Underwater Acoustic Signal Processing and Its Applications

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Title:
Underwater Acoustic Signal Processing and Its Applications
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english
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Zhao, Kexin
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University of Florida
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Gainesville, Fla.
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Degree:
Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Electrical and Computer Engineering
Committee Chair:
LI,JIAN
Committee Co-Chair:
LIN,JENSHAN
Committee Members:
WONG,TAN FOON
DING,MINGZHOU

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Subjects / Keywords:
acoustic -- active -- communication -- localization -- multistatic -- sonar -- source -- underwater -- wideband
Electrical and Computer Engineering -- Dissertations, Academic -- UF
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Electrical and Computer Engineering thesis, Ph.D.
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government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Abstract:
Underwater acoustic signal processing has various applications in the underwater sensing systems, including the underwater acoustic communication (UAC) system and the active sonar system. A well-designed UAC system can achieve reliable and high data-rate communication to facilitate the operation of submarines, undersea sensors, and unmanned undersea vehicles (UUVs) while active sonar system demands effective and efficient signal processing techniques to accurately detect, localize, and even track the target of interest. The focus of this dissertation is to use the appropriate signal processing techniques to design such UAC and active sonar systems. For the former, we focus on designing a mobile multi-input multi-output (MIMO) UAC system over double-selective channels subject to both inter-symbol interference and Doppler scaling effects. Temporal resampling is implemented to effectively convert the Doppler scaling effects to Doppler frequency shifts. By simplifying the assumption on the Doppler frequency shifts imposed on the channel taps across all the transmitter and receiver pairs, two sparse channel estimation algorithms, both as an extension of the original sparse learning via iterative minimization (SLIM) method, are proposed for channel estimation. Regarding symbol detection, we employ Turbo equalization and propose a fast implementation of the standard Turbo equalizer for retrieving the transmitted signal. The effectiveness of the considered mobile MIMO UAC scheme is demonstrated using both simulated data and measurements recently acquired during the MACE10 in-water experiment. For the latter, we consider a multistatic active sonar system that employs multiple stationary transmitters and receivers. Two signal processing aspects related to such a system design are addressed, namely target range-Doppler imaging and target parameter estimation. To enhance the range-Doppler imaging performance, a hybrid dense-sparse method is proposed to improve resolution and reduce sidelobe levels simultaneously while maintaining high accuracy. In the presence of multiple targets, each peak of the range-Doppler images need to be associated with a specified target before the target parameter estimation. To efficiently solve this problem, we develop a generalized K-Means clustering (GKC) method, which iteratively assigns peaks to targets and then estimates the target parameters based on the current association pattern. Moreover, based on fact that different transmitter-receiver pairs have different reflection coefficients, we develop a weighted least-squares method where the target parameters are refined in an iterative manner using weighting. Note that if each of the receivers is equipped with a large array that can provide accurate angle estimates of the targets, the peak association problem becomes an easy problem or even disappears entirely. For such case, the active sonar system demands an advanced source localization method. However, most existing techniques are developed under the narrowband assumption, which becomes invalid due to the nature of the sonar application. We thus develop two extensions of the SLIM algorithm to solve the wideband source localization problem, which will be demonstrated to provide satisfactory performance.
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In the series University of Florida Digital Collections.
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Includes vita.
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Statement of Responsibility:
by Kexin Zhao.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: LI,JIAN.
Local:
Co-adviser: LIN,JENSHAN.

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UNDERWATERACOUSTICSIGNALPROCESSINGANDITSAPPLICATIONSByKEXINZHAOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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c2014KexinZhao 2

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IdedicatethisworktomyMother,LingCao,andmyancee,YixueZhang,withoutthemthiswouldhavebeenimpossibleforme. 3

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ACKNOWLEDGMENTS Firstandforemost,Iwouldliketoexpressmydeepestappreciationtomyadvisor,Dr.JianLioftheElectricalandComputerEngineeringDepartment,whohasledmethroughoutmyPhDtrainingwithherpatience,greatknowledge,andcontinuousencouragement.Ithasbeenanexceptionalexperiencetoparticipateintheunderwateracousticcommunicationandtheactivesonarprojects.Withoutherguidanceandpersistenthelpthisdissertationwouldnothavebeenpossible.IacknowledgemycommitteemembersatUniversityofFlorida:Dr.MingzhouDing,Dr.TanWong,andDr.JenshanLin.Iamtrulythankfulfortheirtimeandeffortsthattheyspentonmydissertation.IamtrulygratefultohaveanicegroupoffellowstudentsatDr.Li'slab.IhighlyappreciateallthehelpthatIreceivedfrommylabmembersduringmytimeatUniversityofFlorida.Researchwouldhavebeenmuchlesscolorfulwithoutthem.Lastbutnotleast,Iamdeeplyindebtedtomyfamily,especiallymymother.Theyhaveprovidedmewithimmenseunderstandingandsupportalltheseyears.Thisdissertationisdedicatedtothem. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 1.1ChallengesofUAC ............................... 13 1.2ChallengesofActiveSonarSystems ..................... 14 1.3Notation ..................................... 16 2ENHANCEDMOBILEMULTI-INPUTMULTI-OUTPUTUAC ........... 22 2.1SystemOutline ................................. 24 2.2Double-SelectiveChannelwithDopplerScalingEffects .......... 26 2.2.1ChannelModel ............................. 26 2.2.2TemporalResampling ......................... 28 2.2.3ResamplingFactorEstimation ..................... 28 2.3ChannelEstimation .............................. 29 2.3.1Training-DirectedMode ........................ 29 2.3.2Decision-DirectedMode ........................ 31 2.3.3ChannelEstimationAlgorithm:GoSLIM ............... 32 2.3.4ChannelEstimationAlgorithm:GoSLIM-V .............. 35 2.3.5ComplexityAnalysis .......................... 36 2.4SymbolDetection ................................ 37 2.4.1Problemformulation .......................... 38 2.4.2PhaseCompensation ......................... 39 2.4.3LMMSEBasedSoft-InputSoft-OutputEqualizer ........... 40 2.4.3.1AprioriLLRpre-processor ................. 40 2.4.3.2LMMSEltering ....................... 41 2.4.3.3AposterioriLLRgenerator ................. 43 2.4.4Low-ComplexityApproximateLMMSEFiltering ........... 44 2.5NumericalandExperimentalResults ..................... 45 2.5.1NumericalResults ........................... 45 2.5.2MACE10In-WaterExperimentationResults ............. 47 2.5.2.1Experimentspecics ..................... 47 2.5.2.2Performanceevaluation ................... 49 5

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3ENHANCEDMULTISTATICACTIVESONARSIGNALPROCESSING ..... 64 3.1SystemDescriptionandProblemFormulation ................ 66 3.2ProposedAlgorithms ............................. 67 3.2.1Range-DopplerImaging ........................ 68 3.2.1.1Imagingproblemformulation ................ 68 3.2.1.2Receiverlterforrange-Dopplerimaging ......... 70 3.2.2GeneralizedK-MeansClustering(GKC)AssociationMethod .... 74 3.2.3EXIP-WLSMethodforTargetPositionEstimation .......... 77 3.2.4EXIP-WLSMethodforTargetVelocityEstimation .......... 80 3.3SimulationResults ............................... 82 3.3.1Range-DopplerImagingResults ................... 83 3.3.2TargetParameterEstimationResults ................. 83 4WIDEBANDSOURCELOCALIZATIONUSINGSLIM .............. 91 4.1DataModel ................................... 94 4.2TheWidebandSLIMAlgorithms ....................... 96 4.2.1WB-SLIM-0 ............................... 96 4.2.2WB-SLIM-1 ............................... 98 4.2.3Discussion ................................ 99 4.3RELAX ..................................... 100 4.4NumericalExamples .............................. 102 5CONCLUSIONSANDFUTUREWORK ...................... 112 APPENDIX ATHEDERIVATIONOFTHEDOPPLERSCALINGFACTOR ........... 115 BTHEDERIVATIONOFTHEWEIGHTMATRIX .................. 116 REFERENCES ....................................... 118 BIOGRAPHICALSKETCH ................................ 127 6

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LISTOFTABLES Table page 2-1CodedBERresultsobtainedbyGoSLIMandGoSLIM-V ............. 53 2-2Complexitycomparison(ins)betweenGoSLIMandGoSLIM-V ......... 53 2-3Asummaryoftheperformanceofthethreedetectionschemes ......... 54 2-4TheaveragecodedBERobtainedbyExactLMMSETurboandApproximateLMMSETurbo .................................... 55 3-1Thenoisepowerandthenormofthetargetreectioncoefcients ........ 85 3-2Systemparameters ................................. 85 3-3RMSEofParameterEstimatesUsingULSandEXIP-WLS ............ 85 4-1WB-SLIMalgorithms ................................. 110 4-2TheRELAXalgorithm ................................ 111 7

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LISTOFFIGURES Figure page 1-1Absorptioncoefcientversusfrequency ...................... 18 1-2Underwaterchannelimpulseresponse(CIR)example .............. 19 1-3Scatteringfunctionsobtainedattwodifferentconditions ............. 20 1-4NormalizedCIRevolutionoverapproximatelyaone-miniteperiod ........ 21 2-1AnNMMIMOUACsystem ............................ 53 2-2ThestructureoftheLMMSEbasedsoft-inputsoft-outputequalizer ....... 56 2-3Simulationaveragedover500Monte-Carlotrials ................. 56 2-4ThestructureofthepackageusedintheMACE10experiment ......... 57 2-5Thestructureofthetransmittedsymbolsforthe412MIMOBLASTschemeusedinMACE10 ................................... 57 2-6ThesuperimposedmodulusoftheCIRestimates ................. 58 2-7Theeffectofresampling ............................... 59 2-8TherelativespeedbetweenthetransmitterandreceiverarraygivenbyGPSandestimatedduringthetemporalresamplingstage ............... 59 2-9Dopplerfrequencyevolutionofthe1stpacketinepochE018obtainedbyGoSLIMandGoSLIM-V ............................... 60 2-10CIRestimationcomparisonbetweenGoSLIMandGoSLIM-V .......... 61 2-11GrayscalemascotobtainedfromGoSLIMandGoSLIM-V ............ 62 2-12GrayscalemascotobtainedfromRELAX-BLASTandTurboequalization .... 63 2-13TheLLRsoftinformationaboutthesourcebitsattheoutputofthedecoder .. 63 3-1Thesimulationgeometry .............................. 86 3-2Agenericactivesonarscenario ........................... 87 3-3Descriptionoftheassociationproblem ....................... 88 3-4Range-Dopplerimagesobtainedattherstreceiver ............... 89 3-5Range-Dopplerimagesobtainedatthesecondreceiver ............. 90 4-1Spatialpseudospectraobtainedwithascalarsensorarray ........... 106 8

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4-2Spatialpseudospectraobtainedwithavectorsensorarray ........... 107 4-3Spatialpseudospectrainthecaseofaweaksource ............... 108 4-4PerformanceenhancementusingRELAX ..................... 109 4-5EmpiricalfailurerateandRMSEsversusSNR .................. 109 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyUNDERWATERACOUSTICSIGNALPROCESSINGANDITSAPPLICATIONSByKexinZhaoMay2014Chair:JianLiMajor:ElectricalandComputerEngineering Underwateracousticsignalprocessinghasvariousapplicationsintheunderwatersensingsystems,includingtheunderwateracousticcommunication(UAC)systemandtheactivesonarsystem.Awell-designedUACsystemcanachievereliableandhighdata-ratecommunicationtofacilitatetheoperationofsubmarines,underseasensors,andunmannedunderseavehicles(UUVs)whileactivesonarsystemdemandseffectiveandefcientsignalprocessingtechniquestoaccuratelydetect,localize,andeventrackthetargetofinterest.ThefocusofthisdissertationistousetheappropriatesignalprocessingtechniquestodesignsuchUACandactivesonarsystems. Fortheformer,wefocusondesigningamobilemulti-inputmulti-output(MIMO)UACsystemoverdouble-selectivechannelssubjecttobothinter-symbolinterferenceandDopplerscalingeffects.TemporalresamplingisimplementedtoeffectivelyconverttheDopplerscalingeffectstoDopplerfrequencyshifts.BysimplifyingtheassumptionontheDopplerfrequencyshiftsimposedonthechanneltapsacrossallthetransmitterandreceiverpairs,twosparsechannelestimationalgorithms,bothasanextensionoftheoriginalsparselearningviaiterativeminimization(SLIM)method,areproposedforchannelestimation.Regardingsymboldetection,weemployTurboequalizationandproposeafastimplementationofthestandardTurboequalizerforretrievingthetransmittedsignal.TheeffectivenessoftheconsideredmobileMIMOUACschemeis 10

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demonstratedusingbothsimulateddataandmeasurementsrecentlyacquiredduringtheMACE10in-waterexperiment. Forthelatter,weconsideramultistaticactivesonarsystemthatemploysmultiplestationarytransmittersandreceivers.Twosignalprocessingaspectsrelatedtosuchasystemdesignareaddressed,namelytargetrange-Dopplerimagingandtargetparameterestimation.Toenhancetherange-Dopplerimagingperformance,ahybriddense-sparsemethodisproposedtoimproveresolutionandreducesidelobelevelssimultaneouslywhilemaintaininghighaccuracy.Inthepresenceofmultipletargets,eachpeakoftherange-Dopplerimagesneedtobeassociatedwithaspeciedtargetbeforethetargetparameterestimation.Toefcientlysolvethisproblem,wedevelopageneralizedK-Meansclustering(GKC)method,whichiterativelyassignspeakstotargetsandthenestimatesthetargetparametersbasedonthecurrentassociationpattern.Moreover,basedonfactthatdifferenttransmitter-receiverpairshavedifferentreectioncoefcients,wedevelopaweightedleast-squaresmethodwherethetargetparametersarerenedinaniterativemannerusingweighting.Notethatifeachofthereceiversisequippedwithalargearraythatcanprovideaccurateangleestimatesofthetargets,thepeakassociationproblembecomesaneasyproblemorevendisappearsentirely.Forsuchcase,theactivesonarsystemdemandsanadvancedsourcelocalizationmethod.However,mostexistingtechniquesaredevelopedunderthenarrowbandassumption,whichbecomesinvalidduetothenatureofthesonarapplication.WethusdeveloptwoextensionsoftheSLIMalgorithmtosolvethewidebandsourcelocalizationproblem,whichwillbedemonstratedtoprovidesatisfactoryperformance. 11

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CHAPTER1INTRODUCTION Underwateracousticsignalprocessinghasvariousapplicationsintheunderwatersensingsystems,includingtheUACsystemandtheactivesonarsystem.InasimpleUACsystem,thetransmittersendsaprobingsequenceintotheunderwatermediumandbyanalyzingthemeasurementatthereceiver,wecandeterminethepropertiesoftheacousticchannelviachannelestimation.Similarlybutstilldifferently,inanactivesonarsystem,theprobingsequenceistransmittedtowardanareaofinterestandatargetcouldreectafractionofthewaveformtothedirectionofthereceiver.Thisfractionofsignalfacilitatesthereceiver'sabilitytodetectpotentialtargetsviarange-Dopplerimaging.ThefocusofthisworkishowweapplyvarioussignalprocessingtechniquestosolvetheproblemthatweencounterinthereceiverdesignofbothanUACsystemandanactivesonarsystem. Waterformsamajorpartofthesurfaceoftheearthandenormousattemptshavebeenmadetoexploretheunderwaterenvironment.Theadvancesintechniquesduringthepastseveraldecadeshaveledtovariousunderwateractivities,includingharbormonitoringandtheexplorationoftheocean.ThesetasksnecessitatetheemploymentoftheunderwatersensornetworksandareliableUACisessentialtoensuregoodcommunicationbetweenthosesensornodes.Asweknow,waterismoresuitableforacousticwavestotransmitthantheelectromagneticwaves.Incontrasttowelldevelopedradiocommunicationsystemswhichalreadyimpactoureverydaylife,UACisstillintheexperimentstagelargelyduetoitsvariouschallenges.Section 1.1 willdescribefourmainchallenges.Similarly,theactivesonarsystemhasmanyapplicationsandthechallengesencounteredarenolessthanthosefortheUACsystem.Section 1.2 willbrieyreviewthethreemajorchallenges.ThenotationsofthiswholedissertationislistedinSection 1.3 12

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1.1ChallengesofUAC Oneofthemainpropertiesoftheacousticchannelisthattheabsorptionrateincreasesasthefrequencyofthesignalrises[ 1 ].TherelationbetweenabsorptioncoefcientandthesignalfrequencycanbefoundinFigure 1-1 .Becauseofthesevereabsorptionathighfrequency,thepowerofasignalwithfrequency120kHzwilldropbyalmost40dBafteran1kmpropagation.Consequently,thepowerofthecorrespondingreceivedmeasurementscouldbeveryweak.Toaddressthisproblem,limitedbandwidthisadoptedintypicalUACsystems[ 1 ].Suchscarcelyavailablebandwidthimposesanupperboundontheattainablesymbolrate.Therefore,thepursuitofhighdatarateinUACleveragesthemulti-inputmulti-output(MIMO)scheme,whichoffersincreaseddataratescomparedtoitssingle-inputcounterpart[ 2 ]. Thetransmittedwaveformcanarriveatthereceiverviamultiplepaths[ 3 ].Figure 1-2A demonstratesasimpleacousticchannelcharacterizedbyadirectpath,abottom-reectedpath,andasurface-reectedpath.Practicalunderwaterchannelisusuallymuchmorecomplicatedthanthisoneduetothepresenceofmorereectioncombinations[ 3 ].Suchmultipathpropagationalongwiththelowunderwatersoundpropagationspeedresultsinlargedelayspread.Thedifferenceinthepropagationtimebetweentheearliestandlatestarrivalscouldspantenstohundredsofsymbolperiods,whichtranslatesintolongchannelimpulseresponse(CIR)andsevereinter-symbolinterference(ISI)atthereceiverside.Figure 1-2B showsapracticalCIRestimateoflength80.SymboldetectioninaMIMOsetupiscumbersomebecauseMIMOschemeintroducesinterferencesacrossallthetransmittersinadditiontothepresenceoftheISIeffects. Besidesthelongdelayspread,UACchannelsalsosufferfromtheDopplereffects[ 3 4 ].ThepresenceofDopplereffects,owingtotherelativemotionsbetweenthetransmitterandreceiverplatformsandthedynamicunderwateracousticmedium,inducestemporalscaling(stretchingorcompression)tothetransmittedsignals. 13

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Doppler-inducedscalingeffectsimpairthereliabilityofUAC,especiallyinthecaseofaphase-coherentdetectionscheme.Apreferabletoolforcharacterizingadoublespreadingchannelisthescatteringfunction(SF),whichdecouplestheacousticchannelintoabankofpathsthatexperiencedifferentdelaysandDopplerfrequencies[ 5 ].TwoSFsfromdifferentseaenvironmentareshowninFigure 1-3 .FortheArcticOceancaseasshowninFigure 1-3A ,bothmainpathsarecenteredaround0Hz,whichsuggeststhatthischannelexperiencesnegligibleDopplereffects.IncomparisonfortheBahamaswindyweathercase,Figure 1-3B showsthatthetwomainpathssufferfromsevereDopplereffects. TheUACchannelisalsoquicklyvaryingovertime.Figure 1-4 showsanevolutionofapracticalCIR.WecanseefromFigure 1-4 thatthechanneltapsafter4msaresignicantlyvariantovertime.Hencesuchtime-varyingpropertyonlyallowsashortcoherenceprocessingtime[ 6 9 ]. Chapter 2 focusesonsingle-carrierUACsystem.WeprovideadetailedmobileMIMOUACsystemdesignbypresentingtechniquesforaccuratetemporalresampling,properchannelmodeling,effectiveandefcientchannelestimationandsymboldetection.TheeffectivenessoftheproposedmobileMIMOUACsystemisveriedusingbothnumericalandexperimentalresults. 1.2ChallengesofActiveSonarSystems Thenalgoalofanactivesonarsystemistodetect,localize,andeventrackthepotentialtargets.Therststeptoachievethisgoalistoformrange-Dopplerimagesbasedonthereceivedmeasurementsandtheknowtransmittedprobingsequences.WecanobtaintherangeandDopplerinformationfromthedominantpeaksoftherange-Dopplerimagesandthenproceedtoestimationthetargetparameters.Onestandardreceiverdesignfortherange-Dopplerimagingapplicationisthecommonlyusedmatchedlter,whichessentiallycorrelatesthereceivedsignalwiththetransmittedone.However,whensevereinterferencesarepresent(thisisquitenormalfora 14

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multistaticactivesonarapplication),themultiplesimultaneouslytransmittedprobingsequencesactasinterferencestooneanother,makingthematchedlterbasedreceiverineffective.Therefore,itisnecessarytodesignmoreadvancedadaptivereceiverlterscapableofprovidingrange-Dopplerimageswithbothlowsidelobelevelsandhighaccuracy. Inthepresenceofmultipletargetsintheeldofview,oncethemultistaticrange-Dopplerimagesareavailable,werstneedtodeterminethenumberofthetargetsintheareaofinterestandsolvethetargetassociationproblembeforewecanproceedwiththetargetparameterestimation.Theassociationapproachaimstodetermineaproperone-to-onecorrespondencebetweenthetargetsandthepeaklocationsofeachrange-Dopplerimage.Thebrute-forceassociationisapplicableonlywhenthenumbersoftransmitters,receivers,andtargetsaresmall;otherwise,thecomputationalcomplexitywillbetoohightoimplement.Thussuchapplicationdemandsanefcientandeffectivetargetassociationapproachandonethatsimultaneouslydealswiththetargetassociationproblemandthetargetpositionestimationchallengeispreferred. Aftertheassociationprocedure,wecanproceedtoestimatethetargetparametersbasedontherangeandDopplerestimatesalreadyobtainedandassignedtoeachtarget.Onestandardapproachwouldbeusingthequasi-Newtonmethod(a.k.a.GaussorGauss-Newtoninterpolation)viaiterativelinearization.Morespecically,thequasi-NewtonmethodmakesuseoftheTaylorexpansiontoapproximateacollectionofnonlinearalgebraicpositionequationsaslinearones,andrenesthetargetpositionestimateinaniterativemanner.Beingconceptuallysimpleandcomputationallyattractive,thequasi-Newtonmethodhasbecomeastandardalgorithmimplementedontheglobalpositioningsystem(GPS)devicesfortheenduser.Oncethetargetpositionisavailable,itsvelocityisdeterminedbysolvingaleast-squares(LS)ttingproblem.However,thisapproachtreattherangeandDopplerestimatesequallywhile 15

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estimatingthetargetparametersdespitethefactthatdifferenttargetshavedifferentreectioncoefcients.ApreferablewaytoaddressthisproblemistoassigndifferentweightstotherangeandDopplerestimatescorrespondingtodifferenttargetsinordertoenhancetheestimationaccuracy. Chapter 3 focusesonamultistaticactivesonarsystem.Weprovidethoroughinvestigationofthesonarsystemdesignbyprovidingadetailedtreatmentofeverystepinvolvedinthereceiverdesignfromrange-Dopplerimagingtotargetparameterestimation.Thisisdonebypresentingapproachesforrange-Dopplerimagingwithimprovedresolutionandhighlysuppressedsidelobelevels,efcienttargetassociationschemethatincorporatesthetargetlocationestimation,andeffectivetargetparameterestimationwithimprovedaccuracyusingcalculatedweights.Simulationresultsvalidatetheeffectivenessoftheproposedoverallreceiverdesignforamultistaticactivesonarsystem. Notethatifeachofthereceiversisequippedwithalargearraythatcanprovideaccurateangleestimatesofthetargets,thepeakassociationproblembecomesaneasyproblemorevendisappearsentirely.Forsuchcase,theactivesonarsystemdemandsanadvancedsourcelocalizationmethod.However,mostexistingtechniquesaredevelopedunderthenarrowbandassumption,whichbecomesinvalidduetothenatureofthesonarapplication.HenceinChapter 4 ,wedeveloptwoextensionsoftheSLIMalgorithmtosolvethewidebandsourcelocalizationproblem,whichwillbedemonstratedtoprovidesatisfactoryperformance. 1.3Notation Vectorsandmatricesaredenotedbyboldfacelowercaseanduppercaseletters,respectively,kkdenotestheEuclideannormofavector,jjisthemodulusand()isthecomplexconjugateofascalar.()Tand()Hdenotethetransposeandconjugatetranspose,respectively,ofamatrixorvector,Idenotesanidentitymatrixofappropriatedimension,and^xdenotestheestimateofx.diag(v)representsadiagonalmatrixin 16

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whichtheelementsofvareonthediagonal.Re()andIm()representtherealandtheimaginarycomponentsofacomplex-valuedscalar,respectively.ABdenotestheKroneckerproductoftwomatricesAandB.Othermathematicalsymbolsaredenedaftertheirrstappearance. 17

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Figure1-1. Absorptioncoefcientversusfrequency.Copyrightimagecourtesyof[ 1 ]. 18

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A B Figure1-2. Underwaterchannelimpulseresponse(CIR)example.A)Anunderwateracousticchannelwiththreemultipaths.B)ApracticalCIRwith80channeltaps. 19

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A B Figure1-3. Scatteringfunctionsobtainedattwodifferentconditions.A)Arcticenvironmentwithfrozenseasurface.B)BahamaIslandsonawindyday.Copyrightimagecourtesyof[ 10 ]. 20

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Figure1-4. NormalizedCIRevolutionoverapproximatelyaone-miniteperiod. 21

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CHAPTER2ENHANCEDMOBILEMULTI-INPUTMULTI-OUTPUTUAC ThischapterfocusesoneffectivemobileMIMOUACoverdouble-selectiveacousticchannelssufferingfrombothISIandDopplerscalingeffects.Convertingthedouble-selectivechannelintoanISIchannelviatemporalresamplingisaneffectivewaytotacklemobileUACdifculties[ 11 ].AlthoughtheDopplerscalingeffectscanbelargelymitigatedviasuchatemporalresamplingprocess,theresidualDopplerstillcausesfrequencyshiftonthereceivedmeasurements.CoherentUACrequiresthereceivertoacquireknowledgeoftheunderlyingchannelaftertemporalresamplingviachannelestimation[ 2 ].Channelestimationcouldbeconductedeitherinthetraining-directedmode,usingknowntrainingsequences,orinthedecision-directedmode,usingthedetectedpayloadsymbols[ 12 15 ].ApreferabletooltocharacterizeachannelsubjecttobothISIandDopplerfrequencyshiftisthescatteringfunction(SF),whichessentiallydecouplestheacousticchannelintoabankofpathsthatexperiencedifferentdelaysandDopplerfrequencies[ 5 ].ThemajorconcerninSF-basedchannelestimationisthattheproblembecomesoverparameterizedwithtoomanydegreesoffreedom.Itispracticallymorebenecialtolookforachannelmodelwiththesmallestnumberofparameters,butonethatstillsufcientlyreectsthedeningcharacteristicsoftheacousticchannelofinterest.Alongthislineofthought,weassumethatateachreceiver,thechanneltapsforallthetransmittersexperiencethesameDopplerfrequency,butdifferentreceiversexperiencedifferentDopplershifts.Thenumberofunknownsinthefrequencydimension,asaconsequence,issignicantlyreduced.Accordingly,weproposethegeneralizationofthesparselearningviaiterativeminimization(GoSLIM)algorithmtoestimatetheCIRsandtheunderlyingDopplerfrequencyinajointmanner. TheaforementionedchannelmodelisfurthersimpliedbyassumingthatthechanneltapsforallthetransmitterandreceiverpairsexperiencethesameDopplerfrequency.Asaconsequence,theimpactoftheDopplerfrequencyshiftonthereceived 22

