Data-Adaptive Spectral Estimation Algorithms and Their Sensing Applications

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Data-Adaptive Spectral Estimation Algorithms and Their Sensing Applications
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Ojowu, Ode, Jr
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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:
ZMUDA,HENRY
FAN,ZHONGHUI HUGH

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Subjects / Keywords:
estimation -- remote -- sensing -- spectral
Electrical and Computer Engineering -- Dissertations, Academic -- UF
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Electrical and Computer Engineering thesis, Ph.D.
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Abstract:
Spectral analysis of signals, or the problem of spectral estimation revolves around estimating the distribution of power over frequency of a random signal. It has useful applications in various fields of study (including Speech analysis, Medicine, RADAR and SONAR) due to the fact that the frequency content of an observed signal can provide very useful information in these fields. A well known method for estimating the spectral content of a signal is the Periodogram (developed by Arthur Schuster), which is a data-independent method of estimation. This method is based on computing the Fourier transform of the signal which can be computed efficiently using the Fast Fourier Transform (FFT) algorithm. This method however, is limited by relatively poor resolution and high side-lobes, which can lead to degradation in retrieval of the desired information present within the signal. Data-dependent (adaptive) techniques both non-parametric and parametric can offer superior performance over data-independent methods like the periodogram at a cost of increased computational complexity. These data-adaptive approaches however, can lead to improved spectral resolution and lower side-lobes, which can reveal more information about the signal under study. These advantages have led to increased interest in data-adaptive approaches to the problem of spectral estimation. This dissertation proposal revolves around applying robust adaptive techniques to real-world problems to achieve superior performance. In Chapter 2, adaptive techniques are used in the problem of frequency estimation (harmonic retrieval) in the presence of strong interference. The focus is on the problem of digital audio forensics, where the goal is to extract the embedded network frequency from a digital recording and compare it to a known database for digital audio verification. In the presence of significant interference, extracting the network frequency using the standard method (Periodogram) is difficult due to poor resolution and high-sidelobes. We therefore use a robust adaptive algorithm (Iterative Adaptive Approach) to improve the spectral resolution and suppress side-lobes hence effectively separating the network frequency from interference. A frequency tracking method based on dynamic programming is used in addition to this data-adaptive method to extract the Network frequency accurately and hence provide more reliability for the verification process compared to the current standard. In Chapter 3, we once again apply an adaptive technique for harmonic retrieval. The goal here is to effectively suppress Radio Frequency Interference (RFI) picked up by an Ultra-wideband (UWB) RADAR (currently being built by the Army Research Lab (ARL) for landmine detection) which samples its returned signals using an equivalent sampling scheme. This equivalent sampling scheme makes RFI suppression difficult (due to under-sampling (aliasing)). The current method for RFI suppression for this UWB RADAR is simply averaging multiple realizations of the measured data. We model the aliased RFI signals as a sum of sinusoids and estimate the aliased frequencies and amplitudes accurately using a robust algorithm - RELAX. These estimates are used to reconstruct the aliased RFI samples accurately and are then suppressed from the data without altering the desired radar signals. Our current research/Future work is focused on applying adaptive techniques to signals measured using this UWB RADAR. The goal is to apply data-adaptive spectral estimation techniques on the desired radar signals (after RFI suppression) for SAR imaging (Chapter 4) in other to get improved resolution compared to the standard data-independent method (Delay-and-Sum approach). This will allow for more effective landmine detection.
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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 Ode Ojowu.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: LI,JIAN.
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Co-adviser: LIN,JENSHAN.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-06-30

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DATA-ADAPTIVESPECTRALESTIMATIONALGORITHMSANDTHEIRSENSINGAPPLICATIONSByODEOJOWUJR.ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013OdeOjowuJr. 2

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IdedicatethistoGod,familyandfriends. 3

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ACKNOWLEDGMENTS Thisdissertationwouldnothavebeenpossiblewithoutthesupportofseveralpeople.Iwouldrstofallliketothankmyparents,mysiblingsandfriendsfortheloveandmoralsupporttheyhavegivenmethroughouttheyears.Iwouldalsoliketothankmyadvisor,Prof.JianLifortakingmeinasastudent,andtakingthetimeandpatiencetoguidemethroughoutthisimportantphaseofmyacademiccareer;Iwillforeverbegrateful.Thisdissertationalsowouldnothavebeenpossiblewithoutthehelpofsomeofmyclosecolleagues,labmatesandfriendsattheSpectralAnalysisLab,whichinclude:WilliamRowe,Dr.JohanKarlsson,Dr.DucVu,ChrisGianelli,KexinZhao,Dr.LuzhouXu,Dr.HaoHe,Dr.JunLing,LimDeoksu,QilinZhangandDr.MingXue.Thedailydiscussionsandadvicehelpedwithmyworktremendously.Iwouldnallyliketothankmycommitteemembers,Prof.HenryZmuda,Prof.JenshanLinandProf.HughFanfortheirguidanceandsupport,andalsofortakingthetimetobeonmycommittee.Iappreciatethesacricesincerely. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1REVIEWOFSPECTRALESTIMATIONANDTECHNIQUES .......... 13 1.1Introduction:SpectralEstimationProblem .................. 13 1.1.1EnergySpectralDensity ........................ 16 1.1.2PowerSpectralDensity ........................ 17 1.1.3PowerSpectralDensityEstimation .................. 17 1.2Periodogram:Non-parametricMethod .................... 18 1.2.1Resolution:Periodogram ........................ 19 1.2.2Filter-bankInterpretation:Periodogram ................ 22 1.3Data-adaptiveApproaches .......................... 23 1.3.1CAPON:Non-parametric ........................ 23 1.3.2AmplitudeandPhaseEstimation(APES):Non-parametric ..... 25 1.3.3IterativeAdaptiveApproach(IAA):Non-parametric ......... 26 1.3.4SLIMandSPICEAlgorithms:Non-parametric ............ 28 1.3.5RELAX:Parametric ........................... 28 1.4Conclusions ................................... 29 1.5Notations .................................... 30 2DATA-ADAPTIVETECHNIQUESFORNETWORKFREQUENCYEXTRACTIONFROMDIGITALRECORDINGS ........................... 31 2.1ChapterSummary ............................... 31 2.2Introduction ................................... 31 2.3NetworkFrequencyCharacteristicsandDatabase ............. 34 2.4ExtractionAlgorithms ............................. 36 2.4.1FrequencyDomainAnalysis(STFT) ................. 36 2.4.2IAAandTRIAA ............................. 39 2.4.3FrequencyTracking ........................... 43 2.4.4MatchingtheExtractedENFtoDatabase .............. 44 2.5ExperimentalResults ............................. 45 2.5.1Data1Analysis ............................. 47 2.5.2Data2Analysis ............................. 49 2.6Conclusions ................................... 51 5

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3DATA-ADAPTIVERELAXFORRFISUPPRESSIONFORTHESYNCHRONOUSIMPULSERECONSTRUCTION(SIRE)RADAR ................. 53 3.1ChapterSummary ............................... 53 3.2Introduction ................................... 54 3.3SIREEquivalentSamplingScheme ..................... 56 3.4ExistingRFISuppressionMethods ...................... 60 3.5ProposedRFISuppressionMethod:RELAXandAveraging ........ 63 3.5.1ModellingofRFI ............................ 63 3.5.2RELAXAlgorithm ............................ 64 3.5.3Multi-snapshotRELAXAlgorithm ................... 69 3.6Autoregressive(AR)Modelling ........................ 72 3.7ExperimentalResults ............................. 74 3.7.1Simulations ............................... 74 3.7.2SniffExperimentalData ........................ 76 3.8Conclusions ................................... 79 4DATA-ADAPTIVE,SPARSESUPER-RESOLUTIONIMAGINGFORTHESIREFLGPRRADAR ................................... 82 4.1ChapterSummary ............................... 82 4.2Introduction ................................... 82 4.3DataModel:SIREImpulseBasedFLGPR .................. 85 4.4Back-projection/Delay-and-sum(DAS)BasedMethods ........... 87 4.4.1Back-projection/DAS .......................... 88 4.4.2Sparse:CLEANMethod ........................ 89 4.5Super-resolutionMethods ........................... 90 4.5.1OrthogonalProjectionandTimeGating ............... 91 4.5.2SLIM ................................... 95 4.5.3SPICE .................................. 96 4.6NumericalandExperimentalResults ..................... 99 4.7Conclusions ................................... 106 5CONCLUDINGREMARKSANDFUTUREWORK ................ 107 REFERENCES ....................................... 109 BIOGRAPHICALSKETCH ................................ 117 6

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LISTOFTABLES Table page 1-1Notations ....................................... 30 2-1Abbreviations ..................................... 34 2-2ParametersfortheExperiment ........................... 45 2-3CorrelationcoefcientsofAlgorithms(Data1) ................... 47 2-4StandardDeviationoferrorforAlgorithms(Data1) ................ 47 2-5CorrelationcoefcientsofAlgorithms(Data2) ................... 48 2-6StandardDeviationoferrorforAlgorithms(Data2) ................ 48 3-1ARLParametersforSynchronousReconstructionRadar. ............ 57 3-2SuppressionAlgorithm:RELAX+Averaging ................... 68 3-3SuppressionAlgorithm:M-RELAX+Averaging .................. 71 3-4RFISuppression(dB):File1(~P=1) ........................ 77 3-5RFISuppression(dB):File2(~P=1) ........................ 77 4-1SLIMAlgorithm .................................... 96 4-2CGSPICEAlgorithm ................................. 97 4-3Subspaceapproximation .............................. 99 7

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LISTOFFIGURES Figure page 1-1Syntheticapertureradar(SAR)imaging ...................... 14 1-2Spectrogram:EstimatingtheElectricNetworkFrequency(ENF)inaudioanrecording ....................................... 15 1-3Bartlettwindowspectrum .............................. 20 1-4Spectraoftwosinusoidswithlargefrequencyspacing .............. 21 1-5Spectraoftwosinusoidswithsmallfrequencyspacing .............. 21 1-6Spectrum:Comparisonofadaptivemethodstotheperiodogram ........ 27 2-1FDRDistributioninNorthAmerica ......................... 35 2-2SegmentationofdataforSTFT ........................... 38 2-3MatchingextractedENFtodatabase(Data1) ................... 46 2-4PowerSpectrumofoneFrame(Data2):poorresolutionofFFT ......... 49 2-5PowerSpectrumofoneFrame(Data2):stronginterferencesignal ....... 49 2-6ExtractedENFviaFrequencyTracking ....................... 50 2-7MatchingextractedENFtodatabase(Data1) ................... 51 2-8AbsoluteerrorofAlgorithms ............................. 51 3-1SynchronousImpulseReconstruction(SIRE)equivalenttimesampling ..... 58 3-2SpectrumofSIREsamplingafterinterleavingcomparedtothespectrumofregularsampling ................................... 59 3-3SpectrumSIREsamplingpattern:Onefasttimepulse .............. 59 3-4SpectrumSIREsamplingpattern .......................... 60 3-5RFISuppression(dB):Averagingmethod(Mrealizations)forsimulatedSIREsampledRFIsignals ................................. 62 3-6RFIsuppression(SIREsampling)-usingRELAXwithP(real-valued)sinusoidsestimated ....................................... 75 3-7RFIsuppression-RELAXalgorithmswithP(real-valued)sinusoidsestimatedandsuppressedfromsniffdata ........................... 78 3-8Echoretrieval(File1)-RELAXwithP(real-valued)sinusoidscomparedtoidealechosignal ................................... 78 8

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3-9Echoretrieval(File1)-RELAXwithP(real)sinusoidscombinedwithM-RELAXwith~P=1realsinusoid,comparedtoidealechosignal ............. 79 3-10RFIsuppression-ARmodellingwithorderqcomparedtoaveragingforsniffdata .......................................... 80 3-11Echoretrieval(File1)-ARmodellingwithorderq,comparedtoidealechosignal 80 4-1Forwardlookinggroundpenetratingradar ..................... 84 4-2SIREFLGPR:2DapertureforSARimaging .................... 86 4-3Timegating ...................................... 91 4-4Subspacedimension(s)forhighresolutionimaging ............... 100 4-5FLGPRSARImaging-detectionofweaktarget .................. 101 4-6FLGPRImaging-resolutionimprovement ..................... 102 4-7Orthogonalprojectioncomparison ......................... 103 4-8Realdata-SIREFLGPRSARImaging ...................... 104 4-9ROCcomparison ................................... 105 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDATA-ADAPTIVESPECTRALESTIMATIONALGORITHMSANDTHEIRSENSINGAPPLICATIONSByOdeOjowuJr.December2013Chair:JianLiMajor:ElectricalandComputerEngineeringSpectralanalysisofsignals,ortheproblemofspectralestimationrevolvesaroundestimatingthedistributionofpoweroverfrequencyofarandomsignal.Ithasusefulapplicationsinvariouseldsofstudy(includingSpeechanalysis,Medicine,RADARandSONAR)duetothefactthatthefrequencycontentofanobservedsignalcanprovideveryusefulinformationintheseelds.AwellknownmethodforestimatingthespectralcontentofasignalisthePe-riodogram(developedbyArthurSchuster),whichisadata-independentmethodofestimation.ThismethodisbasedoncomputingtheFouriertransformofthesignalwhichcanbecomputedefcientlyusingtheFastFourierTransform(FFT)algorithm.Thismethodhowever,islimitedbyrelativelypoorresolutionandhighsidelobeproblems,whichcanleadtodegradationinretrievalofthedesiredinformationpresentwithinthesignal.Data-dependent(adaptive)techniquesbothnon-parametricandparametriccanoffersuperiorperformanceoverdata-independentmethodsliketheperiodogramatacostofincreasedcomputationalcomplexity.Thesedata-adaptiveapproacheshowever,canleadtoimprovedspectralresolutionandlowersidelobes,whichcanrevealmoreinformationaboutthesignalunderstudy.Theseadvantageshaveledtoincreasedinterestindata-adaptiveapproachestotheproblemofspectralestimation.Thisdissertationrevolvesaroundanalyzingandapplyingrobustadaptivetechniques 10

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toreal-worldproblemsinaunique,effectiveandefcientwaytoachievesuperiorperformanceovertheirdata-independentcounterparts.Theintroductionchapterbrieyreviewstheproblemofspectralestimationaswellassomeofthemethodsforspectralestimation.WestartthisdissertationinChapter2withthebasicproblemoffrequencyestimation(harmonicretrieval).Inthischapter,adaptivetechniquesareusedintheproblemofharmonicretrievalinthepresenceofstronginterference.Thefocusisontheproblemofdigitalaudioforensics,wherethegoalistoextracttheembeddednetworkfrequencyfromadigitalrecordingandcompareittoaknowndatabasefordigitalaudioverication.Inthepresenceofsignicantinterference,extractingthenetworkfrequencyusingthestandardmethod(Periodogram)isineffectiveandprovestobechallengingduetopoorresolutionandhighsidelobeproblems.Wethereforeusearobustadaptivealgorithm(IterativeAdaptiveApproach-IAA)toimprovethespectralresolutionandsuppresssidelobeshenceeffectivelyseparatingthenetworkfrequencyfrominterference.Afrequencytrackingmethodbasedondynamicprogrammingisusedinadditiontothisdata-adaptivemethodtoextracttheNetworkfrequencyaccuratelyandhenceprovidemorereliabilityforthevericationprocesscomparedtothecurrentstandard,whichisbasedonthedata-independentFouriertransform.Chapters3and4arethefocusofthisdissertation.Inthesechapters,theremotesensingtoolknownastheSynchronousImpulseReconstruction(SIRE)Ultra-widebandradar(currentlybeingbuiltbytheArmyResearchLab(ARL)forlandminedetection)isanalyzedandstudied.InChapter3,weonceagainapplyanadaptivetechniqueforharmonicretrieval.ThegoalhereistoeffectivelysuppressRadioFrequencyInterference(RFI)pickedupbythisUWBradarwhichsamplesitsreturnedsignalsusinganequivalentsamplingscheme.ThisequivalentsamplingschememakesRFIsuppressiondifcultduetoitsirregularandunder-sampleddata(aliasing).ThecurrentmethodforRFIsuppressionforthisUWBradarissimplyaveragingmultiplerealizations 11

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ofthemeasureddata.Inthischapter,wemodelthealiasedRFIsignalsasasumofsinusoidsandestimatethealiasedfrequenciesandamplitudesaccuratelyusingarobustalgorithm-RELAX.Adirectimplementationofthisalgorithmiscomputationallyintensive,therefore,anefcientmethodforimplementationispresentedinthischapter,whichtakesadvantageofthisequivalentsamplingandimprovescomputation.AsRFIsuppressionisthegoal,theestimatesareusedtoreconstructthealiasedRFIsamplesaccuratelyandarethensuppressedfromthedatawithoutalteringthedesiredradarsignals.InChapter4,wefocusonradarimagingforlandminedetectionforthisSIREUWBradar.Thestandardmethodcurrentlyusedforthisradaristhedata-independentbackprojectionordelay-and-sum(DAS)approach.Thismethodsuffersfromhighsidelobeproblemsandpoorresolution.Arecursivesidelobeminimization(RSM)algorithmwasrecentlyproposedbythearmyresearchlaboratoryforeffectivesidelobereduction.Thisdata-independentapproachhowever,hasthesameresolutionlimitationasthebackprojectionalgorithm.Asimagingresolutionisimportantforseparatingdesiredtargets(mines)fromclutter,thischapter,focusesonsparsesuper-resolutionimagingtechniquesforimaging.Anewtechniqueforimagingbasedonapplyingdata-adaptiveapproachespostsignicantdatareductionaswellasinterferencereductionviaanorthogonalprojectionisproposedinthischapter.Thisapproachisabletoachieveanimprovementinimagingresolutionbyafactorofapproximately2,basedonsimulatedexperiments.Chapter5providestheconcludingremarksandpossiblefuturework.ThecontentsofChapter2arepublishedinIEEEtransactionsoninformationforensicsandsecurityVolume7,no.4.ThecontentsofChapter3arepublishedintheInternationalJournalofRemoteSensingandApplications(IJRSA)vol3Issue1.ThecontentsofChapter5aretobesubmittedforpublication. 12

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CHAPTER1REVIEWOFSPECTRALESTIMATIONANDTECHNIQUES 1.1Introduction:SpectralEstimationProblemMostphenomenaorsignalsthatoccurinnatureorinpracticearetypicallyrandominnature,andarebestmodelledasrandomsignals.Examplesofsuchrandomsignalsincludebutarenotlimitedtospeech/audiosignalsandthermalnoisegeneratedbyelectronicdevices.Duetotherandomuctuationofthesesignals,theyarebestcharacterizedintermsofstatisticalaverages.Theautocorrelationfunctionofarandomprocessisastatisticalaverageusedforcharacterizingtheserandomsignalsinthetimedomain.Thepowerspectraldensity(spectrum)providesthefrequencycontentofsuchsignals.Spectralanalysisofsignalsorthespectralestimationproblem,involvesestimatingthefrequencycontentofarandomsignal.Thisisdonebyestimatingthepowerdistributionoverfrequencyfromastationarysequenceofnitetimesamples,whichisknownasthepowerspectrumofthesignal[ 1 5 ].Schusterinthelate19thcenturypioneeredthemostwell-knownspectralestimationtechniquescalledthePeriodogram.Thisharmonicanalysisapproachallowsfordetectingandmeasuringhiddenperiodicities[ 6 ]intheobserveddata.Spectralestimationcanalsobeperformedonnon-stationarydata,bydividingthedataintosegmentsintime(eachassumedtobestationary)[ 7 ],[ 8 ],[ 9 ].Atime-varyingpowerspectrum(image)canbedisplayedtoprovideinformationaboutthesignal(alsoknownasthespectrogram[ 10 ]).Powerspectralestimationhasapplicationsinmanyelds[ 1 3 11 12 ].Speechsignalswhichareperiodicinnatureareanalyzedusingthespectrogram.Thisfrequencydomainanalysisprovidesusefulinformationthatcanleadtospeechrecognitionandgeneration.InthesensingeldsofRADARandSONAR,thespectralcontentofreceivedsignalsmayprovideinformationaboutthetargetsofinterest[ 11 13 ]ina 13

