.)*, (.)T and 0 denote the direct operator (stacking the columns of a matrix

on top of each other), complex conjugate, transpose and Kronecker product of

m atrices, ,- I/. 1/.:;. /;;

2.3.2 Performance Analysis of the Multivariate Capon Estimator

We now establish the theoretical properties of the multivariate Capon

estimator.

2.3.2.1 Bias analysis

In Section 2.3.1, we know that the multivariate ML estimator is unbiased.

We will investigate the bias of the multivariate Capon estimator by studying the

relationship between the two estimators.

Comparing (2-2) and (2-13), we note that


R T + XSH(SSH) -SXH. (2-45)


Applying the matrix inversion lemma gives


AHT-1XSH(SXHT-1XSH + SSH)-1


AHR-1XSH(SSH)-1


(2-46)









and

(AHR-1A)-1

= [AHT-1A AHT-1XSH(SXHT-1XSH + SSH)-SXHT-1A]-

= (AHT-1A)-1- (AHT-1A)- AHT-1XSH (2-47)

[SXHT-1A(AHT-1A)- AHT-XSH SXHT-XSH SSH]-1

SXHT-1A(AHT-1A)-1.

Substituting (2-46) and (2-47) into (2-8), and after some straightforward

manipulations, we get

BCapon BMLA, (2-48)

where

A [I+ V]-1, (2-49)

with

V SXHT-1XSH(SSH)-1- SXHT-1A(AHT-1A)-'AHT-1XSH(SSH)-1

(2-50)

Then inserting (2-1) into (2-50) gives

V ZH[T-- T-1A(AHT-1A)-AHT-1]Zs(SSH)-1. (2-51)

Note that there are two random matrices, i.e., T and Zs, in (2-26) and (2-51).

Since the columns of Z are statistically independent zero-mean Guassian random

vectors while the columns of SH are orthogonal to those of I SH(SSH)S, by

the property of joint Gaussian distribution we know that Zs and Z' A Z[I -

SH(SSH)S] are two independent Gaussian random matrices. Hence, Zs and

T = Z*(Z*)H are also independent of each other. By Lemmas 1.9 and 1.11 in

[18], which can be readily extended to the complex-valued case, we have Zs ~
CN(0; Q, SSH) and T ~ CW(L K, M; Q).









Since Zs and T are statistically independent of each other and by (2-26) and

(2-51), we know that (BML B)A is an odd function with respect to Zs. Hence

replacing Zs with -Zs yields

E[(BL B)A]lzs=-zs -E[(BML B)A]. (2-52)

On the other hand, since Zs is a zero-mean Gaussian random matrix, -Zs, as a

random matrix transformed from Zs, retains all the statistical properties of Zs.

Hence, replacing Zs by -Zs will not change the expectation of (BML B)A, i.e.,


E[(BML B)A]Iz=-zs = E[(BML B)A]. (2-53)

It follows from (2-52) and (2-53) that

E[(BML- B)A] 0. (2-54)

Therefore, by (2-48) and (2-54) we have


E(Bcapo) BE(A). (2-55)

Now we follow the same technique used in the previous subsection to simplify

A and V via transformation of random matrices.

Following the definitions in (2-29), (2-30) and (2-31) and inserting them into

(2-51), we get

V (SSH)ZH [T-1 T-1A HTA-l)-1 HT-1]Zs(SS)-. (256)

Then we adopt the decomposition in (2 33), the definitions in (2-34) and

(2-35), and the partitions in (2-36), and insert them into (2-56). By the inversion









lemma of partitioned matrices, we obtain


V (SSH) HZ-H 1 F- F-P(PHF-1P) -pHF-]E(SSH)-

{ Fpll pl2 IF11 pl2
=(SSH>) H J- (E(SSH) (2-57)
p21 p22 p21 r21 p1l1 -l1pl2

S(SSH) rZ2 1Z 2(SSH)-

From (2-49) and (2-57) and by the matrix inversion lemma, it follows that

A = (SSH 122 -1(SSH-2
A (SSH) [I + EHF22E2] (ssH)258)
(2-58)
I (SSH) ( -H(F22 2+ ) 32(SSH) -

To calculate the expectation of A, we use the following lemma.

Lemma 2.4. Let Txp and I'n be two independent random matrices, and

T ~ CN(0; ~, Ip), I ~ CW(l, n; I,). (2-59)

Denote H TH( Y + TTH)-IY. Then the expectation and correlation matrices of
the random matrix H are

E(I) = I,, (2-60)
l+p
n(n + 1)
E(vec(H)vec(n)H) = +)( ) vec(I) vec(Ip)H + ID,, (2-61)

where rT is a scalar and ip'i',,.;' i,:,l. l;1 equal to (i- p- ) for 1r''' 1 + p, and Dp is

a p2 p2 matrix with its element at the [(c, )p+ r] th row and the [(c2 1)p+ r]th

column (rl, ci, r2, 1, 2, .. .p) being

1 when r = r2, c1 = c2 but ri / c,

d(c -l)p+rl,(c2-I)p+r2 -1 when r c1, r2 = c2 but ri / r2 (2-62)
0 otherwise.


Proof: Appendix A.5.









Applying the above lemma to E and F, which by construction satisfy the

assumptions in the lemma, we have immediately


E(A) ( (2-63)

Inserting (2-63) into (2-55), we get


E(BCapon) ( M ) B. (2-64)

The above equation shows that the multivariate Capon estimator shares

the same properties as the univariate Capon in [25]. In other words, it is biased

downward for finite snapshot number L. However, for large L, it is .i,-mptotically

unbiased. It is also worth noting that the bias of the Capon estimator is not related

to K, which means that increasing the number of rows in the temporal information

matrix S will not cause higher bias.

Moreover, we note that when M = N, the multivariate Capon estimator

becomes unbiased as the multivariate ML estimator. In this case, both ML and

Capon reduce to the same estimator A-1XSH(SSH)-1. Hence, for the same reason

that we have stated in Section 2.3.1, this unbiasedness of the multivariate Capon

estimator should not be seen as a significant advantage.

2.3.2.2 Mean-squared-error analysis

We investigate the MSE of the multivariate Capon estimator below. Using

the same technique to obtain (2-54), we can prove that ABA and B(I A) are

uncorrelated. Hence, from (2-26) and (2-48), we have

MSE(BCapon)

E[vec(Bc pon- B) vec(Bcapon- B)H]
(2-65)
=E{vec[ABA B(I A)] vec[ABA B(I A)]H

=E[vec(ABA) vec(ABA)H] + E[vec(B(I A)) vec(B(I A))H].








We first calculate E[vec(ABA)vec(ABA)H].
By (2-37) and (2-58), we have

ABA = (AHQ-1A)- [E1 r12rF21 2[I + EZr2 2 -l(SSH)- (2-66)

Using the fact that vec(XYZ) = (ZT 0 X)vec(Y) as well as (2-20) yields

E[vec(ABA)vec(ABA)H] [CRB(B)] F [CRB(B)] (2-67)

where

F E(vec[(Ei 12Fr Z)(I + ZF 2)-1
S+(2-68)
vec[(E F12F12)(I+ r2 22-2) 1

Then using the lemma that vec(XY) (YT 0 I)vec(X) and the fact that E~ and
E2 are independent standard matrix-variate Gaussian distributions, after some
manipulations, we get

F E{ [(I+ (HF 22)- -

[I + Err2 (vec(r12F22 ) vec(r12F 2)H)] (2 69)

[(I + (E-2 222) 1 ] }
Note that

Er 21r22 [vec(i12F1 2) vec( 12F2122)H]

S[(,l 2)T E I]Er,|. [vec(l2) vec(l2) ] [(rlz2)* I]
(2 70)
[(P Jz2) I][*2 I][(P JZ2)* I]

(2 t221i2)* I.

To get the above equation, we have utilized Lemma 2 in [76], i.e., r121 22
CN(O; I, F22). Hence, by the complex-valued counterpart of Lemma 1.8 in [18], we
know that the covariance matrix of vec(F12) given F22 is F2, I.








Inserting (2-70) into (2-69) and recalling (2-58) and (2-63) yield

F E{(I+(EHT-222- j-'1}

S[(SSH) -E(A*)(SSH)] 0I (2-71)

=1 I.
(1 ML N)

From (2-67) and (2-71), the equation

L-M + N
E[vec(ABA) vec(ABA)H] LCRB(B) (2-72)
L

follows directly.
Now we consider the second term in (2-65). By (2-58), we know that

B(I A) = B(SSH) (r22 + z2 )-l12(SSH)- (2-73)

We know that E2 and F22 are independent of each other with CN(0; IM-N, IK) and
CW(L K, M N; IM-N) distributions, respectively. Then using the fact that
vec(XYZ) = (ZT 0 X)vec(Y), the following equation is obtained following Lemma
2.2.

E{vec[B(I A)] vec[B(I A)]H}

S{(S*ST)- [B(SSH)2]}E{v, [E= (F22 + 2H )-1 2
[ET(r22 + -22l -121H} {(S*ST)- [B(SSH) ]} (2 74)
(M- N) (M N+ 1)H
SN(M- N vec(B) vec(B)H
L(L + 1)
+ ( {(S*S)- 0 [B(SSH)] }DK{ (S*S)- 0 [(SSH) BH]},

where DK is a K2 xK2 matrix defined as (2-62), and ( is a scalar and approximately
equal to N(LMN) for large L.









By (2-65), (2-72) and (2-74), we get the MSE of the multivariate Capon

estimator
L-M+N
MSE(Bcapon) L C RB )
(M N)(M N + t)
+( N)(M N+ ) vec(B)vec(B)H
L(L + 1)

(C {(SS)-2 0 [B(SSH) ]}DK {(S*ST)- 0 [(SSH) BH]}

(2-75)

Equation (2-75) gives an approximate closed-form expression of the MSE of

the multivariate Capon estimator. In this equation, we note that the MSE consists

of three terms. The first term is proportional to CRB(B). The second term is

proportional to the outer-product of vec(B) and is not related to the parameter

K and the temporal information matrix S. In the third term, although there is no

explicit dependence of the parameter K, the number of non-zero elements in DK

is dependent of K. Hence, the third term will increase as K increases. Moreover,

the third term is a function of (S*ST), which depends on the the correlation among

the rows of S. As we will see in the following numerical simulations, for S with

correlated rows, the MSE of an element of B increases as K increases and/or as the

other elements in B increase. On the contrary, when K = 1, the third term is zero

because the matrix D becomes a scalar 0 according to its definition. If we further

set N 1= then (2-75) reduces to the conclusion in the univariate case in [25].

We also note that when the number of snapshots L is large, the last two terms

approach zero while the first term approaches CRB(B). Hence, the multivariate

Capon estimator is also .i- iiinil' itically statistically efficient for large L.

Furthermore, we note that when M = N, the MSE of the multivariate Capon

estimator is simplified to CRB(B) like the multivariate ML estimator. This is

consistent with our conclusion in the above subsection that the two multivariate

methods reduce to the same estimator when M = N. For the same reason that we









stated in Section 2.3.1, this efficiency of the multivariate Capon estimator should

not be seen as a significant advantage.

Now we summarize the statistical properties of the multivariate Capon

estimator by the following theorem.