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measurementsacrossallthereceiversistakenintoaccountthroughoneunknowncommonfrequency.Accordingly,avariationofGoSLIM,referredtoasGoSLIM-V(Vstandsforvariation),isalsoproposedforchannelestimation.LikeGoSLIM,GoSLIM-VaddressessparsitythroughahierarchicalBayesianmodel,andbecauseGoSLIM-Visuserparameterfree,itiseasytouseinpracticalapplications.WewilldemonstrateviaexperimentalresultsthattheemploymentofGoSLIM-Vnotonlyreducestheoverallcomplexityinthechannelestimationstage,butalsoslightlyimprovesthedetectionperformancecomparedtoitsGoSLIMcounterpart. Followingthechannelestimationisthedesignofthedetectionschemeforextractingthetransmittedsignals.Thechannel-inducedphaseshiftshouldberstcompensatedoutusingtheDopplerfrequencyestimate[ 10 16 17 ].Suchphasecompensationtask,alongwiththeaforementionedtemporalresamplingprocess,effectivelyconvertsadouble-selectivechannelsubjecttobothDopplerscalingeffectsandISItoanISIchannel,whichallowsfortheemploymentofvariousequalizationtechniquesthatcaneffectivelycombatISI.Weusealinearminimummean-squarederror(LMMSE)basedlterforsymboldetection.InaMIMOsetup,ontopofISI,multiplesimultaneouslytransmittedsignalsactasinterferencestooneanother.Therefore,interferencecancellationschemealsoplaysacriticalroleintheoveralldetectionperformance.Aharddecisionbasedinterferencecancellationscheme,includingverticalBLAST(V-BLAST)[ 17 19 ]andRELAX-BLAST[ 12 ],subtractsouttheharddecisionsofdetectedsignalsfromthereceivedmeasurementstoaidthedetectionoftheremainingsignals.BycombiningV-BLASTwiththecyclicprincipleoftheRELAXalgorithm[ 20 ],RELAX-BLASTprovidessuperiordetectionperformanceoverV-BLASTatthecostofslightlyincreasedcomplexities[ 12 13 16 21 ]. Thedetectionperformancecanbefurtherenhancedbyemployingasoftinterferencecancellationscheme,includingTurboequalization[ 22 25 ].ForareceiveremployingTurboequalization,boththeequalizeranddecoderinvolvedareconguredassoft-input 23

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soft-output.Thedetectionperformanceimprovesasthesoftinformationcyclesbetweentheequalizeranddecoder.ThemaindrawbackoftheTurboequalizationschemeistheincreasedcomputationalcomplexitycomparedtoitsharddecisionbasedcounterparts.Toaddressthisproblem,weproposealowcomplexityapproximationofsoft-inputsoft-outputequalizer.WewillshowvianumericalandexperimentalexamplesthattheemploymentoftheproposedapproximateequalizerenjoysacomputationalcomplexitycomparabletoRELAX-BLASTandprovidesonlyslightlydegradeddetectionperformancecomparedtoadirectlyimplementedequalizer. Therestofthischapterisorganizedasfollows.Section 2.1 presentsasystemoutline.Section 2.2 describesamodelfortheacousticchannelsubjecttobothISIandDopplerscalingeffectsandreviewsthetemporalresamplingprocedure.Section 2.3 formulatesthechannelestimationprobleminbothtraining-directedanddecision-directedmodesandthenintroducesbothGoSLIMandGoSLIM-Vasthechannelestimationalgorithm.Section 2.4 rstformulatesthesymboldetectionproblem,andthendetailstheLMMSEbasedsoft-inputsoft-outputequalizeranditslowcomplexityapproximation.Section 2.5 presentsthesimulationresultsoftheTurboequalizationscheme,followedbytheexperimentalresultsobtainedfromanalyzingtheMACE10in-watermeasureddata. 2.1SystemOutline ConsideranNMmobileMIMOUACsystemequippedwithNtransmittransducersandMreceivehydrophones.Thetransmittedpayloadsequencesaredividedintomultipleblocks,eachofwhichisencodedseparately.Figure 2-1A demonstratestheconstructionofasinglepayloadsymbolblock(theconstructionofotherblocksfollowsthesameprocedure).Denotea(k)2f0,1gasthekthsourcebitfork=1,...,NK.fa(k)gNKk=1arerstfedintoa1=2rateconvolutionalencoderwithgeneratorpolynomials(10011)and(11011).Theencodedbitsfb(k)g2NKk=1arethenpassedtoarandominterleaver,followedbyaquadraturephase-shiftkeying(QPSK)modulationusingGray 24

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codemapping.InFigure 2-1 ,interleaveranddeinterleavermodulesarerepresentedbyand)]TJ /F12 7.97 Tf 6.59 0 Td[(1,respectively.Next,theso-obtainedQPSKpayloadsymbolsfx(k)gNKk=1aredemultiplexedintotheNpayloadblocks,eachconsistingofKsymbols,acrosstheNtransmittersinaround-robinfashion(thisiswheretheDEMUXmoduleinFigure 2-1A comesintoplay).Morespecically,inourdesign,x(n+(q)]TJ /F5 11.955 Tf 11.99 0 Td[(1)N)correspondstotheqthsymbolsentbythenthtransmitter,denotedasxn(q),forq=1,...,Kandn=1,...,N.Accordingly,wedenotecn(2q)]TJ /F5 11.955 Tf 12.18 0 Td[(1)andcn(2q)asthetwoconsecutiveinterleavedbitsinfc(k)g2NKk=1thatmaptoxn(q)accordingtotheformulagivenbelow: xn(q)=1 p 2j()]TJ /F5 11.955 Tf 9.3 0 Td[(1)cn(2q)]TJ /F12 7.97 Tf 6.59 0 Td[(1)+()]TJ /F5 11.955 Tf 9.3 0 Td[(1)cn(2q),q=1,...,K,n=1,...,N.(2) Sincec(k)2f0,1g,thesupportoffxn(k)gisa4-elementalphabetsetS=f(j1)=p 2g. ThestructureofareceiveremployingaTurboequalizationschemeisshowninFigure 2-1B .ThemeasurementsacquiredbytheMreceivehydrophonesarerstresampled,followedbychannelestimationandphasecompensation.Afterphasecompensation,thedouble-selectivechannelisconvertedtoanISIchannel,andtheTurboequalizationschemeisemployedhereintoretrievethetransmittedinformation.ThesuperiordetectionperformancepromisedbyTurboequalizationismainlyduetoitsmechanismofcyclingsoftinformationbetweentheequalizerandthedecoder[ 22 26 27 ].Accordingly,Turboequalizationconsistsoftwokeymodules,namelyasoft-inputsoft-outputequalizerandasoft-inputsoft-outputdecoder[ 28 29 ].Thesoftinformationofagenericbita2f0,1g,commonlyknownasthelog-likelihoodratio(LLR),isdenedas: L(a)=lnP(a=0) P(a=1),(2) whereP(a=0)2[0,1]representstheprobabilityofabeing0.AsshowninFigure 2-1B ,themultiplexedanddeinterleavedversionoffLe(cn(k))gN2Kn=1,k=1,theaposterioriextrinsicLLRgeneratedbytheequalizer,formstheaprioriinputsfLe(b(k))g2NKk=1to 25

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thedecoder.Conversely,theinterleavedanddemultiplexedversionoffLd(b(k))g2NKk=1,theaposterioriextrinsicLLRgeneratedbythedecoder,servesasaprioriinformationtotheequalizer.ThesubscripteordremindsusthattheLLRsaregeneratedbytheequalizerorthedecoder,respectively.Thesoftinformationiscycledbetweentheequalizerandthedecodermultipletimesbeforemakingharddecisionsonthesourcebits.Notethattheinterleaveranddeinterleaverinvolvedatthetransmitterandreceiverhavethesamestructure,whereastheDEMUXmoduleinsidethedashedrectangleinFigure 2-1B isdifferentfromthatinFigure 2-1A inthesensethattheformerandthelatterdemultiplex,respectively,thesoftinformationfLd(c(k))g2NKk=1andQPSKsymbolsfx(k)gNKk=1.Oncef~a(k)gNKk=1,theharddecisionsonthesourcebits,areavailable,wefollowthestepsinthesymbolgenerationprocessshowninFigure 2-1A :f~a(k)gNKk=1arefedintotheconvolutionalencoder,followedbyrandominterleaving,QPSKmapping,anddemultiplexing.Thisway,anerrorfreedecodingensuresaperfectrecoveryofthetransmittedQPSKsymbolsfxn(k)gNKn=1,k=1.Therecoveredpayloadsymbolswillbeusedinthedecision-directedchannelestimationstage;seeFigure 2-1B 2.2Double-SelectiveChannelwithDopplerScalingEffects Inthissection,westartwiththemodelingofthedouble-selectivechannelsufferingfrombothISIandDopplerscalingeffects.ThenwedescribethetemporalresamplingproceduretomitigatetheDopplerscalingeffects.Afterthat,weprovideapracticalapproachtoestimatetheDopplerscalingfactor. 2.2.1ChannelModel Byadoptingasingle-carriercommunicationscheme,atthenthtransmitter,thecontinuousbasebandsignalxn(t)(generatedbypassingthediscretepayloadsymbolsfxn(k)gtoapulseshapingler)anditscorrespondingfrequencymodulatedsignalxn(t)arerelatedthrough xn(t)=Refxn(t)ej2fctg,n=1,...,N,(2) 26

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wherefcrepresentsthecarrierfrequency.Forsimplicity,thepulseshapinglter,frequencymodulation,andrealcomponentextractionoperationarenotshowninFigure 2-1A Duetomultipatheffects,theactualtransmittedsignalsfxn(t)gNn=1canreachthereceivehydrophonesviadifferentpropagationpathswithdifferentdelays.Herein,theunderlyingacousticchannelbetweeneachtransmitterandreceiverpairischaracterizedbyRresolvedpaths.Therthpathbetweenthenthtransmitterandmthreceiverpair(r=1,...,R,n=1,...,N,andM=1,...,M)willaffectthetransmittedsignalxn(t)inthreeaspects,namelyamplitudeattenuation,Dopplerscaling,anddelay,whicharedenoted,respectively,bythreereal-valuedscalarsn,m(r),n,m(r),andn,m(r).Thesignaltransmittedviatherthpathandacquiredbythemthreceivercanbewrittenasn,m(r)xn(n,m(r)t)]TJ /F13 11.955 Tf 12.55 0 Td[(n,m(r)).BytakingintoaccountalloftheNtransducersandRresolvedpaths,thereceivedsignalatthemthhydrophonecanbeexpressedas(forsimplicity,thenoisetermisomittedforthetimebeing): zm(t)=NXn=1RXr=1n,m(r)xn(n,m(r)t)]TJ /F13 11.955 Tf 11.95 0 Td[(n,m(r)),m=1,...,M.(2) WeassumethatthepropagationpathsforallthetransmitterandreceiverpairsexperienceacommonDopplerscalingfactorandtheresolvedpathsaresynchronizedamongallthetransmitterandreceiverpairs,i.e.,n,m(r)=andn,m(r)=(r).(Interestedreadersarereferredto[ 30 ]foradetailedtreatmentofsynchronizationprocedure.)Byusingtheseassumptions,( 2 )reducesto: zm(t)=NXn=1RXr=1n,m(r)xn(t)]TJ /F13 11.955 Tf 11.96 0 Td[((r)),m=1,...,M.(2) Substituting( 2 )into( 2 )yields: zm(t)=Re(NXn=1RXr=1n,m(r)xn(t)]TJ /F13 11.955 Tf 11.96 0 Td[((r))ej2fc(t)]TJ /F18 7.97 Tf 6.59 0 Td[((r))).(2) 27

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2.2.2TemporalResampling Byresamplingthereceivedmeasurementsfzm(t)gusingafactor,theresampledsignalym(t)isgivenby[ 11 31 32 ]: ym(t)=zmt .(2) Then,thebasebandreceivedsignalym(t),whichisrelatedtoym(t)viaym(t)=Reym(t)ej2fct,canbeexpressedas: ym(t)=e)]TJ /F12 7.97 Tf 6.59 0 Td[(2j()]TJ /F19 5.978 Tf 5.76 0 Td[( )fctNXn=1RXr=1hn,m,rxn t)]TJ /F13 11.955 Tf 11.96 0 Td[((r),(2) wherehn,m,r,n,m(r)e)]TJ /F9 7.97 Tf 6.59 0 Td[(jfc(r)representstherthchanneltapbetweenthenthtransmitterandthemthreceiverpair,forr=1,...,R,n=1,...,N,andm=1,...,M.Itcanbereadilyveriedthataslongas=1,wehavexn t)]TJ /F13 11.955 Tf 11.96 0 Td[((r)xn(t)]TJ /F13 11.955 Tf 11.96 0 Td[((r)).Accordingly,( 2 )canbeapproximatedas: ym(t)e)]TJ /F12 7.97 Tf 6.58 0 Td[(2j()]TJ /F19 5.978 Tf 5.75 0 Td[( )fctNXn=1RXr=1hn,m,rxn(t)]TJ /F13 11.955 Tf 11.96 0 Td[((r)).(2) Oneobservesfrom( 2 )thateffectivetemporalresampling(meaning)convertstheDopplerscalingeffectstoDopplerfrequencyshiftswiththefrequencygivenbelow: f=)]TJ /F13 11.955 Tf 11.96 0 Td[( fc.(2) Therefore,thedeterminationoftheresamplingfactorplaysacrucialroleintheeffectivemitigationoftheDopplerscalingeffects. 2.2.3ResamplingFactorEstimation Wetakeadvantageofthepreambleandthepostambleofadatapackettoestimate[ 31 33 ](thestructureofadatapacketwillbediscussedinSection 2.5 ).Bycross-correlatingthereceivedsignalwiththeknownpreambleandpostamble,thereceiverestimatesthetimedurationofapacket^Trx[ 11 34 ].Bycomparing^TrxwithTtx,thedurationofthesamepacketatthetransmitterside,theDopplerscalingfactorcanbe 28

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estimatedas: ^=^Trx Ttx.(2) Althoughthismethodisconceptuallysimpleandeasytoimplement,itsaccuracyissensitivetothesignal-to-noise-ratio(SNR). MoreaccurateDopplerscalingfactorestimatecanbeachievedviachannelestimationinsteadofcross-correlation.BasedontheCIRsestimatedfromthetwomeasurementsegmentsinresponsetothepreambleandpostamble,thechangeinthetimeduration^TdimposedonthepacketcanbeinferredfromthetapshiftoftheprincipalarrivalsofthesetwoCIRs.ThentheDopplerscalingfactorestimatecanbecomputedas ^=Ttx+^Td Ttx.(2) The^obtainedusing( 2 )ismorerobustagainstthenoisecontaminationthantheonefrom( 2 ).WewillshowlateroninSection 2.5 viatheMACE10in-waterexperimentaldatathatthemethodin( 2 )workswellinpractice. 2.3ChannelEstimation Sincethe^obtainedusing( 2 )canneverbeperfectlyaccurate,aftertemporalresampling,Dopplerfrequencyshifts(see( 2 ))stillexist,althoughDopplerscalingeffectsbecomenegligible.Westartbelowwiththeproblemformulationofchannelestimationinbothtraining-directedanddecision-directedmodes[ 12 13 ].Then,weproposetheGoSLIM-ValgorithmforjointlyestimatingtheunderlyingCIRsandDopplerfrequency. Inwhatfollows,fym(t)gMm=1andfxn(t)gNn=1in( 2 )arerepresentedindiscrete-timeform.Unlessotherwisestated,itisassumedthatthechanneltapsforalltheNMtransmitter-receiverpairsexperiencethesameDopplerfrequencyf. 2.3.1Training-DirectedMode Theinitialtaskofthereceiveristoacquireknowledgeoftheunderlyingchannelbetweenalltransmitterandreceiverpairsusingthetrainingsequences.Byadoptingthe 29

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cyclicprexschemein[ 2 ],thetrainingsequenceatthenthtransmitter(n=1,...,N)isgivenby xn=[xn(P)]TJ /F4 11.955 Tf 11.95 0 Td[(LCP+1),...,xn(P)| {z }LCPprexsymbols,xn(1),xn(2),...,xn(P)| {z }Pcoretrainingsymbols],(2) where[xn(1),...,xn(P)]isthecoretrainingsequenceandtheleadingLCPsymbolsformthecyclicprex.Ingeneral,wehaveP>LCPR)]TJ /F5 11.955 Tf 11.97 0 Td[(1.Fromanamplierefciencypointofview,itispracticallydesirabletouseunitmodulus(unimodular)trainingsequences,i.e.,jxn(p)j=1forn=1,...,Nandp=1,...,P. ForMIMOUACoverempiricalacousticchannelssubjecttobothISIandDopplerfrequencyshifts,themeasurementvectorscanbewrittenas[ 5 35 ] ym=mNXn=1Xnhn,m+em,m=1,...,M,(2) where ym=[ym(1),...,ym(P)]T,(2) containsthePsynchronizedmeasuredsymbols(forinstance,fym(1)gmapstofxn(1)g).Xn2CPRisgivenby Xn=266666664xn(1)xn(P)...xn(P)]TJ /F26 10.909 Tf 10.91 0 Td[(R+2)xn(2)xn(1)...xn(P)]TJ /F26 10.909 Tf 10.91 0 Td[(R+3)............xn(P)xn(P)]TJ /F24 10.909 Tf 10.91 0 Td[(1)...xn(P)]TJ /F26 10.909 Tf 10.91 0 Td[(R+1)377777775,(2) wheren=1,...,N,andemrepresentsadditivenoise.Inaddition, hn,m=[hn,m,1,...,hn,m,R]T,(2) characterizesthechanneloflengthRbetweenthenthtransmitterandthemthreceiverforn=1,...,Nandm=1,...,M.Finally,theso-calledDopplershiftmatrixm2CPPin( 2 )hastheform: m=diagh1,e)]TJ /F12 7.97 Tf 6.58 0 Td[(2jfmTs,...,e)]TJ /F12 7.97 Tf 6.58 0 Td[(2jfmTs(P)]TJ /F12 7.97 Tf 6.59 0 Td[(1)i,(2) 30

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form=1,...,M,wherefmandTsrepresenttheDopplerfrequencyandsymbolperiod,respectively. TheISIandDopplershifteffectscanbeviewedseparatelyin( 2 ).Morespecically,thetermPNn=1Xnhn,mcharacterizesthecombinedcontributionsofNISIchannels,whereastheimpactoftheDopplereffectsonthemeasurementscomesthroughmonly,whichcorrespondstotheassumptionthatalltheNRCIRtapsinvolvedatthemthreceiver(recallthatwehaveNtransmittransducersandanR-tapchannelbetweeneachtransmitterandreceiverpair)experiencethesameDopplerfrequencyfm.Thepurposeofsettingtherstdiagonalelementofmto1istoeliminateambiguities.Inourexample,relativetoym(1),agenericmeasurement,sayym(p),experiencesaphaseshiftof)]TJ /F4 11.955 Tf 9.3 0 Td[(fmTs(p)]TJ /F5 11.955 Tf 11.96 0 Td[(1). Weexpress( 2 )inamorecompactform: ym=mXhm+em,(2) whereX=[X1,...,XN]andhm=[hT1,m,...,hTN,m]T.Thenthetraining-directedchannelestimationreducestoestimatinghmandfmfromthemeasurementvectorymandknownXform=1,...,M.Thesubjectofsynthesizingunimodulartrainingsequences,coupledwiththeemploymentofthecyclicprexscheme,tofacilitateISIchannelestimationistreatedin[ 13 ].TheshiftedPeCANwaveforms[ 36 ]areusedasthetrainingsequencesintheMACE10in-waterexperimentations. 2.3.2Decision-DirectedMode Thedecision-directedchannelestimationproblemisonlyaslighttwistofitstraining-directedcounterpart.Fortheformer,weusethepreviouslyestimatedpayloadsymbols,insteadofthetrainingsymbols,toestimatethechannels.Accordingly,( 2 )canstillbeused,where ym=[ym(ti),...,ym(tf)]T,m=1,...,M,(2) 31

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containsthemeasurementsatthemthreceiverbelongingtothetimeindexinterval[ti,tf],and Xn=266666664^xn(ti)^xn(ti)]TJ /F24 10.909 Tf 10.91 0 Td[(1)...^xn(ti)]TJ /F26 10.909 Tf 10.9 0 Td[(R+1)^xn(ti+1)^xn(ti)...^xn(ti)]TJ /F26 10.909 Tf 10.9 0 Td[(R+2)............^xn(tf)^xn(tf)]TJ /F24 10.909 Tf 10.91 0 Td[(1)...^xn(tf)]TJ /F26 10.909 Tf 10.91 0 Td[(R+1)377777775,(2) forn=1,...,N,where^xn(ti)]TJ /F4 11.955 Tf 12.4 0 Td[(R+1)and^xn(tf)representtherstandlastpreviouslyestimatedsymbols(someofthemcouldbetheknowntrainingsymbols),respectively,usedforupdatingthechannel.(Fornotationalsimplicity,Xnisusedinboth( 2 )and( 2 )torepresenttwosimilarbutdifferentquantities.TheuseofXn,however,shouldbeclearfromthecontext.)ThetrackinglengthisrepresentedasLTR=tf)]TJ /F4 11.955 Tf 9.89 0 Td[(ti+1,i.e.,thenumberofrowsofXn.Toconformwiththematrixdimensions,theDopplershiftmatrixmnowhasdimensionLTR,constructedasm=diag\0021,e)]TJ /F12 7.97 Tf 6.58 0 Td[(2jfmTs,...,e)]TJ /F12 7.97 Tf 6.59 0 Td[(2jfmTs(LTR)]TJ /F12 7.97 Tf 6.58 0 Td[(1)form=1,...,M.Similarlytothetraining-directedmode,thechannelestimationprobleminthedecision-directedmodeaimstoestimatehmandfmfromthemeasurementvectorymandknownXformedfromthedecision-directedfXngNn=1in( 2 ),form=1,...,M. 2.3.3ChannelEstimationAlgorithm:GoSLIM Thechannelestimationalgorithmateachreceiver,ineithertraining-ordecision-directedmode,hasthegenericformgivenby(see( 2 )) ym=mXhm+em,m=1,...,M.(2) Thenoisevectoremin( 2 )isassumedtocontaincircularlysymmetricindependentandidenticallydistributedcomplex-valuedGaussianrandomvariableswithzeromeanandvariancem,denotedasemCN(0,mI).TheproblemisthentoestimatefmandhmgivenymandX.Channelestimationcanbeperformedateachreceiverinparallel.InUACsystems,thechannelhmisusuallysparse,i.e.,althoughitcontainsNRunknowns,manyofthemcanbeapproximatedaszero[ 37 ].WeusetheGoSLIMalgorithmtosolve 32

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thissparsechannelestimationproblem.NotethatsincehmcontainstheCIRsforallNtransmitters,GoSLIMwillestimatethemsimultaneously. ConsiderthefollowinghierarchicalBayesianmodel: ymjhm,m,mCN(mXhm,mI),(2) hmjpmCN(0,Pm),(2) where( 2 )followsdirectlyfromtheassumptionemCN(0,mI).Letpn,m,rbethevarianceofhn,m,rforn=1,...,N,m=1,...,M,andr=1,...,R,anddenepn,m=[pn,m,1,pn,m,2,...,pn,m,R]Tandpm=[pT1,m,pT2,m,...,pTN,m]T.ThenthecovariancematrixPmin( 2 )isconstructedasPm=diag(pm). Furthermore,byconsideringaatprioronfm,m,andfpn,m,rgNRn=1,r=1,thechannelvectorhm,Dopplerfrequencyfm,thecovariancematrixPm(ormoreprecisely,itsdiagonalelementspm),andthenoisepowermcanbeestimatedbasedonthemaximumaposterioricriterion( 2 ): maxhm,pm,m,fmp(hm,pm,m,fmjym)=maxhm,pm,m,fmp(ymjhm,m,fm)p(hmjpm)(2) Bycombining( 2 ),( 2 ),and( 2 ),andbytakingthenegativelogarithmofthecostfunction,theoptimizationproblemformulatedin( 2 )becomes( 2 ) minhm,pm,m,fm dylogm+ym)]TJ /F10 11.955 Tf 11.96 0 Td[(mXhm2 m+NXn=1RXr=1logpn,m,r+NXn=1RXr=1jhn,m,rj2 pn,m,r!,(2) whichcanbesolvedusingacyclicoptimizationapproach:ateachiteration,oneoftheparametervectorshm,pm,m,andfmisupdatedwhilekeepingtheotherthreexed.Inthisway,asingledifcultjointoptimizationproblemisdividedintofoursimplerseparatesubproblems.GoSLIMkeepsiteratinguntilapredenediterationnumberisreachedoraconvergencecriterionismet.Undermildconditions,thecyclicoptimizationschemeguaranteesthattheGoSLIMalgorithmconverges,atleasttoalocalminimumof( 2 )[ 38 ]. ThevestepsoftheGoSLIMalgorithmatthetthiterationareoutlinedbelow: 33

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1. Givenh(t)]TJ /F12 7.97 Tf 6.58 0 Td[(1)m,theoptimalP(t)mthatminimizesthecostfunctionin( 2 )isgivenby: p(t)n,m,r=h(t)]TJ /F12 7.97 Tf 6.59 0 Td[(1)n,m,r2,n=1,...,N,r=1,...,R.(2) Forbetternumericalstability,wesetp(t)n,m,r(orequivalentlyh(t)n,m,r)tozeroifp(t)n,m,r<10)]TJ /F12 7.97 Tf 6.58 0 Td[(15. 2. OnceP(t)misavailable,h(t)misobtainedas: h(t)m=XHX+(t)]TJ /F12 7.97 Tf 6.59 0 Td[(1)mP(t)m)]TJ /F12 7.97 Tf 6.59 0 Td[(1)]TJ /F12 7.97 Tf 6.59 0 Td[(1(t)]TJ /F12 7.97 Tf 6.58 0 Td[(1)mXHym.(2) WhileinvertingP(t)m,itszerodiagonalentriesareremoved,andthecorrespondingcolumnsinXarediscarded. 3. Next,usingthemostrecentlyobtainedh(t)min( 2 ),weestimatetheDopplerfrequencyfm.Foreaseofexposition,wedenotez(t)m(i)=ym(i)~x(t)m(i),whereym(i)and~x(t)m(i)represent,respectively,theithelementofthemeasurementvectorymand~x(t)mwith~x(t)m=Xh(t)mfori=1,...,dy.Itiseasytoverifythat ym)]TJ /F23 10.909 Tf 10.91 0 Td[(mXh(t)m2=const)]TJ /F24 10.909 Tf 10.91 0 Td[(2Re0@dyXi=1z(t)m(i)e)]TJ /F12 7.97 Tf 6.58 0 Td[(2jfmTs(i)]TJ /F12 7.97 Tf 6.59 0 Td[(1)1A.(2) Sincetheconstanttermin( 2 )isnotafunctionoffm,minimizingthecostfunctionin( 2 )isequivalenttosolving f(t)m=argmaxfmRe0@dyXi=1z(t)m(i)e)]TJ /F12 7.97 Tf 6.59 0 Td[(2jfmTs(i)]TJ /F12 7.97 Tf 6.59 0 Td[(1)1A.(2) Sincethesummationtermwithintheparenthesisaboveisthediscrete-timeFouriertransform(DTFT)ofthesequencefz(t)m(i)gdyi=1evaluatedatfrequencyfm,f(t)misobtainedasthelocationofthedominantpeakoftherealpartoftheDTFT. 4. Usingh(t)mand(t)mmostrecentlyobtainedin( 2 )and( 2 ),respectively,wenallyestimatethenoisepoweras: (t)m=1 dyym)]TJ /F10 11.955 Tf 11.95 0 Td[((t)mXh(t)m2.(2) 5. Sett=t+1.GobacktoStep1iftislessthanapredenediterationnumber,orterminateotherwise. Inthetraining-directedmode,thechannelcharacteristicsingeneralarenotavailableapriori.Inourexamples,h(0)misinitializedusingthestandardmatchedlter,f(0)misinitializedas0andthenoisepower(0)misinitializedwithasmallpositivenumber, 34

