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Adaptive Radar Signal Processing

Permanent Link: http://ufdc.ufl.edu/UFE0041432/00001

Material Information

Title: Adaptive Radar Signal Processing
Physical Description: 1 online resource (119 p.)
Language: english
Creator: Roberts, William
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: adaptive, array, beamforming, mimo, phased, processing, radar, receive, signal, transmit
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Probing waveform synthesis and receive filter design play crucial roles in achievable performance for many radar applications. A flexible receive filter design approach, at the costs of lower signal-to-noise ratio (SNR) and higher computational complexity, can be used to compensate for missing features of the probing waveforms. A well synthesized waveform, meaning one with good auto-correlation properties, can reduce computational burden at the receiver and improve performance. Herein, we investigate various signal processing strategies to improve the performance of modern day radar systems. We highlight the interplay between waveform synthesis and receiver design. We consider both single antenna systems (referred to as single-input single-output, or SISO, radar) as well as MIMO (multiple-input multiple-output) radar schemes. For SISO radar, we review a novel, cyclic approach to waveform design, and then compare the merit factors of these waveforms to other well-known sequences. Furthermore, we overview several advanced techniques for receiver design, including data-independent instrumental variables (IV) filters, a data-adaptive iterative adaptive approach (IAA), and a data-adaptive Sparse Bayesian Learning (SBL) algorithm. We show how these designs can significantly outperform conventional matched filter (MF) techniques for range compression. We extend our discussion to include MIMO radar systems. We briefly highlight sequence set design, and we motivate the need for sequences with low auto- and cross-correlations. To further reduce clutter effects, we discuss receiver design for MIMO systems. We present a new least squares approach to target estimation. Additionally, we show how IAA can be extended to the MIMO case, both in the negligible and non-negligible Doppler cases. We present a new, regularized version of the algorithm designed to account for interferences outside of an angular region of interest. We provide a theoretical convergence analysis of IAA. Finally, we consider MIMO transmit and receive beampattern design using sparse antenna arrays. We present a cyclic approach to beampattern design. Compared to a uniform linear array, we show that our sparse array design approach can, through larger degrees of freedom, better approximate desired transmit and receive beampatterns.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by William Roberts.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Li, Jian.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041432:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041432/00001

Material Information

Title: Adaptive Radar Signal Processing
Physical Description: 1 online resource (119 p.)
Language: english
Creator: Roberts, William
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: adaptive, array, beamforming, mimo, phased, processing, radar, receive, signal, transmit
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Probing waveform synthesis and receive filter design play crucial roles in achievable performance for many radar applications. A flexible receive filter design approach, at the costs of lower signal-to-noise ratio (SNR) and higher computational complexity, can be used to compensate for missing features of the probing waveforms. A well synthesized waveform, meaning one with good auto-correlation properties, can reduce computational burden at the receiver and improve performance. Herein, we investigate various signal processing strategies to improve the performance of modern day radar systems. We highlight the interplay between waveform synthesis and receiver design. We consider both single antenna systems (referred to as single-input single-output, or SISO, radar) as well as MIMO (multiple-input multiple-output) radar schemes. For SISO radar, we review a novel, cyclic approach to waveform design, and then compare the merit factors of these waveforms to other well-known sequences. Furthermore, we overview several advanced techniques for receiver design, including data-independent instrumental variables (IV) filters, a data-adaptive iterative adaptive approach (IAA), and a data-adaptive Sparse Bayesian Learning (SBL) algorithm. We show how these designs can significantly outperform conventional matched filter (MF) techniques for range compression. We extend our discussion to include MIMO radar systems. We briefly highlight sequence set design, and we motivate the need for sequences with low auto- and cross-correlations. To further reduce clutter effects, we discuss receiver design for MIMO systems. We present a new least squares approach to target estimation. Additionally, we show how IAA can be extended to the MIMO case, both in the negligible and non-negligible Doppler cases. We present a new, regularized version of the algorithm designed to account for interferences outside of an angular region of interest. We provide a theoretical convergence analysis of IAA. Finally, we consider MIMO transmit and receive beampattern design using sparse antenna arrays. We present a cyclic approach to beampattern design. Compared to a uniform linear array, we show that our sparse array design approach can, through larger degrees of freedom, better approximate desired transmit and receive beampatterns.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by William Roberts.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Li, Jian.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041432:00001


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Foremost,Iwouldliketorecognizemyadvisor,Dr.JianLioftheElectricalEngineeringDepartment.Dr.LirstofferedmeapositioninherlabwhenIwasanundergraduate,andthisopportunitytrulyinspiredmetocompletemygraduateworkattheUniversityofFlorida.Withoutherpatienceanddedication,thisdissertationwouldnothavebeenpossible.IfurthermoreacknowledgeDr.PetreStoicaofUppsalaUniversity,whoseassistancehascertainlymademeintoabetterengineer.IrecognizemycommitteemembersattheUniversityofFlorida:Dr.HenryZmuda,Dr.JenshanLin,andDr.RenweiMei.Theirtimeandeffortstowardsmydissertationanddefensehavebeengreatlyappreciated.Finally,IrecognizemyfamilyandmyfriendsintheSignalsAnalysisLaboratory.Theirencouragementhasinspiredmeateverylevelofmygraduatestudies. 3

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page ACKNOWLEDGMENTS .................................. 3 LISTOFTABLES ...................................... 6 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1SINGLEANTENNARADAR ............................. 10 1.1Introduction ................................... 10 1.2ProblemFormulation .............................. 10 1.3TransmitWaveformDesign .......................... 14 1.3.1ChirpSignal ............................... 15 1.3.2FrankCode ............................... 16 1.3.3P4Code ................................. 16 1.3.4GolombCode .............................. 17 1.3.5CyclicAlgorithm ............................ 17 1.3.6PerformanceConsiderations ...................... 19 1.4ReceiverDesign ................................ 20 1.4.1InstrumentalVariablesReceiveFilter ................. 20 1.4.2IterativeAdaptiveApproach ...................... 24 1.4.3SparseBayesianLearning ....................... 26 1.5NumericalExamples .............................. 28 1.5.1NegligibleDopplerExample ...................... 28 1.5.2Non-NegligibleDopplerExample ................... 29 1.6Conclusions ................................... 30 2MIMORADAR .................................... 37 2.1Introduction ................................... 37 2.2Multiple-InputMultiple-OutputSignalModel ................. 39 2.3MIMOTransmitWaveformConsiderations .................. 40 2.4MIMOReceiverDesign ............................ 42 2.4.1MatchedFilterDesign ......................... 42 2.4.2IVFilterDesign ............................. 42 2.4.3Least-SquaresReceiverDesign .................... 43 2.5IterativeAdaptiveApproachforMIMORadar ................ 45 2.5.1NegligibleDopplerCase ........................ 46 2.5.1.1Problemformulation ..................... 46 2.5.1.2IAA .............................. 47 2.5.1.3IncorporatingBIC ...................... 49 4

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..................... 50 2.5.2.1Problemformulation ..................... 50 2.5.2.2IAA .............................. 52 2.5.2.3IncorporatingBIC ...................... 52 2.5.3RegularizedIAA(IAA-R) ........................ 53 2.5.4MaximumLikelihoodBasedIAA-R(IAA-R-ML) ........... 54 2.6NumericalResults ............................... 57 2.6.1MIMOSARImagingwithNegligibleDoppler ............. 58 2.6.1.1Example1 .......................... 58 2.6.1.2Example2 .......................... 59 2.6.1.3Example3 .......................... 61 2.6.2MIMORange-Angle-DopplerImagingwithNon-NegligibleDoppler 62 2.6.2.1Example1 .......................... 63 2.6.2.2Example2 .......................... 64 2.6.3ComplexityAnalysis .......................... 64 2.7Conclusions ................................... 65 3SPARSEARRAYDESIGNFORMIMORADAR .................. 83 3.1Introduction ................................... 83 3.2TransmitBeampatternDesign ......................... 85 3.2.1ProblemFormulation .......................... 85 3.2.2SparseTransmitArrayDesign ..................... 87 3.2.2.1DeterminationofRviaacyclicapproach ......... 87 3.2.2.2Antennaselection ...................... 89 3.3MatrixApproachtoReceiveBeampatternDesign .............. 90 3.3.1ProblemFormulation .......................... 90 3.3.2SparseReceiveArrayDesign ..................... 92 3.4VectorApproachtoReceiveBeampatternDesign .............. 92 3.4.1ProblemFormulation .......................... 93 3.4.2SparseArrayDesign .......................... 94 3.4.2.1Weightdeterminationviaacyclicapproach ........ 94 3.4.2.2Antennaselection ...................... 95 3.5NumericalExamples .............................. 96 3.5.1Example1 ................................ 96 3.5.2Example2 ................................ 98 3.5.3Example3 ................................ 99 3.6Conclusions ................................... 101 REFERENCES ....................................... 113 BIOGRAPHICALSKETCH ................................ 119 5

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Table page 1-1IAAforRange-Dopplerimaging. .......................... 32 1-2SBL-. ......................................... 32 2-1IAAforMIMOSARimaging. ............................. 74 2-2IAAforAngle-Range-DopplerimagingwithaMIMOarray. ............ 74 3-1Section 3.5 Example1:Transmitbeampatterndesign. .............. 107 3-2Section 3.5 Example2:Receivebeampatterndesign. .............. 107 3-3Section 3.5 Example3:Receivebeampatterndesign. .............. 107 6

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Figure page 1-1Receivedsignalalignedwiththereturnfromatargetinrangebin~r. ...... 32 1-2MeritFactorversusthesequencelengthNforFrank,P4,Golomb,andCANsequences(CANisinitializedwithaFranksequence). ............. 33 1-3AmbiguityfunctionsforN=16. ........................... 34 1-4RangeprolesforN=256andSNR=20dB. ................... 35 1-5Range-DopplerimagesforN=36andSNR=10dB. .............. 36 2-1OverlaidcorrelationsforaCAsequencesetwithM=4,N=256,andP=30. 75 2-2SpotlightSARimagesfor~N=10,L=128,JNR=100dBandSNR=15dB. .. 76 2-3SpotlightSARimagesfor~N=1,L=64andSNR=20dB. ............ 77 2-4MIMOSARimagesfor~N=1,L=64andSNR=20dB. .............. 78 2-5MIMOSARimagesfor~N=3,L=32andSNR=15dB. .............. 79 2-6Range-Dopplerimagesat1relativetoabroadsidescanforL=32andSNR=20dB. ........................................ 80 2-7Angle-rangeimagesat1DopplerforL=32andSNR=20dB. ........ 81 2-8PassivearraytargetestimatesforN=10. ..................... 82 3-1Section 3.5 Example1. ............................... 108 3-2AntennapositionsforSection 3.5 Example1. ................... 109 3-3Section 3.5 Example2. ............................... 110 3-4AntennapositionsforSection 3.5 Example2. ................... 111 3-5Section 3.5 Example3:sparsearrayreceivebeampatterns. ........... 111 3-6AntennapositionsforSection 3.5 Example3. ................... 112 7

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Probingwaveformsynthesisandreceivelterdesignplaycrucialrolesinachievableperformanceformanyradarapplications.Aexiblereceivelterdesignapproach,atthecostsoflowersignal-to-noiseratio(SNR)andhighercomputationalcomplexity,canbeusedtocompensateformissingfeaturesoftheprobingwaveforms.Awellsynthesizedwaveform,meaningonewithgoodauto-correlationproperties,canreducecomputationalburdenatthereceiverandimproveperformance.Herein,weinvestigatevarioussignalprocessingstrategiestoimprovetheperformanceofmoderndayradarsystems.Wehighlighttheinterplaybetweenwaveformsynthesisandreceiverdesign. Weconsiderbothsingleantennasystems(referredtoassingle-inputsingle-output,orSISO,radar)aswellasMIMO(multiple-inputmultiple-output)radarschemes.ForSISOradar,wereviewanovel,cyclicapproachtowaveformdesign,andthencomparethemeritfactorsofthesewaveformstootherwell-knownsequences.Furthermore,weoverviewseveraladvancedtechniquesforreceiverdesign,includingdata-independentinstrumentalvariables(IV)lters,adata-adaptiveiterativeadaptiveapproach(IAA),andadata-adaptiveSparseBayesianLearning(SBL)algorithm.Weshowhowthesedesignscansignicantlyoutperformconventionalmatchedlter(MF)techniquesforrangecompression. WeextendourdiscussiontoincludeMIMOradarsystems.Webrieyhighlightsequencesetdesign,andwemotivatetheneedforsequenceswithlowauto-and 8

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Finally,weconsiderMIMOtransmitandreceivebeampatterndesignusingsparseantennaarrays.Wepresentacyclicapproachtobeampatterndesign.Comparedtoauniformlineararray,weshowthatoursparsearraydesignapproachcan,throughlargerdegreesoffreedom,betterapproximatedesiredtransmitandreceivebeampatterns. 9

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1 ].Thesignalisreectedbytargetsinthescene,whichcouldbeatdifferentangularandrangelocationsrelativetotheradar.Thereectedsignals,whichareattenuatedandtime-shiftedversionsofthetransmittedwaveform,arelinearlycombinedatthereceiveantenna(whichcouldbethesameordifferentastheoneusedfortransmission).Signalprocessingofthereceivedsignalisperformedtodetermineunknownpropertiesofthetargets,suchastheirrange,radarcrosssection(RCS),andspeed(orDopplershift). Highrangeresolution,meaningtheradar'scapabilitytoseparateclosely-spacedtargetsindistance(relativetotheradar),canbeachievedbytransmittinganarrowpulse.Thesignal-to-noiseratio(SNR)atthereceiver,ontheotherhand,dependsontheenergyinthesignal,whichisproportionaltothepulse'sdurationandamplitude.Therefore,toachieveacertainresolutionwhilemaintainingaspecicSNR,ahighamplitudepulseisnecessitated.Thepeakpowerofthetransmitpulse,however,islimitedbytheradar'shardwaresystem.Withpulsecompression,peakpowerconstraintsaremediatedbytransmittingaphase-codedpulseoflongerduration.Rangeresolutioncanbeimprovedbydesigningthesignaltoattainalargebandwidthsothat,aftermatchedltering,thewidthofthereceivedpulseiscomparabletothatofashorter,singlepulse.Pulsecompression,inthisway,canbeusedtoproducealowpeak-powersignalwithasufcientlylargetime-bandwidthproduct. 10

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where()Tdenotesthetransposeoperation.Duetohardwareconstraints(suchasthelimitationsofthepoweramplier)inpractice,componentsofthetransmittedwaveformarecommonlyrestrictedtobeingconstantmodulus.Withoutlossofgenerality,weconsiderfx(n)gNn=1beingunimodular,sothat:x(n)=ejn,n=1,...,N, wherenrepresentsthephaseofx(n).IfthesetoftargetsinthescenearerepresentedbytheirRCSsfrgRr=1,withR>N(ingeneral,weassumer=0foranyrsuchthatr=2f1,...,Rg),thenthereceivedsignaly~r(alignedwiththetransmittedwaveform'sreectionfromatargetofinterest~rfor~r=1,...,R)canbemodeledas:y~r=~rx+N1Xr=N+1r6=0~r+rJrx+,~r=1,...,R, where=(1),(2),...,(N)TreferstothenoisecomponentofthereceivedsignalandwhereJr=2666666666664r+1z }| {10...103777777777775=JTr,r=0,...,N1. Weillustratethemodelfory~rinFigure 1-1 .AlthoughDopplereffectsarenotconsideredinthereceivedsignalrepresentationgiveninEquation 1 ,wewillmodifytheequationlatertoaccomodatethenon-negligibleDopplercase. 11