Theorem 2.2. For the data model in (2-1), the multivariate Capon estimate of

B in (2-8) is biased downward. However, for I.i,,. number of data samples, it is

-1i 'lI...1.I: illi. unbiased and .-Il.':.: ..ll.i/ efficient. Its bias and MSE matrices are

given by (2-64) and (2-75), ,, -"'. ,' 1;

2.4 Numerical Examples

In this section, several numerical examples are presented to verify the

performance analysis results of the two multivariate estimators. We consider a

uniform linear array with M = 4 sensors and half-wavelength spacing. We assume

N = 2 signals arriving at the sensor array with DOAs (Direction Of Arrival) of 0

and 150 relative to the array normal. Unless specified otherwise, we assume that

L = 16 and SNR = 10 dB and S is formed by K = 2 complex sinusoids with

unit amplitudes and frequencies 0.10 Hz and 0.125 Hz, respectively. Except in Fig.

2-6, the elements in B are all set to be 1. The interference and noise term in our

data model in (2-1) is temporally white but spatially colored zero-mean circularly

symmetric complex Gaussian with the spatial covariance matrix Q given by


[Q], p (0.9)i-1, (2-76)

where p = 1/SNR and [.]yi denotes the ith row and jth column element of a matrix.

The figure below are all for [B]11. The figures for other elements of B are similar.

We obtain the empirical results in Fig. 2-2 using 10000 Monte Carlo trials while

the others 1000 trails.

We first investigate the bias performance. Fig. 2-1 shows the bias properties

of the two multivariate estimators (denoted by \ V-ML" and \ V-Capon") from





























Figure 2-1: Bias versus L when SNR = 10 dB, K = 2, N = 2.


both theoretical predictions (denoted by "Theo.") and Monte Carlo trials (denoted

by "Empi."). As expected, the multivariate ML is unbiased whereas Capon is

biased downward for finite snapshots. However, when the number of snapshots L

is large, the bias of the multivariate Capon approaches zero, as predicted by our

theoretical analysis.

Fig. 2-2 illustrates the relationship between the bias and the number of rows

of the temporal information matrix S, i.e., K, when the frequency difference of the

complex sinusoids in S is 0.04 Hz. As predicted by our theoretical analysis, the bias

of the multivariate Capon estimator is independent of K.

Fig. 2-3 illustrates the MSEs of the multivariate estimators as well as the

CRB as a function of L. As illustrated, the theoretical and empirical MSEs are

consistent. The performance of the multivariate ML estimator is better than the

multivariate Capon and very close to the corresponding CRB. As we have predicted

in Section 2.3 that both multivariate estimators are .,-i','ii11i.1 ically statistically










0.1 I I I
Theo. bias of MV-Capon
> Empi. bias of MV-Capon
Theo. bias of MV-ML
o Empi. bias of MV-ML





0 ----a----ec--c------a----o-----g----^
U)
Fn


-0.1
..| .. --- .-. ..> . . -- . -L -




-0.2
1 2 3 4 5 6 7 8
K


Figure 2-2: Bias versus K when SNR = 10 dB, L = 16, N = 2.


efficient for large number of snapshots, and the performance curves of the two

estimators approach the CRB as L increases.

Fig. 2-4 shows the relationship between the MSE and SNR. Note that the

error floor occurs at high SNR for the multivariate Capon estimator due to its bias.

As shown in our theoretical analyses, for a fixed M, L, N and K, the MSE of ML

is proportional to CRB(B), and hence no !hi -!i. Il effect" occurs. Note also that,

like in the univariate case, the Capon estimate can provide a smaller MSE than ML

at low SNR. At such a low SNR, though, both ML and Capon perform poorly.

Fig. 2-5 gives the MSEs of the multivariate Capon and ML estimators as well

as the corresponding CRB as a function of K when the frequency difference of the

complex sinusoids in S is 0.04 Hz. As we can see, both the CRB and the MSEs

of the two multivariate estimators increase as K increases. However, due to the

contribution of the third term in (2-75), the MSE of Capon increases more quickly

than the CRB and the MSE of ML.










10 . .
--. Theo. MSE of MV-Capon
o Empi. MSE of MV-Capon
100 -- Theo. MSEofMV-ML
t> Empi. MSE of MV-ML
CRB
10-1 %C,


U 10 -


10-3


10-4


10-5 1
10 102 103 104
L

Figure 2-3: MSE versus L when SNR 10 dB, K = 2, N = 2.


In Fig. 2-6, we consider the case where B has unequal elements. We set

N = K = 2, [B]I = [B]21 = 1 and [B]12 = [B]22 = a- where a is the power

ratio between the two complex sinusoids in S. Fig. 2-6 gives the CRB and MSEs

of [B]n as a varies. As illustrated, the MSE of the multivariate ML estimator

is almost constant with respect to a, while the MSE of the multivariate Capon

estimator increases rapidly when a is decreased to be lower than 0 dB due to its

biased nature.

2.5 Conclusions

We have investigated the theoretical performance of two multivariate

parameter estimators, namely the multivariate Capon and ML estimators. Through

theoretical analysis and numerical simulations, we conclude that the multivariate

ML estimator is unbiased, whereas the multivariate Capon estimator is biased

downward for finite snapshots; both estimators are .,-vmptotically statistically

efficient when the number of snapshots is large.







































SNR(dB)


Figure 2-4: MSE versus SNR when L








0.5
.--- Theo. MSE of MV-Capon
0.45 o Empi. MSE of MV-Capon
-- Theo. MSE of MV-ML
0.4 Empi. MSE of MV-ML
CRB

0.35-

0.3-
LU
0 0.25 -

0.2-

0.15 o

0.1 /,' ''

0.05 -'


16, K = 2, N


Figure 2-5: MSE versus K when SNR = 10 dB, L


16, N = 2.



































-. Empi. MSE of MV-Capon
.45 o Theo. MSE of MV-Capon
-- Empi. MSE of MV-ML
Theo. MSE of MV-ML
0.4 -
CRB

.35-

0.3

.25
to
0.2

.15

0.1 -


5 ....-.. I. 15 2 2.
-5 0 5 10 15 20 25 30


a (dB)


Figure 2-6: MSE versus a when SNR = 10 dB, L


16, K = 2, N = 2.


0


LU




0



0














CHAPTER 3
DIAGONAL GROWTH-CURVE MODEL

3.1 Introduction and Problem Formulation

In this chapter, we consider a variation of the growth-curve (GC) model,

referred to as the diagonal growth-curve (DGC) model,


X ABS + Z with B i -(3), (3-1)


where diag(3) denotes a diagonal matrix with its diagonal formed by the elements

of the vector 3. In (3-1), X E CMXL denotes the observed snapshots with L being

the snapshot number and M the snapshot dimension. The columns in A E CMXN

are the known spatial information vectors, referred to as the steering vectors. The

rows in S E CNxL are the known temporal information vectors, referred to as

waveforms. The elements in 0 E CNX1 are the unknown complex amplitudes.

The columns of the interference and noise matrix Z E CMXL are independently

and identically distributed (i.i.d.) circularly symmetric complex Gaussian random

vectors with zero-mean and unknown covariance matrix Q. Throughout this

chapter, we assume that M + r < L with r being the row rank of S. The problem

of interest is to estimate the unknown complex amplitudes. Note that in the DGC

model we do not need to assume the linear independence of the steering vectors or

waveforms, unlike in the GC model.

The only difference between DGC and GC is that the complex amplitude

matrix B in DGC is constrained to be diagonal while it is arbitrary in GC.

However, this seemingly minor difference makes the derivations of the multivariate

ML estimator in [21] and [22] invalid. In fact, to our knowledge, no closed-form

ML estimator for 3 in (3-1) exists in the literature. In this chapter, we propose









an approximate maximum likelihood (AML) estimator for / in this model. We

also investigate its statistical properties via theoretical analyses and numerical

simulations.

We remark that although in this chapter we focus on the complex amplitude

estimation of signals with known steering vectors and waveforms, the proposed

DGC model and AML estimator can also be used in the case where both the

steering vectors and waveforms are parameterized by unknown parameters. Using

the proposed AML method, we can construct a concentrated (approximate)

likelihood function of the unknown parameters [27] [30] [31]. The unknown

parameters in the steering vectors and waveforms can then be estimated via

maximizing the concentrated likelihood function.

The remainder of the chapter is organized as follows. In Section 3.2, we

introduce the AML estimator for 3 based on the DGC model in (3-1). Section

3.3 presents the performance analysis of the AML estimator and the CRB for the

unknown complex amplitudes. Numerical examples are presented in Section 3.4.

Finally, Section 3.5 contains our conclusions.

3.2 Approximate Maximum Likelihood Estimation

In this section, we derive an approximate maximum likelihood (AML)

estimator for 3 in (3-1). This approach is based on the ML principle, but an

approximation is made to get a closed-form solution.

It follows from (3-1) that the negative log-likelihood function of the observed

data samples (to within an additive constant) is


f(3, Q) = Lln|Q| + tr[Q-(X ABS)(X ABS)H], (3-2)


where tr(-), I | and (.)H denote the trace, the determinant and the conjugate

transpose of a matrix, respectively. Minimizing the negative log-likelihood function









with respect to Q yields

Q (X- ABS)(X- ABS)H. (3-3)
L

Note that Q is nonsingular, i.e., |Q| / 0, with probability one when M < L.

Substituting (3-3) into (3-2), the ML estimate of / can be formulated as


1ML argminlnl(X- ABS)(X- ABS)HI with B diag(/). (3-4)

In general, the optimization problem in (3-4) does not appear to admit a

closed-form solution. Here, we use the technique in [25] and [27] to solve this

problem approximately. Let Is and Hi denote the orthogonal projection matrices

given by

ns A SH(SSH)-S, (3-5)

and

Hnj AI Hs, (3-6)

where (.)- denotes the generalized inverse of a matrix [74]. Note that when the

waveforms, i.e., the rows of S, are not linearly independent of each other, the

matrix (SSH) is singular and hence its generalized inverse is not unique. However,

Hs and Hn are unique [74].

Using (3-5) and (3-6), it follows that

I(X- ABS)(X- ABS)H

I(X ABS)(Hs + H )(X ABS)H
(3-7)
I(xnH ABS)(XHs ABS)H + T

I(XHs ABS)(XHs ABS)HT- + 1 I IT

where the matrix T is defined by


T A XH XH.


(3-8)









Note that the rank of Hi is L r. Hence, T has full rank M with probability one

when M + r < L. Note also that T is an estimate of the covariance matrix Q

(to within a multiplicative constant) obtained by projecting out the desired signal

components from X.

Using (3-7) and Lemma 4 in [25] (or Theorem 1 in [27]), the solution of the

optimization problem in (3-4) can be approximated for a large snapshot number as

follows.

/AML argmintr[(XIs ABS)HT-1(XHs ABS)]
03 (3-9)
argmin I vec(T- Xns) vec(T- ABS) |2


where T- is the Hermitian square root of T-1, I|| 1 | denotes the Euclidean vector

norm and vec(-) denotes the vectorization operator (stacking the columns of a

matrix on top of each other).

To solve the above optimization problem, we first introduce the following

lemmas.

Lemma 3.1. Let U and V be m x p and n x p matrices, ,' i'..1 /;. l; and let G be

a p x p diagonal matrix. Then

vec(UGVT) (V E U) vcd(G), (3-10)

where (.)T denotes the transpose of a matrix, vecd(.) denotes a column vector

formed by the diagonal elements of a matrix, and E denotes the Khatri-Rao matrix

product [77 [78].