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forinstance,10)]TJ /F12 7.97 Tf 6.58 0 Td[(10.OurempiricalexperiencesuggeststhattheGoSLIMalgorithmdoesnotprovidesignicantperformanceimprovementsafternomorethan15iterations. 2.3.4ChannelEstimationAlgorithm:GoSLIM-V Thechannelestimationproblemformulatedin( 2 ),aspreviouslymentioned,correspondstoanassumptionthattheNRchanneltapsseenbyeachreceiverexperiencethesameDopplerfrequency,butthefrequencyvaluecouldvaryatdifferentreceivehydrophones.WeconsiderhereinafurthersimpliedassumptionthattheDopplerfrequencyisthesameacrossMreceivers,i.e.,f=f1==fM.(Thepracticalvalidityofthisassumptionwillbeveriedbyanalyzingin-waterexperimentaldatainSection 2.5.2 .)Accordingly,replacingfmin( 2 )byfyieldsaDopplershiftmatrixthatisindependentofthereceiverindexm: =diagh1,e)]TJ /F12 7.97 Tf 6.59 0 Td[(2jfTs,...,e)]TJ /F12 7.97 Tf 6.59 0 Td[(2jfTs(P)]TJ /F12 7.97 Tf 6.59 0 Td[(1)i.(2) Combining( 2 )with( 2 )givesym=Xhm+em.Thenstackingthemeasurementsfromallthereceiversfollows y=Xh+e,(2) wherey=[yT1,yT2,...,yTM]T,h=[hT1,hT2,...,hTM]T,e=[eT1,eT2,...,eTM]T,=IMM,andX=IMMX.ThentheGoSLIM-ValgorithmaimstoestimatefandhgivenyandX.NotethatunlikeGoSLIM,whichcanbeemployedateachreceivertoestimatethechannelinparallel,GoSLIM-Vimplicitlysuggeststhatthemeasurementsacquiredatdifferentreceiversshouldbeassembledinacentralprocessorbeforeperformingchannelestimation.Moreover,GoSLIM-VsimultaneouslyestimatestheCIRsamongalloftheMNtransmitterandreceiverpairs. GoSLIM-VisdevelopedbasedonthefollowinghierarchicalBayesianmodel,similarlyto( 2 )and( 2 ): yjh,,CN(Xh,I),(2) hjpCN(0,P),(2) 35

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wherep=[pT1,pT2,...,pTM]TandP=diag(p).ThevestepsofGoSLIM-Vatthetthiterationarebrieylistedbelow: 1. Givenh(t)]TJ /F12 7.97 Tf 6.58 0 Td[(1),theoptimalP(t)isgivenby: p(t)n,m,r=h(t)]TJ /F12 7.97 Tf 6.59 0 Td[(1)n,m,r2,(2) forn=1,...,N,m=1,...,M,andr=1,...,R. 2. OnceP(t)isavailable,theCIRisupdatedas: h(t)=XHX+(t)]TJ /F12 7.97 Tf 6.59 0 Td[(1)P(t))]TJ /F12 7.97 Tf 6.59 0 Td[(1)]TJ /F12 7.97 Tf 6.59 0 Td[(1(t)]TJ /F12 7.97 Tf 6.58 0 Td[(1)XHy.(2) 3. TheDopplerfrequencyf(t)isupdatedas: f(t)=argmaxfRe24dyXi=1 MXm=1z(t)m(i)!e)]TJ /F12 7.97 Tf 6.58 0 Td[(2jfTs(i)]TJ /F12 7.97 Tf 6.59 0 Td[(1)35.(2) 4. Thenoisepowerisestimatedas: (t)=1 dyMy)]TJ /F23 10.909 Tf 10.91 0 Td[((t)Xh(t)2.(2) 5. Sett=t+1.GobacktoStep1iftislessthanapredenediterationnumber,orterminateotherwise. TheinitializationofGoSLIM-VissimilartothatofGoSLIM.Inourexamples,h(0)isinitializedusingthestandardmatchedlter,f(0)isinitializedas0andthenoisepower(0)isinitializedwithasmallpositivenumber,forinstance,10)]TJ /F12 7.97 Tf 6.59 0 Td[(10.OurempiricalexperiencesuggeststhattheGoSLIM-Valgorithmdoesnotprovidesignicantperformanceimprovementsafternomorethan15iterations. 2.3.5ComplexityAnalysis EmpiricalexperienceindicatesthatthecomputationalbottleneckofGoSLIMisintheupdateoftheDopplerfrequency;see( 2 ).Previouslyin[ 21 ],ateachGoSLIMiteration,wezeropadthesequencefz(t)m(i)gdyi=1toalengthof220,followedbytheemploymentoffastFouriertransform(FFT)toobtainthefrequencyspectrum.Then,f(t)misobtainedasthelocationofthedominantpeakoftherealpartoftheso-obtainedspectrum.Thisprocedurehasapproximately224oatingpointoperations 36

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(ops).Alternatively,wecanalsoestablishafrequencygridofdggridpointsandcalculatetheDTFToffz(t)m(i)gdyi=1evaluatedateachfrequencygridpointusing( 2 )directly.Inthisway,thecomplexityreducestoO(dydg)perGoSLIMiteration.Inourexample,dyisequalto512inthetraining-directedmode.Thegridisfrom)]TJ /F5 11.955 Tf 9.3 0 Td[(5Hzto5Hzwithastepsizeof0.001Hz,andtherefore,dg=10001.Thissetupcorrespondstoapproximately222ops,whichisfourtimeslowerthanthatofusingFFT.Therefore,theDTFTgridsearchispreferableovertheFFTtechnique,anditisemployedtoanalyzethein-watermeasurements.AlthoughDTFTismoreefcientthanFFT,itstillconstitutesthecomputationalbottleneckoftheGoSLIMalgorithm. SinceGoSLIMisemployedateachreceivertoconductchannelestimationinparallel,theoverallcomplexityoftheDopplerfrequencyupdatestepisO(dydgM)perGoSLIMiteration.Incontrast,thecomplexityoftheDopplerfrequencyupdatestepin( 2 ),bymakinguseofDTFT,isO(dydg)perGoSLIM-Viteration.TheMtimescomplexityreductionismainlyduetothefactthatthefrequencysearchisperformedateachreceiverforGoSLIM,butitisperformedonlyonceforGoSLIM-V.Asaconsequence,GoSLIM-ViscomputationallymuchmoreefcientthanGoSLIMinaMIMOconguration,especiallywithalargenumberofreceiveelements.Duetothereducednumberofunknowns,theGoSLIM-VdatamodelismoreparsimoniousthanthatofGoSLIM,whichcouldenhancethesymboldetectionperformanceifthemodelisreasonablyaccurate.WewillshowviaMACE10in-waterexperimentationresultsthatGoSLIM-VslightlyoutperformsGoSLIM.Hence,GoSLIM-VispreferredandthesubsequentsymboldetectionsectionisbuiltuponemployingGoSLIM-Vinthechannelestimationstage. 2.4SymbolDetection Inthissection,weproceedtostudythedetectionofthepayloadsymbolsgiventheestimatesofCIRsandDopplerfrequencyfobtainedbyGoSLIM-V.Thedetectiontaskisachievedviatwosteps:phasecompensationfollowedbyTurboequalization.As 37

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showninFigure 2-1B ,Turboequalizationconsistsofanequalizerandadecoder,bothconguratedassoft-inputsoft-output.ThedecoderisconventionallyimplementedbytheMax-Log-MAPalgorithm[ 28 ],andourfocushereinisonthesoft-inputsoft-outputequalizer.Werstformulatethesymboldetectionproblemandthendescribethephasecompensationprocedure.Afterthat,weelaboratetheLMMSEbasedTurboequalizationdesignanddiscussitslowcomplexityapproximation. 2.4.1Problemformulation TreatingthetransmittedsymbolsastheunknownsandtheCIRsandDopplerfrequencyasknown,themeasurementvectorin( 2 )canbeexpressedas[ 5 35 ]: ym(k)=^(k)NXn=1^Hn,mxn(k)+em,m=1,...,M,(2) wheretheestimatedCIRmatrix^Hn,m2CR(2R)]TJ /F12 7.97 Tf 6.59 0 Td[(1)isgivenby ^Hn,m=266664^hn,m,R...^hn,m,10............0^hn,m,R...^hn,m,1377775,(2) forn=1,...,Nandm=1,...,M.Theentry^hn,m,rhererepresentstheestimateofhn,m,rin( 2 )givenbyGoSLIM-Vattheconclusionoftheiteration.Also, xn(k)=[xn(k)]TJ /F4 11.955 Tf 11.95 0 Td[(R+1),...,xn(k),...,xn(k+R)]TJ /F5 11.955 Tf 11.96 0 Td[(1)]T,n=1,...,N,(2) and ym(k)=[ym(k),...,ym(k+R)]TJ /F5 11.955 Tf 11.95 0 Td[(1)]T,m=1,...,M.(2) Thevariablekrepresentsthetimeindexcorrespondingtothepayloadsymbolsofcurrentinterest.Althoughymrepresentsdifferentportionsofthereceivedsignalin( 2 ),( 2 ),and( 2 ),itsuseshouldbeclearfromthecontext.Perthediscussionsfollowing( 2 ),once^fisavailable,theestimatedDopplershiftmatrix^(k)in( 2 ) 38

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canbeconstructedas: ^(k)=diaghe)]TJ /F12 7.97 Tf 6.58 0 Td[(2j^fTs(k)]TJ /F12 7.97 Tf 6.59 0 Td[(1),...,e)]TJ /F12 7.97 Tf 6.58 0 Td[(2j^fTs(k+R)]TJ /F12 7.97 Tf 6.58 0 Td[(2)i.(2) Whendetectingsymbols,weusetheestimatesf^hn,mgand^fobtainedfromthepreviouschannelupdateandwetreatf^Hn,mgand^(k)in( 2 )asknown. 2.4.2PhaseCompensation Stackingupallthemeasurements,( 2 )canbewrittenas 266664y1(k)...yM(k)377775=(k)NXn=1266664^Hn,1...^Hn,M377775xn(k)+266664e1...eM377775,(2) or,equivalentlyas, y(k)=(k)NXn=1^Hnxn(k)+e=(k)^Hx(k)+e, (2) wherey(k)ande2CMR1,(k)=IMM^(k),f^HngNn=12CMR(2R)]TJ /F12 7.97 Tf 6.58 0 Td[(1),^H=[^H1,...,^HN],andx(k)=[xT1(k),...,xTN(k)]T.Thephasecompensationtaskissimplyachievedbymultiplying[(k)]Htobothsidesof( 2 ),yielding y(k)=^Hx(k)+e,(2) wherey(k)=[(k)]Hy(k)ande=[(k)]He.GiveneCN(0,I),estillhasthedistributionofCN(0,I)since[(k)]Hisunitary. Phasecompensation,alongwiththeaforementionedtemporalresamplingprocess,effectivelyconvertstheoriginaldouble-selectivechanneltoanISIchannel.Giventhephase-compensatedmeasurementvectory(k),theestimatedCIRmatrix^H,andfLd(cn(k))gN2Kn=1,k=1,weconsiderusinganLMMSEbasedsoft-inputsoft-outputequalizertocomputetheaposterioriextrinsicinformationofthetransmittedsignal. 39

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2.4.3LMMSEBasedSoft-InputSoft-OutputEqualizer AsshowninFigure 2-2 ,anLMMSEbasedsoft-inputsoft-outputequalizercanbefunctionallydividedintofourmodules.TheaprioriLLRpre-processorcalculatesthemeanandthevarianceofeachQPSKpayloadsymbolxn(k),denotedasxn(k)andvn(k),respectively,fromtheaposterioriextrinsicinformationLd(cn(2k)]TJ /F5 11.955 Tf 12.58 0 Td[(1))andLd(cn(2k))generatedbythedecoderforn=1,...,Nandk=1,...,K.Next,thetransmittedsymbolxn(k)isestimatedviaLMMSElteringgiveny(k)and^Hin( 2 ),alongwithfxn(k)gandfvn(k)g.Specically,asdemonstratedinFigure 2-2 ,theLMMSElterisappliedtotheresidualsignalgeneratedbysubtractingouttheso-calledsoftinterferencesfromthephase-compensatedmeasurements.Thesoftinterferencescharacterizethecontributionsofallthepayloadsymbolsexceptxn(k),theoneofthecurrentinterest,intermsofsoftinformation.Basedonthesymbolestimates^xn(k),theaposterioriLLRgeneratorprovidestheextrinsicLLRoutputsLe(cn(2k)]TJ /F5 11.955 Tf 12.58 0 Td[(1))andLe(cn(2k))(n=1,...,N,k=1,...,K),whichwillbefedintothesoft-inputsoft-outputdecoderasaprioriLLR,seeFigure 2-1B .Inthefollowing,thesemoduleswillbeelaboratedfurther. 2.4.3.1AprioriLLRpre-processor Inthistask,wecalculatexn(k)andvn(k)fromLd(cn(2k)]TJ /F5 11.955 Tf 12.98 0 Td[(1))andLd(cn(2k)).Accordingtothedenitions,themeanandvarianceofxn(k)aregivenby[ 22 ]: xn(k)=4Xi=1iP(xn(k)=i), (2) vn(k)=4Xi=1jij2P(xn(k)=i))-221(jxn(k)j2=1)-222(jxn(k)j2, (2) wherefig4i=1denotethefourQPSKconstellationpointsofSandjij=1fori=1,...,4;seethedenitionofSafter( 2 ).Onecanseefrom( 2 )thatvn(k)dependsonxn(k),andtheevaluationofxn(k)in( 2 )requiresP(xn(k)=i)fori=1,...,4.Sincetheinterleavedbitsfc(k)gcanbereasonablyassumedtobeindependentofeachother 40

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duetotheemploymentoftherandominterleaver(seeFigure 2-1A ),P(xn(k)=i)isdeterminedastheproductoftheprobabilitiesofthetwointerleavedbitsthatmaptoi.Forinstance,cn(2k)]TJ /F5 11.955 Tf 12.25 0 Td[(1)=0andcn(2k)=1maptoxn(k)=()]TJ /F5 11.955 Tf 9.3 0 Td[(1+j)=p 2accordingto( 2 ),andtherefore,P(xn(k)=()]TJ /F5 11.955 Tf 9.3 0 Td[(1+j)=p 2)=P(cn(2k)]TJ /F5 11.955 Tf 11.06 0 Td[(1)=0)P(cn(2k)=1),whereforagenericinterleavedbitcn(q)wehave P(cn(q)=0)=eLd(cn(q)) 1+eLd(cn(q)),P(cn(q)=1)=1 1+eLd(cn(q)),(2) forn=1,...,Nandq=1,...,2K.Equation( 2 )followsfrom( 2 )andP(cn(q)=0)+P(cn(q)=1)=1. Plugging( 2 )into( 2 )gives: xn(k)=1 p 2jtanhLd(cn(2k)]TJ /F5 11.955 Tf 11.96 0 Td[(1)) 2+tanhLd(cn(2k)) 2,n=1,...,N,k=1,...,K,(2) which,combinedwith( 2 ),yieldsvn(k). 2.4.3.2LMMSEltering DependingonwhethertheaprioriLLRinformationisincorporatedornot,twotypesofLMMSEltersarestudiedinthefollowing. Intheabsenceofaprioriknowledge .Theequalizerisperformedintheabsenceofaprioriknowledgeattheveryrstiterationbeforeusingthedecoder.ThisscenarioamountstosettingLd(cn(k))=0forn=1,...,Nandk=1,...,2K,whichimpliesthatxn(k)=0andvn(k)=1accordingto( 2 )and( 2 )forn=1,...,Nandk=1,...,K.Inthiscase,theLMMSEltercoefcientvector,denotedasfn,isgivenby[ 12 39 ]: fn=^H^HH+^I)]TJ /F12 7.97 Tf 6.59 0 Td[(1sn.(2) Here,^representsthenoisepowerestimategivenbyGoSLIM-Vattheconclusionoftheiteration,sn=[^hTn,1,...,^hTn,M]Tdenotesthesteeringvectorcorrespondingtoxn(k)in 41

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( 2 )forn=1,...,N,and^hn,mistheestimateofhn,mdenedin( 2 ).Anestimateofxn(k)isobtainedbyapplyingfntothephase-compensatedmeasurementvectorobtainedin( 2 ): ^xn(k)=fHny(k),n=1,...,N.(2) Inthepresenceofaprioriknowledge .Inthiscase,theLMMSEestimateofxn(k)isgivenby[ 22 ]: ^xn(k)=xn(k)+vn(k)f0n(k)Hhy(k))]TJ /F5 11.955 Tf 13.57 2.66 Td[(^HE(x(k))i,(2) where f0n(k)=h^HV(k)^HH+^Ii)]TJ /F12 7.97 Tf 6.58 0 Td[(1sn(2) representstheLMMSEltercoefcientvector.In( 2 ),eachcomponentofE(x(k))istheexpectedvalueofthecorrespondingcomponentofx(k)calculatedin( 2 ).In( 2 ),thecovariancematrixV(k)=diag\002v1(k)T,...,vN(k)T,and vn(k)=[vn(k)]TJ /F4 11.955 Tf 11.96 0 Td[(R+1),...,vn(k),...,vn(k+R)]TJ /F5 11.955 Tf 11.95 0 Td[(1)]T,n=1,...,N.(2) Eachcomponentofvn(k)isobtainedaccordingto( 2 ). Equation( 2 )suggeststhattheestimationofxn(k)dependsonitsownextrinsicLLRinformationLd(cn(2k)]TJ /F5 11.955 Tf 12.27 0 Td[(1))andLd(cn(2k)),whoseimpacton^xn(k)comesthroughxn(k)andvn(k).Fromthebeliefpropagationtheorypointofview,thegenerationofextrinsicinformationofapayloadsymbolneedstoavoidsuchdependency[ 40 ].Toachievethisgoal,wemodify( 2 )as: ^xn(k)=sHnh^HV(k)^HH+^I+(1)]TJ /F4 11.955 Tf 11.96 0 Td[(vn(k))snsHni)]TJ /F12 7.97 Tf 6.58 0 Td[(1hy(k))]TJ /F5 11.955 Tf 13.57 2.66 Td[(^HE(x(k))+xn(k)sni.(2) Comparedto( 2 ),thepresenceofthetwoadditionaltermsin( 2 ),namely(1)]TJ /F4 11.955 Tf 12.88 0 Td[(vn(k))snsHnandxn(k)sn,resemblesascenarioofxn(k)=0andvn(k)=1(orequivalently,Ld(cn(2k)]TJ /F5 11.955 Tf 12.11 0 Td[(1))=Ld(cn(2k))=0)in( 2 ),asifxn(k)isestimatedwithoutincorporatingitsownLLRinformation. 42

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Dene f00n(k)=h^HV(k)^HH+^I+(1)]TJ /F4 11.955 Tf 11.96 0 Td[(vn(k))snsHni)]TJ /F12 7.97 Tf 6.59 0 Td[(1sn,(2) and _yn(k)=y(k))]TJ /F14 11.955 Tf 11.96 13.27 Td[(h^HE(x(k)))]TJ /F5 11.955 Tf 12.14 0 Td[(xn(k)sni.(2) Then,( 2 )canberewrittenas: ^xn(k)=f00n(k)H_yn(k).(2) In( 2 ),thetermswithinthesquarebracketscorrespondtotheoutputofthesoftinterferencegeneratorinFigure 2-2 .Toget^xn(k),theLMMSEltercoefcientvectorin( 2 )isappliedtotheresidualmeasurementvector_yn(k).Notethat( 2 )includes( 2 )asaspecialcasewhennoaprioriknowledgeisavailable,i.e.,Ld(cn(k))=0forn=1,...,Nandk=1,...,2K. 2.4.3.3AposterioriLLRgenerator ThistaskcalculatestheextrinsicLLRLe(cn(2k)]TJ /F5 11.955 Tf 11.35 0 Td[(1))andLe(cn(2k))fromthesymbolestimates^xn(k)obtainedin( 2 )or( 2 )forn=1,...,Nandk=1,...,K. Weassumethatgivenxn(k)=i,^xn(k)isacircularlysymmetrici.i.d.complex-valuedGaussianrandomprocess,i.e.,P(^xn(k)jxn(k)=i)CN(i,2),wherethemeaniandvariance2arecalculated,respectively,asi=if00n(k)Hsnand2=f00n(k)Hsn)]TJ /F10 11.955 Tf 12.73 0 Td[(f00n(k)Hsnsnf00n(k)[ 22 ].Underthisassumption,theoutputLLRofthetwoconsecutivebitsmappingtoxn(k)iscalculatedas[ 22 ]: Le(cn(2k)]TJ /F5 11.955 Tf 11.96 0 Td[(1))=p 8Im(f00n(k)H_yn(k)) 1)]TJ /F10 11.955 Tf 11.95 0 Td[(sHnf00n(k),Le(cn(2k))=p 8Re(f00n(k)H_yn(k)) 1)]TJ /F10 11.955 Tf 11.95 0 Td[(sHnf00n(k),(2) forn=1,...,Nandk=1,...,K. LetR0(k)=^HV(k)^HH+^IandR00n(k)=^HV(k)^HH+^I+(1)]TJ /F4 11.955 Tf 10.07 0 Td[(vn(k))snsHn.Thenf0n(k)in( 2 )andf00n(k)in( 2 )canberewrittenasf0n(k)=R0(k))]TJ /F12 7.97 Tf 6.58 0 Td[(1snandf00n(k)=R00n(k))]TJ /F12 7.97 Tf 6.58 0 Td[(1sn, 43

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respectively.Oneobservesthatthederivationofff00n(k)grequirestheinversionofR00n(k)foreachtransmitterateachtimeindex,whereasthecomputationofff0n(k)gneedstoinvertR0(k)ateachtimeindex.Consequently,byfollowing( 2 )and( 2 )directly,thecomputationalcomplexityofcalculatingff00n(k)gisapproximatelyNtimesmoreexpensivethanobtainingff0n(k)g. SinceR00n(k)=R0(k)+(1)]TJ /F4 11.955 Tf 11.95 0 Td[(vn(k))snsHn,theuseofthematrixinversionlemmagives: R00n(k))]TJ /F12 7.97 Tf 6.59 0 Td[(1=R0(k))]TJ /F12 7.97 Tf 6.58 0 Td[(1)]TJ /F5 11.955 Tf 13.15 8.09 Td[((1)]TJ /F4 11.955 Tf 11.95 0 Td[(vn(k))R0(k))]TJ /F12 7.97 Tf 6.59 0 Td[(1snsHnR0(k))]TJ /F12 7.97 Tf 6.59 0 Td[(1 1+(1)]TJ /F4 11.955 Tf 11.95 0 Td[(vn(k))sHnR0(k))]TJ /F12 7.97 Tf 6.59 0 Td[(1sn.(2) Rightmultiplyingsnonbothsidesof( 2 )yields: f00n(k)=f0n(k) 1+(1)]TJ /F4 11.955 Tf 11.96 0 Td[(vn(k))sHnf0n(k),(2) which,combinedwith( 2 ),follows Le(cn(2k)]TJ /F5 11.955 Tf 11.96 0 Td[(1))=p 8Im(f0n(k)_yn(k)) 1)]TJ /F4 11.955 Tf 11.96 0 Td[(vn(k)sHnf0n(k),Le(cn(2k))=p 8Re(f0n(k)_yn(k)) 1)]TJ /F4 11.955 Tf 11.96 0 Td[(vn(k)sHnf0n(k).(2) Complexity-wise,theLLRcalculationformulain( 2 )ispreferableover( 2 )sinceaswejustremarked,itismoreefcienttocalculateff0n(k)gthanff00n(k)g.Duetothisreason,LLRiscalculatedaccordingto( 2 )inournumericalandexperimentalexamplesprovidedlateron. 2.4.4Low-ComplexityApproximateLMMSEFiltering AlthoughthecalculationofaposterioriLLRaccordingto( 2 )ismoreefcientthan( 2 ),itstillconstitutesthemajorcomputationalbottleneckinTurboequalizationmainlybecauseff0n(k)gneedstobecalculatedateachtimeindex.Tofurtherreducethecomputationalcomplexity,weconsideralow-complexityapproximateLMMSElterwhosecoefcientvectorisgivenby: f0n=^HV^HH+^I)]TJ /F12 7.97 Tf 6.58 0 Td[(1sn,(2) 44

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whereV=1 KPKk=1V(k).Sinceff0ngisconstantforeachtransmitteroveronepayloadblock(hencethetimeindexkisdroppedin( 2 )),theoverallcomplexityofcalculatingff0ngisapproximatelyKtimesfasterthanderivingff0n(k)gaccordingto( 2 ).Substitutingf0n(k)in( 2 )withf0nyields: Le(cn(2k)]TJ /F5 11.955 Tf 11.95 0 Td[(1))=p 8Im(f0n_yn(k)) 1)]TJ /F4 11.955 Tf 11.95 0 Td[(vn(k)sHnf0n,Le(cn(2k))=p 8Re(f0n_yn(k)) 1)]TJ /F4 11.955 Tf 11.95 0 Td[(vn(k)sHnf0n.(2) WehereafterrefertotheTurboequalizationschemethatcalculatestheaposterioriextrinsicinformationaccordingto( 2 )and( 2 )asExact-LMMSE-TurboandApproximate-LMMSE-Turbo,respectively. NotethatmatrixinversionisanindispensablestageincalculatingtheLMMSEltercoefcientsin( 2 ),( 2 ),and( 2 ).Toexpeditethecalculation,wecanmakeuseoftheconjugategradient(CG)methodandfastFouriertransform(FFT)operations,aselaboratedin[ 41 ].Although[ 41 ]focusesontheefcientcalculationoftheLMMSEltercoefcientsintheformof( 2 ),theextensiontoamoregeneralscenarioin( 2 )or( 2 )isstraightforward.Inthework,bothExact-LMMSE-TurboandApproximate-LMMSE-TurboareimplementedusingtheFFT-basedCGmethod. 2.5NumericalandExperimentalResults 2.5.1NumericalResults Considertransmittingfourpayloadblockssimultaneouslyovertime-invariantISIchannelsusingaMIMOUACsystemequippedwithN=4transmittersandM=12receivers.BlocklengthisxedatK=250.ThefourpayloadblocksacrosstheN=4transmittersareconstructedfromarandomlygeneratedbinarysourcesequenceoflengthNK=1000accordingtotheproceduredetailedinSection 2.1 .WesimulateNM=48frequency-selectivechannelsinvolvedintheMIMOUACsystem.ToresemblepracticalUACscenarios,thesesimulatedCIRsareestimatedfromMACE10in-waterexperimentaldataandeachCIRhasR=50taps.CIRshavebeennormalized 45

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to1,i.e.,khn,mk2=1forn=1,...,Nandm=1,...,M.Thereceiveddatasamplesarethenconstructedaccordingto( 2 ).SinceDopplereffectsarenotconsideredinthisexample,=I.Thenoisevectorfemgisassumedtocontaincircularlysymmetrici.i.d.complex-valuedGaussianrandomvariableswithzero-meanandvariance2.ThesimulationofISIchannels,combinedwiththeassumptionthateachreceiverhasperfectknowledgeonthechannelcharacteristicsfhn,mg,suggeststhatwecanbypassthetemporalresampling,channelestimation,andphasecompensationmodulesinFigure 2-1B andapplyExact-LMMSE-Turbo,Approximate-LMMSE-Turbo,andRELAX-BLASTdirectlytothereceivedmeasurements.Figures 2-3A and 2-3B showtheaveragecodedbiterrorrate(BER)givenbyExact-LMMSE-TurboandApproximate-LMMSE-Turbo,respectively,alongwiththeRELAX-BLASTperformanceatdifferentSNRs,whereSNRisdenedas1=2.Eachpointisaveragedover500Monte-Carlotrials.Thebinarysourcesequenceandthenoisepatternvaryfromonetrialtoanother.ThecurvelabeledasNoIterationisobtainedbyemployingtheequalizerandthedecoderonlyonce,i.e.,thefeedbackloopisyettobeformed.Inaddition,theaveragecodedBERgivenbyRELAX-BLASTisobtainedafterthreeiterations.WecanseefromFigure 2-3 thatbothtypesofTurboequalizationschemeseffectivelyreducethecodedBERastheiterationproceedsandsignicantlyoutperformRELAX-BLAST,andExact-LMMSE-TurboprovidesonlyslightlybetterdetectionperformancethanApproximate-LMMSE-Turbo.Complexity-wise,theaveragetimerequiredtonishonetrialis18.64s,0.49s,and0.19sonanordinaryworkstation(IntelXeonE5506processor2.13GHz,12GBRAM,Windows764-bit,andMATLABR2010b)forExact-LMMSE-Turbo,Approximate-LMMSE-Turbo,andRELAX-BLAST,respectively.Consequently,Approximate-LMMSE-TurboispreferredoveritsExact-LMMSE-TurbocounterpartsincetheformerprovidesalmostthesamedetectionperformanceasthelatterbutwithacomputationalcomplexityonthesameorderasRELAX-BLAST. 46

