Spectrally Constrained Active Sensing Waveform and Receiver Filter Design

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Spectrally Constrained Active Sensing Waveform and Receiver Filter Design
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Rowe, William T
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Electrical and Computer Engineering
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LI,JIAN
Committee Co-Chair:
LIN,JENSHAN
Committee Members:
WONG,TAN FOON
BAROOAH,PRABIR

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Subjects / Keywords:
radar -- sar -- waveform
Electrical and Computer Engineering -- Dissertations, Academic -- UF
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Electrical and Computer Engineering thesis, Ph.D.
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Abstract:
Traditional active sensing waveform and receiver filter design has always assumed that a continuous block of spectrum will be allocated for use. Due to growing demand for bandwidth among many different fields large continuous blocks of spectrum are not available. However, large bandwidths are required to meet the system requirements for high performance active sensing systems. A system that has to operate under such a constraint would be a spectrally constrained active sensing system. Traditional waveform design and receiver filters are not optimal in this scenario. In this work, we present a waveform design approach called SHAPE for the spectrally constrained problem.We also present a receiver filter for processing a spectrally constrained waveform in a synthetic aperture radar application. Finally, we present a method of designing spectrally constrained waveform sets that have short-time orthogonality and low correlation zones.
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by William T Rowe.
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Thesis (Ph.D.)--University of Florida, 2014.
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Adviser: LI,JIAN.
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Co-adviser: LIN,JENSHAN.

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SPECTRALLYCONSTRAINEDACTIVESENSING:WAVEFORMANDRECEIVERFILTERDESIGNByWILLIAMT.ROWEADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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c2014WilliamT.Rowe 2

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IdedicatethisworktomyMother,KatherineTaber,andmyancee,KatherineSmalley. 3

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ACKNOWLEDGMENTS Iwouldliketothankmyadvisor,JianLi,andmycommittee,JenshanLin,TanWong,andPrabirBarooahforalloftheircommentsandfeedbackonthiswork.IwouldalsoliketothankPetreStoicaandJohanKarlssonforalloftheinvaluableinsightandhelpingtoansweranyofmyquestions.IwouldalsoliketothankChrisGianelliforthemanyconversationstalkingshopwithmeoverabeerorinthesquatrack.Finally,Iwouldliketothankallofmyfellowmembers(past,present,andfuture)oftheSpectralAnalysisLabforallofyourhelpandhardwork.ThisworkwassupportedinpartbytheOfceofNavalResearch(ONR)underGrantNo.N00014-12-1-0381,theSwedishResearchCouncil(VR),NSFCCF-1218388,theEuropeanResearchCouncil(ERC),andtheSMARTfellowshipprogram.Theviewsandconclusionscontainedhereinarethoseoftheauthorsandshouldnotbeinterpretedasnecessarilyrepresentingtheofcialpoliciesorendorsements,eitherexpressedorimplied,oftheU.S.Government.TheU.S.GovernmentisauthorizedtoreproduceanddistributedreprintsforGovernmentalpurposesnotwithstandinganycopyrightnotationthereon. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 14 1.1ActiveSensing ................................. 14 1.2ReceiverFilterDesignforActiveSensing .................. 17 1.3WaveformDesignforActiveSensing ..................... 25 1.4SpectrallyConstrainedActiveSensing .................... 32 2SPECTRUMSHARINGVIASEQUENCEDESIGN ................ 38 2.1MotivationforSpectralSharing ........................ 38 2.2SHAPEAlgorithm ............................... 40 2.3PracticalIssues ................................. 44 2.3.1Convergence .............................. 45 2.3.2Quantization ............................... 48 2.3.3FrequencySampling .......................... 49 2.3.4Initialization ............................... 50 2.4ExamplesofSequenceDesignforSpectralCompliance .......... 52 2.4.1WidebandRadar ............................ 53 2.4.2RandomPhaseSonarWaveform ................... 55 3MISSINGDATAIAAANDSPECTRUMNOTCHES ................ 62 3.1SpectralEstimationandMissingData .................... 62 3.2SpectralEstimationandIAA .......................... 64 3.2.1DataModel ............................... 64 3.2.2IterativeAdaptiveApproach(IAA) ................... 65 3.3ComputationalComplexitiesandFastCalculationofIAA .......... 66 3.3.1CalculateR)]TJ /F5 7.97 Tf 6.59 0 Td[(1 .............................. 67 3.3.2CalculateN(!) ............................ 68 3.3.3CalculateD(!) ............................ 68 3.3.4Summary ................................ 70 3.4MissingDataIAAandFastCalculations ................... 70 3.4.1FastCalculationofMissingDataIAA ................. 72 3.4.1.1GetR,theGSfactorizationofR)]TJ /F5 7.97 Tf 6.59 0 Td[(1,andevaluateN(!)andD(!) .......................... 74 5

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3.4.1.2GetL=(SmR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SmT))]TJ /F5 7.97 Tf 6.58 0 Td[(1=2andX=)]TJ /F5 7.97 Tf 19.39 4.34 Td[(1=2 ........... 74 3.4.1.3Evaluatea(!))]TJ /F3 11.955 Tf 6.78 0 Td[(yf ...................... 75 3.4.1.4Evaluatea(!))]TJ /F3 11.955 Tf 6.78 0 Td[(a(!) ..................... 75 3.4.1.5GetN(!)andD(!) .................... 76 3.4.1.6Summary ........................... 76 3.5CalculatingtheMissingData,MIAA-t ..................... 77 3.6Applications ................................... 78 3.6.1ComputationalComplexity ....................... 78 3.6.21DSinusoidIdentication ....................... 79 3.6.3Application:Sparse2DSARImaging ................. 80 4SPECTRALLYCONSTRAINEDSARANDMIMOSARGMTIWAVEFORMDESIGN ........................................ 86 4.1MotivationForSpectralConstraintsinSAR ................. 86 4.2KroneckerWaveforms ............................. 89 4.2.1CorrelationPerformance ........................ 90 4.2.2KroneckerWaveformSpectrum .................... 91 4.3SpectrallyConstrainedWaveformSetDesignforSAR ........... 93 4.3.1SpectrallyConstrainedKroneckerWaveformsforSAR ....... 95 4.3.1.1SHAPEalgorithm ...................... 96 4.3.1.2WeCANalgorithm ...................... 98 4.3.1.3ZCZwaveformsetsfromperfectsequences ........ 99 4.3.2SpectrallyConstrainedWaveformsforMIMOSARGMTI ...... 100 4.4SpectrallyConstrainedSARandMIMOSARSimulation .......... 101 5FUTUREWORKANDCONCLUSIONS ...................... 113 5.1Conclusions ................................... 113 5.2FrequencyDiverseandSpectrallyConstrainedWaveformSets ...... 115 5.3TransmittingSHAPEWaveforms ....................... 117 5.4SHAPEReceiverFilter ............................. 117 APPENDIX ACOMPUTATIONALCOMPLEXITYOFMIAAALGORITHM ........... 119 BTOEPLITZMATRICESANDFASTCALCULATIONS ............... 120 REFERENCES ....................................... 122 BIOGRAPHICALSKETCH ................................ 128 6

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LISTOFTABLES Table page 2-1TheSHAPEAlgorithm ................................ 45 3-1FastIAA ........................................ 69 3-2FastMIAA ....................................... 76 4-1ResolutionandBandwidth .............................. 93 7

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LISTOFFIGURES Figure page 1-1Time-DelayEstimationConcept ........................... 15 1-2TheDopplerEffect .................................. 16 1-3TheDopplerEffectAppliedtoaFunction ..................... 20 1-4BasicRFDirectDownConversionReceiver .................... 21 1-5RectangleSignalMatchedFilter .......................... 22 1-6PulseCompression ................................. 23 1-7ClutterEffect ..................................... 24 1-8ChirpAmbiguityFunction .............................. 26 1-9RectangularWaveAmbiguityFunction ....................... 27 1-10FrankCodeAmbiguityFunction ........................... 29 1-11M-SequenceAmbiguityFunction .......................... 30 1-12CANWaveformAmbiguityFunction ........................ 32 1-13SpectrallyConstrainedWaveformDesignProblem ................ 33 1-14GappedLFM ..................................... 34 1-15NotchedLFM ..................................... 35 1-16GappedRandomPhase ............................... 36 1-17SCAN ......................................... 37 2-1ConvergenceFailure ................................. 46 2-2ConvergenceSuccessful .............................. 47 2-316BitsQuantization ................................. 49 2-412BitsQuantization ................................. 49 2-58BitsQuantization .................................. 50 2-64BitsQuantization .................................. 50 2-7InitialSpectrumusingFullBandwidth ....................... 51 2-8InitialSpectrumusingSampleandHold ...................... 52 8

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2-9Initialization(GappedLFMWaveform) ....................... 53 2-10SHAPEOutput .................................... 54 2-11Autocorrelation .................................... 54 2-12AutocorrelationZoom ................................ 55 2-13Stage1Initialization ................................. 56 2-14Stage1Output .................................... 56 2-15Stage2Initialization ................................. 57 2-16Stage2Output .................................... 58 2-17Stage3Initialization ................................. 58 2-18Stage3Output .................................... 59 2-19FinalStageInitialization ............................... 59 2-20FinalStageOutput .................................. 60 2-21RealPartofWaveform ................................ 60 2-22ImaginaryPartofWaveform ............................. 60 2-23Autocorrelation .................................... 61 2-24AutocorrelationZoom ................................ 61 2-25Cross-CorrelationsofTwoSHAPEWaveforms .................. 61 3-1FastMIAA-tSpeedUp ................................ 78 3-2ExampleOfMissingDataForATimeSeries ................... 81 3-3ExampleofMissingDataSpectralEstimates ................... 81 3-4ComputationTimeComparison ........................... 82 3-5FFTSARImagaing .................................. 84 3-6SLIM-1SARImaging ................................ 85 4-1GroundTruthforSARImage ............................ 102 4-2InnerWaveform .................................... 104 4-3OuterWaveform ................................... 105 4-4KroneckerWaveform ................................. 106 9

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4-5SARImaging ..................................... 107 4-6SARImagingWithInterference ........................... 109 4-7MIMOSARWaveformsCross-Correlation ..................... 110 4-8MIMOGMTIResults ................................. 112 5-1FrequencyDiverseSpectra ............................. 115 5-2FrequencyDiverseCross-Correlation ....................... 116 5-3FrequencyDiverseAutocorrelation ......................... 116 5-4TransmittedSignalExample ............................. 117 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySPECTRALLYCONSTRAINEDACTIVESENSING:WAVEFORMANDRECEIVERFILTERDESIGNByWilliamT.RoweMay2014Chair:JianLiMajor:ElectricalandComputerEngineeringTraditionalactivesensingwaveformandreceiverlterdesignhasalwaysassumedthatacontinuousblockofspectrumwillbeallocatedforuse.Duetogrowingdemandforbandwidthamongmanydifferenteldslargecontinuousblocksofspectrumarenotavailable.However,largebandwidthsarerequiredtomeetthesystemrequirementsforhighperformanceactivesensingsystems.Asystemthathastooperateundersuchaconstraintwouldbeaspectrallyconstrainedactivesensingsystem.Traditionalwaveformdesignandreceiverltersarenotoptimalinthisscenario.Inthiswork,wepresentawaveformdesignalgorithmcalledSHAPEforthespectrallyconstrainedproblem.Wealsopresentareceiverlterforprocessingspectrallyconstrainedlinearfrequencymodulatedwaveformsinasyntheticapertureradar(SAR)application.Finally,wepresentanovelmethodofdesigningspectrallyconstrainedwaveformsetsforapplicationinmultiple-inputmultiple-outputSAR.Thisworkbeginsbyexaminingthefundamentalsofactivesensingintheintroduction.Inactivesensingthecommongoalsaretoestimatetherangeandvelocityofobjectsinasensor'seldofview.Thisisaccomplishedbymeasuringtheechoedreturnsofaknowntransmittedsignalandanyfrequencyshiftsappliedtothatsignalbyanobject'svelocity(whichiscausedbytheDopplereffect).Thechoiceofthetransmittedsignaliscrucialbecauseitdeterminesourabilitytoresolvemultipleobjects,anyinterferenceduetoneighbouringobjects,andhowthesignalreactstoDopplershiftsatthereceiver. 11

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Averygooddesignchoiceistouseasignalthathasaverylargebandwidthandhasenergyuniformlydistributedacrosstheentirebandwidth.However,duetothelimitednatureofthefrequencyspectrumandthelargebandwidthrequirementswemaynotbeabletomeetalldesignrequirementsbecauseotheruserswillbeoccupyingspecicbands.Ifweradiateenergywithoutconsideringtheinterferencewewillcause,thenitisanillegalactivity,anditcouldhavecatastrophiceffectssuchasdisruptingcrucialaviationnavigationequipment.Toovercomethisweconsiderspectrallyconstrainedactivesensing.Therstproblemweaddressinspectrallyconstrainedactivesensingishowtodesignaprobingwaveformthatdoesnotcauseinterferenceinsomegivenbands.AdesignalgorithmcalledSHAPEispresentedfordevelopingsuchaprobingsignal.Theeffectsofaspectrallyconstrainedprobingsignalmayhaveunwantedconsequencesinactivesensing.Incertainapplicationssuchassyntheticapertureradar,itwillresultinamissingdataproblem.ThemissingdatawilldegradetheSAR'soutputoftentoanunusablelevel.Toovercomethisweproposetousethemissingdataiterativeadaptiveapproach(MIAA)toestimatethemissingdata.However,SARdataisoftenverylargeandthetraditionalMIAAalgorithmisnotoptimizedforcomputationalefciency.Afastalgorithmwasproposed,butitisonlyefcientwhenover50%ofthedataismissing.Forthisreason,wedevelopedanewfastMIAAalgorithmthatiscomputationallyefcientwhenthemissingdataislessthan50%.WedemonstratetheeffectivenessofthealgorithmonsomesimulatedmissingdataSARexamples.WealsoconsiderthecaseofspectrallyconstrainedwaveformdesignforSARandmultiple-input,multiple-output(MIMO)SARgroundmovingtargetindicator(GMTI).HereweshowthatbyusingKroneckerwaveforms(onewaveformembeddedinanother),itispossibletogeneratespectrallyconstrainedwaveformsthathavelowcorrelationzones.Furthermore,byusingtheseKroneckerwaveformswearegivencontrolofthecross-correlationbetweenthewaveformsintheset.Usingthisapproachwepresenta 12

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novelmethodforgeneratingspectrallyconstrainedwaveformsets,whichhasnotbeenexploredintherecentliterature.WedemonstratetheefcacyofthesewaveformsusingSARandMIMOSARGMTIsimulations.Thisworkisconcludedbyexaminingthepossibilitiesandfuturedirectionsofthespectrallyconstrainedactivesensingapproachesdescribedwithin.Amajormilestoneistoperformtestsusingupdatedsoftwaredenedradars.Bydoingthiswecanbetterunderstandthelimitationsofthedesignedwaveformsandwhatparametersaremoreimportantthanothers.Herewecouldalsounderstandhowthenon-lineareffectsofthetransmitterandreceiverfront-enddistortoursignalandhowtoaccountforthatinthedesignprocess.Finally,wewouldliketostudyimprovedmethodsofdesigningthereceivertoimprovedetectionandperformanceofthesesignalsinextremelyclutteredenvironments.Allofthesearecriticaltodevelopingactivesensingsystemsthatcanmeettheupcomingstringentspectralrequirementsofthefuture. 13

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CHAPTER1INTRODUCTIONWhatdoesthetitleSpectrallyConstrainedActiveSensing:WaveformandReceiverFilterDesignmean?Therstkeyphraseisactivesensing,whichtellsuswhattypeoftechnologywewillbefocusingoninthisdissertation.Waveformandreceiverltersaretwocrucialcomponentsinanactivesensingsystem.Finally,thedescriptionspectrallyconstrainedreferstoanemergingtypeofactivesensingwhichisthemaincontributionofthiswork.Webeginourdiscussionbyreviewingactivesensingandbymotivatingtheneedforspectrallyconstrainedactivesensing. 1.1ActiveSensingAnactivesensingsystemtriestoestimateinformationaboutthesurroundingenvironment.TheyareusedforamyriadoftaskssuchascreatinghighresolutionterrainmapsoftheEarth'ssurface[ 1 ],searchingforobjectshiddenunderground[ 2 ],ortargetdetectionandtrackingusingradarorsonar[ 3 ].Whileeachoneofthesetasksisachievedinadifferentmannertheyallshareacommonconcept;theyintroduceenergyintotheenvironmentandmeasurehowitdisruptstheenvironment.Oneofthemostcommonusesofactivesensingistomeasuredistance.Thesesystemsemitasignal(ortheprobingwave)intotheenvironmentandexpectobjectsofinterest(referredtoastargets)toreectthesignal.Bymeasuringthetimedelayofthereectedsignalsanestimateoftherangeprolecanbemade.Whilethecoredesignconceptissimple,theexecutionofsuchasystemwithahighprecisionisoftenquitecomplicated.Radarisaprimeexampleofanactivesensingsystemthatestimatestimedelay.Figure 1-1 demonstratesthebasicconceptgraphically1 1Figure 1-1 istakenfrom[ 4 ]:C.Wolff.(2014,Feb)Radartutorial.Online.[Online].Available: www.Radartutorial.eu 14

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Figure1-1. Theconceptfortime-delayestimationinactivesensing. Radarstandsforradiodetectionandrangingandwillbecommonlyusedasanexampleinthiswork.Wewillfocusonradarbecauseofitsprevalentusageinmodernsocietyandbecausecurrentradarproblemsmotivatedasignicantportionofworkinthisdissertation.However,themethodswewilldescribearenotlimitedtoonlyradarspecicplatforms.Anactivesensingsystemcaninfermoreinformationabouttheenvironmentfromthereectedreturns.Therelativeradialvelocityisacommonlymeasuredvalue.RadialvelocityismeasuredviatheDopplershiftoftargetsinteractingwithourprobingwave.ADopplershiftisachangeinthefrequencyofawavecausedbythemovementofthetargetrelativetothesensor.Figure 1-2 showsthebasicgeometryandwhytheDopplershiftoccurs2.HereweassumethatpeakAofasinusoidstartspropagatingattimet0.ThetargetisatpositionR0andismovingwithsomeradialvelocityv.Duetothemotion,thetargetisatpositionR0+vtwhenpeakAarrives,wheretisthetimeittakes 2Figure 1-2 istakenfrom[ 3 ]:N.LevanonandE.Mozeson,RadarSignals.Hoboken,NJ:Wiley,2004. 15

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Figure1-2. TiminginaDopplerscene. forthewavetopropagateatvelocityCp.WhenpeakBarrivesthetargetisinpositionR0+vt+vT,whereTistheperiodofthesinusoid.ThereectedsignalthenwillhaveachangeinseparationofthepeaksAandB.Thisappearsasafrequencyshiftofthesinusoidatthereceiver.Whenthemotionistowardsthesensorthefrequencyincreasesandwhenthemotionisawayfromthesensorthefrequencydecreases.Targetmotionisactuallymoredifcultthanthissimpleexampleandwillbediscussedindetailinlatersectionsoftheintroduction.Activesensingcanalsobeusedtoestimateazimuthandelevationinformationoftargetsaswell.Theseparametersarecommonlyestimatedinimagingsensorssuchasasyntheticapertureradar[ 1 ].Traditionallytogeneratehighresolutionintheazimuthandelevationdomainsasensorwouldrequireanarrowilluminationpatternonthesceneinthosedimensions.Foranarrowilluminationpatternthesensorwouldrequireanantennawithalargeaperturewhichisexpensiveandheavy.However,byilluminatingastationaryscenefromalargenumberofvantagepointsasyntheticaperturecanbecreatedandahighresolutionimagecanbemade. 16

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Therearemanytypesofactivesensorswhichwehavenotdiscussedinthisbriefintroductiontoactivesensing.Wehavecoveredthebasicsthatwillbefocusedoninthiswork.Namely,wewillfocusonactivesensorsthatuseecholocationtoestimateparametersrange,velocity,azimuth,andelevation.Theinteractionwiththeenvironmentwillbethroughtheuseofsomesortofprobingsignal.Thesignalcanbegeneratedusingelectromagneticwavessuchasinradarorthoughmechanicalwavessuchasinultrasoundorsonar(soundnavigationandranging).Wewillbeginourdiscussionbyreviewingthefundamentalsofanactivesensorreceiverlterdesign.Wewillfollowthiswithadiscussiononwaveformdesignforactivesensingwhichdrawsonmanyoftheaspectsfromthereceiverlterdesign.Finally,weintroducethespectrallyconstrainedactivesensingproblemanddiscusswhytraditionalwaveformdesignandreceiverltersarenolongeroptimalwhenspectrallyconstrained. 1.2ReceiverFilterDesignforActiveSensingInactivesensingaknownsignalistransmittedoutintotheenvironmentandthenthereectionsoftheknownsignalaremeasured.Thistransmittedsignalcanbewrittenas: sT(t)=Refx(t)ej2fctg,(1)wherej=p )]TJ /F6 11.955 Tf 9.3 0 Td[(1,fcisthecarrierfrequency,andx(t)isourcomplexprobingsignal.Thetransmittedwaveinteractswithobjectsintheenvironmentthatreectbackthesignal.Theseobjectsarecalledscattersandechoedsignalsaregeneratedthoughsevenbasicmechanics:re-entrantstructures,specularscatters,traveling-waves,diffraction,surfacediscontinuities,creepingwaves,andstructureinteractions[ 5 ].Ageneralmodelofthemeasuredsignalarrivingatthereceiverisgivenby sR(t)=RefKXk=1kx(t)]TJ /F7 11.955 Tf 11.95 0 Td[(k)ej2fc(t)]TJ /F9 7.97 Tf 6.59 0 Td[(k)g,(1)whereKisthetotalnumberofscattersintheenvironment,kisthecomplexamplitudeandphaseofthekthscatter,andkisthetotaldelaytothekthscatter[ 6 ].Thismodel 17

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hasassumedthatthereectionprocessisfrequency-independentwhichmaynotbetrueinultra-widebandoperation.Italsoassumedthatthereectionprocessislinearwhichmaynotbetrueinthemorecomplicatedscatteringmechanisms.ThetraditionaldesignapproachassumesthatK=1andtheonlydistortiontoourreceivedsignalisintheformofadditivenoise~e(t)generatedfromthermalnoiseinthereceiver.Inthiscasethemeasuredresponseinthereceiveris ~y(t)=Refx(t)]TJ /F7 11.955 Tf 11.96 0 Td[()ej2fc(t)]TJ /F9 7.97 Tf 6.59 0 Td[()g+~e(t).(1)Wewillnowconsidertheeffectsoftargetmotionusingthesameapproachas[ 6 ].Letatargetmovewithsomeconstantradialvelocityv,startatsomeinitialpositionR0andassumewemeasurethesignalatsometimet.Thetotaldelaytothetargetisgivenby(t)=2R(t)]TJ /F7 11.955 Tf 12.38 0 Td[((t)=2)=CpandR(t)]TJ /F7 11.955 Tf 12.38 0 Td[((t)=2)=R0+v(t)=2.NotethatunlikeEq. 1 ,isnowafunctionoftimeduetothetargetmotion.UsingsubstitutiononthetworelationshipsregardingR(t)and(t)wecanwritethefollowingequationfor(t): (t)=2R0=Cp 1+v=Cp+2v=Cpt 1+v=Cp.(1)Forthemajorityofproblemsinactivesensing,theradialvelocityismuchlessthanthespeedofpropagation,hencev=Cp1.Thedelaycanbeapproximatedas (t)=2R0 Cp+2v Cpt,0+2v Cpt.(1)SubstitutingEq. 1 inEq. 1 gives ~y(t)=Refxt)]TJ /F6 11.955 Tf 11.95 0 Td[((0+2v Cpt)ej2fc(t)]TJ /F19 7.97 Tf 6.59 8.8 Td[(0+2v Cpt)g+~e(t)=Refx(1)]TJ /F6 11.955 Tf 13.17 8.09 Td[(2v Cp)t)]TJ /F7 11.955 Tf 11.96 0 Td[(0)ej2fc(1)]TJ /F20 5.978 Tf 8.43 3.26 Td[(2v Cp)t)]TJ /F8 7.97 Tf 6.59 0 Td[(fc0g+~e(t). (1) Thetargetmotionhastwoeffects:First,itcausestimedomainstretchingorcompressionofourprobingsequencex(t).Second,itcausesafrequencyshiftonthecarrierfrequency. 18