Proof: This result is not new but we include a simple proof in Appendix B.1

for the completeness of this chapter.

Lemma 3.2. Let U, G and V be matrices with dimensions mx p, px n and n x m,

,, ./.,. /:.;, /;/ Then


vecd(UGV) (V E UT)Tvec(G).


(3-11)









Furthermore, if p = n and G is a diagonal matrix, then

vecd(UGV) = (U VT) vecd(G), (3-12)

where ) denotes the Hadamard product (elementwise multiplication) of two

matrices.

Proof: Appendix B.2.

Lemma 3.3. Let U and V be m x p and n x p matrices, ', I/'.. ,t;. l,;q Then

(U E V)H(U E V) (UHU) ) (VHV). (3-13)

Proof: This lemma is a straightforward extension of Lemma A2 in [78].

By Lemma 3.1, it follows from (3-9) that


AML = argmin I vec(T- XHs) F03 2 (3 14)

with r A ST E (T-A). Note that (3-14) is in a quadratic function of 0.

Minimizing (3-14) with respect to f yields


(AML (FHr) -Hvec(T- xns). (3-15)

In (3-15), we have assumed that the columns of F are linearly independence of

each other and hence FHF is invertible. Note that this assumption is much weaker

than that in the GC model.

On the other hand, using Lemmas 3.3 and 3.2, respectively, we have that

FHr (SSH)T) (AHT-A) (3-16)

and


Hvec(T-XnHs) vecd(AHT-IXSH).


(3-17)









Substituting (3-16) and (3-17) into (3-14) yields the following expression for the

AML estimate of f


lAML = [(AHT-1A) (SSH)T]-lvecd(AHT-1XSH). (3 18)

Note that (3-18) does not require the existence of the inverse of AHT-1A.

Moreover, (3-18) does not require SSH to be invertible. Therefore, unlike the

estimators in the GC model [22], the AML estimator in DGC does not require the

linear independence of the steering vectors or of the waveforms.

Appendix B.3 gives another interpretation of AML from a generalized least-

squares (GLS) point of view.

3.3 Performance Analysis

We now establish the theoretical properties of the AML estimator.

3.3.1 Bias Analysis

Substituting (3-1) into (3-18) gives

/AML =[(AHT-1A) 0 (SSH)T]-lvecd(AHT-1ABSSH)
(3-19)
+ [(AHT-1A) ( SSH)T]-lvecd(AHT-1ZSH).

By Lemma 3.2 we get

vecd[(ATT-1A)B(SSH)] = [(AHT-1A) 0 (SSH)T]3. (3-20)

Hence, from (3-19) we obtain the estimation error


3AML = [(AHT-1A) 0 (SSH)T]-lvecd(AHT-1ZSH). (3-21)

On the other hand, substituting (3-1) into (3-8) yields


T = ZHfZH.
S


(3-22)









From (3-22), we see that T is an even function of Z, and hence 3AML / is an odd

function of Z. Moreover, we have assumed that Z is a zero-mean Gaussian random

matrix. Using the statistical properties of the zero-mean Gaussian distribution [75],

we can readily show that the expectation of (3-21) is zero, i.e., 3AML is unbiased.

3.3.2 Mean-Squared-Error Analysis

The exact MSE of /AML is difficult to determine, if not impossible. Herein, we

provide an approximate expression for the MSE which holds for a large snapshot

number.

From the theory on the linear statistical model [16], we know that L is an

unbiased and consistent estimate of the noise covariance matrix Q, which converges

to Q when L approaches infinity. Hence, for a large snapshot number L, the

estimation error in (3-21) can be approximated by


/AML w CRB(/) vecd(AHQ-1ZSH), (3-23)

where

CRB(3) [(AHQ-'A) 0 (SSH)T]-1 (3-24)

is the Cram6r-Rao bound for 3 that is the lowest possible MSE of any unbiased

estimator of / ( see Appendix B.3 for the derivation of CRB).

From (3-23) and Lemma 3.2, it follows that

var(dAML) AE[(AML /)(AML -_ )H

,CRB(3) E[vecd(AHQ-1ZSH) vecd(AHQ-1ZSH)H] CRB(3)

=CRB(3) [SH E (AHQ-)T]T E[vec(Z)vec(Z)H] [SH E (AHQ-1)T]* CRB(0)

=CRB(3) [ST E (Q-'A)]H (I Q) [ST E (Q-'A)] CRB(3)

(3-25)

where (.)* and 0 denote the complex conjugate and the Kronecker matrix product,

respectively. From (3-25), similarly to Equation (B-7) in Appendix B.3, we get the









MSE of 3AML as


var(AML) = [(AHQ-1A) (SSH)T]-1 CRB(/). (3-26)

We see that the AML estimate of 0 is .,-i1 'i11il' ically statistically efficient for a

large snapshot number. This theoretical result is verified by the numerical examples

in Section 3.4.

3.4 Numerical Examples

In this section, several numerical examples of two different applications of the

DGC model are presented to demonstrate the performance of the proposed AML

estimator.

3.4.1 Examples in Array Signal Processing

We consider a uniform linear array with M = 6 elements, and half-wavelength

spacing between .,.i ,i:ent sensors. The sensor elements are assumed to be omni-

directional. The incident signals are quadrature phase shift keyed (QPSK)

sequences, and the complex amplitudes are equal to one. The columns of the

interference and noise matrix Z are i.i.d. circularly symmetric complex Gaussian

random vectors with zero-mean and an unknown covariance matrix Q given by


[Q] ,n pO.99~m- e32 (3-27)

where p = 1/SNR and []m,, denotes the (m, n)th element of a matrix.

In the following examples, we present the MSE of the AML estimator in

(3-18) as well as the corresponding CRB in (3-24). For comparison, we also show

the MSE of the ML estimator in [21] and [22] based on the GC model, which

ignores the diagonal constraint on B, and of the least squares (LS) estimator which

assumes that the interference and noise vectors in Z are uncorrelated both spatially

and temporally. The LS estimator can be obtained immediately via replacing T in










10-1
-U GC-ML
->0 DGC-LS
-0 DGC-AML
10-2 CRB


10-3

10 -





105
10--6





10 102 103
Snapshot Number L

Figure 3-1: Empirical MSE's and the CRB versus L when SNR 10 dB, M
6, and N = 3 with linearly independent steering vectors and linearly independent
waveforms.


(3-18) by an identity matrix, i.e.,


/LS [(AHA) 0 (SSH)T]-lvecd(AHXSH). (3-28)


The empirical estimation performances are all obtained by 1000 Monte-Carlo

simulations.

We first consider the case where the steering vectors are linearly independent

of each other and so are the waveforms. Suppose that N = 3 signals with known

waveforms arrive at the sensor array from 01 = -100, 02 50 and 03 = 10,

respectively. Assume that the DOAs of the signals have been estimated accurately

and therefore they can be considered to be known. The signal waveforms are

generated independently and hence are uncorrelated with each other. Due to space

limitations, we only show below the MSE of the complex amplitude estimate of

the signal arriving from 02 = 5. The performances for other signals are similar.













.I0- ., -u^-MrVIL
10 -0 -- CRB




LU -3 sh hi



10-4

10s n, rse t




10-6




6, and N 3 with linearly independent steering vectors and linearly independent
waveforms.


Fig. 3-1 and Fig. 3-2 show the estimation performance as a function of the

snapshot number L and the SNR, respectively. The SNR is fixed at 10 dB in Fig.

3-1 while the snapshot number is fixed at 128 in Fig. 3-2. Note that the ML

estimator based on the GC model gives relatively poor estimation performance.

This method ignores the a priori information about the diagonal structure of the

complex amplitude matrix B, which increases the number of unknowns from N

to N2 and hence results in much larger estimation variances. In Fig. 3-1, we see

that the AML estimator outperforms LS significantly, and approaches the CRB

rapidly, as predicted theoretically. In Fig. 3-2, we see that the AML estimator

outperforms the other methods and is very close to the CRB in the whole range of

SNR considered.

Fig. 3-3 and Fig. 3-4 show the estimation performance when the waveforms

are linearly dependent of each other. We retain the same simulation parameters as










10- I
->0 DGC-LS
-0 DGC-AML
CRB

10-2



LU -3
U) 10 -




10-4




101 102 103
Snapshot Number L

Figure 3-3: Empirical MSE's and the CRB versus L when SNR 10 dB, M 6,
and N = 3 with identical waveforms.


in Fig. 3-1 and Fig. 3-2 except that the waveforms of the three incident signals

are identical in this case. Since the ML estimator based on the GC model requires

that the waveforms are linearly independent of each other [22], it fails to work

properly in this case. On the other hand, while the MSE's of AML are somewhat

larger than those for linearly independent waveforms, the AML estimator remains

. i-mptotically statistically efficient and provides higher estimation accuracy than

LS.

Next we consider an example where N 13 incident signals impinge on the

sensor array from 0 = -300, -250, ... 300. The other simulation parameters are the

same as those for Fig. 3-1 and Fig. 3-2. Note that in this case N > M, which in

particular means that the steering vectors are linearly dependent and hence it does

not satisfy the assumptions required by the GC model; consequently again the ML

estimator based on the GC model cannot be used. From Fig. 3-5 and Fig. 3-6, we






44



101
^ --> DGC-LS
10 -- DGC-AML
100 CRB





LI. -2
S10 -


10-3


10-4



-20 -15 -10 -5 0 5 10 15 20
SNR (dB)

Figure 3-4: Empirical MSE's and the CRB versus SNR when L = 128, M = 6, and
N = 3 with identical waveforms.


see that the AML estimator has a better estimation accuracy than LS and remains

.i-vmptotically statistically efficient.

3.4.2 Spectral Analysis Examples

In this subsection we consider the problem of estimating the complex

amplitudes of sinusoidal signals from observations corrupted by colored noise.

This problem has been studied, for example, in [24]. We apply the AML estimator

to this 1-D spectral estimation problem. As we show below, the proposed AML

estimator can achieve better estimation accuracy and exhibit greater robustness

than the methods in [24].

Consider the noise-corrupted observations of N complex-valued sinusoids

N
x(l) Z= 3ej z + z(1), 1 0, 1, Lo- 1, (3-29)
n=l

where 3, is the complex amplitude of the nth sinusoid with frequency w,; Lo is the

number of data samples; z(1) is the observation noise, which is complex valued and










10-2
-> DGC-LS
-- DGC-AML
CRB
10 --



LU1-4 0
10-3
lo-- -
10-4 ,





10-6



10-7
102 103
Snapshot Number L

Figure 3-5: Empirical MSE's and the CRB versus L when SNR 10 dB, M = 6,
and N = 13 with linearly dependent steering vectors


assumed to be stationary and possibly colored with zero-mean and unknown finite

power spectral density (PSD). We assume that {wn}7 1 are known, with wa / Lk

for n / k. The problem of interest is to estimate {3N} from the observations

{x(1)} o 1

To solve this problem, we divide the data sequence into overlapping sub-

sequences with shorter lengths [60] [23]. Then we reformulate the data equation

in (3-29) into the form of a DGC model, and use the proposed AML approach to

estimate the complex amplitudes of the sinusoids.

Let
x(0) x(1) ... x(Lo M)


Sx(1) x(2) ... x(Lo- M+1) (330)


x(M- ) x(M) ... x(Lo- 1)










100 I
-0> DGC
-- DGC
10-1 CRB


10 2





10-4


10-


10-6
-20 -15 -10 -5 0 5 10
SNR (dB)

Figure 3-6: Empirical MSE's and the CRB versus SNR when L
N = 13 with linearly dependent steering vectors.