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2.5.2MACE10In-WaterExperimentationResults 2.5.2.1Experimentspecics TheMACE10in-waterexperimentwasconductedbytheWoodsHoleOceanographicInstitution(WHOI)offthecoastofMartha'sVineyard,MAinJune2010.Asourcearrayconsistingof4transducerswasverticallydeployedatadepthof80mandtowedbyavessel.Atthereceiverside,a12-elementhydrophonearraywasmountedonabuoy.Thevesselmovedfromtheminimumrangeof500mawayfromthereceivingarrayoutboundtothemaximumrangeof4000mandtheninboundbacktotheminimumrange.Thecarrierfrequency,samplingfrequency,andsymbolrateemployedintheMACE10experimentwere13kHz,39.0625kHz,and3.90625kHz,respectively.BytransmittingN=4sequencessimultaneouslyandincorporatingthemeasurementsacquiredfromalloftheM=12receiverelementsforanalysis,weestablisheda412MIMOUACsystem. ThestructureofatransmitteddatapackageisshowninFigure 2-4 .Eachpackageconsistsof4packets.The1stpacketconveysagrayscaleGatormascotandthesubsequent3packetscombinedformacoloredmascot.TheRGBcomponentsofthecoloredimageweretransmittedinthe2nd,3rd,and4thpackets,respectively.EachpixeloftheGatorgrayscaleimageisrepresentedby5bits,correspondingto32differentintensities(e.g.,purewhiteandpuredarkpixelsarerepresentedby11111and00000,respectively).The64-pixelby100-pixelgrayscalemascotimage,asaconsequence,isrepresentedbyatotalof32ksourcebits.Accordingly,acoloredmascotimageisrepresentedby96kbits.Thecontrastofthegrayscaleimage,aswellasthehueofthecoloredimage,hasbeencarefullyadjustedsothattheimagecarriesapproximatelyequalnumbersof1'sand0's. AsshowninFigure 2-4 ,eachpacketisconstructedasfollows:time-markingsequencesareplacedatthebeginningofeachpackettofacilitatethetemporalresamplingprocedure;twoguardintervals,eachcontaining500silentsymbols,are 47

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placed,respectively,beforeandafterthesegmentscontainingthepayloadsymbolsandtrainingsequences.ThepayloadsymbolscontaintheinformationoftheGatormascotimage.Wehereinelaboratehowtogeneratethe1stpacketfromthegrayscaleGatormascotimage(thepacketgenerationforeachoftheRGBcomponentsofthecoloredimagefollowsthesameprocedure).Specically,the32ksourcebitsarerstinterleavedsothatthebitsfeedingintotheconvolutionalencodermodulehaveanequalchanceofbeing0or1;seeFigure 2-5 .Theso-obtained32kinterleavedsourcebitsarethendividedinto32groups,eachcontaining1kbits.Thebitsintheithgroup(i=1,...,32)willbeusedtoconstructtheithpayloadsymbolblockacrossthe4transmittedsequences,andtheconstructionprocedurefollowsFigure 2-1A .NotethatinFigure 2-5 ,thedepthoftheinterleavers0andis32kand2k,respectively.Figure 2-5 illustratesascenariowithi=1.TheshiftedPeCANtrainingsequenceswithlengthP=512,inconjunctionwithLCP=99cyclicprexsymbols,formthetrainingsection,whichislocatedbetweenthe16thand17thpayloadblocks.ThisMIMOUACdesignleadstoanetcodeddatarateof11.7kbps.Thedatapackagewastransmittedperiodicallyandrecordedbythereceiverarray.Atotalof120epochswereavailableandtheyarereferredtoasE001)]TJ /F1 11.955 Tf 9.3 0 Td[(E120,respectively. ToestimatetheDopplerscalingfactor,wetreatthetime-markingsequencesatthebeginningofapacketasitspreambleandthoseatthebeginningofthesubsequentpacketasthepostamble.Takethe2ndpacketofepochE002forexample.Forthechannelbetweenthe1sttransmitterandthe1streceiver,thesuperimposedmodulusoftheCIRsobtainedbyGoSLIM-VfromthepreambleandpostambleisshowninFigure 2-6 .Theindexesoftheprincipalarrivalsforthepreambleandpostambleare12and21,respectively.Hence,thetimedurationchangeimposedonthepacketis^Td=(21)]TJ /F5 11.955 Tf 12.01 0 Td[(12)Ts,whereTsisthesymbolperioddenedafter( 2 ).ThentheDopplerscalingfactor^canbeestimatedaccordingto( 2 ). 48

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Toassesstheperformanceoftheresamplingprocess,theCIRandDopplerfrequencyevolutionsobtainedbyGoSLIM-Vbeforeweresamplethe2ndpacketofepochE002areshowninFigures 2-7A and 2-7C ,respectively.Incomparison,Figures 2-7B and 2-7D demonstratethecorrespondingCIRandDopplerfrequencyevolutionsobtainedafterresamplingthepacket,respectively.WecanseefromFigure 2-7 thatthetemporalresamplingproceduresuccessfullyreducestheDopplerscalingeffectstoDopplerfrequencyshifts.Therelativespeedbetweenthetransmitterandthereceiverarrayscanbeestimatedas^v=^)]TJ /F5 11.955 Tf 11.96 0 Td[(1c,usingacommonunderwatersoundspeedofc=1500m/s.ItisinterestingtolookatFigure 2-8 wherethevesselspeedestimatedduringtheresamplingstageisplottedontopoftheGPSreferenceinformationprovidedbyWHOI(theGPSdevicewasequippedonthemovingvessel).Thegoodagreementbetweenthesetwocurvesveriestheeffectivenessoftheresamplingprocedureweemploy.Theanalysispresentedhereafterisbasedontheresampledmeasurements. 2.5.2.2Performanceevaluation Wechoosethe1stpacketofepochE018toverifythechannelmodelbehindGoSLIM-V(otherepochsgivesimilarobservations).TheevolutionoftheDopplerfrequenciesproducedbyGoSLIMforallthe12receivehydrophonesareplottedsuperimposedinFigure 2-9 alongwiththeevolutionoftheDopplerfrequencyobtainedbyGoSLIM-V.Oneobservesthatthecurvesshowgoodagreementwitheachother,whichveriesthevalidityofthekeyassumptionthatdifferentreceiversexperiencethesameDopplerfrequency.Moreover,theCIRevolutionbetweenthe1sttransmitterandthe1st(2nd)receiverobtainedbyGoSLIMisshowninFigure 2-10A (Figure 2-10B ),andtheCIRevolutionbetweenthe1sttransmitterandthe1st(2nd)receiverobtainedbyGoSLIM-VisshowninFigure 2-10C (Figure 2-10D ).BycomparingFigure 2-10A (Figure 2-10B )withFigure 2-10C (Figure 2-10D ),weobservenovisibledifferencebetweentheCIRestimatesobtainedbythetwoalgorithms. 49

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Next,weproceedtoassesstheimpactofthechannelestimationalgorithmontheresultingdetectionperformance.ThetrackinglengthisxedatLTR=450forallofthe120epochs.Thechanneltrackingstartswithtraining-directedchannelestimationusingGoSLIMorGoSLIM-V.Thenweperformphasecompensationseparatelyateachreceivinghydrophoneasdonein( 2 )beforeproceedingtoemployRELAX-BLASTtodetecttherst250payloadsymbolscontainedinthe17thpayloadblockforeachtransmittedsequence;seeFigure 2-5 .Next,thechannelsareupdatedinthedecision-directedmodeusing450symbols(containingthepreviouslydetectedpayloadsymbols,aswellasaportionofthetrainingsequenceaswell).WiththeupdatedCIRsandDopplerfrequency(frequencies),afterphasecompensation,thesubsequent250payloadsymbolscontainedinthe18thblockaredetectedusingRELAX-BLAST.Thisprocesscontinuesuntilallofthe16payloadblockstotheright-handsideofthetrainingsequencesaredetected.Thissametrackingschemecanbeappliedinareversemannertothedetectionofthe16payloadblocksaheadofthetrainingsequences. Thereisatotalof480packetsavailableandwedeemapackettobesuccessfullydetectedifitscodedBERislessthan0.1.WhenGoSLIMisemployedasthechannelestimationalgorithm,wehavesucceededintrackingtheentire32payloadblocksfor391packets.AcodedBERof1.710)]TJ /F12 7.97 Tf 6.58 0 Td[(2isachievedafteraveragingoverthe391successfulpackets.Incomparison,whenGoSLIM-Visused,396packetsaresuccessfullyretrievedwithanaveragecodedBERof1.610)]TJ /F12 7.97 Tf 6.59 0 Td[(2.WhenGoSLIMisemployed,Figures 2-11A 2-11B ,and 2-11C showtherecoveredgrayscalemascotsforE013,E016,andE018,respectively.ThecorrespondingcodedBERsandthecomputationaltimeconsumedatthechannelestimationstageonanordinaryworkstation(twoIntelXeonE5506processors2.13GHz,12GBRAM,Windows764-bit,andMATLABR2010b)arelistedintherstrowofTables 2-1 and 2-2 ,respectively.Incomparison,whenGoSLIM-Visemployed,therecoveredgrayscalemascotsareshowninFigures 2-11D 2-11E ,and 50

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2-11F ,withthecorrespondingBERsandcomputationalcomplexitieslistedinthesecondrowofTables 2-1 and 2-2 ,respectively.OneobservesfromTable 2-1 thatGoSLIM-VyieldsslightlybetterBERresultsthanitsGoSLIMcounterpart.Moreover,Table 2-2 demonstratesthatGoSLIM-Visabout4timesfasterthanGoSLIM.(Notethat,bytakingintoaccountthecomplexityofallthestepsotherthantheDopplerfrequencyupdate,thecomputationalsavingprovidedbyGoSLIM-VagainstGoSLIMisthuslessthan12times.) WethenproceedtoaccesstheperformanceofthevarioussymboldetectionschemeswithapplyingGoSLIM-Vinthechannelestimationstage.Afteranalyzingthe480packetsavailable,Table 2-3 summarizesthesuccessfullydetectedpacketpercentage,thezeroBERpacketpercentage,thecodedBERaveragedoverthesuccessfulpackets,andthetimeratioofthetimeconsumedtoprocessapacketontheworkstationspeciedinSection 2.5.1 toTtx=2.741s(Ttxisdenedin( 2 ))obtainedusingExact-LMMSE-Turbo,Approximate-LMMSE-Turbo,andtheRELAX-BLASTscheme,respectively.Theresultsareobtainedbyapplying3iterationsforallofthethreetypesofdetectionschemesconsidered.OneobservesfromTable 2-3 that1)BER-wise,bothExact-LMMSE-TurboandApproximate-LMMSE-TurbooutperformRELAX-BLASTsignicantly,2)comparedtoExact-LMMSE-Turbo,Approximate-LMMSE-TurbogreatlyreducesthecomputationaltimeatthecostofslightBERperformancedegradation,and3)comparedtoRELAX-BLAST,Approximate-LMMSE-TurboimprovestheBERperformancebytwoordersofmagnitudewithoutsignicantlyincreasingthecomputationalcomplexities.TheseobservationsareinlinewiththosemadefromthenumericalexamplesinSection 2.5.1 .Moreover,weanalyzeepochE054thatleadstoperfectrecoveryofboththegrayscaleandcoloredmascots(seeFigures 2-12B and 2-12D )usingeitherExact-LMMSE-TurboorApproximate-LMMSE-Turbo.Incomparison,thegrayscaleandcoloredmascotsrecoveredfromepochE054usingRELAX-BLASTareshowninFigures 2-12A and 2-12C ,respectively,withthecorrespondingcoded 51

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BERsbeing1.810)]TJ /F12 7.97 Tf 6.58 0 Td[(1and1.510)]TJ /F12 7.97 Tf 6.59 0 Td[(1.WenotethattheTurboequalizationschemesarehighlyeffective. TofurtherillustratethedetectionperformanceofTurboequalization,Table 2-4 showsthecodedBERaveragedoverallofthe480packetsatdifferentiterationnumbersobtainedbyExact-LMMSE-TurboandApproximate-LMMSE-Turbo.WecanseefromTable 2-4 thatthecodedBERimproveswithiteration.EmpiricalexperienceindicatesthatthedetectionperformanceforbothtypesofTurboequalizationconvergesafterthreeiterations.Next,wechooseonepayloadblockanddenotefLd(a(k))g1000k=1astheLLRsoftinformationofthecorresponding1ksourcebitsfa(k)g1000k=1generatedbytheMax-Log-MAPdecoder.Figures 2-13A 2-13D and 2-13E 2-13H showfLd(a(k))g1000k=1obtainedbyExact-LMMSE-TurboandApproximate-LMMSE-Turbo,respectively,atdifferentiterationnumbers.f~a(k)ginFigure 2-1B aretheharddecisionsdeterminedfromfLd(a(k))g.Specically,ifLd(a(k))>0then~a(k)=0,otherwise~a(k)=1(see( 2 )).InFigure 2-13 ,thecirclesindicatebiterrors.WecanseefromFigure 2-13 thattheLLRofsourcebitsaremovingawayfromzeroastheiterationproceeds(therstiterationhasthemostsignicantimpact),whichsuggeststhatwiththehelpofcyclingsoftinformation,thedecoderismoreandmorecondentaboutthecorrespondingsourcebitsbeing0or1. 52

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Table2-1. CodedBERresultsobtainedbyGoSLIMandGoSLIM-V,respectively. E013E016E018 GoSLIM3.610)]TJ /F12 7.97 Tf 6.59 0 Td[(23.610)]TJ /F12 7.97 Tf 6.59 0 Td[(31.610)]TJ /F12 7.97 Tf 6.58 0 Td[(4GoSLIM-V2.310)]TJ /F12 7.97 Tf 6.59 0 Td[(23.110)]TJ /F12 7.97 Tf 6.59 0 Td[(39.410)]TJ /F12 7.97 Tf 6.58 0 Td[(5 Table2-2. Complexitycomparison(ins)betweenGoSLIMandGoSLIM-V. E013E016E018 GoSLIM(15iterations)122.8121.1123.0GoSLIM-V(15iterations)32.131.832.3 A B Figure2-1. AnNMMIMOUACsystem.A)Transmitterstructure.B)ReceiverstructurebyemployingTurboequalization. 53

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Table2-3. Asummaryoftheperformanceofthethreedetectionschemes(3iterationsapplied). successfulpacketzeroBERpacketaveragetimepercentage(%)percentage(%)codedBERratio Exact-LMMSE-Turbo10076.79.210)]TJ /F12 7.97 Tf 6.59 0 Td[(5488.1Approximate-LMMSE-Turbo10074.42.110)]TJ /F12 7.97 Tf 6.59 0 Td[(417.2RELAX-BLAST82.54.81.610)]TJ /F12 7.97 Tf 6.59 0 Td[(216.9 54

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Table2-4. TheaveragecodedBERobtainedbyExactLMMSETurboandApproximateLMMSETurbo,respectively. Noiteration1iteration2iterations3iterations Exact-LMMSE-Turbo2.710)]TJ /F12 7.97 Tf 6.59 0 Td[(18.110)]TJ /F12 7.97 Tf 6.59 0 Td[(41.310)]TJ /F12 7.97 Tf 6.59 0 Td[(49.210)]TJ /F12 7.97 Tf 6.59 0 Td[(5Approximate-LMMSE-Turbo2.710)]TJ /F12 7.97 Tf 6.59 0 Td[(12.210)]TJ /F12 7.97 Tf 6.59 0 Td[(33.210)]TJ /F12 7.97 Tf 6.59 0 Td[(42.110)]TJ /F12 7.97 Tf 6.59 0 Td[(4 55

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Figure2-2. ThestructureoftheLMMSEbasedsoft-inputsoft-outputequalizer. A B Figure2-3. A)CodedBERperformancebyusingExact-LMMSE-TurboalongwithRELAX-BLASTperformance.B)CodedBERperformancebyusingApproximate-LMMSE-TurboalongwithRELAX-BLASTperformance.Eachpointisaveragedover500Monte-Carlotrials.Inthissimulation,N=4,M=12,R=50andK=250. 56

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Figure2-4. ThestructureofthepackageusedintheMACE10experiment. Figure2-5. Thestructureofthetransmittedsymbolsforthe412MIMOBLASTschemeusedinMACE10. 57

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Figure2-6. ThesuperimposedmodulusoftheCIRsobtainedfromthepreambleandpostamble,respectively. 58

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A B C D Figure2-7. A)CIRevolutionofEpochE002beforeresampling.B)CIRevolutionofEpochE002afterresampling.C)DopplerfrequencyevolutionofEpochE002beforeresampling.D)DopplerfrequencyevolutionofEpochE002afterresampling. Figure2-8. TherelativespeedbetweenthetransmitterandreceiverarraygivenbyGPSandestimatedduringthetemporalresamplingstage(courtesyofMilicaStojanovic'sgroup). 59

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Figure2-9. Dopplerfrequencyevolutionofthe1stpacketinepochE018obtainedbyGoSLIMandGoSLIM-V,respectively. 60

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A B C D Figure2-10. A)CIRevolutionbetweenthe1sttransmitterandthe1streceiverobtainedbyGoSLIM.B)CIRevolutionbetweenthe1sttransmitterandthe2ndreceiverobtainedbyGoSLIM.C)CIRevolutionbetweenthe1sttransmitterandthe1streceiverobtainedbyGoSLIM-V.D)CIRevolutionbetweenthe1sttransmitterandthe2ndreceiverobtainedbyGoSLIM-V. 61

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A B C D E F Figure2-11. A)GrayscalemascotrecoveredfromepochE013.B)GrayscalemascotrecoveredfromepochE016.C)GrayscalemascotrecoveredfromepochE018.D)GrayscalemascotrecoveredfromepochE013.E)GrayscalemascotrecoveredfromepochE016.F)GrayscalemascotrecoveredfromepochE018.A)-C)areobtainedbyGoSLIM.D)-F)areobtainedbyGoSLIM-V. 62

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A B C D Figure2-12. A)GrayscalemascotrecoveredfromepochE054usingRELAX-BLAST.B)GrayscalemascotrecoveredfromepochE054usingTurboequalization.C)ColoredmascotrecoveredfromepochE054usingRELAX-BLAST.D)ColoredmascotrecoveredfromepochE054usingTurboequalization. A B C D E F G H Figure2-13. TheLLRsoftinformationaboutthesourcebitsattheoutputofthedecoder.A)-D)areobtainedbyExact-LMMSE-Turbofromnoiterationto3iterations,respectively.E)-H)areobtainedbyApproximate-LMMSE-Turbofromnoiterationto3iterations,respectively. 63

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CHAPTER3ENHANCEDMULTISTATICACTIVESONARSIGNALPROCESSING Multistaticactivesonarsystemsinvolvethetransmissionandreceptionofmultipleprobingsequencesandcanachievesignicantlyenhancedperformanceoftargetdetectionandlocalizationthroughexploitingspatialdiversity[ 42 43 ].ThereectedacousticsignalsacquiredbythereceiverscarrytherangeandDopplerinformationofpotentialtargets,whichprovidesabasisfortheestimationofthetargetparameters,includingtheirpositionsandvelocities[ 43 44 ].Inthischapter,weconsideramultistaticactivesonarsystemthatemploysmultiplestationarytransmittersandreceivers[ 42 ].Twosignalprocessingaspectsrelatedtosuchasystemdesignareaddressed,namelytargetrange-Dopplerimagingandtargetparameterestimation. Thereceiverlterdesignplaysacriticalroleintheoverallperformanceofamultistaticactivesonarsystemsinceitdirectlydeterminesthequalityofrange-Dopplerimagingandaffectstheaccuracyofthesubsequenttargetparameterestimation.Asaclassicalreceiverlter[ 45 46 ],thematchedlteristheoptimallinearlterformaximizingthesignal-to-noiseratio(SNR)forthecaseofasingletargetinthepresenceofadditivewhitenoise.However,inamultistaticactivesonarsystem,themultiplesimultaneouslytransmittedprobingsequencesactasinterferencestooneanother,makingthematchedlter-basedreceiverineffective.Therefore,itisnecessarytodesignmoreadvancedadaptivereceiverlterscapableofprovidingrange-Dopplerimageswithbothlowsidelobelevelsandhighaccuracy.Inthischapter,wepresentahybriddense-sparserange-Dopplerimagingmethod,whichrstappliestheiterativeadaptiveapproach(IAA)[ 37 ]toobtainaccurateanddenserange-Dopplerimages,andthenachievesparsitybyusingonestepoftheSLIMmethod[ 47 ].SinceSLIMisamaximumaposteriori(MAP)approach,werefertothishybridmethodasIAA-MAP.WeshowthatIAA-MAPcanimproveresolutionandreducesidelobelevelssimultaneouslywhilemaintaininghighaccuracy,whichisdesirableforimprovedtargetparameterestimation. 64

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Thetargetparametersareestimatedusingthepeaks(afterthedirectblastsareremoved)oftherange-Dopplerimages.However,inthepresenceofmultipletargetsintheeldofview,eachpeakneedstobeassociatedwithaspecictarget.Thefailureinassociatingthesaidpeakswiththetargetscouldcausesevereperformancedegradations[ 48 ].Thisassociationproblemcanalsobeviewedasarangettingproblem.Thisisacombinatorialoptimizationprobleminvolvingboththepeakassociationandtargetpositionestimation.Toefcientlysolvethisproblem,wedevelopageneralizedK-Meansclustering(GKC)methodforpeakassociation,whichiterativelysolvesanoptimizationproblem.Eachiterationconsistsoftwosteps,theMeansupdatestep,wherethetargetpositionisestimatedforthecurrentassociationpattern,andtheLabelassignmentstep,wheretheassociationpatternisselectedbasedonthecurrenttargetpositionestimates.(Weremarkthatifeachofthereceiversisequippedwithalargearraythatcanprovideaccurateangleestimatesofthetargets,thepeakassociationproblembecomesaneasyproblemorevendisappearsentirely.) Basedonthefactthatdifferenttransmitter-receiverpairshavedifferentreectioncoefcients,wedevelopanextendedinvarianceprinciple-basedweightedleast-squares(EXIP-WLS)methodfortargetpositiondetermination(whichistheaforementionedMeansupdatestep)andvelocityestimation.Morespecically,nonlinearalgebraicpositionequationsareapproximatedaslinearonesviaTaylorexpansionandthetargetpositionandvelocityestimatesarerenedinaniterativemannerusingweighting[ 49 ].TheweightingmatricesweusearetheblocksoftheFisherinformationmatrix(FIM)correspondingtoanunstructureddatamodel. Therestofthischapterisorganizedasfollows.Section 3.1 outlinesthemultistaticactivesonarsystemmodelandformulatestheproblemofinterest.Section 3.2 presentstheIAA-MAPmethodforrange-Dopplerimaging,theGKCmethodforpeakassociation,andtheEXIP-WLSmethodfortargetparameterestimation.InSection 3.3 ,wepresentthesimulationresults. 65

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3.1SystemDescriptionandProblemFormulation Consideratwo-dimensional(2D)multistaticactivesonarsystemequippedwithNstationarytransmittersandMstationaryreceiverswithknownlocations.(Althoughthischapterpaysattentiontothe2Dcase,extensiontothethree-dimensional(3D)caseisstraightforward.)AssumefurtherthatthereareQmovingtargetsintheregionofinterestandthattheirlocationsandvelocitiesaresought.Denotetn=[xtn,ytn]T,rm=[xrm,yrm]T,andq=[xq,yq]TastheCartesiancoordinatevectorsofthenthtransmitter,themthreceiver,andtheqthtarget,respectively,forn=1,...,N,m=1,...,M,andq=1,...,Q.Further,denotesn(t)asthepingsentbythenthtransmitterforn=1,...,N.TheNpingsfsn(t)gNn=1aretransmittedsimultaneouslyandeveryreceiverisassumedtohavetheperfectknowledgeonfsn(t)gNn=1.Figure 3-2 showsagenericsensingscenarioforthenthtransmitter,themthreceiver,andtheqthtarget,inwhichvq=[xvq,yvq]TisthevelocityvectoroftheqthtargetintheCartesiancoordinate.Additionally, q,n=cos)]TJ /F12 7.97 Tf 6.59 0 Td[(1xq)]TJ /F4 11.955 Tf 11.95 0 Td[(xtn kq)]TJ /F10 11.955 Tf 11.96 0 Td[(tnk(3) and 'q,m=cos)]TJ /F12 7.97 Tf 6.59 0 Td[(1xq)]TJ /F4 11.955 Tf 11.95 0 Td[(xrm kq)]TJ /F10 11.955 Tf 11.96 0 Td[(rmk(3) arethebearinganglesmeasuredfromtheeasttothelineconnectingqandtnandthelineconnectingqandrm,respectively. Atransmittedsignalsn(t)isreectedontheqthtargetandtheechoisreceivedbythemthreceiver.Thereectedechosn,q,m(t)andthetransmittedpingsn(t)arerelatedthrough: sn,q,m(t)=n,q,msn(n,q,m(t)]TJ /F13 11.955 Tf 11.95 0 Td[(n,q,m)).(3) Heren,q,misthecomplex-valuedreectioncoefcient,thepropagationtimedelay n,q,m=kq)]TJ /F10 11.955 Tf 11.96 0 Td[(tnk+kq)]TJ /F10 11.955 Tf 11.96 0 Td[(rmk c(3) 66

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isdeterminedastheratioofthetargetrangetotheunderwaterspeedc(thetargetrangeiskq)]TJ /F10 11.955 Tf 12.2 0 Td[(tnk+kq)]TJ /F10 11.955 Tf 12.2 0 Td[(rmk,whichgeometricallyrepresentsthesumofthedistancebetweenthenthtransmitterandtheqthtargetandthedistancebetweentheqthtargetandthemthreceiver),and n,q,m=c+xvqcosq,n+yvqsinq,n c)]TJ /F4 11.955 Tf 11.96 0 Td[(xvqcos'q,m)]TJ /F4 11.955 Tf 11.95 0 Td[(yvqsin'q,m(3) istheDopplerscalingfactor(pleaserefertoAppendix A foritsderivation). Inadditiontothetargetreections,thereceivedmeasurementsarealsosubjecttodirectblasts,whicharethetransmittedsignalspropagatingdirectlyfromthetransmitterstothereceivers[ 43 ].Thedirectblastfromthenthtransmittertothemthreceiver,saysn,m(t),canbewrittenas: sn,m(t)=n,msn(t)]TJ /F13 11.955 Tf 11.96 0 Td[(n,m).(3) NotethatthedirectblastdoesnotsufferfromDopplerscalingforstationarytransmitterandreceiverplatforms,andthusitischaracterizedbyonlythecomplexamplituden,mandtimedelayn,m.Intensity-wise,thedirectblaststendtobemuchstrongerthanthetargetreections[ 43 ]. BytakingintoaccountthecontributionsfromalloftheNtransmittersandQtargets,thereceivedsignalym(t)acquiredatthemthreceivercanberepresentedas: ym(t)=NXn=1QXq=1sn,q,m(t)+NXn=1sn,m(t)+em(t),m=1,...,M,(3) whereem(t)representstheadditivenoise.Themaingoalofthischapteristorecoverthetargetpositionsandvelocitiesfromthetransmittedsignalsfsn(t)gNn=1andthereceivedmeasurementsfym(t)gMm=1. 3.2ProposedAlgorithms Inthissection,werstpresenttheIAA-MAPmethodforrange-DopplerimagingtoobtaintherangeandDopplerestimatesofthetargets.ThentheGKCmethodis 67