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Thetimedomainstretchingandcompressionisofconcernmainlyinwidebandwidthapplicationsorwhenthespeedofpropagationisslow.Ageneralruleofthumbfrom[ 3 ]isthatifthebandwidthisone-tenthofthecarrierfrequency,thenthetimedomainstretchingandcompressioncanbeignored.Anotherapproachforignoringthiseffectistakenfrom[ 6 ],whichcomparestheerrorbetweenx(t)]TJ /F7 11.955 Tf 12.5 0 Td[(0)andxt)]TJ /F6 11.955 Tf 11.95 0 Td[((0)]TJ /F5 7.97 Tf 13.35 4.7 Td[(2v Cpt).Clearly,themaximumerrorwilloccurattheedgesofsomelengthTwherethestretchingorcompressionhasoccurred.Ifthebandwidthofx(t)isB,thensignalwillnotchangesignicantlyinatime1=B.Thisimpliesthat 2vT Cp1 B!BTCp 2v.(1)Hencewhenthebandwidthislargeandtheexpectedtargetvelocitiesarehigh,thetimedomaincompressionandstretchingshouldnotbeignored.Furthermore,ifCpissmall,thenthetimedomaincompressionandstretchingshouldnotbeignoredaswell.Formostofthereceiverlterdesignthatwillbedoneinthiswork,weassumethatEq. 1 issatisedunlessspecicallystatedotherwise.ThesecondeffectistheDopplershift.Typically,thoughthevaluefc2v CpissmallrelativetofcandtheeffectoftheDopplershiftwillappearasaphaseshiftinshortpulses.ToincreasetheDopplerresolution,coherentpulsetrainsareusedtoincreaseourtimeobservationwindow.Figure 1-3 showsagraphicaldescriptionoftheproblemforasimplex(t).3Thetopplotshowsthereceivedsignalanditappearsasthoughthereturnsarephaseshiftedsequentially.ThebottomplotsshowenvelopefunctionandthemodulationcausedbytheDopplershift.AsthelengthofourtimeobservationwindowincreasesourDopplerresolutionincreases. 3Figure 1-3 istakenfrom[ 3 ]:N.LevanonandE.Mozeson,RadarSignals.Hoboken,NJ:Wiley,2004. 19

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Figure1-3. TheDopplereffectonx(t) Ourreceivedsignalwhichaccountsfortargetmotion(andneglectingtimedomainstretchingandcompression)canbewrittenas ~y(t)=RefKXk=1kx(t)]TJ /F7 11.955 Tf 11.96 0 Td[(k)ej2(fc(t)]TJ /F9 7.97 Tf 6.59 0 Td[(k)+fd,kt))g+~e(t),(1)wherefd,k=()]TJ /F6 11.955 Tf 9.3 0 Td[(2vk=Cp)fcistheDopplershiftofthekthtarget.Wewouldliketomanipulatethecomplexbasebandsignalandnottherealvaluereceivedsignal.Wewillassumethatthisisachievedthoughanidealdirectdownconversionprocess.Figure 1-4 showsablockdiagramofwhatadirectdownconversionprocesslookslike4. 4Figure 1-4 istakenfrom[ 7 ]:A.Mashhour,W.Domino,andN.Beamish,Onthedirectconversionreceiver-atutorial,MicrowaveJournal,vol.44,pp.114128,2001. 20

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Figure1-4. ABasicRFdirectdownconversionreceiverfrontend Thesignalisreceivedbytheantenna,passesthroughalargebandwidthband-selectlter,andthenisampliedatthelownoiseamplier.Thesignalisthensplitintothein-phaseyI(t)andquadratureyQ(t)partsviathemultiplierandthenlowpassltered[ 3 ].Ourreceivedcomplexbasebandsignalisgivenby yI(t)+jyQ(t)=y(t)=KXk=1kx(t)]TJ /F7 11.955 Tf 11.96 0 Td[(k)ej2fd,kt)+e(t),(1)wheree(t)isacomplexcircularlysymmetricGaussianrandomvariable.Theconstantphaseoffsetduetothedelaykwasmovedintotheunknowncomplexscalark.Thegoalofthereceiveristoestimatetheparametersk,k,fd,k,andKfork=1,2,...,K.Considerthescenariowherek=1andthenoisee(t)issomecomplexwhitenoiserandomprocess.TheoptimalreceiverlterinthisscenarioisthematchedlterwhichcanbetracedbacktoD.O.Northin1943[ 3 ].ThematchedltercanbederivedusingtheNeyman-Pearsoncriterionasin[ 8 ]orbytryingtodesignalterthatmaximizesthesignal-to-noiseratio(SNR)asin[ 3 ].Amatchedlterh(t)issimplycorrelatingthereceivedsignalwiththecomplexconjugateofthetransmittedprobingsignal.Correlatingasignalwithatime-delayedcomplexconjugate(denotedas())ofitselfisalsoknownastheautocorrelationfunctionofasignal. 21

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A BFigure1-5. Rectanglesignalmatchedlteroutput.A)Resolvedtargetreturns.B)Unresolvedtargetreturns. h(t)=x(t)(1)Thematchedlteralsohelpsovercomeasecondprobleminactivesensing.Itcanbeusedtoimprovetheresolutioninthetimedelayestimateviaamethodcommonlycalledpulsecompression[ 3 ].Assumex(t)=rect(t Tp)whererect()is1whentheargumentisgreaterthan0butlessthan1.Nowlettwostationarytargetsbeinthesceneattimedelays1and2.Alsolete(t)=0forthesimplicityofthisexample.InFigure 1-5 A,twopeaksareclearlyresolvedinthematchedlteroutput.Thepeaksalsoalignwiththetruetargettimedelay.However,inFigure 1-5 Bthepeaksofthetargetsarenotresolvableandthematchedlteroutputshowsonelargetarget.ThisproblemcanbeavoidedbypulsecompressionasshowninFigure 1-6 .PulsecompressionutilizesthefactthattheautocorrelationfunctionandthepowerspectrumofasignalareFouriertransformpairs[ 9 ].Pulsecompressionutilizesthefactthatasignalwithatime-bandwidthproductgreaterthanunitywillhaveamorenarrowautocorrelationmainlobe.Theratioofthetransmittedsignalpulsewidthtotheautocorrelationfunctionmainlobewidthiscalledthepulsecompressionratio.Thepulsecompressionratiois 22

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A BFigure1-6. Pulsecompression.A)Resolvedtargetsduetoalargertime-bandwidthproductintheprobingsignal.B)Unresolvedreturnsfromtherectanglesignal. approximatelyequivalenttothetime-bandwidthproduct[ 10 ].Furthermore,therangeresolutioncanbeshowntobeequaltoCp 2B,whereBisthebandwidthofx(t).Thematchedlteriscomputationallyinexpensive,robust,andoptimalwhenthereisonetargetandthenoiseiswhite.Whenthenoiseisnotwhite,thenawhiteningltermaybeusedinconjunctionwiththematchedlter[ 8 ].Thisapproachrequiresagoodestimateofthecovarianceofthenoisewhichisnotalwaysavailable.Anotherproblemiswhenthesignalisnotnoiselimited,butisinsteadclutterlimited.ConsiderthecasewhenK>1andweareinterestedinsometargetatdelay0.Wecanre-writeEq. 1 thenas y(t)=0x(t)]TJ /F7 11.955 Tf 11.96 0 Td[(0)ej2fd,0t| {z }target+K)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xk=1kx(t)]TJ /F7 11.955 Tf 11.95 0 Td[(k)ej2fd,kt)| {z }clutter+e(t)| {z }noise.(1)Whenthesignalenergyarrivingfromtheclutterismuchstrongerthanthetargetofinterest,thematchedlterisnolongeroptimal.Figure 1-7 showsagraphicalrepresentationofwhattheclutterproblemlookslike.Theunwantedreturnsfromneighbouringdelayscausesenergyspillageintothedelayofinterest.Tomitigatethisacluttersuppressionltercanbedesignedusingtheinstrumentalvariable(IV)method. 23

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Figure1-7. Agraphicalrepresentationofclutter Theseltersareinformallyclassiedasamismatchedlter[ 11 ].[ 11 ]alsoprovidesathoroughreviewofdifferentmismatchedlterapproaches.Twonotablemethodsofalgorithmicpulsecompressionthatcanprovidecluttersuppressionareiterativeadaptiveapproach(IAA)[ 12 ]andDopplercompensatedadaptivepulsecompression(DC-APC)[ 13 ].Thelattermethodisbasedonaminimummeansquareestimateapproachtoreducethecluttercausedbytargets,butitrequirestheuseofahyperparameter.Thereisnosetmethodforhowtoselectthishyperparameterwhichmaybeproblematicinactualsystems.TheothermethodIAAperformsatasimilarleveltoDC-APC(withagoodvalueforthehyperparameter)withoutanyhyperparameters.However,IAAiscomputationallyexpensiveinthepulsecompressionproblemwhencomparedtothematchedlter.5Inthissection,wehavereviewedthefundamentalsofactivesensingfromareceiverlterdesignapproach.Wediscussedtherelationshipbetweenthetransmitted 5DC-APCandIAAaresimilarintheircomputationalcomplexity 24

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waveform,theenvironmentandthereceivedsignal.Thematchedlteristhemostcommonmethodusedtoachievebothpulsecompressionandtimedelayestimates.Inthenextsection,wewillexaminehowthematchedlterhasdrivenwaveformdesign. 1.3WaveformDesignforActiveSensingClassicalwaveformdesignisbasedontheperformanceofawaveforminamatchedlter.Thefunctionthatmodelsawaveform'sperformanceinamatchedlteriscalledtheambiguityfunction[ 3 ].OriginallyproposedbyJamesWoodwardin1953,theambiguityfunctionsimplydescribesasignal'smatchedlterresponsetoadelayedandDopplershiftedcopy.Thisissoimportantinactivesensing(andRadar)becausealargeportionofactivesensingsystemsusethereceivedechostoinferinformationaboutthescene.Similartobefore,letx(t)beourprobingsequencethenweusethefollowingdenitionoftheambiguityfunctionofx(t): j(,)j2=jZ1x(t)x(t+)ej2tdtj2,(1)wherecorrespondstodelayandcorrespondstoDoppler.Figure 1-8 showsanexampleofwhattheambiguityfunctionlookslike.Thezero-Doppler(=0)cutcorrespondstotheautocorrelationfunctionwhichwillbeanimportantfactorinwaveformdesign.Thezero-delay(=0)cutistheFouriertransformofthecomplexenvelopeofourprobingsequence[ 3 ].Beforewediscusswaveformdesign,weexaminefourpropertiesoftheambiguityfunction.Therstpropertystatesthemaximumvalueoftheambiguityfunctionwillalwaysoccurattheorigin(=0,=0).Whenthesignalenergyisnormalized,thenthemaximumvaluewillbeunity.Thesecondpropertyoftheambiguityfunctionstatesthatthevolumeoftheambiguityfunctionisaconstantvalue.Thethirdpropertyisthattheambiguityfunctionissymmetricabouttheorigin.Finally,thefourthpropertyisthatlinearfrequencymodulation(LFM)ofx(t)willresultinashearoftheambiguityfunction.A 25

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Figure1-8. Anexampleoftheambiguityfunction linearfrequencymodulatedwaveformismathematicallyrepresentedby x(t)=ej2(f0t+t2)rect(t Tp),(1)wheref0isthestartingfrequency,isthemodulationrate,andTpisthepulsewidth.ThispropertyisdemonstratedinFigure 1-8 whichistheambiguityfunctionofaLFMwaveform.Alargeridgeisapparentintheplotthatstartsattheorigin.ComparetheLFMwaveformambiguityfunctiontothatofasimplerectangularwave(wherex(t)=rect(t Tp))asshowninFigure 1-9 .WecanseethedrasticdifferenceinmainlobewidthoftheLFMwaveformcomparedtothesquarewaveform.Theidealambiguityfunctionisoftenreferredtoasthethumbtackplot.Itiszeroeverywhere,butithasavalueofoneattheorigin.Clearly,bytheconstantvolumepropertyoftheambiguityfunctionthisidealwaveformisnotachievable.Adesignermaytrytondwaveformsthenthatcanapproximatethisthumbtackshapeoftheambiguityfunction,butthereisnouniversalsolutionforsynthesizingawaveformfromadesiredambiguityfunction[ 14 ].Methodsforwaveformsynthesisbasedonambiguityfunction 26

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Figure1-9. Arectangularwave'sambiguityfunction designsexist,butcanonlycontrolsmallportionsoftheambiguityfunction[ 14 ].Theproblemisquitedifcultbecauseoftheconstantvolumeconstraintontheambiguityfunctionandduetoalackofdegreesoffreedominthewaveformdesign.Thesedifcultieshaveledmostwaveformdesigntofocusmainlyonzero-Dopplercut(autocorrelation)basedoptimization.TheidealautocorrelationisaDiracDeltafunctionwhichhasapowerspectrumwithaninniteatbandwidth.Clearly,thisisnotachievableinanyrealsystem.Designersthentrytomimicthisstructurebyusingwaveformsthathavelargeatbandwidths.Theautocorrelationfunctionisacontinuousfunctionwhichmakesnumericaloptimizationdifcult.However,whenthewaveformisaunimodularphasesequenceintheformof x(t)=1 p TpNXn=1x[n]rectt)]TJ /F6 11.955 Tf 11.96 0 Td[((n)]TJ /F6 11.955 Tf 11.96 0 Td[(1)tb tb,(1)wherex[n]=ejn,Tpisthepulsewidth,andtbisthesub-pulsewidth,thenitcanbeshownthatoptimizationofthediscreteautocorrelationfunctionissufcient[ 3 14 ].Allvaluesofthecontinuousautocorrelationcanbefoundvialinearinterpolationofthe 27

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discreteautocorrelationfunctionevaluatedatintegervaluesofthesub-pulsewidth(orchipwidth).Thediscreteautocorrelationfunctionthencanbewrittenas r[k]=N)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xn=k+1x[n]x[n)]TJ /F14 11.955 Tf 11.96 0 Td[(k]=r[)]TJ /F14 11.955 Tf 9.3 0 Td[(k],fork=0,1,...,N)]TJ /F6 11.955 Tf 11.95 0 Td[(1.(1)Technicallythisistheaperiodicdiscreteautocorrelationfunction.Wecanwritetheperiodicautocorrelationfunctionas ~r[k]=N)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xn=k+1x[n]x[(n)]TJ /F14 11.955 Tf 11.95 0 Td[(k)modN]=~r[)]TJ /F14 11.955 Tf 9.3 0 Td[(k]=~r[N)]TJ /F14 11.955 Tf 11.95 0 Td[(k],fork=0,1,...,N)]TJ /F6 11.955 Tf 11.96 0 Td[(1.(1)Notethesubtledifferencebetweenthetwofunctions.Phase-codedwaveformaretypicallybasedonoptimizingeitherEq. 1 orEq. 1 ,butnotboth.Wewillbedealingmainlywiththediscreteaperiodicautocorrelationfunction(whichwillbereferredtoasACFfromhereon),andhenceanyreferencetotheACFwillimplythediscreteaperiodicautocorrelationfunctionunlessspeciedotherwise.TheACFhasamaximumatlagindexzeroandthevaluewillalwaysbeequivalenttothesignalenergy.Alllagsk6=0arereferredtoassidelobesandareunwanted.ItistrivialtoshowthattheACFwillalwayshavenon-zerosidelobeswhenthewaveformisunimodular[ 3 ].Thiscanbeseenbylookingatthelastlagpositionk=N)]TJ /F6 11.955 Tf 12.32 0 Td[(1whichmusthaveamagnitudeof1.Conversely,fortheperiodicACFzerosidelobesequencesexistforunimodularsequencesofanylength[ 14 ].Thesesequencesarereferredtoasperfectcodesorconstantamplitudezeroautocorrelation(CAZAC)codes.CAZACcodesaremainlypolyphasecodes(canhaveanyphase).Binary(phaseis0or)CAZACcodesexist,butareseverelyrestricted[ 15 ].ExamplesoffamousperfectcodesincludetheFrankcodeandtheZadoff-Chucodes.Figure 1-10 showsanexampleoftheambiguityfunctionforaFrankcodeof 28

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Figure1-10. TheambiguityfunctionforaFrankcodeoflength64. length64.6TheFrankcodeisaquadraticphasesequence,henceitbehavesasasampledlinearfrequencymodulatedwaveform.Thisiswhyweseethelinearshearintheambiguityfunction.Aclassofbinarywaveformscalledmaximumlengthsequences,orM-sequences,arecalledasymptoticallyperfectsequences[ 3 ].ThisisbecausethesidelobeleveloftheperiodicACFofanM-sequenceoflengthNhasavalueof)]TJ /F6 11.955 Tf 9.3 0 Td[(1andapeakvalueofN.AsNgetslargertheratiobetweenthesidelobelevelandthepeakapproaches0.ThesesequencesareverypopularbecauseoftheirsimplicitytogenerateandbecausealargenumberofdifferentcodesexistwhenNbecomeslarge[ 3 ].Figure 1-11 showstheaperiodicambiguityfunctionforalength63M-sequence.Thiswaveformresemblesathumb-tacklikeshapewhichisdesirableinmanyapplications.TheM-sequenceisnotconsideredDopplertolerantsincethereisnoridgealongtheDoppleraxis.This 6Aperiodicambiguityfunctiondenitionexistswhichwouldshowthezeroautocorrelationsidelobes,butwillnotbeusedinthiswork. 29

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Figure1-11. TheambiguityfunctionforaM-sequenceoflength63. meansthatalargenumberofmatchedlterssettospecicDopplershiftvalueswouldberequiredtodetectsignalswithunknownDopplershifts.ThelastclassofwaveformswewilldiscussarecalledCyclicAlgorithmNew(CAN)waveforms[ 14 ].CANwaveformsperformalocalminimizationontheintegratedsidelobelevel(ISL)metric ISL=KXk=1jr[k]j2.(1)Equation 1 canbeimplementedinthefrequencydomainusingthefollowingequation[ 14 ]: ISL=2NXp=1 NXn=1x[n]e)]TJ /F8 7.97 Tf 6.58 0 Td[(j!pn2)]TJ /F14 11.955 Tf 11.96 0 Td[(N!2.(1)ThisdenitionoftheISLisveryintuitivesinceitsaystheoptimalx[n]shouldbeatinthefrequencydomain.However,theISLmetricinEq. 1 ishardtooptimizewithrespecttox[n]duethequarticdependency.CANfocusesinsteadonoptimizingan 30

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almostequivalentmetricgivenby minfx[n]gNn=1,f[p]g2Np=12NXp=1 NXn=1x[n]e)]TJ /F8 7.97 Tf 6.58 0 Td[(j!pn)]TJ 11.95 10.78 Td[(p Nej[p]!2.(1)ThismetricisalmostequivalentbecausewhenEq. 1 issmallthenEq. 1 isalsosmallaswell.Thismetrichasintroducedphasevariables[p]tohelpwiththerelaxationfromthemodulusofthediscreteFouriertransformofx[n].SolvingEq. 1 isthenalocalminimizerofEq. 1 [ 14 ].Equation 1 issolvedusinganiterativeapproachwhichusesonlyfastFouriertransforms(FFT).CANmustbeinitializedwithsomeinitialx[n]=x0[n]though.TheCANwaveformwillhaveanequivalentorbetterISLmetricthentheinputx0[n].AlsosinceCANisalocalminimizer,itoftenwillretainfeaturesintheambiguityfunction.Forexample,ifaFrankcodeisusedtoinitializeaCANwaveform,thentheCANwaveformwillretainsomeaspectsoftheDopplerridge.Figure 1-12 showstwoCANwaveform'sambiguityfunctions:Figure 1-12 AisaCANwaveforminitializedwiththeFrankcodeusedtogenerateFigure 1-10 andFigure 1-12 BisaCANwaveforminitializedwiththeM-sequenceusedtogenerateFigure 1-11 .TheISLmetricontheCANwaveformsisbetterthantheinitialwaveforms,butthetwoCANwaveformsresembletheirinitializingwaveformmorethaneachother.Inthissectionwehavebrieyreviewedsomeoftherichhistoryandfundamentalsofwaveformdesign.Wediscussedtheambiguityfunctionanditsrelationshiptothematchedlter.Wehavealsoexaminedtheambiguityfunctionsofthecontinuoustimerectangleandlinearfrequencymodulatedwaveforms.Wediscussedseveralofthemostcommonphase-codedwaveforms(FrankCodes,Zadoff-Chu,andM-sequences)andexaminedtheirambiguityfunctions.Finally,weexaminedtherecentCANwaveforms.Eachwaveformisusedbecausetheyareeithersimpletoimplementortheyareoptimalinsomesenserelatedtothezero-Dopplercutintheambiguityfunction.Wehaveseenthatbeingoptimalinthezero-Dopplercutistheequivalenttohavingalargeatbandwidthaswell.Whenthewaveformsarespectrallyconstrained,theymaynothave 31

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A BFigure1-12. TheambiguityfunctionofaCANwaveformA)InitializedwithaFrankCode.B)InitializedwithanM-sequence accesstolargeatportionsofspectrum.Thismeansallthewaveformdesignmethodsdiscussedinthissectionarenotapplicable.Wefocusonthisproblemnext. 1.4SpectrallyConstrainedActiveSensingWehavereviewedthefundamentalsoftraditionalwaveformdesignandreceiverlterdesigninthiswork.Thetraditionalmethodsallassumethatthesystemhasaccesstosomecontinuousblockofspectrumwhichitmustutilize.Recall,thattherangeresolutioninanactivesensingsystemisafunctionofthetotalbandwidthituses,sohighresolutionactivesensingsystemsareinherentlybandwidthgreedy.AreasonableproblemtoaskisWhathappenswhenIneedtouseXamountofspectrum,butonlysomesubsetofthetotalbandwidthisavailable?.Thisisthequestionweseektoanswerinspectrallyconstrainedactivesensing.AnexampleisdemonstratedinFigure 1-13 whereFigure 1-13 Ashowstheupper(redline)andlower(greenline)boundsofthespectrumandFigure 1-13 Bshowstherequiredtimedomaincomplexenvelope.ThisisagrowingprobleminRFsensingdevicessensehighdataratecommunicationdevicesarebeingallocatedlargerbandsthatusedtobeallocatedforactivesensingapplications[ 16 ].Theproblemexistsinothereldsaswellthough.Considertheactivesensingproblemforsonar.Herewestillrequirealargebandwidth,butoftentimes 32