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givensceneofinterest(seeFig. 1-1 ).Alsothepowerspectrumofsignalsmayprovideinformationaboutradiofrequencyinterferenceinsuchasignalandhenceleadtoeffectivesuppressionoftheinterference.IntheeldofMEDICINE,thepowerspectrumofelectroencephalogram(EEG)signalscanbeusedtoevaluatethedifferentsleepcyclesinhumans[ 14 15 ].Thesecanareusedtoinvestigateandstudynarcoleptic(diseasecharacterizedbytheinabilitytoproperlyregulatesleep-wakecycles)patients[ 15 ].Morerecentlyinaudioanalysis,thespectrogramoftheaudiosignalcanindicatethepresenceoftheelectricnetworkfrequency(seeFig. 1-2 ),whichcanbeusedfordigitalaudioauthentication[ 16 ]. A BFigure1-1. Syntheticapertureradar(SAR)imaging:(A)Photographofobjectat45o(B)SARimageformedusingSpectralestimation(FFT) Therearetwobroadapproachestospectralestimation.Therstapproachiscalledthenon-parametricmethodandtheotheriscalledtheparametricmethod.Thenon-parametricmethodsassumesnopriorinformationaboutthedata,whereastheparametricmethodsassumesaspecicmodelofthedata,whichthenresultsinaproblemofparameterestimation.Theparametricmethodsaremoreaccuratethantheclassicalnon-parametrictechniques,whentheassumedmodelisaccurate.However,theyperformpoorlywhenthereareinaccuraciesinthedata-model. 14

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Figure1-2. Spectrogram:EstimatingtheElectricNetworkFrequency(ENF)inaudioanrecordingforforensicanalysis(seechapter1formoredetails) Inthischapter,theproblemofpowerspectraldensityestimationofsignalsisbrieydescribed.Commonlyusedtechniquesforspectralestimationwithinthesetwobroadmethods(non-parametricandparametricmethods)forestimatingthespectrumofasignal,willbebrieydiscussed.Thelimitationsofthesemethodsinpracticewillalsobebrieydiscussed.Somedata-dependent(adaptive)algorithms(Capon,APES,IAA,SLIMandSPICEwhicharenon-parametricandRELAXwhichisparamteric)willbementionedalongwiththeirbenets[ 1 2 17 ]overtheclassical(data-independent)approachesinpracticalscenarios.Thecoreofthisdissertationiseffectiveandefcientapplicationofdata-adaptivetechniquestosolvingreal-worldproblems.Beforedelvingintotheproblemofspectralestimationofrandomsignals,letusconsiderthecaseofspectralestimationofnitelengthdeterministicsignals.Thisanalysisisfairlystraightforwardasdeterministicsignalsarepredictableovertime[ 18 ].Theresultswillthenbeextendedtothecaseofrandomsignals. 15

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1.1.1EnergySpectralDensityConsiderasignalx[n](discrete)withniteenergy,thatis,E=1Xn=jx[n]j2<1 (1)thenitsdiscretetimefouriertransform(DTFT)existsandisgivenby:X(!)=1Xn=x[n]e)]TJ /F8 7.97 Tf 6.58 0 Td[(j!n (1)where!istheangularfrequencyvariablemeasuredinradianspersample.FromParseval'stheoremequation( 1 )canbewrittenas:E=1Xn=jx[n]j2=1 2Z)]TJ /F8 7.97 Tf 6.59 0 Td[(jX(!)j2 (1)Fromtheequationabovetheenergyspectraldensityofx[n]whichisthedistributionoftheenergyofthesignaloffrequencyisthereforedenedas:Sxx(!)=jX(!)j2 (1)NotethattheenergyspectraldensitySxx(!)canbewrittenastheFouriertransformoftheautocorrelationsequencerxx(k)ofthesignalx[n]:Sxx(!)=1Xn=rxx(k)e)]TJ /F8 7.97 Tf 6.59 0 Td[(j!kd! (1)whererxx(k)=1Xn=x[n]x[n)]TJ /F4 11.955 Tf 11.95 0 Td[(k] (1)Theanalysisabove,isspecicallyforsignalswithniteenergy(deterministicsignals).However,signalstypicallyencounteredinapplicationsarecharacterizedasstochasticprocessesanddonothaveniteenergyandhencedonotpossesaFouriertransform.Theserandomsignalshowever,possesanaveragepowerandcanbedescribedbytheirpowerspectraldensity. 16

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1.1.2PowerSpectralDensityConsiderastationarystochasticprocessy[n],whereEfy[n]g=0foralln.Theautocovariancefunction(sameasautocorrelationfunctionforstationarystochasticprocesswithmeanzero)ofy[n]isgivenbyryy(k)=Efy[n]y[n)]TJ /F4 11.955 Tf 11.96 0 Td[(k]g (1)whereEfgisthestatisticalaverageoverallrealizations.Thepowerspectraldensity(PSD)ofy[n]isdenedas(Wiener-Khintchinetheorem[ 1 ]):yy(!)=1Xn=ryy(k)e)]TJ /F8 7.97 Tf 6.58 0 Td[(j!k (1)Thissimplythefouriertransformoftheautocorrelationfunction.NotethattheinversetransformofthisPSDgivesryy(k)asshownbelow1 2Z)]TJ /F8 7.97 Tf 6.59 0 Td[(yy(!)ej!kd!=1Xs=ryy(s)1 2Z)]TJ /F8 7.97 Tf 6.58 0 Td[(ej!(k)]TJ /F8 7.97 Tf 6.58 0 Td[(s)d!=1Xs=ryy(s)ks=ryy(k)weredenotestheKroneckerdeltafunction.Notethat,theaveragepowerofthestochasticprocessy[n]isgivenbythezerolagoftheautocorrelationfunctionryy(0):Efjy[n]j2g=ryy(0)=1 2Z)]TJ /F8 7.97 Tf 6.59 0 Td[(yy(!)d! (1)Thisequation( 1 )leadstothemotivationfordeningthepowerspectraldensityin( 1 ).ThePSDcanalsobedenedas:yy(!)=limN!1E8<:1 NNXn=1y[n]e)]TJ /F8 7.97 Tf 6.59 0 Td[(j!n29=; (1)whichisequivalenttothedenitionin( 1 )undertheassumptionthattheautocovaraincesequence(ACS)ryy(k)decaysquickly. 1.1.3PowerSpectralDensityEstimationObtainingthetruepowerspectraldensity(PSD)yy(!)ofarandomprocessisimpossiblefromanitesetofmeasurements.Thisisduetothefactthatonewillneed 17

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tocomputeananinnitenumberofvaluesfromanitesetofdata,whichisanill-posedproblem[ 1 2 ].Theproblemofspectralestimation,thenbecomesgettinganestimate^yy(!)ofthetruePSDyy(!)ofarandomprocessfromanitesequenceofobservationsofthesignal.Ifthesignalisstatisticallystationary,thelongertheobservedsequencethemoreaccuratetheestimate.Howeverifthesignalisstatisticallynon-stationary,thenonecannotselectandarbitrarilylongdatalengthforestimation.ThisisamajorlimitationonthequalityofthePSDestimate.RecallthatthePSDdescribeshowthepowerofasignalisdistributedinfrequency.Thiscanthenbeinterpretedphysicallyaslteringtherandomsignalthroughanarrowbandlteraroundaspecicfrequencyofinterest(!o).Thisprocessisthenrepeatedforallthefrequenciesofinterest()]TJ /F4 11.955 Tf 9.3 0 Td[(!o).Fourierbasedmethods(computedefcientlyusingtheFastFourierTransform(FFT))ofspectralestimationarebasedonthistechnique[ 1 ]andarediscussednext. 1.2Periodogram:Non-parametricMethodAsmentionedinthesectionabove,thenon-parametricmethodsofspectralestimationprovideanestimateofthepowerspectralassumingnopriorinformationofthedatamodel.TheperiodogramwhichwasintroducedbySchusterin1898todetecthiddenperiodicitiesinasignal,isaclassicalnon-parametricmethodwhichiswidelyusedforspectralestimation.Thisfourierbasedmethod,alongwithitsmodiedversionsarebaseddirectlyonthedenitionin( 1 ).TheperiodogramofasetofNsamplesofrandomprocessfy[n]gNn=1isgivenas(thesubscriptyyinyy(!)hasbeendroppedfornotationalsimplicity):^p(!)=1 NNXn=1y[n]e)]TJ /F8 7.97 Tf 6.59 0 Td[(j!n2 (1)Notethat( 1 )isessentiallythesameasthe( 1 )withtheexpectationandlimitoperationremoved.ThisommissionisduetothefactthatonlyNsamplesofthesignal 18

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areavailable.Theperiodogramcanbecomputedusingthediscretefouriertransformoftheavailablesamples(whichcanbeefcientlycomputedusingthefastfouriertransform(FFT).ThisyieldssamplesofthePSDestimateatfrequencies!k=2k=Nfork=0;1;:::;N)]TJ /F3 11.955 Tf 11.95 0 Td[(1).Notethatequation( 1 )canbewrittenas:^p(!)=N)]TJ /F11 7.97 Tf 6.58 0 Td[(1Xk=)]TJ /F8 7.97 Tf 6.58 0 Td[(N+1^r[k]e)]TJ /F8 7.97 Tf 6.59 0 Td[(j!k (1)where^r[k]=1 NN)]TJ /F11 7.97 Tf 6.58 0 Td[(1Xk=)]TJ /F8 7.97 Tf 6.58 0 Td[(N+1y[n]y[n)]TJ /F4 11.955 Tf 11.96 0 Td[(k] (1)correspondstothebiasedestimatesoftheACSsequence.Thisisreferredtoasthecorrelogram.TheunbiasedestimateoftheACScanalsobeusedtocomputethecorrelogram.Onemajorlimitationoftheperiodogramislimitedspectralresolution,whichisdiscussednext. 1.2.1Resolution:PeriodogramOnekeyconceptinspectralestimationisspectralresolution,whichistheabilitytoresolveorseperatecloselyspacedfrequencycomponentswithinasignal.Theresolutionoftheperiodogramisonemajordrawbackofthisdata-independentmethodofspectralestimation.Notethattheexpectedvalueoftheperiodogramcanbewrittenas:Ef^p(!)g=N)]TJ /F11 7.97 Tf 6.59 0 Td[(1Xk=)]TJ /F8 7.97 Tf 6.59 0 Td[(N+1Ef^r[k]ge)]TJ /F8 7.97 Tf 6.58 0 Td[(j!k=N)]TJ /F11 7.97 Tf 6.59 0 Td[(1Xk=)]TJ /F8 7.97 Tf 6.58 0 Td[(N+1w[k]r[k]e)]TJ /F8 7.97 Tf 6.59 0 Td[(j!k (1)where(basedon( 1 ))w[k]=8><>:1)]TJ /F9 7.97 Tf 13.15 5.7 Td[(jkj Nforn=1;2;:::;N0otherwise (1) 19

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istheBartlettwindowandr[k]isthetruePSD.Equation( 1 )istheFouriertransformoftheproductoftwotimesequences,whichcorrespondtotheconvolutionoftheirindividualFouriertransformsasgivenin( 1 ).Ef^p(!)g=1 2Z)]TJ /F8 7.97 Tf 6.58 0 Td[(()W(!)]TJ /F4 11.955 Tf 11.95 0 Td[() (1)whereW(!)istheFouriertransformoftheBartlettwindow.W(!)=1 Nsin(!N=2) sin(!=2)2 (1)Figure3belowshowsW(!)forN=10andN=20.The3dB(half-power)mainlobe Figure1-3. Bartlettwindowspectrum:resolutionlimitationperiodogram(windowlength=N) widthisapproximatelyequalto4=2N=2=Nradianspersample(1=Ncyclespersample).Thespectralestimateofperiodogram^(!)willnotbeabletoresolvepeaksinthetruePSD(!)thathavelessthan1=Ncyclespersampleseparation.Increasingthenumberofobservedsampleswillimprovethespectralresolution(notbeconfusedwithzero-padding).TheestimatedspectrumcanbecomputedusingtheDFT(andefcientlyusingtheFFTasmentionedearlier).Increasingthenumberofavailablesamplesbyzero-padding(addingzerostotheendofthesignal)canprovidemoredetailinthe 20

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spectrumcomputedusingtheFFT.Thisresultsintheinterpolationofspectrum,howeveritdoesnotchangethespectralresolutionasshowninFigure 1-4 and 1-5 .Figures 1-4 and 1-5 showthespectrumofsinusoids(N=20samples)withfrequencyspacing!=2(0:06)and!=2(0:02)respectively.Eachgureshowsdifferentzero-paddingfactors.Theperiodogramsuffersfromrelativelypoorresolutionandhigh Figure1-4. Spectraoftwosinusoidswithfrequencyspacing!=2(0:06) Figure1-5. Spectraoftwosinusoidswithfrequencyspacing!=2(0:02) sidelobeproblemsasseeninFigure 1-3 .Thesereasonshaveledtoinvestigationintodata-adaptivemethodsofspectralestimationthatcanprovideimprovedresolutionandsidelobesuppressioncapabilities.Inthenextsubsectionsomeofthesedata-adaptive 21

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algorithmsarediscussed.Priortothisdiscussionwewillreviewtheperiodogramindifferentlightwhichleadstooneofthewellknowndata-adaptivealgorithmsknownastheCAPONalgorithm[ 19 ]. 1.2.2Filter-bankInterpretation:PeriodogramRecallthatthePSDisthepowerdistributionoverfrequencyofthesignal,whichasmentionedearliercanbeinterpretedaspassingthesignalthroughabankofnarrowbandlters(atdifferentfrequencies)andcomputingtheoutputpower(whichisthendividedbythebandwidthofthelter).Inthislight,theperiodogramestimator^p(!)atagivenfrequency!canbewrittenas:^p(!)=1 NNXn=1y[n]ej!(N)]TJ /F8 7.97 Tf 6.59 0 Td[(n)2=NN)]TJ /F11 7.97 Tf 6.59 0 Td[(1Xn=0h![k]y[N)]TJ /F4 11.955 Tf 11.95 0 Td[(k]2=Njz(N)j2 (1)wherez(N)=1Xn=0h![k]y[N)]TJ /F4 11.955 Tf 11.96 0 Td[(k] (1)andh![k]=8><>:ej!kfork=0;1;:::;N)]TJ /F3 11.955 Tf 11.95 0 Td[(10otherwise (1)Notethattheperiodogramcanbeinterpretedaslteringthesignaly=fy[k]gNk=0throughanarrowbandpasslterh!=fh![k]gNk=0andselectingjustasingleoutputz(N)ofthelteringprocesshH!yforpowercalculationatthespeciedfrequency(fgandfgHcorrespondtotheconjugate(scalar)andconjugatetranspose(vector)operation).Thisfactleavestheperiodogramwithalargevarianceirregardlessofthedatalength(N).Theoutputpowerdividedbythebandwidth(PSD)isthencalulcatedasEjz[n]j2==jz[N]j2=,where=1=Ncyclespersampleisthelter'sbandwidth.ModiedversionsoftheperiodogramsuchastheBartlettandWelchwhichsegment(non-overlappingandoverlappingrespectively)thestationarysequence 22

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inquestionandaveragetheperiodogramsofthesegmentscanbeusedtoreducethevariance[ 2 ].Intermsofthelter-interpretation,thesemethodscanbeseenascomputingthepowerwithmorethanonesample(numberofsegments).However,theperiodogramiscomputedusingareducedlengthofthedata,hencethereisatrade-offbetweenstatisticalvarianceandresolution.Somedata-adaptivenon-parametricmethodshaveaddressedthelimitationsoftheperiodogrambydesigningadata-adaptivelter,toprovidemoreaccuratePSDestimateswithbetterresolution.InthenextsectionmethodsliketheCAPON,APES(AmplitudeandPhaseEstimation),IAA(IterativeAdaptiveApproach)whicharedatadata-adaptivenon-parametricapproachesarediscussed.Thedata-adaptiveparametricapproachknownasRELAX(strictlyforsinusoidalparameterestimation)isalsodiscussed. 1.3Data-adaptiveApproachesInthissection,wediscusssomewellknownnon-parametricdata-adaptiveapproaches(CAPON,APES)aswellasrecentnon-parametricspectralestimators(IAA,SLIMSPICE).Thesealgorithmsimproveupontheperiodogramspectralestimatorintermsofresolutionandsidelobereduction.Aparametricapproachspecicallyforestimatingparametersoflinespectra(sinusoids)knownasRELAXisalsomentionedanddiscussedindetaillateroninChapter3. 1.3.1CAPON:Non-parametricFromthelastsub-section,theperiodogramoutputataspecicfrequency!canbeinterpretedasusingadata-independentlter(bandpasslter)withanimpulseresponsefh![k]=e)]TJ /F8 7.97 Tf 6.59 0 Td[(j!kgN)]TJ /F11 7.97 Tf 6.59 0 Td[(1k=0correspondingsimplytotheFourierTransformvector.Unlikethedata-independentlterusedintheperiodogram,theCAPONmethod[ 19 21 ](alsoknownastheminimumvariancemethod)designsadata-dependent(adaptive)bandpasslterh!=fh![k]gl)]TJ /F11 7.97 Tf 6.59 0 Td[(1k=0toachievesomespecicdesiredproperties 23

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(TheCAPONmethodusesoverlappingsegmentsoflength(l1)ofthedatatoimprovestatisticalvariance).Thesepropertiesinclude: 1. Designabankofltersh!thatpassthefrequencycomponent(orsinusoidwithfrequency)!undistorted. 2. Filtershouldalsoeffectivelysuppress(orminimize)allout-of-bound(anyotherfrequencies)powerwithinthesignal.Thisprocesscanbeexpressedasfollows.Lettheoutputofthelteratanyinstantn=[0;1;:::;N)]TJ /F3 11.955 Tf 11.96 0 Td[(1]begivenby:z[n]=l)]TJ /F11 7.97 Tf 6.58 0 Td[(1Xk=0h![k]y[n)]TJ /F4 11.955 Tf 11.96 0 Td[(k]=hH!yn (1)whereyn=[y[n];y[n)]TJ /F3 11.955 Tf 12.86 0 Td[(1];:::;y[n)]TJ /F4 11.955 Tf 12.85 0 Td[(l+1]]T.ThetotaloutputpowerofthelteristhengivenasEfjz[n]j2=hH!Rh!.WhereR=EfynyHngisthecovariancematrixofthedatavector.TheCAPONlterisdesignedtomeetthepropertiesintheaforementionedstepsbyminimizingthetotaloutputpoweroftheltersubjecttotheconstraintthatthefrequency!islteredwithoutdistortiongivenbytheoptimizationequation( 1 ).minh!hH!Rh!subjecttohH!a(!)=1 (1)wherea(!)=fe)]TJ /F8 7.97 Tf 6.59 0 Td[(j!gln=0isthesinusoidcomponentwithfrequency!tobepassedundistorted.Theresultinglterisgivenby:h!=R)]TJ /F11 7.97 Tf 6.59 0 Td[(1a(!) aH(!)R)]TJ /F11 7.97 Tf 6.58 0 Td[(1a(!)(CAPONlter) (1)ThePSDestimatecanthenbecalculatedaslteroutputpowerEfjz[n]j2dividedbythebandwidth1=(l).^CAPON(!)=Efjz[n]j2 =l aH(!)R)]TJ /F11 7.97 Tf 6.59 0 Td[(1a(!)(CAPONspectralestimate) (1) 24

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Thesamplecovariancematrix^RbasedontheM=N)]TJ /F4 11.955 Tf 12.47 0 Td[(l+1overlappingsegments(eachoflengthl)ofthedataisusedtoestimatethecovariancematrixandisgivenby:^R=1 MM)]TJ /F11 7.97 Tf 6.59 0 Td[(1Xn=0ynyHn (1)AverysimilaralgorithmtotheCAPONalgorithmknownastheAmplitudeandPhaseEstimationalgorithm(APES)isdescribednext. 1.3.2AmplitudeandPhaseEstimation(APES):Non-parametricNotethatinthedescriptionoftheCAPONalgorithm,thelterdesignwasbasedonpassingasinglefrequency,whilesuppressingallotherout-of-boundfrequencies.CAPONachievesthesuppressionbyminimizingthetotaloutputpower.APESalgorithm[ 22 ],[ 23 ],[ 24 ]usesthesameideabutsuppressingout-of-boundfrequenciesisachievedbydesigningaltersuchthatthelteredsequenceisascloseaspossibletotheasinusoidalsignalatthegivenfrequency!intheleastsquares(LS)sense.Theoptimizationequationisgivenby:min(!);h!M)]TJ /F11 7.97 Tf 6.59 0 Td[(1Xn=0jhH!yn)]TJ /F4 11.955 Tf 11.95 0 Td[((!)ej!nj2subjecttohH!a(!)=1 (1)Thecostfunctionin( 1 )canbere-writtenas:1 MM)]TJ /F11 7.97 Tf 6.58 0 Td[(1Xn=0jhH!yn)]TJ /F4 11.955 Tf 11.95 0 Td[((!)ej!nj2=jhH!^Rh!)]TJ /F4 11.955 Tf 11.95 0 Td[((!)hH!~y!)]TJ /F4 11.955 Tf 11.95 0 Td[((!)~yH!h!+(!)j2=j(!))]TJ /F12 11.955 Tf 11.95 0 Td[(hH!~y!j2+hH!^Rh!)-222(jhH!~y!j2 (1)Notethatthesecondandthirdtermsin( 1 )donotdependon(!)andthereforetheminimizationofthiscostfunctionwithrespectto(!)isgivenby^(!)=hH!~y!where~y!=(1=M)PM)]TJ /F11 7.97 Tf 6.59 0 Td[(1n=0yne)]TJ /F8 7.97 Tf 6.59 0 Td[(j!n.Theoptimizationproblemfordesigningthelterh!isgivenas:minh!hH!^Q!h!subjecttohH!a(!)=1 (1) 25