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introducedtoassociatetheseorderlessrangeestimateswiththecorrespondingtargetsandtheEXIP-WLSmethodispresentedtoestimatethetargetpositionsandvelocities. 3.2.1Range-DopplerImaging Inthissubsection,wefocusontheestimatingtherangesandDopplersofthetargets,i.e.,therange-Dopplerimagingproblem,wheretargetsarecharacterizedbythetimedelay(range)-Dopplerpairinsteadoftheposition-velocitypair. 3.2.1.1Imagingproblemformulation ThetimedelayandDopplerregionsofinterestaredividedintoRandLpointsrespectively,hencethereareRLpixelsintherange-Dopplerimage.Letfr,lgbethetimedelay-Dopplerpairofthepotentialtargetandn,m,r,lbethetargetreectioncoefcientassociatedwiththefr,lgpairandwithrespecttothenthtransmitterandthemthreceiver.Thenthereceivedsignalym(t)acquiredatthemthreceivercanberewrittenaccordingtofn,m,r,l,r,lgas ym(t)=NXn=1RXr=1LXl=1sn,m,r,l(t)+em(t),(3) where sn,m,r,l(t)=n,m,r,lsn(l(t)]TJ /F13 11.955 Tf 11.95 0 Td[(r))(3) isthereectedechocorrespondingtothenthtransmittedpingsn(t)andthefr,lgpair.Thedirectblasts(see( 3 ))arealsocontainedin( 3 ),wheretherelatedDopplerscalingfactorsareequalto1.NotethatingeneralRLismuchlargerthantheactualnumberofthetargetsQandthusonlyafewcomponentsoffn,m,r,lgwillbenon-zero. Let~sn,l=[sn,l(1),,sn,l(Pl)]TbethesampledsequencewithlengthPloftheDoppler-scaledsignalsn(lt)forthelthDopplerbin.Inaddition,letdmisthelengthofthesampledreceivedsignalym(t)and~ristheroundedratioofrtothesampling 68

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period.Thus,thesampledversionofsn(l(t)]TJ /F13 11.955 Tf 11.95 0 Td[(r))canbeexpressedintermsof~sn,las sn,m,r,l=[sn,m,r,l(1),sn,m,r,l(2),...,sn,m,r,l(dm)]T=2666640~rPl0~r(dm)]TJ /F9 7.97 Tf 6.59 0 Td[(Pl)IPlPl0Pl(dm)]TJ /F9 7.97 Tf 6.59 0 Td[(Pl)0(dm)]TJ /F9 7.97 Tf 6.58 0 Td[(Pl)]TJ /F12 7.97 Tf 6.95 0 Td[(~r)Pl0(dm)]TJ /F9 7.97 Tf 6.59 0 Td[(Pl)]TJ /F12 7.97 Tf 6.94 0 Td[(~r)(dm)]TJ /F9 7.97 Tf 6.59 0 Td[(Pl)377775264~sn,l0(dm)]TJ /F9 7.97 Tf 6.59 0 Td[(Pl)1375 (3) Thenthesampledversionofym(t)isgivenby ym=[ym(1),ym(2),...,ym(dm)]T=NXn=1RXr=1LXl=1n,m,r,lsn,m,r,l+em, (3) whereem2Cdm1isthesampledversionofem(t).Inordertowritethismorecompactly,dene n,m,l=[n,m,1,l,,n,m,R,l]T,n,m=[Tn,m,1,,Tn,m,L]T,m=[T1,m,,TN,m]T, (3) andsimilarly,dene Sn,m,l=[sn,m,1,l,,sn,m,R,l],Sn,m=[Sn,m,1,,Sn,m,L],Sm=[S1,m,,SN,m]. (3) Thus,ymcanberewrittenas ym=NXn=1Sn,mn,m+em=Smm+em, (3) whereSm2CdmNRLandm2CNRL1. 69

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Equation( 3 )maybeviewedasasparserepresentationproblem,whereasparsevectormissoughtgiventhemeasurementvectorymandthedictionarySm.Oncethevectorn,mdenedin( 3 )isobtained,therange-DopplerimageofsizeRLisformedwithrespecttothenthtransmitterandthemthreceiver.Hence,wecanformatotalofMNrange-DopplerimagesfromfymgMm=1.TheMsetsofNrange-Dopplerimagescanbeformedinparallel. 3.2.1.2Receiverlterforrange-Dopplerimaging Hereinweaddresstherange-Dopplerimagingproblematthemthreceiver,thesamemethodologycanbereadilyappliedtootherreceiversdirectly.Fourdifferentreceiverlters,namelymatchedlter,IAA,SLIM,andIAA-MAParepresentedbelowtoestimatem. MatchedFilter(MF):Asoneoftheclassicalreceiverltersfortheactivesensingapplications[ 45 46 ],thematchedlter(MF)correlatesthereceivedsignalwiththetime-alignedandDoppler-scaledversionofthetransmittedping,i.e., ^n,m,r,l=sHn,m,r,lym sHn,m,r,lsn,m,r,l,n=1,...,N,r=1,...,R,l=1,...,L.(3) Inasingle-transmittersingle-targetcase,MFistheoptimalreceiverlterformaximizingtheSNRinthepresenceofadditivewhitenoise.However,inmultistaticactivesonarsystems,themultiplesimultaneouslytransmittedprobingsequencesactasinterferencestooneanother,makingtheperformanceofMFunsatisfactory. IterativeAdaptiveApproach(IAA):Asoneofthenonparametricmethods,theiterativeadaptiveapproach(IAA)[ 37 ]hasbeenshowntopossesssuperiorperformanceinawidevarietyofapplications,includingpassivearrayprocessing[ 37 ],underwateracousticcommunications[ 50 ],andMIMOradarimaging[ 51 ]. 70

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TheIAAalgorithmobtainsmbyiterativelysolvingthefollowingweightedleastsquaresproblem: minn,m,r,ljjym)]TJ /F13 11.955 Tf 11.95 0 Td[(n,m,r,lsn,m,r,ljj2Q)]TJ /F34 5.978 Tf 5.76 0 Td[(1n,m,r,l,n=1,...,N,r=1,...,R,l=1,...,L,(3) wherejjujj2Q)]TJ /F34 5.978 Tf 5.76 0 Td[(1=uHQ)]TJ /F12 7.97 Tf 6.59 0 Td[(1u.In( 3 ),theinterferenceandnoisecovariancematrixQn,m,k,lisgivenby: Qn,m,r,l=Rm)-222(jn,m,r,lj2sn,m,r,lsHn,m,r,l,(3) where Rm=NXn=1RXr=1LXl=1jn,m,r,lj2sn,m,r,lsHn,m,r,l.(3) Theminimizationof( 3 )withrespectton,m,r,lgives: ^n,m,r,l=sHn,m,r,lQ)]TJ /F12 7.97 Tf 6.59 0 Td[(1n,m,r,lym sHn,m,r,lQ)]TJ /F12 7.97 Tf 6.59 0 Td[(1n,m,r,lsn,m,r,l,n=1,...,N,r=1,...,R,l=1,...,L,(3) ByusingthedenitionofQn,m,r,lin( 3 )andthematrixinversionlemma,( 3 )canberewrittenas: ^n,m,r,l=sHn,m,r,lR)]TJ /F12 7.97 Tf 6.58 0 Td[(1mym sHn,m,r,lR)]TJ /F12 7.97 Tf 6.58 0 Td[(1msn,m,r,l,n=1,...,N,r=1,...,R,l=1,...,L,(3) whichavoidsthecomputationofQn,m,r,lforn=1,...,N,r=1,...,R,andl=1,...,L(i.e.,NRLtimesintotal),andthussignicantlyreducesthecomputationalcomplexity.OnceRmisavailable,n,m,r,lforn=1,...,N,r=1,...,R,andl=1,...,Lcanbecomputedinparallel.Since( 3 )requiresRm,whichinturndependsontheunknowntargetparameters(see( 3 )),IAAneedstobeimplementedinaniterativemanner.EmpiricalresultsshowthatIAA,initializedwiththeMFoutputin( 3 ),typicallyconvergesinnomorethan15iterations(alocalconvergenceproofforIAAisgivenin[ 51 ]). Sparselearningviaiterativeminimization(SLIM):Asoneofthesparsesignalrecoveryapproaches,thesparselearningviaiterativeminimization(SLIM)method 71

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(referredtoasSLIM-1in[ 47 ])considersthefollowinghierarchicalBayesianmodel: ymjm,mCN(Smm,mI), (3) mjpmCN(0,Pm), (3) pn,m,r,lG(2,1),n=1,...,N,r=1,...,R,l=1,...,L, (3) wherepm=[p1,m,1,1,...,pN,m,R,L],Pm=diagfpmg,andtheaprioriprobabilitydensityfunctionofmisassumedtobeat,i.e.,f(m)/1. Thus,theobjectivefunctionforSLIMcanberepresentedas: maxm,pm,mp(m,pm,mjym)=maxm,pm,mp(ymjm,m)p(mjpm)NYn=1RYr=1LYl=1p(pn,m,r,l).(3) Bycombining( 3 )( 3 ),theoptimizationproblemformulatedin( 3 )canberewritteninanegativelogarithmformas: minm,Pm,m dmlogm+kym)]TJ /F10 11.955 Tf 11.96 0 Td[(Smmk2 m+NXn=1RXr=1LXl=1jn,m,r,lj2 pn,m,r,l+NXn=1RXr=1LXl=1pn,m,r,l!.(3) Besidesthetargetreectioncoefcientvectorm,thecostfunctionintroducesthecovariancematrixPm(orequivalently,itsdiagonalelementsinpm)andthenoisepowerm.Wesolve( 3 )byusingthefollowingcyclicapproach:ateachiteration,oneoftheparametervectorsm,pm,andmisupdatedwhilekeepingtheothertwoxed.SLIMkeepsiteratinguntilapredenednumberofiterationsisreachedoruntilconvergence.ThethreestepsoftheSLIMalgorithmattheithiterationareoutlinedbelow: 1. GivenP(i)]TJ /F12 7.97 Tf 6.59 0 Td[(1)mand(i)]TJ /F12 7.97 Tf 6.58 0 Td[(1)mfromthepreviousSLIMiteration,settingthepartialderivativeof( 3 )withrespecttoHmto0yieldstheoptimal(i)m (i)m=P(i)]TJ /F12 7.97 Tf 6.59 0 Td[(1)mSHm(i)]TJ /F12 7.97 Tf 6.58 0 Td[(1)mI+SmP(i)]TJ /F12 7.97 Tf 6.59 0 Td[(1)mSHm)]TJ /F12 7.97 Tf 6.59 0 Td[(1ym.(3) 2. Once(i)misavailable,bytakingthepartialderivationof( 3 )withrespecttopn,m,r,landsettingtheresulttozero,theoptimalpn,m,r,lisobtainedas p(i)n,m,r,l=(i)n,m,r,l,n=1,...,N,r=1,...,R,l=1,...,L.(3) 72

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3. Usingthemostrecentlyobtained(i)m,wenallyestimatethenoisepowerbytakingthepartialderivationof( 3 )withrespecttomandthesolutionisgivenby: (i)m=1 dmym)]TJ /F10 11.955 Tf 11.95 0 Td[(Sm(i)m2.(3) ItgenerallytakesSLIMnomorethan15iterationstoconverge[ 47 ]. IAA-MAP:IAAandSLIMmethodspossessvariousmeritsandlimitations,withIAAbeingdenseandaccurateandSLIMbeingsparseandbiaseddownward.Below,weconsiderahybridmethodthattakesadvantagesofthesemeritswhileovercomingthelimitationsoftheseparatemethods.Indeed,IAAisanonparametric,robust,anduserparameterfreealgorithm,whichhasalsobeenfoundtobemoreaccuratethanthecorrespondingSLIMestimates,althoughwithanotablyhighersidelobelevel.InordertoachievesidelobelevelscomparabletothoseofSLIM,onemayinsteadformahybridapproachthatrstusesIAAtocomputeadenserange-Dopplerimage,whichisthen,uponconvergence,followedbyasinglestepofSLIM-0[ 47 ]: ^n,m,r,l=^(IAA)n,m,r,l2sHn,m,r,lR(IAA)m)]TJ /F12 7.97 Tf 6.59 0 Td[(1ym,n=1,...,N,r=1,...,R,l=1,...,L,(3) where^(IAA)n,m,r,landR(IAA)mdenotethecorrespondingestimatesobtainedattheconclusionoftheIAAiterations. SinceSLIMachievessparsitybasedonsolvingahierarchicalBayesianmodelthroughmaximizingaposterioriprobabilitydensityfunction[ 47 ],thissinglestepofSLIM-0isreferredtoasaMAPstep,andtheresultingalgorithmastheIAA-MAPalgorithm.ComparedtoSLIM-1,SLIM-0providessparserresultsbutismoresensitivetonoiseanddisturbances.IAAtendstobemorerobustagainstnoiseanddisturbancesthanSLIM.DuetotheaccurateandrobustIAAresultandasinglestepofSLIM-0,IAA-MAPisrobust,sparseandaccurate.ThemeritsofIAA-MAParedesirableforachievingimprovedtargetparameterestimation. 73

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3.2.2GeneralizedK-MeansClustering(GKC)AssociationMethod Giventherange-Dopplerimages,weuseamodel-orderselectiontool,i.e.,theBayesianinformationcriterion(BIC)[ 52 53 ],toestimatethetargetnumberandlocatethecorrespondingpeaksbeforemovingontothetaskoftargetparameterestimation.Foreaseofexposition,weassumethatwecansuccessfullylocatetheQpeaks(eachcorrespondstoonetarget)oneachoneoftheNMrange-Dopplerimagesandtherangeestimatesobtainedfromthosepeaksareutilizedtoestimatethetargetpositions.Weremarkthattheproposedassociationschemecanbeeasilymodiedtosuitmorecomplicatedcases,i.e.,whentheestimatedtargetnumbersforalltransmitter-receiverpairsarenotequaltothetruetargetnumberQ. AsshowninFigure 3-2 ,thetargetpositioncannotbeuniquelydeterminedfromtherangevalueandthepositionsofasingletransmitterandreceiverpair.Rather,thetargetcouldbeatanypointonanellipse.Ina2DmultistaticactivesonarsystemequippedwithNtransmittersandMreceivers,itiswell-knownthatuniquetargetpositionestimationingeneralrequiresatleastthreerangevalues(orequivalently,threeellipses)[ 49 ],i.e.,NM3.Additionally,morerangemeasurementscanyieldmoreaccuratetargetpositionestimates.Therefore,wecollectrangeinformationfromalloftheNMimagestodeterminethepositionofatarget. However,whentherearemultipletargetsintheeldofinterest,weneedtosolvethetargetassociationproblem,whichaimstodetermineaproperone-to-onecorrespondencebetweentheQtargetsandtheQpeaksofeachrange-Dopplerimage.Todescribetheproblemmoreclearly,wetaketheimagingresultsshowninFigure 3-3 asanexample,wheretherearetwotransmitters,tworeceivers,andtwotargets,i.e.,fN=2,M=2,Q=2g.IfweassignLabel1,i.e.,the1sttarget,tothepeakattheright-handsideoftherstsubgure,thenweneedtodeterminewhichthreepeaksintherestofthethreesubgurescorrespondtothesametarget.Givenanassumedassociationpattern,alltherangevaluesobtainedfrompeaksthatareassignedtoa 74

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specictargetarecollectedtoestimatethetargetpositionandtheincorrectassociationassumptionwouldleadtosevereperformancedegradations.Therefore,solvingtheassociationproblemplaysacriticalroleintheoverallperformanceofthesonarsysteminthepresenceofmultipletargetsandintheabsenceoftargetangleestimates. Apossibleansweristhebrute-forceassociation(BFA),whichestimatesthetargetpositionsforeverypossibleassociationpattern(tobemoreexact,thereareatotalof(Q!)NM)]TJ /F12 7.97 Tf 6.59 0 Td[(1associationpatternsthatneedtobechecked),andthenselectstheassociationpatternthatyieldstheminimumcostvalueastheoptimalone.Althoughconceptionallysimple,theBFAapproachiscomputationallyintensive(NP-hard). Toalleviatethecomputationalburden,wedevelopanewassociationmethod,inspiredbytheK-Meansclustering(KMC)ideainthemachinelearningeld[ 54 ],asageneralizationoftheconventionalKMCmethod,andreferredtoastheGKCmethod.Thisclustering-basedmethodaimstopartitionalltheNMQrangemeasurementsintoQclasses,andensuresthat:i)eachclasscontainsNMsamples(i.e.,rangevalues);ii)theQrangemeasurementsobtainedfromthepeaksofeachrange-Dopplerimagehavedifferentlabels(1,2,...,Q);andiii)therangeestimatesassignedtothesamelabelcorrespondtoauniquetarget.Letfn,q,mgdenoteacollectionoftargetrangeestimatesobtainedfromtherange-Dopplerimages,theGKCmethodcanthenbeformulatedtosolvethefollowingoptimizationproblem: minQXi=1QXq=1NXn=1MXm=1(i,q,n,m)jn,q,m)]TJ /F5 11.955 Tf 11.95 -.17 Td[((ki)]TJ /F10 11.955 Tf 11.96 0 Td[(tnk+ki)]TJ /F10 11.955 Tf 11.96 0 Td[(rmk)j,(3) where(i,q,n,m)=1ifandonlyifn,q,misclassiedintotheithclass;otherwise,(i,q,n,m)=0.Equation( 3 )impliesthatthisisactuallyarangettingproblem,andonlythecorrectassociationpatternandtargetpositionestimationareabletoprovidethebestttotheserangeestimates.Therefore,itisacombinedoptimizationproblemofthepeakassociationandtargetpositionestimation.Fromtheclusteringpointofview,therangeestimaten,q,mandki)]TJ /F10 11.955 Tf 12.84 0 Td[(tnk+ki)]TJ /F10 11.955 Tf 12.84 0 Td[(rmk(afunctionofthe 75

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targetposition)representavirtualsampleandaparticularmean,respectively.IntheclassicalK-Meansalgorithm,themeanofaclassisupdatedbyaveragingallsamplesinthisclass.However,intheclusteringproblemdescribedherein,themeanofaclassdependsontherelatedtargetposition,whichneedstobeestimatedbeforere-assigningtheselabels.Foreaseofexposition,anewtargetpositionestimationalgorithmwillbeelaboratedinthenextsubsectionandtheoutlineoftheproposedGKCmethod,whichomitsthedetailsoftargetpositionestimation,isasfollows: 1. Initialization:RandomlyassignQrangesestimates(peaks)ofeachimagewithlabels1,2,...,Q; 2. UpdateofMeans: Fori=1toQ FromtheNMrangeestimates(orequivalently,samplesinthemachinelearningeld)assignedtoLabelicurrently,wedeterminetheithtargetpositiondenotedas^i(theestimationalgorithmwillbedevelopedinthenextsubsection); Plugging^iintoki)]TJ /F10 11.955 Tf 12.02 0 Td[(tnk+ki)]TJ /F10 11.955 Tf 12.02 0 Td[(rmkyieldsthenewmeansk^i)]TJ /F10 11.955 Tf 12.02 0 Td[(tnk+k^i)]TJ /F10 11.955 Tf 12.02 0 Td[(rmkforn=1,...,N,andm=1,...,M. 3. Re-assignmentofLabels: Forn=1,...,N,m=1,...,M,andq=1,...,Q Assigntherangeestimaten,q,mtotheithclass(Labeli)ifandonlyif n,q,m)]TJ /F14 11.955 Tf 11.95 13.27 Td[(k^i)]TJ /F10 11.955 Tf 11.96 0 Td[(tnk+k^i)]TJ /F10 11.955 Tf 11.96 0 Td[(rmk=minp2(1,2,...,Q)n,q,m)]TJ /F14 11.955 Tf 11.96 13.27 Td[(k^p)]TJ /F10 11.955 Tf 11.96 0 Td[(tnk+k^p)]TJ /F10 11.955 Tf 11.95 0 Td[(rmk(3) Intheactualimplementation,weswapn,q,mandn,i,m(afterthere-assignmentofn,q,mtotheithclass),forq=1,...,Qsuccessively,toensurethatn,q,malwaysrepresentstherangeestimatecorrespondingtotheqthtargetafterStep3. 4. RepeatSteps2and3untilconvergence. TheproposedGKCapproachismoreefcientthantheBFAmethodbecausethelatterconsidersallpossibleassociationswhiletheformerinitializeswithonecandidateandconvergetothecorrectassociationpatternafterafewiterations.WeremarkthatifthetargetnumberestimatesobtainedviaBICaredifferentfordifferentrange-Dopplerimages,thenaltargetnumberestimate^Qcanbedeterminedaccordingtosome 76

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add-hoccriterionandthentheGKCapproachcanbeeasilymodiedtobeapplicabletothiscase. 3.2.3EXIP-WLSMethodforTargetPositionEstimation Inthissubsection,weapplytheEXtendedInvariancePrinciple(EXIP)[ 55 ]toestimateqfromtheNMrangeestimatesn,q,m(q)assignedtoLabelq.Wewillfocusonestimatingthepositionparametersoftheqthtargetthroughoutthissubsectionandthesamemethodologycanbereadilyappliedtodealwithothertargetsinastraightforwardmanner. Theorem1:Assumethataone-to-onefunctionfexistsandsatises =f()2D,82D,(3) wherethesetsDandDrepresentsthedomainofthegenericparametervectorsand,respectively. If lim!1^=lim!1f(^),(3) then ^^=argmin^)]TJ /F4 11.955 Tf 11.95 0 Td[(f()TW^)]TJ /F4 11.955 Tf 11.95 0 Td[(f()(3) isasymptotically(whenthenumberofdatasamplesislarge)equivalenttotheestimate^,with W=Eh@2V() @@Ti=^,(3) whereE[]denotestheexpectationoperationandV()representsalossfunctionthatcanbeparameterizedintermsof.Therelatedproofcanbefoundin[ 55 ]. TheweightingmatrixW2R(NM)(NM)canbeobtainedfromthecorrespondingblockofFIMthatisrelatedtothesignalparametervector;seeAppendix B formoredetailsonthissubject. 77

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BasedonEXIP,wedene 1=1,q,1...N,q,11,q,2...N,q,MT(3) astheparametervectoroftheunstructuredmodel,whichcanbeobtainedfromrange-Dopplerimagingresults.Inaddition, f1(q)=2666666666666664kq)]TJ /F10 11.955 Tf 11.95 0 Td[(t1k+kq)]TJ /F10 11.955 Tf 11.96 0 Td[(r1k...kq)]TJ /F10 11.955 Tf 11.96 0 Td[(tNk+kq)]TJ /F10 11.955 Tf 11.95 0 Td[(r1kkq)]TJ /F10 11.955 Tf 11.95 0 Td[(t1k+kq)]TJ /F10 11.955 Tf 11.96 0 Td[(r2k...kq)]TJ /F10 11.955 Tf 11.96 0 Td[(tNk+kq)]TJ /F10 11.955 Tf 11.96 0 Td[(rMk3777777777777775(3) istheone-to-onefunctionoftheparametervectorqintheunstructuredmodel(seethedenitionofthestructuredandunstructuredmodelsinAppendix B ,orreferto[ 55 ]formoredetails). Equation( 3 )providesanestimateofqthatisasymptoticallyequivalenttothemaximumlikelihoodestimateofthestructuredmodel.However,itisanonlinearfunctionofqandthusasearchoveratwo-dimensionalspaceisrequired. Toavoidsuchacomputationallyintensivesearch,weapproximatethesenonlinearequationsusingonlythelinearpartsoftheirTaylorexpansiontorenethetargetpositionestimateinaniterativemanner,anddevelopanEXIP-basediterativeandweightedleastsquare(EXIP-WLS)methodfortargetpositionestimation. Givenaninitialguessofthetargetlocation,denotedas^q=[^xq^yq]T,andtheerrorqbetweenthetruetargetpositionqandtheestimate^qcanbegivenby q=q)]TJ /F5 11.955 Tf 12.36 2.66 Td[(^q.(3) 78

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Thus,thetruetargetrangekq)]TJ /F10 11.955 Tf 12.93 0 Td[(tnk+kq)]TJ /F10 11.955 Tf 12.92 0 Td[(rmkcanberepresentedinitsTaylorexpansionformas:kq)]TJ /F10 11.955 Tf 11.96 0 Td[(tnk+kq)]TJ /F10 11.955 Tf 11.95 0 Td[(rmk=k^q)]TJ /F10 11.955 Tf 11.96 0 Td[(tnk+k^q)]TJ /F10 11.955 Tf 11.95 0 Td[(rmk+)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(qThrk^q)]TJ /F10 11.955 Tf 11.95 0 Td[(tnk+k^q)]TJ /F10 11.955 Tf 11.96 0 Td[(rmk)i+...k^q)]TJ /F10 11.955 Tf 11.96 0 Td[(tnk+k^q)]TJ /F10 11.955 Tf 11.95 0 Td[(rmk+)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(qThrk^q)]TJ /F10 11.955 Tf 11.95 0 Td[(tnk+k^q)]TJ /F10 11.955 Tf 11.96 0 Td[(rmk)i (3) wherer()denotesthegradientvector.In( 3 ),thenonlineartermsaftertherst-orderderivativesaretruncated,andrk^q)]TJ /F10 11.955 Tf 11.96 0 Td[(tnk+k^q)]TJ /F10 11.955 Tf 11.95 0 Td[(rmk)=24@k^q)]TJ /F10 11.955 Tf 11.95 0 Td[(tnk+k^q)]TJ /F10 11.955 Tf 11.96 0 Td[(rmk) @^xq,@k^q)]TJ /F10 11.955 Tf 11.96 0 Td[(tnk+k^q)]TJ /F10 11.955 Tf 11.95 0 Td[(rmk) @^yq35T=hcos^q,n+cos^'q,m,sin^q,n+sin^'q,miT, (3) wherethebearinganglesf^q,ngandf^'q,mgaredenedin( 3 )and( 3 ),respectively,onlywiththetruetargetpositionqreplacedbythecurrentestimate^q. Dene D1=2666666666666664cos^q,1+cos^'q,1sin^q,1+sin^'q,1......cos^q,N+cos^'q,1sin^q,N+sin^'q,1cos^q,1+cos^'q,2sin^q,1+sin^'q,2......cos^q,N+cos^'q,Msin^q,N+sin^'q,M3777777777777775(3) and a1=1)]TJ /F4 11.955 Tf 11.95 0 Td[(f1(^q).(3) LetW1betheblockmatrixofFIMthatisrelatedtothesignalparametervector1(seeAppendix B formoredetailsonhowtoobtainW1).Then( 3 )canbeapproximately 79