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SimilarestimatescanbegeneratedfortheothertargetsinthescenebyreformulatingthemodelforthereceivedsignalinEquation 1 (sothaty~risalignedwiththereturnfromatargetofinterest~rfor~r=1,...,R). Neglectinginterferencesfromotherrangebins,amatchedlterprovidesoptimalperformance(thehighestSNR)inthepresenceofstochasticadditivewhitenoise.Inmostpracticalradarapplications,however,detectionperformanceishamperedmorebycluttercomponents,i.e.reectionsreceivedfromtargetsinneighboringrangebinstotheoneofinterest[ 2 ].Forthematchedltertoprovideaccuratedetectionthen,theauto-correlationfunctionofthetransmittedwaveformneedstobeimpulse-like.Inotherwords,waveformswithhighmeritfactors(MF)aredesirable[ 3 4 ],whereweletMF=jr0j2 withrk=NXn=k+1x(n)x(nk)=rk,k=0,...,N1, andwhereISLreferstotheintegratedsidelobeleveloftheauto-correlationfunction. 12

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5 ].However,thelongestknownBarkersequenceisoflength13,andresearcherscontendthatnolongerwaveformscansatisfytheBarkercriteria[ 6 8 ].ToidentifybinarysequencesthatyieldamaximumMF(foragivenN)requiresanexhaustivesearchwhosecomputationalcomplexityincreasesexponentiallywiththelength,whichquicklyprovesintractableasNincreases.Atthecostofincreasedhardwarecomplexity,thebinaryrestrictioncanberelaxedtodesignunimodularsequences(whichmayormaynotuseanite-alphabet)withlowersidelobelevelsandhighermeritfactors.Perhapsthemostwell-knownpolyphasesequenceistheFrankcode,whichisderivedfromthephasehistoryofachirpwaveform[ 9 ].Frankcodesexhibitrelativelylowsidelobes,butareonlydenedforwaveformsofsquarelength(N=H2,whereHisaninteger).Numerousothertypesofsequenceshavebeenproposed,eachofwhichpossessescertainadvantagesandlimitations(see,e.g.,[ 10 13 ]). Forpolyphasesequences,theMFequationinEquation 1 canprovehighlymultimodal,meaningthefunctioncouldhavenumerouslocalmaxima.StochasticoptimizationalgorithmshavebeenpresentedtoidentifysequenceswithhighMFvalues,butthesealgorithmscansufferfrompoorconvergencepropertiesandhighcomputationalburdensforN103orlarger.Recently,aseriesofcylicalgorithms(CA)werepresentedtoprovidelocalmaximizationoftheMFmetric[ 14 17 ].Notably,CAN(CA-new),whichisbasedonFFToperations,canbeusedtodesignsequences(uptolengthN106orlargeronanordinaryPC)withcorrelationlevelsthataresignicantlylowerthanthoseofanypre-existingwaveforms[ 15 ]. AlthoughCANisabletoprovidesignicantreductionofcorrelationlevels,insomesituationsthisapproachtowaveformsynthesis(or,infact,anyotherapproach)mightnotprovideasmuchsidelobeattentuationasdesired.Inthiscase,amatchedlter, 13

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2 ].WhenDopplereffectsareconsidered,IVltersarenotquiteaseffective,andmoreadvancedalgorithmsareneeded.Theiterativeadaptiveapproach(IAA),anon-parametricanduserparameter-freeweightedleast-squaresalgorithm,wasrecentlyshowntoofferhighresolution(forrange-Dopplerimaging)andexcellentinterferencerejectioncapability[ 18 ].TheBayesianinformationcriterion(BIC)(see,e.g.,[ 19 ]),amodelorderselectiontool,couldfurtherbeappliedtoobtainasparseestimatefromtheIAAresult.Alternatively,topromotesparsityinthetargetestimatesandtoproducehigherresolutionthanIAA,anExpectationMaximization(EM)basedSparseBayesianLearningalgorithm,referredtoasSBL-[ 20 ],couldbeadoptedatthereceiver. Herein,weprovideanoverviewonrecentdevelopmentsoftransmitwaveformsynthesisandreceiverdesignsforSISOradarapplications(thereaderinterestedintheextensionofsomeofthetopicsdiscussedheretomultipleantennasystems,alsoknownasMIMOradars,isreferredto[ 21 ]andthemanyreferencestherein).InSection 1.3 ,wediscusstheaforementionedcyclicapproachtowaveformdesign,andwecontrasttheperformanceofthisalgorithmtothatofmoretraditionalradarsequences.Then,inSection 1.4 ,wehighlighttheneedforadvancedprocessingtechniquesattheradarreceiverandwereviewseveralnovelapproaches(mentionedabove)toreceiverdesign.WepresentnumericalexamplesinSection 1.5 tofurtheremphasizethevariousmethods,andweofferconcludingremarksinSection 1.6 1 ,referstothematchedltertimeresponseofatargetwithnegligibleDopplershift(zerovelocityrelativetotheradar).Moregenerally,theambiguityfunction(AF)j(k,d)j

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1 ]:j(k,d)j=Z1s(t)s(t+k)ej2dtdt. AlthoughweprovidethecontinuousAFmodelinEquation 1 (whichisatypicaldescriptionoftheAF),adiscrete(sampledversionofEquation 1 )AFmodelcanbesimilarlyattained.Wefollowtheprocedurein[ 22 ]togeneratetheAFplotsinFigure 1-3 Ideally(foramatchedltertoprovideaccuratetargetestimates),theAFofasequenceshouldbeshapedsimilartoathumbtack,withahighpeakattheoriginandlowsidelobeselsewhere.Designingawaveformtoapproximateadesiredtime-frequencyAF,however,canprovechallenging.Thus,althoughtheAFofawaveformcertainlyholdsimportance(especiallywhenmotionofthetargetand/orradarisinvolved),researchershaveoftenfocusedondesigningsignalswithgoodauto-correlationproperties(MFmetricgiveninEquation 1 ). Inthissection,wereviewseveraltraditionalphase-codedunimodularwaveformsforradarsystems.Additionally,wediscussanew,cyclicapproachtowaveformsynthesis,whichcanbeusedtodesignsequenceswithsignicantlyhigherMFs.WecontrasttheperformanceofeachapproachbyexaminingtheAFandbycomputingtheMFmetrics. 6 ].Inaddition,chirpsignalshavespectralefciency,meaningthepowerofthewaveformsisdispersedevenlythroughoutthefrequencyspectrum,whichallowsforhighrangeresolution.Highresolution,however,comesatacostofrange-Dopplercoupling,whichisevidentbytheridgethatappearsintheAF(see,e.g., 15

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1-3 A).Asaresult,targetsmovingtowardsaradarcouldappearcloserinrangethanstationarytargetsatthesamedistance. Thechirpsignals(t)isgivenby[ 6 ]:s(t)=1 whereisthechirprateofthesignal(=B=). 9 ],andwasdesignedtoapproximatethephasehistoryofalinearsteppedfrequencywaveform,orchirpsignal.Franksequencespossessrelativelylowaperiodicauto-correlationsidelobes(frkgN1k=1),butaredenedonlyforsequencesofsquarelength(sothatN=H2,whereHissomeinteger). Explicitly,thecomponentsinEquation 1 aredened,foraFranksequenceoflengthN,as:x((m1)H+p)=ejm,p, where1mH,1pH,N=H2,andm,p=2(m1)(p1) 6 ].ThesesequencesexistforanylengthN,andexhibitbettertolerancetoDopplerfrequencyshifts(meaning,withrelativelylargeDopplershifts,theperformanceofthewaveformsdoesnotdegradesignicantly)andslightlyhighersidelobelevelsthantheaforementionedFrankcode.P3andP4sequencescanbeobtainedascyclicallyshiftedanddecimatedversionsoftheZadoff-Chucodes[ 13 ].Inthischapter,weconsidertheP4code,whichhashigherbandwidthtolerancethanthe 16

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6 ]. P4codesexistforanylengthN,andthephaseelementsaregivenby:n=2 10 ].GolombsequencesaredenedforanylengthN.ThecomponentsofxaregivenbyEquation 1 ,wheren=2(n1)n 1 forunimodularphase-codedsignals.Inthischapter,wewillfocusonCAN(cyclicalgorithm-new);wepresentasynopsisofthealgorithmhere(see[ 15 ]formoredetails). MinimizationofEquation 1 wasshowntobealmostequivalenttothefollowingsimplerproblem(whosecriterionisaquadraticfunctionoffx(n)gNn=1):minfx(n)gNn=1;fpg2Np=12NXp=1NXn=1x(n)ejpnp ThetermsfpgdenotethefollowingFourierfrequencies:p=2 Towithinamultiplicativeconstant,thecriteriainEquation 1 canberewrittenas:AHwv2, 17

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andwhereAH=1 Iff=AHwrepresentstheFouriertransformofw,thenforxedf,theminimizerofEquation 1 isgivenby:p=arg(f(p)),p=1,...,2N. Similarly,foragivenv,andifg=AvdenotestheinverseFouriertransformofv,thentheminimizingsequencefx(n)gNn=1ofEquation 1 isgivenby:x(n)=ejarg(g(n)),n=1,...,N. ThestepsofCAN,whichprovidethecyclicminimizationoftheISL-relatedmetricinEquation 1 ,canbesummarizedasfollows: Setfx(n)gNn=1tosomeinitialvalues(e.g.,fx(n)gNn=1canberandomlygeneratedorgivenbyagoodexistingsequence,suchasaFrankorGolombsequence). Computethefpg2Np=1thatminimizethemetricforfx(n)gNn=1xedattheirmostrecentvalues(seeEquation 1 ). Computethesequencefx(n)gNn=1thatminimizesthemetric,undertheunimodularconstraint,forfpg2Np=1xedattheirmostrecentvalues(seeEquation 1 ). 18

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RepeatSteps1and2untilapre-speciedstopcriterionissatised,e.g.,kx(i)x(i+1)k<,wherex(i)isthesequenceobtainedattheithiterationandisapredenedthreshold,suchas103. DuetoitssimpleFFToperations,CANcanbeusedforwaveformsynthesisonanordinaryPCwithverylargevaluesofN,suchasN106. 1-2 ,wecomparethemeritfactorsoftheFrank,P4,Golomb,andCANsequences(CANwasinitializedwithaFranksequence)forthefollowinglengths:N=32,52,102,152,202,302,702and1002.Theresultsareshownusingalog-logscale.WedonotshowtheMFforachirpwaveform,sincethesesequencesperformratherpoorlyforhighervaluesofN.TheCANsequenceprovidesthehighestmeritfactorforeachvalueofNconsidered.WhenN=1002,theCANsequenceprovidesthelargestmeritfactorof1769.05,whichisseveraltimeslargerthanthatgivenbytheFranksequence(whichis246.39).AlthoughaFranksequencewasusedheretoinitializeCAN,asimilarresultwouldhavebeenobtainedbyinitializingthealgorithmwithaP4orGolombsequence(sincethesechirp-basedwaveformsarecloselyrelated). WeplottheAFforeachsequenceinFigure 1-3 ,whereweletN=16(chosentoallowafaircomparisonsinceN=H2forFranksequencesandchosensufcientlysmallsuchthatthefeaturesoftheAFareclearlyvisible).Asalreadymentioned,adiscrete,sampledversionofthecontinuousAFfunctioninEquation 1 isusedtoprovidetheAFplotsinFigure 1-3 (see[ 22 ]foracompletedescriptionofthemethodologyused).Onlythersttwoquadrantsareshown,sincetheAFinEquation 1 issymmetricabouttheorigin(j(k,d)j=j(k,d)j).InFigure 1-3 A,weshowtheAFforthechirpwaveform,whichcontainstheaforementioneddiagonalridgeextendingfromthedominantpeak.TheAFsfortheFrank,P4,andGolombsequencesareshowninFigures 1-3 B,C,andD,respectively.Asthesesequencesarecloselyrelatedtothechirpsignal,theydisplayasimilarridge,sothatonlytheslopeoftheridgediffers.InFigures 1-3 EandF,we 19

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1-3 E,apseudo-noise(PN)sequenceisusedtoinitializeCAN.InFigure 1-3 F,aFranksequenceisused,sothatadiagonalridge(correspondingtotheoneoccurringintheGolombsignal)becomesapparent.Asevidenced,thesidelobelevelsfortheCANsequencesappeartobelowerthanthoseoftheotherwaveforms(althoughthetotalareaunderanAFisalwaysconstrainedtoone),especiallyforthezero-Dopplerslice(sinceCANmaximizestheMFmetric). 1.3 ,wedescribedseveraldifferentwaveforms,alldesignedtoprovideahighMFandthusallowforbettercluttersupressionatthereceiver.WeshowedthatCANwasabletoproducewaveformswithsigncantlylowersidelobesthantraditionaltransmitsequences.Forsomecases,however,evencarefulconstructionoftheradar'stransmitwaveforms,whencoupledwithamatchedlteratthereceiver,stillmightnotprovidesufcientclutterreduction.Toaddressthesesituations,wenowturnourattentiontothereceiverstageofaradarsystem.Inthissection,wereviewseveralapproachestorangecompression,andwediscussthemeritsofeachmethod. 1 (i.e.,if~r+r=0foranyr6=0),thenthematchedlterwouldprovideahighlyaccurateestimateof~r.Wheninterferenceterms(clutter)arepresentinthereceivedsignal,whichiscommonlythecaseinpractice,thematchedltercanperformpoorly.Theinstrumentalvariables(IV)method(alsoknownasthemismatchedlterapproach),amoregeneralmethodforestimating~r,canbeusedtosignicantlylowertheclutteratthecostofareducedSNR[ 2 23 24 ].TheIVestimateof~risgivenby:^~r=zHy~r 20

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1 reducestothematchedlterestimateof~r.Ingeneral,andunlikethematchedlter,theelementsofzarenotrestrictedtobeunimodular,sincethisvectorisonlydesignedforthepurposesofestimation.WeassumehereinthatzisavectoroflengthN,although,bypaddingthetransmitwaveformwithzeros,alongerIVvectorcouldbedesignedtoimproveclutterreductionevenmore(atacostoffurtherreducedSNR). WeconsidertheIVformulationgivenin[ 2 ].ThegoaloftheIVapproachistondavectorzthatminimizestheISL,which,inthenegligibleDopplercase,isgivenby:ISLIV=PN1k=(N1),k6=0zHJkx2 ByapplyingtheCauchy-Schwartzinequality,theminimumvalueofISLIVwasshowntobeachievedwhenz=R1IVx,where:RIV=N1Xk=N+1,k6=0JkxxHJTk. WhenDopplereffectscannotbeneglected,wemustreformulatethemodelinEquation 1 as:y~r=~r,~l~x~l+N1Xr=N+1LXl=1~r+r,lJr~xl+,(r,l)6=(0,~l) wherefr,lg(forr=1,...,Rdenotingtherangebinandl=1,...,LdenotingtheDopplerbin)denotestheRCSsofthecollectionoftargets,~r,~lreferstothetargetofinterest,LdenotesthenumberofDopplerbins,~xl=xaldenotestheDopplershiftedwaveform(referstotheHadamardproductoperation),andal=1,ej!l,...,ej!l(N1)T. TheDopplershiftsofthetargetsinthescenef!lgLl=1areassumedtoliewithinanintervaldenotedby=[!a,!b](where!b>!aandwherewechooseLsuchthat 21