1





.. (M-1)yN


128, M = 6, and







(3 31)


and
1 gewl j(Lo-M)wj

1 gej2 j(Lo-M)w2
S (3-32)


1 gJWN ... j(Lo-M)wN

Then, (3-29) can be readily written in the DGC form in (3-1) with =

[/1, /02, .." N]T, L = L M + 1 and the noise matrix Z defined similarly

to X in (3-30). Hence, the AML estimator in (3-18) can be applied directly. In

doing so we ignore the fact that the columns of the interference and noise matrix Z


1

j 1)w



gj(M-1)Wi


1

j(W2



6j(M 1)W2









are correlated due to the overlapping of data samples, which is commonly done in

the literature to retain the simplicity of the parameter estimation algorithm.

We consider a numerical example used in [24]. The observed data with Lo = 32

consists of N = 3 complex sinusoids with frequencies f = 0.10, f2 0.11,

f3 = 0.30 (fk k=/2r) and complex amplitudes 31 = ed, 32 = e3 and /3 = e3,

respectively. The colored noise z(1) is described by the following autoregressive

(AR) equation

z(1) 0.99(1 1) + e(l), (3-33)

where e(l) is a complex white Gaussian noise with zero-mean and variance oa2. The

PSD of the test data is shown in Fig. 1 of [24] when a2 = 0.01.

For comparison, we provide the estimation performance of the proposed

AML method and of the matched-filter bank (i\AFI) approach [24], which is

the most competitive one among the algorithms presented in [24], as well as the

corresponding CRB. Note that in this application the columns of the interference

and noise matrix Z are not statistically independent, and hence the CRB in (3-24)

is not applicable. Instead, we utilize the CRB formula presented in Equation (9) of

[24]. The MSE values presented in the following examples are all obtained via 1000

Monte-Carlo simulations. Since the PSD of the AR noise varies in the frequency

domain, we utilize the local SNR as a measure of the signal quality for a particular

sinusoid [24] [79].

Fig. 3-7 shows the MSE's of the two amplitude estimators for /3 and 31

along with the corresponding CRB as the corresponding local SNR varies, when

M = Lo/4 = 8. (The results for 32 are omitted because they resemble those for

31.) In Fig. 3-7(a), MAFI and AML both provide high estimation accuracy and
are very close to the CRB. However, the estimation performance in Fig. 3-7(b)

is significantly poorer than that in Fig. 3-7(a) for both estimators due to the

interference from the second sinusoid at f2. (Note that f2 f = 0.01, which is



































10-7L


30 35 40 45 50 55 60 25 30 35 40
local SNR (dB) local SNR (dB)

(a) (b)


Figure 3-7: Empirical MSE's and the CRB versus local SNR when Lo
8, and the observation noise is colored. (a) For /3 and (b) for 31.


10 i
>- MAFI
-e- AML
10- CRB

S-2

10 ,

10-3


10-4



2 4 6 8 10 12 14 16
Subvector Length M


2 4 6 8 10 12 14
Subvector Length M


Figure 3-8: Empirical MSE's and the CRB versus M when Lo = 32, a2 = 0.01, and

the observation noise is colored. (a) For 33 and (b) for 31.


- MAFI
-*- AML
- CRB .


32, M


0
w


U,
C
a)


->- MAFI
-e- AML
- CRB


\ ,' '
.^ .
\.. ^ 9


I I I I i i i









smaller than the Fourier resolution limit, i.e., 1/Lo a 0.03.) As we can see, MAFI

deviates away from the CRB at high SNR because of the interference at f2, which

introduces a bias into the estimate of 31 that dominates the MSE at high SNR.

The AML estimator provides better estimation accuracy than MAFI, especially at

high SNR. Note that in the presented spectral wi", i, -i application, the columns of

the interference and noise matrix Z are not i.i.d. and hence the theoretical analysis

in Section 3.3 showing that AML is .. -mptotically statistically efficient is no longer

valid. However, as we can see from Fig. 3-7(a) and Fig. 3-7(b), the MSE of AML

remains close to the CRB which shows the high interference-resistant capability of

AML.

Intuitively, we can expect that as M increases, the AML estimator as well as

MAFI (and also other methods presented in [24]) can deal better with the case of

closely spaced sinusoids, but their statistical accuracy, in general, decreases [71].

Hence, there is a tradeoff when choosing M. The following example examines the

effect of M on the performance of these estimators. The scenario is similar to the

example in Fig. 3-7, except that we fix a2 = 0.01, which corresponds to a local

SNR of 30.8 dB for the first sinusoid (at fli 0.1) and 39.2 dB for the third sinusoid

(at f3 = 0.3). The subvector length M is varied from 1 to 16 for AML and from 3

to 16 for MAFI (\ AFI requires that M > N = 3). The MSE's of the amplitude

estimates of /3 and 31 and the corresponding CRB's are shown in Fig. 3-8(a) and

Fig. 3-8(b). As we can see, when no sinusoids are close to the one being estimated,

such as the third sinusoid in this example, both AML and MAFI perform quite

well for a wide range of M values. For the more difficult case shown in Fig. 3-8(b),

the choice of M becomes critical. As we can see, the AML estimator outperforms

MAFI over a wide range of M values and demonstrates its robustness to the choice

of M.









3.5 Conclusions

We have presented an approximate maximum likelihood estimator for the

diagonal growth-curve model where the steering vectors and the waveforms of the

signals are known and the unknown complex amplitude matrix is constrained to

be diagonal. Via a theoretical analysis, we have shown that the AML estimator is

unbiased and .,-imptotically statistically efficient for a large snapshot number. We

have applied the AML estimator to complex amplitude estimation problems in both

array signal processing and spectral ,in 1, -i- The presented numerical examples

provide compelling evidence that the AML method can achieve better estimation

accuracy and exhibit greater robustness than the best existing methods.















CHAPTER 4
BLOCK DIAGONAL GROWTH-CURVE MODEL

4.1 Introduction and Problem Formulation

In this chapter, we consider a general variation of the GC model, referred to as

the block diagonal growth-curve (BDGC) model


X ABS + Z with B Diag(Bi, B2, 3 ,Bj), (4-1)


where

A [A1 A2 ... Aj], (4-2)

and

S [ST S ... ST]T. (4-3)

In (4-1), X E CMXL contains the observed data samples with M being the

snapshot dimension and L being the snapshot number. The columns in Aj E

CMxNj and the rows in Sj E CK3xL are known and assumed to be linearly

independent of each other. The matrices Bj E CNjxK3 contain the unknown

regression coefficients. Throughout this chapter, we assume that M < Nj and

M + r < L with r being the row rank of S. The columns of the error matrix Z are

assumed to be i.i.d. circularly symmetric complex Gaussian random vectors with

zero-mean and unknown covariance matrix Q. The problem of interest herein is to

estimate the unknown block-diagonal matrix B.

Note that in the BDGC model the linear independence among the columns

of Aj and among the rows of Sj is only required within the submatrices. In

other words, the columns from different submatrices of A and the rows from

different submatrices of S can be linearly dependent on each other. Note also that









the BDGC model unifies the GC model [18] [21], and the DGC model in [36].

We remark that in this dissertation we focus on the complex-valued parameter

estimation problem; however the proposed AML estimator can be used directly for

real-valued parameter estimation as required in some statistical applications [11].

The remainder of this chapter is organized as follows. We first give some

preliminary matrix results in Section 4.2, and then derive the AML estimator in

Section 4.3. The theoretical performance analysis and numerical examples are

provided in Sections 4.4 and 4.5, respectively. Finally, Section 4.6 contains the

conclusions.

4.2 Preliminary Results

In this section, we introduce several partitioned matrix operations and lemmas,

which will be used frequently in the derivation and performance analysis of the

AML estimator.

Definition 4.1. Let G be a partitioned matrix with Gij being the (i, j)th (i,j

1, 2, J) submatrix of G. Then the block-diagonal vectorization operation is

1, /7,.,1 by

vecb(G) A [vec(Gi,1) vec(G2,2) vec(Gjj)], (4 4)

where vec(.) denotes the matrix vectorization operator (stacking the columns of a

matrix on top of each other).

Definition 4.2. Let E and Q be two partitioned matrices with conformal

,,II:l/:..'i':ij and with ESi and fQj being the (i, j)th submatrices of E and 2,

" -i"'. /;' /I; Then the generalized Khatri-Rao product is 1, I,, ,1 by



[E O 0],,, A SE,, 0 QJ, (4-5)

where [.]ij denotes the (i,j)th submatrix of the given partitioned matrix, and 0

denotes the Kronecker matrix product (the generalized Khatri-Rao product was also

used, e.g., in [80).









Note that the block-diagonal vectorization vecb(-) and the generalized Khatri-

Rao product O are defined based on a particular matrix partitioning, i.e., different

matrix partitionings will lead to different results. Note also that the standard

vectorization vec(.) [74] and the diagonal vectorization vecd(-) [36] are both special

cases of the block-diagonal vectorization vecb(.), while the Kronecker product, the

Hadamard product and the Khatri-Rao product [77] [78] are all special cases of the

generalized Khatri-Rao product based on different matrix partitionings. It is also

worth pointing out that matrix partitioning may be inherited through matrix

operations. For example, for the partitioned matrix A given by (4-2), AHA is a

partitioned matrix with the (i, j)th (i,j = 1, 2 .. J) submatrix being AHAj;

hereafter, the superscript H denotes the conjugate transpose of a matrix.

Next, we give two lemmas on partitioned matrix operations.

Lemma 4.1. Let E and f2 be two partitioned matrices with K block rows and J

block columns, and let G be a block-diagonal matrix with compatible dimensions and

conformal 1'ri.:/..' ,.,ii with E and f2. Then


vecb(EGFT) = (0 E)vecb(G). (4-6)


Proof: We note that EGQT is a partitioned matrix with the (k, k)th (k

1, 2, .. ,K) submatrix being

J
[EG ]k,k [1k,J[G],,j[]r (4-7)
j=1

Hence,

J
vec([EG ]kk) vec([]kj [G]k,,,[n] )
j 1
J
= ([kj 0 [E]k, )vec([G],j) (4-8)
j= 1

= ([]k,1 0 []k,1) ... ([ k,J 0 [S]k,j) vecb(G),









where we have used the fact that vec(ABC) = (CT 0 A)vec(B) [74]. Arranging the
vectors vec([EG'T]k,k) (k = 1, 2, .. J) in (4-8) into a column vector yields (4-6).
Lemma 4.2. Let U and V be two partitioned matrices with 1 block row and J
block columns, and let H and F be two partitioned matrices with 1 block row and K
block columns and with compatible dimensions with U and V, /".. /,:'; l Then,


(U V)H(H F) = (UHH) (VHF). (49)

Proof: Note that U V is a partitioned matrix with 1 block row and J block
columns, and with the (1,j)th submatrix being

[U V]i,j [U],j 0 [V]l,j. (4-10)

Similarly, H F is a partitioned matrix with 1 block row and K block columns,
and with the (1, k)th submatrix being

[H F]1,k [H]i,k [F],k. (4 11)

Hence, (U V)H(H F) is a partitioned matrix with J block rows and K block
columns, and with the (j, k)th submatrix being

[(U V)H(H F)]j,k ([U]1,j 0 [V]i,j)H([H] ,k [F]1k)
(4-12)
([U]H [H]i,k) 0 ([V], [F]I,k),

where we have used the fact that (A 0 B)H(C 0 D) = (AHC) 0 (BHD) [74].
On the other hand, UHH and VHF are two partitioned matrices with J block
rows and K block columns, and with the (j, k)th submatrices being

[UHH]i,k [U]H [H],k (4 13)


(4 14)


[VHF]j,k = [V]H, [F]1,k.