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where^Q!=^R)]TJ /F3 11.955 Tf 11.9 0 Td[(~y!~yH!TheAPESlterisgivenby:h!=^Q)]TJ /F11 7.97 Tf 6.59 0 Td[(1!a(!) aH(!)^Q)]TJ /F11 7.97 Tf 6.59 0 Td[(1!a(!)(APESlter) (1)TheamplitudespectrumofAPESalgorithmisgivenby:^(!)=a(!)H^Q)]TJ /F11 7.97 Tf 6.59 0 Td[(1!~y(!) aH(!)^Q)]TJ /F11 7.97 Tf 6.59 0 Td[(1!a(!)(APESamplitudespectrum) (1)TheAPESandCAPONalgorithmshavebeenshowntoprovidehigherresolutioncomparedtotheclassicalnon-parametricmethods.TheCAPONalgorithmminimizesthetotaloutputpowersubjecttoaconstraintwhichtendstoprovidespectralestimatesthatarebiaseddownwardduetothenoisegainofthelter[ 23 ].TheAPESalgorithmminimizesaleastsquarefunctionrequiringthelteroutputtobeascloseaspossibletotheasinusoid.Thisprovidesmoreaccuratespectralestimates.However,inthecaseswherethedataisnotstationaryforalongperiodoftime(onlyfewsnapshotsareavailable),theAPESandCAPONmethodsyieldundesirableresults.TheIterativeAdaptiveApproach(IAA)algorithmimprovesonthesealgorithmsbybeingabletogivegoodspectralestimatesforafewsnapshots(evenasinglesnapshot),whileprovidinghighspectralresolution,makingitverysuitableforpracticalapplications.ThisalgorithmisdiscussedbrieyinthenextsubsectionandalsoinChapter1whereitisused. 1.3.3IterativeAdaptiveApproach(IAA):Non-parametricTheIAAalgorithm[ 25 ],[ 26 ],[ 27 ],[ 28 ]forspectralestimationisderivedbyminimizingaweightedleastsquarescostfunction(describedinChapter1).ThespectralestimatefortheIAAalgorithmforasinglesnapshotyisgivenbelow:^(!)=a(!)H^Q)]TJ /F11 7.97 Tf 6.58 0 Td[(1!y aH(!)^Q)]TJ /F11 7.97 Tf 6.58 0 Td[(1!a(!)(IAAamplitudespectrum) (1)ThisestimatelookssimilartotheAPESestimate,withthemaindifferencesbeingthattheIAAalgorithmisiterativeandalsothecomputationofthecovariancematrixof 26

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thenoiseisgivenas^Q!l=^R)]TJ /F10 11.955 Tf 12.57 8.97 Td[(PKi=0;i6=lpia(!i)a(!i)H,whereR=APAHandPisadiagonalmatrixwithelementscorrespondingtofpigKi=0=fj(!i)j2gKi=0(powersateachindividualfrequencies).NotethatunliketheCAPONandAPESalgorithmswherethecovariancematricesarebasedonthedatasamplesandcomputedonce.ThecovariancematrixoftheIAAalgorithmisdependentonthespectralestimateandhencerenediteratively,withtheinitialestimatesofthespectralpowerscomputedusingtheperiodogram.ThisrenementallowstheIAAalgorithmtoproduceaccuratespectralestimateswithasinglesnapshotandhencemakesitusefulforpracticalapplications(wheretheavailabledataforestimationisusuallylimitedtoasinglesnapshot).Fig. 1-6 showsthespectrumof Figure1-6. Spectrum:Comparisonofadaptivemethodstotheperiodogram threesinusoidsinwhitenoise(SNR=30dB)withfrequencies!1=0:63rad/samp,!2=1:26rad/sampand!3=1:33rad/samp.TheCAPONandIAAestimatesarepoor,duetoill-conditioningofthematrices.HoweverwithasinglesnapshottheIAAspectraiscapableofpickingoutthesinusoids.AcomparisonoftheperiodogramtotheIAAalgorithminthisgureshowshowthisadaptivetechniqueimprovesovertheperiodgoramintermsofspectralresolutionandsidelobesuppression. 27

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1.3.4SLIMandSPICEAlgorithms:Non-parametricTheSparseLearningviaIterativeMinimization(SLIM)[ 29 ]andtheSparseIterativeCovariance-basedEstimation(SPICE)[ 30 ]algorithmsaretwosuper-resolutionalgorithmscapableofprovidinghighresolutionestimateseveninthepresenceofasinglesnapshotandcoherentsourcessimilartotheIAAalgorithm.TheybothestimatethecovariancematrixRiterativelysimilartotheIAAalgorithmandarehencealsousefulforpracticalapplications.TheSLIMapproachisamaximumaposterioriapproach(MAP)basedonthehierrachialmodel.Thegoalistouseasparsepriortopromotesparsityintheestimateswhichisusefulforcertainapplications.TheSPICEalgorithmontheotherhandminimizesacovariancecostfunctionthatyieldssparseestimates.ThesetwoalgorithmsempiricallyyieldlessaccuratethattheIAAapproach.HowevertheyprovidesparseandhigherresolutionestimatescomparedtotheIAAalgorithmandcanbeusefulincertainapplications.TheyaredescribedinmoredetailinChapter4.Thealgorithmsmentionedaboveareallnon-parametricmethodsthatdonotassumeaspecicmodelforthedata.Nextwebrieymentionparametricmethods,whichassumeaspecicdatamodelforPSDestimation.Arobustalgorithm(whichislaterdiscussedinmoredetailinChapter3)RELAX[ 31 ];whichisspecicforestimatingparametersofline-spectra(sinusoids)isdiscussednext. 1.3.5RELAX:ParametricParametricmethodsunlikethenon-parametricmethodsassumeaspecicmodelfortheobserveddata.ThesemethodsessentiallyestimatethePSD,byassumingthedatatakesonaspecicmodelandthenestimatestheparametersofthemodel.Auto-regressive(AR)methodssuchasYule,Prony,Forward-BackwardPronymethodsareusedforestimatingparametersforcontinuousspectraandEigen-analysismethods(MUSIC,ESPRIT)areusedforestimatingparametersoflinespectra(sinusoids).TheARmethodsmodelthedataastheoutputofalinearsystemdrivenbywhitenoiseandproceedtoestimatetheparametersofthatsystem.Onemajorlimitationofthese 28

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parametricmethodsisthattheyasubjecttoerrorsduetopoormodelspecications.TheEigen-analysismethods(forlinespectra)estimatefrequencycomponentsofsinusoidsburiedinnoisebyaneigen-decompostionoftheautocorrelationmatrix.Thesemethodstendtoperformpoorlyinpracticalapplicationsduetodatamodelinaccuracies.TheRELAXalgorithmisanalgorithmthatisrobust,andthatestimatesparametersofsinusoidsinaniterativemanner.Itestimatestheparametersofthesinusoidaccuratelyevenwithmodellingerrorsandcolorednoise[ 31 ].ThisalgorithmisdescribedinmoredetailinChapter3whereRadioFrequencyInterference(RFI)ismodeledassumofsinusoids.TheRELAXalgorithmisusedthereforidentifyingandsuppressingtheRFIsignals. 1.4ConclusionsInthissection,abriefdiscussionontheproblemofspectralestimationispresented.Theperiodogramwhichisadata-independentalgorithmforspectralestimationandalsowidelyappliedinpracticalapplicationsisbrieydiscussed.Thisalgorithmisthenre-interpretedasalteringprocesswithadata-independentlter.Thisre-interpretationhasledtosomedata-dependent(adaptive)lterswhichprovideimprovedspectralestimates.Wediscussedsomewell-knowndata-adaptive(non-parametric)algorithms(CAPON,APES)andtheadvantagesprovidedbythesedata-adaptiveapproachesovertheclassicalnon-parametricmethods.Howeverthesealgorithmsperformpoorlyinthecasewhenonlyonesnapshotofdataisavailable.Morerecentdata-adaptive(non-parametric)algorithms,whicharerobustandperformwelleveninthesinglesnapshotcasewerebrieymentionedandwillbediscussedinmoredetailinlaterchapters.Theimprovedrobustnessofthesealgorithmsallowsforusefulapplicationsinapracticalsetting,whileprovidingbetterspectralproperties(highresolution,lowersidelobes)overthecommonlyusedperiodogram. 29

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ArobustparametricalgorithmknownastheRELAXforsinusoidalparameterestimationisalsomentionedbriey(discussedinmoredetailinChapter3).Thisalgorithmiscapableofaccuratesinusoidalparameterestimationeveninthepresenceofcolorednoisemakingitsuitableforpracticalapplications.Inthisdissertationwefocusonsolvingspecicrealworldproblemsbyanalyzingthesedata-adaptivetechniquesandcomingupwitheffectiveandefcientwaystoapplythemtogivesuperiorperformancetothestandarddata-independentapproaches. 1.5NotationsNotation:Throughoutthisdissertation,Boldfaceupper-caseandlower-caselettersareusedtodenotematricesandvectors,respectively.SeeTable 1-1 formoredetailsonnotation. Table1-1. Notations xavectorXamatrixdiag(x)adiagonalmatrixwithelementsofxonthediagonal()Hconjugatetransposeofamatrixorvector()Ttransposeofamatrixorvector()(n)nthiterationofascalar,vectorormatrixinalgorithmjjjj2`2norm^xestimateofscalarx,denition 30

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CHAPTER2DATA-ADAPTIVETECHNIQUESFORNETWORKFREQUENCYEXTRACTIONFROMDIGITALRECORDINGS 2.1ChapterSummaryAnovelforensictoolusedforassessingtheauthenticityofdigitalaudiorecordingsisknownastheElectricNetworkFrequency(ENF)criterion.Itinvolvesextractingtheembeddedpowerline(utility)frequencyfromsaidrecordings,andmatchingittoaknowndatabasetoverifythetimetherecordingwasmade,anditsauthenticity.Inthischapter,anon-parametric,adaptive,andhighresolutiontechniqueknownastheTime-RecursiveIterativeAdaptiveApproach(TRIAA),ispresentedasatoolfortheextractionoftheENFfromdigitalaudiorecordings.Acomparisonismadebetweenthisdatadependent(adaptive)lter,andtheconventionalShort-timeFourierTransform(STFT).ResultsshowthattheadaptivealgorithmimprovestheENFestimationaccuracyinthepresenceofinterferencefromothersignals.TofurtherenhancetheENFestimationaccuracy,afrequencytrackingmethodbasedondynamicprogrammingwillbeproposed.ThealgorithmusestheknowledgethattheENFisvaryingslowlywithtimetoestimatewithhighaccuracythefrequencypresentintherecording. 2.2IntroductionTheuseofdigitalrecordershasbecomemoreprevalentintheworldtodayduetotheadvancementindigitaltechnologyandthesignicantprogressmadeintheeldofdigitalsignalprocessing(DSP).Priortotheincreaseduseofdigitalrecorders,forensicaudioanalysisreliedondifferenttechniquesofaudioauthentication.Forinstance,themagneticsignaturesthatareleftbytheerase,recordorplayheadsonthemagnetictapeofanalogrecorders,canbeusedtoverifytheauthenticityofsuchrecordings.Whenitcomestodigitalrecordings,alterationscanbemadeveryeasilywithoutleavingbehindsuchimprints,becausedigitalrecordersproducearecordingbyconvertingsoundvariationstoaseriesofnumbers,makingauthenticationoftheserecordingsalotmoredifcult[ 32 ].Theimportanceofbeingabletoverifytheauthenticity 31

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ofarecordingcanbeseeninlitigationcases[ 16 ][ 33 ],wheredigitalrecordingsarebroughtforwardasevidenceinatrial.Therefore,morereliablemethodsofverifyingtheauthenticityofdigitalrecordingsneedtoberesearched.TheElectricNetworkFrequency(ENF)CriterionwasproposedbyGrigoras[ 16 ],[ 34 ]toaddresstheissueofdigitalaudioauthentication.TheENFcriterionisbasedonextractingtheutilityfrequencyorENFfromadigitalaudiorecording,andmatchingtheextractedfrequencyestimatetoareferencedatabaseinordertodeterminetheauthenticity,andalsotimeofthedigitalrecording.Thisprocessispossiblebecause,insomecases,digitalrecorders(evensomebatterypoweredrecorders[ 35 ]),canpickuptheaudiblesoundthatisgeneratedbytheoscillationofapowergrid'salternatingcurrentatthisfrequency.Thefrequencyofoscillationisapproximately60HzintheUnitedStates,whereasinEuropeitoscillatesatapproximately50Hz.Thecorrespondingharmonicsofthisfrequencymightalsobepresentinthedigitalrecording.TheENFcriterionisbasedontwoassumptions.Firstly,theENFforinterconnectednetworksisthesameatallpointswithinthenetwork.Secondly,thefrequencyvariesrandomlywithinagiveninterconnection,andhence,arenotrepeatableoveralongperiodoftime.[ 33 ]TherearethreeknownmethodsofextractingtheENFovertimefromadigitalrecording[ 16 ],[ 34 ].Theyaretermed: time/frequencydomainanalysis-Thismethodisbasedoncomputingthespectrogramofthesignalandvisuallycomparingittothedatabase. frequencydomainanalysis-Thismethodisbasedonselectingthefrequencylocationcorrespondingtothemaximumamplitudeofthepowerspectrumofsegments(frames)ofthedataafterapplyingaband-passlter. timedomainanalysis-Thismethodisbasedonmeasuringthezerocrossingsofthesignalinthetimedomainafterabandpasslterhasbeenappliedtotherecording.Recentlyin[ 36 ],aquadraticinterpolationschemewasappliedtothefrequencydomainanalysismethodtoestimatethespectralpeaklocations(frequencies)more 32

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accurately.Thisreducestheestimationerrorresultingfromtheuseofaxedgridsizeinthespectralestimationprocess.Besidesthetime-domainanalysis,theabovemethodsestimatetheENFbasedoncomputingtheFastFourierTransform(FFT)ofoverlappingsegments(frames)ofthedataknownastheShort-TimeFourierTransform(STFT)whichislimitedbythetrade-offbetweentimeresolutionandfrequencyresolution[ 2 ].ParametricmethodssuchastheFrequencySelectiveESPRIT,whichgivesuperiorresolutioncomparedtotheFFT,canalsobeusedsuccessfullytoextracttheENFfromoneframetoanother.However,inthepresenceofsignicantinterferencewithinagivenframe,theparametricmethodsyieldpoorfrequencyestimatesbecauseoftheirsensitivitytoanassumeddatamodel.Thischapterfocusesontwomethodsofextraction.Therst,buildsuponthefrequencydomainanalysiswithquadraticinterpolation.However,inplaceoftheFFT,thespectrumisestimatedforeachsegmentofthedatausinganon-parametricandhighresolutionadaptivealgorithmknownastheIterativeAdaptiveApproach(IAA)[ 25 ].InthepresenceofinterferingsignalswithfrequencieswithintherangeofvaluestheENFcantakeon,IAAyieldsmoreaccurateestimatesoftheENFcomparedtotheFFTasaresultoftheimprovedspectralresolutionandinterferencesuppressioncapability.Thesecondmethodinvolvesapplyingafrequencytrackingalgorithmbasedondiscretedynamicprogramming[ 37 ],whichtakesintoaccounttheslowlyvaryingnatureoftheENFovertime.Thistrackingalgorithmisnecessarybecause,insomeframesofthedata,themaximumspectralpeakmightcorrespondtoaninterferencesignalratherthanthenetworkfrequencysignalevenwithintheacceptableENFlimits.TheENFisthenestimatedinaccurately,whichcanresultinafalsediagnosisthattherecordinginquestionhasbeenedited.Itisworthwhiletopointoutthat,inorderfortheproposedmethodstowork,theENFmustbeembeddedintherecording,whichisnotalwaysthecaseespeciallyinsomebatteryoperatedrecorders[ 35 ].ThisiscertainlyadrawbackofusingtheENFcriterion 33

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fordigitalauthentication.However,iftheENFisembeddedinadigitalrecording,morereliablemethodsofextractionneedtobesought. Table2-1. Abbreviations APESAmplitudeandPhaseEstimationENFElectricNetworkFrequencyESPRITEstimationofSignalParametersbyRotationalInvarianceFDRFrequencyDisturbanceRecorderFIAAFastIterativeAdaptiveApproachIAAIterativeAdaptiveApproachQN-IAAQuasi-NewtonIterativeAdaptiveApproachSTFTShort-timeFourierTransformTRIAATime-RecursiveIterativeAdaptiveApproach ExtractioncanalsobecarriedoutusingtheharmonicsoftheENFsignalforthefrequencyestimationprocess.Insomecases,theharmonicsmaygivebetterestimatesbecauseofahighersignal-to-interference-and-noiseratiocomparedtothefundamentalfrequency.Theremainingsectionsofthischapterareorganizedasfollows.InSection 2.3 ,thenetworkcharacteristicsandthenetworkfrequencydatabasearedescribed.InSection 2.4 ,theIAAandTRIAAalgorithmsaredescribedalongwiththefrequencytrackingalgorithmforENFextraction.InSection 2.5 ,theexperimentalresultsbasedonasetofdigitalaudiorecordingsarepresented.Finally,Section 2.6 containstheconclusionsdrawnfromtheresults.Abbreviations:TheabbreviationsarepresentedforeasyreferenceinTable 2-1 2.3NetworkFrequencyCharacteristicsandDatabaseThefrequencyatwhichalternatingcurrentisdistributedtovariouscustomersfrompowerstations,correspondstotheutilityfrequencyorENF.ForEuropeanandmostAsiancountriesthevalueofthisfrequencyis50Hz,whilethevalueis60HzinNorthAmerica,andseveralcountriesinSouthAmerica.Japanusesbothfrequencies(50and 34

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Figure2-1. FDRDistributioninNorthAmerica 60Hz)forelectricitydistribution.Thisfrequencyisdeterminedbythespeedofrotationoftheturbinesusedtodrivethegeneratorsatthevariouspowerplants[ 38 ].Naturally,therotationspeedisnotconstantandvarieswithinacertainlimit(approximately0.05Hz)dependingontheamountofloadconnectedtothenetwork,andamountofpowergeneratedatagiventime.ExperimentscarriedoutinsomeEuropeancountries[ 16 ],[ 39 ],haveshownthatthisfrequencyvariationisrandomanduniquewithinspecicgeographiclocations.Thisuniquenessinfrequencyvariationwithinaregion,coupledwiththefactthatnetworkfrequencyisnotrepeatableoveralongperiodoftimeiswhatmakestheaforementionedENFcriterionpossible.AdatabaseofthenetworkfrequencyisneededinordertomatchtheextractedENFfromarecordingforverication.In[ 16 ],suchadatabaseiscreatedbyconnectingthesoundcardofacomputertoatransformerwhichisthenconnecteddirectlytoanACpoweroutlet.ThedatabasecurrentlybeingbuiltinNorthAmericainvolvesdeployingseveralsensorstermedfrequencydisturbancerecorders(FDRs),whichperformaccurateENFmeasurements,uptoabout0.0005Hz.ThemeasureddatacollectedbytheFDRsistransmittedovertheinternettoservers,whereitcanbeanalyzedandstoredinasystemtermedtheInformationManagementSystem(IMS)[ 40 ].ThiscollectionformstheFrequencyMonitoringNetwork(FNET). 35