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1 mustbeadjusted[ 2 ]:ISLIV,D=1 AsimilaranalysistothatinthenegligibleDopplercaseshowsthat,toachievetheminimumISLIV,D,weshouldchoosez~l=(RIV,D)1~x~l,whereRIV,D=N1Xk=N+1,k6=0Jk1 Theestimatefor~r,~l(correspondingtoatargetatarangebin~randatDopplerbin~l)isgivenby:^~r,~l=zH~ly~r WeconsidertheIVformulationgivenin[ 2 ].Assumingaspeciedsidelobeattentuationlevelandtransmitsequencex,thegoaloftheIVapproachistondasignalzthatminimizestheISNR(Inverse-SNR).Wecanrestatethisgoalinthefollowingoptimizationproblem,whichneglectsDopplereffects:minzISNRs.t.ISLIV, whereISNR=kzk2 andISLIV=PN1k=(N1),k6=0zHJkx2 22

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2 ],theyshowedthattheprobleminEquation 1 canbereformulatedasaconvexoptimizationproblem(whichcanbesolvedefciently),subjecttorestrictionsontheminimumvalueof(andhencetheminimumachievableISLIV). WhenDopplereffectscannotbeneglected,wemustreformulatethemodelinEquation 1 as:y=0,~l~x(!~l)+N1Xk=N+1LXl=1k,lJk~x(!l)+,(k,l)6=(0,~l) wherefk,lg(fork=K+1,...,K1denotingtherangebinandl=1,...,LdenotingtheDopplerbin)denotesthecollectionoftargets,0,~ldenotesthetargetofinterest,LdenotesthenumberofDopplerbins,~x(!l)=xa(!l)denotestheDopplershiftedwaveform,anda(!l)=1,ej!l,...,ej!l(N1). TheDopplershiftsofthetargetsinthescenef!lgLl=1areassumedtoliewithinanuncertaintyintervaldenotedby=[!a,!b](where!b>!aandwherewechooseLsuchthatf!lgLl=1covers).Sincenoknowledgeisassumedofthetargets'Dopplershifts,otherthanthattheybelongto,thecriteriainEquation 1 mustbeadjusted:ISLDIV=N1Xk=(N1)k6=01 TheIVltervectorsz(!l)(forl=1,...,L)canbepre-computedbysolvingaconvexoptimizationproblemsimilartotheoneinEquation 1 .Thetargetestimatefor0,~l(correspondingtoatargetatarangebinofinterestandatthe~lthDopplerbin)isgivenby:^0,~l=z(!~l)Hy z(!~l)H~x(!~l), 23

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1.5 ,theIVapproachdoesnotperformwellwhenDopplereffectsarenon-negligible. Asalreadymentioned,IVlterscanbeprecomputedofine.Fromacomputationalstandpoint,therefore,IVcertainlyoffersminimalburdentothereceiver,asthecomplexityofitsapplicationiscomparabletothatofthematchedlter. 18 ],wasshowntoofferimprovedresolutionandinterferencerejectionperformance.IAAisanon-parametricanduserparameter-freeweightedleast-squaresalgorithmthatwasshowntoperformwelleveninthesingledatavectorcaseofinteresthere.Webrieysummarizethealgorithmbelow. Considerthemodelfory~rinEquation 1 .ThegoalofIAAistominimizethefollowingweightedleast-squarescostfunctionwithrespecttotheRCSofatargetofinterest~r,~l:y~r~r,~l~x~l2Q1~r,~l, wherekuk2Q1,uHQ1u.TheinterferencecovariancematrixforthetargetofinterestwithRCS~r,~lisdenotedbyQ~r,~l,andisdenedas:Q~r,~l=RIAA(~r)j~r,~lj2~x~l~xH~l, 24

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Theweightedleast-squaresestimateof~r,~l,aftersomesimplication,isgivenby:^~r,~l=~xH~l(RIAA(~r))1y~r SincetheestimateinEquation 1 dependsonthecovariancematrixRIAA(~r),whichinturndependsonthetargetamplitudes,thealgorithmusesaniterativeapproach,whichissummarizedinTable 1-1 .Thetargetcoefcientsareinitializedusingamatchedlter.Toestimatetargetsinotherrangebins,wesimplyredeney~r,whichrepresentstheNlengthsignalvectoralignedwiththereceivedreectionfromarangebinofinterest~r. ThecomputationalcomplexityofIAAisdeterminedbyEquation 1 andisO(N2(2N1)RLTIAA)forrangecompression,whereTIAAisthenumberofiterationsneededbyIAAtoconverge.IAAtypicallyconvergesafteraboutteniterations(TIAA=10);alocalconvergenceproofforIAAisofferedin[ 25 ]. ToobtainasparseestimatefromtheIAAresult,wecanapplytheBayesianinformationcriterion(BIC),amodelorderselectiontool(see,e.g.,[ 19 ]).GiventheIAAestimateforascene,theBICapproachselectsatargetwithrange-Dopplerindices(~r,~l),whichminimizesthefollowingcriterion: BIC(~r,~l)()=2(N+R1)ln24yX(r,l)2fJ()S(~r,~l)g^r,lJrxl2235+4ln(2(N+R1)), wheredenotesthenumberoftargetscurrentlyselected(=1fortherststep),y2C(N+R1)1representstheentirereceivedsignal,xl=[~xl,0R1]T,Jr2C(N+R1)(N+R1)isdenedsimilarlytoEquation 1 ,andwhere4intherightmosttermwaschosentorepresentthenumberofunknownsforeachtarget(range,Doppler 25

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1 doesnotdecreaseanymore.ReectioncoefcientsnotpresentinJ()attheendofthisprocedurearesettozeroandareassumednottorepresenttruetargetsinthescene. 26 ]).Recently,amodiedExpectationMaximization(EM)basedSBLalgorithm,calledSBL-,wasproposedin[ 20 ].SBL-usesathree-stagehierarchialBayesianmodeltorepresentthesignalcomponentsiny.Attherststage,itisassumedthatCN(0,I),i.e.,yCN(X,I),whereCN(,)representsthemultivariatecomplexGaussiandistributionwithmeanandcovariancematrix,2C(N+R1)1representsthereceivednoise,X=J0x1,J1x1,...,JR1x1,J0x2,J1x2,...,JR1xL, and2CRLrepresentsthevectoroftargetcoefcients.Atthesecondstage,itisassumedthatr,lCN(0,pr,l)andf()/1 26

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whereIG(a,b)denotestheinversegammadistributionforsomea>0andb>0.Itwasfurthershownin[ 20 ]that:fjy,p,(jy,p,)/1 ()RLjje()H1(), whereP=diagfpr,lg,=(1XHX+P1)1,and=1XHy. Weletp=[p1,1,p2,1,...,pR,1,p1,2,p2,2,...,pR,L]T.ThegoalofSBL-thenistoobtainanestimateforp,,and.AnEM-basedapproachisused,sothattheestimatesforpandatiterationtaregivenby:p(t)=argmaxpElnfp,jy,(p,jy,)jy,p(t1),(t1)(t)=argmaxElnfp,jy,(p,jy,)jy,p(t1),(t1), whereistreatedasmissingdataandErepresentstheexpectationoperation.TheSBL-algorithm,whichisiterative,issummarizedinTable 1-2 .Theinitialestimatesp(0)and(0)canbegeneratedusingamatchedlter,althoughamoresophisticatedinitialization,suchasusingIAA,wasshowntoprovidemoreaccurateresults.AttheendoftheiterationT,thetargetestimatesaregivenby^=(T). SBL-(with=1)wasshowntoofferlowerreconstructionerrorscomparedtootherSBLapproaches,aswellascomparedtoIAA.ThecomputationalcomplexityofSBL-isdeterminedbytheestimatorsforandgiveninTable 1-2 ,andisO((N+R1)2RLTSBL)forrangecompression,whereTSBListhenumberofiterationsneededforSBL-toconverge.Finally,becauseSBL-isbasedonEMoperations,itsconvergencetoalocalmaximuminEquation 1 isassured. 27

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Foreachexample,wewillassumecircularlysymmetricindependentandidenticallydistributed(i.i.d.)additivecomplexGaussiannoisewithzero-meanandvariance2noise.Thesignal-to-noiseratio(SNR),indB,isdenedasSNR=10log101=2noise. WeshowtheresultusingaFranksequenceandamatchedlteratthereceiverinFigure 1-4 A.Asevidenced,thetwostrongertargetsaresuccessfullyidentiedusingthisscheme.Thethird,weakertarget,however,appearswithinthesidelobesofthestrongesttarget,andthematchedlterdoesnotproduceapeakatthetruetargetlocation.InFigure 1-4 B,weagainuseamatchedlter,butnowtransmitaCANwaveform.Forthiscase,sidelobesarereduced,andapeakisnowdiscernibleatthelocationoftheweakesttarget.WeuseCANwaveformsfortheremaininggures. 28

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1-4 C.ComparedtothematchedlterresultinFigure 1-4 B,IVproduceslowerleakageandfurthermoreshowswell-separatedpeaksatthetruetargetlocations.Asmentioned,IVlterscanbeprecomputedofine,sothistechniqueoffersnofurthercomputationalburdenthanthematchedlter.TheIAAresultisshowninFigure 1-4 D.Forthiscase,IAAachievessimilarperformancetotheIVlter,butatthecostofincreasedcomputationaleffortsatthereceiver.WeapplyBICtotheIAAresultinFigure 1-4 E,whichservestoproduceamoresparseresultbyisolatingthetargetpeaks.TheSBL-resultwith=1isshowninFigure 1-4 F,which,exceptforunderestimatingtheamplitudeoftheweakesttarget,performssimilarlytoIAAwithBIC.Again,SBLisperformedwithincreasedcomputationalburdencomparedtoamatchedlterandIVreceivelter(forthisparticularcase,andasevidencedbythepreviouslystatedcomputationalcomplexitiesofthetwoalgorithms,SBL-performsfasterthanIAA).Wechose=1basedontherecommendationin[ 20 ]. 29

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1-5 AandB,respectively,withamatchedlteratthereceiver.Asevidenced,thematchedlterperformspoorlyforthisnon-negligibleDopplercase,asthetargetestimateshavehighsidelobesandtheweakertarget(10dB)isnotidentied.TheresultwiththeCANsequenceappearstohaveslightlylowersidelobes.Fortheremaininggures,then,weagainuseonlyCANsequencesfortransmission. ThematchedlterisreplacedwithanIVlteratthereceiverinFigure 1-5 C.Asexpected,theIVlterdoesnotperformquiteaswellformobiletargets.Thesidelobesareslightlyreducedfromthematchedlterresult,butlargesidelobescontinuetodominatetheimage.TheIAAresultisshowninFigure 1-5 D.ComparedtothematchedlterandIVlter,IAAsignicantlyreducessidelobesandproducesapeakateachofthetruetargetlocations(again,atthecostofincreasedcomputation).TheIAAwithBICresultisshowninFigure 1-5 E.BICisabletoidentifythetwostrongesttargets,althoughtheweakesttargetismissedintheestimate.Finally,theSBL-1resultisshowninFigure 1-5 F.SBL-1identieseachofthetruetargetswithnofalsealarms.AlthoughSBL-1outperformedIAAwithBICforthisparticularsimulation(withaweakertargetspacedcloselyinDopplerfrequencytotwostrongertargets),thetwoapproachestypicallyachievesimilarresultsforrange-Dopplerimaging. 30

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31

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IAAforRange-Dopplerimaging. SBL-. 1+(t1)r,l2+(t1)r,l(t)=1 Figure1-1. Receivedsignalalignedwiththereturnfromatargetinrangebin~r. 32

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MeritFactorversusthesequencelengthNforFrank,P4,Golomb,andCANsequences(CANisinitializedwithaFranksequence). 33

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CD EF Figure1-3. AmbiguityfunctionsforN=16.A)Chirp.B)Frank.C)P4.D)Golomb.E)CANinitializedwithaPNsequence.F)CANinitializedwithaFranksequence. 34

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CD EF Figure1-4. RangeprolesforN=256andSNR=20dB.A)AFranksequencewithamatchedlteratthereceiver.B)ACANsequencewithamatchedlteratthereceiver.C)ACANsequencewithanIVreceivelter.D)aCANsequencewithIAA.E)ACANsequencewithIAAandBIC.F)ACANsequencewithSBL-1.`O'denotesatruetargetlocationand`X'denotesanestimatedtargetlocation. 35

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CD EF Figure1-5. Range-DopplerimagesforN=36andSNR=10dB.A)AFranksequencewithamatchedlteratthereceiver.B)ACANsequencewithamatchedlteratthereceiver.C)ACANsequencewithanIVreceivelter.D)ACANsequencewithIAA.E)ACANsequencewithIAAandBIC.F)ACANsequencewithSBL-1.`O'denotesatruetargetlocationand`X'denotesanestimatedtargetlocation. 36

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27 ]-[ 37 ]andthereferencestherein).ThewaveformdiversityaffordedbytheMIMOradarcanservetoincreasetheexibilityatthetransmitter[ 32 ]-[ 37 ].Forexample,probingsignalscanbedesignedtoapproximateadesiredtransmitbeampatternandtominimizeundesirablecross-correlationterms(see,e.g.,[ 38 ]-[ 41 ]andthereferencestherein).Whenorthogonalwaveformsaretransmitted,theparameteridentiabilityoftheradar,meaningthemaximumnumberoftargetsthatcanbeuniquelyidentied,isvastlyimproved.Withcarefulconstructionoftheradar'santennastructure,infact,theparameteridentiabilitycanbeincreasedbyafactorofM,whereMisthenumberoftransmittingantennas,overthecorrespondingphased-arraysystem[ 42 ].Furthermore,whentheradartransmitsMorthogonalwaveforms,thevirtualarrayoftheradarsystemisalledarraywithanaperturelengthuptoMtimesthatofthereceivearray[ 35 36 ].ThisadvantageofMIMOradarcanbeexploitedtoachieveanM-foldimprovementinthespatialimagingresolutionovertheconventionalphased-arrayradar[ 35 36 ]. Irrespectiveofthearraysystem,data-independentapproaches,suchasdelay-and-sum(DAS)(ormatchedltering),canbeusedforradarimaging.However,DASsuffersfromlowresolutionandhighsidelobelevelproblems.WithanarrowbandMIMOradar,orthogonalwaveformsfromthetransmitantennashittargetsinthesceneofinterestatdifferenttimeinstants,thusundergoingdifferentphaseshifts.Therefore,thereectedsignalsatthereceiverarelinearlyindependentofeachotherwhenthenumberoftargetsperrangeandDopplerbinislessthanorequaltothenumberoforthogonalprobingwaveforms.Duetothisindependence,adaptivebeamformingtechniques, 37