From (4-12), (4-13) and (4-14), we obtain


[(U V)(H F)]j,k [UHH]j,k 0 [VHF]j,k, (4-15)

which yields (4-9) immediately.

We remark that Lemmas 3.1 and 3.2 in C'! lpter 3 are special cases of Lemma

4.1 in this chapter, while Lemmas Al and A2 in [78] and Lemma 3.3 in [36] are

special cases of Lemma 4.2 in this chapter.

4.3 Approximate Maximum Likelihood Estimation

In this section, we derive an approximate maximum likelihood (AML)

estimator for the unknown block-diagonal regression coefficient matrix B in

(4-1). Our approach is based on the ML principle, but an approximation is made

to get a closed-form solution. In the following derivation, we utilize the partitioned

matrix operations and lemmas of Section 4.2. Note that the partitioned matrix

operations used in the derivations are based on the matrix partitionings of A, S

and B in (4-2), (4-3) and (4-1), respectively. The matrices X and Z are both

treated as non-partitioned matrices.

For convenience, we arrange the unknowns in B into a column vector, i.e.,

3 = vecb(B). It follows from (4-1) that the negative log-likelihood function of the

observed data samples is (to within an additive constant)


f(0, Q) = Lln|Q| + tr[Q-1(X ABS)(X ABS)H], (416)

where tr(-) and | I denote the trace and the determinant of a matrix, respectively.

Minimizing the negative log-likelihood function with respect to Q yields

Q (X- ABS)(X- ABS)H. (4 17)
L


For M < L, Q is nonsingular with probability one.









Substituting (4-17) into (4-16) yields the ML estimate of 3


OML = argminln (X ABS)(X ABS)H. (4 18)

In general, the optimization problem in (4-18) does not appear to admit

a closed-form solution. Here, we use the technique in [25] and [27] to solve this

problem approximately. Let H1s and H respectively, denote the orthogonal

projection matrices given by

ns A SH(SSH)-S, (4-19)

and

Hn A I Hs, (4-20)

where (.)- denotes the pseudo or generalized inverse of a matrix [74]. Note that

when the rows of S are not linearly independent of each other, the matrix (SSH) is

singular and its generalized inverse is not unique. However, 1s and H1 are unique

[74].

Using (4-19) and (4-20), we get

|(X- ABS)(X- ABS)H

-(X ABS)(Hs + Hf)(X ABS)H
(4-21)
(xnHs ABS)(XHs ABS)H + TI

-(XHs ABS)(Xns ABS)HT-1 + II ITI,

where the matrix T is defined by

TA XnIXH. (4-22)

Note that the t t rank of H is L r. Hence, T has full rank M with probability one

when M + r < L.









Using (4-21) and Lemma 4 in [25] (or Theorem 1 in [27]), the solution of the

optimization problem in (4-18) can be approximated, for a large snapshot number

L, as follows.

/AML arg mintr[(Xns ABS)HT-(XIIs ABS)]
0/ (4-23)
= argmin I vec(T-XXHs) vec(T- ABS) |2,


where T-2 is the Hermitian square root of T-1 and || || denotes the Euclidean

vector norm .

By using Lemma 4.1 (with K = 1), we have

vec(T- ABS) = vecb(T-ABS) [ST (T-A)]/. (4-24)

Substituting (4-24) into (4-23) yields a quadratic function of 3, whose minimizer is

given by

OAML (FHF) -FHvec(T- XHs) (4-25)

where

F A ST (T-IA). (4-26)

To guarantee that the matrix FHF is invertible, we assume that the columns of

F are linearly independent of each other, which requires the linear independence

among the columns within each submatrix of A and among the rows within each

submatrix of S. However, the columns from different submatrices of A and the

rows from different submatrices of S can be linearly dependent on each other.

By Lemmas 4.2 and 4.1, respectively, we have that

fHr (SSH)T (AHT-1A) (4-27)


and


rHvec(T- Xns) = rHvecb(T- Xns) = vecb(AHT-1XSH).


(4-28)









Substituting (4-27) and (4-28) into (4-25) yields the AML estimate of /


/AML [(SSH)T (AHT-1A)]-vecb(AHT-1XSH). (4-29)

We note that in (4-29) when J = 1 the generalized Khatri-Rao product

reduces to the Kronecker product and vecb(-) reduces to vec(.). Then, using the

fact that vec(ABC) = (CT 0 A)vec(B), it can be readily shown that the AML

estimator in (4-29) reduces to


BGC-ML (AHT-1A)- AHT-1XSH(SSH)-1, (430)

which is the exact ML estimator based on the GC model [17] [18] [21] [22]. On the

other hand, when Nj = Kj = 1 for j = 1, 2, .. J, the generalized Khatri-Rao

product reduces to the Hadamard product and vecb(-) reduces to vecd(.). Hence,

(4-29) reduces to the AML estimator for the DGC model in [36], namely


fDGC-AML [(AHT -A) (SSH -vecd(AHT-1XS), (431)

where ) denotes the Hadamard product (elementwise multiplication) between two

matrices, and vecb(-) denotes a column vector formed by the diagonal elements of a

matrix.

4.4 Performance Analysis

In this section, we establish the theoretical properties of the AML estimator.

4.4.1 Bias Analysis

Substituting (4-1) into (4-29) and using Lemma 4.1, we have that


/AML 3 = [(SSH)T (AHT-1A)]-vecb(AHT-1ZSH). (4-32)

On the other hand, substituting (4-1) into (4-22) yields


T = ZnHZH.
S


(4-33)









From (4-33), we see that T is an even function of Z, and hence SML / is an odd

function of Z. Moreover, we have assumed that Z is a zero-mean Gaussian random

matrix. Using the statistical properties of the zero-mean Gaussian distribution, we

can readily show that the expectation of (4-32) is zero, i.e., the AML estimator

based on the BDGC model is unbiased.

4.4.2 Mean-Squared-Error (MSE) Analysis

Before calculating the MSE of the AML estimate, we first discuss the best

possible performance for any unbiased estimate of 3, i.e., the Cramir-Rao bound

(CRB). By utilizing Lemma 4.1, (4-1) can be rewritten as

vec(X) = (ST A)3 + vec(Z). (4-34)

Note that (4-34) is a linear statistical model with unknown noise covariance matrix

I 0 Q. It can be easily verified that the Fisher information matrix for this model is

a block-diagonal matrix with respect to / and Q [73]. Hence, the unknowns in Q

do not affect the CRB for 3. It follows that the CRB for 0 can be readily written

[73] as
CRB(3) [(ST A)H(I Q)-(ST A)]-1. (4-35)

Then, by using Lemma 4.2, (4-35) can be simplified as

CRB(3) = [(SSH)T (AHQ-1A)]-1. (4-36)

Next, we turn to the MSE analysis of )AML. The exact MSE of /AML is difficult

to determine, if not impossible. Herein, we provide an approximate expression for

the MSE of ,AML, which holds for a large snapshot number L.

From the theory of the linear statistical model [16], we know that LT is an

unbiased and consistent (in L) estimate of the noise covariance matrix Q. Hence,

for a large snapshot number L, the estimation error in (4-32) can be approximated









by

fAML P w CRB(3) vecd(AHQ-1ZSH). (437)

Using Lemmas 4.1 and 4.2 (and viewing 0 as a special case of *), it follows

from (4-37) that

MSE(CAML) AE[(fAML f)(AML 3)H1

,CRB(3) E[vecb(AHQ-1ZSH) vecb(AH -1ZSH)H] CRB(3)

=CRB(3) [S* o (AHQ-1)] E[vec(Z)vec(Z)H] [ST O (AHQ-1)H] CRB(3)

=CRB(3) [S* o (AHQ-1)] (I 0 Q) [ST O (Q-'A)] CRB(3)

CRB(3),

(4-38)

where (.)* denotes the complex conjugate.

We see from (4-38) that the AML estimate of f3 is .,-mptotically statistically

efficient for a large snapshot number L. This theoretical result is illustrated

numerically in Section 4.5. It can be easily verified that the CRB for the GC model

in Equation (13) of [22] and the CRB for the DGC model in Equation (24) of [36])

are special cases of (4-36).

4.5 Numerical Results

In this section, numerical examples illustrating the application of the BDGC

model in wireless communications are presented to demonstrate the performance of

the AML estimator and to verify the theoretical results in Section 4.4.

We consider a DS-CDMA receiver with a uniform linear array consisting of

M = 6 antennas with half-wavelength spacing between .,.i i:'ent antennas. The

array elements are assumed to be omni-directional. Consider J = 2 transmitters,

whose signals are modulated by two different pseudo-noise (PN) sequences [81],

respectively. The PN sequences are known a priori by the receiver. Suppose that

the signal from the 1st transmitter arrives at the array through N1 = 2 paths









with directions of arrival (DOAs) of 01 = -100 and 02 = 5, while the signal

from the 2nd transmitter arrives at the array through N2 = 1 path with DOA of

03 = 10. We assume that the DOAs have been estimated accurately using, e.g.,

the method of direction estimation (\ ODE) algorithm [82] [83], and therefore they

can be considered to be known. The problem of interest is to estimate the unknown

complex amplitudes f0 and 32 for the two paths of the 1st transmitter, and 3 for

the 2nd transmitter, which contain the transmitted information. The estimates of

{ pj}j can be used for symbol detection. The error matrix Z contains the noise
and interference, whose columns are assumed to be i.i.d. circularly symmetric

complex Gaussian random vectors with zero-mean and an unknown covariance

matrix Q given by

Qn" p0.91- 1y ( (4-39)

where p = 1/SNR with SNR being the signal-to-noise ratio and Qm,, denotes the

(m, n)th element of Q.
Let

A, [a(01) a(02)] and A2 [a(03)], (4-40)

where a(0) = [1 e-jwsin(O) e-j2sin(O) ... e-j(M-1)sin(O)]T is the steering vector of the

signal with DOA of 0. Let


B1= [li 2]T and B2 = [3]. (4 41)

Let Si e ClxL and S2 C C1xL be the known PN sequences for the two transmitters,

respectively, with L being the length of each PN sequence. Under the previous

assumptions we can describe the received signal using a BDGC model with M = 6,

J = 2, N1 = 2, N2 = 1 and K1 = K2 = 1. Hence, the AML estimator in (4-29) can

be applied directly.

We present the MSE of the AML estimator in (4-29) as well as the corresponding

CRB in (4-36). For comparison purposes, we also show the performances of the GC









method, the least squares (LS) and the exact ML estimators. In the GC method,

we estimate the full matrix of B using (4-30) [17] [18] [21] [22], and then pick up

the corresponding block-diagonal submatrices of BGC-ML as the estimate of Bj. The

LS estimator assumes that the interference and noise vectors in Z are uncorrelated

both spatially and temporally, and can be obtained immediately from (4-29) by

replacing T there by an identity matrix, i.e.,


/3LS [(SSH)T (AHA)]-vecb(AHXSH). (4-42)

The exact ML estimates are obtained by applying the cyclic maximization (C'l)

technique [84] to the cost function in (4-18) with respect to various Bj. In

each step of this iterative algorithm, we assume that all regression coefficient

submatrices, except for Bk, are known, which means that Bk can be readily

estimated by using (4-30). We use the AML estimates as the initial values of the

regression coefficient matrices in the exact ML estimator. The empirical estimation

performances are all obtained from 500 Monte-Carlo simulations.