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transformedintothefollowingproblem: ^q=argminqa1)]TJ /F10 11.955 Tf 11.95 0 Td[(D1q]TW1a1)]TJ /F10 11.955 Tf 11.95 0 Td[(D1q.(3) Thesolutionto( 3 )isgivenby ^q=DT1W1D1)]TJ /F12 7.97 Tf 6.58 0 Td[(1DT1W1a1.(3) Once^qisavailable,thetargetpositionestimateisupdatedas^q+^q.Torenetheestimate,werepeattheaboveprocedurefrom( 3 )-( 3 )untilconvergence(e.g.,whenk^qkbecomesessentiallyzero). WeremarkthatwhenW1=I,theEXIP-WLS-basedtargetpositionestimationapproachdegradesintoanUnweightedLeastSquare(ULS)one,whichtreatsallrangemeasurementsequally.However,inpracticedifferenttransmitter-receiverpairsencounterdifferentreectioncoefcients.TheEXIP-basedweightingschemeexploitsthefactthatnotalltransmitter-receiverpairsarecreatedequallyforaparticulartargetandthusshouldimprovetheaccuracyofthetargetpositionestimation. 3.2.4EXIP-WLSMethodforTargetVelocityEstimation ByapplyingtheproposedGKCmethod,wecanjointlyobtaintheoptimalassociationpatternandthetargetpositionestimateswhichalsofacilitatethesubsequenttargetvelocityestimation.SimilarlytothemethodologydescribedinSection 3.2.3 ,wecanalsoapplytheextendedinvarianceprincipletodeterminethetargetvelocitiesgiventheassociatedDopplerscalingfactormeasurementsn,q,mandthetargetpositionestimate^q. Dene 2=1,q,1...N,q,11,q,2...N,q,MT(3) 80

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astherelatedparametervectoroftheunstructuredmodel,whichcanbeobtainedfromtherange-Dopplerimagingresults,and f2(vq)=2666666666666664c+xvqcosq,1+yvqsinq,1 c)]TJ /F9 7.97 Tf 6.59 0 Td[(xvqcos'q,1)]TJ /F9 7.97 Tf 6.59 0 Td[(yvqsin'q,1...c+xvqcosq,N+yvqsinq,N c)]TJ /F9 7.97 Tf 6.59 0 Td[(xvqcos'q,1)]TJ /F9 7.97 Tf 6.59 0 Td[(yvqsin'q,1c+xvqcosq,1+yvqsinq,1 c)]TJ /F9 7.97 Tf 6.59 0 Td[(xvqcos'q,2)]TJ /F9 7.97 Tf 6.59 0 Td[(yvqsin'q,2...c+xvqcosq,N+yvqsinq,N c)]TJ /F9 7.97 Tf 6.59 0 Td[(xvqcos'q,M)]TJ /F9 7.97 Tf 6.59 0 Td[(yvqsin'q,M3777777777777775,(3) whichisaone-to-onefunctionoftheparametervectorvqintheunstructuredmodel. Givenaninitialguessofthetargetvelocity^vq=[^xvq^yvq]T,andtheerrorbetweenthetruetargetvelocityvqandtheestimate^vqisvq=vq)]TJ /F5 11.955 Tf 11.52 0 Td[(^vq.LetW2betheblockmatrixofFIMthatisrelatedtothesignalparametervector2(seeAppendix B formoredetailsonhowtoobtainW2).ThenwecansolvethevelocityestimationproblemsimilarlytothatinSection 3.2.3 ): ^vq=argminvqa2)]TJ /F10 11.955 Tf 11.95 0 Td[(D2vq]TW2a2)]TJ /F10 11.955 Tf 11.95 0 Td[(D2vq,(3) where a2=2)]TJ /F4 11.955 Tf 11.95 0 Td[(f2(^vq),(3) and D2=266666666666666664(cos^q,1+cos^'q,1)c+sin(^q,1)]TJ /F12 7.97 Tf 7.83 0 Td[(^'q,1)^yvq (c)]TJ /F12 7.97 Tf 6.7 0 Td[(^xvqcos^'q,1)]TJ /F12 7.97 Tf 6.78 0 Td[(^yvqsin^'q,1)2(sin^q,1+sin^'q,1)c+sin(^'q,1)]TJ /F12 7.97 Tf 7.59 1.77 Td[(^q,1)^yvq (c)]TJ /F12 7.97 Tf 6.7 0 Td[(^xvqcos^'q,1)]TJ /F12 7.97 Tf 6.77 0 Td[(^yvqsin^'q,1)2......(cos^q,N+cos^'q,1)c+sin(^q,N)]TJ /F12 7.97 Tf 7.84 0 Td[(^'q,1)^yvq (c)]TJ /F12 7.97 Tf 6.7 0 Td[(^xvqcos^'q,1)]TJ /F12 7.97 Tf 6.78 0 Td[(^yvqsin^'q,1)2(sin^q,N+sin^'q,1)c+sin(^'q,1)]TJ /F12 7.97 Tf 7.59 1.77 Td[(^q,N)^yvq (c)]TJ /F12 7.97 Tf 6.7 0 Td[(^xvqcos^'q,1)]TJ /F12 7.97 Tf 6.77 0 Td[(^yvqsin^'q,1)2(cos^q,1+cos^'q,2)c+sin(^q,1)]TJ /F12 7.97 Tf 7.83 0 Td[(^'q,2)^yvq (c)]TJ /F12 7.97 Tf 6.7 0 Td[(^xvqcos^'q,2)]TJ /F12 7.97 Tf 6.78 0 Td[(^yvqsin^'q,2)2(sin^q,1+sin^'q,2)c+sin(^'q,2)]TJ /F12 7.97 Tf 7.59 1.77 Td[(^q,1)^yvq (c)]TJ /F12 7.97 Tf 6.7 0 Td[(^xvqcos^'q,2)]TJ /F12 7.97 Tf 6.77 0 Td[(^yvqsin^'q,2)2......(cos^q,N+cos^'q,M)c+sin(^q,N)]TJ /F12 7.97 Tf 7.84 0 Td[(^'q,M)^yvq (c)]TJ /F12 7.97 Tf 6.7 0 Td[(^xvqcos^'q,M)]TJ /F12 7.97 Tf 6.78 0 Td[(^yvqsin^'q,M)2(sin^q,N+sin^'q,M)c+sin(^'q,M)]TJ /F12 7.97 Tf 7.59 1.77 Td[(^q,N)^yvq (c)]TJ /F12 7.97 Tf 6.69 0 Td[(^xvqcos^'q,M)]TJ /F12 7.97 Tf 6.77 0 Td[(^yvqsin^'q,M)2377777777777777775.(3) 81

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Notethatin( 3 )n,q,misreplacedbytheactualDopplermeasurementandfcos'q,m,sin'q,m,cosq,n,sinq,ngisobtainedbyplugging^qinto( 3 )and( 3 ).In( 3 ),onlythelineartermsoftheTaylorexpansionarekeptandD2istherelatedgradientvectoroff2()at^vq.Similariterativeandweightedleast-squaresmethodisappliedtorenethevelocityestimate.Specically,thesolutionto( 3 )canbegivenby^vq=DT2W2D2)]TJ /F12 7.97 Tf 6.59 0 Td[(1DT2W2a2,andthetargetvelocityestimateisthenupdatedas^vq+^vq.WhenW2=I,theEXIP-WLS-basedtargetvelocityestimationapproachalsodegeneratesintoanULSone.SincetheDopplerandrangemeasurementsarepaired,thevelocityandpositionestimatesarepairednaturally. 3.3SimulationResults ConsideramultistaticactivesonarsystemequippedwithN=2transmittersandM=2receivers.ThesystemgeometryisillustratedinFigure 3-1 .ThecoordinatevectorsofthetworeceiversRx1andRx2arer1=[2000,0]Tandr2=[0,2000]T,respectively(theunitofdistanceismeter).Twotransmitters,Tx1andTx2,arelocatedatt1=[0,0]Tandt2=[2000,2000]T,respectively,andtransmittworandomphase(RP)sequencessimultaneously.TheRPsequencesareunimodularwithphasesindependentlyanduniformlydistributedover[0,2).ThereareQ=2targetsmovingintheledofview.Thersttarget,locatedat1=[1000,995]T,ismovingatavelocityofv1=[)]TJ /F12 7.97 Tf 6.58 0 Td[(1.8 p 2,)]TJ /F12 7.97 Tf 6.59 0 Td[(1.8 p 2]Tknots.Thesecondtargetislocatedat2=[1050,965]Tandismovingatv2=[0,)]TJ /F5 11.955 Tf 9.3 0 Td[(2]Tknots.TheDopplerbinscorrespondtoDopplerscalingfactorsrangingfrom0.9976to1.0024withastepsizeof0.0003.Thezero-meanwhiteGaussiannoisewithapowerof1or2isaddedtothemeasurementsacquiredatRx1orRx2,respectively.Thenoisepower,thenormofthetargetreectioncoefcients,andthenormoftheamplitudeandphasemodicationsassociatedwiththedirectblastsarelistedinTable 3-1 ,andtheremainingsystemparametersaresummarizedinTable 3-2 82

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3.3.1Range-DopplerImagingResults ThereceiveroutputsareprocessedusingMF,IAA,SLIM,andIAA-MAP.Theintensityofallrange-Dopplerimagesisnormalizedsothatthepeakisat0dBandisclippedat)]TJ /F5 11.955 Tf 9.3 0 Td[(40dB.Forthesakeofclarityandalsoduetothefactthatthelocationofthedirectblastintherange-Dopplerimagesispredictablegiventhepositionsofthetransmittersandreceivers,therange-Dopplerimagespresentedhenceforthshowthetargetrangeonly.Figure 3-4 andFigure 3-5 showtherange-Dopplerimagesproducedbythefourreceiverlterdesigns.TheMFimageswithrespecttothetwotransmittersformedbyRx1(Rx2)areshowninFigures 3-4A and 3-4B (Figures 3-5A and 3-5B ),respectively.Oneobservesthatduetothemutualinterferencesofthetargetreectionsandstrongdirectblasts,theMFimagesaremiredwithbackgroundnoise,makingitdifculttodetectthetwotargets,whichisinagreementwiththeanalysisinSection 3.2 .Therange-DopplerimagesproducedbyIAA,SLIMandIAA-MAPareshowninFigures 3-4C 3-4D (Figures 3-5C and 3-5D ),Figures 3-4E 3-4F (Figures 3-5E 3-5F )andFigures 3-4G 3-4H (Figures 3-5G 3-5H ),respectively.WecanseethatIAA,SLIM,andIAA-MAPallpossessexcellentinterferencesuppressioncapabilitiesandproducemuchsharperimagesthanMF.Inparticular,IAAcanprovidequiteaccurateestimationresults,buttheresultingdensesidelobesmayburytargetswithweakreectioncoefcients.Incontrast,SLIMenforcessparsityandprovidesrange-DopplerimageswithmuchlesssidelobesthanIAA.AsthehybridofIAAandSLIM,theIAA-MAPmethodtakesadvantagesoftheirmeritswhileovercomingtheirlimitations.Specically,IAA-MAPprovidesmoreaccurateestimatesthanSLIM,whilemaintainingasignicantlylowersidelobelevelthanIAA.IAA-MAPprovidesthecleanestrange-Dopplerimagesamongallmethodsconsideredherein. 3.3.2TargetParameterEstimationResults Fromthetwopeaksofeachrange-DopplerimagegivenbyIAA-MAP,weobtainfourpairsofrangemeasurements(inunitofkilometer),i.e.,f2.7825,2.8200g,f2.7600,2.8275g, 83

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f2.8275,2.9025g,andf2.8350,2.8800g.Forthismultistaticactivesonarsystemequippedwith2transmittersand2receivers,thereare8possibleassociationsforthecaseof2targets.Thebrute-forceapproachneedstoconsiderallpossibilities,andobtainsthecorrectassociationpatternatthecostof1.23secondsonanordinaryworkstation(IntelXeonE5506processor2.13GHz,12GBRAM,Windows764-bit,andMATLABR2010b).Incomparison,theproposedGKCmethodonlyrequires0.56secondsduetoitsefcientsearch.(Themoretargetsarepresentintheeldofview,themorecomputationalsavingtheGKCmethodcanprovide.Forexample,whenthereare3targets,GKCrequires3.47secondswhilethebrute-forcemethodneedsamuchlonger26.01seconds.)Finally,twogroupsofassociatedrangemeasurementsareobtainedas:f2.7825,2.7600,2.9025,2.8800gandf2.8200,2.8275,2.8275,2.8350g.Actually,theMeansupdatestepintheassociationprocedureinvolvestheestimationofthepositionsofthetargetsunderthecurrentassociationassignment.Therefore,theassociationpatternandthecorrespondingtargetpositionestimatesareobtainedsimultaneouslyattheconclusionoftheGKCiterations.Oncethetargetpositionestimatesareavailable,wecandeterminetheirvelocityestimatesasspeciedinSection 3.2.4 .ToevaluatetheperformanceoftheproposedEXIP-WLSalgorithm,therootmean-squarederror(RMSE)oftheestimatedtargetpositionsandvelocitiesobtainedviatheEXIP-WLSandULSmethods(forbothmethods,theso-obtainedtargetpositionestimatesareutilizedtofacilitatethesubsequentvelocityestimation)from100MonteCarlotrialsarelistedinTable 3-3 ,fromwhichwecanseethattheEXIP-basedweightingschemesignicantlyimprovestheestimationaccuracy. 84

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Table3-1. Thenoisepowerandthenormofthetargetreectioncoefcients. WithrespecttoRx1 j1,1,1jj1,2,1jj2,1,1jj2,2,1jj1,1jj2,1j10.080.20.20.30.90.9)]TJ /F5 11.955 Tf 9.3 0 Td[(10dB WithrespecttoRx2 j1,1,2jj1,2,2jj2,1,2jj2,2,2jj1,2jj2,2j20.240.10.160.30.90.9)]TJ /F5 11.955 Tf 9.3 0 Td[(10dB Table3-2. Systemparameters. cunderwatersoundspeed1500m/sor2915.77knotsPlengthofthetransmittedpings400Wbandwidthofthetransmittedpings200HzLnumberofDopplerbins17carrierfrequency900Hzsamplingfrequencyattransmitterandreceiver8000HzP=Wdurationofthetransmittedpings2s Table3-3. RMSEofParameterEstimatesUsingULSandEXIP-WLS. Target1Target2Method^1(dB)^v1(dB)^2(dB)^v2(dB) ULS-9.8561-19.7429-5.7636-15.6197EXIP-WLS-13.6826-22.3587-8.8532-18.1237 85

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Figure3-1. Thesimulationgeometry. 86

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Figure3-2. Agenericactivesonarscenarioforthenthtransmitter,themthreceiver,andtheqthtarget. 87

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Figure3-3. Descriptionoftheassociationproblem. 88

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A B C D E F G H Figure3-4. Range-Dopplerimagesobtainedattherstreceiverproducedbyamultistaticactivesonarsystemusingvariousreceiverlters.Circlesanddiamondsindicatethetruelocationsoftherstandsecondtargets,respectively. 89

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A B C D E F G H Figure3-5. Range-Dopplerimagesobtainedatthesecondreceiverproducedbyamultistaticactivesonarsystemusingvariousreceiverlters.Circlesanddiamondsindicatethetruelocationsoftherstandsecondtargets,respectively. 90

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CHAPTER4WIDEBANDSOURCELOCALIZATIONUSINGSLIM Sourcelocalizationusingasensorarrayplaysanimportantroleinalargevarietyofsignalprocessingapplicationsinvolvingelectromagnetic,acoustic,andseismicsensing.ForexampleasremarkedinChapter 3 ,thetargetassociationproblembecomesaneasyproblemorevendisappearsentirelyprovidedthatthesourcelocalizationtechniquecanaccuratelyestimatetheangleofthetargetsforamultistaticactivesonarsystemofwhicheachreceiverisequippedwithalargearray.Manyadvancedsourcelocalizationtechniques,suchasMUSIC,Capon,andESPRIT(see[ 27 56 57 ]andreferencestherein),havebeendevelopedinthepastdecades.However,mostalgorithmsintheliteratureweredevelopedunderthenarrowbandassumption,underwhichthetime-differenceofarrival(TDOA)ofasignalatvarioussensorsisnegligible.Inotherwords,itisassumedthatasignalarrivesatvarioussensorssimultaneouslywithdifferentphaseshifts.However,inmanypracticalapplications,suchasaeroacousticarrayprocessing[ 58 61 ]andsonar[ 62 63 ],thisassumptionisnotvalid,andthechallengingwidebandarrayprocessingproblemarises. Severaladaptivewidebandsourcelocalizationapproaches,includingthespatialtimefrequencydistributionbasedmethod(e.g.,[ 64 ])andthefocussing-matrixbasedcoherentsignal-subspace(CSS)method(e.g.,[ 65 ]),havebeendevelopedintheliterature.However,theformerisproposedforangleestimationintheparticularcaseofchirpsources,whileforthelattertherequirementofpreliminaryangleestimatesandtheneedforthedesignofafocusingmatrixforeachfrequencybinarerathercomplicatedtasks. Ontheotherhand,thesparsesignalrecovery(SSR)techniqueshaveattractedmuchinterestamongresearchersrecently(e.g.,[ 66 68 ]).Sparsealgorithms,suchas`1normminimization(e.g.,[ 67 69 ])andFOCalUnderdeterminedSystemSolution(FOCUSS)[ 68 ],havebeenappliedtothenarrowbandsourcelocalizationproblem, 91

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resultinginbetterlocalizationperformanceandenhancedcapabilityofresolvingclosely-spacedsources[ 68 71 ].AmongthenumerousadaptiveSSRalgorithms,therecentlydevelopeduser-parameterfreeSparseLearningviaIterativeMinimization(SLIM)algorithm[ 13 47 ]hasshownsatisfactoryperformancesinvariousapplications,suchasnonparametricspectralestimation,radarimaging,andchannelestimationforunderwateracousticcommunications[ 13 ].TheSLIMalgorithmcanworkwithevenasinglesnapshot,arbitraryarraygeometry,coherentornon-coherentsources,andoffershighresolutionspatialestimationresultsatarelativelylowcomputationalcomplexity[ 13 72 ]. ToreaptheSSRbenets,severalSSR-basedwidebandsourcelocalizationmethodshavebeendeveloped.In[ 73 ],theauthorsformulatethewidebandsourcelocalizationproblemasajoint/groupsparsityproblem,andthenproposeamodiedCoSaMP[ 74 ]methodtosolvethejointsparsesignalrecoveryproblem.However,itisknownthatCoSaMPanditsorthogonalmatchingpursuit(OMP)variationsfailtoprovidesatisfactoryperformanceinthesourcelocalizationandspectralestimationapplications,especiallyinthepresenceofcloselyspacedsources(see,e.g.,[ 75 ]).Amethod,namedwidebandcovariancematrixsparserepresentation(W-CMSR),isproposedin[ 76 ],byusingthecovariancematrixttingtechnique.Thismethodrequiresaprioriinformationonthesourcecorrelationfunctions,whichisnotavailableinmanypracticalapplications,suchasaeroacousticandpassivesonar. ThegroupLASSO(gLASSO)algorithm[ 77 ]canalsobeappliedtothewidebandsourcelocalizationproblem.AstheconventionalLASSOalgorithm,gLASSOcontainsauserparameterbalancingthettingerrorandthesparsitypromotingterm.Selectingthisuserparameterisadifculttask.Furthermore,in[ 77 ]theshootingalgorithm[ 78 ]isutilizedtosolvethegLASSOoptimizationproblem,whichessentiallyminimizesthecostfunctionwithrespecttovariousgroupsofcoefcientscyclically.Aswewillshowvianumericalexamples,thisalgorithmsuffersfromaslowconvergenceproblem,and 92

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failstoprovidesatisfactoryresults.WefurtherremarkthatmanytheoreticalanalysesofgLASSOhavebeenperformedintheliterature[ 79 82 ]toinvestigateitsuniquenessandrecoveryconditions.However,mostoftheirconditionsarenotsatisedinthesourcelocalizationapplications,wheresteeringvectorscouldbehighlycorrelatedtoeachotherdependingontheselectedanglesearchinggranularity.Toourbestknowledge,itisstillanopenproblemtotheoreticallyanalyzethebehaviorandperformanceofgLASSO,aswellasmanyotherSSRalgorithms,whenthebasevectorsaresampledfromacontinuousmanifoldfunction. Inthischapter,wepresenttwoextensionsoftheSLIMalgorithmin[ 13 47 ]towidebandsourcelocalization,referredtoasWB-SLIM-0andWB-SLIM-1.Bothalgorithmsexploitthejoint/groupsparsestructure.Weconsiderwidebandsourceswhichemitwidebandsignalsoccupyingallfrequenciesofinterest.Hence,oncewendasignalatacertainangleandaspecicfrequency,wecanexpectsignalsatthesameangleandotherfrequencies.Toexploitthejointspatialsparsestructure,weutilizeahierarchicalBayesianmodelandassumethesamestatisticaldistributionofspatialpseudospectraforvariousfrequencies.Thecyclicminimization(CM)approach[ 53 ]isthenusedtosolvetheMaximum-a-Posteriori(MAP)problem.AswewilldiscussinSection 4.2.3 ,theWB-SLIM-1algorithmcanbealsoformulatedinasimilarformasgLASSO.However,unliketheexistinggLASSOalgorithm,WB-SLIM-1isuserparameterfreeandhencedoesn'trequirenetuningofuser-parameter(s).Moreover,theproposedalgorithmcanconvergemuchfasterthantheexistinggLASSOalgorithms[ 77 ]. WedemonstratetheeffectivenessoftheproposedWB-SLIMalgorithmsforacousticsourcelocalizationusingboththeacousticvectorsensorarrays(VSA)[ 83 85 ]andtheconventionalscalarsensorarrays(SSA).Acousticvectorsensorsmeasurescalarpressurealongwithparticlemotion,andhaveattractedmuchattentionfromresearchersandpractitionersalike.Thistechnologyfeaturesmanyadvantagesovertheomnidirectionalhydrophonesensor,includingresolvingofspatialleft-rightambiguityand 93

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theabilitytoundersampleanacousticwavewithoutspatialaliasing.However,wehavenoticedthatduetothewidebeamofasinglevectorsensor,theambiguitylobecannotbeeliminatedeffectivelyusingtheconventionaldata-independentdelay-and-sum(DAS)approach,especiallywhenthesteeringangleisnearendre.ThisfactimpliesthatDAScannotdetectandlocalizetargetsnearendreuniquely,evenwithavectorsensorarray.Inthischapter,wewilldemonstratethattheproposedWB-SLIMalgorithms,alongwithVSA,caneffectivelyresolvetheleft-rightambiguityproblem. 4.1DataModel ConsiderKfar-eldwidebandsourcesignalsarrivingatasensorarrayfromdirectionsfkgKk=1.LetLbethenumberofsamplesofthereceivedsignal.Atthepre-processingstage,anL-pointdiscreteFouriertransform(DFT)isappliedtothetime-domaindatareceivedateachsensortoconvertthewidebandsignalintoLnarrowbandfrequencysignals.Then,thearrayoutputvectorfylginthepresenceofadditivenoisecanberepresentedas(see,e.g.,[ 65 73 ]): yl=KXk=1al(k)xk,l+nl,forl=1,2,,L(4) whereal(k)istheM1steeringvectorforthesignalarrivingfromkatthelthfrequencybin,withflasthecenterfrequency,andMisthenumberofsensors.Specically,forauniformlineararray(ULA)formedbyscalarsensorswithinter-elementspacing,thesteeringvectorcanbewrittenas: al(k)=1e)]TJ /F9 7.97 Tf 6.59 0 Td[(j2flcos(k) ce)]TJ /F9 7.97 Tf 6.58 0 Td[(j4flcos(k) ce)]TJ /F9 7.97 Tf 6.58 0 Td[(j2(M)]TJ /F12 7.97 Tf 6.59 0 Td[(1)flcos(k) cT,(4) wherecdenotesthepropagationspeed,andkisdenedrelativetotheendre.Foravectorsensorarray(e.g.,[ 83 85 ]),al(k)canbewrittenas: al(k)=al,array(k)aVS,(4) 94

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whereal,arrayisthearraysteeringvectorforascalarsensorarraywithanequivalentspatialcongurationandplanewaveinput,aVSdenotestheresponseofasinglevectorsensor(see[ 85 ]),anddenotesthematrixKroneckerproduct.In( 4 ),xk,ldenotesthecomplex-valuedamplitudeofthesignalarrivingfromkatthelthfrequencybin,andnldenotesthenoiseandinterference. Inpractice,thenumberofwidebandsourcesKisusuallyunknown.Weuseaneanglegrid,denotedbyfngNn=1,coveringthesetoflocationsofsources.Eachpointofthisgridisconsideredtobeapotentialsourcelocation.Viaestimatingthesignalpowerarrivingateachpotentialangle,weobtainspatialpseudospectraforvariousfrequencybins,fromwhichthesourcescanbedetectedandlocalized.Therefore,thefollowingSSRproblemarises:yl=alxl+nl,l=1,...,L, (4) whereal=hal(1),...,al(N)i2CMNwithal(n)denotingthesteeringvectorforthenthpointofthescanninggridatthelthfrequencybin,xl=hx1,l,x2,l,...,xN,liT2CN1denotesthepseudospatialspectrumatthegridpointsandthelthfrequencybin.NotethatusuallyNK.Hence,thecolumnsofalformanovercompletebasisforthesignalyl.Generally,xlcannotbedetermineduniquelyfrom( 4 ).However,inrealapplications,thenumberofsourcesisrelativelysmall.Thisleadstoasparsepropertyofxlthatcanbeexploitedtoidentifyxluniquely. Notethatthedatamodelin( 4 )isdifferentfromthemulti-snapshotmodelin[ 70 86 ],wherethesteeringmatricesarethesameforallsnapshots,i.e.,a1=a2==aL.Notealsothat( 4 )canbere-writtenintheformofthestandardSSRdatamodely=ax+nbystackingthecolumnvectorsfylgontopofeachotherandwiththesteeringmatrixabeingablock-diagonalmatrixformedbyfalg.Thegroup/blockSSRtechniques[ 87 88 ]canthenbeapplied.However,mostoftheexistinggroup/blockSSRalgorithms, 95

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suchasgLASSO[ 77 ],areformulatedusingtheregularizationtechnique,whichrequiresnetuningofuser-parameter(s). 4.2TheWidebandSLIMAlgorithms InspiredbytheSLIMalgorithmin[ 13 47 ],wepresentinthissectiontwoalgorithms,referredtoasWB-SLIM-0andWB-SLIM-1. 4.2.1WB-SLIM-0 Toexploittheaforementionedsparsitystructure,weutilizethehierarchicalBayesianmodel[ 89 ].First,weassumethatthenoisevectorsfnlgareindependentlyidenticallydistributed(i.i.d.)circularlysymmetriccomplexGaussianrandomvectorswithzeromeanandcovariancematrixIwithbeinganunknowndeterministicparameter.Wefurtherassumefxlgtobei.i.d.circularlysymmetriccomplexGaussianrandomvectorswithzeromeanandadiagonalcovariancematrixP,diagfp1,p2,...,pNg.Inthissubsection,fpngareassumedtobedeterministicunknowns.NotethatthejointsparsitystructureoffxlgacrossfrequencybinsisimposedviaassumingthesamecovariancematrixPforallfxlg. Fromtheaboveassumptions,wehavethefollowingprobabilitydensityfunctions(pdf):f(fylgjfxlg,)=LYl=1f(yljxl,)=LYl=11 ()Me)]TJ /F34 5.978 Tf 8.16 3.25 Td[(1 kyl)]TJ /F12 7.97 Tf 6.59 0 Td[(alxlk2, (4) andf(fxlgjfpng)=LYl=11 NQNn=1pne)]TJ /F12 7.97 Tf 6.58 0 Td[(xHlP)]TJ /F34 5.978 Tf 5.75 0 Td[(1xl. (4) TheWB-SLIM-0estimatesoffxlg,fpngandareobtainedbysolvingthefollowingmaximuma-posteriori(MAP)problem:minfxlgfpng,gWB-SLIM-0(fxlgfpng,), (4) 96