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A BFigure1-13. Thespectrallyconstrainedactivesensingwaveformdesignproblem.A)Thespectrumbounds.B)Thetimedomainenvelopeconstraint. thefrequenciesweusearealsousedbymarinemammals.Itisproposedthatthesonariscausingadverseeffectsoncertainmarinemammals[ 17 ].Ifwerestrictedourspectrumusagetonotincludebandsthatcausesignicantproblems,itmayreducetheinterferenceonthemarinelife.Traditionalwaveformdesigndoesnotconsiderthespectralrestrictiononthedesignofanactivesensingsystem.Traditionalreceiverlterdesignwillalsodegradeinthisscenario.AnywaveformthatsatisestheboundsshowninthelefthandplotofFigure 1-13 willhavemoreenergyintheautocorrelationsidelobesthanawaveformthatwasatacrosstheentireband.Thisincreasedenergyinthesidelobescausesthesignal-to-clutter-plus-noiseratio(SCNR)toincreaseinthematchedlter.Designingawaveformthatcanmeetspectralconstraintscanalsobeconsiderspectralshaping.Oneoftherstattemptsatspectralshapingwasfornon-linearfrequencymodulation[ 18 ].Thismethodwasusedtodesignwaveformsthathadashapedspectrumtohelpedreducesidelobelevels.Itusedtheconceptofstationaryphasewhichsaysthattheenergyataspecicfrequencyislargertheslowerthefrequencymodulationrateisatthatfrequency[ 3 ].Amodernapproachonthiswould 33

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Figure1-14. GappedLinearFrequencyModulation bethefrequencyjumpburstwaveformsthatattempttohopoverunwantedbands[ 16 ].Whenthefrequencyrateisconstantinotherbands,thisfrequencyjumpburstwaveformissimplyagappedlinearfrequencymodulationwaveform.Theproblemwithanapproachlikethisisthatthereisnocontrolinthedesignonthelimitsofthespectrum.Figure 1-14 showsanexampleofgappedlinearfrequencymodulationappliedtotheconstraintsshowninFigure 1-13 .Weseethatthepowerisreducedinthebands,butthewaveformisnotspectrallycompliant.Anotherapproachtofrequencyjumpburstwaveformsistonotchafullchirpwaveforminthetimedomainsuchasdescribedin[ 16 ].ThiswillresultinasignalsimilartoFigure 1-14 ,butitwillalsoincludeamplitudemodulationwhichisnon-idealanddoesnotsatisfythetimedomaincomplexenvelope.Anotherpopularapproachtogeneratingspectrallyconstrainedwaveformsistonotchalinearfrequencymodulatedwaveform.Thedifferencebetweenthegappedandthenotchisverysubtle.Notchedlinearfrequencymodulationplacesnarrownotchesinthetransmittedwaveformatunwantedfrequencies.Anexampleofanotchedlinear 34

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A BFigure1-15. NotchedLinearFrequencyModulationA)NarrownotchesB)Widenotches frequencymodulationwaveformthatwasgeneratedusinganapproachdescribedin[ 19 ]isshowninFigure 1-15 .Figure 1-15 Ashowsthatthisnotchingisgoodwhenfocusedinnarrowbands.However,whennotchingisappliedtolargenotchessuchasrequiredbyourexampletheapproachperformsverypoorlyasshowninFigure 1-15 B.Thisapproachstilldoesnotallowformatchingtoaspectralshape,itonlyallowsustotryandremoveenergyfromspecicnarrowbands.Thenotchedandgappedlinearfrequencymodulatedwaveformscanonlybeappliedtolinearfrequencymodulatedwaveforms.Whilelinearfrequencymodulatedwaveformsarepossiblythemostcommontypeofwaveformsused,theymaynotbeappropriateforeveryscenario.Amoregeneralapproachtothespectrallyconstrainedwaveformdesignproblemwasproposedin[ 20 ].Herethegoalwastominimizethefollowingfunction: J=X!2sw(!)ZsjNXn=1x[n]e)]TJ /F8 7.97 Tf 6.59 0 Td[(j2!nj2d!,(1)wheresisthesetofallfrequenciesinthestopband(frequencieswedonottoradiateenergyinto)andw(!)isaweightingfunction.ThecostfunctionshowninEq. 1 isjustaweightedsumofthepowerinthestopband.AminimizationofEq. 1 canbeperformedwithrespecttox[n]usingabasicgradientdecentoptimizationapproach. 35

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Figure1-16. GappedRandomPhaseWaveform Figure 1-16 showstheoutputoftheapproachfrom[ 20 ]whenappliedtoourexampleproblem.Theapproachisabletogeneratewideanddeepnotches,butitdoesnotallowustospecicallyshapethespectrum.Anotherapproachthatextendedthiswasproposedin[ 21 ],wheretheyuseaweightedcombinationofthecostfunctionsEq. 1 andEq. 1 .Thisapproachsuffersthoughbecauseoftheconictinggoalsofthetwocostfunctions.Thenalspectrallyconstrainedwaveformdesignmethodwewilldiscussiscalledspectrallyconstrainedcyclicalgorithmnew(SCAN)[ 14 22 ].SCANissimilartothemethodproposedin[ 21 ].InsteadofusingthecostfunctionEq. 1 tominimizetheenergyinthestopband,SCANtriestondasequencex[n]thathasaFouriertransformwhichmapstozerointhestopband.AmajordifferenceisthatSCANcanbecomputedquicklyusingFFToperationsandelement-wisecomparisons.Figure 1-17 showsanexampleofSCAN.SCANstillsuffersfromtheproblemofconictinggoalsinthecostfunction.SCANalsodoesnotallowustospecicallyshapethespectrum.Itonlyallowsustopromotenullsandaatpassband. 36

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Figure1-17. SCANRandomPhaseWaveform Wehavereviewedcurrentapproachestospectrallyconstrainedwaveformdesign.Alloftheapproachesdonotallowadesignertocontroltheactualspectralshapeoftheoutputwaveform.Furthermore,allofthedesignswillhaveadegradedautocorrelationresponseduetothenotching.7Inthiswork,wewillshowamethodcalledSHAPEthatallowsfordesigningspecicspectrallyshapedwaveformswithdesiredtimedomaincomplexenvelopefunctions.Wethenwilldiscussafastmethodofspectralestimationforspectrallyconstrainedwaveformsthatusesamissingdataapproach.Finally,wewillproposenewavenuesofresearchtohelpimprovethestate-of-the-artinspectrallyconstrainedactivesensing. 7In[ 20 ],amethodofmismatchedlterdesignwasproposedtomitigatethiseffect. 37

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CHAPTER2SPECTRUMSHARINGVIASEQUENCEDESIGN 2.1MotivationforSpectralSharingElectronicdevicesusuallyinteractwiththeenvironmentthroughelectromagneticormechanical(acousticorultrasonic)waves.Thefunctionofthesedevicescouldbeascommonplaceasstreamingmusictoamobilephoneviaawirelesslinkoritcouldbeanationalsecurityfunctionsuchassearchingforelusivesubmarinescarryingnuclearweapons.Thecapabilitiesofthesedevicesusedtobelimitedbytheircomputationalabilitytohandleincomingdatastreams,butthisisnolongerthecase.Insteaddevicesarebeinglimitedbytheamountofthecorrespondingspectrumthattheyareutilizing[ 23 ],[ 3 ].Asaresultdevelopersaredesigningdevicestoutilizemorespectrumtosatisfythedemandforenhancedperformance.Theproblemwiththisincreaseddemandonbandwidthisthatthespectrumisalimitednaturalresourcethatmustbeshared,butonlycertainbandsinthespectrumaregoodforspecictasks.Ifanytwodevicestrytoaccessthesamebandatthesametimetheycouldpotentiallyinterferewitheachother.Theconsequencesofinterferencecanrangefrommildlyirritating(e.g.,temporarylossofmobilephonesignal)toseverelydangerous(e.g.,environmentaldamagetomarinelife,failureofaviationequipment,ormisseddetectionofincomingenemyre).Thetraditionalmethodforpreventinginterferencehasbeenthepartitioningofthespectrumintobands.Thesebandsarethenstaticallyassignedtospecicpartiesandtaskswhichareenforcedandregulatedbygovernmentagencies.Thestaticassignmentmethoddoesnotefcientlyusethespectrumunlessallbandsareutilizedatagiventimeandarea[ 24 ].ThishasledtotheconceptofDynamicSpectrumManagement(DSM)incommunicationswheredevicescanmakeuseoftemporarilyunusedportionsofthespectrum[ 25 ].TheInstituteofElectricalandElectronicEngineers(IEEE)hasalreadyformedastandardscommitteethatiscurrently 38

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determininghowDSMshouldbeimplementedfornarrowbandcommunicationdevices[ 26 ].TheU.S.DefenseAdvancedResearchProjectAgency(DARPA)isactivelyseekingDSMstrategiesthatextendbeyondjustcommunicationsintothefullrealmoftheirelectronicwarfareequipment[ 27 ]andtheNationalScienceFoundation(NSF)isalsoseekingDSMtechniquestoincreasewirelessbroadbandaccess[ 28 ].ThereisnoreasonforDSMstrategiestobelimitedtoonlyRFsystems.Onnavalvesselsaroundtheworld,sonarsystemsareusedforamyriadofcriticaltasksandhavecomeunderscrutinylatelyduetotheeffectonmarinemammals[ 29 ],[ 17 ].However,itmaybepossibletosharethespectrumwithmarinemammalsbyutilizingsonarsignalsthatdonotcauseadversephysiologicalorbehavioralreactionsonmarinemammals[ 30 ].Tosharethespectrum,waveformsneedtobedesignedthatwoulddistributetheenergyappropriatelywhilestillprovidingtherequiredbandwidth.Theproblemofdesigningwaveformsisanumericalsequencedesignproblem.Formostpracticalapplications,theproblemisexacerbatedbyaconstantamplitudeconstraintonthesequencewhichsignicantlyincreasestheproblemcomplexity.Thereasonforthisconstraintisthattransmittersoperatemoreefcientlywhenusingahighpoweramplierinthesaturationregionratherthaninthelinearregion.Thetraditionalwaveformdesignapproacheswerefocusedonoptimizingtheautocorrelationofthesequences.Anymethodthatoptimizestheautocorrelationfunctionisimplicitlyperformingspectralshaping[ 3 ],[ 14 ].Researchintospectrallyshapedwaveformshasincreasedoverthepastdecadeduetothedesireforwidebandapplications.Mostofthesemethodshavefocusedonplacingnotchesinthespectrumwhilemaintaininggoodautocorrelationproperties.Theexistingmethodsfallshortbecausetheyareeitherlimitedbynotchdepth,notchwidth,orthenumberofnotches.Othermethodsalsocannotguaranteethattheywillmeettheirspectralrequirements.Furthermore,mostmethodsarebasedonlinearfrequencymodulation(LFM)waveformswhichmightnotbeidealforallapplications.See[ 14 20 21 31 32 ]forpreviousworkinthisarea. 39

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InthispaperwepresenttheSHAPEalgorithm,acomputationallyefcientmethodofdesigningsequenceswithdesiredspectrumshapes.ThealgorithmusesalternatingfastFouriertransforms(FFTs),element-wisecomparison,andelement-wisearithmeticsuchthatitcaneasilybeimplementedusingparallelcomputingmethods.SHAPEsolvestheproblemofndingasequencewithanarbitrarytimedomaincomplexenvelopeandanarbitrarilyshapedspectrum.TheubiquitousconstantmodulussequencedesignproblemisaspecialcasethatSHAPEcansolve.Inthispaperweshalldenoteavectorwithboldfacelowercaseletter(x)andwedenotetheelementsofthevectorasx=[x1,x2,...,xN]T.Similarlyamatrixisdenotedbyaboldfaceuppercaseletter(X).Here()Trepresentsthetransposeoperationandtheconjugatetransposeoperationisdenotedby()H.Asuperscriptonavariablesuchasxkrepresentsaniterationcounterandnotapoweroperation.Ahatsymbolonavariable(^x)representsthesolutiontoaminimizationproblem. 2.2SHAPEAlgorithmAleast-squaresttingapproachforthespectralshapingproblemcanbeformulatedas: minimizexjjjFHxj)]TJ /F3 11.955 Tf 17.93 0 Td[(yjj22subjecttojxij2=hi,fori=1,2,...,N,(2)wherex2CN1isoursequenceweseektodesign,y2R+N1isthedesiredspectrummagnitude,andF2CNNisaunitaryFouriermatrix.ThedesiredsequencelengthcanbesmallerthanN,inwhichcasethesequenceiszeropaddedinthevectorxtoutilizeoptimalFFTimplementations.Theenvelopeconstraintonthetimedomainsequenceisrepresentedbyh=[h1,h2,...,hN]T.Ingeneralweallowhtohaveanyvalue,butitcommonlyonlytakesonvaluesrelatingtowindowfunctions(e.g.,box,Hamming,Taylor,orTukey)toallowforgradualriseandfadeoftheprobingsequences[ 9 ]. 40

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ThemodulusoperationinEq. 2 makesadirectsolutiondifcult.Thisproblemcanberesolvedbytheinclusionofthephasevariable2RN1ofFHx.Thisresultsinthefollowingminimizationproblem: minimizex,jjFHx)]TJ /F3 11.955 Tf 11.96 0 Td[(yejjj22subjecttojxij2=hi,fori=1,2,...,N,(2)whererepresentstheentry-wiseproductoperation.Thisproblemiswell-knownandeasytosolveinaniterativemannersincethematrixFisunitary(see[ 14 ],[ 33 ]): Givenx:^=argfFHxgGiven:^x=p hiejargfF(yej)g.UsingthisapproachforarbitraryspectrumshapesandarbitraryenvelopefunctionsmayresultinconvergenceofthecostfunctioninEq. 2 tosignicantnon-zerovalueswhichimpliesthatthespectralrequirementswillnotbemet.Evenwhentheconvergenceistoavaluenearzero,theresultingspectrummayoscillatearoundthedesiredspectrumshape,y,whichmaynotmeetstrictspectrumrequirementseither.Thisapproachwillalsogivepoorresultsaswelliftheenergycontainedintheenvelopeandthedesiredspectrumarenotequivalent.Weovercometheproblemsofthedirectttingapproachbyrelaxingourspectralshapetointroducemoredegreesoffreedom.Insteadofttingtoaspecicshape,wesearchforanysequencewithaspectrumcontainedwithinsomeupperandlowerbounds,f(!)andg(!),respectively.Wedenotethevectorsf=[f(!1),f(!2),...,f(!N)]Tandg=[g(!1),g(!2),...,g(!N)]Tastheboundingfunctionssampledonourfrequencygridpoints.Thenwecansearchinsteadforsomespectrumzwithmoduluscontainedwithinthebounds.Wealsointroduceascalerfactortoaccountforanypossibleenergymismatchand/orconstantphaseoffsetbetweenthesequencexandthe 41

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spectrumz.Thisresultsinthefollowingminimizationproblem: minimizex,z,jjFHx)]TJ /F7 11.955 Tf 11.95 0 Td[(zjj22subjecttojxij2=hi,fori=1,2,...,Njzijfifori=1,2,...,Njzijgifori=1,2,...,N,(2)wherefi=f(!i)andgi=g(!i).AgainsincethematrixFisunitarythisproblemcanbesolvedinaniterativemannerviatheSHAPEalgorithm.TherearethreemainstepstosolveineachiterationoftheSHAPEalgorithm:given(,x)minimizewithrespectto(w.r.t.)zsuchthatzsatisestheconstraints,given(x,z)minimizew.r.t.,andgiven(,z)minimizew.r.t.xsuchthatxsatisestheconstraints.Westartbyexaminingtherstminimizationsub-problemthatndsasatisfactoryspectrumvectorzgivensometime-domainsequencexandenergyscalar: minimizezjjFHx)]TJ /F7 11.955 Tf 11.95 0 Td[(zjj22subjecttojzijfifori=1,2,...,Njzijgifori=1,2,...,N.(2)Iftherewerenoconstraintsonz,theanswerissimplyzopt=~x=or(zi)opt=j~xij jjej(~xi)]TJ /F9 7.97 Tf 6.59 0 Td[()where~x=FHx.TheconstrainedproblemposedinEq. 2 hasasolutionsimilartotheunconstrainedproblem.Letc1(z)bethecostfunctionfromEq. 2 : c1(z)=(FHx)]TJ /F7 11.955 Tf 11.96 0 Td[(z)H(FHx)]TJ /F7 11.955 Tf 11.96 0 Td[(z)=xHFFHx+jj2zHz)]TJ /F6 11.955 Tf 11.96 0 Td[(2RefHzHFHxg(2)whichwerewriteasc1(jz1j,z1,...,jzNj,zN)=NXi=1)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(j~xij2+jj2jzij2)]TJ /F6 11.955 Tf 11.96 0 Td[(2jjjzijj~xijRefej(~xi)]TJ /F9 7.97 Tf 6.58 0 Td[()]TJ /F9 7.97 Tf 6.59 0 Td[(zi)g. 42

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Sincej~xij,jj,jzij0and)]TJ /F6 11.955 Tf 9.3 0 Td[(1fithentheminimumvalueofthecostfunctionoccursatjzij=fi.ThenthesolutiontoEq. 2 isgivenby^zi=8>>>><>>>>:fiej(~xi)]TJ /F9 7.97 Tf 6.59 0 Td[():j~xij=jj>fi,giej(~xi)]TJ /F9 7.97 Tf 6.59 0 Td[():j~xij=jj
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whereweutilizethefactthatFisunitarytorewriteourcostfunctionas: c2(x)=(x)]TJ /F7 11.955 Tf 11.95 0 Td[(Fz)H(x)]TJ /F7 11.955 Tf 11.96 0 Td[(Fz)=xHx+jj2zHFFHz)]TJ /F6 11.955 Tf 11.95 0 Td[(2RefHzHFHxg(2)andthereforec2(jx1j,x1,...,jxNj,xN)=NXi=1)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(jxij2+jj2j~zij2)]TJ /F6 11.955 Tf 11.96 0 Td[(2jjj~zijjxijRefej(xi)]TJ /F9 7.97 Tf 6.59 0 Td[()]TJ /F9 7.97 Tf 6.59 0 Td[(~zi)g.Herewehaveusedthesubstitution~z=Fz.Similartobeforetheminimumofc2(x)willoccuratRefej(~xi)]TJ /F9 7.97 Tf 6.59 0 Td[()]TJ /F9 7.97 Tf 6.58 0 Td[(zi)g=1.Thisresultsinthesamephaseastheunconstrainedoptimalanswerxi=~zi+.Theamplitudeofxiisxedbytheconstraintssothentheresultingminimizerisgivenby ^xi=p hiej(~zi+)(2)fori=1,2,...,N.Astep-by-stepguidetoimplementingSHAPEispresentedinTable 2-1 .SHAPEiscomputationallyefcientsinceitutilizesonlyFFTs,innerproducts,andelement-wiseoperations(magnitude,scalardivision,scalarmultiplication,scalarcomparisons).Alloftheforloopsinthealgorithmcanbeunrolledandimplementedusingparallelprocessingfurtherincreasingthespeed. 2.3PracticalIssuesTosuccessfullyutilizetheSHAPEalgorithmforaspecicapplicationseveralpracticalissuesneedtobeaddressed.Therstproblemistheselectionoftheboundingfunctionsf(!)andg(!)andhowtheyaffecttheconvergenceofSHAPE.Thesecondproblemisthatinanyrealsystemthesequencewillbequantizedwhichdistortsthespectrum.Thethirdproblemdealswithsamplinginthefrequencydomain.Finally,theinitializationoftheSHAPEalgorithmisalsoacriticalconsideration. 44

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Table2-1. TheSHAPEAlgorithm initialize: 1)Set0=1 2)Choosex0suchthatjxij2=hi,fori=1,2,...,N 3)Settheiterationindexk=1andsetmaxiterationK repeat 1)Calculatezkgivenk)]TJ /F5 7.97 Tf 6.59 0 Td[(1,xk)]TJ /F5 7.97 Tf 6.58 0 Td[(1usingu=FHx= fori=1toNdo ifjuij>fithen zki=fiui=juij elseifjuijK 2.3.1ConvergenceApriorithereisnoguaranteethatasolutionexistsforagivenspectralshapeandtimedomainwindow,andwhenSHAPEencountersscenarioslikethisitwillconvergetoanon-zerovalueofthecostfunction.However,wehavefoundthatwegreatlyincreaseourchancesofndingsequencesxthatsatisfytheessentialconstraintsbyallowingsomebands()tobeunrestricted(f()=1andg()=0).Howtochooseisdependentupontherestrictionsthatareplacedonthespectrumshape.Thereisnoformulaormethodforanoptimalselectionof,butadesignercantypicallyndasolutionbyincludingasmallunrestrictedregion. 45

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Figure2-1. ConvergenceFailure Wedemonstratetheproblemandtheresolutionusingasimpleexample.Considerdesigninganotchedrandomphasesequencewithbandwidth1MHzandunimodularcomplexenvelope.Thepowerspectrumshapeallows3dBrippleinthepassbandandthatstopbandnotchestobe)]TJ /F6 11.955 Tf 9.3 0 Td[(40dBdownwithwidthsof10,1and100kHz.Figure 2-1 showsthepowerspectrumoftheSHAPEwaveformwhenusingthestrictlimits.Theredlinecorrespondstotheupperboundf(!)andthegreenlinecorrespondstothelowerboundg(!).After60000iterationsSHAPEwasterminatedandexaminationofthecostfunctionshowsitwasoscillatingaroundanon-zerovalue.Thisismostlikelyduetothesharptransitionsinthespectrumshapeintroducingalargeamountofringingwhichwasconictingwiththespectralbounds.Toovercomethisweintroducetherelaxationf()=1andg()=0for15kHzguardbandsonbothsidesofeachspectralnotch.Figure 2-2 showstheSHAPEoutputwheninitializedwiththesameunimodularrandomphasesequenceasFigure 2-1 .Whenthespectrumconstraintswererelaxed,SHAPEwasabletoconvergetoazerocostvaluewithinthe60000iterations.Weseethatbyallowingsomefreedominthepassbandthedeepnotchescanbeachievedatthecostofadegradedpassband,whichshouldbeacceptableinmanypracticalapplications. 46

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Figure2-2. ConvergenceSuccessful AnotherfactorthathelpsimprovetheconvergencetozerocostofSHAPEistheuseofanoffsetforthespectralshapes.Supposethatwehavethedesiredspectralshapingfunctionsf(!)andg(!).WhenSHAPEsolvestheminimizationstepitusesaleastsquarescriteriattingsolution.Thismeansthattheanswermaytendtooscillatearoundthedesiredboundary.Thisproblemisespeciallyprevalentinthenotches,butitiseasilyaccountedforbyusingasmalloffset.Inallexamplesinthispaperweuse=1.5dBrelativetof(!)andg(!).ThenthespectralfunctionsweapplytoSHAPEaregivenby: ~f(!)=f(!)(1)]TJ /F7 11.955 Tf 11.96 0 Td[()~g(!)=g(!)(1+)=10=10(2)IfthereisaregionthatisunrestrictedthenEq. 2 isnotapplied.Toimplementthisidea,Table 2-1 shouldbemodiedtouse~fand~gwhenlimitingthespectralvaluesofzi.Notethesubtledifferencethatthedesiredboundsarestillfandg,butwettooffsetbounds~fand~g.Thismodicationforcesthettoaslightlymorerestrictiveconstraint.Thisallowsforripplestobecontainedwithinthedesiredvalueswhichhelpsstrictlymeetthespectrumrequirements. 47