Figs. 4-1 and 4-1 show the estimation performance as functions of the length

of the PN sequences L and the SNR, respectively. The SNR is fixed at 10 dB in

Fig. 4-1 while L is fixed to be 128 in Fig. 4-2. Note that the ML estimator based

on the GC model has a relatively poor estimation performance. This method

ignores the a priori information about the block-diagonal structure of the complex

amplitude matrix B, which doubles the number of unknowns and hence results

in much larger estimation variances. Note also that due to the (quasi-)orthogonal

property of the PN sequences we have Rss = LI (approximately), and hence the

CRB in (4-36) reduces to *[I (AHQ-1A)]-1 in this example. As shown in Figs.

4-1(a)-4-1(c), the CRB decreases linearly as the snapshot number L increases,

when both CRB and L are presented in log-scales. From Figs. 4-1(a) 4-1(c), we

can also see that the AML estimator achieves a very similar performance as the















10 CRB 10 CRB


10 0 10
1 L- --L- oI




S10' -., 10 -


10 10-
10 1 10101 102
Snapshot Number L Snapshot Number L

(a) (b)
102
-y GC
-* BDGC-LS
S- BDGC-AML
-m BDGC-ML
10 CRB


U10
to0-,

10 -



101 102
Snapshot Number L

(c)

Figure 4-1: Empirical MSE's and the CRB versus L when SNR 10 dB. (a) For

31, (b) for /3, and (c) for 33.


exact ML; and it outperforms LS and GC significantly. As predicted theoretically,

both the exact ML and AML are .,-i!', 1'ill ically statistically efficient for a large

snapshot number, and they approach the corresponding CRB rapidly. From Fig. 4

2, we note that AML and the exact ML provide almost identical performances, and

their estimates are very close to the CRB for the entire range of SNR considered.

Again, AML outperforms the LS and GC estimators significantly.

4.6 Conclusions

We have presented an approximate maximum likelihood estimator for the

block-diagonal growth-curve model where the unknown regression coefficient











I l
-V GC
-0 BDGC-LS
*-- BDGC-AML
r -B BDGC-ML
C- RB


-30 -25 -20 -15 -10 -5
SNR(dB)

(a)


0 5


25 -20 -15 -10 -5 0
SNR(dB)

(b)


SNR (dB)

(c)

Figure 4-2: Empirical MSE's and the CRB versus SNR (dB) when L = 128. (a)
For /3, (b) for 32, and (c) for /33.



matrix is constrained to be block-diagonal. Via a theoretical analysis, we have

shown that the AML estimator is unbiased and .,-!' ,, I..1l ically statistically efficient

for a large snapshot number. We have applied the AML estimator to a complex

amplitude estimation problem in wireless communications. The numerical examples

provide compelling evidence that the AML method can achieve excellent estimation

accuracy.















CHAPTER 5
ITERATIVE GENERALIZED LIKELIHOOD RATIO TEST FOR MIMO RADAR

5.1 Introduction and Signal Model

In C'!I pter 1, we have discussed several multiple-input multiple-output

(\!I\ O) radar systems. According to their antenna configurations, the MIMO

radars discussed can be grouped into two classes. One is the conventional radar

array, in which both transmitting and receiving antennas are closely spaced

for coherent transmission and detection [44] [57]. The other is the diverse

antenna configuration, where the antennas are separated far away from each

other to achieve spatial diversity gain [40] [43]. To reap the benefits of both

schemes, in this chapter, we consider a general antenna configuration, i.e., both

the transmitting and receiving antenna arrays consist of several well-separated

subarrays with each subarray containing closely-spaced antennas. We establish

the growth-curve models in C'!i lters 2 4 and devise several estimators for the

proposed MIMO radar system.

Consider a narrow-band MIMO radar system with N and MI subarrays

for transmitting and receiving, respectively. The nth transmit and mth receive

subarrays have, respectively, N, and i., closely-spaced antennas, n = 1, 2, N,

m 1, 2, .. M. We assume that the subarrays are sufficiently separated, and

hence for each target its radar cross-sections (RCS) for different transmit and

receive subarray pairs are statistically independent of each other. Let v,(0) and

a,(O) be the steering vectors of the nth transmitting subarray and the mth

receiving subarray, respectively, where 0 denotes the target location parameter, for

example its angular location. Let the rows of ), be the waveforms transmitted

from the antennas of the nth transmit subarray. We assume that the arrival time is









known. Then, the signal received by the mth subarray due to the reflection of the

target at 0 can be written as

N

n=l

where f, r,o is the complex amplitude proportional to the RCS for the (m, n)th

receive and transmit subarray pair and for the target at the location 0, and the

matrix Z, denotes the residual term containing the unmodelled noise, interference

from targets other than 0 and at other range bins, and intentional or unintentional

jamming. For notational simplicity, we will not show explicitly the dependence of

Z, on 0.

Let

X [XT .. X ]T e CMxL, (5-2)

A(0) Di ,[at(0),... a(0)] e CMxM, (5-3)

V(0) Diag[vi(0), ... vN(0)] e CNXN, (54)

and

4 = [T ... T] CNxL, (5-5)

where M = 1M + .. + MM and N N= + ... + N are the total numbers of

receive and transmit antennas, respectively, L is the number of data samples of the

transmitted waveforms, (.)T denotes the transpose operator, and Diag(al, .- ay)

is a block-diagonal matrix with al, aa being its diagonal submatrices. Then,

(5-1) can be readily rewritten in the growth-curve (GC) model in C'! plter 2, i.e.,


X = A(0)BoS(0) + Z (5-6)


where the (m, n)th element of the M x N matrix BE is f3m,o,0 Z is defined similarly

to X in (5-2), and the rows of S(0) are the reflected waveforms by the target at









location 0, i.e.,

S(0) = VT(). (5-7)

Note that when N = AM 1, the signal model in (5-6) reduces to the

MIMO radar model in [55] [57], whereas when N = N and M = M it reduces

to the diversity data model in [40] [42]. Based on this data model, We below

propose two classes of nonparametric methods, i.e., spatial spectral estimation and

generalized likelihood ratio test (GLRT), for target detection and localization.

The remainder of this C'! plter is organized as follows. In Section 5.2, we

introduce several adaptive spatial spectral estimators including Capon [60]

and APES [23]. In Section 5.3, we describe a generalized likelihood ratio test

(GLRT) and a conditional generalized likelihood ratio test (cGLRT), and we

then propose an iterative GLRT (iGLRT) procedure for target detection and

parameter estimation. Numerical examples are provided in Section 5.4. We first

compare the Cram6r-Rao bounds (CRBs) for MIMO radars with different antenna

configurations, and then present the detection and localization performance of the

proposed methods. Finally, Section 5.5 contains the conclusions.

5.2 Several Spatial Spectral Estimators

We introduce several spatial spectral estimators for the proposed MIMO

radar system. We use these methods to estimate the complex amplitudes in Bo for

each 0 of interest from the observed data matrix X. The Frobenius norm of the

so-obtained B0 forms a spatial spectrum in the 1D case or a radar image in the 2D

case. We can then estimate the number of targets and their locations by searching

for the peaks in the so-obtained spectrum (or image).

A simple way to estimate B0 in (5-6) is via the Least-Squares (LS) method,

i.e.,


BLS, = [AH(0)A(O)]-IA(O)XSH(0)[S(O)SH(O)]-1


(5-8)









where (.)H denotes the conjugate transpose. However, as any other data-

independent beamforming-type method, the LS method suffers from high-

sidelobes and low resolution. In the presence of strong interference and jamming,

the method completely fails to work. We introduce below two adaptive spatial

spectral estimation approaches that offer much higher resolution and interference

suppression capabilities.

5.2.1 Capon

The Capon estimator for Be in (5-6) consists of two main steps [60] [85] [22].

The first is a generalized Capon beamforming step. The second is a LS estimation

step, which involves basically a matched filtering to the known waveform S(O).

The generalized Capon beamformer can be formulated as


mintr(WHRW) subject to WHA(0) I, (5-9)
w

where W E CMXM is the weighting matrix used to achieve noise, interference and

jamming suppression while keeping the desired signal undistorted, tr(-) denotes the

trace of a matrix, and

R = XXH (5-10)
L
is the sample covariance matrix with L being the number of data samples.

Solving the optimization problem in (5-9), we can readily have


WCapon R--A(O)[AH(OR--1A(O)]-1. (5-11)

By using (5-11) and (5-6), the output of the Capon beamformer can be written as


[AH(O)R-1A(O)]-1AH(O)-X = BoS(O)+[AH(O)R-1A(0)1]-AH(O)R-1Z. (5-12)









By applying the least-squares (LS) method to (5-12), the Capon estimate of B0
follows readily, i.e.,

Bcapon,0 [AH(O)R-1A(O)]-AH ()R-1XSH(0)[S(0)SH()]-1. (5-13)

5.2.2 APES

The generalized APES method is a straightforward extension of the APES

method [23] [24], which can be formulated as

min I| WHX BeS(O) 12 subject to WHA(0) = I, (5-14)
W,B

where I|| || denotes the Frobenius norm, and W is the weighting matrix.

Minimizing the cost function in (5-14) with respect to Be yields


BAPES, WHXSH(O)[S(O)SH(O)]-1. (5-15)

Then the optimization problem reduces to

mintr(WHQW) subject to WHA(0) = I, (5-16)

with
1
Q R -XSH(o)[S()SH()-1S()XH. (5-17)
L
For notional simplicity, we have omitted the dependence of Q on 0.

Solving the optimization problem of (5-16) gives the generalized APES

beamformer weighting matrix


WAPES,O [AH(0)QA(0)]- 1Q A(0). (5-18)

Inserting (5-18) in (5-15), we readily get the APES estimate of Be as


BAPES, = [AH(O)Q-1A(O)]-1AH(O)Q-1XSH(O)[S(O)SH(O)]-1


(5-19)









Interestingly, we note that (5-19) has the same form as the ML estimate in

C'!I pter 2. However, the APES estimate is derived based on the beamforming

method, and, unlike the ML in C'!i pter 2, it does not need probability density

function (pdf) of Z.

5.3 Generalized Likelihood Ratio Test

Generalized likelihood ratio test (GLRT) has been used widely for target

detection and localization. We derive below a GLRT and a conditional generalized

likelihood ratio test (cGLRT) for the proposed MIMO radar, and then propose an

iterative GLRT (iGLRT) procedure for improved performance.

5.3.1 Generalized Likelihood Ratio Test (GLRT)

Throughout this section, we assume that the columns of the interference

and noise term Z in (5-6) are independently and identically distributed (i.i.d.)

circularly symmetric complex Gaussian random vectors with mean zero and an

unknown covariance matrix Q.