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43 ].Theexistingadaptiveapproaches,suchasCapon[ 44 ]andamplitudeandphaseestimation(APES)[ 45 ],canbeusedtomitigatecluttereffectsandimproveresolutionoverconventionalDAS.Whennoise(andclutter)levelsinthereceivedsignalarehigh,however,theperformanceofdata-dependentapproachesmightdegradesignicantlyunlessthenumberofsnapshotsisratherlarge. AtacostofreducedSNR,anIVlter,asinSection 1.4.1 ,canbeappliedtothereceivedsignaltoreduceclutterandinterferenceeffects[ 46 ].Inthisway,anIVreceiveltercanservetoreduceburdenatthetransmitterbynotimposinganystrictconditionsonthetransmitwaveform(althoughgoodcorrelationpropertiesarestilldesirable).InthepresenceofDopplereffects,asbefore,theIVltercanfailtoprovidesatisfactoryresults,asthedegreesoffreedomofthelterlimitthenumberofinterferencetermsthatcanbecompensatedfor.Alternatively,advancedestimationtechniquescanbeadoptedatthereceiverstage.Theiterativeadaptiveapproach(IAA),whichweinitiallydescribedinSection 1.4.2 ,isanon-parametricanduserparameter-freealgorithmwhichwasshowntoperformwellforcasesoffeworevenasinglesnapshotandclosely-spacedsources[ 18 ].InChapter 1 ,weshowedthatIAAcanbeused,atacostofincreasedcomputation,toproducehigheraccuracyandincreasedsparsitycomparedtoothermethods(includingmatchedlter,IV,etc),inboththenegligibleandnon-negligibleDopplercases.PreviousdescriptionsofIAAforradarimaging,however,havebeenrestrictedtothesingleantennacase. Inthischapter,wewillhighlighttheadvantagesofaMIMOradarsystem.First,thesignalmodelwillbeestablishedinSection 2.2 .Next,wewillbrieydescribesequencedesignforMIMOradarinSection 2.3 .InSection 2.4 ,wewillmotivatetheneedforadvancedreceiverdesignsinMIMOapplications.Wewilldescribethematchedlter,anewleast-squaresapproachtotargetestimation,andalsotheIV 38

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2.6 ,demonstratetheperformanceofaMIMOsystemcomparedtothatofasingle-inputmultiple-output(SIMO)antennaarraysystem,aswellastheperformanceoftheaforementionedreceiverdesigns.ConclusionsareprovidedinSection 2.7 .Intheappendix,weprovideatheoreticallocalconvergenceanalysisofIAA. representthelengthLtransmittedsignalfromthemthtransmitantenna.LetX=x1x2...xM consistofthetransmittedsignalsfromallthetransmitantennas(X2CLM). Thereceivedsignal,synchronizedwiththereectionfromarangebinofinterestp0,canbeexpressedas:DHp0=KXk=1p0,kakbTkXH+QHp0+EH, wherethecomplexscatteringcoefcientsofthetargets,whicharedirectlyproportionaltotheircorrespondingradarcrosssection(RCS),arerepresentedbyfp,kg(p=1,...,Pandk=1,...,K),kdenotestheangleindex,Kisthenumberofpotentialscatterersin 39

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Thetermsfakgandfbkgrefertothereceiveandtransmitsteeringvectors,respectively.Foruniformlineararrays,theycanbedescribedby:ak=ej2(0)sin(k) andbk=ej2(0)sin(k) wheredtanddrrefertothedistancesbetweenadjacenttransmittingandreceivingantennas,respectively,0representsthecarrierwavelengthoftheradarsystem,andkdenotestheimpingingangle(relativetothearraynormal)oftargetsinthekthanglebin.Finally,Jp2RLLisashiftmatrixusedtodescribethereceivedsignalsfromdifferentrangebins,anditcanbewrittenas:Jp=26666666666664jpp0j+1z }| {10...1037777777777775=JTp. 40

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ForasetofMunimodularsequencesfxmg(m=1,...,Mandn=1,...,N),thecross-correlationbetweenthekthandsthsequenceattimelaglisdenedas:rks(l)=NXn=l+1xk(n)xs(nl)=rsk(l) Asinthesinglesequencecase,sequencesetswithlowISLMIMOvaluesaredesirable.MinimizationoftheISLMIMOcanbeperformed,foralldelays,usinganFFT-basedapproach,whichallowsforefcientcomputationandpermitsthedesignoflongersequences.ThisapproachparallelstheCANformulationreviewedinSection 1.3.5 ,andwereferthereaderto[ 47 ]forfurtherdetail.Whenthemaximumlagconsideredislessthanthesequencelength,theCAapproachdescribedin[ 47 ][ 16 ][ 14 ]canalsobeappliedtogeneratewaveformswithlowcorrelationlevelsoverthetime-delayregionofinterest. InFigure 2-1 ,weprovidetheauto-andcross-correlationsforasetofM=4CAsequenceswithlengthN=256.Weconsider30correlationlags(weareonlyinterestedinminimizingfrks(l)ginEquation 2 forjlj<30).ForFigure 2-1 A,weoverlaythesetof4auto-correlations.InFigure 2-1 B,weoverlaythesetof7cross-correlations(i.e.,betweentherstandsecondwaveform,therstandthirdwaveform,etc.).Asevidenced,theauto-andcross-correlationsarewellbelow-250dBintheregionofinterest. 41

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2.4.1MatchedFilterDesign 2 areuncorrelatedwithX.Notethatwhenthisassumptionistrue,thenpost-multiplicationofDHp0byX(i.e.,thematchedlteringoperation)willhavethebenecialeffectofsignicantlyattenuatingthesecond(out-of-range)terminEquation 2 .However,forthistohappen,weneedthatforp=1,...,P,p6=p0,XHJpX2=small(ideally,zero). Inpractice,thisoftenmeansthatXHJpX2ismuchlowerthanXHX2.However,enforcingorthogonalityamongthetransmittedwaveformsaswellasrequiringsuchgoodauto-correlationpropertiesatthesignalgenerationphasesignicantlycomplicatesthewaveformsynthesisproblem.Itmayevenbepossiblethatsuchaconstrainedoptimizationproblemisinfeasible. 46 ],aninstrumentalvariable(IV)approachistakentodealwiththereceiverdesignproblem.Tomitigatethelimitationsonsignaldesignencounteredinthematchedlterproblem,theIVmethoddoesnotimposeanystrictconstraintsonthetransmittedsignal,althoughgoodcorrelationpropertiesarestilldesirable.InsteadofmultiplyingthereceivedsignalbythetransmittedwaveformX,anIVlterW2CLNisinsteadapplied:DHp0W=KXk=1p0,kakbTkXHW+QHp0W+EHW. Ifthesingularvaluedecomposition(SVD)ofXisrepresentedbyX=UXX~UHX, 42

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where~W=UX1X~UHX, 2 :kXHJp(~W+B)k20,p=1,...,P.,p6=p0 AsthenumberofdegreesoffreedominBtendstobemuchhigherthanthoseinX[ 46 ],theIVmatrixWcanachievesuperiorinterferencesuppressioncomparedtothematchedlterapproach.Thisimprovedclutterrejection,however,comesatthecostofadecreasedSignal-to-NoiseRatio(SNR).TheSNRloss,relativetothematchedlterresult,canbeeffectivelyminimizedinpolynomialtimeviasolvingaconvexoptimizationproblem,namely,aQCQP(quadraticallyconstrainedquadraticprogram)(see[ 46 ]fordetails). 43

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whereisa(P1)Mmatrixofzeros,whichisneededtowritethereceivedsignalcompactly.Indeed,thereceivedsignal~DH2CN(L+P1)(nowcontainingtheentirereceivedsignal)canbemodeled,similartoEquation 2 :~DH=PXp=1KXk=1pkakbTk~XH~Jp+~EH, where~E2C(L+P1)Ndenotesthecomplexjammingandnoiseand~JpisdenedsimilarlytoEquation 2 ,exceptthatnow~Jp2R(L+P1)(L+P1).WecanrewriteEquation 2 as: whereSp=KXk=1pkakbTk,p=1,,P, andZH=266664~XH~J1...~XH~JP377775. 44

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wherekkdenotestheFrobeniusmatrixnorm.Theaboveminimizationgivestheestimate^)]TJ/F1 11.95 Tf 10.26 0 TD[(of)]TJ/F1 11.95 Tf 10.26 0 TD[(as:^)]TJ/F5 11.95 Tf 10.26 0 TD[(=~DHZ(ZHZ)1. Inthisway,therangebinestimatesareattainedinasinglestep.Inaddition,thematrixZ(ZHZ)1canbecalculatedpriortoevaluation,whichfurtherreducesruntimecomputations.NotethatZHinEquation 2 isaMP(L+P1)matrix,andhenceanecessaryconditionfor(ZHZ)1in(18)toexistisMP
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1 canbeextendedtotheMIMOsystemcase.Inthefollowingsections,wediscusstheMIMOradarimagingnegligibleDopplerandnon-negligibleDopplerextensions. 1 ]imagingproblemanddescribeshowIAAcanbeextendedtoestimatethetargetparameters,specicallytheamplitude,range,andangleofeachtargetpresent.Thetargetsareassumedtobestationaryandthereforetheintra-pulseDopplereffectsareneglected.Thenon-negligibleDopplercasewillbeanalyzedinthenextsection. 2 canberewrittenasafunctionofthecollectionpositionn(forn=1,...,~N):~DH(n)=PXp=1KXk=1pkak(n)bTk(n)~XH~Jp+~EH(n). Tofurthersimplifynotation,weaccumulatethereceivedsignalsfromeachofthelookpositionsas:d=266664vec(~DH(1))...vec(~DH(~N))377775. Inasimilarway,wecandeneamatrixYthatcontainstheknownquantities(thesteeringvectors,thetransmitwaveformsandtheshiftmatrix)inthereceivedsignalasfollows:Y=y1,1...y1,Ky2,1...yP,K, 46

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and~yp,k(n)=vechak(n)bTk(n)~XH~Jpi,p=1,...,P,k=1,...,K,n=1,...,~N. Consequently,byusingEquations 2 and 2 ,Equation 2 canbeexpressedas:d=Y+e, where=1,1...1,K2,1...P,KT, ande=266664vec(~EH(1))...vec(~EH(~N))377775. TheDASestimatesofthetargetparametersaregivenby:^p,k=yHp,kd yHp,kyp,k,p=1,...,P,k=1,...,K. 2 (westillrefertotheextendedalgorithmasIAAforsimplicity). 47

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Thedatacovariancematrixisdenedas:R=PXp=1KXk=1jp,kj2yp,kyHp,k, whereR2C(N~N(L+P1))(N~N(L+P1)). IAAminimizesthefollowingweightedleast-squarescostfunction(see,e.g.,[ 48 ],[ 49 ])withrespecttothereectioncoefcient,p,k,ofthetargetofinterest:(dp,kyp,k)HQ1p,k(dp,kyp,k). TheminimizationofEquation 2 yieldsthefollowingestimateforthetargetparameters:^p,k=yHp,kQ1p,kd yHp,kQ1p,kyp,k,p=1,...,P,k=1,...,K. NotingthatQp,k=Rjp,kj2yp,kyHp,k, andapplyingthematrixinversionlemmatoEquation 2 (see,e.g.,[ 49 ])yield,yHp,kQ1p,k=yHp,kR1 Consequently,Q1p,kinEquation 2 canbereplacedwithR1,whichneedstobecomputedonlyonce.TheIAAestimateofp,kthenbecomes:^p,k=yHp,kR1d yHp,kR1yp,k,p=1,...,P,k=1,...,K. 48

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18 ].ThealgorithmissummarizedinTable 2-1 .AlocalconvergenceanalysisofIAAisprovidedintheappendix.NotethatRinIAA(seeEquation 2 )iscomputedusingthepreviousestimatesofthetargetparametersfp,kgandnotfromthemeasurementsdasdoneinadaptivearrayprocessingalgorithms.NotealsothatsettingR=IN~N(L+P1)inEquation 2 resultsintheDASestimate. 19 ],canbeused.GiventheIAAestimateforascene,theBICapproachselectsatargetwithrange-angleindices(~p,~k),whichminimizesthefollowingcriterion: BIC(~p,~k)()=2N~N(L+P1)ln24dX(p,k)2fJ()S(~p,~k)gyp,k^p,k2235+4ln(2N~N(L+P1)), wheredenotesthenumberoftargetscurrentlyselected(=1fortherstiteration).Thevalueof4inthepenaltytermofEquation 2 ischosentoreectthenumberofunknownsforeachtarget:amplitude(complexvalued),range,andangle.Also,J()referstothesetoftargetindicesalreadyselectedatthecurrentiteration(J(1)=f;g,wheref;gdenotestheemptyset),and(~p,~k)representsaremainingtargetpointinthescene(i.e.,(~p,~k)=2J()).Ateachiterationofthealgorithm,anewtargetpoint(~p,~k)isselectedsuchthat,alongwiththesetofindicesJ()currentlyselected,BIC(~p,~k)()isminimized.ThisprocedureisrepeateduntilthefunctioninEquation 2 doesnotdecreaseanymore.ReectioncoefcientsnotpresentinJ()attheendofthisprocedurearesettozeroandareassumednottorepresenttruetargetsinthescene. 49

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2 doesnotmattertopeakselection;itmattersonlywhenEquation 2 isusedtoselectthenumberofpeakstoretain. whereDisusedtodenoteDoppler,pistherangeindex,kistheangleindex,histheDopplerindex,HdenotesthenumberofbinsintheDopplerintervalofinterestand!hdenotestheangularDopplerfrequencycorrespondingtothehthDopplerbin,h=1,...,H.Thesteeringvectors,fakgandfbkg,andthenoiseterm,EHD,aredenedsimilarlytotheircorrespondingtermsinEquation 2 ,exceptthatthedependencyonnisremovedsincetheradarisassumedtobestationary.Thecomplexscatteringcoefcientsofthetargets,fp,k,hg,nowincludeathirddimension(indexedbyh)tomodelthetargets'speeds.Welet~xm(!h)=xmd(!h),m=1,...,M,h=1,...,H, 50

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2 ,andd(!h)=1ej!h...ej!h(L1)T,h=1,...,H. Wethendene~XD(!h)similarlytoEquation 2 byletting~XD(!h)=264XD(!h)0(P1)M375, whereXisreplacedbytheDopplershiftedsignalXD(!h)2CLM:XD(!h)=~x1(!h)~x2(!h)...~xM(!h). TowritethesignalmodelgiveninEquation 2 morecompactly,letz=vec(ZH).zcanthenberepresentedinthefollowingform:z=YDD+eD, whereYD=y1,1,1y1,1,2...yP,K,H, withyp,k,h=vechakbTk~XHD(!h)~Jpi,p=1,...,P,k=1,...,K,h=1,...,H. ThecomplexnoiseinEquation 2 issimplydenedaseD=vec(EHD). SimilarlytoEquation 2 ,theDASestimateofthetargetparametersisgivenby:^p,k,h=yHp,k,hz yHp,k,hyp,k,h,p=1,...,P,k=1,...,K,h=1,...,H. 51