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wheregWB-SLIM-0,)]TJ /F5 11.955 Tf 11.29 0 Td[(log[f(fylgjfxlg,)f(fxlgjfpng)]. (4) From( 4 ),( 4 )andafterdiscardingirrelevantconstants,gWB-SLIM-0canbeexpressedas:gWB-SLIM-0=LMlog+1 LXl=1kyl)]TJ /F10 11.955 Tf 11.95 0 Td[(alxlk2+LNXn=1logpn+LXl=1NXn=1jxn,lj2 pn. (4) Notethatwhenpn!0or!0,thecostfunctionin( 4 )canapproachforcertainvaluesoffxlg.Inotherwords,thiscostfunctiondoesnothaveaglobalminimumovertheunconstrainedparameterset.Toaddressthisproblem,weconstrainpnand,withbeingasmallpositivenumber(=10)]TJ /F12 7.97 Tf 6.59 0 Td[(16inournumericalexamples).Weremarkthatundertheconstraintthatpn,minimizing( 4 )willnotleadtoastrictlysparsesolution,i.e.,pn,aswellaselementsofxl,willnotbeexactlyzeros.However,aswewillshowvianumericalexamples,theso-obtainedsolutioncanbeconsideredsparsepractically,inthesensethatmostoftheobtainedfpngaremuchsmallerthantherestandhencethesourcescanbeeasilyseparatedfromnoise. Theoptimizationproblemof( 4 )canbesolvedbyusingthecyclicminimization(CM)technique[ 53 ].First,givenfxlgLl=1and,theminimizationproblemcanbedecoupledandsimpliedasfollows: minpngn(pn),Llogpn+PLl=1jxn,lj2 pn,s.t.pn,(4) forn=1,2,...,N.Differentiatinggn(pn)withrespectto(w.r.t.)pnyields:@gn(pn) @pn=L pn)]TJ /F5 11.955 Tf 15.76 8.09 Td[(1 p2nLXl=1jxn,lj2. (4) Wecaneasilyverifythat@gn(pn) @pn=0whenpn=1 LPLl=1jxn,lj2.Furthermore,thecostfunctiongn(pn)ismonotonicallydecreasingwhen0
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( 4 )is:pn=max(1 LLXl=1jxn,lj2,),n=1,...,N. (4) Similarly,givenfxlgLl=1andfpng,theoptimizationproblemreducesto ming(),LMlog+1 LXl=1kyl)]TJ /F10 11.955 Tf 11.96 0 Td[(alxlk2,s.t.,(4) whosesolutioncanbeobtainedeasilyasfollows:=max(1 LMLXl=1kyl)]TJ /F10 11.955 Tf 11.96 0 Td[(alxlk2,). (4) Finally,forgivenandpn,differentiatinggWB-SLIM-0w.r.t.xlandsettingthederivativetozeroyield:xl=aHlal+P)]TJ /F12 7.97 Tf 6.59 0 Td[(1)]TJ /F12 7.97 Tf 6.59 0 Td[(1aHlyl=PaHlalPaHl+I)]TJ /F12 7.97 Tf 6.58 0 Td[(1yl,l=1,...,L. (4) Thesolutionof( 4 )canbeobtainedviaiterating( 4 ),( 4 ),and( 4 ). 4.2.2WB-SLIM-1 Aswewillshowbynumericalexamples,WB-SLIM-0providesasparsesolution.However,thedownsideofthisisthatitmayfailtodetectweaktargets,especiallyatlowSNR.Inthissubsection,weproposeavariationofWB-SLIM-0,calledWB-SLIM-1,whichprovidesalesssparsebutmorerobustsolution. InadditiontotheGaussiandistributionassumptionsonfnlgandfxlgintroducedinSection 4.2.1 ,wefurtherassumethatfpngNn=1arei.i.d.Gamma(L+1,1 L)distributedthatisf(pn)/pLne)]TJ /F9 7.97 Tf 6.59 0 Td[(Lpn,n=1,...,N. (4) 98

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SimilarlytoWB-SLIM-0,aftertakingthenegativelogarithmofthejointpdf,weobtainthecostfunction:gWB-SLIM-1=LMlog+1 LXl=1kyl)]TJ /F10 11.955 Tf 11.96 0 Td[(alxlk2+LNXn=1pn+LXl=1NXn=1jxn,lj2 pn. (4) Weapplyasimilarcyclicoptimizationprocedureto( 4 ),asdetailedbelow.Forxedfxlgand,theoptimizationproblemreducesto: minpnLpn+LXl=1jxn,lj2 pns.t.pn.(4) Byusingthesametechniqueasfor( 4 ),wecaneasilygettheoptimalsolutionfor( 4 )asfollows:pn=max8<:vuut 1 LLXl=1jxn,lj2,9=;,forn=1,...,N. (4) TheupdatingofxlandisexactlythesameasforWB-SLIM-0(see( 4 )and( 4 )). 4.2.3Discussion TheWB-SLIM-0andWB-SLIM-1algorithmsaresummarizedinTable 4-1 .Bothalgorithmsareinitializedusingtheconventionaldelay-and-sum(DAS)estimates. AsshowninTable 4-1 ,thedifferencebetweenWB-SLIM-0andWB-SLIM-1liesinthefpngupdates,duetothedifferentpriorsassumedforfpng.Weremarkonthefactthatundertheconstraintsandpnforn=1,...,N,thecostfunctionsofWB-SLIM-0andWB-SLIM-1in( 4 )and( 4 )areboundedfrombelow.Byacyclicminimization(CM)property,thecostfunctionsaremonotonicallynon-increasingateachiteration.ThisimpliesthatbothWB-SLIM-1andWB-SLIM-0areconvergentintermsofcostfunction.OurempiricalexperiencesuggeststhattheproposedWB-SLIMalgorithmsdonotprovidesignicantperformanceimprovementsafterabout20iterations.Wefurtherremarkthatthecostfunctionin( 4 )isnotconvex.Hence,convergencetotheglobaloptimumisnotguaranteedforanyalgorithm,includingthose 99

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proposedones.However,aswewilldemonstratenumerically,agoodestimationresultcanbeachievedviachoosingtheinitializationappropriately. Notealsothatignoringtheconstraintpnandminimizingthecostfunctions( 4 )and( 4 )withrespecttofPng,theconcentratedcostfunctionscanbewritten(towithinanadditiveconstant),respectively,asfollows:gWB-SLIM-0=LMlog+1 LXl=1kyl)]TJ /F10 11.955 Tf 11.95 0 Td[(alxlk2+NXn=1log"LXl=1jxn,lj2#, (4) andgWB-SLIM-1=LMlog+1 LXl=1kyl)]TJ /F10 11.955 Tf 11.95 0 Td[(alxlk2+2p LNXn=1"LXl=1jxn,lj2#1 2. (4) Therefore,theWB-SLIM-0andWB-SLIM-1algorithmscanbereformulatedastheminimizationof( 4 )and( 4 ),respectively,whichareextensionsoftheSLIM-qformulation(withq=0or1)in[ 47 ].Theoriginalcostfunctions( 4 )and( 4 )canbeinterpretedasaugmentedfunctionsof( 4 )and( 4 ),whichintroduceadditionaloptimizationvariables(i.e.,fpng)tofacilitatetheCMtechnique.ThisaugmentationoptimizationtechniquecanalsobeusedtosolvethegLASSOoptimizationproblem,whichyieldsanalgorithmsimilartoWB-SLIM-1(Table 4-1 )butwithxed.Inourexperience,thisnewgLASSOalgorithmcanconvergemuchfasterthantheexistingoneintheliterature[ 77 ],wheretheshootingalgorithmisused. 4.3RELAX TheparametricRELAX[ 20 90 ]algorithmisadoptedheretoobtaintherenedangleandpowerestimates.TheWB-SLIMalgorithmsproducesparsespatialangleestimatesandareabletoresolveclosely-spacedsources.HencethenumberofsourcescanbereliablydeterminedbyasimplethresholdingprocedureorbyorderselectionalgorithmssuchastheBayesianinformationcriterion(BIC)[ 52 ].TheRELAXalgorithmrequiresinformationonthenumberofsourcesbutdoesnotdependonthescanninggrid 100

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thatcoverstheregionofpossiblesourcelocations,andhenceitcanbeusedtorenetheWB-SLIMestimates. TheRELAXalgorithmcanbeformulatedasajointnonlinearleast-squareproblem: minfk,xk,lgLXl=1kyl)]TJ /F12 7.97 Tf 18.38 16.72 Td[(^KXk=1al(k)xk,lk2,(4) where^Kistheestimatednumberofsources.InitializedwiththeWB-SLIMresults,RELAXsolvestheoptimizationproblem( 4 )cyclically.Toupdatetheestimatesofthekthsource,wedene: ~yk,l=yl)]TJ /F14 11.955 Tf 11.96 11.36 Td[(Xi6=kal(^i)^xi,l,(4) wheref^i,^xi,lgi6=karethemostrecentestimatesoftheothersources.Then,theoptimizationproblemisreducedto: minfk,xk,lgLXl=1k~yk,l)]TJ /F10 11.955 Tf 11.95 0 Td[(al(k)xk,lk2.(4) SolvingthisoptimizationproblemyieldstheDAS-typeestimates: ^k=argmax^kLXl=1jaHl(^k)~yk,lj2,(4) and ^xk,l=aHl(^k)~yk,l kal(^k)k2forl=1,2,...,L.(4) TheRELAXalgorithmupdatesf^k,^xk,lgforallsourcesiterativelyusing( 4 )and( 4 ).Weterminatetheiterationwhenthenormofthedifferencebetweentwoconsecutiveestimatesfallsbelowapredenedsmallthreshold(10)]TJ /F12 7.97 Tf 6.59 0 Td[(4inournumericalexamples).TheRELAXapproachissummarizedinTable 4-2 OwingtotheCMapproachemployed,theRELAXalgorithmislocallyconvergent.NotethatRELAXrequiresonly1DmaximizationsaroundtheWB-SLIMestimates.Thismaximizationoperationcanbeefcientlyimplementedusingderivative-freeuphillsearch 101

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methodssuchastheNelder-Meadalgorithm[ 91 ],whichisincorporatedintheMATLABoptimizationtoolboxasthefunctionfminsearch. 4.4NumericalExamples Inthissection,wepresentseveralnumericalexamplestodemonstratetheexcellentangleestimationperformanceoftheproposedmethods.Weconsidertwoclosely-spacedwidebandsourceslocatednearthearrayendreat10.11and11.02.Weassumethatthesignalsemittedbythetwosourcesarestatisticallyindependentwithatpowerspectrarangingfrom)]TJ /F5 11.955 Tf 9.3 0 Td[(10KHzto10KHz.A128-pointFFTisperformedatthepreprocessingstagetodecomposetheentirefrequencybandintoL=128narrowfrequencybins.TheSNRsofthetwosourcesare25dBand20dB,respectively,unlessotherwisespecied.Weassumethatthespeedofsound(inthesea)is1530m/s. Werstconsidera20-elementscalar-sensorULAwiththeinter-elementspacingequalto=3meters.WeapplytheproposedWB-SLIM-0andWB-SLIM-1tothereceivedsignal.Forcomparisonpurposes,severalconventionalangleestimationalgorithms,namelyDAS,NB-SLIM-0andNB-SLIM-1,arealsoconsidered.ThesemethodsapplytheDASorthenarrowbandSLIMalgorithmsof[ 13 47 ]toeachfrequencybin,andthencombinetheobtainedspatialpseudospectranon-coherently.Forallthealgorithmsinthissection,thescanninggridisuniformwith0.25incrementbetweenadjacentpoints.Throughoutthischapter,theiterationnumberofgroupLASSOissetto5000,andtheiterationnumbersofNB-SLIMandWB-SLIMarexedto50. Figure 4-1 showsthespatialpseudospectra,i.e.,fpngNn=1,obtainedbyvariousalgorithms.Thedashedlinesindicatethetrueangles.FromFigures 4-1A 4-1F ,wecanseeclearlythatthescalar-sensorarraysuffersfromtheleft-rightambiguityproblem.Furthermore,wenotethatthedata-independentDASalgorithmsuffersfromthelow-resolutionproblem:itisunabletoresolvethetwoclosely-spacedtargets.Figure 4-1B showsthespatialpseudospectraobtainedbygLASSO,whichrepresentsthebestresultswegotvianetuningofthegLASSOuser-parameter.Aswecan 102

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see,thegLASSOalgorithmfailstodetectthetwosourcescorrectly.NotethattheshootingalgorithmhasbeenusedtoobtainFigure 4-1B .Aswediscussedabove,theaugmentationoptimizationtechniquecanalsobeusedtosolvethegLASSOproblem.ThisnewgLASSOalgorithmyieldssimilarresultsastheWB-SLIM-1inFigure 4-1E .ThespatialpseudospectraofNB-SLIM-1andNB-SLIM-0areshowninFigures 4-1C and 4-1D ,respectively.Thesetwonarrow-bandSLIMalgorithmscan(althoughonlybarely)resolvethetwoclosely-spacedtargets.However,bothprovidenoisyspatialpseudospectra,whichmayleadtohighfalsedetectionrates.ThereasonforthisisthatbothmethodsapplythenarrowbandSLIMalgorithmtoeachfrequencybinindependently.Whenthefrequencyislargerthan510Hz,thecorrespondinginter-elementspacingislargerthanhalfwavelength.ThenarrowbandSLIMalgorithmwillthengenerateambiguouslobesforfrequencybinswithfrequencylargerthan510Hz.Afterthenon-coherentcombinationofspatialpseudospectra,theseambiguouslobesbecomehighsidelobes.Incontrast,asshowninFigures 4-1E and 4-1F ,bothWB-SLIM-0andWB-SLIM-1areabletoeffectivelysuppressthesehighsidelobes,providecleanandsparsespatialpseudospectra,andneatlyresolvethetwoclosely-spacedsources.NotethatWB-SLIM-0performsbetterthanWB-SLIM-1inthisexample. Figure 4-2 correspondstoavectorsensorarrayexample.ThesimulationparametersareexactlythesameasthoseusedtoobtainFigure 4-1 ,exceptthatthescalarsensorsarereplacedbyvectorsensors.FromFigures 4-2A and 4-2B ,wecanseethat,onceagain,DASandgLASSOcannotresolvethetwoclosely-spacedsources.Theambiguouslobearound-10degreesisabout0.3dBlowerthanthemainlobe.Thismeansthatthevectorsensorarrayinprinciplehasthecapabilitytoresolvetheleft-rightambiguityproblem.AswecanseefromFigures 4-2C and 4-2D ,NB-SLIM-1andNB-SLIM-0suppresstheambiguouslobeevenmoretoabout10dBundermainlobe.Figures 4-2E and 4-2F showsthattheproposedWB-SLIM-0andWB-SLIM-1algorithmsoutperformtheirnarrowbandcounterpartssignicantly.Bothareabletoprovideclean 103

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andsparsespatialpseudospectra,resolvethetwoclosely-spacedsources,andsupresstheambiguouslobes.WeconsideramorechallengingexampleinFigure 4-3 ,wheretheSNRofthesecondsourceisdecreasedto0dB,i.e.25dBlowerthanfortherstone.TheothersimulationparametersarethesameasthoseusedinFigure 4-2 .Forabetterillustration,weshowthezoomed-inspatialpseudospectraintheregion[0,20]degrees.Aswecansee,onlyWB-SLIM-1isabletoidentifythetwosourcescorrectlyinthischallengingcase. Figure 4-4 showstheperformanceimprovementachievedbyusingtheRELAXalgorithm.ThesimulationparametersarethesameasthoseusedinFigure 4-1 .WerstutilizetheWB-SLIM-1orWB-SLIM-0algorithmtodetectthenumberofsourcesandobtaininitialangleandamplitudeestimates,whicharethenrenedusingRELAX.InFigures 4-4A and 4-4B ,theredcirclesindicatethetrueanglesandpowersofthesources,andtheblackstarsshowtheRELAXestimates.Forcomparisonpurposes,wealsoshowthespatialpseudospectraofDASandWB-SLIM,whosepeaksindicatethecorrespondingangleandpowerestimates.Wedisplaytheangleandpowerestimatesobtainedin20Monte-Carloruns.FromFigures 4-4A and 4-4B ,weseethatRELAXeffectivelyrenestheangleandpowerestimatesofWB-SLIM-1orWB-SLIM-0,andobtainsquiteaccurateestimates. Finally,wedemonstratetheperformanceofvariousalgorithmsintermsofthedetectionrateandtheroot-mean-squared-error(RMSE).Forperformancecomparisonpurposes,wecomputethewidebanddeterministicCramer-Raobound(CRB)fortheangleestimates,whichisgivenby[ 92 ],asfollows:CRB()= 2"LXl=1Ren(_aHl?al_al)(xlxHl)To#)]TJ /F12 7.97 Tf 6.59 0 Td[(1 (4) where_al,"dal() d=1,...,dal() d=K#, (4) 104

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and?al,I)]TJ /F10 11.955 Tf 11.95 0 Td[(al)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(aHlal)]TJ /F12 7.97 Tf 6.59 0 Td[(1aHl. (4) WeconsiderthesamearrayandwidebandsourcesforFigure 4-1 .Thex-axesofFigures 4-5A 4-5C showtheSNRoftherstsource.TheSNRthesecondsourceis5dBsmallerthanthatoftherst.Weperform100Monte-Carlosimulations.AMonte-CarlotrialisdeemedfailedwhenoneormoresourcesarenotdetectedoreitheroftheDOAestimatesismorethan0.5degreesawayfromthecorrespondingtruevalue.Thedetectionrateisdenedastheratioofthenumberoftrialsinwhichbothsourcesaredetectedandlocalizedcorrectlyoverthetotaltrialnumber.FromFigure 4-5A ,wecanseethatbothWB-SLIM-0andWB-SLIM-1achieve100%detectionratewhenSNRislargerthanorequalto5dB.However,atlowSNR,WB-SLIM-1outperformsWB-SLIM-0signicantly.Figures 4-5B and 4-5C showthatRELAX,initializedbyeitherWB-SLIM-1orWB-SLIM-0,canachievegoodestimationperformance.NotethatwhenSNRislessthan5dB,theRMSEoftheWB-SLIM-0&RELAXestimatesisnotdenedduetothedetectionfailure.Hence,weonlyprovideRMSEwhenSNR5dBforthisalgorithm.TheRMSEsoftheangleestimatesareapproximatelyequaltoRCRBforthestrongersource,andareabouttwotimestheRCRBfortheweakone(notethatthedeterministicCRBisnotnecessarilyachievable,see,e.g.,[ 92 ]). 105

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A B C D E F Figure4-1. Spatialpseudospectraobtainedwithascalarsensorarrayandvariousalgorithms:A)DAS,B)gLASSO,C)NB-SLIM-1,D)NB-SLIM-0,E)WB-SLIM-1,andF)WB-SLIM-0. 106

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A B C D E F Figure4-2. Spatialpseudospectraobtainedwithavectorsensorarrayandvariousalgorithms:A)DAS,B)gLASSO,C)NB-SLIM-1,D)NB-SLIM-0,E)WB-SLIM-1,andF)WB-SLIM-0. 107

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A B C D E F Figure4-3. Spatialpseudospectra,inthecaseofaweaksource,obtainedbyvariousalgorithms:A)DAS,B)gLASSO,C)NB-SLIM-1,D)NB-SLIM-0,E)WB-SLIM-1,andF)WB-SLIM-0. 108

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A B Figure4-4. PerformanceenhancementusingRELAX.A)WB-SLIM-1andRELAX,B)WB-SLIM-0andRELAX. A B C Figure4-5. EmpiricalfailurerateandRMSEsversusSNR.A)Detectionrate,B)RMSEofSource1,andC)RMSEofSource2. 109

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Table4-1. WB-SLIMalgorithms. InitializefxlgLl=1andwiththeDASestimatesasfollows: x(0)n,l=aHl(n)yl=kal(n)k2,forn=1,...,N;l=1,...,L;(0)=max(1 10LMLXl=1kx(0)lk2,) Repeatthefollowingstepsfort=0,1,2,: p(t+1)n=8><>:maxn1 LPLl=1jx(t)n,lj2,o(WB-SLIM-0)maxq 1 LPLl=1jx(t)n,lj2,(WB-SLIM-1),forn=1,...,N;x(t+1)l=P(t+1)aHl(alP(t+1)aHl+(t)I))]TJ /F12 7.97 Tf 6.58 0 Td[(1yl,l=1,...,L;(t+1)=max(1 LMLXl=1kyl)]TJ /F10 11.955 Tf 11.96 0 Td[(alx(t+1)lk2,). untilconvergence 110

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Table4-2. TheRELAXalgorithm. ^K:NumberofsourcesobtainedfromWB-SLIM f^kg^Kk=1:LocationsofthesourcesobtainedfromWB-SLIM f^xk,lg^K,Lk=1,l=1:CorrespondingwaveformsobtainedfromWB-SLIM repeat fork=1,...,^K ~yk,l=yl)]TJ /F14 11.955 Tf 11.96 8.96 Td[(P^Ki=1,i6=kal(^i)^xi,ll=1,...,L ^k=argmaxkPLl=1jaHl(k)~yk,lj2 ^xk,l=aHl(^k)~yk,l kal(^k)k2,l=1,...,L endfor until(convergence) 111

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CHAPTER5CONCLUSIONSANDFUTUREWORK VarioussignalprocessingtechniqueshavebeenproposedandappliedtothereceiverdesignoftheMIMOUACsystemandthemultistaticactivesonarsystem,respectively.InChapter 2 ,wearemainlyinvolvedinndingthepropermodelfortheacousticchannelssubjecttobothISIandtimescalingeffectswhicharelargelyinducedbytherelativemotionsbetweenthetransmitterandreceiverarrays.WendoutthatapreferablewayistoparsimoniouslymodelthechannelbyassumingacommonDopplerscalingfactorimposedonthepropagationpathsamongallthetransmitterandreceiverpairs.TemporalresamplinghasbeenusedtoeffectivelyconverttheDopplerscalingeffectstoDopplerfrequencyshiftsandadata-adaptivesparsechannelestimationalgorithm,referredtoasGoSLIM-V,isusedtoestimatetheunderlyingCIRsandDopplerfrequencyinajointmanner.Forsymboldetection,wehaveinvestigatedtheturboequalizationschemesimplementedbytheLMMSE-basedsoft-inputsoft-outputequalizeraswellasitslowcomplexityapproximation.Thelatterprovidesonlyslightlydegradeddetectionperformancebutatasignicantlylowercomputationalcomplexitycomparedtotheformerandisthuspreferred. InChapter 3 ,wehavefocusedontwosignalprocessingaspectsofamultistaticactivesonarsystem:Range-Dopplerimagingandtargetparameterestimation.Fortheformer,wehavepresentedtheIAA-MAPmethodtoenhancetheresolutionandsuppresssidelobelevelssimultaneously.Forthelatter,wehaveintroducedageneralizedK-Meansclusteringapproachfortargetpeakassociationandappliedtheextendedinvarianceprincipletodeterminethetargetpositionandvelocityestimatesviaweightedleast-squarestting.Moreover,onepossiblewaytomitigatethetargetassociationproblemistoaccuratelyestimatetheangleofthetargetsbyapplyingadvancedsourcelocalizationmethodtoreceiverseachequippedwithalargearray.Sincemostexistingtechniquesaredevelopedunderthenarrowbandassumption,we 112

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areinspiredtoderivetheWB-SLIM-0andWB-SLIM-1algorithmsforwidebandsourcelocalization,basedontwodifferenthierarchicalBayesianstatisticalmodels.Thesetwoalgorithmsprovidehigh-resolutionangleestimateswithouttheneedoftuninganyuserparameters.WehavealsoproposedawidebandRELAXalgorithmtorenetheangleandpowerestimatesobtainedwiththeWB-SLIM-0andWB-SLIM-1beyondtheaccuracyallowedbythenenessofthegridusedinthelatter.BothnumericalandexperimentalexampleshavebeenprovidedinChapters 2 3 ,and 4 todemonstratetheeffectivenessandtheexcellentperformanceoftheproposedUACandactivesonarsystemdesign. NotethatconventionalsonarsystemasdiscussedinChapter 3 isapulsedactivesonar(PAS)system,whereapowerfulpulseissentoutbeforearelativelylonglistingtime[ 42 43 93 ].ThePASsystemhoweverhasseveraldisadvantages.Firstly,atargetatlongrangerequireslonglisteningtimebecauseofthelowunderwaterpropagationspeedofsound.Consequently,thereisonlyashortperiodoftimefordetectingtargetsduetotheshortpulseduration.Secondly,thepoweroutputofthetransmittersishighenoughtoinduceenvironmentpollution.Forinstance,suchpollutioncoldpossiblyleadtothemassstrandingofmarineanimals[ 94 ].Thirdly,theDopplerresolutionisinverselyproportionaltothepulsedurationandisthereforerelativelylow.Incomparison,multistaticcontinuousactivesonar(CAS)systemsoffersomeattractiveadvantages[ 95 96 ].Firstly,byexploitingthecontinuoustransmissionandthespatialdiversity,multistaticCASsystemscanprovidesignicantlyenhancedtargetdetectionandparameterestimationperformance.Secondly,lowerpeakpowerlevelisachievedbecauseofcontinuoustransmission,whichalleviatesdetrimentalnoisepollution.Thirdly,theDopplerresolutionofthemultistaticCASsystemismuchhigherthanthepulsedsystem.TheseadvantagesmotivateustoinvestigatetheimplementationofthemultistaticCASsysteminthefuture. 113

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Wealsowanttodesignawaveformsetwhichpossessessatisfactoryauto-andcross-correlationpropertiestoimprovethetargetdetectionperformanceandalsosatisesspectrumcontainmentrestrictionsrequiredbytheoceanicenvironmentandthehardwaresystem[ 97 100 ].Also,goodrange-Dopplerimagingperformanceisdesiredforthesubsequenttargetparameterestimation.Theperformanceofvariousadvancedadaptivereceiverlterdesignsinprovidingrange-Dopplerimageswithbothlowsidelobelevelsandhighaccuracywillbeinvestigated.Wewanttocomparethesparsityoftherange-DopplerimagesobtainedfromthesevariousmethodsandtheircomputationalcomplexitiestodecidethemostappropriatemethodfortheMCASapplication.Moreover,theimplementationoftheMCASsystemfacilitatesthetargettracking.ThelengthoftheCPIcanbeadjustedtodeterminehowoftenwewouldliketotrackthetargetandthedetailedtargetmaneuvercanbemonitoredwitharathershortCPI.Inanutshell,thedesignoftheprobingsequencesetanditscorrespondingreceiverlterforanMCASsystemformthefocusofmyfuturework. 114

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APPENDIXATHEDERIVATIONOFTHEDOPPLERSCALINGFACTOR Thetransmittedsignalisreectedfromthenthtransmittertothemthreceiverbytheqthmovingtarget,andthustheDopplerscalingfactorbetweenthesignaltransmittedfromthenthtransmitterandtheonereceivedatthemthreceivercanberepresentedas[ 93 101 ]: n,q,m=c+vts(n) c)]TJ /F4 11.955 Tf 11.96 0 Td[(vt(n)c+vr(m) c)]TJ /F4 11.955 Tf 11.95 0 Td[(vrt(m),(A) wherevts(n)isthevelocityofthetargetrelativetothenthtransmitter,andvrt(m)isthatofthemthreceiverrelativetothetarget.Inaddition,vt(n)andvr(m)arethevelocitiesofthenthtransmitterandmthreceiver,respectively. Denethephaseangleofthetargetvelocityvectorvq=[xvq,yvq]Tas q=arctan(yvq=xvq),wecanrepresentvts(n)andvrt(m)asvts(n)=kvqkcos(q,n)]TJ /F13 11.955 Tf 12.57 0 Td[( q)andvrt(m)=kvqkcos( q)]TJ /F13 11.955 Tf 12.19 0 Td[('q,m),respectively.Inaddition,vt(n)andvr(m)equalzerosinceboththetransmitterandreceiverarestationary.Therefore,theDopplerscalingfactorn,q,mcanberewrittenas: n,q,m=c+vts(n) c)]TJ /F4 11.955 Tf 11.95 0 Td[(vrt(m)=c+p (xvq)2+(yvq)2cos(q,n)]TJ /F13 11.955 Tf 11.95 0 Td[( q) c)]TJ /F14 11.955 Tf 11.95 9.58 Td[(p (xvq)2+(yvq)2cos( q)]TJ /F13 11.955 Tf 11.96 0 Td[('q,m)=c+p (xvq)2+(yvq)2cos qcosq,n+p (xvq)2+(yvq)2sin qsinq,n c)]TJ /F14 11.955 Tf 11.95 9.58 Td[(p (xvq)2+(yvq)2cos qcos'q,m)]TJ /F14 11.955 Tf 11.95 9.58 Td[(p (xvq)2+(yvq)2sin qsin'q,m=c+xvqcosq,n+yvqsinq,n c)]TJ /F4 11.955 Tf 11.95 0 Td[(xvqcos'q,m)]TJ /F4 11.955 Tf 11.96 0 Td[(yvqsin'q,m. (A) 115