Consider the following hypothesis test problem

Ho : X Z
(5-20)
H: X A(0)BoS(0) + Z,

i.e., we want to test if there exists a target at location 0 or not. Similarly to [65]

and [86], we define a generalized likelihood ratio (GLR)

t maxQ f(X Ho) L
p(L) maxB0,Q f( H) (5-21)
maXBo,Q f(X|Hi) \L

where f(X|Hi) (i = 0, 1) is the pdf of X under the hypothesis Hi. From (5-21),

we note that the value of the GLR, p(O), lies between 0 and 1. If there is a target

at a location 0 of interest, we have maxB,Q f(XH1I) > maxQ f(XlHo), i.e., p t 1;

otherwise p 0.









Under Hypothesis Ho, we have


f(XHo) LM L exp{tr(Q-XXH)}, (5-22)
wLM QIL

where I denotes the determinant of a matrix. Maximizing (5-22) with respect to

Q yields

max f(X|Ho) (re)-LMI -L, (5-23)
Q
where R is defined in (5-10).

Similarly, under Hypothesis Hi, we have

1
f(Xl|H) 7MI L exptr{Q-[X A(O)BoS(O)][X A(0)BoS()]H}. (5-24)

Maximizing (5-24) with respect to Q yields

1 -L
maxf(X |H) (re)-LM -[X A(0)BoS(0)][X A()BoS()]H (5-25)
Q L

Hence, the optimization problem in the denominator of (5-21) reduces to


min [X A(0)BoS(0)][X A()BoS()]H .(5-26)
B0 L








Following [21] and [22] and dropping the dependence of A, S and B on 0 for
notional convenience, we have

-[X- ABS][X-ABSH
L
S[ABo XSH(SSH)-] (SSH) [ABo XSH(SSH)-l] H
Li
= |Q I+ Q [AB XSH(SSH) -](SSH)[ABo XSH(SSH)-1]Q-

l |Q I+ L(SSH) [ABo XSH(SSH)l]H
Q- [ABo- XSH(SSH)-1](SSH)2 (5-27)
S|Q| I (SSH)- SXH [Q -_ --A(AHQ -A)- AHQ-l]XSH(SSH)-
+(SSH) [Bo (AHQ-1A)l-AH Q-XSH(SS-H)I H(AH -A)

[Bo (AHQ- A)-IAHQ-IXSH(SSH)-I](SSH

> |Q| I (SSH)- SXH [Q- -1A(AHQ -A) -AH -1]

XSH(SSH)- (5-28)

where Q is defined in (5-17). To get (5-27), we have used the fact that |I + XY
II + YX| [74], and the equality in (5-28) holds when B equates to the APES
estimate in (5-19). Note that

|Q I (SSH) -SXH [Q-_ -1A(AH Q-A)-IAHQ-1]XSH(SSH)-
= |Q I+ [Q-- Q- A(AHQ1A)- AHQ-R Q)
I- A(AHQ-1A)-'AHQ-1(R- Q)

IRl I- AH(AHQ-1A)-1AH(Q-1 R -1)
Rl I (AHQ-1A)- (AHR-1A) (5-29)

From (5-25), (5-28) and (5-29), it follows that

maxf(X|Hi) = (re)-LM RI-LIAH(0)Q-1A()ILI AH()R-1A(O) -L. (5-30)
Bo,Q









Substituting (5-23) and (5-30) into (5-21) yields

AH(O)R -A(0)|
p(0) A-1 5-Ai( (5-31)
|AH Q-IA(0)|
We remark that when there are multiple targets, and the number of targets
(-w K) are known a priori, the GLRT in (5-31) can be extended to a multivariate

counterpart by considering the following hypothesis testing problem

{ Ho : X = Z
Ho X (5-32)
HK: X- E 1 A(0k)BoS(0k)+Z.

As a parametric method, this multivariate GLRT can provide better target

detection and parameter estimation performance than its univariate counterpart.

However, the multivariate GLRT is computationally intensive due to the fact that

it needs to search in the K-dimensional parameter space {Ok} 1. Moreover, the
number of targets is hardly known a priori in practice.

We propose below an iterative GLRT (iGLRT), which require only one-

dimensional search (like the univariate GLRT), but provides a target detection and

parameter estimation performance close to the multivariate GLRT.

5.3.2 Conditional Generalized Likelihood Ratio Test (cGLRT)

Before we describe the iGLRT procedure, we first consider the following

hypothesis testing problem, referred to as the conditional generalized likelihood

ratio test (cGLRT). Suppose that we have known (or detected) p targets at the

estimated locations {Ok} =1, and we want to determine if there are any additional

targets. This problem can be formulated in the following hypothesis testing

problem
H : X = E A(Ok)B S(Ok) + Z
SH+1 : X A(0)BoS(0) + EL 1 A(Ok)B0 S(Ok) + Z.









Note that both the equations in (5-33) are in the form of the block diagonal
growth curve (BDGC) model studied in C'!i pter 4. For convenience, we rewrite

(5-33) as
H, : X = A,BS + Z
(5-34)
Hp+1 : X Ap+1Bp+1Sp + Z,
where

B, Diag(B01, ., Bp ) (5-35)

Ap [A(01) A(0)], (5-36)

S, [ST(0^) ... ST(,)]T, (5-37)

Bp+ Diag(Bo, B, .. B ) (5-38)

A,+I = [A(0) A(01) ... A(0)], (5-39)

Sp+i [ST(O) ST(01) ... ST(P)]T, (5-40)

and Diag(Qi, .. QK) denotes a block diagonal matrix formed from fi, QK.
Similarly, we define a conditional generalized likelihood ratio (cGLR)


maxbp,, f (X|H )


where f(X|He) is the pdf of X under the HZ hypothesis, and Q is the covariance
matrix of the columns of Z.
We first consider the optimization problem of the numerator in (5-41).
Maximizing f(X Hp) with respect to Q yields

maxf(X Hp) (wre)-ML -(X ApBSp))(X- ApBpS)H (5-42)
Q L

Hence, the optimization problem reduces to

min (X ApBpSp)(X- ApBpSp)H with B = Diag(B 0, B ). (5 43)









The optimization problem of (5-43) does not appear to admit a closed-form
solution, due to the constraint that B, is a block-diagonal matrix. Herein, we
adopt a technique used in C'!h pters 3 and 4 to get approximate closed-form
solution.
Note that

-(X AApBS)(X ApBpSp)H

S -(X ABpS,)(Hn + n~)(X AP SP)H

S -(Xln A P),, 8,,,,pBpS) + QP84
9Xt ABS)A(XH A B S)H +I
S -(XH-% AB S)(XH A% BsA)HQ- l+I Q,, (5-44)

where

n1H = H(S H)-S,, HI- I- (5-45)

and
1
Qp XII XH (5-46)

with (.)- denoting the generalized matrix inverse.
Consider the idempotent matrices H1g and H Assume that the number of
data samples is large enough, i.e., L > Np. Note that S, is an Np x L matrix.
Hence, we have

rank(nH) < Np and rank(H ) > L Np, (5-47)

with rank(.) denoting the rank of a matrix. Then, we have


Qp O(1),


(5-48)









-(xnHp APBS)(xn~ ABS,)
1 1T o
(X ABS,) (X ABpSp)T 1 (5-49)

Therefore, we get


L-(X H APBPS)(XnH APBPSP)Q 0 ( L 1. (5-50)

Let {Ai}l" be the eigenvalues of the matrix in (5-50), which obviously satisfy that

0 < Ai < 1. By using some matrix manipulations, we obtain


-(Xn, A,B,S,)(Xn, ABS Q;1 + I
M M
H(l+ Aj) t1+ A
i= 1 i= 1
1 + F1
= 1+ -tr (Xr -AB )(XH P-A,BPS)Qp ;
1 1 1
S1+ || vec(Qp, XHs) vec(Qp ABS,) |2, (5-51)

where || || and vec(.) denote the Euclidean norm and vectorization operator

(stacking the columns of a matrix on top of each other), respectively, and Qp 2 is

the Hermitian square root of QOP. In (5-51), we have omitted the high-order terms

of {Ai} for the approximation.

Hence, for a large number of data samples, the optimization problem in (5-43)

can be approximated as

1 1
mmin vec(Qp XnXH) vec(Qp ApBpSp) |2
Bp
with B, Diag(Bl, ... B0). (5-52)

To solve the above optimization problem, we need the block-matrix operations

and lemmas 4.1 and 4.2 in C'!I pter 4. Throughout this chapter, the partitioned

matrix operation are all based on the partitionings in (5-35) (5-40).









Now, let


P/ = vecb(Bp),


(5-53)


where vecb(.) denotes the block-diagonal vectorization operator (Definition 4.1),
i.e.,


p-, [vecT(B) ... vecT(Bp)]T.


(5-54)


By using Lemma 4.1 (with K = 1 and J= p) of C'!I iter 4, we obtain


vec(Qp` ApBpSp)


[S (Q7 'A)]/3 rA3p,


where denotes the generalized Khatri-Rao product ( Definition 4.2). Hence,

|| vec(Q;p x H ) vec(Q;pApBpSp) 12

II vec(QP XnH ) p- rF 2
>1 H( 1 v ~HnH -1H 1vec
> I| vec(Q, 'Xll) I| -vec (q xn ) (> r ) r> vec(Q~ 'Xn)


Ltrl(QfR) LM vecbH (A -XSH)

[(SSH)T (A'HQ 1A,)] vecb(AH' 1XSH),
p \p P P P


where we have used Lemmas 4.1 and 4.2, and the equality holds when


= (rfHf) r vec(Qp 2XHn ).


By using (5-42), (5-44), (5-51) and (5-56), it follows that


max f (X|H )]
QBp


L 1
(7e) 01, "'", ) Qp


(5-58)


g(01, Op)


1 M + tr(Q 1R)


vecbH H -1XS'H)
(P qP xP


(5-59)


[(SSH)T (AHQA,)] -k vecb(AH QplXSH).


(5-55)


(5-56)


(5-57)


where








Similarly, we have


1^ 1
maxf(X|Hpi)
Q,Bp+I] (rge)M g(, 1,' ) p)I Q+1

where Qp+l and g(0, 01,.. 0,) are defined similarly to Qp in (5-46) and

g(O1, *- p) in (5-59), respectively.
Substituting (5-58) and (5-60) into (5-41) yields the conditional GLR

p(0{0j}) { (1 9( 0 ',) Q,,
g(Ol, p) |OpQ


(5-60)








(5-61)


5.3.3 Iterative Generalized Likelihood Ratio Test (iGLRT)
The basic idea of the iterative generalized likelihood ratio test (iGLRT)
is to detect and localize targets sequentially. In each step of the iteration, the
results from the previous iterations and steps are exploited for the detection and
localization of new targets by calculating cGLR. Specifically, we first perform
GLRT to get the location of the dominant target, and the following targets are
detected and localized by using cGLRT conditioned on the most recently available
estimates. The detailed steps of iGLRT are described as follows.
Step I:
Calculate p(O) in (5 31) for each 0.
Compare p(0) to a threshold ('.', po). If p(O) < po for all 0, then Stop;
otherwise, 01 = argmaxo p(O), go to Step II.
Step II: For k = 1, 2, .. do the following substeps:
Calculate p(0 {Oi}~ ) in (5-61) for each 0.

If p(01{0i}i 1) < po for all 0, then go to Step III; otherwise, Ok+1
argmax p(0 ({i) ) -
Step III: Repeat the following substeps until convergence









for k = 1, 2, .. K (suppose that K i .ii. I are detected

in Steps I and II)

Calculate p(0O{0i}i k) for each 0.