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2.5.1.2 willbeused.AsinEquation 2 ,wecanmodelthecovariancematrixofthereceivedsignalby:RD=PXp=1KXk=1HXh=1jp,k,hj2yp,k,hyHp,k,h, whereRD2C(N(L+P1))(N(L+P1)).Werepresentthecovariancematrixoftheinterferencetotheparticulartarget,p,k,h,as:Qp,k,h=RDjp,k,hj2yp,k,hyHp,k,h. Weagainconsidertheweightedleast-squarescostfunction:(zp,k,hyp,k,h)HQ1p,k,h(zp,k,hyp,k,h). MinimizationofEquation 2 withrespecttotheRCSrelatedamplitudeofthetargetofinterestp,k,hyieldstheupdateformula:^p,k,h=yHp,k,hQ1p,k,hz yHp,k,hQ1p,k,hyp,k,h,p=1,...,P,k=1,...,K,h=1,...,H. ThematrixinversionlemmacanagainbeusedtoreplaceQ1p,k,hinEquation 2 withR1Dtodecreasecomputationtime.TheresultingalgorithmissummarizedinTable 2-2 2.5.1.3 .ThecriterioninEquation 2 canberewrittenfortherevisedsignalmodelas: BIC(~p,~k,~h)()=2N(L+P1)ln24zX(p,k,h)2fJ()S(~p,~k,~h)gyp,k,h^p,k,h2235+5ln(2N(L+P1)). 52

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2.5.1.3 withtheonlydifferencebeingtheinclusionoftheDopplerdimensionintheanalysis.Weuse5inthepenaltytermofEquation 2 toaccountfortheunknowntargetparameters:amplitude(complexvalued),range,angle,andDoppler. 2 andRDinEquation 2 couldincreasetounfavorablelevels.InthenegligibleDopplercase,RisinvertibleonlyifPKN~N(L+P1).Forthenon-negligibleDopplercase,invertibilityofRDrequiresthatPKHN(L+P1).Moreover,RinEquation 2 andRDinEquation 2 donotexplicitlyconsiderthecontributionofthenoisetermseandeD,respectively.Toresolvetheseissues,weregularizetheRandRDwithdiagonalmatricesandD,respectively,whosediagonalelementsrepresenttheunknownnoisepowersandarecomputedautomatically.TheregularizedIAAapproach,referredtohereafterasIAA-R,tsnaturallywithintheuserparameter-freeframeworkoftheexistingIAAalgorithm. Whenintra-pulseDopplereffectsareneglected,wecanwritetheregularizedversionofEquation 2 as:R=PXp=1KXk=1jp,kj2yp,kyHp,k+, wherethenoisepowerestimatesalongthediagonalofaredenotedbyf2lgN~N(L+P1)l=1.IAA-RisimplementedasinTable 2-1 ,exceptthatthenumberofunknownsisnow 53

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vHlR1vl2,l=1,...,N~N(L+P1). Thenoisepowerestimatescanbeinitializedasallzerosorasasmall(relativetothestrengthofthetargets)constantnumber. Inthenon-negligibleDopplercase,verysimilarly,wecanwritetheregularizedversionofEquation 2 as:RD=PXp=1KXk=1HXh=1jp,k,hj2yp,k,hyHp,k,h+D, wherethenoisepowerestimates,f2lgN(L+P1)l=1,areagaincontainedalongthediagonalofD.TheIAA-Restimateforlisthen:^2l=vHlR1Dz wherevldenotesthelthcolumnofIN(L+P1). IAA-Rperformswellwithirregularlysampledscanninggridsandwitharbitrarysensorspacings,whereastheinversionofRinEquation 2 andRDinEquation 2 withouttheregularizingtermmightbeproblematicinsuchcases.Certainly,thetechniquecouldbesimilarlyappliedtootheractiveandpassivesensingapplicationsconsideredin[ 18 ]ifdesired. 54

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Weassumethereceivedsignal,z,tohaveacomplexmultivariateGaussiandistributionwithzeromeanandcovariancematrix,RD,giveninEquation 2 ,sothatthelikelihoodofzhastheform:p(zjRD)=1 MaximizationofthelogarithmofthelikelihoodwithrespecttotheunknowntermsinRDisequivalenttominimizationofthefollowingcostfunction:zHR1Dz+lnjRDj. Usingthematrixinversionlemma,therelationshipinEquation 2 ,andthepropertiesofthedeterminantoperation(specically,thatjI+ABj=jI+BAj),weobtain:jRDj=jQp,k,hj(1+jp,k,hj2yHp,k,hQ1p,k,hyp,k,h), andR1D=Q1p,k,hjp,k,hj2Q1p,k,hyp,k,hyHp,k,hQ1p,k,h where(p,k,h)representsanytargetofinterest.Fornotationalconvenience,letp,k,h=jp,k,hj2.ByusingEquations 2 and 2 ,theminimizationofEquation 2 withrespecttop,k,h(forxedQp,k,h)becomesequivalenttominimizing:f(p,k,h)=ln(1+p,k,hyHp,k,hQ1p,k,hyp,k,h)p,k,hzHQ1p,k,hyp,k,hyHp,k,hQ1p,k,hz 55

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2 ,wesettherstderivativeofEquation 2 withrespecttop,k,htozeroandsolveforp,k,h:~p,k,h=yHp,k,hQ1p,k,h(zzHQp,k,h)Q1p,k,hyp,k,h TakingthesecondderivativeofEquation 2 andinsertingtheaboveestimateofp,k,h,wendthatd2f(p,k,h) whichisstrictlypositive.Hence,theestimateofp,k,hinEquation 2 istheglobalminimizerofEquation 2 .Thoughunlikely,itispossiblethattheestimateforp,k,hgiveninEquation 2 couldbenegative.Toenforcenonnegativityofthepowerestimates,theIAA-R-MLestimateisthengivenby:^p,k,h=max0,~p,k,h. Since~p,k,histheuniqueminimizeroff(p,k,h)andsincetherstderivativeoff(p,k,h)isgreaterthanzeroforp,k,h>~p,k,h,wecanconcludethat^p,k,hminimizesf(p,k,h)subjecttop,k,h0. Aswedidintheprevioussections,wereplaceQ1p,k,hinEquations 2 and 2 withR1Dviathematrixinversionlemma(toreducecomputation):^p,k,h=max0,yHp,k,hR1D(zzHRD)R1Dyp,k,h SincethisestimatedependsonRDandp,k,h=jp,k,hj2,wemustagainadoptaniterativeapproach;fp,k,hgcanbeinitializedusingDAS.Furthermore,werecalculateRDaftereachp,k,hisupdated. WecanrewritetheestimateinEquation 2 as:^p,k,h=max0,yHp,k,hR1D(zzH)R1Dyp,k,h yHp,k,hR1Dyp,k,h. 56

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2 isjusttheestimateforp,k,hattainedwithastandardCaponbeamformer(SCB)[ 49 ](assumingRDisinvertible).Ifp,k,hisclosetotheSCBestimate(whichshouldbethecaseatleastlocallyaroundthetruevalues),thentheestimategiveninEquation 2 isapproximatelyequaltotheIAA-Restimate.Inthisway,wemayviewIAA-RasanapproximationtoIAA-R-ML. IAA-R-MLisacyclicalgorithmthatmaximizesthelog-likelihoodfunction,andisthereforelocallyconvergent.SinceIAA-RisanapproximationtoIAA-R-ML,weexpectthatitsharessimilarconvergenceproperties.AlocalconvergenceanalysisforIAA-R,liketheoneforIAApresentedintheappendix,isnotpossiblesinceIAA-Rtakesintoaccountthenoisecovariancematrix,whereastheanalysisintheappendixisconcernedwiththeconvergencetothetrueparametersinthenoise-freecase. 35 36 42 ].TheSIMOsystemunderconsideration,ontheotherhand,containsN=5receiveantennasspacedatdr=0.50andM=1transmitantenna.ThetotaltransmittedpoweroftheMIMOradaristhesameasthatoftheSIMOsystem.Wewillassumecircularlysymmetricindependentandidenticallydistributed(i.i.d.)additivecomplexGaussiannoisewithzero-meanandvariance2throughouttheexamples.Wedenethesignal-to-noiseratio(SNR)as10log10tr(XHX) Foreachoftheguresshownintheseexamples,wextheamplitudescaleusedtorepresentthetargets;weperformthresholdingtothetargetsthatfalloutsideofthese 57

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Inseveraloftheexamples,weusedata-adaptivebeamformingmethods(includingCapon,APES,andCAPES)toproducetheradarimage.ToformaMIMOradarimageusingthesetechniques,werstformasyntheticapertureusingthereturnsfromthe~Npositions,andthenperformrangecompressiontoformthematrix~Dp:~Dp=DHp(1)YMF,DHp(2)YMF,...,DHp(~N)YMF,p=1,...,P, whereYMFrepresentsthematchedlter(YMF=X(XHX)1)andDHp2CNLdenotestheportionofthereceivedsignal(frompositionn)synchronizedwiththereturnfromthepthrangebin.Wecanthenapplyabeamformerto~Dptoobtainanestimatefortargetsatthepthrangebin[ 14 43 ]. 50 51 ]).Ifthemotionoftheradarcannotbeneglected,thenwecouldadjustthesignalmodeltoaccountfortheradar'svelocity. 2.4 .SincetheIVreceivelterYischosensothatXHY=I,withIdenotingtheidentitymatrix,toensureafaircomparison,thefollowingmatchedlterwillbeused:X(XHX)1[ 46 ]. Wewillassume,forthisexample,thatastrongjammeriscontainedinthereceivedsignal.Thestrongjammerwillbeplacedat10,withitswaveformuncorrelatedwiththeMIMOradartransmittedwaveforms.Thejamming-to-noiseratio(JNR),denedasthe 58

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ThetruetargetimageconsistsofP=24rangebins,with2separationbetweentargetsinasinglebin.Theimagecontains30randomlyplacedtargetswithamplitudesselectedrandomlyfromauniformdistributionbetween0and1.WeshowthetruetargetinFigure 2-2 A.WewilltransmitasignaloflengthL=128.ThenumberofSARpositions,~N,willbesetat10.Weletdn,theseparationbetweencollectionpositions,be12.50.TheSNRwillbesetat15dB.AHadamardsequencescrambledbypseudo-noise(PN)willbeusedasthetransmitsignal,whichpossessesgoodauto-correlationandorthogonalityproperties[ 52 ].ToformtheSARimages,wewilluseGLRTtorstidentifythelocationsofthetargets,andthenCAPES(anapproachthatcombinesCaponandAPES)toestimatetheamplitudeandrenethelocationestimates(see[ 46 53 ]forfurtherdetails).Figures 2-2 B,C,DandEshowtheresultingSARimages.NotethatbyusingCAPESandGLRT,wecangetridofthestrongjammerinalltheSARimages.However,duetothepresenceofthestrongjammerat10,severaltargetsnearthejammeranglearemissed. Asevidenced,theLSreceiverdesignperformssimilarlyastheIVlterdesignandoutperformsthematchedlterdesign.UnliketheMFmethod(Figure 2-2 B),fortheIVandLSmethods(Figures 2-2 Cand 2-2 D),theresultingSARimagesdonotcontainasignicantnumberoffalsealarms.WeobservethatthefalsealarmsoftheIVandLSapproachesarenotlocatedatthesamepositions,whichmeansthatwecanreducethefalsealarmratefurtherbyonlykeepingthetargetspresentinboththeIVandLSimages,asshowninFigure 2-2 E,whereeachtargetamplitudeistakenastheaverageofthoseestimatedbyIVandLS. 59

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Figures 2-3 A,B,andCdemonstratetheperformanceofDAS,IAA-R,andIAA-RwithBICusingaSIMOradar.InFigure 2-3 A,aPNsequence[ 54 ],whichhasreasonablygoodauto-correlationproperties,istransmittedandthetargetparametersareestimatedatthereceiverusingDAS.ThesamePNsequenceistransmittedforthecaseinFigure 2-3 B,butnowIAA-RisusedforestimatingthetargetparametersinsteadofDAS.InFigure 2-3 C,BICisappliedtothesetofIAA-Restimates.TheresultsobtainedusingaMIMOradarareshowninFigures 2-3 D,E,andF.Foreachoftheseexamples,orthogonalHadamardwaveforms,scrambledwithaPNsequence,aretransmitted.InFigures 2-3 DandE,DASandIAA-R,respectively,areusedtoestimatethetargetparameters.TheIAA-RestimatewhentheBICalgorithmisappliedisshowninFigure 2-3 F. FromFigure 2-3 ,weobservethattheMIMOsystemoffersmuchimprovedresolutionovertheSIMOsystem,evenwhenusingDASasthereceivelter.Ontheotherhand,itisobservedthatIAA-RsuccessfullyrenestheDASestimatesinboththeSIMOandMIMOcasestoeffectivelyreducethesidelobelevels.IAA-RshowsquitegoodestimationaccuracyintheMIMOcase.ToevaluatetheperformanceofIAA-RwithBICfortheSIMOandMIMOsystems,wecomparetheprobabilityofdetection(PD)andtheprobabilityoffalsealarm(PFA)foreachcase;weletPD=(numberoftargetsdetected)/(totalnumberoftruetargets)andPFA=(totalnumberoffalsepositives)/ 60

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2-3 C,PD=65.5%andPFA=1.0%.FortheMIMOresultinFigure 2-3 F,PD=96.6%andPFA=0%.AsBICdoesnotprovidesatisfactoryresultsfortheIAA-RresultfromtheSIMOsystem,weommittheseimagesinthenextexamples.WefurthermoreneglecttoshowtheresultsofapplyingBICtotheDASresults,asthepoorresolutionoftheseimages,withBICapplied,wouldleadtoanunacceptablylowPD[ 55 ]. InFigure 2-4 ,weshowtheresultsofusingIAAwithoutregularizationfortheMIMOcase.ForFigure 2-4 A,theradarscansanangularintervalofinterestrangingfrom30to30,with1angulargridsize,asinthepreviousimages.Since,inthiscase,theconditionnumberofRinEquation 2 reachesunfavorablelevels(duetoKbeingsmallrelativetotheentireangularregion),IAAsuffersfrompoorperformance.InFigure 2-4 B,ontheotherhand,thescanningregionisincreasedto90through90(theentirescanningregion),againwith1angulargridsize.Inthisimage,however,onlytheangularregionofinterest(30to30)isshown.TheconditionnumberofRinEquation 2 issignicantlyreducedinthiscase(sinceKismuchlarger),andtheperformanceofIAAimprovesdrastically.Thus,weconcludethatIAAworkswellwhentheentireangularregionisconsidered.Ifthetargetsareknowntoexistwithinasmallerregionrelativetothewholescanninggrid,thentheactualscanninggridcanbedecreasedtoreducecomputationalcostsbyusingIAA-RinsteadtoavoidproblemsthatmightarisewiththeinversionofR. 61