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2.3.2QuantizationTheSHAPEalgorithmassumesthatwehaveaccesstoinniteprecisionnumberswhichisnottrueinanyactualsignalprocessingapplication.Theuseofdoubleprecision(64bit)numbersismorethanadequateforndingsequencesusingSHAPE,buttheproblemliesintruncationtolowerprecisionintegersequences.Thespectralshapeisdeterminedbythephaseandamplitudeofthecomplexsequence,andbittruncationmaydistortthespectralshape.Similarlythecomplexenvelopemaybecomedistortedwhichwillintroduceamplitudeuctuationsfromthedesiredenvelope.Thequantizationproblemisrelevantbecausedigital-to-analogconverters(DAC)ofmodestbandwidthwilltypicallyhavebitprecisionlessthan32bitsandutilizexedpointrepresentations.Theuseofthexedpointvaluesresultsinadiscretesetofphasesthatcanactuallyberepresented.HencetheDACisgoingtoquantizeourphasesequenceanddistortourspectrum.Fromourempiricalresults,thequantizationbyitselfdoesnothaveaseriousimpactontheresultingspectrumuntilapproximately8to10bitsareusedtorepresenteachnumber.ConsiderthesameSHAPEwaveformfromFigure 2-2 ,butnowtherealandimaginarypartshavebeenquantizedtoxedprecisionsignedintegers.Figure 2-3 andFigure 2-4 showtheresultingspectrumwhentheSHAPEwaveform'srealandimaginarypartsarequantizedto16bitand12bitintegersrespectively.Thereissomeslightdegradationuponcloseexamination,butingeneralitisminor.InFigure 2-5 and 2-6 therealandimaginarypartshavebeenquantizedto8bitand4bitsignedintegers.Herethereissomesignicantdegradationofthespectralshape.Alsoofsomeconcernisthatthequantizedsequencesnolongerperfectlymeetthetime-domaincomplexenvelopeconstraint,buttheintroducedrippleissmallrelativetothemagnitudeoftheintegersequence. 48

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Figure2-3. 16BitsQuantization Figure2-4. 12BitsQuantization 2.3.3FrequencySamplingIfweconsidernon-periodicwaveforms,thenthespectrawillbecontinuous[ 9 ].SHAPEwouldhavetoapproximatesuchaspectrumbyusingtheDFT(FFTinSHAPE).Stilltherewouldhardlybeanyguaranteethatspectralcompliancecouldbemetateverypointinthefrequencydomain.Ontheotherhand,ifweconsiderperiodicwaveforms,thentheresultingspectraareactuallydiscreteandcanbematchedperfectlybySHAPE.Forexample,consideranactivesensingsystemthatutilizesapulsedmodeofoperation.Wecanconsiderthistobeaperiodicwaveformdesignbyaccountingforthe 49

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Figure2-5. 8BitsQuantization Figure2-6. 4BitsQuantization listeningtimebetweentransmissions.Thatis,wedesignoursequenceaccordingtotheappropriateamountofzerosneededduringthelisteningtime.Usingthisapproachresultsinadiscretespectrumwhichcanmatchthedesignedspectrumandbeinfullspectralcompliance. 2.3.4InitializationSHAPEisdependentontheinitialsignalx0.RecallthatSHAPEsearchesforapair(z,x)thatsatisesthetimedomainandspectralconstraintsandthereisanunknownnumberofpairsthatactuallysatisestheconstraints.Thendifferentinitializationscan 50

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Figure2-7. InitialSpectrumusingFullBandwidth leadtodifferentoutputsfromtheSHAPEalgorithm.Howtochooseaninitialsignalisdependentuponasystem'srequirements,butwhentheinitialsignal'sspectrumhasfeaturessimilartothedesiredspectrum,SHAPEseemstoconvergefastertozerocostvalue.Theinitializationproblemisbestdemonstratedwithanexample.Asystemdesignerwishestoutilizeaunimodularrandomphasesequenceforanapplicationthatrequiresapassbandof1kHz,butitwillbeoversampledat100kHz.WethenwanttouseSHAPEtondarandomphasewaveformthatmeetssteepspectralroll-offrestrictions.OnepossibleoptionfortheinitializationofSHAPEistousearandomphasewaveformsampledatthemaximumbandwidth100kHz.Anotheroptionistouseasequencethatissampledat1kHzandthenupsampledto100kHz.Theupsamplingtechniqueusedisasampleandholdmethod.Ifx(n)isouroriginalsequenceoflengthNandweupsamplebyMthenxu((n)]TJ /F6 11.955 Tf 11.96 0 Td[(1)M+m)=x(n)forn=1,2,...,Nandm=1,2,...,M.Figure 2-7 showsthespectrumwhenweutilizearandomphasesequenceandaspectralroll-offrequirement.Theinitialspectrumisnotsimilartothedesiredspectrumatallandusingthisinitialspectrumwouldbedifcult.Figure 2-8 showsthespectrumwhenweutilizetheupsamplingapproachtotherandomphasewaveform.Theinitialwaveformsspectrumismodulatedbyasincfunctionduetothesampleandhold 51

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Figure2-8. InitialSpectrumusingSampleandHold method.Thiscausesthespectrumtoroll-off,butitisnotenoughroll-offtomeettherequirements.ThisspectrumwouldbeabetterchoicetoinitializeSHAPEwithsincetheenergyisalreadyfocusedtowardsthepassband.ThereisanotheroptionforinitializationthatusesaSHAPEwaveformforinitializationofSHAPE.WecallthisthetieredapproachtoSHAPEbecausewesolvemultiplespectralshapingproblemstondagoodinitializationforthenalproblem.InthisapproachwestartbyutilizingSHAPEtondawaveformatsomeinitialsamplingratethatislowerthanourdesiredsamplingratethatalsosatisessomespectrumconstraintforthisbandwidth.TheSHAPEoutputwaveformisthenupsampledusingthesampleandholdapproach.TheupsampledwaveformisagoodinitializerforSHAPEbecauseitalreadymaintainsfeaturessimilartothedesiredspectralconstraints.ThereforewefeedtheupsampledwaveformbackintoSHAPEwithanewsetofspectralrestrictionsfortheexpandedbandwidth.Thisprocessisrepeateduntilthedesiredsamplingrateismet.Thetieredwaveformapproachwillbedemonstratedinthesonarexampleinthefollowingsection. 2.4ExamplesofSequenceDesignforSpectralComplianceThebenetoftheSHAPEalgorithmisthatitcanndarbitrarilyshapedwaveformsintheboththetimeandfrequencydomain.Thisopensrealopportunitiesintheactive 52

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Figure2-9. Initialization(GappedLFMWaveform) sensingandthecommunicationseld.Thestrictspectrumrequirementsposedbyaregulatoryagencycannowbemetwhilemaximizingavailablebandwidth. 2.4.1WidebandRadarWedemonstratetheutilityoftheSHAPEalgorithmbyconsideringapracticalsequencedesignexampleforwidebandradarimaging.Weassumethatwehaveaccesstothe225-328.6MHzand335-400.15MHzbandsallocatedfortheUSFederalgovernment.Thebandsarequitelimitedforhighresolutionimaging;therefore,weseektoutilizethenearbywhitespaceintheunusedportionsofthespectrumallocatedforlicensedtelevisionbroadcastswhichoccurfrom470-698MHz.Eachtelevisionstationisallocated6MHzofbandwidththatwemustnotinterferewithand10stationsarelicensedforoperationinAlachuaCounty,Florida.Wethenwishtoexploitthe473MHzofavailablespacebyplacingnotchesinthebands(MHz):(328.6-335),(400.15-470),(482-486),(500-506),(554-560),(572-578),(590-596),(602-608),(638-644),(650-656),(674-680),and(680-686).Fortheradarwaveform,wealsoshapethespectrumroll-offfactoraccordingtotheNationalTelecommunicationsandInformationAdministration(NTIA)standardsforgroupCradar[ 34 ].Finallyinradaritisparticularlyimportanttohaveaconstantamplitudesequenceandhencehi=1fori=1,2,...,N. 53

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Figure2-10. SHAPEOutput Figure2-11. Autocorrelation SHAPEisinitializedwithagappedLFMwaveformthatdoesnotmeetthespectralshapingrequirementsf(!)asshowninFigure 2-9 .Inthisexampleg(!)=0sincetheLFMwaveformhasafairlyatpassbandalreadyandthenotchrequirementsarenumerousanddeep.TheLFMpulsewidthis25secwithatotalbandwidthof473MHz.Thesamplingfrequencyissettobeexactlytwicethetotalbandwidthinthisexampleresultingin23645timedomainsamplesandtheFFTsizeusedwas65536(216).TheSHAPEwaveform'spowerspectrumwhichstrictlymeetsspectrumrequirementsisshowninFigure 2-10 .Figure 2-11 showstheautocorrelationsoftheinitialgapped 54

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Figure2-12. AutocorrelationZoom LFMsequenceandtheSHAPEwaveform.TheSHAPEwaveformmaintainsasimilarautocorrelationfunctionshapetotheinitialsequence,butthesidelobelevelsarehigher. 2.4.2RandomPhaseSonarWaveformTheapplicationofSHAPEusingthetieredapproachwillbedemonstratedusingasonarexamplewithverystrictspectralrequirementstomitigatetheeffectsonmarinemammalsandtoreducemanmadeacousticnoiseintheocean.Thetimedomainsequenceinthisparticularexampleisa1secondsignalwindowedwithaTukeywindowwithwindowparameter0.1.Thewindowshapeisallowsforriseandfadetimesinthecomplexenvelopeandwasspeciedbyasystemrequirement.TherampratecanbeseeninFigure 2-21 .Thespectrumpassbandisgivenas1050to1950Hz,thenthepowerspectrummustdecreaseby100dBby5000Hz,andnallythespectrummustlinearlydecreaseto)]TJ /F6 11.955 Tf 9.3 0 Td[(150dBat50KHz.Therearesequencesthatcanmeetthisspectralrequirement,buttheytypicallyresembleLFMwaveforms.Anadversarywouldlikelyrecognizeoneofthesesequencesasasonarwaveformandsinceaprimarygoalinsonarisalsotobecovertwewouldlikeoursequencetobesimilartoarandomphasesequencetopreventrecognitionbyanadversary.ThedesignofthisSHAPEwaveformwitharandomphasesignalinitializationisdifcultasdiscussedinthesectiononinitialization.Figure 2-7 and 2-8 areexamples 55

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Figure2-13. Stage1Initialization oftwodifferentrandomphaseinitializationapproaches.Tosolvethisproblemweusea4stagetieredSHAPEapproach.Weutilizesamplingratesof1,5,10,and100kHzinstages1-4.Forstage1,wehavetorstshapethespectrumtosatisfythe900Hzpassband.WeinitializeSHAPEwitharandomphasewaveformasshowninFigure 2-13 andgettheoutputwaveformshowninFigure 2-14 .Noticethatwehavemadeourrandomphasewaveformsatisfythe900Hzpassbandrequirement. Figure2-14. Stage1Output Forthesecondstageweupsamplethestage1outputbyafactorof5usingthesampleandholdapproach.Wethenshiftthewaveformtothecarrierfrequencyof1500 56

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Figure2-15. Stage2Initialization Hz.ThisresultsinthethepowerspectrumshowninFigure 2-15 .Wedonothavearequiredspectralconstraintuntil5000Hzsowesimplyrestrictthestopbandto)]TJ /F6 11.955 Tf 9.3 0 Td[(30dBandthestage2outputisshowninFigure 2-16 .Forstage3,thestage2outputisupsampledbyafactorof2usingthesampleandholdapproachagainresultinginthespectrumshowninFigure 2-17 .Nowweenforcealinearsuppressionof)]TJ /F6 11.955 Tf 9.29 0 Td[(30dBto)]TJ /F6 11.955 Tf 9.3 0 Td[(100dBfrom2550Hzto5000Hzand450to)]TJ /F6 11.955 Tf 9.3 0 Td[(5000Hz.Noticethatbetween1950to2550Hzand450to1040Hztherearenorestrictions.ThisgivesSHAPEsomefreedomwhenshapingthespectrum(seethediscussiononthecostfunction'sconvergencetozero).Thestage3outputwaveformisshowninFigure 2-18 .Forthenalstage,weutilizeanupsamplingfactorof10usingthesampleandholdmethodagain.WethenintroducethenalandcompletespectralrequirementsasshowninFigure 2-19 .Thespectrumoftheupsampledwaveformispeaky,butthemajorityoftheenergyliesinthepassband.SHAPEisabletomovetheenergyintothepassbandandunrestrictedregionssuchthatthespectralrestrictionscanbemet.Figure 2-20 istheSHAPEwaveformoutput.Figure 2-21 showstherealpartandFigure 2-22 showstheimaginarypartsoftheoutputwaveforminthetimedomain.Theoutputwaveformstillmaintainssimilarcorrelationpropertiestotheinitialrandomphasesignal.Thenormalizedmagnitudesquaredvaluesoftheautocorrelations 57

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Figure2-16. Stage2Output Figure2-17. Stage3Initialization oftheinitialsequenceandtheSHAPEsequenceareplottedinFigure 2-23 .Themainlobewidensandthesidelobelevelsrise,butthewaveformisperformanceisstillsimilartotheinputrandomphasesequence.Ifanotherrandomphasesignalisgeneratedinthismanner,thenthecross-correlationperformanceissimilartothecross-correlationoftheinputsignals.WecreatedtwoSHAPEwaveformsutilizingthesamespectralrequirements,buttheywereinitializedwithtwodifferentrandomphasesequences.Weshowthecross-correlationperformanceoftheinputandSHAPEsequencesinFigure 2-25 ,wherethecross-correlationisnormalizedbythetotal 58

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Figure2-18. Stage3Output Figure2-19. FinalStageInitialization spectrumpower.Thecross-correlationisslightlylargerfortheSHAPEwaveforms,buttheyarestillonthesameorderofmagnitudeastheinputsequences. 59

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Figure2-20. FinalStageOutput Figure2-21. RealPartofWaveform Figure2-22. ImaginaryPartofWaveform 60

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Figure2-23. Autocorrelation Figure2-24. AutocorrelationZoom Figure2-25. Cross-CorrelationsofTwoSHAPEWaveforms 61

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CHAPTER3MISSINGDATAIAAANDSPECTRUMNOTCHES 3.1SpectralEstimationandMissingDataAfundamentaltaskinspectralestimationistoestimatenoisysinusoids'frequenciesandamplitudesfromasetofmeasurements[ 9 ].Thecommonsolutionistousetheperiodogramtoestimatethespectra,amethodwhichingeneralsuffersfromlargesidelobesandpoorresolution.Inthecaseofmissingdata,thesidelobeproblembecomesevenworse.ThesidelobesincreaseduetomodulationinthesamplingdomainbytheincompletesamplingpatternrelativetotheuniformsamplingintheFouriermatrix.Itisthereforeoftendesirabletointerpolatethedataontoauniformsamplingpatternorestimatethemissingdataifpatchesofdataaremissing.Arecentlydevelopedhighresolutionnonparametricspectralestimationtechnique,theiterativeadaptiveapproach(IAA)[ 12 ],canalsobeusedinthecasewithmissingdata(MIAA)[ 35 ].Thisisamethodbasedoniterativeweightedminimization,wheretheweightisupdatedtoincreasetheresolutionandsuppresssidelobes.IAAprovidesresolutionsuperiortotheperiodogram,andhastheadvantagethatonlyasinglesnapshotisrequired.ThemajordrawbackofIAAisthecomputationalcostsforadirectimplementation.Intworecentpapers([ 36 ],[ 37 ])implementationsofIAAweredevelopedbasedonFFToperations,whichconsiderablyspeedsupthealgorithmandmakesitapplicableforlargerproblems.ThesefastimplementationsutilizeTopelitzandVandermondestructuresthatarisewhenthesamplinggridisuniformandcomplete.WhenthesamplinggridisnotcompletetheToeplitzstructureofthecovariancematrixislost,andthefastimplementationsarenotefcientwhenthenumberofmissingsamplesissmall.Toresolvethis,weusealowrankcompletiontotransformtheproblemtothestructuredproblemwherethecovariancematrixisToeplitz.Thisleadstothemaincontributionofthispaper,afastimplementationofMIAAwhichisconsiderably 62

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fasterthanthestate-of-the-artforthecasewhenthemissingdataissmall.Finally,anapplicationtosyntheticapertureradar(SAR)imagingisexamined.InmanypracticalSARscenariosthesetofmeasureddataisincomplete,dueto,e.g.,interference,jammingordatadropouts[ 38 ],[ 39 ],resultinginanincompletedataset.Hereweconsiderthecasewhereaproportionofthesamples(<50%)aremissingduetofrequencynotchingbeingusedtosuppressinterference.Thisisasituationwhichisnotuncommoninultrahighfrequency(UHF)orveryhighfrequency(VHF)SAR,wherethespectrumisoftencrowded.HereweapplyMIAAfortherecoveryoflostfasttimesamplesduetofrequencynotchingintheoccupiedbands.ThisisshowntosignicantlyimprovetheresultingSARimagequality.InSection 3.2 wesetupthedatamodel,discussthespectralestimationproblem,andintroducethealgorithmIAA.Section 3.3 describescomputationalcomplexitiesoftheIAAalgorithmandhowtoutilizetheToeplitz/Vandermondestructuresintheproblemtosignicantlydecreasethecomputationalcomplexity.InSection 3.4 themissingdataalgorithmisdiscussedandwepresentthenewfastalgorithmformissingdataIAA.InSection 3.5 itisshownhowtoutilizetheresultsfromSection 3.4 inordertoprovideacomputationallyefcientrecoveryofthemissingdata.InSection 3.6 wepresentexamplesthatillustratethecomputationalbenetsofthenewalgorithmsandthenapplythealgorithmtodatarecoveryinsparseSARimaginginthepresenceofspectrumnotches.Thenotationusedinthispaperwillbebrieydened.Avectorisrepresentedbyaboldfacelowercaseletter(x)andamatrixisrepresentedbyaboldfaceuppercaseletter(X).Thetransposeofamatrixis()Tandtheconjugatetransposeis().Theconjugateofacomplexnumberisgivenby ().AmatrixXisHermitianifX=X.ForHermitianmatricesX,Y,letX>Y(XY)denotethatthematrixX)]TJ /F3 11.955 Tf 12.58 0 Td[(Yispositivedenite(semidenite).AfastFouriertransformoperation(FFT)ofsizeNisdenotedbyF()N,whereappropriatezeropaddingisperformedifnecessarywithoutdiscussion. 63

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Asubscriptonavectorormatrixofgdenotesthedatawasmeasured(given)whileasubscriptofmrepresentsdatathatwasnotmeasured(missing).Adenotationof()1:Krepresentsanindexingoperation,i.e.elements1toKofavector. 3.2SpectralEstimationandIAA 3.2.1DataModelConsidertheproblemofrecoveringthespectralcontentfromameasuredsignal.Lety=(y0,y1,...,yN)]TJ /F5 7.97 Tf 6.59 0 Td[(1)TdenoteasampleddatasequenceoflengthNandlet A=(a(!0),...,a(!K)]TJ /F5 7.97 Tf 6.59 0 Td[(1))(3)beanoversampledFouriermatrixsuchthatK>N.ThecolumnsofAarea(!k)=)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(1,ej!k,...,ej(N)]TJ /F5 7.97 Tf 6.58 0 Td[(1)!kTwhichcorrespondtothefrequencyvectorsand!kcorrespondstothefrequencygridpoint!k=2k K,fork=0,...,K)]TJ /F6 11.955 Tf 10.94 0 Td[(1.Letx=(x0,x1,...,xK)]TJ /F5 7.97 Tf 6.59 0 Td[(1)Twherexkdenotesthecomplexspectralcontentatfrequency!kofthesignaly.Thedatamodelcanthenbeformulatedas y=Ax+e,(3)wherethenoisecontributionise.Thegoaloftheproblemistoestimatex.Themostcommonmethodforsolvingthisistheperiodogram(matchedltermethod), xk=a(!k)y a(!k)a(!k),k=0,1,...,K)]TJ /F6 11.955 Tf 11.95 0 Td[(1,(3)whichmaybecalculatedusingtheFFT.Thisiscomputationallyefcient,butsuffersfromhighsidelobesandpoorresolution.Onewaytoovercometheseissuesistousedata-adaptivemethodssuchastheCaponmethod[ 40 ],[ 41 ],AmplitudeandPhaseEstimation(APES)[ 42 ],[ 43 ],orIterativeAdaptiveApproach(IAA)[ 12 ].HerewewillfocusexplicitlyonIAAasithasshownpromiseintheeldsofradarimaging,sonar,communications[ 12 ],medicaldiagnostics[ 44 ],informationforensics[ 45 ],andgeneralspectralestimation[ 46 ]. 64

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3.2.2IterativeAdaptiveApproach(IAA)IAAseekstondaspectralestimatexkinEq. 3 bymodelingtherestofthespectrumx`,`6=k,asinterference[ 12 ].Theinferencereferstoallthesignalsatfrequencygridpointsotherthanthegridpointofinterest!kandismodelledasx`2N(0,p`)inEq. 3 for`6=k.ThecovariancematrixQkoftheinterferenceisthengivenbyQk=R)]TJ /F14 11.955 Tf 11.95 0 Td[(pka(!k)a(!k),where R=K)]TJ /F5 7.97 Tf 6.59 0 Td[(1X`=0p`a(!`)a(!`)=APA.(3)HereRisthecovariancematrixofthedataandP=diag(p),wherep=(p0,p1,...,pK)]TJ /F5 7.97 Tf 6.59 0 Td[(1)T,andp`=jx`j2denotesthepowerestimateatthefrequencygridpoint!`,for`=0,1,...,K)]TJ /F6 11.955 Tf 13.24 0 Td[(1.Maximizingthelikelihoodofxkthenresultsinminimizationoftheweightedquadraticcostfunction (y)]TJ /F3 11.955 Tf 11.95 0 Td[(a(!k)xk)Q)]TJ /F5 7.97 Tf 6.59 0 Td[(1k(y)]TJ /F3 11.955 Tf 11.95 0 Td[(a(!k)xk),(3)wheretheoptimalsolutionisgivenby xk=a(!k)Q)]TJ /F5 7.97 Tf 6.58 0 Td[(1ky a(!k)Q)]TJ /F5 7.97 Tf 6.59 0 Td[(1ka(!k),k=0,1,...,K)]TJ /F6 11.955 Tf 11.95 0 Td[(1.(3)Usingthematrixinversionlemma,Eq. 3 equals xk=a(!k)R)]TJ /F5 7.97 Tf 6.59 0 Td[(1y a(!k)R)]TJ /F5 7.97 Tf 6.59 0 Td[(1a(!k),k=0,1,...,K)]TJ /F6 11.955 Tf 11.95 0 Td[(1.(3)ThisconsiderablyspeedsupthecalculationsinceEq. 3 doesnotrequirethecomputationoftheinverseoftheinterferencecovariancematrixQkforeachfrequencygridpoint.NotethatRdependsonx,hencesolvingEq. 3 andEq. 3 isanon-trivialtask.IAAhandlesthisinaniterativemanner.ThealgorithmstartswithaninitialsolutionwhichisoftentakenasthePeriodogramEq. 3 .Thefollowingstepsarethentaken: 65

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1. ThecovariancematrixRiscalculatedusingEq. 3 2. xkiscalculatedusingEq. 3 fork=0,1...,K)]TJ /F6 11.955 Tf 11.96 0 Td[(1.Steps1)and2)arerepeateduntilconvergence,andthespectralestimateinthepoint!kisgivenbypk=jxkj2.Fromempiricalresultsusually10)]TJ /F6 11.955 Tf 12.24 0 Td[(15iterationsaresufcientforthealgorithmtoconverge[ 12 ].Inthescenariosconsideredhere,thesteeringmatrixAisanoversampledFouriermatrix.However,IAAperformsequallywellforotherapplicationssuchasimagingandchannelestimationincommunicationswherethecolumnsofAconsistsofotherbasisfunctionssuchasdelayedorDopplershiftedversionsofprobing/trainingsignals[ 47 ].ComparingEq. 3 toEq. 3 weseethattheonlydifferencebetweentheperiodogramandIAAistheweightingmatrixQk.AssumingthattheinterferencecovarianceQkisknown,Eq. 3 givesabetterestimateofxkthanEq. 3 underquitegeneralconditions[ 46 ].IAAutilizesthisinordertoachievebetterresolutionthantheperiodogram.ThematricesQkandRaretypicallywell-conditionedifthesignalcontainsnoise.1ThisisalsothereasonthatIAAcanworkwithaslittleasasinglesnapshot,whichisbenecialcomparedtoexistinghighresolutiontechniquesthattypicallyrequiremanysnapshotstoestimatethecovariancematrix. 3.3ComputationalComplexitiesandFastCalculationofIAAIAAprovideshighresolutionestimateswithlowsidelobes,butitisalsorathercomputationallydemanding.IneachiterationitrequiresevaluationofthenumeratoranddenominatoroftheexpressionEq. 3 ,denotedby N(!)=a(!)R)]TJ /F5 7.97 Tf 6.58 0 Td[(1y, (3) D(!)=a(!)R)]TJ /F5 7.97 Tf 6.58 0 Td[(1a(!), (3) 1ForhighSNRscenarioswithsparsesignalspectrumthematricesQkandRmaybeillconditioned.ThismaybehandledbyregularizingIAA[ 47 ]. 66