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APPENDIXBTHEDERIVATIONOFTHEWEIGHTMATRIX Foreaseofexposition,weassumethatinthesaidactivesonarsystem,therearetwotransmitters,tworeceivers,andonemovingtarget.BasedonthesignalmodelinSection 3.2 ,thereceivedsignaly1(t)andy2(t)atthe1stand2ndreceiverscanberepresentedinanunstructuredmodelas: y1(t)=1,1,1s1(1,1,1(t)]TJ /F13 11.955 Tf 11.96 0 Td[(1,1,1))+2,1,1s2(2,1,1(t)]TJ /F13 11.955 Tf 11.96 0 Td[(2,1,1))+1,1s1(t)]TJ /F13 11.955 Tf 11.96 0 Td[(1,1)+2,1s2(t)]TJ /F13 11.955 Tf 11.95 0 Td[(2,1)+e1(t), (B) and y2(t)=1,1,2s1(1,1,2(t)]TJ /F13 11.955 Tf 11.96 0 Td[(1,1,2))+2,1,2s2(2,1,2(t)]TJ /F13 11.955 Tf 11.96 0 Td[(2,1,2))+1,2s1(t)]TJ /F13 11.955 Tf 11.96 0 Td[(1,2)+2,2s2(t)]TJ /F13 11.955 Tf 11.95 0 Td[(2,2)+e2(t), (B) respectively,wheretheparametervectoroftheunstructuredmodelis 1=[Ref1,1,1g,Imf1,1,1g,Ref2,1,1g,Imf2,1,1g,Ref1,1,2g,Imf1,1,2g,Ref2,1,2g,Imf2,1,2g,Ref1,1g,Imf1,1g,Ref2,1g,Imf2,1g,Ref1,2g,Imf1,2g,Ref2,2g,Imf2,2g,1,1,1,2,1,1,1,1,2,2,1,2,1,1,2,1,1,2,2,2,1,1,1,2,1,1,1,1,2,2,1,2]T. (B) Inthesampleddiscreteform,( B )and( B )canberepresentedas y1(k)=1(k)+e1(k),(B) and y2(k)=2(k)+e2(k),(B) respectively,fork=1,2,...K,where 1(k)=1,1,1s1(1,1,1(k)]TJ /F4 11.955 Tf 11.95 0 Td[(k1,1,1))+2,1,1s2(2,1,1(k)]TJ /F4 11.955 Tf 11.95 0 Td[(k2,1,1))+1,1s1(k)]TJ /F4 11.955 Tf 11.95 0 Td[(k1,1)+2,1s2(k)]TJ /F4 11.955 Tf 11.96 0 Td[(k2,1), (B) 116

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and 2(k)=1,1,2s1(1,1,2(k)]TJ /F4 11.955 Tf 11.95 0 Td[(k1,1,2))+2,1,2s2(2,1,2(k)]TJ /F4 11.955 Tf 11.95 0 Td[(k2,1,2))+1,2s1(k)]TJ /F4 11.955 Tf 11.95 0 Td[(k1,2)+2,2s2(k)]TJ /F4 11.955 Tf 11.96 0 Td[(k2,2). (B) In( B )and( B ),Kisthelengthofthereceivedmeasurementsands(k)=s(t)jt=kT(Tisthesamplingperiod).WeassumethatE[eieHi]=2I,fori=1and2.AccordingtotheSlepian-Bangsformula[ 102 ],theFIMisgivenasfollows: F1=2K 2Re(2Xi=1KXk=1DHi,k(1)Di,k(1)),(B) where Di,k(1)=@i(k) T1,fori=1,2,andk=1,2,...,K.(B) Sincethetruevaluesof1isunavailableintheactualimplementation,wereplacethemwiththeestimatesobtainedfromtherange-Dopplerimagingstep.Thesequencess1(k)ands2(k)areknownatthereceiversides. OncetheapproximateFIMisobtained,weextractthe44blockofF1relatedtof1,1,1,2,1,1,1,1,2,2,1,2gastheweightingmatrixW1,andtheblockwithsize44correspondingtof1,1,1,2,1,1,1,1,2,2,1,2gasW2. Wenallyremarkthatbyplugging( 3 )and( 3 )into( B )and( B ),whicharefunctionsoff1,1,1,2,1,1,1,1,2,2,1,2gandf1,1,1,2,1,1,1,1,2,2,1,2g,theso-obtainedequationsbecomefunctionsoffx1,y1,xv1,yv1gandarereferredtoasthestructuredmodel.Therefore,thecorrespondingparametervectorofthisstructuredmodelis 2=[Ref1,1,1g,Imf1,1,1g,Ref2,1,1g,Imf2,1,1g,Ref1,1,2g,Imf1,1,2g,Ref2,1,2g,Imf2,1,2g,Ref1,1g,Imf1,1g,Ref2,1g,Imf2,1g,Ref1,2g,Imf1,2g,Ref2,2g,Imf2,2g,1,1,2,1,1,2,2,2,x1,y1,xv1,yv1]T. (B) 117

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REFERENCES [1] M.StojanovicandJ.Preisig,Underwateracousticcommunicationchannels:Propagationmodelsandstatisticalcharacterization,IEEECommunicationsMagazine,vol.47,no.1,pp.84,January2009. [2] D.TseandP.Viswanath,FundamentalsofWirelessCommunication,CambridgeUniversityPress,NewYork,NY,2005. [3] J.Preisig,Acousticpropagationconsiderationsforunderwateracousticcommunicationsnetworkdevelopment,SIGMOBILEMobileComputingCommu-nicationsReview,vol.11,no.4,pp.2,2007. [4] J.Catipovic,Performancelimitationsinunderwateracoustictelemetry,IEEEJournalofOceanicEngineering,vol.15,no.3,pp.205,July1990. [5] W.LiandJ.C.Preisig,Estimationofrapidlytime-varyingsparsechannels,IEEEJournalofOceanicEngineering,vol.32,no.4,pp.927,October2007. [6] W.Li,Estimationandtrackingofrapidlytime-varyingbroadbandacousticcommunicationchannels,PhDthesis,MassachusettsInstituteofTechnology,Cambridge,MA,2005. [7] N.Weste,M.Bickerstaff,T.Arivoli,P.J.Ryan,J.W.Dalton,D.J.Skellern,andT.M.Percival,A50MHz16-pointFFTprocessorforWLANapplications,ProceedingsoftheIEEE1997CustomIntegratedCircuitsConference,pp.457,May1997. [8] H.WangandL.Cai,OnadaptivemultibandsignaldetectionwithGLRalgorithm,IEEETransactionsonAerospaceandElectronicSystems,vol.27,pp.225,Mar.1991. [9] M.Wax,Detectionandestimationofsuperimposedsignals,March,1985. [10] D.Kilfoyle,J.C.Preisig,andA.B.Baggeroer,Spatialmodulationexperimentsintheunderwateracousticchannel,IEEEJournalofOceanicEngineering,vol.30,no.2,pp.406,April2005. [11] B.Li,S.Zhou,M.Stojanovic,L.Freitag,andP.Willett,MulticarriercommunicationoverunderwateracousticchannelswithnonuniformDopplershifts,IEEEJournalofOceanographicEngineering,vol.33,no.2,pp.198,April2008. [12] J.Ling,T.Yardibi,X.Su,H.He,andJ.Li,Enhancedchannelestimationandsymboldetectionforhighspeedmulti-inputmulti-ouputunderwateracousticcommunications,JournalofAcousticSocietyofAmerica,vol.125,no.5,pp.3067,May2009. [13] J.Ling,X.Tan,T.Yardibi,J.Li,H.He,andM.L.Nordenvaad,EnhancedchannelestimationandefcientsymboldetectioninMIMOunderwateracoustic 118

PAGE 119

communications,in43thAsilomarConferenceonSignals,SystemsandComput-ers,PacicGrove,CA,November1-4,2009. [14] M.Stojanovic,J.Catipovic,andJ.Proakis,Phasecoherentdigitalcommunicationsforunderwateracousticchannels,IEEEJournalofOceanicEngineering,vol.19,no.1,pp.100,January1994. [15] M.Stojanovic,L.Freitag,andM.Johnson,Channel-estimation-basedadaptiveequalizationofunderwateracousticsignals,IEEE/MTSOceansConference,vol.2,pp.985,September1999. [16] A.Song,M.Badiey,andV.K.McDonald,MultichannelcombiningandequalizationforunderwateracousticMIMOchannels,inProceedingsofMTS/IEEEOceansconference,QuebecCity,Canada,September2008. [17] S.Gray,J.Preisig,andD.Brady,Multiuserdetectioninahorizontalunderwateracousticchannelusingarrayobservations,IEEEJournalofOceanicEngineering,vol.21,no.3,pp.148,January1997. [18] P.W.Wolniansky,G.J.Foschini,G.D.Golden,andR.A.Valenzuela,V-blast:Anarchitectureforrealizingveryhighdataratesovertherich-scatteringwirelesschannel,Proc.ISSSE-98,Sept.29,1998. [19] R.L.Cupo,G.D.Golden,C.C.Martin,K.L.Sherman,N.R.Sollenberger,J.H.Winters,andP.W.Wolniansky,Afour-elementadaptiveantennaarrayforIS-136PCSbasestations,47thIEEEVehicularTechnologyConference,vol.3,pp.1577,May1997. [20] J.LiandP.Stoica,Efcientmixed-spectrumestimationwithapplicationstotargetfeatureextraction,IEEETransactionsonSignalProcessing,vol.44,pp.281,February1996. [21] J.Ling,K.Zhao,J.Li,andM.L.Nordenvaad,Multi-inputmulti-outputunderwatercommunicationsoversparseandfrequencymodulatedacousticchannels,JournaloftheAcousticalSocietyofAmerica,vol.130,pp.249,July2011. [22] M.Tuchler,A.Singer,andR.Koetter,Minimummeansquarederrorequalizationusingapriorinformation,IEEETransactionsonSignalProcessing,vol.50,no.3,pp.673,March2002. [23] M.Tuchler,R.Koetter,andA.Singer,Turboequalization:principlesandnewresults,IEEETransactionsonCommunications,vol.50,no.5,pp.754,May2002. [24] R.Koetter,A.Singer,andM.Tuchler,Turboequalization,IEEESignalProcess-ingMagazine,vol.21,no.1,pp.67,January2004. [25] K.Zhao,J.Ling,andJ.Li,OnestimatingsparseandfrequencymodulatedchannelsforMIMOunderwateracousticcommunications,Proceedingsof 119

PAGE 120

49thAnnualAllertonConferenceonCommunication,Control,andComputing,Monticello,IL,pp.15,September2011. [26] M.LundbergandT.Oberg,IterativereceptionforacousticunderwaterMIMOcommunications,inProceedingsofMTS/IEEEOceansconference,Boston,USA,September2006. [27] S.Roy,T.Duman,V.McDonald,andJ.Proakis,High-ratecommunicationforunderwateracousticchannelsusingmultipletransmittersandspace-timecoding:Receiverstructuresandexperimentalresults,IEEEJournalofOceanicEngineering,vol.32,pp.663,July2007. [28] P.Robertson,E.Villebrun,andP.Hoeher,Acomparisonofoptimalandsub-optimalMAPdecodingalgorithmsoperatinginthelogdomain,IEEEIn-ternationalConferenceonCommunications,Seattle,WA,pp.1009,June1995. [29] C.Douillard,C.M.Jezequel,C.Berrou,A.Picart,P.Didier,andA.Glavieux,Iterativecorrectionofintersymbolinterference:Turbo-equalization,EuropeanTransactionsonTelecommunications,vol.6,pp.507,September1995. [30] J.Ling,H.He,P.Stoica,J.Li,andW.Roberts,Covertunderwateracousticcommunications,JournalofAcousticSocietyofAmerica,vol.128,pp.2898,November2010. [31] B.S.Sharif,O.H.J.Neasham,andA.E.Adams,AcomputationallyefcientDopplercompensationsystemforunderwateracousticcommunications,IEEEJournalofOceanicEngineering,vol.25,no.1,pp.52,January2000. [32] P.BeaujeanandL.R.LeBlanc,Adaptivearrayprocessingforhigh-speedacousticcommunicationsinshallowwater,IEEEJournalofOceanicEngineering,vol.29,no.3,pp.807,July2004. [33] S.L.MillerandR.J.O'Dea,Peakpowerandbandwidthefcientlinearmodulation,IEEETransactionsonCommunications,vol.46,no.12,pp.1639,December1998. [34] J.A.Hogbom,Aperturesynthesiswithanon-regulardistributionofinterferometerbaselines,AstronomyandAstrophysicsSupplements,vol.15,pp.417,1974. [35] J.C.Preisig,Performanceanalysisofadaptiveequalizationforcoherentacousticcommunicationsinthetime-varyingoceanenvironment,JournaloftheAcousticalSocietyofAmerica,vol.118,no.1,pp.263,July2005. [36] H.He,D.Vu,P.Stoica,andJ.Li,Constructionofunimodularsequencesetsforperiodiccorrelations,2009AsilomarConferenceonSignals,SystemsandComputers,PacicGrove,CA,November1-4,2009. 120

PAGE 121

[37] T.Yardibi,J.Li,P.Stoica,M.Xue,andA.B.Baggeroer,Sourcelocalizationandsensing:Anonparametriciterativeadaptiveapproachbasedonweightedleastsquares,IEEETransactionsonAerospaceandElectronicSystems,vol.46,no.1,pp.425,Jan.2010. [38] W.I.Zangwill,NonlinearProgramming:AUniedApproach,Prentice-Hall,Inc.,EnglewoodCliffs,NJ07632,1969. [39] N.Wiener,Extrapolation,Interpolation,andSmoothingofStationaryTimeSeries,Wiley,NewYork,NY,1949. [40] R.J.McEliece,D.J.C.MacKay,andJ.-F.Cheng,TurbodecodingasaninstanceofPearl'sBeliefPropagationalgorithm,IEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS,vol.16,no.2,pp.140,February1998. [41] J.Ling,X.Tan,J.Li,andM.L.Nordenvaad,EfcientchannelequalizationforMIMOunderwateracousticcommunications,Proceedingsof6thSensorArrayandMultichannelSignalProcessingWorkshop,Ma'aleHahamisha,Israel,pp.73,October2010. [42] M.Swift,J.Riley,S.Lourey,andL.Booth,AnoverviewofthemultistaticsonarprograminAustralia,Proc.Int.Symp.onSignalProcessinganditsApplications,Brisbane,Australia,pp.321,August1999. [43] H.Cox,Fundamentalsofbistaticactivesonar,ProceedingsoftheNATOAdvancedStudyInstituteonUnderwaterAcousticDataProcessing,Kingston,Canada,pp.3,July1988. [44] W.C.Knight,R.G.Pridham,andS.M.Kay,Digitalsignalprocessingforsonar,ProceedingsoftheIEEE,vol.69,no.11,pp.1451,November1981. [45] N.LevanonandE.Mozeson,RadarSignals,Wiley,NY,2004. [46] S.Coraluppi,D.Grimmett,andP.deTheije,Benchmarkevaluationofmultistatictrackers,9thInternationalConferenceonInformationFusion,Florence,Italy,pp.1,July2006. [47] X.Tan,W.Roberts,J.Li,andP.Stoica,SparselearningviaiterativeminimizationwithapplicationtoMIMOradarimaging,IEEETransactionsonSignalProcessing,vol.59,no.3,pp.1088,March2011. [48] K.R.Pattipati,S.Deb,Y.Bar-Shalom,andR.B.J.Washburn,Anewrelaxationalgorithmandpassivesensordataassociation,IEEETransactionsonAutomaticControl,vol.37,no.2,pp.198,February1992. [49] E.D.KaplanandC.J.Hegarty,UnderstandingGPS:principlesandapplications,(ArtechHouseMobileCommunications,Norwood,MA),pp.1,2006. 121

PAGE 122

[50] J.Ling,T.Yardibi,X.Su,H.He,andJ.Li,Enhancedchannelestimationandsymboldetectionforhighspeedmulti-inputmulti-outputunderwateracousticcommunications,JournaloftheAcousticalSocietyofAmerica,vol.125,no.5,pp.3067,May2009. [51] W.Roberts,P.Stoica,J.Li,T.Yardibi,andF.A.Sadjadi,IterativeadaptiveapproachestoMIMOradarimaging,IEEEJournalofSelectedTopicsinSignalProcessing,vol.4,no.1,pp.5,Feb.2010. [52] G.Schwarz,Estimatingthedimensionofamodel,TheAnnalsofStatistics,vol.6,pp.461,May1978. [53] P.StoicaandY.Selen,Cyclicminimizers,majorizationtechniques,andexpectation-maximizationalgorithm:Arefresher,IEEESignalProcessingMagazine,pp.112,January2004. [54] S.P.Lloyd,LeastsquaresquantizationinPCM,IEEETransactionsonInforma-tionTheory,vol.28,no.2,pp.129,March1982. [55] A.L.SwindlehurstandP.Stoica,Maximumlikelihoodmethodsinradararraysignalprocessing,ProceedingsoftheIEEE,vol.86,pp.421,February1998. [56] H.L.VanTrees,OptimumArrayProcessing:PartIVofDetection,Estimation,andModulationTheory,JohnWiley&Sons,NewYork,NY,2002. [57] J.G.Proakis,DigitalCommunications,McGraw-HillInc.,thirdedition,1995. [58] T.Yardibi,J.Li,P.Stoica,andL.N.Cattafesta,Sparsityconstraineddeconvolutionapproachesforacousticsourcemapping,TheJournaloftheAcousticalSocietyofAmerica,vol.123,no.5,pp.2631,May2008. [59] R.RoyandT.Kailath,Espritestimationofsignalparametersviarotationalinvariancetechniques,IEEETransactionsonAcoustics,Speech,andSignalProcessing,vol.ASSP-37,no.7,pp.984,July1989. [60] A.Ruiz,J.M.Ciof,andS.Kasturia,Discretemultipletonemodulationwithcosetcodingforthespectrallyshapedchannel,IEEETransactionsonCommunications,vol.40,no.6,pp.1012,June1992. [61] N.Al-Dhahir,Single-carrierfrequency-domainequalizationforspace-timeblock-codedtransmissionsoverfrequency-selectivefadingchannels,IEEECommunicationsLetters,vol.5,no.7,pp.304,July2001. [62] A.B.BaggeroerandH.Cox,Passivesonarlimitsuponnullingmultiplemovingshipswithlargeaperturearrays,33thAsilomarConferenceonSignals,SystemsandComputers,vol.1,pp.103,1999. 122

PAGE 123

[63] A.Baggeroer,W.Kuperman,andP.Mikhalevsky,Anoverviewofmatchedeldmethodsinoceanacoustics,IEEEJournalofOceanicEngineering,vol.18,no.4,pp.401,October1993. [64] A.B.GershmanandM.G.Amin,Widebanddirection-of-arrivalestimationofmultiplechirpsignalsusingspatialtime-frequencydistributions,IEEESignalProcessingLetters,vol.7,no.6,pp.152,June2000. [65] H.WangandM.Kaveh,Coherentsignal-subspaceprocessingforthedetectionandestimationofanglesofarrivalofmultiplewide-bandsources,IEEETrans-actionsonAcoustics,Speech,andSignalProcessing,vol.ASSP-33,no.4,pp.823,August1985. [66] P.Stoica,P.Babu,andJ.Li,SPICE:Asparsecovariance-basedestimationmethodforarrayprocessing,IEEETransactionsonSignalProcessing,vol.59,no.2,pp.629,Feb2011. [67] D.L.DonohoandM.Elad,Optimallysparserepresentationingeneral(nonorthogonal)dictionariesvia`1minimization,ProceedingsoftheNa-tionalAcademyofSciencesoftheUnitedStatesofAmerica,vol.100,no.5,pp.2197,March2003. [68] I.F.GorodnitskyandB.D.Rao,SparsesignalreconstructionfromlimiteddatausingFOCUSS:Are-weightedminimumnormalgorithm,IEEETransactionsonSignalProcessing,vol.45,no.3,pp.600,1997. [69] J.A.Tropp,Justrelax:Convexprogrammingmethodsforidentifyingsparsesignals,IEEETransactionsonInformationTheory,vol.51,no.3,pp.1030,March2006. [70] D.Malioutov,M.Cetin,andA.Willsky,Asparsesignalreconstructionperspectiveforsourcelocalizationwithsensorarrays,IEEETransactionsonSignalProcess-ing,vol.53,no.8,pp.3010,Aug.2005. [71] F.ClassenandH.Meyr,FrequencysynchronizationalgorithmsforOFDMsystemssuitableforcommunicationoverfrequencyselectivefadingchannels,IEEE44thVehicularTechnologyConference,Stockholm,Sweden,pp.1655,June1994. [72] S.Suthaharan,A.Nallanathan,andB.Kannan,Space-timecodedMIMO-OFDMforhighcapacityandhighdata-ratewirelesscommunicationoverfrequencyselectivefadingchannels,4thInternationalWorkshoponMobileandWirelessCommunicationsNetwork,2002,pp.424,September2002. [73] P.T.Boufounos,P.Smaragdis,andB.Raj,Jointsparsitymodelsforsidebandarrayprocessing,SPIEWaveletsandSparsityXIV,SanDiego,CA,2011. 123

PAGE 124

[74] D.NeedellandJ.Tropp,CoSaMP:Iterativesignalrecoveryfromincompleteandinaccuratesamples,AppliedandComputationalHarmonicAnalysis,vol.26,pp.301,May2009. [75] D.Vu,L.Xu,M.Xue,andJ.Li,NonparametricmissingsamplespectralanalysisanditsapplicationstointerruptedSAR,IEEEJournalofSelectedTopicsinSignalProcessing,vol.6,no.1,pp.1,Feb2012. [76] Z.-M.Liu,Z.-T.Huang,andY.-Y.Zhou,Direction-of-arrivalestimationofwidebandsignalsviacovariancematrixsparserepresentation,IEEETranscationsonSignalProcessing,vol.59,no.9,pp.4256,September2011. [77] M.YuanandY.Lin,Modelselectionandestimationinregressionwithgroupedvariables,JournaloftheRoyalStatisticalSociety:SeriesB(StatisticalMethodol-ogy),vol.68,no.1,pp.49,February2005. [78] W.J.Fu,Penalizedregressions:Thebridgeversusthelasso,JournalofComputationalandGraphicalStatistics,vol.7,no.3,pp.397,February1998. [79] F.R.Bach,Consistencyofthegrouplassoandmultiplekernellearning,TheJournalofMachineLearningResearch,vol.9,no.6,pp.1179,June2008. [80] X.Lv,G.Bi,andC.Wan,Thegrouplassoforstablerecoveryofblock-sparsesignalrepresentations,IEEETransactionsonSignalProcessing,vol.59,no.4,pp.1371,April2011. [81] E.Appleton,Automaticsynchronizationoftriodeoscillators,Proc.CambridgePhil.Soc.,vol.21,pp.231,1922. [82] G.HsiehandJ.Hung,Phase-lockedlooptechniques-asurvey,IEEETransac-tionsonIndustrialElectronics,vol.43,pp.609,December1996. [83] A.NehoraiandE.Paldi,Acousticvector-sensorarrayprocessing,IEEETransactionsonSignalProcessing,vol.42,no.9,pp.2481,Sep1994. [84] M.HawkesandA.Nehorai,Acousticvector-sensorbeamformingandCapondirectionestimation,IEEETransactionsonSignalProcessing,vol.46,no.9,pp.2291,Sep1998. [85] A.J.Poulsen,Robustvectorsensorarrayprocessingandperformanceanalysis,Ph.D.dissertation,MassachusettsInstituteofTechnology,Boston,Massachusetts,2009. [86] M.HyderandK.Mahata,Direction-of-arrivalestimationusingamixedl2,0normapproximation,IEEETransactionsonSignalProcessing,vol.58,no.9,pp.4646,Sept.2010. 124

PAGE 125

[87] J.Huang,Structuredsparsity:Theorems,algorithmsandapplications,Ph.D.dissertation,TheStateUniversityofNewJersey,NewBrunswick,NewJersey,2011. [88] Y.C.Eldar,P.Kuppinger,andH.Bolcskei,Block-sparsesignals:Uncertaintyrelationsandefcientrecovery,IEEETransactionsonSignalProcessing,vol.58,no.6,pp.3042,June2010. [89] M.E.Tipping,SparseBayesianlearningandtherelevancevectormachine,JournalofMachineLearningResearch,vol.1,pp.211,2001. [90] J.Li,P.Stoica,andD.Zheng,AngleandwaveformestimationviaRELAX,IEEETransactionsonAerospaceandElectronicSystems,vol.33,pp.1077,July1997. [91] J.A.NelderandR.Mead,Asimplexmethodforfunctionminimization,ComputerJournal,vol.7,pp.308,1965. [92] P.StoicaandA.Nehorai,MUSIC,maximumlikelihood,andCramer-Raobound,IEEETransactionsonAcoustics,Speech,andSignalProcessing,vol.ASSP-37,no.5,pp.720,May1989. [93] S.Coraluppi,Multistaticsonarlocalization,IEEEJournalofOceanicEngineer-ing,vol.31,no.4,pp.964,October2006. [94] Richardson,J.Greene,Malme,andThomson,MarineMammalsandNoise,AcademicPress,1998. [95] H.A.DeFerrari,H.B.Nguyen,andA.Rogers,Continuousactivepulsecompressionsonar,ProceedingsoftheInternationalConferenceUnderwaterAcousticMeasurements:TechnologiesandResults,pp.1,June2005. [96] R.vanVossen,S.P.Beerens,andI.E.vanderSpek,Anti-submarinewarfarewithcontinuouslyactivesonar,SeaTechnologyMagazine,pp.33,November2005. [97] W.Roberts,H.He,J.Li,andP.Stoica,Probingwaveformsynthesisandreceiverlterdesign,IEEESignalProcessingMagazine,vol.27,no.4,pp.99,July2010. [98] H.He,J.Li,andP.Stoica,WaveformDesignforActiveSensingSystemsAcomputationalapproach,CambridgeUniversityPress,2012. [99] D.KilfoyleandA.Baggeroer,Thestateoftheartinunderwateracoustictelemetry,IEEEJournalofOceanicEngineering,vol.25,pp.4,January2000. 125

PAGE 126

[100] A.B.MacKenzieandL.A.DaSilva,Applicationofsignalprocessingtoaddresswirelessdatademand,IEEESignalProcessingMagazine,vol.29,no.6,pp.166,2012. [101] H.Cox,Fundamentalsofbistaticactivesonar,Proc.NATOAdvancedStudyInst.UnderwaterAcousticDataProcess.,1989. [102] H.L.VanTrees,Detection,Estimation,andModulationTheory,PartI,JohnWileyandSons,Inc.,NewYork,NY,1968. 126

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BIOGRAPHICALSKETCH KexinZhaoreceivedthedegreeofBachelorofSciencefromtheUniversityofScienceandTechnologyofChina,Hefei,China,in2009,andthedegreeofMasterofSciencefromUniversityofFlorida,Gainesville,FL,in2009,bothinelectricalengineering.HewillgraduatewiththedegreeofDoctorofPhilosophyfromtheDepartmentofElectricalandComputerEngineeringatUniversityofFloridainMay,2014.Hisresearchinterestsincludesignalprocessinganditsapplicationtomulti-inputmulti-outputunderwateracousticcommunicationsandmultistaticactivesonarsystems. 127