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TherearetwomajorinterconnectionsinNorthAmericaandthreeminorinterconnections.Theseregionshaveunsynchronizednetworks(frequencyandphase)andarethereforeconnectedviaHighVoltageDirectCurrentLines(HVDC)[ 41 ].TheEasternandWesternInterconnectionsformthemajorinterconnections,whiletheQuebec,TexasandAlaskaInterconnectionsformtheminor.TheAlaskaInterconnectionisisolated,inthesensethatitisnotconnectedtoanyoftheotherinterconnections.ItisthereforegenerallynotconsideredtobepartoftheNorthAmericangrid.Fig. 2-1 showsthedistributionoftheFDRsinWestern,Eastern,QuebecandTexasInterconnections.FrequencymeasurementscollectedbytheFDRsintheseinterconnectionsshowthatthefrequencypatternisdifferentatagiventimefromoneinterconnectiontoanother.However,thefrequencypatternisuniqueatdifferentlocationswithineachinterconnection[ 42 ].TheFNETsystem,therefore,providesaviableENFdatabase. 2.4ExtractionAlgorithms 2.4.1FrequencyDomainAnalysis(STFT)DuetothefactthattheENFvarieswithtime,theextractionprocessinvolvesanalysinganon-stationarydatasequence.STFTisacommonmethodfortime-frequencyanalysisofsignals.Thisanalysisassumesthesignalofinterestisstationarywithinshorttimewindows(frames);theFFTofthesignalisthencomputedforeachframe.Thefrequencydomainanalysis[ 16 ]methodofextractionisbasedonthisidea.Theprocessinvolvesre-samplingtheaudiosignaltoalowersamplingrate,toreducethecomputationalcomplexityoftheanalysis.Abandpasslterwithanarrowbandwidthisappliedtothesignalwithcenterfrequency50/60Hzasapreprocessingstep.Therestoftheanalysisisdescribedasfollows.Let,z=[z0;z1:::zN)]TJ /F11 7.97 Tf 6.59 0 Td[(1]T (2)denotethere-sampledandltereddiscrete-timesignal.ThissignalisthensplitintoRoverlappingframesasshowninFig. 2-2 ,witheachframehavinglengthMandashift 36

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fromframetoframeoflengthT.Usingthefrequencydomainanalysismethod,theENFoftherthframeisestimatedbyndingthefrequencythatmaximizesthespectrumofeachframewhichiscomputedusingtheFFTbasedperiodogram.Inordertogetamoreaccurateestimateofthefrequency,quadraticinterpolationisused[ 36 ],[ 43 ].Thisinterpolationscheme,involvesttingaquadraticmodeloftheformlog^(!)=m(!)]TJ /F4 11.955 Tf 11.95 0 Td[(!kmax)]TJ /F3 11.955 Tf 11.95 0 Td[()2+c (2)aroundthefrequencypointthatmaximizesthepowerspectrum:!kmax=argmax!kr(!k) (2)where!k=2k=K;k=0;1;:::;K)]TJ /F3 11.955 Tf 12.6 0 Td[(1correspondstothefrequencygridpointofafrequencygridwithsizeK,andr(!k)ispowerspectrumoftherthframe.Thevalueof!thatmaximizesthemodel( 2 )istakenastheestimatedpeakofthespectrum.Thisvalueisdeterminedbyttingthemodeltothehighestsampleofthepowerspectrumandthetwoadjacentpointswithcorrespondingfrequencies(!kmax)]TJ /F11 7.97 Tf 6.59 0 Td[(1;!kmax;!kmax+1).Thisvalueof!thatmaximizesthemodelis:!=!kmax+ (2)where=1 2)]TJ /F11 7.97 Tf 6.59 0 Td[(1)]TJ /F4 11.955 Tf 11.95 0 Td[(1 )]TJ /F11 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(20+1(!kmax+1)]TJ /F4 11.955 Tf 11.95 0 Td[(!kmax) (2)`,logr(!kmax+`);`=)]TJ /F3 11.955 Tf 9.3 0 Td[(1;0;1: (2)ThecorrespondingfrequencyestimateoftherthframeinHzisgivenby:^f(r)=2(!kmax+)Fs (2)whereFsisthesamplingfrequency(inHz)ofthesignal. 37

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Figure2-2. SegmentationofdataforSTFT TheuseofSTFTwillresultinatrade-offbetweenfrequencyresolutionandtimeresolution.Foragivenframelength,thistrade-offcanbeoptimizedbyapplyingarectangularwindowtoeachframe,whichwillprovidethebestspectralresolutionatacostofhighersidelobescomparedtootherspectralwindows.InordertogetimprovedspectralresolutionoverFFT,onehastoresorttousingparametricmethodsordata-dependent(adaptive)non-parametricmethodsforspectralestimation.Parametricmethods,ontheonehand,arenotrobustagainstdatamodelerrors.Ontheotherhand,non-parametricadaptivemethodsaremorerobust,sincetheydonotassumeaspecicparametricdatamodel.Well-knownadaptivemethodsincludetheCaponalgorithmandtheAmplitudeandPhaseEstimation(APES)algorithm.Thesealgorithmsalsoprovidehigherresolutionandlowersidelobesthantheperiodogram.However,thesemethodsareinadequatebecausetheyrequiremultiplerealizations(snapshots)oftherandomsignal,whichisnotthecasewiththecurrentdata,asonlyonesnapshotisavailableforfrequencyestimation.Spatialsmoothing(segmentingandspectralaveragingofthedata)canbeusedtoimprovethespectralestimatesoftheCaponandAPESalgorithmsintheone-snapshotcase;butthecostofdoingthiswillbeadegradationinthespectralresolution,whichisnotdesirable.Thewavelettransformisalsoacommontoolfortime-frequencyanalysis.ContrarytotheSTFT,whichusesa 38

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xedwindowsize,thewavelettransformusesshortwindowsathighfrequenciesandlongerwindowsatlowfrequencies.Thewavelettransformisthereforenotsuitableforourproblembecauseweareinterestedonlyinasmallrangeoffrequencies.IAAisanon-parametricdata-dependentalgorithmbasedonWeightedLeastSquares(WLS),originallypresentedin[ 25 ]forDirectionofArrival(DOA)estimationinarrayprocessing.TheIAAalgorithmiscapableofyieldinghighresolutionandlowsidelobeseveninthecaseofasinglesnapshot[ 25 ],hencemakingitsuitableforestimatingtheENFinthepresenceofinterferences. 2.4.2IAAandTRIAATheENFcanbeextractedwithhighaccuracyinthepresenceofinterferenceusingtheIAAalgorithmforagivenframe.TheproposedENFextractionprocessfollows( 2 )-( 2 ),withtheFFTspectralestimaterreplacedbytheIAAspectralestimate.TheIAAandTRIAA[ 44 ]usedforspectralestimationofnon-stationarydatawillbediscussedinthissection.Thespectralestimationproblemcanbeset-upasfollows.Lety=[y0;y1:::yM)]TJ /F11 7.97 Tf 6.58 0 Td[(1]TdenoteauniformlysampledstationarydatasequenceandA=[a(!0);a(!1):::a(!K)]TJ /F11 7.97 Tf 6.59 0 Td[(1)],wherea(!k)=[1;ej!k;:::;e(M)]TJ /F11 7.97 Tf 6.58 0 Td[(1)j!k]Tcorrespondstoasteering(frequency)vector,and!k=2k=K;k=0;1;:::;K)]TJ /F3 11.955 Tf 12.32 0 Td[(1,correspondstoafrequencygridpointofafrequencygridwithsizeK.Alsolet=[(!0);(!1);:::;(!K)]TJ /F11 7.97 Tf 6.59 0 Td[(1)]T,with(!k)denotingthecomplexspectralestimatesofyat!k.Thefollowingdatamodelcanbeformulated:y=A (2)wherethenoisecontributionsofyaretakenintoaccountimplicitly[ 25 ]. 39

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TheIAAalgorithmsolvesforthespectralestimatesbyminimizingthefollowingquadraticcostfunctionin( 2 )usingweightedleastsquares(WLS):jjy)]TJ /F12 11.955 Tf 11.95 0 Td[(a(!k)(!k)jj2Q)]TJ /F18 5.978 Tf 5.76 0 Td[(1(!k) (2)wherejjxjj2Q)]TJ /F18 5.978 Tf 5.76 0 Td[(1(!k),xHQ)]TJ /F11 7.97 Tf 6.59 0 Td[(1(!k)x,Q(!k)=R)]TJ /F4 11.955 Tf 11.95 0 Td[(pka(!k)aH(!k) (2)R=APAH (2)andP,diag[p0;p1;:::pK)]TJ /F11 7.97 Tf 6.58 0 Td[(1],withpkfork=0;:::;K)]TJ /F3 11.955 Tf 11.41 0 Td[(1,denotingthepowerestimateateachfrequencygridpoint,givenbyj(!k)j2.R1isthecovariancematrixofthedataandQ(!k)isthecovariancematrixoftheinterferenceandnoise,whereinterferencereferstoallthesignalsatfrequencygridpointsotherthanthecurrentgridpointofinterest!k.Minimizingthecostfunctionin( 2 )withrespecttothe(!k)fork=0;:::;K)]TJ /F3 11.955 Tf 12.11 0 Td[(1givesthefollowingsolution:^(!k)=aH(!k)Q)]TJ /F11 7.97 Tf 6.59 0 Td[(1(!k)y aH(!k)Q)]TJ /F11 7.97 Tf 6.59 0 Td[(1(!k)a(!k);k=0;1;:::;K)]TJ /F3 11.955 Tf 11.96 0 Td[(1 (2)Thesolutionin( 2 )canbere-writtenas^(!k)=aH(!k)R)]TJ /F11 7.97 Tf 6.59 0 Td[(1y aH(!k)R)]TJ /F11 7.97 Tf 6.58 0 Td[(1a(!k);k=0;1;:::;K)]TJ /F3 11.955 Tf 11.96 0 Td[(1 (2)usingtheWoodburymatrixidentity2and( 2 ).ThispreventsthecomputationoftheinterferencecovariancematrixQ)]TJ /F11 7.97 Tf 6.58 0 Td[(1(!k)foreachfrequencygridpoint.NotethatthecomputationofR)]TJ /F11 7.97 Tf 6.59 0 Td[(1requirestheknowledgeof(!k)andviceversa.Hencethisalgorithmissolvedinaniterativemanner,withtheestimateofinitializedusingthe 1R=APAH+2Iforill-conditionedmatrices[ 45 ]2matrixinversionlemma 40

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FFT.Thisiterativealgorithmtakesabout10to15iterationstoconvergebasedonexperimentalandnumericalresults.Notealsothatwithoutaccountingfortheinterferencefromotherfrequencygridpoints(withoutweighting),minimizingthecostfunctionin( 2 )forK=MgivestheDiscreteFourierTransform(DFT)ofthesignal:^(!k)=aH(!k)y M;k=0;1;:::;M)]TJ /F3 11.955 Tf 11.95 0 Td[(1: (2)TheIAAalgorithmdescribedaboveisusedforspectralestimationofstationarydata.AnalogoustotheSTFT,thespectralcontentofanon-stationarydatasequence,suchas(1),canbeestimatedusingtheTRIAA[ 44 ].ThesignalissplitintooverlappingframessimilartoFig. 2-2 andtheIAAspectralestimateiscomputedforeachframe.However,toreducethecomputationalcomplexity,eachsubsequentframeaftertherstframeisinitializedwiththespectralestimateofthepreviousframeinsteadoftheFFTbasedperiodogramasdescribedintheIAAalgorithm.TheresultingalgorithmyieldsbetterspectralresolutionandlowersidelobesthantheSTFT.ThereisstillasignicantincreaseinthecomputationalcomplexitywhenusingtheTRIAAalgorithmcomparedtousingSTFTforspectralestimation.ThiscomputationalcomplexityisreducedslightlybyreducingthenumberofiterationsinsubsequentframesfortheTRIAA.Thisisbecauseconvergenceoftheestimatedspectrumwilloccurinfeweriterationsgiventhecurrentframeisinitializedbythespectralestimateofthepreviousframe.Whenthedatasetissignicantlylarge,theuseofthisalgorithmisstillimpractical.Thebottle-neckoftheTRIAAalgorithmisinthecomputationofthedenominatorin( 2 )foreachframe.In[ 46 ],[ 47 ]theToeplitzstructureofthecovariancematrixRisexploitedandthecomputationofR)]TJ /F11 7.97 Tf 6.58 0 Td[(1isperformedusingtheGohberg-Semencul(GS)factorizationofthismatrix[ 2 ].Moreover,thedenominatorisobtainedviaevaluatingapolynomial.Thisreducesthecomputationalcomplexityofthedenominatorin( 2 )(whichisthe 41

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bottleneckoftheIAAalgorithm)fromO(M2K)toO(M2)oatingpointoperations(ops)[ 46 ]foragivenframe,withoutalossinperformance.ThealgorithmistermedtheFastIAA(FIAA),whichisasignicantimprovementbutstillcomputationallyexpensiveforlargedatasets.ThecomputationalcomplexityofIAAandFIAAareO(M2K)andO(M2+KlogK),respectively,whereMisthedatalengthandKisthegridsize,withK>>M.AnapproximatealgorithmtotheIAAalgorithmwithsignicantlyfastercomputationaltimeisdescribedin[ 48 ]andreferredtoastheQuasi-NewtonIAA(QN-IAA).TheQN-IAAalgorithmestimatesthecovariancematrixasifitwerefromalow-order(L)autoregressive(AR)process,whereL<
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2.4.3FrequencyTrackingAmethodofestimatingtheENFbytrackingitfromoneframetoanotherisformulatedherefromamathematicalpointofview.Theproposedmethodusesdiscretedynamicprogramming[ 37 ]tondaminimumcostpath.Acostfunctionasshowninthissectionisselectedwhichtakesintoaccounttheslowlyvaryingnatureoftheactualnetworkfrequency.ThiscostfunctionpenalizessignicantjumpsinfrequencyfromframetoframeandthecorrespondingpathisusedtoestimatetheENF.Thisalgorithminvolvesndingthepeaklocationsfromthespectrumofeachframeandassigningcostsbasedonthedifferencebetweenapeaklocationinoneframeandapeaklocationinanotherframe.Themagnitudeoftheassignedcostisrelatedtothedifferenceinthefrequencyfromoneframetoanother.TheminimumcostpathfromtherstframetothelastframeiscomputedtoestimatetheENF.Toestimatethenumberofrelevantpeaks(sinusoids)inagivenframe,amodelorderselectiontoolknownastheBayesianInformationCriterion(BIC)isused.TheBICforcomplexsinusoidsinnoiseisgivenby(referto[ 2 ][ 49 ]forafullderivation):BIC(nr)=Mln jjy)]TJ /F11 7.97 Tf 13.73 14.94 Td[(2nrXk=1a(!k)^(!k)jj2!+5(2nr)lnM: (2)Thenumberofpeaks(realsinusoids)nr,isestimatedastheminimizingargumentoftheaboveBICcriterion.Thersttermin( 2 )isaLeast-Squaresdatattingterm,whichdecreasesasthenumberofestimatedpeaksnrincreases,whereas,thesecondtermisapenaltytermthatprevents'over-tting'ofthedatamodel.Oncethenrlargestpeaksandcorrespondinglocationsaredeterminedineachframe,thefrequencytrackingproblemisformulatedandsolvedasfollows.Assumethatforagivenframer,asetofestimatedpeaklocations(frequencies)isdenotedbyr=fPr1;Pr2;:::Prnrg.WewouldliketondapathffrgRr=1,suchthatfr2randwherethedifferencefr)]TJ /F4 11.955 Tf 12.65 0 Td[(fr)]TJ /F11 7.97 Tf 6.58 0 Td[(1isassmallaspossibleforr=1;2;:::;R.ThissetcorrespondstotheestimatedENFoverallframesandcanbeobtainedasthe 43

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minimizingargumentinthefollowingoptimizationproblem:J=minfr2rr=1;:::;RRXr=2(fr)]TJ /F4 11.955 Tf 11.95 0 Td[(fr)]TJ /F11 7.97 Tf 6.59 0 Td[(1)2: (2)Calculatingthiscostusinganexhaustivesearchisimpractical.However,usingdynamicprogramming[ 37 ]thepaththatminimizesthiscostcanbecomputedrecursivelyandefcientlybyminimizingthecostfromagivenframej
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matching.Amethodofcorrelationmatchingproposedin[ 50 ]forshortdigitalrecordings(10-15minutes)isusedinplaceofthisMSEmethod.Theprocessofcorrelationmatchingisdescribedasfollows.Assumethatf=[f1;f2;:::;fR]istheextractedENFsignalandd=[d1;d2;:::;dL]correspondstothedatabasesignalwithL>R.Thematchingprocessrequiresndinglmaxsuchthat:lmax=argmaxlc(l);l=1;2;:::;L)]TJ /F4 11.955 Tf 11.95 0 Td[(R (2)wherec(l)isthecorrelationcoefcientbetweenfandthevector[dl;dl+1;:::;dl+R)]TJ /F11 7.97 Tf 6.59 0 Td[(1].Animportantpointtomakeisthat,themaximumcorrelationcoefcientc(lmax)isusedhereonlyformatchingtheestimatedENFtothedatabaseandcomparingtheaccuracy(reliability)ofthedifferentalgorithmspresented.Onceamatchhasbeenmade,determininglocationsofeditstoarecordingshouldbebasedonthedifferencesbetweentheENFestimateandthedatabase. Table2-2. ParametersfortheExperiment PARAMETERSData1Data2 T(TimeShift)1s1sM(LengthofFrame)20s33sR(NumberofFrames)18001800 2.5ExperimentalResultsThealgorithmspresentedintheprevioussectionareappliedtotwodifferentdigitalaudiodatasetsreferredtoasData1andData2.Thetwodatasetsarerecordedsimultaneouslyandtherefore,shouldcontainthesameENFpatternovertime.Therstdataset(Data1)isacquiredbyconnectinganelectricoutletviaavoltagedividerdirectlytotheinternalsoundcardofadesktopcomputer,resultinginanENFsignalwitharatherhighsignal-to-interference-and-noiseratio.Ontheotherhand,theseconddataset(Data2)isanactualspeechrecordingplayedfromaspeakerandpickedupbytheinternalmicrophoneofalaptopcomputer. 45

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Eachoftheserecordingsareoriginallysampledat44.1kHzatabitrateof16bitspersample.Eachdatasetisre-sampledto441Hz,hencekeepingonlythefundamentalfrequency(1stharmonic)andthetwohigherharmonicsoftheENF.AbandpasslterwithanarrowbandwidtharoundthenetworkfrequencyisappliedtothedatatoeliminateasmuchinterferenceaspossiblewithoutdistortingtheENFsignal.BasedonFig. 2-2 eachdatasetissplitusingthevaluesshowninTable 2-2 .Thisset-upresultsinanENFestimateeverysecondforatotalof30minutesforeachdataset.Anincreaseintheframelengthimprovesthesignal-to-noiseratioofthesignal[ 36 ]andthespectralresolutionatthecostoflowertimeresolution.Therefore,alargerframelengthisusedforData2whichhasaweakENFsignalcomparedtoData1whichhasastrongENFsignal. Figure2-3. MatchingextractedENFtodatabase(Data1-scaledto60Hz) Fig. 2-3 showstheextractedENFsignal(shiftedby0.05Hzforillustrationpurposes)fromData1,matchedwiththetruthobtainedfromtheFDRs,whenthedatasethasnotbeenalteredinanyform(usingSTFTand( 2 )).Fig. 2-7 showstheextractedENFusingtheSTFTbasedmethodandourproposedmethod(alsoshiftedforcomparisonpurposes).Tables 2-3 and 2-5 givethemaximumcorrelationcoefcientc(lmax)ofthevariousmethodsforData1andData2,respectively,alsowhenthesignalshavenotbeenaltered.Themaximumcorrelationcoefcientvaluesareusedtocompare 46