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Figures 2-5 A,B,andCshowtheDAS,Capon,andAPESestimatesofthescene,respectively.Data-adaptivemethods,suchasCaponandAPES,performpoorlywhenthenumberofdatasnapshots,whichisrepresentedbythenumberofcolumnsinEquation 2 (forthisexample,M~N=15),isnotsignicantlygreaterthanthenumberofarraysensors(N=5)[ 45 ].Inthiscase,theestimatedsignalcovariancematrix,whichdependsdirectlyonthenumberofdatasnapshots,candiffersignicantlyfromthetruecovariancematrix.Whenonlyasinglesnapshotisattained(M=~N=1inEquation 2 ),adaptivebeamformingmethodsfail,sincethesamplecovariancematrixbecomessingular.IAA-R(andIAA),ontheotherhand,canperformwellevenwhenthenumberofsnapshotsisone.Forthisexample,weseemuchbetterresultsusingIAA-RandIAA-RwithBIC(comparedtoCaponandAPES),asshowninFigures 2-5 Dand 2-5 E,respectively. InFigure 2-5 F,aplotofthemean-squarederror(MSE)ofIAA-Rversustheiterationnumberisshown,withMSEdenedas:kB0B(i)IAA-Rk2F whereidenotestheiterationnumber,B0denotesthegroundtruthandB(i)IAA-RdenotestheIAA-Rrange-angleimageestimateatiterationi.Iteration0denotestheMSEofDAS(asIAA-RisinitializedwithDAS).NotethattheMSEvaluedecreasesmonotonicallywiththeiterationnumberandappearstoconvergeafteronlyafewiterationsofIAA-R. 2.5 ,wedonotconsidertheIVandLSlterdesignsforthisnon-negligibleDopplercase. 62

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2-6 AandB.Inthesegures,weexaminearange-Dopplersliceofthetargetestimatestakenat1relativetoabroadsidescan.TheresultsobtainedusingaMIMOarrayareshowninFigures 2-6 C,D,andE,usingDAS,IAA-R,andIAA-RwithBIC,respectively.Fortheseimages,weagaintransmitHadamardwaveformsscrambledwithPNsequences.Fromtherange-Dopplerresults,wecanobservethattheDASimagesforboththeSIMOandMIMOarrayshavesignicantlylowerresolutionthanthecorrespondingIAA-Rimages,andnoneofthetargetscanbeidentiedusingDAS.Furthermore,theIAA-RimageobtainedusingaMIMOarraydemonstratessuperiorperformanceoverthatobtainedusingaSIMOarray;theIAA-RwithBICalgorithmproducesasparseresultandidentiesallofthetargetsinthescene. 63

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2-6 .TheDASandIAA-RresultswithaSIMOarrayareshowninFigures 2-7 AandB,respectively.FortheMIMOradarcase,theDASandIAA-RresultsareshowninFigures 2-7 CandD,respectively.Duetothestrongpresenceofthegroundreturninthisangle-rangeslice,theBICresultwasnotsatisfactoryandthustheresultisnotshown.Asevidenced,thepoorresolutionandaccuracyofDASisfurtheremphasizedfromtheangle-rangeperspective.Inaddition,theimprovedangularresolutionthatresultsfromusingtheMIMOarray,ascomparedtotheSIMOantennaarray,isdemonstratedintheIAA-Rresult.TheIAA-RresultagainshowsnotablybetterperformancethanDAS. InFigures 2-7 Aand 2-7 C,anull(approximatelyzeroamplitude)occursintheamplitudeestimatesat14,acrossallrangebins.WhenweapplyaDASreceiveltermatchedto14,azerooccursinthelterresponseatprecisely9.Thus,thecontributionfromthecluttergroundreturnisremovedfromtargetestimatesat14.InFigure 2-7 E,weshowthebeampatternresponseoftheDASlter,whichissteeredto14.Asevidenced,thegroundclutterreturnoffersnocontributiontothesetargetestimates. 64

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Further,wehavedescribedtheextensionofIAAforMIMOradarimagingapplicationswithbothnegligibleintra-pulseDoppler(e.g.,spotlightSARstationarytargetimaging)andwithnon-negligibleintra-pulseDoppler(mobiletargetimaging).WefurtherprovidedalocalconvergenceanalysisofIAA.AregularizedversionofIAA,IAA-R,wasproposedinordertoimprovetheperformanceofIAAandtomakeitworkwithincompletescanningregions,whichisusuallythecaseinradarapplications.WealsoderivedthemaximumlikelihoodbasedIAA-R-ML,whichassumesastatisticalmodelonthereceivedsignalandisguaranteedtobelocallyconvergent.WeshowedhowIAA-RcanbeviewedasanapproximationtoIAA-R-ML. Inthenegligibleandnon-negligibleDopplercases,IAA-Rdemonstratedsuperiorperformanceforbotharraysystems,ascomparedtoDAS,adaptivebeamformingtechniques,andtheIAAalgorithmwithincompletescanningregionsandwithoutregularization.ByincorporatingtheBICalgorithm,wewereabletoproducefurthersparsityintheIAA-Rresult. 65

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2 :d=A+e, whered2CM1,A2CMN(N>M),2CN1,e2CM1,andA=a1...aN. Theanalysisbelowassumesthateisnegligible.Intheabsenceofnoise,disassumedtohavetheform:d=~A~, where~A2CMK,(K
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Letidenotetheiterationindex.Then(seeEquations 2 and 2 ),forn=1,...,N,i+1n=aHn(Qin)1d aHn(Qin)1an, andQin=PNk=1,k6=njikj2akaHk+I=Rijinj2anaHn. 2 canbechosenonlyslightlylargerthanthevalueneededtoavoidanill-conditioningmessageduringthenumericalimplementation.Inpracticalapplications,onemaychoose=0rstuntiltheill-conditioningwarningoccurs,atwhichpointonemayreplacethe=0withasmall>0.(b)Usingthematrixinversionlemma,wecanshowthat(Qin)1inIAAcanbereplacedby(Ri)1(seeEquation 2 andEquation 2 ).However,assuggestedbythecalculationsbelow,thecomputationaladvantagethatfollowsfromthisreplacementisoffsetbythefactthat,closetoconvergence,theuseof(Qin)1inIAAturnsouttoleadtoanumericallymorestablealgorithm.Inpracticalapplications,onemayuse(Ri)1atthebeginningoftheiterationsuntiltherstill-conditioningwarningoccurs,atwhichpointonemayreplace(Ri)1with(Qin)1(seeEquation 2 below). ConsiderrstthestationarypointsofIAA.Thesepointssatisfytheequation:n=aHnQ1nd aHnQ1nan,n=1,...,N. 67

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2 yieldsn=aHnd whichisgenerallyacontradiction.Thisresultstarklycontrastswhathappensformostsparsealgorithms(suchasFOCUSS,etc.),forwhichn=0isastationarypoint[ 56 ].Moreimportantly,wenotethatthetrue~givesastationarypoint(as!0),i.e.,n=8><>:~k,n=nk,k=1,...,K0,n6=nk,k=1,...,K. Toseethis,rstobservethatthematricesQncorrespondingtoEquation 2 aregivenby:Qn=8><>:~Ak~Pk~AHk+I,n=nk,k=1,...,K~A~P~AH+I,n6=nk,k=1,...,K, where~Akisthematrix~Awithoutthekthcolumn,~P=266664j~1j20...0j~Kj2377775, and~Pkistheabovematrixwithoutthekthrowandcolumn. Next,letusconsiderageneralmatrixhavingtheformofEquation 2 :Q=BBH+I, 68

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whereP?BistheorthogonalprojectorontothenullspaceofBH. ItfollowsfromEquation 2 andEquation 2 ,aswellastheassumptionsmade,that(fortendingtozero):n=8><>:aHnkP?~Ak~A~ NotethataHnkP?~Akank6=0becauseank62Range(~Ak),andsimilarlyforaHnP?~Aan.Withtheseobservations,theproofthatEquation 2 isastationarypointofIAA(for!0)isconcluded. 2 wouldhavebeenaHnkP?~Aank=0,whichpointstotheill-conditioningthatmightbecausedbysuchareplacementasbrieymentionedinRemark(b)followingEquation 2 Next,weprovethelocalconvergenceofIAAtothetruevalues.LetusdenotethetruevaluesinEquation 2 byf0ng:0n=8><>:~k,n=nk,k=1,...,K0,n6=nk,k=1,...,K. AssumethatIAAisinitializedasfollows:n=8><>:ck,n=nk,k=1,...,KO(),n6=nk,k=1,...,K, 69

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whichimpliesthatIAAislocallyconvergentwithatleastaquadraticrate(notethattheattributelocalherereferstotheinitializationofthezerocomponentsoff0ng;thenon-zerocomponentsofthisvectorcanbeinitializedarbitrarily!). ToproveEquation 2 ),rstobservefromEquation 2 that:jnj2=O(2),n6=nk,k=1,...,K. LetQn=NXk=1,k6=njkj2akaHk+I, letSdenoteanorthonormalbasisofRange(~An),andletGcompriseanorthonormalbasisofthenullspaceof~AHn.(WeomitthedependenceofSandGonntosimplifythenotation.)Usingthisnotation,wecanwrite(undertheassumptionsmade):Qn=~AnC~AHn+BnDBHn+2I, where~An=~Aforn6=nk(k=1,,K)byconvention,Bnisthematrixwhosecolumnsareequaltothevectorsfakgk6=ninEquation 2 thatdonotappearin~A,CisadiagonalmatrixwithdiagonalelementsfckgKk=1,k6=ncorrespondingtothefakgin~An,andDisadiagonalmatrixmadefromthetermsfjkj2gcorrespondingtothefakginBn;henceD=O(2)(seeEquation 2 ). 70

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andBn=SG264EF375. InsertingEquations 2 and 2 intoEquation 2 ,wecanrewritetheexpressionforQnas:Qn=SHCHHSH+SG264EDEHEDFHFDEHFDFH375264SHGH375+2SG264SHGH375=SG264W1W3WH3W2375264SHGH375, whereW1=HCHH+EDEH+2I=HCHH+O(2) tendstoanonsingularmatrix,namelyHCHH,as!0,W2=FDFH+2I=O(2) isanonsingularmatrixforany>0,andW3=EDFH=O(2). 71

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2 existsandisgivenby:264W1W3WH3W23751=264(W1W3W12WH3)1(W1W3W12WH3)1W3W12W12WH3(W1W3W12WH3)1(W2WH3W11W3)1375. Observethatthe(1,1),(1,2),and(2,1)blocksoftheabovematrixtendtoconstantmatricesas!0,whereasthe(2,2)blocktendstoinnityas1=2.ItfollowsfromthisobservationalongwithEquation 2 andEquation 2 that:2Q1n=SG264O(2)O(2)O(2)375264SHGH375=GGH+O(2), where=2(W2WH3W11W3)1=I+F1 tendstoaconstantpositive-denitematrixas2!0. MakinguseofEquation 2 inthemainequationoftheIAAalgorithmyields:^n=aHn2Q1nd aHn2Q1nan=aHnGGHd aHnGGHan+O(2), fromwhichwecanobtainEquation 2 byacalculationsimilartoEquation 2 .TheproofofEquation 2 isthusconcluded. InFigures 2-8 AandB,weprovideanexampletoillustratetheconvergencebehaviorofIAAanditsdependenceon,asdiscussedabove.Wesimulatea1-DpassivearraywithN=10receivingantennasseparatedatdr=0.50andwithonlyasingledatasnapshot.Thisantennaarrayscansfrom90to90,relativetothearraynormal,with1separationbetweenadjacentscanningpoints.Twotargetsignals,eachwithunitpower,arelocatedat7and14,respectively.Weneglectnoiseinthisexample.InFigure 2-8 A,weshowtheDASestimateforthiscase.InFigure 2-8 B,we 72

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73

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IAAforMIMOSARimaging. yHp,kyp,k,p=1,...,P,k=1,...,KrepeatR=PXp=1KXk=1j^p,kj2yp,kyHp,k^p,k=yHp,kR1d yHp,kR1yp,k,p=1,...,P,k=1,...,Kuntil(acertainnumberofiterationsisreached) Table2-2. IAAforAngle-Range-DopplerimagingwithaMIMOarray. yHp,k,hyp,k,h,p=1,...,P,k=1,...,K,h=1,...,HrepeatRD=PXp=1KXk=1HXh=1j^p,k,hj2yp,k,hyHp,k,h^p,k,h=yHp,k,hR1Dz yHp,k,hR1Dyp,k,h,p=1,...,P,k=1,...,K,h=1,...,Huntil(acertainnumberofiterationsisreached) 74

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B Figure2-1. OverlaidcorrelationsforaCAsequencesetwithM=4,N=256,andP=30.A)Auto-correlations.B)Cross-correlations. 75

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CD E Figure2-2. SpotlightSARimagesfor~N=10,L=128JNR=100dBandSNR=15dB.A)Truetargetdistribution.B)MF.C)IV.(D)LS.E)CombinedIVandLS. 76

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CD EF Figure2-3. SpotlightSARimagesfor~N=1,L=64andSNR=20dB.A)DASwithaSIMOarray.B)IAA-RwithaSIMOarray.C)IAA-RwithaSIMOarrayandBICapplied.D)DASwithaMIMOarray.E)IAA-RwithaMIMOarray.F)IAA-RwithaMIMOarrayandBICapplied.AlllevelsshownareindB.`O'denotesatruetargetand`X'representsatargetestimate. 77

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Figure2-4. MIMOSARimagesfor~N=1,L=64andSNR=20dB.A)IAAwithoutregularizationandwithascanninggridfrom30to30.B)IAAwithoutregularizationandwithascanninggridfrom90to90.AlllevelsshownareindB.`O'denotesatruetargetand`X'representsatargetestimate. 78

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CD EF Figure2-5. MIMOSARimagesfor~N=3,L=32andSNR=15dB.A)DAS.B)Capon.C)APES.D)IAA-R.E)IAA-RwithBICapplied.F)MSEvs.iterationnumberforIAA-R.ThelevelsshownareindB.`O'denotesatruetargetand`X'representsatargetestimate. 79

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CD E Figure2-6. Range-Dopplerimagesat1relativetoabroadsidescanforL=32andSNR=20dB.A)DASwithaSIMOarray.B)IAA-RwithaSIMOarray.C)DASwithaMIMOarray.D)IAA-RwithaMIMOarray.E)IAA-RwithaMIMOarrayandBICapplied.AlllevelsshownareindB.`O'denotesatruetargetand`X'representsatargetestimate. 80

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CD E Figure2-7. Angle-rangeimagesat1DopplerforL=32andSNR=20dB.A)DASwithaSIMOarray.B)IAA-RwithaSIMOarray.C)DASwithaMIMOarray.D)IAA-RwithaMIMOarray.E)ReceivelterresponseforimagesAandC.LevelsshownareindB.`O'denotesatruetarget. 81

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Figure2-8. PassivearraytargetestimatesforN=10.A)DAS.B)IAA.`O'denotesatruetargetlocation. 82