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ateachofthepoints!k,k=0,1,...,K)]TJ /F6 11.955 Tf 13.5 0 Td[(1.Usingabruteforceapproach,ineachiterationthematrixRwouldbeinvertedandthenR)]TJ /F5 7.97 Tf 6.59 0 Td[(1a(!k)computedfork=0,1,...,K)]TJ /F6 11.955 Tf 12.83 0 Td[(1.ThistakesO(N2K)whichistoocomputationallydemandinginmanyapplications[ 48 ].InsituationswhenNandKarelarge,memoryrequirementsmaypreventusfromevencreatingthematricesAandR.However,sincethematriceshaveknownstructure(Toeplitz/Vandermonde),thematricescanberepresentedbyvectorsandoperationsperformedonthosevectorsinsteadofthefullmatrix.Nextwewillreviewthealgorithmproposedin[ 36 ],wherethesestructuresareusedtocalculateIAAseveralordersofmagnitudesquickerbyutilizingFFToperations.InthefollowingdiscussionwefrequentlyusefastcalculationswithToeplitzmatrices.ThesearebrieyreviewedinAppendixB. 3.3.1CalculateR)]TJ /F5 7.97 Tf 6.59 0 Td[(1TherststepinthealgorithmistocalculatethematrixR.BynotingthatPisdiagonalandAisaVandermondematrix,R=APAisaHermiteanToeplitzmatrixanditissufcienttocalculatetherstcolumnofR.TherstcolumnofPAisp=[p1,...,pK]T,andsinceAisthepartialFouriermatrixEq. 3 ,therstcolumnofRisobtainedbytakingtherstNelementsintheinverseFouriertransformofp,i.e.,R1:N,1=K(F)]TJ /F5 7.97 Tf 6.58 0 Td[(1(p)K)1:N.NotethatRisuniquelydenedbyitsrstrowsinceitisToeplitzandHermitean.ThenextstepinIAAistoobtaintheGohberg-Semencul(GS)factorization,R)]TJ /F5 7.97 Tf 6.59 0 Td[(1=L(u,D)L(u,D))-222(L(~u,D)L(~u,D),whichallowsforexpressingR)]TJ /F5 7.97 Tf 6.59 0 Td[(1intermsofthelowertriangularToeplitzmatricesL(u,D)=(u,Du,D2u,...DN)]TJ /F5 7.97 Tf 6.59 0 Td[(1u)andL(~u,D)=(~u,D~u,D2~u,...DN)]TJ /F5 7.97 Tf 6.58 0 Td[(1~u).HereD=0B@01(N)]TJ /F5 7.97 Tf 6.59 0 Td[(1)0IN)]TJ /F5 7.97 Tf 6.59 0 Td[(10(N)]TJ /F5 7.97 Tf 6.58 0 Td[(1)11CA 67

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denotetheshiftmatrixandthevectorsuand~uarecalculatedusingtheLevinson-Durbin(LD)algorithminO(N2)(seeAppendixBand[ 9 ]fordetails).UsingtheGohberg-Semenculfactorization,wecansolvetheequationRx=yforanyyinO(NlogN).Next,thenumeratoranddenominatoroftheIAAestimateareexamined. 3.3.2CalculateN(!)ExaminingthenumeratorEq. 3 ,therstproblemistheevaluationofR)]TJ /F5 7.97 Tf 6.59 0 Td[(1y.ThisisdoneusingtheGSfactorizationandefcientToeplitzmatrix-vectormultiplicationinO(NlogN)time(seeAppendixB).Thentheestimate,N(!k),canbecomputedforallkwithanFFToperationonR)]TJ /F5 7.97 Tf 6.58 0 Td[(1y, 1)]TJ /F5 7.97 Tf 6.58 0 Td[(N=F(R)]TJ /F5 7.97 Tf 6.59 0 Td[(1y)K,(3)where1)]TJ /F5 7.97 Tf 6.58 0 Td[(N=(N(!0),N(!1),...,N(!K)]TJ /F5 7.97 Tf 6.59 0 Td[(1))T.HencethenumeratorcanbecalculatedinO(KlogK)time. 3.3.3CalculateD(!)ThedenominatormayalsobecalculatedusingFFT/IFFTbyconsideringthetrigonometricpolynomial(z)=(z)]TJ /F5 7.97 Tf 6.59 0 Td[(1)R)]TJ /F5 7.97 Tf 6.59 0 Td[(1(z)=N)]TJ /F5 7.97 Tf 6.59 0 Td[(1X`=)]TJ /F8 7.97 Tf 6.59 0 Td[(N+1c`z`,forwhere(z)=(1,z,z2,...,zN)]TJ /F5 7.97 Tf 6.58 0 Td[(1)T,andc`isthecoefcientforz`.Sincez)]TJ /F5 7.97 Tf 6.59 0 Td[(1=zwheneverjzj=1,itfollowsthat(z)isrealvaluedattheunitcircleandc)]TJ /F9 7.97 Tf 6.58 0 Td[(`=c`.EvaluationofthedenominatorD(!)correspondstoevaluationof(z)attheunitcircleD(!)=(z)jz=ei!=N)]TJ /F5 7.97 Tf 6.59 0 Td[(1X`=)]TJ /F8 7.97 Tf 6.59 0 Td[(N+1c`ej`!.Sincethegridf!kgK)]TJ /F5 7.97 Tf 6.59 0 Td[(1k=0isuniform,D(!k)maybecomputedusingFFTfromthecoefcientsfcngN)]TJ /F5 7.97 Tf 6.59 0 Td[(1n=)]TJ /F8 7.97 Tf 6.59 0 Td[(N+1.Thismaybeexpressedas 1)]TJ /F5 7.97 Tf 6.59 0 Td[(D=KF)]TJ /F5 7.97 Tf 6.59 0 Td[(1(c)(3) 68

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wherec=(c0,c1,...,cN)]TJ /F5 7.97 Tf 6.58 0 Td[(1,0TK)]TJ /F5 7.97 Tf 6.58 0 Td[(2N+1,c)]TJ /F8 7.97 Tf 6.58 0 Td[(N+1,...,c)]TJ /F5 7.97 Tf 6.58 0 Td[(1)Tand1)]TJ /F5 7.97 Tf 6.59 0 Td[(D=(D(!0),D(!1),...,D(!K)]TJ /F5 7.97 Tf 6.58 0 Td[(1))T.UsingtheGSfactorizationofR)]TJ /F5 7.97 Tf 6.59 0 Td[(1andnotingthattheelementsofccorrespondtosummingupthediagonalsofR)]TJ /F5 7.97 Tf 6.58 0 Td[(1,thevectorcmaybeobtainedfromuas0BBBBBBB@c)]TJ /F8 7.97 Tf 6.59 0 Td[(N+1...c)]TJ /F5 7.97 Tf 6.59 0 Td[(1c01CCCCCCCA=0BBBBBBB@uN)]TJ /F5 7.97 Tf 6.59 0 Td[(1002uN)]TJ /F5 7.97 Tf 6.58 0 Td[(2uN)]TJ /F5 7.97 Tf 6.59 0 Td[(1...............0Nu02uN)]TJ /F5 7.97 Tf 6.58 0 Td[(2uN)]TJ /F5 7.97 Tf 6.58 0 Td[(11CCCCCCCA0BBBBBBB@u0u1...uN)]TJ /F5 7.97 Tf 6.59 0 Td[(11CCCCCCCA)]TJ /F17 11.955 Tf 11.96 49.13 Td[(0BBBBBBB@u10.........(N)]TJ /F6 11.955 Tf 11.96 0 Td[(1)uN)]TJ /F5 7.97 Tf 6.59 0 Td[(1.........0(N)]TJ /F6 11.955 Tf 11.95 0 Td[(1)uN)]TJ /F5 7.97 Tf 6.59 0 Td[(1u11CCCCCCCA0BBBBBBB@0uN)]TJ /F5 7.97 Tf 6.59 0 Td[(1...u11CCCCCCCA,whichisasetofToeplitzmatrix-vectorproducts,andhencemaybecomputedinO(NlogN)usingFFT. Table3-1. FastIAA Step0-Initialize:Calculatex(0)=F(y)KStep1:CalculateRfrom(F)]TJ /F5 7.97 Tf 6.58 0 Td[(1(p)K)1:N,wherep(i)k=jx(i)kj2Step2:CalculateGSFactorizationofR)]TJ /F5 7.97 Tf 6.59 0 Td[(1viaLDStep3:CalculateR)]TJ /F5 7.97 Tf 6.58 0 Td[(1yusingToeplitzmatrix-vectorcalculationsStep4:Calculate1)]TJ /F5 7.97 Tf 6.58 0 Td[(N=F(R)]TJ /F5 7.97 Tf 6.59 0 Td[(1y)KStep5:Calculate1)]TJ /F5 7.97 Tf 6.58 0 Td[(D=KF)]TJ /F5 7.97 Tf 6.59 0 Td[(1(c)KobtainedfromuStep6:Calculatex=1)]TJ /F5 7.97 Tf 6.59 0 Td[(N=1)]TJ /F5 7.97 Tf 6.58 0 Td[(D(elementwisedivision)Step7:RepeatSteps1)]TJ /F6 11.955 Tf 11.96 0 Td[(6untilapre-speciednumberofiterationsisreachedorapre-speciedthresholdissatised. 69

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3.3.4SummaryThestepsforFastIAAaresummarizedinTable 3-1 .TheoverallcomputationalcostisO(N2+KlogK).ThismethodologyforcomputingfastIAAwasindependentlydevelopedin[ 36 ]and[ 37 ],andtheyarebasedonearlierfastimplementationsofAPES[ 49 ]. 3.4MissingDataIAAandFastCalculationsIn[ 35 ],IAAwasappliedtoproblemswheredatasamplesaremissing.2InthissectionwetreatthespectralestimationpartofmissingdataIAAandprovideanewalgorithmthatisfastwhenthenumberofmissingdatasamplesissmall.Inthenextsectionweutilizetheseresultsforrecoveringthemissingpartofy.Considertheproblemofestimatingxfromavectorofavailabledatayg,whichisasubsetofthefulldatavectory.Theavailableandmissingpartofymayberepresentedasyg=Sgy,ym=Smy,whereSg2RNgNandSm2RNmNaretheselectionmatricescorrespondingtotheavailableandmissingsamples,respectively.HereNgandNmdenotethenumberofavailabledataandmissingdata,andhenceN=Ng+Nm.ThedatamodelEq. 3 isnowreplacedby yg=Sgy=SgAx+Sge(3)whereSgAisthesteeringmatrixandthevectorxissought.SinceIAAisapplicableforthemissingdatacase,eachiterationofMIAAisnowtoevaluate xk=ag(!k)Rg)]TJ /F5 7.97 Tf 6.59 0 Td[(1yg ag(!k)Rg)]TJ /F5 7.97 Tf 6.59 0 Td[(1ag(!k),(3) 2Thiscasemaybeseenasaproblemwithnonuniformdatasampling,henceIAAstillapplies. 70

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where ag(!k)=Sga(!k),Rg=SgAPASgT=SgRSgT, (3) andasbeforeP=diag(pk)wherepk=jxkj2,fork=0,...,K)]TJ /F6 11.955 Tf 12.06 0 Td[(1.DenotethenumeratoranddenominatorofEq. 3 byN(!k)=ag(!k)Rg)]TJ /F5 7.97 Tf 6.59 0 Td[(1yg,D(!k)=ag(!k)Rg)]TJ /F5 7.97 Tf 6.59 0 Td[(1ag(!k).Forthefulldatacase,whereSg=IN,thesepolynomialswouldbeidenticaltoNandDandmaybecalculatedefcientlyasinSection 3.3 .Inthemissingdatacase,however,RgisnotToeplitzandseveralofthestepsintheprevioussectionrequiredforfastcomputationsbreaksdown,includingtheGSfactorization.AbruteforcesolutionwouldincludeamatrixinversionofRgandevaluationsofRg)]TJ /F5 7.97 Tf 6.58 0 Td[(1ag(!k)forallk,whichhavecomputationalcomplexityO(Ng2K).AfastapproachwasproposedreducingthenumberofoperationstoO(Ng3+KlogK)[ 37 ].ThemainburdenhereistheinversionofthematrixRg.IfthenumberofavailablesamplesNgissmall,thenthisisnotaproblem.However,ifthenumberofavailablesamplesNgislarge,thenthisinversionwillbethebottleneck.Nextweconsiderthecasewherethenumberofmissingdatapointsissmallcomparedtothetotalsetofdata.InthiscaseinversionofRgisthebottleneckandcouldevenbeinfeasible.Instead,wewillshowthatalowrankcompletioncanbeusedtotransformtheproblemtothefulldatacase.Usingthis,NandDinMIAAmaybecalculatedbyadjustingNandDfromthefulldatacasebytermswithlowrankstructures. 71

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3.4.1FastCalculationofMissingDataIAAConsiderthecasewherethenumberofmissingsamplesNmissmallcomparedtoallsamplesN.TheproblemishowtoutilizethestructureofRgandSgforevaluatingthetrigonometricpolynomialsN(!)andD(!).Thecalculationsrelyonthefollowingkeyproposition,whichallowsustoexpressthematrixproductSgTRg)]TJ /F5 7.97 Tf 6.59 0 Td[(1SgasasumofalowrankmatrixandtheinverseofaToeplitzmatrix. Proposition1. LetR>0andRgbedenedbyEq. 3 whereAisthesteeringmatrixdenedinEq. 3 .ThenSgTRg)]TJ /F5 7.97 Tf 6.59 0 Td[(1Sg=R)]TJ /F5 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F10 11.955 Tf -283.19 -30.39 Td[(where)]TJ /F10 11.955 Tf 10.26 0 Td[(isgivenby)]TJ /F6 11.955 Tf 10.26 0 Td[(:=R)]TJ /F5 7.97 Tf 6.58 0 Td[(1SmT(SmR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SmT))]TJ /F5 7.97 Tf 6.59 0 Td[(1SmR)]TJ /F5 7.97 Tf 6.59 0 Td[(1. Proof. LetSdenotethepermutationmatrixpartitioningallsamplesintotheavailableandthemissingsamples S=0B@SgSm1CA(3)andnotethattheselectionmatricessatisfy STS=SgTSg+SmTSm=IN, (3) SgST=(SgSgT,SgSmT)=(INg,0NgNm). (3) FromEq. 3 ,Eq. 3 andthenbyinsertingEq. 3 itfollowsthat(INg,0NgNm)=(SgRST)(SR)]TJ /F5 7.97 Tf 6.59 0 Td[(1ST) (3)=(Rg,Rgm)0B@SgR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SgTSgR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SmTSmR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SgTSmR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SmT1CA, 72

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whereRgm:=SgRSmT.ByrstusingtherstblockandthenthesecondblockofEq. 3 ,weget3Rg)]TJ /F5 7.97 Tf 6.59 0 Td[(1=SgR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SgT+Rg)]TJ /F5 7.97 Tf 6.58 0 Td[(1RgmSmR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SgT=SgR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SgT)]TJ /F3 11.955 Tf 11.95 0 Td[(SgR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SmT(SmR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SmT))]TJ /F5 7.97 Tf 6.58 0 Td[(1SmR)]TJ /F5 7.97 Tf 6.58 0 Td[(1SgT=Sg(R)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F6 11.955 Tf 6.94 0 Td[()SgT.FinallynotethatSmT=R)]TJ /F5 7.97 Tf 6.58 0 Td[(1SmT(SmR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SmT))]TJ /F5 7.97 Tf 6.59 0 Td[(1SmR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SmT=R)]TJ /F5 7.97 Tf 6.58 0 Td[(1SmT,andhenceSgTRg)]TJ /F5 7.97 Tf 6.59 0 Td[(1Sg=SgTSg(R)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F6 11.955 Tf 6.94 0 Td[()SgTSg=(IN)]TJ /F3 11.955 Tf 11.95 0 Td[(SmTSm)(R)]TJ /F5 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F6 11.955 Tf 6.94 0 Td[()(IN)]TJ /F3 11.955 Tf 11.96 0 Td[(SmTSm)=R)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F6 11.955 Tf 6.94 0 Td[(. (3) Denotebyyf=SgTyg,thevectorywiththemissingsampleszeroedout.UsingProposition 1 ,weseethatNandDmaybewrittenasN(!)=ag(!)Rg)]TJ /F5 7.97 Tf 6.59 0 Td[(1yg=a(!)SgTRg)]TJ /F5 7.97 Tf 6.59 0 Td[(1Sgyf=a(!)(R)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F6 11.955 Tf 6.94 0 Td[()yf=N(!))]TJ /F3 11.955 Tf 11.95 0 Td[(a(!)yf (3) 3Alternatively,use(0.12)in[ 50 ]withSR)]TJ /F5 7.97 Tf 6.59 0 Td[(1STintheplaceofG,inwhichcaseRg)]TJ /F5 7.97 Tf 6.58 0 Td[(1correspondstoAonpage4in[ 50 ]. 73

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andD(!)=ag(!)Rg)]TJ /F5 7.97 Tf 6.58 0 Td[(1ag(!)=a(!)SgTRg)]TJ /F5 7.97 Tf 6.59 0 Td[(1Sga(!)=a(!)(R)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F6 11.955 Tf 6.94 0 Td[()a(!)=D(!))]TJ /F3 11.955 Tf 11.95 0 Td[(a(!)a(!). (3)HereNandDmaybeevaluatedefcientlyasinSection 3.3 whereyfreplacesy,usingtheToeplitzstructureofR.Therankof)]TJ /F1 11.955 Tf 10.27 0 Td[(isequaltoNm,afactwhichmaybeusedforcalculatingtheremainingpartsofNandD.NextwewillutilizethisstructureinordertoevaluateEq. 3 andEq. 3 efciently. 3.4.1.1GetR,theGSfactorizationofR)]TJ /F5 7.97 Tf 6.58 0 Td[(1,andevaluateN(!)andD(!)ThisisdoneexactlyasinSection 3.3 whereyfisusedinsteadofy. 3.4.1.2GetL=(SmR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SmT))]TJ /F5 7.97 Tf 6.58 0 Td[(1=2andX=)]TJ /F5 7.97 Tf 19.4 4.34 Td[(1=2AsarststeptogetLwecalculatethematrixR)]TJ /F5 7.97 Tf 6.58 0 Td[(1.LetR)]TJ /F5 7.97 Tf 6.59 0 Td[(1=:[rinv1,...,rinvN]andrecursivelycalculaterinvkusingthedisplacementstructurer(R)]TJ /F5 7.97 Tf 6.59 0 Td[(1)(seeAppendixB,c.f.,(18)-(20)in[ 51 ]) rinvk=8><>:u u(1))]TJ /F3 11.955 Tf 12.02 0 Td[(~u ~u(1)k=1Drinvk)]TJ /F5 7.97 Tf 6.59 0 Td[(1+u u(k))]TJ /F3 11.955 Tf 12.02 0 Td[(~u ~u(k)k>1,(3)inO(N2).ThengetSmR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SmTbyselectingtherowsandcolumnscorrespondingtothemissingdata.LetL2CNmNmbetheinverseoftheCholeskyfactorofSmR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SmT(calculatedinO(Nm3)),i.e.,LL=(SmR)]TJ /F5 7.97 Tf 6.58 0 Td[(1SmT))]TJ /F5 7.97 Tf 6.59 0 Td[(1.Finally,getX2CNNmsatisfyingXX=)]TJ /F6 11.955 Tf 6.95 0 Td[(,bymultiplicationX=R)]TJ /F5 7.97 Tf 6.58 0 Td[(1(SmTL)using4theGSfactorizationofR)]TJ /F5 7.97 Tf 6.58 0 Td[(1,inO(NmNlogN). 4Forsmalldatasizes,multiplication(R)]TJ /F5 7.97 Tf 6.58 0 Td[(1SmT)LisfasterO(N2mN). 74

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3.4.1.3Evaluatea(!))]TJ /F3 11.955 Tf 6.77 0 Td[(yfTheremainingpartofthenumeratorN,i.e.,a(!)yf,maybecalculatedbynotingthat a(!)yf=a(!)XLSmR)]TJ /F5 7.97 Tf 6.59 0 Td[(1yf.(3)InthisexpressionrsttheR)]TJ /F5 7.97 Tf 6.59 0 Td[(1yfmultiplicationiscarriedoutusingtheGS-factorization,Sm(R)]TJ /F5 7.97 Tf 6.59 0 Td[(1yf)byselectingtherowscorrespondingtothemissingdata,andthenXL(SmR)]TJ /F5 7.97 Tf 6.58 0 Td[(1yf)bystandardmatrix-vectorcalculations(inO(NmN+NlogN)).FinallyEq. 3 isevaluatedat!kfork=0,1,...,K)]TJ /F6 11.955 Tf 11.95 0 Td[(1inO(KlogK)bynotingthat 0BBBBBBB@a(!0)yfa(!1)yf...a(!K)]TJ /F5 7.97 Tf 6.59 0 Td[(1)yf1CCCCCCCA=F(XLSmR)]TJ /F5 7.97 Tf 6.58 0 Td[(1yf)K.(3) 3.4.1.4Evaluatea(!))]TJ /F3 11.955 Tf 6.77 0 Td[(a(!)Theremainingpartofthenumerator,i.e.,a(!)a(!)maybecalculatedbynotingthata(!)a(!)=a(!)XXa(!)=N)]TJ /F5 7.97 Tf 6.59 0 Td[(1X`=)]TJ /F8 7.97 Tf 6.58 0 Td[(N+1d`ej`!isatrigonometricpolynomial.Herethecoefcientd`isthe`thdiagonalofXX(d`=d)]TJ /F9 7.97 Tf 6.58 0 Td[(`),andmaybecalculatedinO(NmNlogN)usingToeplitzmatrices(c.f.,AppendixB[ 52 ])0BBBB@dN)]TJ /F5 7.97 Tf 6.58 0 Td[(1...d01CCCCA=NmX`=10BBBB@X(1,`)......X(N,`)X(1,`)1CCCCA0BBBB@ X(N,`)... X(1,`)1CCCCA.Sincethegridf!kgK)]TJ /F5 7.97 Tf 6.58 0 Td[(1k=0isuniform,a(!k)a(!k)maybecomputedusingFFTfromthecoefcientsfdngN)]TJ /F5 7.97 Tf 6.59 0 Td[(1n=)]TJ /F8 7.97 Tf 6.59 0 Td[(N+1.Thismaybeexpressedas )]TJ /F6 11.955 Tf 8.64 2.82 Td[(=KF)]TJ /F5 7.97 Tf 6.59 0 Td[(1(d)(3) 75