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theaccuracyofthealgorithmsandhencedeterminewhichismorereliableforENFestimation.WehavealsoincludedsimilarMSE(actuallystandarddeviation)analysisinTables 2-4 and 2-6 forthedatasets,wheretheMSEiscomputedbyaveragingthesquareddifferencebetweentheTrueENFandtheestimatedENF.ItisimportanttopointoutthattheestimatedENFcansometimeshaveaconstantoffset[ 39 ],[ 50 ].Therefore,thecorrelationisthepreferredmethodforaccuracymeasure.Thedatasetsusedforthisexperimentdonothavesuchanoffset.Theyhavealsobeenmadeavailableathttp://www.sal.u.edu/download.html. Table2-3. CorrelationcoefcientsofAlgorithms(Data1) AlgorithmHarmonicSTFTSTFT(Track)TRIAATRIAA(Track)F-ESPRIT 60Hz0.99120.99170.98950.99000.9800120Hz0.99110.99490.99020.99460.9470180Hz0.99680.99680.99610.99610.9962 Table2-4. StandardDeviationoferrorforAlgorithms(Data1) AlgorithmHarmonicSTFTSTFT(Track)TRIAATRIAA(Track)F-ESPRIT 60Hz2.772e)]TJ /F11 7.97 Tf 6.58 0 Td[(32.650e)]TJ /F11 7.97 Tf 6.59 0 Td[(33.032e)]TJ /F11 7.97 Tf 6.58 0 Td[(32.919e)]TJ /F11 7.97 Tf 6.59 0 Td[(35.364e)]TJ /F11 7.97 Tf 6.59 0 Td[(3120Hz2.774e)]TJ /F11 7.97 Tf 6.58 0 Td[(32.145e)]TJ /F11 7.97 Tf 6.59 0 Td[(32.822e)]TJ /F11 7.97 Tf 6.58 0 Td[(32.198e)]TJ /F11 7.97 Tf 6.59 0 Td[(36.570e)]TJ /F11 7.97 Tf 6.59 0 Td[(3180Hz1.900e)]TJ /F11 7.97 Tf 6.58 0 Td[(31.851e)]TJ /F11 7.97 Tf 6.59 0 Td[(31.999e)]TJ /F11 7.97 Tf 6.58 0 Td[(31.999e)]TJ /F11 7.97 Tf 6.59 0 Td[(32.830e)]TJ /F11 7.97 Tf 6.59 0 Td[(3 2.5.1Data1AnalysisFig. 2-3 showstheextractedharmonic(180Hz)oftheENFsignalscaledto60Hzandmatched(usingthelocationcorrespondingtothemaximumcorrelation( 2 ))totheactualdatabasefrequencyobtainedfromtheFDRs.Foreachofthealgorithmsused,thethirdharmonicgavethemostaccurateresultsforthisdatasetasshowninTable 2-3 .Thisisbecauseforaxedgridsize,theestimationerrorwhenusingthethirdharmonicisreducedbyafactorofthreecomparedtothefundamentalfrequency.Harmonicswithfrequencieshigherthan180Hzcanbeusedfortheestimationprocess 47

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atacostofincreasedcomputationalcomplexityduetotheincreasedsamplingrate.AlsofromTable 2-3 ,ItcanbeseenthateachoftheSTFTandTRIAAalgorithms,produceaccurateestimatesoftheENFusing( 2 )becauseoftheratherstrongENFsignal.Thesignalatthesecondharmonicisweakrelativetotherstandthirdharmonics,andinafewframestheestimatewasinaccurate.However,thefrequencytrackingalgorithmmitigatedtheseinaccuraciessuccessfullybytrackingthecorrectspectralpeaks.Theparametricmethod,frequencyselective(F-ESPRIT)[ 2 ],[ 51 ]alsoyieldsaccurateestimatesoftheENFforData1whenthesignalmodelassumesthereisonlyonesinusoidperframe.However,thismethodandotherparametricmethodsarenotappropriateforENFestimationinthepresenceofinterference,becausetheyaresensitivetomodelassumptions.Forthisdataset,theSTFTyieldsslightlybetterresults,comparedtotheadaptivemethod(TRIAA).Thiscanbeexplainedbythefactthattheperiodogramisoptimalforestimatingspectrallines(sinusoids)inthepresenceofwhitenoisewhentheyarewellresolved[ 2 ].However,whenthereareinterferingsignalspresent,thepoorresolutionoftheperiodogramwillyieldinaccurateestimatesasisthecasewithData2,atypicaldigitalrecording. Table2-5. CorrelationcoefcientsofAlgorithms(Data2) AlgorithmHarmonicSTFTSTFT(Track)TRIAATRIAA(Track)F-ESPRIT 120Hz0.91250.98570.93050.99070.8446 Table2-6. StandardDeviationoferrorforAlgorithms(Data2) AlgorithmHarmonicSTFTSTFT(Track)TRIAATRIAA(Track)F-ESPRIT 120Hz7.948e)]TJ /F11 7.97 Tf 6.59 0 Td[(33.369e)]TJ /F11 7.97 Tf 6.59 0 Td[(37.225e)]TJ /F11 7.97 Tf 6.59 0 Td[(32.914e)]TJ /F11 7.97 Tf 6.59 0 Td[(31.086e)]TJ /F11 7.97 Tf 6.59 0 Td[(2 48

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Figure2-4. PowerSpectrumofoneFrame(Data2):poorresolutionofFFT 2.5.2Data2AnalysisForData2,thesecondharmonic(120Hz)isusedtoestimatetheENF,becausetherstandthirdharmonicsaretooweaktobeusedforestimation.Table 2-5 showsthemaximumcorrelationcoefcientvaluesfortheSTFTandTRIAAusing( 2 ),thefrequencytrackingalgorithmusingthespectralpeaksoftheFFTandIAAandtheparametricmethod(F-ESPRIT)withoneassumedsinusoid.TheENFestimationaccuracyisimprovedusingtheadaptivemethod(IAA)becauseofimprovedspectralresolutionforseveralframes.Fig. 2-4 showsacomparisonofthespectrumofoneframeoftheData2,wherethepoorfrequencyresolutionoftheFFTresultsinarelativelypoorestimateofthenetworkfrequencycomparedtotheIAAalgorithm. Figure2-5. PowerSpectrumofoneFrame(Data2):stronginterferencesignal 49

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Figure2-6. ExtractedENFviaFrequencyTracking(Data2-scaledto60Hz) Fig. 2-7 showsthisextractedENFharmonicusingtheSTFTand( 2 )matchedwiththedatabase.Fromthisgure,thereareseveralframeswheretheENFisestimatedinaccurately,duetothefactthatthefrequencycorrespondingtothemaximumspectralpeakforthoseframesdonotcorrespondtotheENF.ThiscanoccurifthereisanothersignalpresentwithfrequencywithinthelimitsoftheacceptablerangeoftheENFasillustratedinFig. 2-5 .Thisgureshowsthatforbothspectralestimationtechniquesused(IAA,FFT)theENFharmonicestimateusing( 2 )willbe120Hz,whereasthetruefrequencyisapproximately119.95Hz.Thisproblemcanberectiedusingourdynamicprogrammingbasedfrequencytrackingalgorithmpresentedabove.Fig. 2-6 showsthespectralpeaklocationscomputedusingtheTRIAAandthecorrespondingENFestimateusingdynamicprogramming.TheestimateofthenetworkfrequencyusingthistrackingalgorithmisthenmatchedtothedatabaseinFig. 2-7 ,whichprovidesabettermatchwhencomparedtousing( 2 ),whichcanalsobeseeninthisgure,Fig. 2-8 (absoluteerror)andalsofromTable 2-5 .Afewimportantpointstomakearethatthefrequencytrackingalgorithmusesthepeaklocationsforeachframeestimatedeitherbytheadaptivealgorithm(IAA)ortheFFT.TheresultsshowthattheestimatedENFismoreaccuratewhenthepeaklocations 50

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Figure2-7. MatchingextractedENFtoDatabase(Data2-scaledto60Hz) Figure2-8. AbsoluteerrorofAlgorithms:STFTandTRIAA(Track) ofIAAareused.ThisisasaresultoftheinaccurateestimatesinsomeframescausedbythepoorresolutionofusingFFT.Also,allthenumberspresentedcanbeimproveduponslightlybyusingtheentiredataset(44.1kHz)foranalysis.Forexample,theSTFTmaximumcorrelationof0.9125willbeimprovedto0.9158withoutre-sampling,whichmaynotbeworththeincreasedcomputationalcomplexity. 2.6ConclusionsWhenitcomestodigitalaudioverication,thereliabilityofthemethodusedforauthenticationcannotbeoveremphasized.ThischapterdemonstratesareliablemethodofextractingthenetworkfrequencyfromadigitalrecordingwhentheENFcannotbe 51

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extractedfromsomeoftheframesusingtheFFTbasedperiodogrameitherbecauseofpoorspectralresolutionorastrongerinterferencesignalwithinsaidframe.Theseproblemsweresolvedbyusinganiterativeadaptivemethod(IAA),whichprovidesbetterspectralresolutionthantheFFTbasedapproach.AlsoafrequencytrackingmethodbasedondynamicprogrammingwasusedforaccurateextractionoftheENFeveninthepresenceofastronginterferencesignalswithinENFlimits.Fromtheresultspresented,theFFTgivesslightlybetterestimatesofthenetworkfrequencywhenthesignal-to-interference-plus-ratioisveryhighasisthecasewiththerstdataset.However,inmostdigitalrecordings,therewillbesignicantinterferencesfromtherecordedspeechsignalsandothersurroundingsoundsthatcouldleadtopoorestimationperformanceusingtheFFTduetoitspoorresolutionandhighsidelobeproblems.Astheresultshaveshown,theadaptivetechniquesandfrequencytrackingmethodshouldbeadoptedforENFestimation,especiallyinchallengingenvironments. 52

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CHAPTER3DATA-ADAPTIVERELAXFORRFISUPPRESSIONFORTHESYNCHRONOUSIMPULSERECONSTRUCTION(SIRE)RADAR 3.1ChapterSummaryThisnexttwochaptersfocusonremotesensingapplications,specicallyontheSynchronousimpulsereconstructionradar(SIRE)UWBradarbuiltbytheArmyresearchlaboratoryforlandminedetection.ThischapterfocusesonsuppressionofRadioFrequencyInterference(RFI)forultra-wideband(UWB)radarsignals,sampledusingthissynchronousimpulsereconstruction(SIRE)timeequivalentsamplingscheme.ThisequivalentsamplingschemeisbasedontheArmyResearchLab's(ARL)effortstobuildanultra-wideband(UWB)radarinforwardlookingmodethatsamplesreturnedradarsignalsusinglowrateandinexpensiveanalog-to-digital(A/D)converters.ThecosteffectivenessofthisSIREUWBradarmakesitplausibleforactualgroundmissionsfordetectingburiedexplosivedevices.However,theequivalenttimesamplingschemecomplicatesRFIsuppressionastheRFIsamplesarealiasedandirregularlysampledinrealtime.Inthischapter,thedata-dependentRELAXandmulti-snapshotRELAXalgorithmsarepresentedasanintermediatesteptothepreviouslyproposedaveragingschemebytheArmyResearchLaboratory,inordertoenhanceRFIsuppressionforthissamplingscheme.AdirectapplicationoftheRELAXalgorithmiscomputationallyintensivesoanefcientmethodforgeneratingthespectrumofthisequivalentlysampleddataisproposedinthischapterthatprovidesafactorof10improvementincomputation.TheproposedsuppressiontechniqueinvolvesmodellingthenarrowbandRFIsignalsasasumofsinusoidsandapplyingtheaforementionedalgorithms.TheRELAXalgorithmimprovestheRFIsuppressionperformancewithoutalteringthetargetsignaturescomparedtoARmodelling.Themulti-snapshotRELAXalgorithmwhichprovidesamoreaccuratesinusoidalmodelthantheRELAXalgorithm,improvesontheRELAXalgorithmintermsofsuppression.However,thetargetsignaturesaresuppressedasthenumberofsinusoidsincreases.Theanalysisofthealgorithmsisperformedusingsniff(passive) 53

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datacollectedusingtheSIREradarinadditiontosimulatedwide-bandechosignals(point-targetsignatures). 3.2IntroductionUltra-wideband(UWB)radarisacommonlyusedtoolforvariousremotesensingapplications.Suchapplicationsincludebutarenotlimitedtotheuseoflowfrequency,highbandwidthpulsesfordetectingimprovisedexplosivedevices(IEDs)andlandminetargets.TheeffectivedetectionoflandminesandotherIEDscouldleadtoincreasedsafetyforvariousgroundrelatedmissions[ 52 ].TheuseoflowfrequenciesinUWBradarisnecessaryforfoliageorgroundpenetration,whereastheuseofwidebandpulsesarenecessaryforgoodresolution(abilitytodetecttargetsfromclutter)[ 53 ].However,becauseoftheserequirements,thedata(targetreturns)collectedbytheUWBradarwillbecorruptedbysignalsintheradiofrequencyspectrum(specicallytheUHF/VHFbands).ThesesignalsincludeFMRadio,TVbroadcastsandothernarrowbandandwidebandcommunicationsignals.Theabilitytoeffectivelydetecttargetsisreducedbythepresenceoftheseradiofrequencysignals.GeneralmethodsofsuppressingRFIandtheirlimitationsarediscussedindetailin[ 53 ]forconventionalUWBradar,whichisthecasewhenthereturnedsignalsaresampledregularlyatorabovetheNyquistrate.Duetothelargebandwidthofthereturnedradarsignals,conventionalsamplingwillrequirehighrateanalog-to-digital(A/D)converterstodigitizethereturnedsignals.ThesehighspeedA/Dconvertersareexpensivetobuildandmakespracticalapplicationsimprobable.InothertoimproveonthecostofUWBradars,theArmyResearchLaboratory(ARL)iscurrentlyworkingonanequivalenttimesamplingUWBradarinforwardlookingmode,referredtoastheSynchronousImpulseReconstruction(SIRE)radar[ 54 ].Thisradaruseslowrate(inexpensiveandcommerciallyavailable)A/Dconverterstosamplethereturnedsignals(approximately3GHzbandwidth),whichmakestheradarmorefeasibleforadoptioninpractice.Thisequivalenttimesampling 54

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schemetakesadvantageofthefactthatthesceneisnotchangingwithtime,hencealiasingofthereturnedtargetsignalscanbeprevented.However,thisisnotthecasewiththeradiofrequencysignalswhicharechangingwithtime.Aliasingandtheirregularsamplingcausedbythetime-equivalentschemebecomesanissuewhenitcomestothesubjectofRFIsuppressionasdiscussedinthenexttwosections.ThischapterfocusesonsuppressionofRFIsignalsforthisequivalenttime-samplingscheme.Thegoalistomodelthenarrowbandinterferenceasasumofsinusoidsinrealtimeandestimateandsubtractthesinusoidsbeforeaveragingtoachievefurthersuppression.AcyclicoptimizationalgorithmknownasRELAX[ 31 ]isproposedforestimatingtheparametersofthesinusoids(inaniterativemanner).Thisalgorithmisanasymptoticmaximumlikelihoodapproach[ 55 ]andiscomputationallyandconceptuallysimple.Ithasbeenappliedtoproblemslikenon-contactvitalsigndetectionformoreaccurateestimatesofrespiratoryratesandheartrates[ 56 ].Ithasalsobeenshowntoestimatetheparametersofsinusoidsaccuratelyeveninthepresenceofcolorednoise[ 55 ].Themulti-snapshotRELAX[ 57 ]algorithm,whichisanextensionoftheRELAXalgorithm,willbeusedtoprovideamoreaccuratesinusoidalmodelfortheSIREsamplingscheme.TheRELAXalgorithmormulti-snapshotRELAXareimplementedasanintermediatesteptothealreadyproposedaveragingmethod[ 58 ]toachievefurthersuppression.InSection 3.3 ,thetimeequivalentSIREsamplingschemeisdescribed.Section 3.4 brieydescribesthelimitationsofsomeconventionalmethodstotheSIREsampleddataforRFIsuppression.Theaveragingmethodproposedin[ 58 ]forRFIsuppressionofSIREsampleddataisalsodiscussedinthissection,alongwithitsperformance.InSection 3.5 ,theRELAXalgorithm,alongwithafastcomputationofthespectrumofirregularlysampledSIREdataforthisalgorithmispresented;themulti-snapshotRELAXalgorithmisalsodescribedinthissection.TheresultsarepresentedinSection 3.7 ,startingwithsimulationsthatshowhowtheRELAXalgorithmsuppressesaliased 55

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sinusoidsforsimulateddata.Thesniff(passivedatacollectedusingtheSIREUWBradar)isthenusedtotesttheeffectivenessoftheproposedalgorithms,whicharecomparedtoARmodellingoftheinterferencebasedonthissamplingscheme(seeAppendix).Finally,theconclusionsofthischapterarepresentedinSection 3.8 3.3SIREEquivalentSamplingSchemeInthissection,theSynchronousImpulseReconstruction(SIRE)equivalenttimesamplingtechniqueasdetailedin[ 54 ]isbrieydescribed.ThistimeequivalentsamplingschemeposessomechallengesonidentifyingandhencesuppressingRFIsources,duetothefactthattheRFIsourcesarechangingwithtimeaswillbediscussed.TheSIREsamplingschemeinvolvessamplingthereturnedradarsignalsfromasceneatasignicantlylowersamplingratefs;(withcorrespondingsamplingperiods),thantheNyquistrate,whichleadstoaliasedsamples.Naliasedsamplesarecollectedperpulserepetitioninterval(PRI)orfasttime,andforeachsubsequentPRI,Nmoresamplesarecollectedwiththerangeproleshiftedbye(intime).AfterKpulserepetitionintervals(PRIs)orslowtime,atotalofKNaliasedsamplesarecollected.ThesesamplesareinterleavedasshowninFig. 3-1 ,whichgivesaneffectivesamplingrateoffe=1=ethatisequalto,orgreaterthantheNyquistrate.Becausethesceneofinterestinnotchangingwithtime,thereturnedsamplesfromagivenrangebintheoreticallyshouldalsoremainunchangedintime.Therefore,theinterleavedsamplesaretheoreticallyeffectivelysampledabovetheNyquistrateandshouldbeunaliased.ThemeasurementsfromeachrangeprolearetypicallyrepeatedMtimesandaddedcoherentlytoimprovethesignal-to-noiseratio(SNR).Fig. 3-1 showsthespecialcaseofM=1.Table1summarizestheparametersusedbyARLintheSIREradarpertainingtoRFIsuppression[ 54 ].TheRFIsignals,whicharecollectedinadditiontothedesiredtargetreturns,ontheotherhand,arechangingwithtime.Therefore,whenthecollecteddataareinterleaved,theydonotrepresentthetruetimesamplesoftheRFIsignals. 56

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Table3-1. ARLParametersforSynchronousReconstructionRadar. RadarA/Dsamplingratefs=40MHzRadarA/Dsamplingperiods=25nsPulserepetitionfrequencyPRF=1MHzPulserepetitionintervalPRI=1sNumberofrangeproles(perslowtime)N=7InterleavingfactorK=193TotalnumberofrangeprolesKN=1351Effectivesamplingperiode=129:53psEffectivesamplingratefe=7:72GHz Forinstance,consideracomplexsinusoidsampledatfe(seeTab. 3-1 ),withtimesamplesh[n]=ej!on.Theperiodogramestimateofthespectrumofh[n]isgivenby(!)=(1=L)jH(!)j2,whereH(!)=LXi=1ej(!o)]TJ /F8 7.97 Tf 6.59 0 Td[(!)n (3)isthediscrete-timeFouriertransform(DTFT)ofh[n]andL=KNisthetotalnumberofsamples.IfthiscomplexsinusoidissampledusingtheSIREtechnique(M=1),thetimesamplesoftheinterleavedsignalwillbegivenby:~h[l]=8>>>>>>><>>>>>>>:h[l(T+1)]forl=0;1;:::;K)]TJ /F3 11.955 Tf 11.95 0 Td[(1h[(l)]TJ /F4 11.955 Tf 11.95 0 Td[(K)(T+1)+K]forl=K;:::;2K)]TJ /F3 11.955 Tf 11.96 0 Td[(1h[(l)]TJ /F3 11.955 Tf 11.95 0 Td[(6K)(T+1)+6K]forl=6K;:::;7K)]TJ /F3 11.955 Tf 11.95 0 Td[(1whereT=(fe=PRF)(theothervariablesaredescribedinTab. 3-1 ).Thecorrespondingperiodogramisgivenby~(!)=(1=L)j~H(!)j2,where~H(!)istheDTFTof~h[l]andis 57