Update Ok by argmaxop {0 (i}i#k)-

Once the locations of the targets are determined, the amplitudes of the

reflected signals can be estimated by using the AML estimator in [37], i.e.,


k [(A Q A ) (S k)] ) vecb(A Q S), (5-62)

where A SK and QK are defined similarly to Ap, Ap and Qp in (5-37), (5-38)

and (5-46), respectively.

We note that Step III of the above iGLRT algorithm actually minimize

the function g(01, ,0 ) with respect to {Ok} k by using the cyclic minimization

(Ci\!) technique [84]. Under a mild condition, i.e., L > NK, we have g(01, 0) >

0. Furthermore, we know that the C\ I algorithm monotonically decreases the cost

function. Hence the iGLRT algorithm is convergent. When K is the true number

of targets, iGLRT reduces to an approximate parametricc) maximum likelihood

estimator. As we will show via numerical examples, the mean-squared-error (i\l ;)

of the estimate of iGLRT approaches the corresponding Cram6r-Rao bound (CRB)

for a large number of data samples. On the other hand, we note that iGLRT needs

only one-dimensional search and hence is computationally efficient.

5.4 Numerical Examples

5.4.1 Cram6r-Rao Bound

We first study the Cramir-Rao bound under various antenna configurations.

Consider a MIMO radar system with M = N = 8 antennas for transmitting

and receiving. We assume that the receiving and transmitting antennas are









grouped into multiple subarrays (each being a uniform linear array with half-

wavelength spacing between .i1i i:ent elements). We consider the following antenna

configuration schemes.

MIMO Radar A: 1 subarray with 8 antennas for transmitting and receiving;

MIMO Radar B: 2 subarrays each with 4 antennas for transmitting, and 1

subarray with 8 antennas for receiving;

MIMO Radar C: 8 subarrays each with 1 antenna for transmitting, and 1

subarray with 8 antennas for receiving;

MIMO Radar D: 2 subarrays each with 4 antennas for transmitting and

receiving.

We assume that the transmitted waveforms are linearly orthogonal to each other

and the total transmitted power is fixed to be 1, i.e., R = _I1.

We consider a scenario in which K = 3 targets are located at 01 = -40,

02 = -4 and 03 = 00, and the elements of {Be }3_ are independently and

identically distributed (i.i.d.) circularly symmetric complex Gaussian random

variables with zero mean and unit variance. There is a strong jammer at 100 with

amplitude 100, i.e., 40 dB above the reflected signals. The received signal has

L = 128 snapshots and is corrupted by a zero-mean spatially colored Gaussian

noise with an unknown covariance matrix. The (p, q)th element of the unknown

noise covariance matrix is N 0.901P-qe J Figs. 5-1(a) and 5-1 (b) show the

cumulative density functions (CDFs) of the CRBs for MIMO radar with various

antenna configurations when SNR 20 dB. (The CRB of 02 is similar to that of 03

and hence is not shown.) The CDFs are obtained by 2000 Monte-Carlo trials. In

each trial, we generate the elements of {Bo }3 randomly, and then calculate the

corresponding CRBs using (C-18) given by in the Appendix C. For comparison

purposes, we also provide the CDF of the phased-array (single-input multiple-

output) counterpart, i.e., the special case of the above MIMO radar when N = 1,

























10 10 10 10
S- Phased-Array
--- MIMO RadarA
MIMO Radar B
s MIMO Radar C
MIMO Radar D





10' 105 104 10
CRB


z-?--"" :--



Phased-Array
-- MIMO Radar A
-6- MIMO Radar B
,' MIMO Radar C
S MIMO Radar D





103 102 10 10o
CRB


Figure 5-1: Cumulative density functions of the Cramdr-Rao bounds for (a) 01 and
(b) 03.



with the same total transmission power. As expected, the MIMO radar provides

much better performance than the phased-array counterpart. Due to the fading

effect of the elements of {Bo} 1,, the CRB of MIMO Radar A varies within a

large range. Within a 95'. confidence interval (i.e., when CDF varies from 2.5'. to

97.5'.), its CRB for 01 varies approximately from 5 x 10-7 to 5 x 10-5. The CRBs

for MIMO radar C varies within a small range.

To evaluate the CRB performance, we define an outage CRB [43] for a given

probability p, denoted by CRBp, as


P(CRB > CRB) = p.


(563)


Figs. 5-2(a) 5-2(d) show the outage CRBo.o0 and CRBo.1 of 01 and 03, as

functions of SNR. As expected, the SNR gains depend on the probability p. As we

can see, when p = 0.01, MIMO radar C outperforms the other radar configurations,

and provides around 20 dB and 12 dB improvements in terms of SNR compared to

the phased-array and MIMO radar A, respectively. On the other hand, Fig. 5-2(d)

shows that MIMO radars A and B outperform others when p = 0.1.
































(a) (b)


SNR (dB)
(c)

Figure 5-2: Outage CRB versus SNR. (a) CRBo.o0 for 01,
CRBo.1 for 01, and (d) CRBo.1 for 02.


SNR(dB)
(d)

(b) CRBo.ol for 02, (C)









5.4.2 Target Detection and Localization

We focus below on MIMO radar B, i.e., a MIMO radar system with 2

subarrays (each with 4 antennas) for transmitting and 1 subarray (with 8 antennas)

for receiving.

We first consider a scenario in which 3 targets are located at 01 = -40,

02 = -200 and 03 = 0 with the corresponding elements in Bo,, B02 and B03 being

fixed to 2, 2 and 1, respectively. The other simulation parameters are the same as

for Fig. 5-1. The Frobenius norm of the spatial spectral estimates of B0 versus 0,

obtained by using LS, Capon and APES are given by Figs. 5-3(a) 5-3(c). For

comparison purposes, we show the true spatial spectrum via dashed lines in these

figures. As seen from Fig. 5-3, the LS method suffers from high-sidelobes and poor

resolution problems. Due to the presence of the strong jamming signal, the LS

estimator fails to work properly. Capon and APES possess excellent interference

and jamming suppression capabilities. The Capon method gives very narrow peaks

around the target locations. However, the Capon estimates of Bo,, B02 and B03

are biased downward. The APES method gives more accurate estimates around

the target locations but its resolution is worse than that of Capon. Note that in

Figs. 5-3(a), 5-3(b) and 5-3(c), a false peak occurs at 0 = 100 due to the presence

of the strong jammer. Despite the fact that the jammer waveform is statistically

independent of the waveforms transmitted by the MIMO radar, a false peak still

exists since the jammer is 40 dB stronger than the weakest target and the number

of data samples is finite. Figs. 5-3(d) and 5-3(e) give the GLRT, and the iGLRT

results, as functions of the target location parameter 0. For convenience, in Fig.

5-3(e), we have included all cGLR functions obtained by iGLRT, each indicating

one target. As expected, we get high GLRs (cGLRs) at the target locations and

low GLRs (cGLRs) at other locations including the jammer location. By comparing

the GLR with a threshold, the false peak due to the strong jammer can be readily





































Angle (deg)

(a)


Angle (deg)

(c)





8

6

4

2

c 0 1


5-

2

5-



1 1
5-


-60 -40 -20 0 20 40 6(
Angle (deg)

(b)
III I I


041-


-60 -40


-20 0
Angle (deg)

(d)


20 40


-BU -4U -2U u
Angle (deg)

(e)


and GLR and cGLR Pseudo-Spectra ,when 01 = -400,

LS, (b) Capon, (c) APES, (d) GLRT, and (e) iGLRT.


06


Figure 5-3: Spatial spectra,

02 = -200, and 03 = 00. (a)


20 4U









detected and rejected, and a correct estimate of the number of the targets can be

obtained by both methods.

Next we consider a more challenging example where 02 is -4 while all the

other simulation parameters are the same as before. As shown in Fig. 5-4(c),

in this example, the APES, Capon and GLRT methods fail to resolve the two

closely spaced targets at 02 -4 and 03 00. On the other hand, iGLRT gives

well-resolved peaks around the true target locations. To illustrate the procedure

of the iGLRT algorithm, we give the GLR, and cGLRs obtained in Steps I and II

of iGLRT in Figs. 5-5(a) 5-5(d). Figs. 5-5(a) and 5-5(b) show the GLR p(0)

and the cGLR p(0131), respectively, where 01 is the estimated location of target 1

from p(O). As we can see, there is no peak at around 03 = 0 in both figures. Yet

a clear peak is shown in p(0810, 02) in Fig. 5-5(c), which indicates the existence

and location of target 3. The cGLR p( 01, 02, 03) in Fig. 5-5(d) shows that no

additional target exists other than the targets at 01, 82 and 03. In other words, the

iGLRT method correctly estimates the number of targets to be 3.

Now we consider the elements in B0,, B0, and B03 as i.i.d complex Gaussian

random variables with mean zero and unit variance. The other parameters are the

same as those in Fig. 5-6. The Figs. 5-6(a) and 5-6(b) present the CDFs of the

MSEs of 01 and 03 as well as the CRBs, when SNR = 20 dB and L = 128. As we

can see, the MSEs of the iGLRT are very close to the corresponding CRBs. Figs.

5-7(a) and 5-7(b) show the outage MSEo.1 and CRBo.1 when p = 0.1 as functions

of SNR when L = 128. Again, the MSEs are very close to the corresponding the

CRBs, and decreases almost linearly as SNR increases. Fig. 5-8 givse the outage

MSEo.1 and CRBo.1 as functions of L when SNR 20 dB. As expected, the outage

MSEo.1 approaches the corresponding CRBo.1 as L increases.












































60 -40 -20 0 20 40 6(
Angle (deg)

(a)


-60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40
Angle (deg) Angle (deg)

(c) (d)

Figure 5-4: Spatial spectra, and GLR and cGLR Pseudo-Spectra, when 01

02 -40, and 03 00. (a) Capon, (b) APES, (c) GLRT, and (d) iGLRT.


8-

6

4

2
I I ,


0' Iil lIa


400,


3, ,-, ,

















1


8-

6


4-

2


-0 -40 -20 0 20 40 6(
Angle (deg)

(a)





8-


6-

4

2

0 ,-I -


1
8-

6


4 -

2


-60 -40 -20 0 20 40 61
Angle (deg)

(b)





8 -


6

4


2-

0


-60 -40 -20 20 0 40 60 60 -40 -20 0 20 40 60
Angle (deg) Angle (deg)

(c) (d)


Figure 5-5: GLR and cGLR Pseudo-Spectra obtained in Steps I and II of iGLRT,

when 01 -400, 02 -40, and 03 00. (a) p(O), (b) p(0811), (c) p(I081,02), and

(d) p(0811,82, 3).


10- 10 10 10 10 10 10 10 10u 10
MSE MSE

(a) (b)


Figure 5-6: Cumulative density functions of the CRBs and MSEs for (a) 01 and (b)

03.







88










10 2 10
I-ORB II-OCRB
-- IGLRT I IGLRT|


10 ~ 10 .





10 10




0 5 10 15 20 25 30 0 5 10 15 20 25 30
SNR (dB) SNR (dB)

(a) (b)

Figure 5-7: Outage CRBo.1 and MSEo.1 versus SNR for (a) 01 and (b) 03.















-,-- j CRB









10 -I 110 -





102 102
L L

(a) (b)

Figure 5-8: Outage CRBo.1 and MSEo.1 versus SNR for (a) 01 and (b) 03.