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57 58 ].Whenthelocationsoftargetsinasceneareunknown,phaseshiftscanbeappliedtothetransmittingantennastosteerthefocalbeamacrossanangularregionofinterest.Incontrast,multiple-inputmultiple-output(MIMO)systems,bytransmittingdifferent,possiblyorthogonalwaveforms,canbeusedtoilluminateanextendedangularregionoverasingleprocessinginterval.Waveformdiversitypermitshigherdegreesoffreedom,whichenablestheMIMOsystemtoachieveincreasedexibilityfortransmitbeampatterndesign(see,e.g.,[ 36 ]-[ 60 ]). Thetransmitbeampatternforanyactivesensingsystemdepends,inpart,onthecovariancematrix(whichwewilldenotebyR)ofthetransmittedsignals.Whenaphased-arrayisadopted,whosewaveformsarefullycoherent,rank(R)=1. UsingaMIMOsystemwithorthogonalwaveforms,conversely,Rcorrespondstoadiagonalmatrixwhoseentriesrepresenttheelementalpowersofeachtransmittingantenna.Beampatterndesignusingpartiallycoherentwaveformshasbeenextensivelydiscussedin[ 39 41 ].Therein,theauthorsdescribedhowconvexoptimizationcanbeusedtodetermineanR(undercertainpowerconstraints)whichbestapproximatesadesiredbeampattern.Additionally,[ 41 ]presentsbeampatternmatchingandminimumsidelobeleveldesignapproaches,andfurtheraddresseshowadesiredbeampatterncanberstdetermined.Theauthorsin[ 39 41 ]assumedthatthepositionsofthetransmittingantennas,whichalsoaffecttheshapeofthebeampattern,werexedprior 83

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Atthereceiver,sparse,orthinned,arraydesignhasbeenthesubjectofanabundanceofliteratureduringthelast50years(see,e.g.,[ 61 ]-[ 74 ]andthereferencestherein).Thepurposeofsparsearraydesignhasbeentoreducethenumberofantennas(andthusreducethecost)neededtoproducedesirablespatialreceivingbeampatterns.Furthermore,previousresearcherssoughttoavoidthetaperingofuniformarraysforsidelobelevelcontrol,whichresultsinanincreasedmainbeamwidth[ 62 65 ].Naturally,Dolph-Chebychevdesignapproacheshavebeentraditionallyadopted,wherebysidelobelevelsareminimizedwithrespecttomainbeamwidthrestrictions(see,e.g.,[ 63 74 ]). Priortotheadventofhigh-speedcomputing,manypreviousauthorsattemptedtomodelthespatialreceivebeampatternusingseriesexpansions,whichcouldonlyapproximateadesiredfunction[ 61 64 ].Athoroughreviewofolderworkscanbefoundin[ 66 ].Morerecentapproacheshavesoughttodesignreceiveantennapositionsandbeampatternweightsviaoptimizationstrategies,includingasimplex-typesearch[ 75 ]andbranch-and-boundtechniques[ 73 ].Geneticalgorithms[ 71 ]andsimulatedannealing[ 72 ]approacheshavealsobeenproposedforreceivebeampatterndesign.Additionally,iterativeoptimizationstrategiesarepresentedin[ 67 ]-[ 70 ].In[ 67 ],theauthorcyclicallyminimizesanl2costfunctiontomatchadesiredreceivepatternwithanactualone.Similarly,thealgorithmdescribedin[ 69 ]involvesthesequentialadditionofantennastoasystem,suchthatantennaweightsandpositionsareoptimizedateachaddition.Theauthorsin[ 70 ]insteadproposestartingwithafullarraystructure,andtheniterativelyremovingtheantennaswhichproducethehighestsidelobelevelsinthereceivebeampattern.Duetononconvexdesignconstraints(specically,thatthenumberofantennasinasystemisrestrictedtosomeconstant),onlylocally,ratherthanglobally,optimalsparsearraydesignapproacheshavebeenshowntoexist. 84

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76 77 ])viathedesignofsparseantennaarrays.Ouralgorithmscanbeseenasextensionstotheiterativereceivebeampatterndesignsdescribedin[ 67 ]-[ 70 ].InSection 3.2 ,wewillpresentourcyclicalgorithmfortransmitbeampatterndesignusingsparseantennaarrays.Wewilldescribesimilarapproaches,inSections 3.3 and 3.4 ,toprovidematrixandvectorreceivebeamforming,respectively,viasparsearrays.WepresentnumericalsimulationsinSection 3.5 ,andweofferconclusionsinSection 3.6 where0denotesthecarrierwavelengthofthesystemanddenotesthesteeringangle,relativetoarraybroadside.IftheangularscanninggridisspannedbyfkgKk=1,thenwecancollectthesetofsteeringvectorsintothesteeringmatrixAT2CNTK:AT=aT(1)aT(2)aT(K). 85

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andwhereEfgreferstotheexpectationoperator.Thetransmitpoweratsomeanglek(k2f1,...,Kg)isgivenby:p(k)=aHT(k)RTaT(k). Thetransmitbeampattern,nowforeachanglefkgKk=1,canberepresentedinitsvectorformas:p=p(1)p(K)T,=diagAHTRTAT. Furtherdiscussionon( 3 )-( 3 )canbefoundin[ 39 41 ].Inasimilarwayto( 3 ),wecandeneadesiredreal-valuedtransmitbeampatternaspd:pd=pd(1)pd(K)T. Forsparsearraydesign,weassumethatonlyMT
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3 )closelyapproximatesthedesiredonein( 3 ). 3 ))andaninitialvalueforoursynthesizedbeampatternp(see( 3 )),wenowproposeaniterativeapproachtotransmitbeampatterndesign.Ineachofourexamples,wewillassumethataULA,withantennaspacing0=2,andorthogonalwaveforms(sothatR=IMTMT)areusedtoinitializethealgorithm.WewillrstdescribeanapproachtocyclicallyupdateR,givenasetofantennapositions.Then,inthenextsubsection,wewillexplaintheprocedureforantennaselection. wherethepositivesemideniteconstraintin( 3 )impliesthatRisindeedatruecovariancematrix.Further,byconstrainingaH(c)Ra(c)=1,weensurethatthebeampatternhasunitheightatsomeuser-determined,centralangle(weassumethatc2fkgKk=1).ThematrixZisadiagonalmatrixcontaininguser-assignedweights,denotedbyfz(k)0gKk=1:Z=266666664z(1)000z(2).........000z(K)377777775. 87

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ByintroducingamatrixU2CMTK,wecanreformulatetheproblemin( 3 )as:minR,UkZdiag[AHRA]diag[UHU]k2s.t.diag[UHU]=ZpdaH(c)Ra(c)=1R0. Finally,wecanapproximatetheoptimizationproblemin( 3 )usingthefollowing,similarproblem:minW,UkWHAZ1=2Uk2s.t.diag[UHU]=ZpdkWHa(c)k2=1, whereW2CMTMTandR=WWH.By( 3 )beingsimilarto( 3 ),wemeanthatanysolutionWandUthatmakes( 3 )smallwillleadtoasmallvaluefortheoptimizationcriterionin( 3 ),andviceversa.Forfurtherdetailsregardingthetypeofproblemin( 3 )anditsrelationshipto( 3 ),see,e.g.,theexplanationofferedin[ 15 ]. AssumingthataninitialvalueforRhasbeenprovided,wecaninitializeWbylettingW=R1=2.OnceaninitialvalueforWhasbeenattained,wecandirectlyapplyacyclicapproachtominimizetheoptimizationcriterionin( 3 ).Thestepsareasfollows: 3 )are:uk=WHa(k) whereukrepresentsthekthcolumnofU(fork=1,...,K).Aderivationof( 3 )isprovidedinAppendixA. 88

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3 )canbedeterminedasfollows.ChooseanMT(MT1)matrixV,suchthatVHa(c)=0andVisfullcolumn-rank.Then,W=a(c)^dopt+V^Yopt, where^dopt=aH(c)AZ1=2?Z1=2AHVUH and?Z1=2AHV=IZ1=2AHV(VHAZAHV)1VHAZ1=2. WemotivatetheexpressionforW,givenin( 3 ),inAppendixB. 3 )). Considerthemthantennainthetransmitarray.Ifweassumethattheremaining(MT1)antennasinourarrayarecurrentlyxedinposition,then(NTMT)positions(inourcandidatesetofpositions)currentlydonotcontainanantenna.Thus,werepositionthemthantennainoneofthe(NTMT+1)availablepositions(whichincludesthecurrentpositionofthemthantenna),suchthat,uponapplyingthecyclicprocedureinSection 3.2.2.1 ,theoptimizationmetricin( 3 )isbestminimized.Torepositionthemthantenna,wesimplyimplythatthemthrowofAin( 3 )hasbeendesignedtoreect 89

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Inpracticalapplications,thephysicalsizeofantennasandmutualcouplingeffects(see,e.g.,[ 78 ])canlimittheminimumpermissiblespacingbetweenantennas.InlieuofusingalargerdT,weinsteadseektopromotedesignexibilitybymaintainingadensegridofcandidatepositions(andthusasmalldT).Toenforceaminimumantennaseparationdistance,wesimplyomitcandidatepositionsthatneighborexistingelements(thosepositionsthataretooclosetoxedantennas)whenwerepositiontheantennasduringaniteration.Inthisway,wemediatedesignconstraintswhilestillallowingforsufcientfreedomduringantennaselection.Forfurtherdetails,seeExample3inSection 3.5 3.2 ,wecanalsoconsiderapplyingourcyclicsparsearraydesignapproachtoMIMOreceivebeamforming.Sincereceivebeamformingdoesnotdependonthecorrelationpropertiesofasystem'stransmittedwaveforms,thefollowingdiscussioncanfurthermorebeappliedtoanyradarapplicationinvolvingmultiplereceiveantennas.Traditionalreceivebeamformingdesignshaveadoptedvectorweightingapproaches.However,asinthetransmitbeampatternformulationinSection 3.2 ,wecouldalsoconsideramatrixweightingapproachtoreceivebeamforming(see,e.g.,[ 79 ]). 3 ),thereceivesteering 90

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wheredRreferstotheminimumseparationbetweenreceivingantennas,NRreferstothetotalnumberofreceiveantennas,anddenotesthereceivearraysteeringdirectionrelativetothearray'sbroadside(with2fkgKk=1).Additionally,wecandenethesteeringmatrixBR2CNRKby:BR=bR(1)bR(K). IfXR2CNRNRreferstoamatrixbeamformer,thenwedenethematrixVR2CNRNRas:VR=XRXHR. Inthisway,thereceivebeampatterncanberepresented:p=diagBHRVRBR. Toobtainasparsearray,werestrictthenumberofavailableantennasforbeamformingdesigntosomeMR
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whereZagainreferstoadiagonalmatrixofweightvalues(see( 3 ))andc2fkgKk=1representsauser-dened,centralangleofthedesiredbeampattern. Thereceivebeampatterndesignproblemin( 3 )closelyresemblesthetransmitproblemdescribedin( 3 ).Thus,wecanconsiderapplyingthesparsearraydesignapproachdescribedinSection 3.2.2 tonowconstructasparsereceivearrayusingmatrixweighting.Asin( 3 ),weconstrainVtobepositivesemidenite(V0),whichisimpliedbythedenitionofVRin( 3 ).SincethemethodologyforsparsereceivearraydesignusingmatrixweightingdirectlyparallelsthealgorithminSection 3.2.2 ,weomitthedetailsofthisapproach.Fordetailsontheantennaselectionscheme,pleaseseeSection 3.4.2.2 61 ]-[ 74 ]).Infact,receivebeamformingusingvectorweightingcanbeviewedasamoreconstrainedversionofthegeneralmatrixweightingapproachdescribedinSection 3.3 .Next,weshowhowthecyclicapproachtosparsereceivearraydesignviamatrixweightingcanbemodiedtoperformreceivebeamformingwithvectorweights. 92

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3.3 ,withtheadditionalconstraintthat(see( 3 )):rank(V)=1. From( 3 ),wecanwriteVinthefollowingform:V=vvH, wherev2CMR1.Usingtheresultin( 3 ),thesynthesizedbeampatternwithvectorweightingcanberepresentedby:p=diagBHvvHB. 3.3 ,wedescribedhowreceivebeamformingusingmatrixweightingcanbeformulatedintoanidenticaloptimizationproblemtothatofthetransmitbeampatterndesigncaseofSection 3.2 (withoutanyrankconstraintsonthecovariancematrixR).Similarly,vectorweightingforreceivebeamforming,whichincludestheadditionalconstraintin( 3 ),isanalogoustotransmitbeampatterndesignwiththerestrictionthatrank(R)=1.Underthisassumption,wearereducedtoaphased-arrayactivesensingsystem(see( 3 )).Althoughweneglecttoaddresssparsetransmitarraydesignforphased-arraysystems(sincewefocus,herein,onapplicationsinwaveformdiversity),certainlythisformulationwoulddirectlyconformwithourdiscussiononvectorweightingforreceivebeamforming. 93

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3.4.2.1Weightdeterminationviaacyclicapproach 3 ),nowforthevectorweightingcase,canberewrittenas:minvkZ(diag[BHvvHB]pd)k2s.t.bH(c)vvHb(c)=1. Sincethedesiredbeampatternpdrepresentsasetofpowers,wecaninsteadconsiderthefollowingsimilaroptimizationproblem[ 80 ](seealso[ 76 77 ]):minv,kZBHvZp1=2dk2s.t.bH(c)v=1, where=266666664ej1000ej2...............000ejK377777775, andwherefk2[0,2)gKk=1areauxiliaryvariables.Further,thevectorp1=2distheelementwisesquarerootofpd,suchthatpd=p1=2dp1=2d.Notethat,in( 3 ),werestrictbH(c)v=1,ratherthanjbH(c)vj=1(anonlinearconstraint),sincecertainlyauniformphasechangetotheelementsinvwouldnotaffectthesynthesizedbeampatternin( 3 ). Assumingthatwearegivenaninitialvalueforthevectorv(fortherstiterationofthealgorithm,wewillassumethatthecomponentsofvare1=MR),wecanapplyacyclicminimizationapproachtotheoptimizationcriterionin( 3 )toobtainanupdatedweightvector.Thealgorithmcanbesummarizedinthefollowingsteps: 94

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3 )withrespecttofkgcanbedirectlyobtained:diag[]=ejarg(BHv). 3 )isgivenby:v=(BZHZBH)1"BZHZp1=2dbH(c)(BZHZBH)1BZHZp1=2d1 Weprovideaproofof( 3 )inAppendixC. 3.2.2.2 .WeagainassumethatMRinitialantennapositionsarespeciedfromasetofNRcandidatepositions.Assumingthattheother(MR1)antennasarexedinposition,weconsiderplacingthemthantennainanyoftheavailable(NRMR+1)availablepositions(againincludingtheantenna'scurrentposition).WeapplytheiterativeprocedureinSection 3.4.2.1 foreachcandidatelocation,andwethenmovethemthantennatothatpositionwhichbestminimizesthecriterionin( 3 ).RepositioningthemthantennacorrespondstoadjustingthemthrowinB(andthus,themthcomponentofeachsteeringvector)toreecttheantenna'scurrentposition.ThisoperationisrepeatedforeachoftheMRantennas.Finally,weperformtheprocedureiterativelyuntilconvergence,suchthattheantennasarenolongerrepositioned. Asbefore(seeSection 3.2.2.2 ),wecouldalsoconsiderenforcingaminimumseparationdistancebetweenantennasduringtheselectionstage(toaccountformutualcoupling,etc.).TokeepdRsufcientlysmall(andthusensureadensegridofcandidate 95