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whered=[d0,d1,...,dN)]TJ /F5 7.97 Tf 6.59 0 Td[(1,0TK)]TJ /F5 7.97 Tf 6.59 0 Td[(2N+1,d)]TJ /F8 7.97 Tf 6.59 0 Td[(N+1,...,d)]TJ /F5 7.97 Tf 6.58 0 Td[(1]Tand)]TJ /F6 11.955 Tf 8.64 2.83 Td[(=[a(!0)a(!0),...,a(!K)]TJ /F5 7.97 Tf 6.58 0 Td[(1)a(!K)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]T,whichisevaluatedinO(KlogK). 3.4.1.5GetN(!)andD(!)GettheevaluationsofN(!)andD(!)fromEq. 3 andEq. 3 usingEq. 3 andEq. 3 .NotethatN(!)isindependentofthemissingsamplesymsince(R)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F6 11.955 Tf 6.94 0 Td[()SmT=0. 3.4.1.6Summary Table3-2. FastMIAA Step0-Initialize:Calculatex(0)=F(yf)KStep1:CalculateRfrom(F)]TJ /F5 7.97 Tf 6.58 0 Td[(1(p)K)1:N,wherep(i)k=jx(i)kj2Step2:CalculateGSFactorizationofR)]TJ /F5 7.97 Tf 6.59 0 Td[(1viaLDStep3:CalculateR)]TJ /F5 7.97 Tf 6.58 0 Td[(1yfusingToeplitzmatrix-vectorcalculationsStep4:Calculate1)]TJ /F5 7.97 Tf 6.58 0 Td[(N=F(R)]TJ /F5 7.97 Tf 6.59 0 Td[(1yf)KStep5:Calculate1)]TJ /F5 7.97 Tf 6.58 0 Td[(D=KF)]TJ /F5 7.97 Tf 6.59 0 Td[(1(c)KobtainedfromuStep6:CalculateL=(SmR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SmT))]TJ /F5 7.97 Tf 6.59 0 Td[(1=2andX=)]TJ /F5 7.97 Tf 6.94 5.15 Td[(1=2Step7:Calculatea(!)yffromEq. 3 Step8:Calculatea(!)a(!)fromEq. 3 Step9:Calculatexk=N(!k)=D(!k)Step10:RepeatSteps1)]TJ /F6 11.955 Tf 11.96 0 Td[(9untilapre-speciednumberofiterationsisreachedorapre-speciedthresholdissatised. ThestepsforfastMIAAareshowninTable 3-2 .EachiterationiscalculatedinO(Nm3+NmNlogN+N2+Klog(K)).Theproposedalgorithmcanhandlethecaseswherethenumberofmissingdataissmall(Nm
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3.5CalculatingtheMissingData,MIAA-tThespectralestimatecanbeutilizedforestimatingthemissingsamples.Thiswasstudiedin[ 35 ],wherethemissingdatavectoroftheform^ym=Tygissoughtwhichminimizesthemeansquarederrorof^ym)]TJ /F3 11.955 Tf 12.71 0 Td[(ym.ThisisreferredtoasMIAA-t,andtheminimizingTis(see[ 35 ])T=SmRSg(SgRSg))]TJ /F5 7.97 Tf 6.59 0 Td[(1.Alsoherethecomputationalburdenofcalculating(SgRSg))]TJ /F5 7.97 Tf 6.58 0 Td[(1ygwouldbeconsiderablewhenNgislarge.However,usingProposition 1 ,weseethat^ymcanbewrittenas ^ym=SmRSg(SgRSg))]TJ /F5 7.97 Tf 6.59 0 Td[(1yg=Sm(IN)]TJ /F3 11.955 Tf 11.95 0 Td[(R)]TJ /F6 11.955 Tf 15.35 0 Td[()yf. (3) Thereforetheestimateddatavector^ycanbewrittenas^y=SgTyg+SmT^ym=(IN)]TJ /F3 11.955 Tf 11.96 0 Td[(SmTSm)yf+SmTSm(IN)]TJ /F3 11.955 Tf 11.96 0 Td[(R)]TJ /F6 11.955 Tf 15.34 0 Td[()yf=yf)]TJ /F3 11.955 Tf 11.95 0 Td[(SmTSmRyf=yf)]TJ /F3 11.955 Tf 11.95 0 Td[(SmTLLSmR)]TJ /F5 7.97 Tf 6.59 0 Td[(1yf,whichmaybeevaluatedinO(Nm2+NlogN).NotethatthematrixIN)]TJ /F3 11.955 Tf 11.61 0 Td[(R)]TJ /F1 11.955 Tf 18.67 0 Td[(isaprojectionalongspan(Sm)ontospan(RSgT).Thatis,ittakesanyvaluesfrommissingdatato0,andanyvectorinspan(RSgT)isunaltered.Then^yminEq. 3 (andconsequently^y)onlydependontheavailabledatayg,andhenceyfcouldbeanyvectorwhoseavailabledatacoincideswithyg,i.e.,yg=Sgyf. 77

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Figure3-1. Thespeedup(measuredbycomparingthenumberofops)forthemissingdataIAAalgorithm. 3.6Applications 3.6.1ComputationalComplexityAdetailedstudyoftheproposedalgorithmshowsthattheleadingtermsofthecomputationalcostare(seeAppendixA) 2 3Nm3+8NmNlog2N+3 2N2+3 2Klog(K).(3)Thisshouldbecomparedto(2=3)Ng3+(3=2)Klog(K)in[ 37 ].Theasymptoticimprovementinthecomputationalcomplexity,asthenumberofsamplesNgotoinnitywhiletheproportionNm=Nisxed,isgivenbyN Nm)]TJ /F6 11.955 Tf 11.96 0 Td[(13ThespeedupmeasuredforthemissingdataIAAalgorithmcomparedtoinvertingRg,forN=(500,1000,2000,4000,8000,1).Theasymptoticspeedup(N=1)isdepictedwithboldline.isdepictedinFigure 3-1 .HereK=8Nisused.NotehoweverthatthechoiceofKisirrelevantfortheplotsincethelasttermofEq. 3 isnegligibleforrelevantcases(K<15N). 78

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3.6.21DSinusoidIdenticationConsideranexampleofidenticationof1)]TJ /F1 11.955 Tf 9.3 0 Td[(Dsinusoidsinnoiseandrecoveryofthemissingdata.Letthesignalynbe yn=6X`=12sin(n^!`+v`)+wn,n=0,1,....N)]TJ /F6 11.955 Tf 11.96 0 Td[(1,(3)wherewnisGaussianwhitenoisewithvariance1,v`isarandomvariablewithuniformdistributionon[0,2],and^!`denotethefrequencies(0.8,1.2,1.4,1.5,1.55,1.575)oftherealsinusoids.Intherstexample,letthenumberofsamplesbeN=200andthenumberoffrequencypointsbeK=8N=1600.Furthermore,let10%(=20)ofthesamplesbemissingintwogapsconsistingofthesamples[101,110]and[121,130].SpectralestimatesarebasedonthemissingdataIAAandtheperiodogram.Figure 3-2 showsthedatasequence(upperplot)andacloseupontheregionwiththemissingsamples(lowerplot).Herewecanseethattherecoveryisquitegoodactuallytheestimateofthemissingdataisclosertotheoriginalsignal(withoutnoise=yn)]TJ /F14 11.955 Tf 12.79 0 Td[(wn)thanthesignalynitself.Figure 3-3 showsthespectralestimates,whereitcanbeseenthatMIAAhasconsiderablylowersidelobesandbetterresolutionthantheperiodogram.SincethemainfocusinthisworkisonthecomputationalcomplexityofMIAA,wereferto[ 48 ]and[ 35 ]foracomprehensivecomparisonofMIAAwithothermethodssuchasMAPES,GAPES[ 53 ],CoSaMP[ 54 ],andSLIM[ 55 ].InsteadwecomparethecomputationalcomplexityoftheproposedimplementationofMIAAwiththeonesproposedin[ 37 ]and[ 48 ].Tocomparethecomputationalcomplexityoftheproposedimplementationwiththeonesin[ 37 ]and[ 48 ],notethatthecomputationalbottleneckin[ 37 ]and[ 48 ]aretheinversionofthematrixRg.Toavoiddiscussionsregardingspecicsintheirimplementation,wewillsimplycomparetheproposedalgorithmwiththeinversionofthematrixRg.ConsiderthesignalydenedbyEq. 3 forthecasesN2f2000,4000,8000gandK=8N.Foreachofthosecaseswecomparetheaverage 79

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timetoperformaMIAAiterationover10totaliterationsoftheproposedalgorithmwiththeaveragetimetoinvertRgineachiteration.Thisisdoneforthemissingdataratiogoingfrom5%to50%.TheresultsareshowninFigure 3-4 .ThetimesfortheproposedalgorithmareconsiderablyshorterthanthetimerequiredtoperformthematrixinversionofRgwhentheproportionofmissingdataislow.Atmissingdataratioof10%,theproposedalgorithmis9,15,and30timesfaster5thanthematrixinversionofRg,fortherespectivedatasizes2000,4000,and8000.Theproposedalgorithmisaboutasfastasthematrixinversionwhentheproportionofmissingdataisaround40%)]TJ /F6 11.955 Tf 12.43 0 Td[(47%,dependingonN.Thisisconsistentwiththeclaimthattheproposedalgorithmisasymptoticallyfasterthanthestate-of-the-art[ 37 ]and[ 48 ]whentheproportionofmissingdataislessthan50%. 3.6.3Application:Sparse2DSARImagingSARimagingcomesdowntoa2Dspectralestimationprobleminthefasttimeandslowtimedomainsafterallofthepreliminarysteps[ 1 ].IfaSARplatformisworkinginacongestedspectrumsuchastheUHForVHFbands,radiofrequencyinterference(RFI)comesfromrelativelynarrowbandsourcessuchasFMradio(commercialandamateur)andtelevisionbroadcasts[ 56 ].WhenRFIinterferenceissuppressedbyfrequencynotching,missingdataisintroducedintothefasttimephasehistorydataresultinginincreasedsidelobeenergy[ 57 ].WeusetheGOTCHAvolumetricSARdatainthisexamplefromtheU.S.AirforceSensorDataManagamentSystem(SDMS)6.TheightpatternwasacircularSAR 5NotethattheserunningtimesbasedonMatlabdifferfromthetheoreticalspeedupfromSubsection 3.6.1 .ThisisbecauserunningtimesinMatlabdependheavilyonprogrammingdetails.Forexample,manyintrinsicMatlabfunctionsareveryefcientandarecapableoftakingadvantageofmulti-coreprocessors.MoreinvolvedalgorithmsimplementedusingMatlabscripts,ontheotherhand,arenotalwayscapableofmakinguseofallthecoresavailable.6Thisdataispubliclyavailablebyrequest. 80

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A BFigure3-2. MissingdataforatimeseriesA)Thefulltimeseries.B)Anenlargementonthemissingdatasectionwherethepunctuatedlineshowstheestimateddata. A BFigure3-3. MissingdataspectralestimatesA)ThemissingdataIAAspectralestimate.B)Theperiodogrambasedontheavailabledata. 81

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A B CFigure3-4. AveragetimeperiterationforrunningthemissingdataIAAcomparedtoinvertingRgforvariousdataandfrequencygridsizes. imagingpattern,andthesensordataavailablehadabandwidthofapproximately600MHz.7Wesimulatefrequencynotching,topreventinterferencefromnarrowbandsources,byremovingthree20MHzbands(approximately10%oftotalbandwidth)fromeachfasttimepulse.Thisresultsinthreeannuliofdatatobemissingfromthepolarannulus(c.f.,[ 58 ]).Thisrepresentsascenariowherethetransmitsignalisachirpandthereferencesignalisanotchedchirp.TheimagesareformedusingstandardSARtechniques.Thedataisreformattedusingpolarreformattingtechniques,andanimageisformed(viaFFTorSLIM-1)usingthedatafromfourdegreesofazimuthdataacrossall360degreeswithnooverlappinglookangles.Theimagesarethenrotatedintoalignmentandeachimageisfused 7TheGOTCHAdataisX-band,butthefrequencynotchedsimulatedscenariocouldequallywellrepresentlowfrequencyinterferenceforUHFandVHFdata. 82

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together.Thefusionisdonebytakingthemaximumofeachpixelfromthesetofalltheimagescreated.Therstsubgureisformedfromthefulldataandisusedasabenchmark.Thesecondimageisformedfromfrequencynotcheddata,i.e.,themissingdataiszeroedout.Thethirdimageisformedfromfrequencynotcheddatawherethemissingdataisestimatedforeachfast-timepulse.InthesimulationswhereMIAAwasusedtorecovermissingdata,itwasappliedtoeveryfasttimepulse(intheslantrange)beforepolarreformatting.8Foreachfourdegreeaperture(90totalapertures),thisresultsin469fasttimepulsesandaMIAAproblemofsizeN=424andK=900.Forthisdatasize,thecomputationaltimeisapproximatelyhalfcomparedtothestateoftheart,resultinginaconsiderablereductionintheimagegenerationtime.IntherstsetofimagesshowninFigure 3-5 theimagesareformedbyapplying2DFFTstothedataontherectangulargrid.ThefulldatasetproducesabaselineimageinFigure 3-5 A.ThemissingdatacaseoccurswhenfrequencynotchingisutilizedandthesidelobesintheimageincreaseasshowninFigure 3-5 B.WhenMIAAisappliedtothedatabeforetheimaging,thereconstructionisnearlyasgoodasthefulldatasetasdemonstratedinFigure 3-5 CcomparedtoFigure 3-5 A.Next,weillustratetheeffectivenessofMIAAwhenappliedasapreprocessingsteptoasparsitypromotingimagingmethod.Inparticular,weuseMIAAtoestimatemissingfrequencysamples,andthesparsitypromotingmethodtoreconstructanunderlyingSARimage.ThesparsitypromotingimagingmethodusedinthisexamplewasSparseLearningviaIterativeMinimizationwith`1norm(SLIM-1)[ 55 ].SLIM-1isamaximumaposteriorimethodthatisbasedoffaBayesianhierarchealmodel.Ithasbeenshown 8Wedonotconsideranyjoint2-Destimationofmissingdata.Ina2-Ddatarecoveryscenario,themissingdatashouldbeestimatedontherectangulargridratherthanthepolargridsincethesinusoidsshoulddecouplebetweenthespatialdomains. 83

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tobeaneffectivemethodforSARimaginginfulldata[ 55 ]andmissingdatascenarios[ 48 ].TheSLIMimagingalgorithmhereisthe2Dextension[ 55 ]wherethemissingdataiszeroedout(notthemissingdataSLIMasshownin[ 48 ]).Figure 3-6 AshowstheoutputfromSLIM-1whenfulldataisavailable.ThespecklenoiseandanysidelobesaresignicantlysuppressedcomparedtoFigure 3-5 A.TheuseoffrequencynotchingtothebadbandsreducesthesparsemethodseffectivenessatsuppressingsidelobesasshowninFigure 3-6 B.WhenthemissingdatawasestimatedwithMIAAbeforeutilizingtheSLIM-1imagingalgorithm,thisresultsintheimageshowninFigure 3-6 Cwhichisnearlyidenticaltotheoriginalfulldataimage.Thisexamplewasnotdesignedtocomparedifferentmissingdataalgorithms,buttoillustratethat A B CFigure3-5. FFTimagingA)Theimageformedwithfulldata.B)Theimageformedwithnotched(missing)data.C)TheimageformedwhenthenotcheddataisestimatedviaMIAA. 84

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A B CFigure3-6. SLIM1imagingA)Theimageformedwithfulldata.B)Theimageformedwithnotched(missing)data.C)TheimageformedwhenthenotcheddataisestimatedviaMIAA. reconstructionusingsparsitypromotingmethodscanbeimprovedbyrstestimatingmissingdata,e.g.,byMIAA.TheseempiricalresultsshowthatusingMIAAtoestimatemissingdatainnotchedbandscanbequiteeffective.ThefastimplementationallowsforthismethodtobemorepracticalforrealworldapplicationssuchasSARimaging.InthisapplicationwehaveshownthatfastMIAAmaybeutilizedsuccessfullytoimproveimagequalityinthecasewherefrequencynotchinghasbeenusedtosuppressinterference. 85

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CHAPTER4SPECTRALLYCONSTRAINEDSARANDMIMOSARGMTIWAVEFORMDESIGN 4.1MotivationForSpectralConstraintsinSARSyntheticapertureradar(SAR)isanimagingtechniquethatgeneratestwo-dimensionalorthree-dimensionalimagesofanilluminatedscene[ 1 59 60 ].Thetraditionalapplicationinairandspaceborneplatformsistocreatetwo-dimensionalimagesoftheEarthforscienticandmilitarypurposes.Theresolutionoftheimagesinthecrossrangeaxisisinverselyproportionaltothelengthofthesyntheticaperturewhiletheresolutionoftheimageinthedownrangeaxisisinverselyproportionaltothebandwidthoftheprobingwaveform[ 1 ].ThedesireforhighresolutionimagesmotivatesthedesireforSARsystemswithlargebandwidth.ThisisproblematicbecausethereisrarelylargeenoughbandwidthfreelyavailabletoaccommodatetheSARsystem'srequirements.ThesystemcannotradiateenergyintobandsoccupiedbylicensedusersduetothepossiblecatastrophicinterferencetheSARmaycause.Thishasmotivatedresearchintospectrallyconstrained(SC)SARsystemssuchasGeoSAR[ 61 ].TheproblemofhowtoappropriatelydesignaspectrallyconstrainedSARsystemisstillapressingproblem[ 16 ].Ingeneralthisproblemcanbeovercomebyusingaspectrallyconstrainedwaveform.Therearetwonon-idealwaveformsthatarecommonlyconsideredforthistask.Therstistousetheso-calledfrequencyjumpburstwaveforms.Theamplitudeisonewhenthewaveform'sinstantaneousfrequencyisinthepassbandandzerowhentheinstantaneousfrequencyisinthestopband[ 58 ].Oftenthewaveformisactuallybrokendownintosub-pulses,buttheconceptisthesame.Therearetwomainreasonswhythisisnotideal:First,thesignal'sspectrummaynothaveadequatenotchdepthinthestopbands.Second,thesignal'speaktoaveragepowerratioisverypoorwhichisundesirablewhenutilizinghighpowerampliersthatoperateinthesaturationregion. 86

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Thesecondcommonapproachistoplacenarrownotchesintothespectrumofalinearfrequencymodulated(LFM)waveform[ 58 ].Thisapproachislimitedinitsabilitytoplacewidenotchesinthespectrumoftheprobingsignalwhichmaynotbeadequateinalloperatingenvironments.Inthischapterweinvestigatethedesignofspectrallyconstrainedwaveforms,usingtherecentlyproposedSHAPEalgorithm[ 62 ](seealsoChapter2),forapplicationinspectrallyconstrainedSAR.TheSHAPEalgorithmisatoolfordesigningsequences/waveformswitharbitraryspectraanddesiredtimedomainenvelopes.SHAPEfocusesonlyonthedesignofthespectrum,buttheautocorrelationoftheprobingsequencegeneratedbySHAPEisoftennon-ideal.Toovercomethisproblem,weutilizeKronecker(orembedded)sequences[ 63 ].Inthiswaywecandesignsequencesthathaveaperiodiclowcorrelationzones,canmeetspectrumconstraints,andcanbeusedtoformgoodSARimagery.Thetrade-offisthatthesesequencesarenotofLFMtypewhichimpliesthatthereceiverdesignismoredifcult.ThisisbecausemanyofthecomputationaltrickscommonlyusedinSAR(e.g.,dechirpingonreceivewhichallowsforadecreasedsamplingrate)requiretheuseofLFMwaveforms[ 1 ].WealsoexpandbeyondtraditionalSARimagingbyexploringthepossibilityofspectrallyconstrainedmultiple-inputmultiple-output(MIMO)SARgroundmovingtargetindication(GMTI).IthasbeenshownthatbyutilizingaMIMOcongurationinaSARplatformmovingtargetscanbedetectedwithoutusingdisplacedphasecenterantenna(DPCA)oralong-trackinterferometry(ATI)[ 64 65 ].Themethodproposedin[ 65 ]utilizesthefactthatbymanipulatingthereceivedsignalinthefrequencydomainthestationaryclutterisprojectedintoasubspacewhichcanbesuppressed.ThismethodisalsoabletosimultaneouslygenerateSARimagesofthesceneaswell.WewillextendtheconceptofusingKroneckersequencesintothedesignofwaveformsetsfortheMIMOSARGMTIparadigm. 87

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ThedesignofMIMOSARwaveformsisdifcultduetotheneedforverylowcross-correlations.Thepreviouslyproposedmethodsincludechirpsets(up,down,andtrianglemodulation),Doppler-divisionmultipleaccess(DDMA)approach,andshorttimefrequencydiversetypewaveformsets[ 14 66 68 ].However,noneoftheproposedmethodsintheliteratureconsidersthedesignofspectrallyconstrainedwaveformsets.InthisworkwewillextendtheconceptoftheKroneckerwaveformsetstoshowthatwecandevelopsetsthathavelowautoandcrosscorrelationzoneswhilemaintainingspectralconstraints.WhilethedesignedwaveformsetisabletoachievegoodGMTIperformance,theSARimagingisslightlydegradedintheslantrangeduetowideningofthemainresponseoftheautocorrelationpeak.WewillexaminetheperformanceofthespectrallyconstrainedwaveformsusingSARandGMTIsimulations.Wewillmakeacomparisontotheperformanceofwaveformsthatdonothavespectralconstraintstogivearelativemeasureofperformancedegradationduetothespectralconstraints.Weshowempiricallythatthespectrallyconstrainedwaveformsareabletoperformadequatelywhenusingdataindependentreceiverprocessingtechniques(matchedlter).Furtherinvestigationintousingdataadaptivelteringismeritedespeciallyinthefasttime(rangecompression).However,thecomputationaltimerequiredforfasttimeprocessingisimportantanditlimitstheusageofsomeoftheslowerapproaches[ 13 47 ].WebeginbydiscussingtheKroneckerwaveformdesignforspectrallyconstrainedwaveformsets.WefollowthatupwithareviewoftechniquesthatcanbeusedtogeneratesequencesusedintheKroneckerwaveform.WethenexaminetheperformanceofthewaveformswithcomputationalexamplesusingsimulatedSARdata(wherethestationarygroundsceneisgeneratedfromrealSARdata).Finally,wepresentourconclusionsonthetopics.Forclaritywewillbrieyreviewthenotationthatwillbeusedinthischapter.Aboldfacelowercaseletterxrepresentsacolumnvectorwhileaboldfaceuppercaseletter 88

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Xrepresentsamatrix.ThenotationRx,y()willbeusedtodenotethediscretetimecorrelationbetweentwovectors.Theoperationjjjj2representstheEuclideannormand()Tand()Hrepresentthetransposeandconjugatetransposeofavector.TheoperatorFrepresentsadiscreteFouriertransform(DFT)operationonavectorandanyzeropaddingisimplicitlyappliedwhennecessary. 4.2KroneckerWaveformsAKroneckerwaveformisacompositeofmultiplewaveforms.Ingeneralmultiplewaveformscanbeembeddedintooneoutputwaveform,butwewillfocusonthecaseoftwowaveforms.AKroneckerwaveformgeneratedbytwovectors(discretetimewaveformsorsequences)xandyisdenedas: x=x1x2...xNT,y=y1y2...yMT,z=x1yTx2yT...xNyTT,z=xy, (4) whereisdenedastheKroneckerproduct.SincetheKroneckerproductisnotacommutativeoperationwechosetorefertothevectorinthesamepositionasxinEq. 4 astheouterwaveformandthevectorinthesamepositionasyinEq. 4 astheinnerwaveform.Thedesignofsequences/waveformsusingthistypeofapproachisnotanewidea.TheconceptofusingembeddedBarkercodeswasstudiedinthelate1960'sforphasecodedradarapplications[ 69 ].Theideawasalsoappliedtoexamineamplitudeandphasemodulatedwaveformsforradarin[ 70 ].Kroneckersequenceshavealsobeenstudiedforuseincodedivisionmultipleaccess(CDMA)systems[ 63 ].In[ 63 ]theaperiodicandperiodiccorrelationofKroneckersequenceswasstudiedindetail.We 89