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givenby: ~H(!)=K)]TJ /F11 7.97 Tf 6.59 0 Td[(1Xl=0ej!ol(T+1)e)]TJ /F8 7.97 Tf 6.58 0 Td[(j!l+K)]TJ /F11 7.97 Tf 6.59 0 Td[(1Xl=Kej!o((l)]TJ /F8 7.97 Tf 6.59 0 Td[(K)(T+1)+K)e)]TJ /F8 7.97 Tf 6.59 0 Td[(j!l+7K)]TJ /F11 7.97 Tf 6.59 0 Td[(1Xl=6Kej!o((l)]TJ /F11 7.97 Tf 6.59 0 Td[(6K)(T+1)+6K)e)]TJ /F8 7.97 Tf 6.59 0 Td[(j!l (3) Therefore,~H(!)= 6Xs=0ej(!o)]TJ /F8 7.97 Tf 6.59 0 Td[(!)sK! K)]TJ /F11 7.97 Tf 6.59 0 Td[(1Xr=0ej(!o(T+1))]TJ /F8 7.97 Tf 6.58 0 Td[(!)r! (3)Fig. 3-2 showstheperiodogramspectralestimateoftheregularlysampledsinusoid(!)andtheinterleavedSIREsampledsignal~(!).Thespectrumofthecomplexsinusoidisnotonlydistorted,butitpeaksatadifferentfrequency.Notethat~H(!)canbere-writtenas:~H(!)= 6Xs=0ej(!o)]TJ /F8 7.97 Tf 6.59 0 Td[(!)sK! K)]TJ /F11 7.97 Tf 6.59 0 Td[(1Xr=0ej(!o)]TJ /F8 7.97 Tf 6.59 0 Td[(!)rej!oTr! (3) Figure3-1. SynchronousImpulseReconstruction(SIRE)equivalenttimesampling. Therefore,if!o=2m=T,wherem2Z,then~H(!)reducestoH(!).Thisconditionimpliesthatthefrequency(inHz)ofthecomplexsinusoidf=fe!o=2=mPRI,isanintegermultipleofthepulserepetitionfrequency.Unlessthisconditionistrue,interleavingwillleadtodistortionofthecomplexsinusoid. 58

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Figure3-2. SpectrumofSIREsampledcomplexsinusoidsampledafterinterleavingcomparedtothespectrumofsampledregularlyabovetheNyquistrate. Figure3-3. SpectrumSIREsamplingpattern:Onefasttimepulse(N=7samples). Asinglecomplexsinusoid,sampledregularlybelowtheNyquistrate(fs),shouldconsistofasinglepeakatanambiguousfrequencyinthefrequencydomain(inabandwidthoffs).However,duetotheirregularsamplingpatternoftheSIREsamplingtechnique,asinglesinusoidwillbeseenasmultiplepeakswithinthisbandwidth.Fig. 3-3 showstheSIREsamplingpattern(inrealtime)anditscorrespondingspectrumforasinglefasttimepulse(Nsamples).Asexpected,thiswillresultinasinclikefunctionevery40MHz(fs)inthefrequencydomain.HoweverrepeatingthissamplingpatternKtimeswillcorrespondtosamplinginthefrequencydomainasseeninFig. 3-4 .Therefore,thespectrumofasinglesinusoidsampledusingtheSIRE 59

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samplingschemewillcorrespondtoconvolvingthespectrumofthissamplingschemewiththatofasinusoidresultinginmultiplepeaks.Inthenextsection,wediscusssomeofexistingalgorithmsforRFIsuppressionaswellasthelimitationsposedbythissamplingscheme,basedontheanalysisabove. Figure3-4. SpectrumSIREsamplingpattern(NK=1351samples). 3.4ExistingRFISuppressionMethodsOnepopulartechniqueforRFIsuppressionbasedonconventionalsamplinginvolvestheuseofnotchlters.Thismethodinvolvesestimatingthespectrumofthecorruptedsignalandremovingthespikesinthisspectrumusinganotchlter.Thismethodworkswellfornarrowbandinterferencesources.However,itwillintroducesidelobesinthetime-domain[ 59 61 ].Filteringtechniquesingeneral,sufferfromltertransientsandreduceddatalength.ThenotchlteringproblemisevenmoreseverebecauseoftheambiguityinfrequencyfortheSIREsamplingschemebasedontheanalysisintheprevioussectionandFig. 3-2 fortheinterleavedsignals.Also,iftheanalysisofthecorruptedsignalisperformedinreal-time(beforeinterleaving),oneinterferencesourcewillappeartohavemultiplepeaksinthespectrumduetoirregularsamplingasseeninFig. 3-4 ,whichmakesthismethodnotapplicable.ModellingtheRFIusingARmodelscanalsobeusedforsuppression(seeAppendix).TheirregularsamplingoftheSIREdatamakesthisendeavourchallenging. 60

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However,theSIREsampleddataissampledregularlyinfasttimeandslowtime,andthiscanbeexploitedforARmodelling.ThereareonlyN=7samplessampledregularlyinfasttime,whereasthereareK=193regularlysampledsamplesinslowtime.TheseslowtimesamplescanbeusedforARmodellingwithN=7snapshotsallowingformorefreedominthechoiceoftheARmodelorder(seeAppendixforARmodellingofSIREsampleddata).ModellingthenarrowbandRFIasasumofsinusoids,andestimatingtheirparametershasbeenshowntobeeffectiveforsuppressingRFIwithlittlesignaldistortion[ 59 ],[ 62 ].Thismethodinvolvesestimatingtheamplitude,frequencyandphaseofeachinterferingsinusoidandsubtractingtheresultingsinusoidfromthecorrupteddata.Theeffectivenessdependsonhowaccuratetheseparametersareestimated,andisreducedifthesinusoidalmodelfortheRFIsignalsstartstobreakdown[ 62 ].Thisoccurswhenthedurationofdataisgreaterthanthemodulationtime(inverseofmodulationbandwidth)oftheRFIsignals.Forinstance,a3kHznarrowbandvoicechannelwillhaveamodulationtimeofapproximately0.3ms,whereaswidebandTVsignalswithbandwidthofseveralkHzwillhaveamuchsmallermodulationtime[ 62 ].Ifthedurationoftheprocesseddataisgreaterthanthismodulationtime,theestimatedparameterswillchangeduringtheacquisitiontime,leadingtolesseffectivesuppression.Thesemethodsarealsocomputationallyexpensivewhenmanyinterferencesourcesareestimated.WhentheRFIsignalsaresampledusingtheSIREequivalentsamplingscheme,estimationoftheseparametersbecomesevenmorechallengingduetotheirregularsamplingpatternandaliasingintroduced,evenifthemodelisaccurate.AnothertechniqueforRFIsuppressionisusingpassivedatatoadaptivelysuppressRFIfromactiveradardatabyprojectingthemeasuredactivedatatoasignalsubspacecreatedbythepassivedata.ThismethodassumesorthogonalitybetweenthedesiredtargetsignaturesandtheRFI,andhasbeenshowntobeeffectiveforRFIsuppression 61

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in[ 63 ]forconventionallysampleddata.However,asnotedin[ 54 ],thismethodisinadequateforsuppressionofSIREsampleddata,duetotheirregularsamplingpatternandaliasedsamplesoftheRFI.ThesechallengeshavepromptedtheneedfornewRFIsuppressiontechniquesfortheSIREsamplingscheme.Theaveragingmethodproposedin[ 58 ]andalsodetailedtherein,hasbeenshowntosuppresswidebandandnarrowbandinterferers.ThemethodisbasedonrepeatingthemeasurementsfromthesamerangeproleMtimesandaveragingtherepeatedmeasurements.TheaveragedsamplesaretheninterleavedandusedforgeneratingSARimages.Fig. 3-2 showstheamountofsuppressionasafunctionofthenumberofrepeatedmeasurementsbasedonsimulatedRFIsources.Asimilarplotcanbeseenin[ 58 ].AnimportantpointtonoteisthatthismethodofsuppressiondoesnottakeintoaccountanypropertiesoftheRFIsignal,whichisthemotivationforimprovingtheperformance. Figure3-5. RFISuppression(dB):Averagingmethod(Mrealizations)forsimulatedSIREsampledRFIsignals. Inthischapter,theaveragingmethodisimprovedbyanalyzingthedatain'real-time'(beforeinterleaving).Thealiasedsamplesofthedatain'real-time'aremodelledasasumofsinusoids,inothertoachievefurthersuppressionwithlittlesignaldistortion.Theparametersofthesinusoidsareestimatedandtheresultingsinusoidsaresubtracted 62

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fromthedatausingtheRELAXalgorithms[ 31 ],[ 57 ](toprovideaccurateestimatesoftheparameters)beforeaveraging.Basedontheanalysisintheprevioussection,asinglesinusoidappearsasmultiplepeaksduetotheirregularSIREsamplingpatternin'real-time'.However,intheory,estimatingtheparametersofasinglesinusoidfromthemaximumpeaklocationofthespectrum,andsubtractingthisfromthedata,willcorrespondtoitsremovalfromthespectrum.Thiswilleliminateallthemultiplealiasedpeaks(Fig. 3-4 ).Thisanalysiswillbeshownonsimulatedsinusoidsintheresultssection.TheRELAXalgorithmanditsmulti-snapshotcounterpartaredescribedinthenextsectionandthestepsforRFIsuppressionarealsopresented. 3.5ProposedRFISuppressionMethod:RELAXandAveraging 3.5.1ModellingofRFITheproposedsuppressionmethod,entailsmodellingRFIsignalsoflengthLcollectedinrealtime(beforeinterleaving)asasumofPcomplex-valuedaliasedsinusoidsasdescribedinEq.( 3 ):z=PXp=1pa(fp) (3)wherepandfparethecomplexamplitudeandfrequencyofthepthsinusoidanda(fp)=1ej2fpej2(L)]TJ /F11 7.97 Tf 6.59 0 Td[(1)fpTThereceivedmeasurementsignalcanbewrittenasy=z+s+n,wherez,s,andn,aretheRFIsignal,desiredtargetreturns,andreceivernoise,respectively.ThetargetreturnshaveawidebandwidthrelativetotheRFIsignalsandcanbemodelledaswhitenoise[ 53 ],[ 64 ].RFIsuppression,then,becomesacaseofestimatingtheparametersofmultiplesinusoidsinthepresenceofwhitenoise.Thenon-linearleastsquares(NLS)approach(anasymptoticMaximumLikelihoodapproach[ 55 ],[ 2 ])estimatesthese 63

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parametersbyminimizingthefollowingnon-linearleastsquarescostfunctioninEq.( 3 ),n^p;^fpoPp=1=argminf^p;^fpgPp=1y)]TJ /F8 7.97 Tf 17.32 14.94 Td[(PXp=1pa(fp)2 (3)wherePisthenumberofsinusoids,whichcanbeestimatedusingamodel-orderselectiontoolliketheBayesianInformationCriterion(BIC)[ 65 ].ThismethodcanapproachtheCramer-Raoboundinperformance,butitinvolvesamulti-dimensionalsearchandhenceinvolvescomplexcomputationsforthecaseofmultiplesinusoids.Itcanalsobesensitivetoinitializations[ 2 ],[ 66 ].TheRELAXalgorithmcanbeusedforsolvingtheprobleminaniterativemannerreducingthecomputationalcomplexitysignicantly[ 31 ].Thisconceptuallyandcomputationallysimplealgorithmwasshowntoestimatesinusoidalparametersaccuratelyandrobustlyeveninthepresenceofcolorednoise[ 55 ].Theparametersareestimatedfortheabovenon-linearleastsquaresttingprobleminaniterativemannerasdescribedbelow. 3.5.2RELAXAlgorithmTheRELAXalgorithmestimatestheparametersasfollows:Letyp,y)]TJ /F8 7.97 Tf 23.68 14.95 Td[(PXi=1;p6=i^ia(^fi) (3)Thefrequencyandcomplexamplitudeestimatesofthepthsinusoidare,respectively,estimatedby:^fp=argmaxfpjaH(fp)ypj2 (3)and^p=aH(fp)yp L| {z }DTFTofypjfp=^fp (3) 64

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TheRELAXalgorithmstepsaregivenby: Step1:AssumeP=1.Estimate^f1and^1fromy. Step2:AssumeP=2.Computey2basedonestimatesfromthepreviousstepandestimate^f2and^2.Computey1andre-estimate^f1and^1.Re-iteratepreviousstepsuntilpracticalconvergence. Step3:AssumeP=3.Computey3andestimate^f3and^3.Re-computey1andre-estimate^f1and^1from^f2;^2;^f3,^3.Re-iterateuntilconvergenceoraxednumberofiterations. RemainingSteps:ContinueuntilP=^P,whichisanestimatedordesirednumber.Notethat,thefrequenciesandcomplexamplitudesin( 3 )and( 3 ),respectively,areestimatedusingtheDTFTofthesignalsyp.ThiscanbeefcientlycomputedusingtheFFTandzero-paddingforconventionally(regularly)sampleddata.BasedonFig. 3-3 andtheanalysisleadingtoEq.( 3 ),aspreviouslydiscussed,theinterleavingprocessofaSIREsampledsinusoidleadstoadistortionofthatsignalexceptforaspeciccase,beingthatthefrequencyofthesinusoidisanintegermultipleofthePRI.TheanalysisoftheRFIusingtheRELAXalgorithmwill,therefore,beperformedonthedatainreal-time(beforetheinterleavingprocess).Aswillbeshownintheresultssection,theestimatedcomplexamplitudesandfrequencies(althoughpossiblyambiguous),canbeusedtoaccuratelyreconstructthealiasedRFIsamplesandyieldeffectiveRFIsuppressionusingtheRELAXalgorithm.TheRELAXalgorithmrequiresthecomputationofthespectrumofthereceivedsamples.ForirregularlysampledSIREdata,thisspectrumcanbecomputedusinganFFTafterre-samplingthedata(interpolatingwithzeros).Re-samplingthisdatatogivearegularlysampleddatawitheffectivesamplingfrequencyoffe,willleadtoasignicantlylongdatasequencewithmostofthesamplesbeingzero.Forinstance,onerealization(M=1)ofaSIREsampleddata,sampledatfs=40MHz,containsN=7aliasedsamplesperPRI.Afterre-samplingtoaneffectiverateoffe=7:72GHz,eachPRIwillconsistofT=fePRI=7720samples.HenceatotalofTK=77201931:5 65

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millionsamplesperrealization.Therefore,applyingadirectFFT(withzero-padding)tothisre-sampleddatatoestimatethefrequenciesandcomplexamplitudesbecomescomputationallyintensivewithacomputationalcomplexityofO(TKlogTK).NotethatsimilartoFig. 3-4 ,asinglesinusoidsampledusingtheSIREsamplingtechniqueandre-sampledasdiscussedabovetogiveaneffectivesamplingrateoffe=7:72GHzwillrepeatitselfapproximatelyevery40MHz(A/Drate)inthefrequencydomain,duetoaliasing.Inordertoreducethecomputationalcomplexityofthisre-samplingscheme,theregularsamplingofthedatainbothfastandslowtimecanbeexploitedandthespectrumcanbecomputedonlyona40MHzbandwidthtosaveoncomputations.Theanalysisisperformedasfollows:ThespectralestimateforSIREsampleddatainrealtime(beforeinterleaving)basedonparametersinTab. 3-1 isgivenby:X(f)=XnXmxm;ne)]TJ /F8 7.97 Tf 6.59 0 Td[(j2f(mm+nn) (3)wheren=0;1;2;;N)]TJ /F3 11.955 Tf 11.96 0 Td[(1(N=7)m=0;1;2;;K)]TJ /F3 11.955 Tf 11.96 0 Td[(1(K=193)n=s=25ns,(ADCsamplingrate);m=PRI+e:AdirectcomputationofthespectruminEq.( 3 )isobviouslycomputationallyintensive,especiallyforanegridsizeinfrequency.Assumingf=k1 m+k2f (3) 66

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wherek1=0;1;2;;K1)]TJ /F3 11.955 Tf 11.96 0 Td[(1K1=T=7720k2=0;1;2;;K2)]TJ /F3 11.955 Tf 11.96 0 Td[(1K2=1=(mf)f=xedgridsize(inHz)Eq.( 3 )isthefrequencygrid(inHz)onwhichthespectruminEq.( 3 )willbecomputed.NotethatthechoiceofK2determinesthegridspacingfandthechoiceofthek1valuesdeterminestheportionofthebandwidthinwhichthespectrumistobeestimated.Forinstance,k1=0;1T=7720computesthespectrumovertheentire7.72GHz(effectivesamplingrate)bandwidth.Forthefrequencygridspeciedin( 3 ),thespectrumin( 3 )canbere-writtenasfollows:X(f)=XnXmxm;ne)]TJ /F8 7.97 Tf 6.58 0 Td[(j2(k1 m+k2)(mm+nn) (3)whichsimpliesto:X(k1;k2)=Xne)]TJ /F8 7.97 Tf 6.58 0 Td[(j2(k1 m+k2f)nnXmxm;ne)]TJ /F8 7.97 Tf 6.58 0 Td[(j2k2 K2mX(k1;k2)=Xne)]TJ /F8 7.97 Tf 6.58 0 Td[(j2(k1 m+k2f)nnXn(k2) (3)FromEq.( 3 ),thespectrumiscomputedbysummingupmultipleFFTs.Alsobecausethesignalisaliased,thespectrumneedsonlytobecomputedoverasmallportion(40MHz-A/Dsamplingrate)oftheentirebandwidth.ThecomputationalcomplexityofthisalgorithmisO(NK2logK2+K1K2N).Notethatthebottleneckofthisalgorithmisinthesecondterm.Whenthespectrumiscomputedovertheentirefrequencygrid,K1=T=7720,thecomputationalcomplexityisonthesameorderasre-samplingandapplyinganFFT.However,whenthespectrumiscomputedovera40 67

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MHzbandwidth(K1=40),thisalgorithmdrasticallyimprovesonthecomputation.Forexample,forafrequencygridwithspacing(f)ofapproximately2kHz,thespectruminFig. 3-4 wascomputedin0.26secswhentheSIREsampleddatawasre-sampledandtheFFTwasapplieddirectlyusingtheMATLABsoftware.However,thespectrumbasedonEq.( 3 )wascomputedin0.035secsona40MHzgrid. Table3-2. SuppressionAlgorithm:RELAX+Averaging Step1:RELAX(Psinusoidsestimated).-ComputetheDTFTofthemeasureddatayfrom( 3 )andestimate^f1and^1,using( 3 ),( 3 )and( 3 ).-Computey2using( 3 )anditsDTFTusing( 3 ).Estimate^f2and^2.Re-estimate^f1and^1fromy1anditerate.Continueforyp,^fpand^p(3pP)(SectionIV.B).Step2:ReconstructaliasedRFIsamplesusingf^figPi=1andf^igPi=1.Subtractfromeachrealization(y).Step3:Averageresiduefromeachrealizationandinterleave. ThisspectrumisusedintheRELAXalgorithm( 3 )and( 3 ),toestimatetheparametersofthesinusoidspresent.TheRELAXalgorithmisappliedheretoonerealization(M=1)ofSIREsampleddata(whichcorrespondtoadatawithanacquisitiontimeof0.193msbasedonTab. 3-1 ).Thereforeanarrowbandinterferencesourcewithamodulationbandwidthof5kHzorlesscanbeaccuratelyapproximatedasasingletone,whereasmultiplesinusoidsareneededtomodelaninterferencesourcewithwiderbandwidth.Thesinusoidalmodelbeginstobreakdownforverywidebandinterferers.Inthenextsubsectionweproposethemulti-snapshotRELAXalgorithmforSIREsampleddatathatprovidesamoreaccuratesinusoidalmodelfortheRFIsignalsbyusingfewersamples(smallermodulationtime),forsuppression.TheoverallproposedRFIsuppressionalgorithmcanbesummarizedinthefollowingstepsasshowninTab. 3-2 forRELAX. 68