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3-1 (a),andisgivenby(see( 3 )):pd()=8><>:1,2[9,13],0,otherwise. Inpractice,wecanselectadesiredbeampatternbyrsttransmittingomnidirectionalwaveforms,sothattheapproximateangularlocationoftargetsinascenecanbedetermined.Then,weareabletofocusthetransmittedenergytowardtheestimatedtargetlocation.Forfurtherdetails,see[ 41 ]. Thediagonalcomponentsfz(k)gKk=1oftheweightmatrixZ(see( 3 )),forthisexample,werechosenas:z()=8>>>>>>><>>>>>>>:1,=11,0.5,2[1,10]S[12,23],0.25,2[21,2]S[24,43],0.15,otherwise. 96

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3 )).Theweightvaluesareplotted,asafunctionofangle,inFigure 3-1 (b).Clearly,usingtheweightsspeciedin( 3 ),wesoughttoconstructabeampatternthatcloselymodeledthepulseinFigure 3-1 (a),andsidelobesfurtherawayfromthemainbeamwerelesspenalizedinthecostfunction(see( 3 )).Certainly,differentsparsearraysanddifferentbeampatternscanbeachievedbyadjustingtheweightparametersin( 3 ). UsingaULAwith10antennasandinter-elementspacing0=2,weappliedthecyclicapproachinSection 3.2.2.1 togenerateanoptimizedcovariancematrixR.ThetransmitbeampatternfortheULA,withoptimizedcovariance,isshowninFigure 3-1 (c).ForoursparseapproachoutlinedinSection 3.2.2 ,weassumedthatNT=200candidateantennapositionswereavailable,andthatthepositionswereseparatedbydT=0=10(seethearraysteeringvectordenitiongivenin( 3 )),sothatthetotalaperturelengthisgivenby19.90.OftheNTcandidatepositions,MT=10antennaswereusedinthedesignapproach.TheULAwasusedtoinitializethealgorithm,and5iterationsoftheprogramwereperformed(after5iterations,noneoftheantennaswererepositionedbythealgorithm).ThetransmitbeampatternofthesparsearrayisshowninFigure 3-1 (d).Thechosenantennapositions,alongwiththoseoftheULA,areillustratedinFigure 3-2 .Forclarity,weshowtheantennasalongthefullcandidateaperture(19.90).Weindicatethepositionsofthedesignedarray'santennasusingcircles,andweusesquarestorepresenttheULA'santennas(weadoptasimilarrepresentationstyleforeachoftheremainingexamples). Allottingtothelargeroccupiedaperturelength,thesparsearrayisabletoproduceanarrowermainbeamregioncomparedtothatoftheULA(withoptimizedcovariancematrix).ThetwoarraysarefurthercomparedinTable 3-1 ,wherethemean-squarederrorMSETisdenedby:MSET=kZ(ppd)k2. 97

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3-1 ,3dBcorrespondstothe3dBmainbeamwidth(the3dBmainbeamwidthofthedesiredresponseis4)andPSLreferstothepeaksidelobelevelofthebeampattern.WeidentiedthePSL,foreachbeampattern,astheamplitudeofthehighestpeakoccurringoutsideofthe3dBmainbeam.Inadditiontoanarrowermainbeam,thesparsearrayachievesasignicantlylowerMSETatthecostofonlyaslightlyhigherPSLcomparedtothatoftheULA(withoptimizedR). 3-3 (a),isdened:pd()=8>>>>>>><>>>>>>>:1,=10,1 62 3,2[5,9],1 6+8 3,2[11,15],0,otherwise. Asevidenced,thedesiredreceivebeampatternhasatriangularshape,andthe3dBwidthofthemainbeamis6.Further,wesetc=10(sothatunitpowerismaintainedinthedesignedbeampatternsatc).Thefollowingweightswerechosenforthisexample:z()=8>>>><>>>>:1,2[5,15],0.7,2[6,4]S[16,26],0.2,otherwise. Usingtheweightsin( 3 ),weemphasizedagainonachievingagoodapproximationtothemainbeamshape.Thereceivebeampatternfora10-elementULA(with0=2elementspacing)isshowninFigure 3-3 (c).TheantennaweightsfortheULAweredeterminedusingthevectorapproachinSection 3.4.2.1 .Foreachoftheremainingsimulations,wemaintainedNR=200anddR=0=10.UsingtheULAresultasaninitialization,wewereabletogenerateasparseantennaarrayusingthevector 98

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3.4 .Thereceivebeampattern,usingthevectorapproach,isshowninFigure 3-3 (d)(5iterationsofthealgorithmwereagainperformed). Forcomparison,wehavealsogeneratedasparsearrayusingthemethodproposedin[ 68 ].UsingthissparsealgorithmproposedbyKishietal.,antennasaresequentiallyplacedintothearray,givenasetofcandidatepositions.Antennasareplacedsuchthat,alongwiththeantennasthatarecurrentlyinthesystem,themean-squarederrorbetweenthedesiredreceivepatternandthegeneratedoneisminimized.Unlikeourvectorweightingapproach,themethodin[ 68 ]isnotiterative,andtheantennaweightsarenotoptimizedcyclically,asinSection 3.4.2.1 .Thereceivebeampatterngeneratedusingtheapproachin[ 68 ]isshowninFigure 3-3 (e). Finally,weshoweachoftheselectedantennapositions,foralloftheapproaches,inFigure 3-4 (werefertothearraygeneratedusingtheapproachin[ 68 ]asaKishiarray,forsimplicity).ThereceivebeampatternsinFigures 3-3 (c)-(e)arecomparedinTable 3-3 .Again,3dBcorrespondstothe3dBangularwidthofthemainbeam(the3dBmainbeamwidthofthedesiredresponseis6),andPSLdenotesthepeaksidelobelevel.AlthoughtheKishiarrayachievesamorenarrow3dBwidth(actually,morenarrowthanthedesiredresponse)owingtoitslongeroccupiedaperture,thePSLissignicantlyhigherusingthisarray.OursparsearrayapproachyieldsthelowestPSL.Themean-squarederrorMSERisdenedsimilarlyto( 3 ),wherepdandpnowrefertothedesiredandsynthesizedreceivebeampatterns,respectively.Asevidenced,oursparsearrayalsoproducesthelowestMSERcomparedtotheULAandKishiarray. 99

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3.4.2 ).Althoughweconsiderreceivearraydesignforthisexample,certainlytheseresultscanbeextendedtotransmitarraysynthesis. Wewillagainadoptthedesiredreceivebeampatternprovidedin( 3 )andthebeampatternweightsdenedin( 3 ).UsingNR=200anddR=0.10(whichcorrespondstotheresultprovidedinExample2),wegeneratethereceivebeampatternshowninFigure 3-5 (a).ForthepatternshowninFigure 3-5 (b),weletNR=80anddR=0.250.Finally,lettingNR=40anddR=0.50,wegeneratethereceivebeampatternshowninFigure 3-5 (c).Foreachoftheseexamples,thetotalcandidatearraylengthwasmaintainedbetween19.50and200(again,thecandidatelengthisequalto(NR1)dR).AsdRisincreased,thecandidateantennapositionsbecomelesscloselyspaced(lessdense).Weshowthechosenantennalocations,foreachcase,inFigure 3-6 .Further,wecomparetheMSER,PSL,and3dBvaluesforeachsimulationinTable 3-3 ,wherethearraysareidentiedaccordingtotheirrespectivedRvalues. Althougheacharrayyieldedsimilarvaluesfor3dB(thearraywithdR=0.250wasclosesttothedesired3dBvalue),theMSERandPSLvaluesclearlydecreaseasthecandidatearrayismademoredense(asdRismadesmaller).Wereiterate,however,thatforeachconstructedarray,theminimumseparationbetweenplacedantennaswasnosmallerthan0.50(whichisconrmedinFigure 3-6 ).Thus,givenxedconstraintsonminimumantennaseparationandtotalaperturelength,wecanconcludethatamoredensearrayofcandidateantennapositionspermitshigherdegreesoffreedomandbettermatchingperformance(lowerMSER).Asexpected,3dBdoesnotseemtodependonthedensityofthecandidatepositions(sincebeampatternwidthmorecloselyrelatestothetotallengthoftheaperture[ 63 ]). 100

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68 ].Finally,wedemonstratedthatbyallowingforamoredensecandidatearray(bydecreasingtheseparationbetweencandidatepositions),weareabletobetterapproximatedesiredbeampatternswhilestillmaintainingaspecieddistancebetweenplacedantennas.AppendixA 3 ).Considertheoptimizationproblemin( 3 ).GivenamatrixW,wewillseektoprovideaclosed-formupdateforU.RecallingthatukrepresentsthekthcolumnofU,werstrewritetheoptimizationproblemin( 3 )as:minUPKk=1kWHa(k)p Considerthat,forsomek2f1,,Kg:kWHa(k)p 101

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3 ),wecanfurtherrepresenttheoptimizationproblemin( 3 ):maxUPKk=1ReWHa(k)p ApplyingtheCauchy-Schwarzinequality,wecanconcludethefollowing:ReWHa(k)p wherejjdenotestheabsolutevalueoperation.Theleftandrightsideoftheinequalitiesin( 3 )areequalifuk=ckWHa(k)p whereckreferstoanarbitraryscalarfactor.Tosatisfytheconstraintin( 3 ),wemusthaveck=p 3 ),weobtaintheupdateformulain( 3 ):uk=WHa(k) 3 )isobtained.GivenU,weseektondamatrixWthatsolvesthefollowingoptimizationproblem(see( 3 )):minWWHAZ1=2U2s.t.WHa(c)2=1. WerstdetermineamatrixV2CMT(MT1)suchthatthecolumnsofVspanthesubspaceorthogonaltoa(c),i.e.,VHa(c)=0andthematrixVisfullcolumn-rank 102

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whered2C1MTandY2C(MT1)MT.Then:WHa(c)2=(a(c)d+VY)Ha(c)2=ka(c)k4kdk2. Hence,theoptimizationproblemin( 3 )isequivalentto:mind,Y(a(c)d+VY)HAZ1=2U2s.t.kdk2=1 Solving( 3 )withrespecttoYyields:^Y=(VHAZAHV)1VHAZ1=2(UHZ1=2AHa(c)d). Substituting( 3 )into( 3 ),theoptimizationproblemisreducedto:minddHpH2s.t.kdk2=1 wherep=a(c)HAZ1=2?Z1=2AHV, and?Z1=2AHV=IZ1=2AHV(VHAZAHV)1VHAZ1=2 103

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3 )canbewrittenas:dHpH2=kpk2kdk22trdHpHH+kHk2=kpk2kd^dLSk2+kHk2pHHHpH where^dLS=pHH Omittingtheconstraint,^dLSrepresentstheleast-squaressolutiontotheoptimizationproblemin( 3 ). 3 ),kpk26=0whenthecolumnsofthematrixZ1=2AHarelinearlyindependent,whichistrueingeneral.Considerthefollowingexplanation.Ifkpk2=0,thenthecolumnvectorZ1=2AHa(c)belongstothesubspacespannedbythecolumnsofthematrixZ1=2AHV.Inotherwords,thereexistsacolumnvectorsuchthat:Z1=2AHa(c)=Z1=2AHV. IfthecolumnsofZ1=2AHarelinearlyindependent,then( 3 )impliesthata(c)=V,whichcannotbetruebasedonthedenitionofV(VHa(c)=0).Thus,wemusthavethatkpk26=0. Usingtheresultin( 3 ),werewritetheoptimizationproblemin( 3 )asfollows:mindkd^dLSk2s.t.kdk2=1 104

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3 )isgivenbythevector^dopt,where: 1. ChooseanMT(MT1)matrixV,suchthatVHa(c)=0andVisfullcolumn-rank. 2. Compute where^dLSisdenedin( 3 ). 3. Compute^Wopt=a(c)^dopt+V^Yopt. 3 ).Then,itcanbeeasilyshownthat^Woptisgivenby:^Wopt=(AZAH)1AZ1=2UH, whichcorrespondstotheleast-squaressolution,againneglectingtheconstraint,totheoptimizationproblemin( 3 ). 105

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3 ).TheLagrangefunctiong(v,)fortheoptimizationproblemin( 3 )canbewritten:g(v,)=kZBHvZp1=2dk2+2bH(c)v, whereisaLagrangemultiplier.Theminimizationof( 3 )withrespecttovyields:v=(BZHZBH)1(BZHZp1=2db(c)). Wecantheninserttheresultin( 3 )intotheconstraintbH(c)v=1toobtain:bH(c)(BZHZBH)1(BZHZp1=2db(c))=1. Solving( 3 )foryields=bH(c)(BZHZBH)1BZHZp1=2d1 whichcanthenbeinsertedinto( 3 )toobtaintheresultin( 3 ):v=(BZHZBH)1"BZHZp1=2dbH(c)(BZHZBH)1BZHZp1=2d1 106

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Section 3.5 Example1:Transmitbeampatterndesign. CriteriaULASparseArray3dB13.525.15MSET0.7200.168PSL0.2300.281 Table3-2. Section 3.5 Example2:Receivebeampatterndesign. CriteriaULAKishiArraySparseArray3dB9.433.136.23MSER0.7550.4730.049PSL0.1850.5130.115 Table3-3. Section 3.5 Example3:Receivebeampatterndesign. CriteriadR=0.10dR=0.250dR=0.503dB6.236.066.08MSER0.0490.0630.089PSL0.1150.3190.326 107

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CD Figure3-1. Section 3.5 Example1.A)Desiredtransmitbeampattern.B)Designweights.C)ULAtransmitbeampattern.D)Sparsearraytransmitbeampattern. 108

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AntennapositionsforSection 3.5 Example1.SquaresareusedtoindicatethepositionsoftheULA'santennas,andthedesignedsparsearray'santennasareindicatedbythecircles. 109

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CD E Figure3-3. Section 3.5 Example2.A)Desiredreceivebeampattern.B)Designweights.C)ULAreceivebeampattern.D)Sparsearrayreceivebeampattern.E)Kishiapproach. 110

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AntennapositionsforSection 3.5 Example2.SquaresareusedtoindicatethepositionsoftheULA'santennas,`X'sareusedtoindicatethepositionoftheantennasinKishi'sarray,andourdesignedsparsearray'santennasareindicatedbythecircles. ABC Figure3-5. Section 3.5 Example3:sparsearrayreceivebeampatterns.A)NR=200anddR=0.10.B)NR=80anddR=0.250.C)NR=40anddR=0.50. 111

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AntennapositionsforSection 3.5 Example3.X'sareusedtoindicatethepositionoftheantennasinthearraywithdR=0.10,circlesforthearraywithdR=0.250,andsquaresforthearraywithdR=0.50. 112

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Asanundergraduate,WilliamRobertswasarecipientoftheNationalMeritScholarship.HereceivedBachelorofScienceandMasterofSciencedegreesinelectricalengineeringfromtheUniversityofFloridain2006and2007,respectively.HegraduatedwithaDoctorofPhilosophyfromtheElectricalandComputerEngineeringDepartmentattheUniversityofFloridainthespringof2010.WilliamisalsoarecipientoftheScience,MathematicsAndResearchforTransformation(SMART)Scholarship.Hisgraduatestudieshavefocusedonradarsignalprocessing,andhewillworkfortheU.S.ArmyinNewMexicoupongraduation. 119