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beginourdiscussionaboutthedesignofspectrallyconstrainedKroneckerwaveformsetsbyreviewingsomeofthefundamentalpropertiesofthesewaveforms. 4.2.1CorrelationPerformanceWebeginwiththeexplicitdenitionoftheaperiodiccross-correlationbetweenanytwosubsequencesuandxoflengthLas: Ru,x(m)=8><>:PL)]TJ /F5 7.97 Tf 6.58 0 Td[(1k=)]TJ /F8 7.97 Tf 6.59 0 Td[(L+1ukxk)]TJ /F8 7.97 Tf 6.59 0 Td[(m,ifjmj
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Rz,z()=Rx,x(a)Ry,y(b)+Rx,x(a+1)Ry,y(b)]TJ /F14 11.955 Tf 11.96 0 Td[(M) (4) for=aM+b,0bM)]TJ /F6 11.955 Tf 11.95 0 Td[(1,)]TJ /F14 11.955 Tf 9.29 0 Td[(NaN)]TJ /F6 11.955 Tf 11.95 0 Td[(1.TheautocorrelationRz,z()isingeneralRy,y(b)repeatedandscaledasafunctionofa.Theresultingoutputautocorrelationisthengoingtocontainnumerousspikeswhentheinnersequencesarealigned(whenb=0)whichisundesirable.However,ifthevaluesthoughofRx,x(a)areverylow,thentheeffectsoftherepeatedspikescouldbenegligible.IntheapplicationswewillconsiderforusingKroneckerwaveformstherewillbeasubsetoflagsthatarerelevant.TosuppressthepeakynatureoftheKroneckerwaveforms'autocorrelationattherelevantlags,wewillconsidersequencesxthathaveanappropriatelowcorrelationzone(LCZ).WebrieynotethatweconsiderLCZsequencesratherthanzerocorrelationzone(ZCZ)sequencesbecausewefocusonthepropertiesoftheaperiodicautocorrelation.ZCZsequencesarecommonlydesignedforperiodicautocorrelationwhichisnotapplicableintheSARparadigm.WealsoseeaninterestingphenomenonwhenjjM.IfweassumethenthatxisalowcorrelationzonesequencesuchthatjRx,x(1)j2jRx,x(0)j2,thenRz,z()Ry,y()forjjM.Whenwetryanddesignsignalswithlowaperiodicautocorrelationzonesthisdependenceontheinnersequenceautocorrelationwillcausespoilageofthelowautocorrelationzone.Fromawaveformdesignperspective,thisimpliesthatthechoiceofMandNisnottrivialandwecannotdesignatrueLCZwaveformunlessjRy,y()j2<,forjj>1whererepresentsthedesiredLCZvalue. 4.2.2KroneckerWaveformSpectrumThefocusofourproblemisthedesignofspectrallyconstrainedwaveformswithdesirableautoandcross-correlationproperties.Intheprevioussectionwediscussed 91

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thecorrelationofKroneckerwaveformsandwewillnowdiscusshowtodesignthespectrumofaKroneckersequence.ConsiderthefollowingKroneckerwaveformz=xywherez2CMN1,x2CN1,andy2CM1.Wecanwritetheelementsofzasz(n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)M+m=xnymforn=1,2,...Nandm=1,2,...,M.ThenthediscreteFouriertransform(DFT)ofsizeMNofzcanbewrittenas Z(p)=PNn=1PMm=1xnyme)]TJ /F8 7.97 Tf 6.59 0 Td[(j2 MN((n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)M+m)p,=PNn=1xne)]TJ /F8 7.97 Tf 6.59 0 Td[(j2 N(n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)pPMm=1yme)]TJ /F8 7.97 Tf 6.59 0 Td[(j2 MNmp,=ej2p NPNn=1xne)]TJ /F8 7.97 Tf 6.59 0 Td[(j2 NnpPMNm=1yme)]TJ /F8 7.97 Tf 6.59 0 Td[(j2 MNmp,=X(p)Y(p)ej2p N. (4) HereY(p)representsthepthelementofthezero-paddedDFTofsizeMNandX(p)denotesthepthelementoftheNpointDFTofx.Clearly,X(p)isperiodicwithperiodNandrepeatsMtimessincep=1,2,...,MN.ThepowerspectrumoftheKroneckerwaveformcanbewrittenas jZj2=jXj2jYj2,(4)whererepresenttheHadamardproductofvectors,X=[X(1),X(2),...,X(MN)]T,andY=[Y(1),Y(2),...,Y(MN)]T.ThepowerspectrumoftheKroneckerwaveformissimplythespectrumofymultipliedbyrepeatedcopiesofthespectrumofx.WewillnowdiscusstwooptionsfordesigningspectrallyconstrainedKroneckerwaveformsusingEq. 4 .Therstoptionwaspresentedheuristicallyin[ 62 ].InthisapproachNMandxwasdesignedtohaveadesiredspectrum.ywasthenchosentobeavectorofoneswhichactedasanupsamplingtypeapproach.Thespectrumofywasthenjustasincfunctionandthespectrumofzwastheproductofrepeatedcopiesofthespectrumofxmodulatedbythesincfunction.Inthisapproachtheoutputspectrumwouldnotmeetanystrictspectralconstraintsanditwasfedintothespectralshapingalgorithmtobeusedasaninitializingwaveform.Thistypeofapproachisuseful 92

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Table4-1. Variousslantrangeresolutionsandthecorrespondingrequiredbandwidths. ResolutionBandwidth 1m150MHz12in491MHz6in983MHz1in5.9GHz whendesigningwaveformsthatcontainthemajorityoftheirenergynearzerofrequencyorcanbecontainedinthemainlobeofthesincfunction.Thesecondoptionistodesignxandysuchthattheproductoftheirspectraresultsinthedesiredspectrum.AnaturalchoicehereistochoosejX(p)j=1,8p.Thenthespectrumofycontrolsthespectrumofz.OfcoursetheselectionofjX(p)j=1,8pisnotexactlyrealizableforanypracticalapplication.However,manywaveformdesignalgorithmsspecicallytrytooptimizetheautocorrelationwhichimplicitlytriestoattenthespectrum[ 14 ].IftherippleinXissmall,thenweshouldexpectthespectrumofztobeclosetoY.Thisistheapproachwewillexplorefordesigningspectrallyconstrainedwaveformsets. 4.3SpectrallyConstrainedWaveformSetDesignforSARTheslantrangeresolutionofaSARsystemisdependentontheprobingsignalbandwidth.HighresolutionSARapplicationsoftenrequireverylargebandwidthstoachievetherequiredimagingresolution.Table 4-1 listssomehighresolutionvaluesandtherequiredbandwidths.Aresolutionrequirementaslargeasonefootrequires500MHzofbandwidth.Furthermore,manytimesSARsystemsliketousespecicbandsforcertaintasks.Forexample,theultrahighfrequency(UHF)bandiscommonlydesiredbecauseitallowsforpenetrationthroughfoliage[ 61 ].TheUHFbandisextremelycongestedandanynon-spectrallyconstrainedwaveformwouldinterferewithotherUHFsystemsoperatingneartheSARplatform.TheSARplatformstillmusttryandmeetitsoperatingrequirementswhichcanbeachievedbyusingaprobingsignalthathasapowerspectrumthatmeetssome 93

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specicconstraintsontheamountofenergyallowedineachband.Thespectrallyconstrainedwaveformdesignproblemistodesignawaveformsuchthatitoccupiesaxedbandwidth,anditsspectrum,jFfzgj2,mustbeconstrainedsuchthatjFfzgj2
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4.3.1SpectrallyConstrainedKroneckerWaveformsforSARInasyntheticapertureradarwedesiretousesignalswithlargebandwidths.AspreviouslydiscussedLFMwaveformstraditionallyareusedinthisrolebecauseoftheirmanybenetssuchaseaseofimplementation,goodspectrumcontainment,andgoodautocorrelationfunction.However,notchedLFMorfrequencyjumpburstwaveformsdonotstrictlymeetspectrumconstraints.InSARapplicationsthereceiverisoftenonlyconcernedwithsomesubsetofthelisteningtimein-betweenpulses.Thisistruebecausethemaximumandminimumrangesilluminatedinthescenecanresultinaspanoftimedelaysofinterestthatarelessthanthetotalpulsewidth.ItisthereforereasonabletodesignLCZwaveformsforuseinaSARsystemwheretheLCZcorrespondstothedelayswithinthescene.ByusingKroneckerwaveformswecandesignsignalsthathaveaLCZandalsomeetspectrumcontainment.WehaveseenthatEq. 4 allowsustohavesomecontroloverthespectrumofzandthatEq. 4 andEq. 4 giveussomecontrolovertheautoandcross-correlationofz.Theproblemisthenstatedasdesigningxandysuchthatzmeetsthedesiredspectrumlimitf(!)andthatjRz,z()j2
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sequencetoyieldanacceptablespectrumfortheKroneckerwaveform.SHAPEisnottheonlyspectrallyconstrainedwaveformdesignalgorithm.Ifthegoalistojustminimizetheenergyinspecicbands,thenalgorithmssuchas[ 20 ]orSCANfrom[ 14 ]couldbeusedaswell.Theseotheralgorithmsthoughhavenocontroloverthenotchdepthwhichcouldberelevantintheoverallwaveformdesignproblem.NowassumethatwehavedesignedsomeysuchthatitsatisesEq. 4 .Thenweneedtodesignxsuchthatwesatisfythesecondwaveformdesignconstraint:jRz,z()j2
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Here,yisthetimedomainvectorwewishtodesign,disaspectruminthefeasiblesetweseek,andisascalarthataccountsforanyconstantenergymismatchbetweendandy.FistheunitaryFouriermatrix,histherequiredenvelopevector,andnally,fandgaretheupperandlowerspectralboundvectors,respectively.Theminimizationproblemin( 4 )issolvedusinganiterativeprocess.TheSHAPEalgorithmaccomplishesthisbysolvingthreesub-problemsperiteration.Therstsub-problemcanbeformulatedas minimizezjjFHy)]TJ /F7 11.955 Tf 11.96 0 Td[(djj22subjecttojdijfifori=1,2,...,Njdijgifori=1,2,...,N.(4)Weassumethatwestartwithanythathasadesiredenvelopeh,andtypicallyassigntoberealvalued.Ifiscomplexvalued,thenitresultsinaconstantphaseoffsetbetweenthewaveformandthespectrum.Let~y=FHy.Thenthesolutionto( 4 )maybewrittenas:^di=8>>>><>>>>:fiej(~yi)]TJ /F9 7.97 Tf 6.59 0 Td[():j~yij=jj>fi,giej(~yi)]TJ /F9 7.97 Tf 6.59 0 Td[():j~yij=jj
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HerewehaveusedthefactthattheEuclideannormisinvariantunderaunitarytransformtorearrangethecostfunction.Theminimizerforthisfunctionis ^yi=p hiej(~di+),(4)where~d=Fd.Thesesub-problemsarerepeatedlysolved(intheprecedingorder)untilthecostfunctionreacheszero,effectivelystopsdecreasing,oraniterationcountisexceeded. 4.3.1.2WeCANalgorithmWealsobrieyreviewtheWeightedCyclicAlgorithmNew(WeCAN)forcompleteness.ThecodefortheWeCANalgorithmisavailableatwww.sal.u.edu/bookintheMATLABccodesection.Adiscussionofthealgorithmanditsvariants,suchasMulti-WeCAN,canbefoundin[ 14 73 ].WeCANminimizestheweightedintegratedsidelobelevelmetric(WISL)whichisgivenas WISL=2N)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xk=12kjRx,x(k)j2, (4) wherekisapositiverealvaluedscalarrepresentingtheweightingfortheparticularlag.BymanipulatingEq. 4 ,itwasshownin[ 14 ]thattheWISLmetriccanbewritteninthefrequencydomainas WISL=20 2N2NXp=1)]TJ /F3 11.955 Tf 5.18 -9.68 Td[(~xHp)24(~xHp)]TJ /F14 11.955 Tf 11.96 0 Td[(N2,(4)where~xp=hx1e)]TJ /F8 7.97 Tf 6.59 0 Td[(j2 2Np,x2e)]TJ /F8 7.97 Tf 6.59 0 Td[(j2 2N2p,...,xNe)]TJ /F8 7.97 Tf 6.58 0 Td[(j2 2NNpiTand)]TJ /F1 11.955 Tf 10.26 0 Td[(isaToeplitzmatrixformedbythevectorofweights1 0[0,1,...,N)]TJ /F5 7.97 Tf 6.58 0 Td[(1].TheproblemofminimizingEq. 4 w.r.t.xisdifcultsinceitisaquarticfunctionofx.Howerver,asshowninthecitedpapers,analmostequivalentrelaxedminimization 98

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problemisgivenas minimizefxigNi=1,fpg2Np=1jjC~xp)]TJ /F23 11.955 Tf 11.95 0 Td[(pjj22subjecttojjpjj22=N,forp=1,2,...,2Njxij=hi,fori=1,2,...,N.(4)HereCisasquarerootof)]TJ /F1 11.955 Tf 10.26 0 Td[((i.e)]TJ /F6 11.955 Tf 11.84 0 Td[(=CTC).Thisproblemisquadraticwithrespecttoxandcanbesolvedusinganiterativeapproach.WeCANsolvesEq. 4 usingFFTbasedoperationswhichmakestheoptimizationcomputationallyefcient.In[ 14 ]theWeCANalgorithmisalsoextendedforthesynthesisofwaveformsetssuchthatthecross-correlationisconsideredaswellastheautocorrelation. 4.3.1.3ZCZwaveformsetsfromperfectsequencesAcommonwaveformclassofpracticalinterestistheconstantamplitudezeroautocorrelation(CAZAC)class.ManyquadraticphasecodesequencessuchasFrankorZadoff-ChuareCAZACsequences[ 74 ].Astheirnameimpliestheirperi-odicautocorrelationhaszeroamplitudeforallnon-zerotimedelays.Thesetypesofsignalshaveapplicationinvariouscommunicationsystemswherethecodeisrepeatedindenitelyataveryhighdutycycle.Inactivesensingapplicationstheperiodicpropertiesarenotapplicableduetothesensingparadigm,however,inacontinuousactivesensingsystemthisisnottrue(duetothehighpulserepetitionfrequency).Thegoalistoformulateawaveformthathasazerocorrelationzoneforsomespeciedlagsintheprobingwaveform.Thiscanbeachievedbymanipulatingthestructureofaperfectsequencebyusingcyclicprexes.CyclicprexesarecommonlyusedincommunicationssystemstoovercometheeffectsofmultipathonasystembecausetheuseofacyclicprexgeneratesaonesidedZCZ.IftheZCZislongerthanthemultipathchannelthenthemultipathcanbeeffectivelysuppressed[ 75 ].However,intheactivesensingcaseweoftenareconcernedwithpositiveandnegativelagsinthecorrelationresponse.Byusingthepsuedoperiodicapproachproposedin[ 76 ]theZCZ 99

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canbemadetobetwo-sided.Thedifferencebetweenthepseudoperiodicsequenceapproachandthecommoncyclicprexisthepsuedoperiodicsequenceappendsacyclicprexandacyclicpostxtothesequence.Thisgeneratesthetwo-sidedZCZwhenlteredwiththeoriginalperfectsequence.AsinglepseudoperiodicsequencecanalsobeusedtogeneratepseudoperiodicsequencesetsthathaveZCZinautoandcross-correlation[ 76 ].Thisisachievedbyapplyingcyclicshiftstotheoriginalsequencefollowedbytheadditionofthenewprexandpostx.Asdescribedin[ 76 ]andexpandedindetailin[ 77 ],thetrade-offisthelossofsignal-to-noiseratioduetotheuseofmismatchedltering.Furthermore,thenumberofsequencesyoucangenerateislimitedbythesizeoftheoriginalperiodicsequence,thenumberofsequencesneeded,andthelengthoftheZCZ. 4.3.2SpectrallyConstrainedWaveformsforMIMOSARGMTIThedesignofnon-spectrallyconstrainedwaveformsforuseinMIMOSARandMIMOGMTIhasbeenanactiveareaofresearch[ 14 66 67 ].In[ 66 ]itwasshownthattheuseoforthogonalwaveformswithdisjointspectraarenotidealfortheMIMOGMTIbecausetheyincreasetherankofthecluttercovariancematrix.IncreasingtherankofthecluttercovariancematrixresultsincluttersuppressionbeingdependentonlyontheoriginalreceivephasecentersandnotthevirtualphasecenterswhichnegatesthebenetsofusingMIMOprocessing[ 66 ].Thewaveformsetdesignmethodproposedinthisworkcanformwaveformsetsthathavezerocorrelationzonesandcantakeadvantageofallthevirtualphasecentersforcluttersuppression.TheuseofspectrallyconstrainedwaveformsforMIMOSARimagingisalsoproblematic.Asaddressedin[ 67 ]manyoftheproposed(non-spectrallyconstrained)methodsonlyconsiderorthogonalityattimelagzerowhichisunacceptable.Toachievetrulyorthogonalwaveformswewouldneeddisjointspectra(orwaveformsthataredisjointintime),whichisoftenimpracticalgivensystemrequirementsandisnotidealforGMTI[ 66 ].Toovercomethisproblem[ 67 ]proposedtousewaveformsthathad 100

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zerocross-correlationforsomesmalltimeinterval.Whenthesewaveformswerecombinedwithdigitalbeamformingonreceivethentheinterferenceduetotheotherwaveformswasnegligible.TheZCZKroneckerwaveformsproposedinthisworkcanalsobeclassiedasshorttimeorthogonalwaveformswhichsatisestheconcernsfororthogonalitydiscussedin[ 67 ].InthespectrallyconstrainedMIMOSARandMIMOSARGMTIscenariowedesiretodesignasetofwaveformsthatmeetthespectralconstraints,havelowautocorrelationzones,andalsohaveazerocross-correlationzoneforsomesubsetoflags.Thisproblemisexceedinglydifcult,butwithKroneckerwaveformswecanachievethesegoals.Todesignaspectrallyconstrainedwaveformsetwerstdesignasetofzerocorrelationzonewaveforms.ThiscanbeachievedbyusingeithertheMulti-WeCANalgorithmorbyusingaperfectsequenceinconjunctionwithcyclicshifting,pre-xes,andpost-xes.Thesedesigntopicsandtherelevantreferenceswerediscussedintheprevioussection.ThesecondstepistouseaspectrallyconstrainedwaveformtodesigntheinnersequenceoftheKroneckerwaveform.Inthiswork,wesuggesttheSHAPEalgorithmtoperformthistask.TheuseofoneinnerwaveformforalltheKroneckersignalsintheouterwaveformsetresultsinKroneckerwaveforms'correlationmatrixhavingtheidealresponseasdescribedin[ 66 ].TheuseoftheKroneckerwaveformsresultsindegradationtotheautocorrelationzerocorrelationzone.TheeffectofthisonaresultingSARimageisincreasedclutterduetosidelobesintheslantrangedomain.Thismotivatestheuseofmismatchedlteringsuchasthemethodproposedin[ 78 ].IntheMIMOscenariothough,thistypeofmismatchedlteringdoesnotdecouplethewaveformsandcannotbeused.FurtherstudiesintothedesignofmismatchedltersfortheMIMOwaveformcaseismerited. 4.4SpectrallyConstrainedSARandMIMOSARSimulationToexaminetheperformanceofthespectrallyconstrainedSARandMIMOSARGMTI,weconsiderasimulatedscenario.Inthisscenariothebackgroundsceneis 101

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Figure4-1. Thegroundtruthdatausedtogeneratethebackgroundclutter formedfromrealdatageneratedfromthetestsetprovidedfortheAFRLGMTIchallengeproblem[ 79 ].Figure 4-1 showsthedatathatisusedtosimulatethebackgroundstationaryclutter.ThisalsorepresentsthetruthintheMIMOSARimagingproblem.TosimulatemoversintothesceneweinjectsimulateddataintotherealdatasetcorerspondingtoFigure 4-1 .ThescenerepresentedinFigure 4-1 isusedasthemagnitudeoftheradarcrosssection(RCS)forthereturnateachpixel'srepresentativegroundlocation.Togeneratethereceiveddatausinganewprobingwaveform,thegeometryisrstcalculatedusingthemeasurementsprovidedinthetestset.Thenusingthegeometryofthescenetheappropriatephaseshiftsarecalculatedforeachpixelincross-rangeateachrangebinforeachpulse.Finallyforeachrangebin,giventhevectorofdatarepresentingthephasehistoryforallthepulsesatthespecicbin,thefasttimeresponseisaddedintothereturnsignalattheappropriatedelay.Theradarparametersareasfollows.TheplatformisoperatinginX-bandat9.6GHzandtheprobingwaveformhasabandwidthof640MHz.Thisresultsinapproximately.25metersofslantrangeresolution.Furthermoreweassumethattherearethreerestrictedbandsintheoperatingbandwidththatexistat(9.456,9.48),(9.64,9.688),and(9.752.9.768)GHz.Ourradiatedsignal'sspectrummustnotexceed)]TJ /F6 11.955 Tf 9.3 0 Td[(40dBintheserestrictedbands(relativetothemaximumpoweratanyfrequency).Thephasecodedwaveformwewillradiatemusthave4096chipswhichresultsin 102

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apulsewidthofapproximately6.4sec.Theregionofinterestoccupiesthelagscorrespondingto362chips.Therefore,werequireourlowcorrelationzonetobeforlagsjj<0.57s.Theaircraftwilloperateintwocongurations.IntherstcongurationweexaminetheperformanceforjustspectrallyconstrainedSARimaging.Heretheradaroperateswith1transmitand4receiveantenna.IntheMIMOSARGMTIcase,theradaroperateswith4transmitand4receiveantennas.Theantennasareassumedtolieinalineararrangementwithnormalizedantennaelementspacingof1.09metersfortransmittersand5.43metersforreceivers[ 65 ].Wecollect1024pulseswhiletheaircraftmovesataconstantvelocity(magnitudeanddirection).Thecenterrangetoscenecenterisapproximately10.4km.Weassumeweoperateinaclutterdominatedenvironmentandsetthecluttertonoiseratioat50dB.Thersttaskistodesigntheprobingwaveform.WeusetheSHAPEalgorithmtodesigntheinnerwaveform.Wemustaddresstwoconcernsregardingtheinnerwaveform.Therstisthatitmustmeetthespectralconstraints.Thesecondisthatitshouldbequiteshorttopreventthespoilageofthelowcorrelationzone.Thetotalnumberofchipsintheoutputwaveformisgiventobe4096,hencethelengthoftheinnerandoutersequencesmustbesomefactorsofthatnumber.WechoseM=32andN=128tobethelengthsoftheinnerandoutersequencesrespectively.ThechoiceofM=32wasbasedonthefactthatthiswouldresultinaspoilageofapproximately10%oftheLCZwhileallowingSHAPEtondaspectrumthatsatisestheconstraints.Wesetthelowerboundtobezeroforallfrequencies(whichimpliesthereisnolowerbound)andonlyplaceupperboundsonthespectrumintherestrictedbands.Therestrictedbandsexistat()]TJ /F6 11.955 Tf 9.3 0 Td[(144,)]TJ /F6 11.955 Tf 9.3 0 Td[(112),(40,88),(152,168)MHzrelativetothecarrierfrequency.Thesebandscorrespondtoanormalized(by)radialfrequencyof()]TJ /F6 11.955 Tf 9.29 0 Td[(0.45,)]TJ /F6 11.955 Tf 9.3 0 Td[(0.35),(.125,275),(.475,.525)whichweusewhendesigningtheSHAPEwaveform.Wechooseanoffsetof5dBtoallowforripplewhichimpliestheupperbound 103