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3.5.3Multi-snapshotRELAXAlgorithmThemulti-snapshotRELAXalgorithm[ 57 ]usesN=7samples(150nsacquisitiontime)forRFIsuppression.Interferencesourceswithmodulationbandwidthof6.7MHzorlesscanbeaccuratelymodelledassinusoids,whichincludeswidebandinterfererslikeTVbroadcastsetc.Themulti-snapshotRELAXalgorithm(M-RELAXforshort)proposedforangleandwaveformestimationin[ 57 ]isamodicationoftheoriginallyproposedRELAXalgorithm[ 31 ].Thealgorithmestimatestheangleofarrival(usingmultiplesnapshotsofthedata)andthecorrespondingwaveformforeachsnapshot.ThisalgorithmisproposedhereforRFIsuppressionofSIREsampleddatatoprovideamoreaccuratesinusoidalmodelfortheRFIsignals.Here,eachsetofN=7fasttimesamplesistreatedasasnapshot.ThedataissplitintoK=193totalsnapshotsbasedontheparametersinTab. 3-1 .Thefrequencyofasingletoneisestimatedbyaveragingtheperiodogramofeachsnapshotandndingthefrequencythatmaximizestheaverage.Thecomplexamplitudesofeachsnapshotisestimatedbyndingthecomplexvalueofthespectrumofeachsnapshotattheestimatedfrequency.Notethatforasinglecomplexsinusoid,K=193complexamplitudesareestimatedfromeachsnapshot,whereasonlyonefrequencyisestimated.TheparametersareestimatedasgiveninEq.( 3 )(modicationofNLSforthemulti-snapshotcase[ 57 ]):n^p;^fpoPp=1=argminfpfpgPp=1KXm=1x(m))]TJ /F8 7.97 Tf 17.31 14.94 Td[(PXp=1p(k)a(fp)2 (3)wherep=[p(1);p(2);:::;p(K)]containstheestimatedcomplexamplitudesofthepthsinusoidforeachoftheKsnapshots,^fpistheestimatedfrequencyofthepthsinusoidforallsnapshotsandx(m)isthemthsnapshot.Theseparametersareestimatedasfollows.Letxp(m),x(m))]TJ /F8 7.97 Tf 23.68 14.94 Td[(PXi=1;p6=i^i(m)a(^fi) (3) 69

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Theestimatesdescribedabovearegivenby:^fp=argmaxfpKXm=1jaH(fp)xp(m)j2 (3)and^p(m)=aH(fp)xp(m) L| {z }DTFTofxp(m)jfp=^fp;m=1;2;:::;K (3)Themulti-snapshotRELAXalgorithmstepsareasfollows: Step1:AssumeP=1.Estimate^f1and^1(m)fromx(m),form=1;2;:::;K. Step2:AssumeP=2.Computex2(m)basedontheestimatesfromthepreviousstepandestimate^f2and^2(m),form=1;2;:::;K.Computex1(m)andre-estimate^f1and^1(m),form=1;2;:::;K.Re-iteratepreviousstepsuntilpracticalconvergence. Step3:AssumeP=3.Computex3(m)usingf^p;^fpg2p=1andestimate^f3and^3(m),form=1;2;:::;K.Re-computex1(m)andre-estimate^f1and^1(m)fromf^p;^fpg3p=2,form=1;2;:::;K.Thenre-computex2(m)andre-estimate^f2and^2(m)fromf^p;^fpgp=1;3,form=1;2;:::;K.Re-iterateuntilconvergence. RemainingSteps:ContinueuntilP=^P,whichisanestimatedordesirednumber.FortheSIREsampleddata,thespectrum(DTFT)ofeachsnapshotcanbedescribedasfollows.Letx(m)=fdm(n)gN)]TJ /F11 7.97 Tf 6.59 0 Td[(1=7n=0correspondtothemthsnapshot,wheredm(n)denotesthenthsampleofthemthfasttimepulse(m=1;2;:::K).Notethateachsnapshotisregularlysampled(attheA/Drate).ThespectralestimatecanthereforebecomputedusinganFFTmultipliedbyacorrespondingphaseshift(overa40MHzbandwidth).Thespectrumofeachsnapshotisgivenby:Xm(f)=6Xn=0dm(n)e)]TJ /F8 7.97 Tf 6.58 0 Td[(j2f(nn+mm) (3)where(f2(0;fs))isthefrequency(inHz),n=1=fsandmarethesamplingperiodandthetimedifferencefromonesnapshottothenext,respectively.Equation( 3 ) 70

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abovecanbesimpliedasfollows:Xm(f)=e)]TJ /F8 7.97 Tf 6.59 0 Td[(j2f(mm)6Xn=0dm(n)e)]TJ /F8 7.97 Tf 6.59 0 Td[(j2f fsn (3)whichsimpliesto:Xm(r)=e)]TJ /F8 7.97 Tf 6.59 0 Td[(j2r R(mm n)6Xn=0dm(n)e)]TJ /F8 7.97 Tf 6.58 0 Td[(j2r Rn (3)foradiscretefrequencygridr=0;1:::R)]TJ /F3 11.955 Tf 12.65 0 Td[(1.Itisimportanttonotethatparameteridentiability(maximumnumberofsinusoidsthatcanbeuniquelyidentied)[ 67 69 ],becomesanissuewiththisapproach.GivenN=7realvaluedsamples,onlyuptoP=2sinusoids(amplitude,frequency,andphase),canbeuniquelyidentied.EstimatingmorethanP=2sinusoidswillsignicantlydistortthetargetsignatures.TheoverallproposedRFIsuppressionalgorithmcanbesummarizedinthestepsasshowninTab. 3-3 formulti-snapshotRELAX.WealsoconsiderAuto-regressive(AR)modellingoftheRFIdataforsuppression,howeverduetodesiredsignaldistortion,thisapproachisnoteffective.ARforRFIsuppressionfortheSIREradarisdescribedinthenextsection. Table3-3. SuppressionAlgorithm:M-RELAX+Averaging Step1:M-RELAX(Psinusoidsestimated)-ComputetheDTFTofthemthsnapshotx(m)ofthemeasureddatayfrom( 3 )andestimate^f1and^1(m),using( 3 ),( 3 ).-Computex2(m)using( 3 )foreachsnapshotanditsDTFTusing( 3 ).Estimate^f2and^2(m).Re-estimate^f1and^1(m)anditerate.Continueforyp,^fpand^p(3pP).(SectionIV.C).Step2:ReconstructaliasedRFIsamplesusingf^figPi=1andf^i(m)gPi=1foreachsnapshot.Subtractfromeachrealization(y).Step3:Averageresiduefromeachrealizationandinterleave. 71

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3.6Autoregressive(AR)ModellingAuto-regressive(AR)models,whichiscommonlyusedformodellingnarrowband(peaky)signals,canbeusedforestimatingandsuppressingRFIsignals.Themeasuredsignal(RFI,desiredtargetreturns,andthermalnoise)ismodelledasanARprocess[ 62 ].TheARmodelling(linearpredictionmodelling)equationiswrittenas:y[tn]=)]TJ /F8 7.97 Tf 17.9 15.43 Td[(qXi=1a[i]y[tn)]TJ /F4 11.955 Tf 11.96 0 Td[(i]+u[tn] (3)where,y[tn]isthemeasureddatasequence,u[tn]correspondstothewhitenoisetermatatimeinstanttnandqcorrespondstotheARorder,whichisdeterminedbythenumberofspectralpeaksandtheirwidths.TheassumptionisthatthersttermontherighthandsideofEq.( 3 )correspondstotheRFIsignal.Thesuppressionprocessthereforeinvolvesestimatingfa[i]gqi=1andusingthecoefcientstosuppresstheRFIsignals.NotethatEq.( 3 )canbere-writtenas:y[tn]=H(z)u[tn] (3)whereH(z)=1=A(z)=1=(1+a[1]z)]TJ /F11 7.97 Tf 6.58 0 Td[(1+:::+a[q]z)]TJ /F8 7.97 Tf 6.59 0 Td[(q),withz)]TJ /F11 7.97 Tf 6.59 0 Td[(1beingthedelayoperator.TheRFIsuppressionprocessinvolvespassingthemeasureddatathroughtheinverselter1=H(z)=^A(z)(fromtheestimatedARcoefcientsf^a[i]gqi=1).Thewell-knownmethodsforsolvingfortheARcoefcientsin( 3 )includetheYule-Walker(YW)method,PronymethodandthemodiedPronymethod[ 2 ].TheYWandPronymethodsgivesimilarresultsforlargedatasamples.However,forsmallerdatarecordsthePronymethodtendstogivegivesmoreaccurateARestimates[ 2 ].Ifbothsidesoftheforwardlinearpredictionequation( 3 )aremultipliedbyy[tn)]TJ /F4 11.955 Tf 12.92 0 Td[(tm],andtheexpectationistaken,thewell-knownYule-Walkerequationsareobtained. 72

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266664r(1)...r(n)377775=)]TJ /F10 11.955 Tf 11.3 38.38 Td[(266664r[0]r[)]TJ /F4 11.955 Tf 9.3 0 Td[(q+1]......r[q]r[0]377775266664a[1]...a[q]377775 (3)whichcanbere-writtenasr=)]TJ /F12 11.955 Tf 9.3 0 Td[(Ra.WhererandRarethecovariancevecotrandmatrixofthedata.TheARcoefcients(a)areestimatedbysolvingEq.( 3 ).TheYule-Walkermethodestimatesthecoefcientsbyreplacingrwiththestandardbiasedautocorrelationsequence(ACS)estimator[ 2 ].ThePronymethodsolvestheforwardlinearpredictionequation( 3 )usingleastsquares(LS).Theproblemreducesto( 3 ),withthecovariancesequenceestimatedbythestandardunbiasedACSestimator[ 2 ].TheModiedcovariance(Prony)method(whichimprovesonthePronymethod)combinestheforwardlinearpredictionin( 3 )andthebackwardlinearequationgivenbelowin( 3 )tosolvefortheARcoefcientsusingleastsquares:y[tn]=)]TJ /F8 7.97 Tf 17.89 15.43 Td[(qXi=1ab[i]y[tn+i]+ub[tn] (3)whereab[i]=a[i].ThisModiedcovariancemethodisappliedtotheSIREsampleddatawhichissampledregularlyinfast-timeandslow-time.N=7setsoftheslow-timeregularsamples(K=193samplesperset)areusedforARmodelling.TheModiedcovariance 73

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equationscanbewritteninmatrixformasfollows(foreachsetofslowtimesamples):2666666666666664y(q)...y(K)]TJ /F3 11.955 Tf 11.96 0 Td[(1)y(0)...y(K)]TJ /F4 11.955 Tf 11.95 0 Td[(q)]TJ /F3 11.955 Tf 11.95 0 Td[(1)3777777777777775=)]TJ /F10 11.955 Tf 11.29 74.24 Td[(2666666666666664y[q)]TJ /F3 11.955 Tf 11.96 0 Td[(1]y[0]......y[K)]TJ /F3 11.955 Tf 11.96 0 Td[(2]y[K)]TJ /F4 11.955 Tf 11.95 0 Td[(q)]TJ /F3 11.955 Tf 11.96 0 Td[(1]y[1]y[q]......y[K)]TJ /F4 11.955 Tf 11.96 0 Td[(q]y[K)]TJ /F3 11.955 Tf 11.96 0 Td[(1]3777777777777775266666664a[1]a[2]...a[q]377777775 (3)whereqistheARorder.Theequationcanbere-writtenasyn=)]TJ /F12 11.955 Tf 9.3 0 Td[(Yna;forn=1;2;:::NwithN=7 (3)Theleast-squaressolutionofthisoverdeterminedlinearsystemofequationsisgivenby:a=)]TJ /F3 11.955 Tf 9.3 0 Td[((YTnYn))]TJ /F11 7.97 Tf 6.58 0 Td[(1YTnynforn=1;2;:::NwithN=7 (3)where(YTnYn))]TJ /F11 7.97 Tf 6.59 0 Td[(1estimatesthecovaraincematixandYTnynestimatestheACSin( 3 ).Amoreaccurateestimateofthecovariancematrixin( 3 )isderivedbyaveragingtheN=7snapshots. 3.7ExperimentalResults 3.7.1SimulationsInthissection,asignalconsistingofthreesinusoidsinwhitenoise(SNR=10dB)issimulatedandsampledusingtheSIREequivalentschemebasedontheparametersinTab. 3-1 ,withnorepeatedmeasurements(M=1).Thesinusoidshavefrequenciesf1=111:111MHz,f2=300MHzandf3=650:255MHz,allwithamplitudesof1.Notethatthesamplesobtainedwillalsocorrespondtoasignalcontainingsinusoidswithfrequenciesfa1=f1modfs,fa2=f2modfsandfa3=f3modfs,aswellasasignalcontainingsinusoidsfa1+kfs,fa2+kfsandfa3+kfs(wherek2ZandfsistheA/Drate).ThisambiguityinfrequencyiscausedbyaliasingduetothelowA/Drateoftheradar.TheRELAXalgorithmcanbeusedtoaccuratelyestimatethecomplexamplitudes 74

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ofthesesinusoidsaswellastheambiguousfrequencies.Thisisachievedusingthespectrumin( 3 )estimatedonlyona40MHzbandwidthtosaveoncomputations.Theestimatedparametersarethenusedtoreconstructthealiasedsamples,inordertosuppressthesinusoidsthroughsubtraction. A B C DFigure3-6. RFIsuppression(SIREsampling)-Signalandspectrumofsimulateddatacontaining3real-valuedsinusoidsinwhitenoiseaftersuppressionusingRELAXwithP(real-valued)sinusoidsestimated(A)Originaldata,(B)P=1,(C)P=2,(D)P=3. Fig. 3-6 ,showstheoriginalsignal,itsspectrumandtheprogressionofsuppressionasthenumberofestimatedaliasedparametersincreases.FromFig. 3-6 ,weobservethatthespectrumofthethreesinusoidscontainsmultiplepeaks,duetotheirregularsamplingasdescribedpreviously.Byestimatingtheambiguousfrequencyandcomplexamplitudesofeachofthesinusoids(ononlya40MHzbandwidth),multiplealiasedpeaksaresuppressed.Thepurposeoftheabovesimulationsistoshowtheabilityofthe 75

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RELAXalgorithmtoestimatetheambiguousfrequencyandcomplexamplitudesofthesinusoidsonasmallbandwidthcorrectly,toeffectivelysuppressthesinusoids(includingthemultiplealiasedpeaks).InthenextsubsectionthesniffdatasetcollectedusingARL'sSIREUWBradarisanalyzedandtheRFIissuppressedusingboththeRELAXandmulti-snapshotRELAXalgorithms.ComparisonwithARmodellingoftheRFIisalsoprovided. 3.7.2SniffExperimentalDataThesniffdatatobeanalyzed,wascollectedbyARLusingtheSIREUWBradarinpassivemodebasedontheparametersinTab. 3-1 .EachsetofdataconsistsofL=KN=1351samples.InthissubsectionthisRFIdatawillbeanalyzedusingtheproposedalgorithms.TwosetsofRFIdatawithdifferentenergylevelsareanalyzed.ForsimplicitytheywillbereferredtoasFile1andFile2.Eachsetofthedata,consistsofM=88realizations.TheamountofsuppressionachievedarepresentedinTable 3-4 .Awidebandechosignalwhichrepresentsareturnfromasinglepointtargetissimulated.Thissignalisaddedtoeachrealizations(M=88)ofthesniffdata,inawaythattheechosignaladdsupcoherently.ThegoalistoshowhowmuchdistortionisintroducedtothedesiredsignalsaftertheapplicationoftheRFIsuppressionalgorithms.Fig. 3-7 showstheamountofsuppressionachievedwhentheRELAXalgorithmwithPreal-valuedsinusoidsaresuppressedforeachrealizationandtheresiduesareaveraged(File1).Theseresultsarecomparedtostraightforwardaveraging,alsointhisgure.Asimilaranalysisisperformedforthemulti-snapshotRELAXalgorithmandtheamountofsuppressioncanbeseeninTable 3-4 and 3-5 .Theaveragepowerofthesignalsf^s(i)gLi=1,arecomputedusing( 3 ):10log10 LXi=1j^s(i)j2!=L (3)FromTable 3-4 and 3-5 ,itisclearthattheamountofsuppressionincreasesasthenumberofreal-valuedsinusoidsincreasesfortheRELAXalgorithm.Thisimprovement 76

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Table3-4. RFISuppression(dB):File1(~P=1) Avg.RELAXM-RELAXRELAX/M-RELAX(~P)AR(q) 20.8923.46(P=1)24.19(P=1)27.04(P=1)23.21(2)26.61(P=4)*PI(P=4)29.11(P=4)27.79(P=7)*PI(P=7)30.38(P=7)24.52(20)28.06(P=10)*PI(P=10)31.27(P=10) Table3-5. RFISuppression(dB):File2(~P=1) Avg.RELAXM-RELAXRELAX/M-RELAX(~P)AR(q) 18.4920.02(P=1)20.22(P=1)22.40(P=1)20.03(2)21.55(P=4)*PI(P=4)23.47(P=4)22.30(P=7)*PI(P=7)23.65(P=7)20.37(20)22.61(P=10)*PI(P=10)25.97(P=10) *PI-Parameteridentiabilitynotmet. comesatacostofincreasedcomputationalcomplexity.However,thetargetsignaturesareleftbasicallyunalteredascanbeseeninFig. 3-8 .Themulti-snapshotRELAXalgorithmshowsasignicantamountofsuppressionofthedataasthenumberofsinusoidsincreases.Duetotheissueofparameteridentiabilitydiscussedintheprevioussection,estimatingmorethanP=2real-valuedsinusoids(intheory),usingonlyN=7real-valuedsamplesper-snapshotwilleffectivelysuppressallthesamplestozero.Thisleadstothesuppressionofthetargetenergy,ascanalsobeseeninFig. 3-8 .Themulti-snapshotRELAXalgorithmcanbeseentoimproveonthesuppressionwithlittletargetdistortionforP=1basedontherealRFIdatacollectedusingtheSIREradar.ThisalgorithmiscombinedwiththeRELAXalgorithmtoeffectivelysuppressbothwidebandandnarrowbandinterferers.ThisimprovementisseeninTable 3-4 and 3-5 andFig. 3-9 showsthereconstructedechoaftersuppression.AsimilaranalysisisperformedforARmodelling.TheARmodellingimprovesonthesuppressioncomparedtoaveragingascanbeseeninFig. 3-10 .However,thisinverse 77

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A BFigure3-7. RFIsuppression-RELAXalgorithmswithP(real-valued)sinusoidsestimatedandsuppressedfromsniffdata(File1)comparedtoaveraging.(A)P=1,and(B)P=10. A B CFigure3-8. Echoretrieval(File1)-RELAXwithP(real-valued)sinusoidscomparedtoidealechosignal.(A)P=1,(B)P=2,and(C)P=10. 78

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A B CFigure3-9. Echoretrieval(File1)-RELAXwithP(real)sinusoidscombinedwithM-RELAXwith~P=1realsinusoid,comparedtoidealechosignal.(A)P=1,(B)P=2,and(C)P=10. lteringtechniqueleavesthedesiredsignaldistorted.Thisdistortionisincreasedasthemodelorderincreases(duetolteringtransients)asseeninFig. 3-9 .HencethecombinedRELAXandmulti-snapshotRELAXoutperformstheARapproachintermsofbothRFIsuppressionanddesiredtargetechopreservation. 3.8ConclusionsInthischapter,wehaveproposedamethodforRFIsuppressionfortheSIREUWBradar,whichisacostefcientsystemofsamplingreturnedradarsignalsusedfordetectinglandminesandIEDsdevelopedbyARL.ThelowsamplingrateandirregularsamplingpatternofthisradarposesachallengeforRadioFrequencyInterference(RFI)suppressionasthemeasuredRFIsignalswillbeseverelyaliased.Inthischapter,wehavediscussedthechallengesofRFIsuppressionforthisradarandproposed 79

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A BFigure3-10. RFIsuppression-ARmodellingwithorderqcomparedtoaveragingforsniffdata(File1).(A)q=2,and(B)q=20. A BFigure3-11. Echoretrieval(File1)-ARmodellingwithorderq,comparedtoidealechosignal.(A)q=2,and(B)q=20. usingtheRELAXalgorithmanditsmulti-snapshotcounterpartasanintermediatesteptothealreadyproposedaveragingschemeforRFImitigation,fortheSIREUWBradar.TheresultsshowthattheRELAXalgorithmcansuppressRFIfurtherthanjustaveragingwithoutalteringdesiredtargetechosignals.TheRELAXalgorithmsareeasytoimplementsincetheyjustinvolveFFTs.TheyhavebeenshowntooutperformARmodellingoftheRFIsingals.Themulti-snapshotRELAXusesashortertime-duration(andfewersamples)forsuppression,whichyieldsamoreaccuratewidebandmodeloftheRFIassumofsinusoidscomparedtotheRELAXalgorithm.However,thisalgorithmsignicantly 80

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suppressestargetsignaturesasthenumberofsinusoidsincreasesandislimitedtoestimatingonlyonesinusoid.Combiningthisalgorithmassumingjustonesinusoid,withtheRELAXalgorithmincreasesthesuppressionperformancewithlittlesignaldistortion. 81