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A BFigure4-2. InnerWaveform.A)Thenormalizedpowerspectrumoftheinnerwaveform.B)Thenormalizedautocorrelationfunctionoftheinnerwaveform intherestrictedbandswassetto)]TJ /F6 11.955 Tf 9.3 0 Td[(45dB.WeuseanFFTsizeof128inSHAPEandensurethatSHAPEsuccessfullyexitsthealgorithm.InFigure 4-2 theinnerwaveform'sspectrum(formedfroma512pointFFT)andautocorrelationareshown.Thesecondtaskistodesignthezerocorrelationzonewaveformwhichwewillgeneratefromaperfectsequence.TogeneratethekerneloftheouterZCZwaveformweuseaZadoff-Chusignalwhichisaquadraticphasecodesignalandhasaperfectperiodicautocorrelation[ 74 ].Thelengthoftheoutersignalis128andwerequireaZCZofsize764intheoutputKroneckerwaveform.ThiscorrespondstoaZCZofsize24intheouterwaveform.WesettheZCZsizeto26toplaceabufferintheregion.Theresultingouterwaveformkernelisthenalength102Zadoff-Chusignal.Weappendacyclicprexandpostxofsize13eachwhichresultsinalength128signal.TogeneratetheZCZtherequiredlteristhekernelZadoff-Chusignalwhichmeansadecreaseof20%processinggainintheoutputwhichdecreasesSNR.Wewillrefertoalllteringofthesewaveformswiththeirkernelasmatchedlteringfortherestofthischapter.Wenotethatthesignalandthekernalarethesamesequence,butthesignalhascyclicallyappendedsectionstomimictheperiodicautocorrelationbehaviorinspecicregions. 104

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A BFigure4-3. OuterWaveform.A)Thenormalizedpowerspectrumoftheouterwaveform.B)Thenormalizedcorrelationoftheouterwaveformwiththekernel. ThedesiredsignalistheKroneckerproductoftheouterandinnersignals.Figure 4-4 showstheresultingpowerspectrum,autocorrelation,andaclose-upofthelowcorrelationzone.InFigure 4-4 AwecanseethattheKroneckerwaveformmaintainsa)]TJ /F6 11.955 Tf 9.3 0 Td[(40dBnotchdepthrelativetothepeakspectralpower.Theverticalredlinescorrespondtotherestrictedbandsandthehorizontaldashedlinecorrespondswiththedesirednotchdepth.InFigure 4-4 BweseethattheautocorrelationmaintainstheLCZstructurewheretheverticalredlinescorrespondtoourdesiredLCZandthegreenlinerepresentsadesireddepth.Furthermore,inFigure 4-4 CtheLCZspoilageisshownduetothestructureoftheautocorrelationofKroneckersequences.Asexpectedthespoilageextendstothedelaycorrespondingtothe31stlag.Thisresultshowsthatwecandesignspectrallyconstrainedwaveformswithlowaperiodiccorrelationzones.WenowexaminetheSARimagingresultsusingthedesignedprobingwaveform.Figure 4-5 Ashowsthematchedlter(MF)SARimagingoutputforthespectrallyconstrainedwaveform.TheresultingspectrallyconstrainedmatchedlterimageisdegradedduetothespoilageintheLCZandhighclutterpowerinthesceneconsequently.Toovercomethisweuseofthemodiedmismatchedlter(MMF)proposedin[ 78 ].Themismatchedlterisaninversetypelterappliedinthefrequency 105

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A B CFigure4-4. KroneckerWaveform.A)ThenormalizedpowerspectrumoftheKroneckerwaveform.B)ThenormalizedcorrelationoftheKroneckerwaveformwiththekernelKroneckerwaveform.C)AcloseupofthespoilageofthelowcorrelationzoneoftheKroneckerwaveform. domain.However,sinceweknowthatourspectrumcontainsdeepnotchesweapplyanidealdigitalbandstopltertothedataatthosefrequencies.Thiswouldalsoallowforsuppressionofanyunwantedinterferenceduetoalicensedradiatoroperatinginthatband.Figure 4-5 Bshowsthemismatchedlteroutput.Theresultingimagefromthebandstopltereddatainconjunctionwiththemismatchedlterhaslowersidelobesandacleareroutput.ForreferenceFigure 4-5 Cshowsthematchedlteroutputofanon-spectrallyconstrainedZCZZadoff-Chuwaveformoflength4096generatedusing 106

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A B CFigure4-5. TheSARimagegeneratedforA)theSCwaveformandaMF.B)theSCwaveformandaMMF.C)thenon-SCwaveformandaMF. thesamemethodologyusedtogeneratetheouterwaveform.ThissignalhasidealcorrelationresponseintherangeswathwhichresultsinaSARimageclosetothetruth.Thepreviousexampledidnotcontainanyunwantedinterferencefromalicensedradiatorintherestrictedbands.Tosimulatethisweinjectednarrowbandinterferencethatvariedwithslowtimefromthreeradiatorsthatoperatedintheirrespectivebandsintherestrictedregion.Theradiatorswere10dBstrongerthantheaveragegroundreturn 107

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power.Figure 4-6 A-Dshowsthematchedlterandmismatchedlterresultsforthespectrallyconstrainedwaveformandthenon-spectrallyconstrainedwaveform.InFigure 4-6 Aweseethattheinterferencedoesnotaffectthespectrallyconstrainedwaveform.Thespectrallyconstrainedwaveform'smatchedlteractsasabandstoplterduetothedeepnotchesintheinterferencebands.Thematchedlterthereforesuppressestheinterferenceandtheresultingimageissimilartothecasewithnointerference.However,inFigure 4-6 Bthenon-spectrallyconstrainedwaveform'sSARimageisseverelydegradedduetotheinterference.Furthermorewhenapplyingthemismatchedlteringwithbandpasslteringtobothwaveforms,weseethatthespectrallyconstrainedwaveforminFigure 4-6 Cproducessimilarqualitycomparedtothenon-spectrallyconstrainedwaveforminFigure 4-6 D.Next,weexaminetheperformanceintheGMTIapplicationwiththespectrallyconstrainedwaveformset.WeconsidertheproblemofdesigningawaveformsetoffourshorttimeorthogonalwaveformsthatmeetspectralconstraintsandhaveaLCZ.WeusethesameKroneckerwaveformdesignprocessasweusedforthesinglewaveformcase.TheinnersignalisthesameSHAPEwaveformusedinthesinglewaveformcaseandisusedforall4Kroneckerwaveforms.TodesigntheoutersequencewestartwiththesameZadoff-Chusignalusedinthesinglewaveformcase.Wecyclicallyshiftthesignalby)]TJ /F6 11.955 Tf 9.3 0 Td[(14,14,and28shiftstocreatethreeseparatekernelsignals.Eachsignalisthenappendedwiththeappropriatecyclicprexandpostx.Figure 4-7 Ashowstheaveragemagnitudesquaredofallthecross-correlationsofthewaveformswiththekernelsintheset.Theverticalredlinesrepresenttheextentoftherequiredzerocorrelationzone.Thewaveformsareabletoachievetherequiredzerocross-correlationzone.Forreferenceweincludethesameplotforasetof4Zadoff-Chuwaveformsgeneratedusingthesamecyclicshifttechnique.ThissetisabletotightlyapproachtheexactwidthoftheZCZsinceitcanselectanylag.TheKroneckermethodisconstrainedinthismannerduetotherepetitionoftheinnerwaveform. 108

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A B C DFigure4-6. TheSARimagegeneratedinthepresenceofinterferenceforA)theSCwaveformwithaMF.B)thenon-SCwaveformwithaMF.C)theSCwaveformwhenwithaMMF.D)thenon-SCwaveformwithaMMF. 109

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A BFigure4-7. Theaverageofthe6cross-correlationsbetweenthe4waveformsintheset.A)TheSCKroneckersetB)Anon-SCZCZset. WeexaminetheperformanceoftheMIMOGMTIusingtheprocessingprocedureproposedin[ 65 ]withdataindependentprocessing.Weinjectfourtargetsintothescenewithradialvelocitiesof6.235,)]TJ /F6 11.955 Tf 9.3 0 Td[(4.9,4.9,and)]TJ /F6 11.955 Tf 9.29 0 Td[(5.213meterspersecondrespectivelywhichresultsinnormalizedDopplerfrequenciesof0.154,)]TJ /F6 11.955 Tf 9.3 0 Td[(0.056,0.086,and)]TJ /F6 11.955 Tf 9.3 0 Td[(0.065.Eachmoverhasasignalstrengthof)]TJ /F6 11.955 Tf 9.3 0 Td[(30dBrelativetothegroundclutterpower.Thegoalistosuccessfullydetectallthemoversandcorrectlyestimatetheirrangeandvelocities.WewilltestwithoutinjectedinterferenceandwithinterferenceusingthespectrallyconstrainedKroneckerwaveformsetandthenon-spectrallyconstrainedZCZwaveformset.Figure 4-8 showstherange-DopplerimageformedforMIMOGMTI.Herewehavereducedfromtheangle-Doppler-rangespacebytakingthemaximumreturnateachrangeacrossallangles.Figure 4-8 AandCshowtherange-DopplerimagefortheSCwaveformsetwithandwithoutinterference.NoticethattheLCZspoilageeffectoftheKroneckerwaveformisshownquiteclearlyintheseimages.Figure 4-8 BandDshowtherange-Dopplerimageusingthenon-SCwaveformsetwithandwithoutinterference.Withoutinterferencebothmethodsareabletosuccessfullydetectthetargetswithin 110

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closeproximitytotheirtruelocations.Inthepresenceofinterference,thenon-spectrallyconstrainedwaveformsfailstodetectallthetargetswhileintroducingunwantedreturns.WehavedemonstratedthatthespectrallyconstrainedKroneckerwaveformscanbeusedforSARImagingandMIMOSARGMTI.Furthermore,wehaveshownthatinthepracticalscenariowhereradiatorsexistintherestrictedbandsthespectrallyconstrainedwaveformsusingstandardmatchedlterprocessingperformsbetterthanthenon-spectrallyconstrainedwaveforms. 111

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A B C DFigure4-8. TheMIMOGMTIrange-DopplerimagegeneratedforA)theSCwaveformswithaMF.B)thenon-SCwaveformswithaMF.C)theSCwaveformsinthepresenceofinterferencewithaMF.D)thenon-SCwaveformsinthepresenceofinterferencewithaMF. 112

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CHAPTER5FUTUREWORKANDCONCLUSIONS 5.1ConclusionsInthisdissertationwehavereviewedthefundamentalsofactivesensingwaveformandreceiverlterdesign.Wealsointroducedthespectrallyconstrainedactivesensingproblemanddiscussedwhytraditionalmethodsarenolongerideal.Themajorcontributionsofthisworkwere:1)TheintroductionandanalysisoftheSHAPEalgorithmforwaveformdesign.2)Afastmissingdataiterativeadaptiveapproachwhenmissinglessthan50%ofthedata.ItisdirectlyapplicableinspectrallyconstrainedSARsystemsthatusesfrequencynotchedLFMwaveformsorfrequencyjumpburstwaveforms.3)AnewmethodofdesigningspectrallyconstrainedwaveformsthathavelowcorrelationzonesbyusingKroneckerwaveformswithapplicationtoSARsystems.Wealsoshowedthatthemethodcanbeextendedtodesignshort-timeorthogonalspectrallyconstrainedwaveformsets.TheSHAPEalgorithmisacomputationallyefcientmethodofquicklydesigningspectrallyconstrainedwaveformswithdesiredtimedomainenvelopefunctions.ItisimplementedusingFFTsandelement-wiseoperationswhichimpliesitisaprimecandidateforparallelprocessing.SHAPEwaveformsarebetterthanexistingspectrallyconstrainedwaveformdesignmethodsatmeetinghardspectralconstraints.However,theuseofaSHAPEwaveformimpliestheuseofareceiverthatcanprocessphase-codedsignalswhichismorecomplicatedthanreceiversthatuseLFMbasedwaveforms.TheproposedfastmissingdataIAAisdesignedforcaseswheretheamountofmissingdataissmall.ThismethodutilizesstructuresintheMIAAalgorithmandreplacesanonstructuredproblemwithastructuredproblembyalowrankcompletion.ThisallowsforreducingthecomputationalcomplexityfromO((N)]TJ /F14 11.955 Tf 12.66 0 Td[(Nm)3+KlogK)(see[ 37 ],[ 48 ])toO(Nm3+NmNlogN+N2+Klog(K)).Thisisanimprovement 113

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oftheasymptoticcomputationalcomplexityof(N=Nm)]TJ /F6 11.955 Tf 12.77 0 Td[(1)3,whichisanincreaseinperformancewhenevertheproportionofmissingdataislessthan50%.Wehavealsonumericallyobservedthattheimprovementinperformanceissignicantcomparedtothestate-of-theart.ThemethodisappliedtoSARimagingwithfrequencynotchingforinterferencesuppression.Frequencynotchingcausesmissingdataresultinginincreasedsidelobeswhichdecreasesimagequality.WedemonstratedthattheuseofmissingdataIAAcanimprovetheimagequalitysignicantly.WehaveexaminedthedesignofspectrallyconstrainedwaveformsforSARandMIMOSARGMTI.WeproposedtodesignKroneckerwaveformsthatmeetdifcultspectralconstraintswhilemaintainingalowcorrelationzone.ThestructureoftheKroneckerwaveformsallowsustocontrolthespectrumoftheoutputwaveformbydesigningtheinnerwaveformtohaveadesiredspectrumwhiledesigningtheouterwaveformtohaveaatspectrum.Theautoandcross-correlationoftheKroneckerwaveformcanalsobecontrolledbydesigningtheouterwaveformtohaveadesirablecorrelationfunction.ThisapproachdoesleadtodegradationofthelowcorrelationzoneintheKroneckerwaveformduetothespoilageatlagsnearzero.WeproposedtodesigntheinnerandouterwaveformsusingtheSHAPEandZCZperfectsequenceswithcyclicshiftswhichprovidedtherequiredspectralshapes,lowcorrelationzones,andzerocross-correlationzones.WedemonstratedtheefcacyoftheproposedspectrallyconstrainedKroneckerwaveformsusingsimulatedSARandMIMOGMTIexamples.InthesesimulationsthebackgroundclutterwasfromanactualSARcollectionwhilethemovingtargetsandinterferenceweresyntheticallyinjected.AsexpectedwesawthatthespectrallyconstrainedwaveformsprovidedaslightlydegradedSARimagecomparedtothenon-spectrallyconstrainedwaveformsduetothespoilageofthelowcorrelationzone.However,whenusedinconjunctionwithmismatchedltertheresultingimagewassimilartotheimageestimatedusingthenon-spectrallyconstrainedwaveformandmismatchedlter.Wealsoshowedthatthespectrallyconstrainedwaveformis 114

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Figure5-1. Anexampleofpicketfencestylewaveformsetspectra. robustagainsttheinterferencethatwouldbepresentintherestrictedbandsduetothedeepspectralnullsintheprobingsequences.IntheMIMOGMTIscenariothespectrallyconstrainedwaveformswereabletosuccessfullydetectallthemoversinthepresenceofinterferencewhilethenon-spectrallyconstrainedwaveformsfailedtodetectany. 5.2FrequencyDiverseandSpectrallyConstrainedWaveformSetsTheproposedMIMOSARwaveformscouldhavebetterorthogonalityiftheywerefrequencydiverse.Traditionalfrequencydiversemethodsrequireaccesstolargerportionsofthespectrumwhichisofteninfeasible.However,byusingtheSHAPEalgorithmitispossibletodesignapicketfencetypesetofcomplementarysequencesthatcanalsomeetspectralconstraints.Figure 5-1 isanexampleofsuchaset(fourunimodularwaveformsoflength4096).Heretheredverticallinesrepresentsomerestrictedbandsandthehorizontalgreendashedlinerepresentsrequirednotchdepth.Therearefourwaveformspresenteachwaveformspectrumrepresentedbyadifferentcolor.Thewaveformsintheexamplehaveexcellentcross-correlationperformanceduetothepartiallydisjointspectra.Themaximumofthemagnitudeateachlagforeachcross-correlationisshowninFigure 5-2 .Themaximumofthecrosscorrelationofthewaveformsisontheorderof)]TJ /F6 11.955 Tf 9.3 0 Td[(32dB.TheautocorrelationperformanceisshowninFigure 5-3 .Whiletheroll-offlooksacceptableinFigure 5-3 Athepeaksidelobelevels 115

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Figure5-2. Anexampleofcross-correlationforpicketfencestylewaveforms. A BFigure5-3. Anexampleoftheautocorrelationofpicketfencestylewaveform'sautocorrelation:a)Foralllagsb)Azoom-innearlagzero aretoohighatapproximately)]TJ /F6 11.955 Tf 9.3 0 Td[(5dB.Thiswouldbedifculttouseinactivesensingwithonlyamatchedlterbecausethehighsidelobelevelwouldintroducesignicantclutter.However,theproblemmightbepossibletoovercomebytheuseofdigitallteringandmismatchedlters.However,eachwaveformonlyusesasmallportionofthespectrumandafterdigitallteringtherewillbemissingdata.Theresultingestimateofthereturnedsignalfromthemismatchedlterwouldhavesignicantsidelobeswhichisunacceptable.Usingafastmissingdataalgorithm,suchasfastmissingdataIAA,couldpossiblybeabletoreducetheeffectsofthesesidelobes. 116

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Figure5-4. ASHAPEwaveformtransmittedusingaUSRPSDR.TheverticalaxisisthenormalizedpowerindBandthehorizontalaxisistheindexofthedatavectorwhichgoesfrom0to65536. 5.3TransmittingSHAPEWaveformsAnywaveformdesignisalltheoreticaluntilthewaveformisradiatedintotheenvironment.SofarwehavetestedSHAPEusingalowcostsoftware-denedradio(SDR)fortransmissionandreception.Figure 5-4 showsapreliminarytestdoneusingaUSRPSDR.Theresultslookpromising,butthetestingneedstobemovedtoabetterplatform.ThegoalistotestSHAPEusinganarbitrarywaveformgeneratorandavectorspectrumanalyzer.Usingabetterpieceofhardware,theeffectsofthephysicalsystemandhowitdistortsaSHAPEwaveformcanbestudied.Furthermore,ifthetransmitter'stransferfunctioncanbeaccuratelyfound,thenitwouldbeenlightingtoinvestigateifaSHAPEwaveformbepre-distortedtomitigatethetransmitter'sdistortion. 5.4SHAPEReceiverFilterTheonlyworkdonesofaronreceiverlterdesigntookamissingdataapproachtoanotchedlinearfrequencymodulatedwaveform.Thisprocessassumedthataspecictypeofreceiverwasused(ade-chirponreceive)togeneratetheinputwaveform.ThequestionstillremainsforaneffectivemethodoflteringgeneralSHAPEwaveforms.ItmaybepossibletoturneverySHAPEwaveformintoamissingdataprobleminthe 117

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frequencydomainbyusingatypeofinverselteringoriginallyproposedforSARcluttersuppression. 118

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APPENDIXACOMPUTATIONALCOMPLEXITYOFMIAAALGORITHMThecomputationalcostforoneiterationoftheproposedalgorithmis2 3Nm3+8NmNlog2N+3 2N2+3 2Klog(K)inadditiontolowerorderterms.ThersttermcomesfromcalculatingL,theinverseoftheCholeskyfactorofSmR)]TJ /F5 7.97 Tf 6.59 0 Td[(1SmT.ThesecondtermisforcalculatingX=R)]TJ /F5 7.97 Tf 6.58 0 Td[(1(SmTL)whichtakes6NmFFT'sofsize2NandforcalculatingdfromXwhichtakes2NmFFT'sofsize2N.ThethirdtermresultsfromLevinson-Durbin(N2,see[ 9 ])andthecomputationofR)]TJ /F5 7.97 Tf 6.58 0 Td[(1from( 3 )(N2=2,alsoutilizingthatR)]TJ /F5 7.97 Tf 6.59 0 Td[(1isHermitianandpersymmetric1).Thenaltermsis(3=2)KlogKandresultsfromthethreeFFT/IFFTcalculationsofsizeK:1)R1:N,1=K(F)]TJ /F5 7.97 Tf 6.58 0 Td[(1(p))1:N,2)Nbysynchronizing( 3 )and( 3 ),3)Dbysynchronizing( 3 )and( 3 ). 1Inthiscase( 3 )reducestotheTrenchsalgorithmfortheestimationoftheinverseofToeplitzmatrices[ 80 ,p.132],notinghoweverthatTrenchsmethodrequiresO(N2)memorystoragewhile(21)onlyO(N),attheexpenseof(3=2)N2extracomputations. 119

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APPENDIXBTOEPLITZMATRICESANDFASTCALCULATIONSThecelebratedfastFouriertransformhasbeenwidelyusedfordecreasingthecomputationalcomplexityofalgorithmswithstructure.Inparticular,thiscanbeusedwhenthematricesinvolvedarecyclicorToeplitz.Herewesummarizesomeoftheconceptsrelevanttothispaper.Foramorecomprehensivereviewonthis,see[ 81 ].MultiplicationbetweenaToeplitzmatrixR2CNNandanyvectory2CNmaybecomputedbyembeddingtheToeplitzmatrixintoacirculantmatrix0B@RUUR1CA.SincetheFouriermatrixdiagonalizesanycyclicmatrix,Rymaybeobtainedfrom0B@RUUR1CA0B@y0N11CA=0B@RyUy1CA,whichiscomputedefcientlyusingFFTinO(NlogN)[ 81 ].Toeplitzsystemsmaybesolvedefcientlyusingthedisplacementstructures[ 82 ].Morespecically,letD=0B@01(N)]TJ /F5 7.97 Tf 6.59 0 Td[(1)0IN)]TJ /F5 7.97 Tf 6.59 0 Td[(10(N)]TJ /F5 7.97 Tf 6.58 0 Td[(1)11CAdenotetheshiftmatrixanddenethedisplacementofaHermiteanmatrixRasr(R)=R)]TJ /F3 11.955 Tf 12.6 0 Td[(DRDT.IfRisanonsingularToeplitzmatrix,thenthedisplacementsofRanditsinverseR)]TJ /F5 7.97 Tf 6.59 0 Td[(1bothhaverankboundedby2.ThedisplacementofR)]TJ /F5 7.97 Tf 6.59 0 Td[(1maybeexpressedas r(R)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=uu)]TJ /F3 11.955 Tf 12.02 0 Td[(~u~u,(B)whereu=1 0B@11CA,~u=1 0B@0~1CA, 120

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=(1,...,N)]TJ /F5 7.97 Tf 6.58 0 Td[(1)TarethevectorofARcoefcientsconsistentwithR,and2isthecorrespondingpredictionerror,i.e.,R0B@11CA=0B@201CA,and~=(N)]TJ /F5 7.97 Tf 6.59 0 Td[(1,...,1)T(see,e.g.,[ 9 ]).Levinson-Durbin'salgorithmmaybeusedtondand,andconsequentlyuand~u,inO(N2)[ 9 80 ].ThismaybeusedforsolvingToeplitzsystemsefcientlyusingthesocalledGohberg-Semencul(GS)factorizationR)]TJ /F5 7.97 Tf 6.59 0 Td[(1=L(u,D)L(u,D))-222(L(~u,D)L(~u,D),whereL(u,D)=(u,Du,D2u,...DN)]TJ /F5 7.97 Tf 6.58 0 Td[(1u)isalowertriangularToeplitzmatrix.SinceL(u,D)andL(~u,D)areToeplitzmatrizes,anynumberofsystemsRxj=yjmaybesolvedbyfourToeplitz-matrixvectormultiplications. 121

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BIOGRAPHICALSKETCH WilliamRowereceivedhismaster'sdegree(2010)andbachelor'sdegree(2009)fromtheUniversityofFlorida.HewasanactivedutyenlistedmemberoftheUnitedStatesCoastGuardfrom2001-2005andareservistfrom2005-2008.HewasawardedtheSMART(Science,Mathematics,andResearchforTransformation)Fellowshipin2010.HissponsoringagencywastheUnitedStatesArmy,SpaceandMissileDefenseCommandwhichishisexpectedemployeruponcompletionofhisdoctoraldegree. 128