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CHAPTER4DATA-ADAPTIVE,SPARSESUPER-RESOLUTIONIMAGINGFORTHESIREFLGPRRADAR 4.1ChapterSummaryInthepreviouschapter,theproblemofRFIsuppressionwaspresentedfortheForwardLookingGroundPenetratingRadar(FLGPR)knownastheSIREradarbuiltbytheArmyResearchLaboratory.FLGPRhasmultipleapplications,oneofwhichincludesitsusefordetectinglandminesandotherburiedimprovisedexplosivedevices(IEDs).Inthischapter,wefocusondata-adaptivehighresolutionimagingforthisSIREFLGPR.ThestandardmethodforgeneratingSARimagesforthisradaristheback-projectionalgorithm,whichislimitedbypoorresolutionandhighside-lobesproblems.Inthischapter,weconsiderusingtheSparseIterativeCovariance-basedEstimation(SPICE)andtheSparseLearningviaIterativeMinimization(SLIM)algorithmsforgeneratingsparsehigh-resolutionimagesforFLGPR.Thepre-processinginvolvesanorthogonalprojectionofthereceivedmeasurementstoasubspacerelatedtotheregionofinterestfordataandclutterreduction.Theseuser-parameterfreealgorithmsarecapableofprovidingsparseresultsaswellasimprovedresolutionsyntheticapertureradar(SAR)images.Wealsoexaminethewell-knownCLEANapproachbasedonasignalmodelinthetimedomainforimaging.WeshowusingsimulateddatathattheSPICE/SLIMalgorithmsprovidehigherresolutionthanCLEANandstandardbackprojectionalgorithm.ImagingusingrealdatacollectedviatheSynchronousImpulseReconstruction(SIRE)radar,amultiple-inputmultiple-output(MIMO)FLGPRradardevelopedbytheArmyResearchLaboratory(ARL))isalsousedforanalysis. 4.2IntroductionTheglobalproblemoflandminesandotherburiedimprovisedexplosivedevices(IEDs)isaffectingbothmilitaryandciviliansalike[ 70 74 ],andeffectiveaswellasefcientmethodsfordetectingthesedevicesisveryimportantintheworldtoday.Methodsfordetectinglandminesincludebutarenotlimitedtotheuseofmetal 82

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detectors,infraredsensingofthelandsurface[ 75 ],biologicalsensorssuchasanimals(dogs)[ 76 ]andmorerecentlydetectingfumesoftheminesusinglaserstoionizetheair[ 74 ].Radarisanexcellenttoolforremotesensingapplications[ 77 ].Groundpenetratingradar(GPR)whichtransmitsanelectromagneticwaveintothegroundandexaminestheback-scatteredreturnstodetermineburiedobjectshasbecomeausefultoolforeffectivelydetectinglandminesandIEDs[ 78 80 ].Byoperatinginforwardlookingmode,GPRcanbeappliedtotheproblemoflandminedetectionasitinspectsthegroundsurfacewithasafestand-offrangeascanbeseeninFig 4-1 .Impulsebasedforwardlookinggroundpenetratingradar(FLGPR)typicallytransmitsamono-cyclepulsewithtypicaloperatingfrequencyrangespanningtheUHF,andLbands[ 54 79 ].ThelowfrequencyofGPRprovidesthenecessarygroundpenetratingpropertiesoftheradarandthelargebandwidthprovidesthenecessarydown-rangeresolution.Thecross-rangeresolutionontheotherhandislimitedbytheantennabeamwidth[ 81 ].Increasingtheantennaphysicalsizecanimprovecross-rangeresolution.However,thisislimitedbyphysicalantennasizeconstraints.Side-lookingsyntheticapertureradartechniquesimprovecross-rangeresolutionbysynthesizingavirtualaperturemuchlargerthanthephysicalaperture[ 11 82 84 ].However,inforwardlookingmode,thecross-rangeresolutionislimitedbythephysicalradarsize.Amulti-inputmulti-output(MIMO)radar[ 85 ]canbeusedtoenhancethisresolution[ 86 ][ 87 ].Forexample,Fig. 4-1 showstheFLGPRforlandminedetectionbuiltbytheArmyResearchLaboratory(ARL)knownasthesynchronousimpulsereconstruction(SIRE)radar[ 54 ].Thisradarconsistsof2transmittersand16receiversandexploitswaveformdiversity[ 54 86 88 ]toenhancecross-rangeresolutionbyalternativelytransmittingbetweenitstwotransmitters.Thewell-knownconventionalmethodforimagingforthistypeofradaristhestandardback-projectionmethod[ 89 90 ].Thisapproachalsoknownasthedelay-and-sum(DAS)approach,suffersfromhighsidelobeproblemsandislimitedbypoorresolution. 83

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Figure4-1. Forwardlookinggroundpenetratingradar[ 54 ] Highresolutionimagingisimportantforseparatingcloselyspacedtargetsaswellasdistinguishingtargetsfromclutterandsuchimagingtechniquesshouldbeinvestigated.Inthischapter,wefocusonsparsehigh-resolutionimagingforimpulsebasedFLGPR.Asignalmodelinthetimedomainisestablishedsincethetransmittedimpulseiswelllocalizedintime.Basedonthismodel,thewell-knownCLEAN[ 91 ],[ 92 ]approachisanalyzedforimagingcomparedtothestandardbackprojectionalgorithm.Thistechniqueeliminatesside-lobes,butitprovidesnoimprovementinimagingresolutionoverthestandardbackprojectionalgorithm.Tworecentlyproposed,user-parameterfreeanddata-adaptivemethodsareconsideredhereforimaging.TheSparseLearningviaIterativeMinimization(SLIM)[ 29 ]andtheSParseIterativeCovariance-basedEstimation(SPICE)[ 93 ]methodsarecapableofprovidingsparse,aswellashighresolutionimagingresults.ThesemethodsareappliedtothedataofsignicantlylowerdimensionforimpulsedbasedFLGPRtoachievesparsehighresolutionimagingresults.Thedata-reductionisachievedviaapre-processingtechnique,whichinvolvesanorthogonalprojectionofthereceiveddatatothesubspacespannedbythedominantsingularvectorsofasteeringmatrixcorrespondingtotheimagingROI.Anefcientdecompositionofthesteeringmatrixisperformedusinganeigenvaluedecompositionofamatrixofmuchreduceddimension. 84

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AconjugategradientSPICE(CG-SPICE)algorithmisalsointroducedinthispapersimilartotheCG-SLIMalgorithm[ 29 ]tospeedupthecomputationofSPICE.Theimprovementinimagingresultsoverthestandardbackprojectionmethodareshownviasimulatedresultsandalsorealmeasuredexperimentaldata.Forrealexperimentaldata,theSIREradardevelopedbyARL[ 54 94 ]isusedforanalysis.ThisradarwasalsopresentedasaMIMOradarin[ 88 ].Theremainingsectionsofthischapterareorganizedasfollows.InSection 4.3 ,aproposeddatamodelispresentedforforwardlookingGPRbasedontheARL'sSIREradar.ThisMIMOradarisalsobrieydescribedtherein.InSection 4.4 ,thestandardback-projectionmethodforimagingisanalyzedaswellARL'sRecursiveSidelobeMinimization(RSM)algorithmwhichisbasedontheBPalgorithmanditerativelyandeffectivelysuppressessidelobes[ 54 95 ].BasedontheproposedmodelwealsoshowthattheCLEANapproachcanbeusedforsparseimagingforimpulsebasedFLGPR.InSection 4.5 wepresentthesparseandadaptivemethodsforimprovedresolutionaswellasthepre-processingstepoforthogonalprojectionforclutteranddatareduction.Section 4.6 containsthenumericalresultsbasedonsimulatedandrealdata.FinallytheconclusionsaredrawninSection 4.7 4.3DataModel:SIREImpulseBasedFLGPRForimpulsebasedFLGPR,weconsidertheSIREradarwhichisdesignedbyARLandmountedonanSUVforlandminedetection[ 54 ].TheradargeometryasseeninFig. 4-1 consistsoftwotransmittersandsixteenreceivers.Eachtransmittertransmitsanimpulsewithafrequencyrangeof0.3-3.0GHz,whichdeterminesthedownrangeresolution.Thecrossrangeresolutionisdeterminedbythephysical2mapertureofthisradar.ThisradarcanbedescribedasapracticalexampleofaMIMOradarwhichexploitswaveformdiversitybytransmittingorthogonalwaveformsfromthetwotransmitantennaslocatedattheedgesofthereceivearray[ 88 ].Theseorthogonalwaveformsareachievedbyalternativelytransmittingnarrowpulses(inping-pongmode[ 96 ])from 85

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Figure4-2. SIREFLGPR:2DapertureforSARimaging eachtransmitter.Thiscreatesavirtualaperturewhichiseffectivelyalmostdoublethephysical2mapertureoftheradarhenceimprovingcross-rangeresolution[ 97 ].The2D-apertureofreceivedmeasurementsshowninFig 4-2 isusedforimageformationwheretherearek=1Kreceivemeasurementsforadesiredimagingareaconsistingofi=1Ltargets(pixels).Letrk(t)denotethekthreceivemeasurement.Thismeasurementcandescribedbythefollowingequation: rk(t)=L+MXi=1k;izis(t)]TJ /F4 11.955 Tf 11.96 0 Td[(k;i)+nk(t)(4)wherek;i=1 Rt(k;i)1 Rr(k;i)isthepropagationpath-loss,ziisthereectivityoftheithtarget,k;i=Rt(k;i)+Rr(k;i) cisthetriptimedelayfromtransmittertotargettoreceiver,whereRt(k;i),Rt(k;i)arethedistancesoftransmittertotargetandthetargettoreceiver,respectively.Thespeedofpropagationisgivenbycandnk(t)isthermalnoiseassociatedwithkthmeasurement.Notethatthemodelofthereceivedmeasurementin( 4 )takesintoaccountthecontributionoftheMscatterersoutsidetheROI,i.e., 86

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desiredimagingareawhichconsistsofLpixels.Themodelin( 4 )canbesimpliedintothefollowinglinearequation: (4) y=Cz+n=264AB375264375+nwherey=[r1(0);:::;r1(T)]TJ /F3 11.955 Tf 12.04 0 Td[(1);:::;rK(0);:::;rK(T)]TJ /F3 11.955 Tf 12.04 0 Td[(1)]T,whichisavectorofreceivedmeasurementsstackedtogether;=figLi=1arethepixelvaluesinthedesiredimaginggridtobeestimated,=figMi=1correspondstothepixelvaluesoutsidetheROIandnisthenoisevector.ThematrixAofdimensions(TK)Lconsistsofdelayedandscaledversionsofthetransmittedsignal,givenby: A=2666641;1s(1;1)1;Is(1;I).........K;1s(K;1)K;Is(K;I)377775(4)wherethevectors(k;i)=fs(t)]TJ /F4 11.955 Tf 12.67 0 Td[(k;i)gT)]TJ /F11 7.97 Tf 6.59 0 Td[(1t=0isthetransmittedimpulsedelayedbyk;i.ThisdatamodelisusedforFLGPRSARimaginginthispaper.InthenextsectionthebackprojectionalgorithmaswellastheCLEANalgorithmaredescribedforSARimaging. 4.4Back-projection/Delay-and-sum(DAS)BasedMethodsThestandardbackprojection(BP)algorithmisawell-knownandwidelyusedalgorithmforFLGPRSARimaging(alsoknownasthedelay-and-sum(DAS)algorithm).Thisalgorithmislimitedindownrangeresolutionbythebandwidthofthetransmittedimpulseandincross-rangeresolutionbythephysical(orvirtual)apertureoftheradar.Oneotherlimitationofthisalgorithmisthatitproducesimageswithhighsidelobes.ARecursiveSidelobeMinimization(RSM)algorithmbasedontheBPalgorithmwasproposedin[ 54 ],[ 95 ]foreffectivelysuppressingsidelobes.TheCLEANapproach[ 91 ](whichisalsobasedonthisBP/DASalgorithm)canalsobeusedforeliminating 87

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sidelobes[ 92 ]aswellasaccuratelyestimatingweaktargetsbyiterativelysubtractingthecontributionsofstrongertargetsfromthereceiveddatabasedontheproposeddatamodel.Inthissection,wedescribethesealgorithmsaswellasanalyzetheCLEANapproachbasedontheproposeddatamodelforimpulsebasedFLGPRSARimaging. 4.4.1Back-projection/DASBasedonFig 4-2 ,thebackprojectionalgorithmisdescribedasfollows:Considertheithpixelinthisgurewithlocation(xi,yi,zi)relativetoapredenedreferencepointororigin.Foraspecictransmit-receivepair,let(xr;k,yrk,zrk)and(xt;k,ytk,ztk)denotethetransmitterandreceiverlocationsinthiscoordinatesystemfork=1;:::;K.ThedelayduetothetransmittedEMpulsefromthetransmittercorrespondingtothekthreceivemeasurementtotheithpixelbacktothecorrespondingreceiverisgivenas: (4) k;i=acq+1 c(q (xt;k)]TJ /F4 11.955 Tf 11.95 0 Td[(xi)2+(yt;k)]TJ /F4 11.955 Tf 11.95 0 Td[(yi)2+(zt;k)]TJ /F4 11.955 Tf 11.96 0 Td[(zi)2+q (xr;k)]TJ /F4 11.955 Tf 11.96 0 Td[(xi)2+(yt;k)]TJ /F4 11.955 Tf 11.96 0 Td[(yi)2+(zt;k)]TJ /F4 11.955 Tf 11.95 0 Td[(zi)2)whereacqistheacquisitiontimedelayassociatedwiththeradarsystem.Theestimateofthereectioncoefcientattheithpixelgivenbyiisgivenas: ^i=1 KKXk=1wkrk(t)]TJ /F4 11.955 Tf 11.96 0 Td[(k;i)(4)Thisestimateissimplyasummationofdelayedreceivemeasurementswiththepropagationlosscompensatedbyaweightingfactorwk.ThisisreferredtoascoherentprocessingofthereceivedmeasurementswhichimprovesSNRbyafactorKcomparedtousingasinglereceivedmeasurement.Thebackprojection/DASalgorithmislimitedinresolutionandsuffersfrompoorsidelobes.Therecursivesidelobeminimization(RSM)algorithmproposedbyARLforsidelobesuppressioneffectivelysuppressessidelobesbygeneratingmultipleDASimageswithapertureswithrandomlymissingmeasurementsandselectingtheminimumvalueacrossallimages[ 54 ].InthenextsubsectionweanalyzeaCLEANapproach 88

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basedonthedatamodelin( 4 )forsparseimaging.TheseDASbasedalgorithmsdonotimproveimagingresolutioncomparedtothestandardDASalgorithm.WethenpresentnewapproachestoimagingusingtheSLIMandSPICEalgorithmsforimprovedresolutioninimagingafterpreprocessingviaanorthogonalprojectionofthedata. 4.4.2Sparse:CLEANMethodThedata-dependentCLEANalgorithm[ 91 ](alsoknownasmatchingpursuit)forimageformationbasedonthedatamodelin( 4 )isbrieydescribedandanalyzedhereforimaging.Thistechniquewasintroducedtoproduce'CLEANer'images(wherepriorknowledgeofthepointspreadfunctionwasrequired)[ 92 ].ThestandardDASalgorithmsuffershighsidelobeproblems.CLEANcanbeusedtoeliminatetheside-lobesofstrongreturnssothatweaktargetscanberevealedbyeliminatingcontributionsofstrongtargetsfromthereceivemeasurements.TheCLEANalgorithmcanthereforebeusediterativelytondthepixellocationofthestrongesttargetandthensubtractallthecontributionsofthattargetfromthedata.Thenextstrongestpointisthencomputedmoreaccuratelybasedontheupdatedmeasurements.TheRELAXalgorithm[ 31 ]canthereforebeusedtogetevenmoreaccurateestimatesaswellasimprovedimagingresolution.TheRELAXapproachinvolvesestimatingthestrongesttarget,subtractingthecontributionsofthistargetandestimatingthenextstrongesttarget.Theinitialpixelvalueofthestrongesttargetisthenre-estimatedbasedonthenewestimateofthenextstrongesttarget.Thesetwoestimatesaretheniteratedbackandforthtoachievemoreaccurateestimates.Theprocessisthenrepeatedforallthetargetsinthesceneofinterest.Duetotheexponentialincreaseincomputationasthenumberoftargetsincreases,thisprocessprovestobetoocomputationallyintensiveforpracticalpurposeswhentherearemanyscatterersintheROI.ThemuchfasterCLEANapproachisdescribedinthefollowingstepsbasedonthedatamodelin( 4 ). 89

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Step1 :DeterminethebrightestestimateP(io)andcorrespondinglocationiofromthebackprojectionimage(basedon 4 ). Step2 :Subtractthecontributionofbrightestpointfromthereceivedsignals(updatereceivemeasurements). rk up=rk)]TJ /F4 11.955 Tf 11.95 0 Td[((io)s(^k;i0)k=1K (io)=P(io) K1 k;io Step3 :GenerateanewimagebyllingintheiothpixelwithP(i0). Step4 :Useupdatedreceivedsignalstoregenerateback-projectionimage. Iteration :Repeatpreviousstepswithregeneratedimageuntilreachingapredenedthreshold.>P(io) 2. threshold(typicallychosenas1). 2Noisevariance.ThisCLEANapproach,althougheffectiveforaccurateandsparseimaging,islimitedinresolution,whichissimilartothestandardbackprojection/DASalgorithm.Wetherefore,investigatesuper-resolutionmethodsforFLGPRSARimaging. 4.5Super-resolutionMethodsInthissection,anewapproachforhigh-resolutionimagingispresentedforimpulsebasedFLGPR.The2DapertureinFig. 4-2 forimagingofthespeciedgridwillresultinadatavectorin( 4 )y2RTK1thatislarge(ontheorderof106),makingpracticalapplicationsofadaptivetechniquesinfeasible.Alsotheavailabilityofasingledatavectormakeswellknownhighresolutiondataadaptiveapproaches,suchastheCAPON,APES,aswellassubspacebasedmethods[ 2 ],notdirectlyapplicable.AnotherchallengeisthatthedatavectorwillcontainclutterreectionsfromregionsoutsidetheimagingROI,whichneedtobeeffectivelysuppressedpriortoimaging.WeproposeadatalteringandreductionapproachviatimegatingandorthogonalprojectiontoreduceinterferencefromscatterersoutsidetheROI.Thisapproachinvolvesasingularvaluedecompositionofthesteeringmatrixandaprojectionofthe 90

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Figure4-3. Timegating datausingthedominantsingularvectors,whicheffectivelyreducesthedatadimensiontopracticallevels.ThisprojectionalsoltersthedatatoeffectivelyutilizereceivedenergyfromonlytheROI.Acomputationallyefcientmethodofthisprojectionisperformedviaaneigenvaluedecompositiontoobtainanupdateddatamodel.Usingtheupdatedmodel,tworecentlyproposedalgorithms(SPICEandSLIM)areusedforimagingtoproducesparse,accurateandhighresolutionimagingresults,evenwithasingledatavector.Theprocessisdescribedinthenextsubsections. 4.5.1OrthogonalProjectionandTimeGatingFortimegating,considerthekthreceivemeasurementfrk(t)gT)]TJ /F11 7.97 Tf 6.59 0 Td[(1t=0.Basedon( 4 ),thedelaysofthepixelsintheROIimaginggrid,k=[k;1;k;2;:::;k;I],tothecorrespondingtransmit-receivepairforthismeasurementcanbecomputed.Theminimumandmaximumdelaysbetweenthistransmit-receivepairandthepixelsintheROIimaginggridaregivenbykmin=min(k)andkmax=max(k),respectively.Thekthreceivemeasurementcanthenbeupdatedtofrk(t)gkmaxt=kminbydiscardingdataoutsidethesecomputeddelays.Withoutlossofgenerality,thisprocedureisshowninFig. 4-3 foracolocatedtransmit-receivepaircenteredbelow(withapre-speciedstandoff-range)theimagingROIgrid.Interferencefromtheregionsbelowtheminimumdelaylineandabovethemaximumdelaylinearediscarded.Thedatavectorin( 4 )canthenbeupdatedto 91

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yg=[r1(1min);:::;r1(1max);:::;rK(Kmin);:::;rK(Kmax)]T,andtheupdateddatamodelis: yg=Ag+Bg+ng:(4)withAg2RQLandBg2RQM(Q
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becomes: (4a) ^=argminjjUAUAT(y)]TJ /F12 11.955 Tf 11.95 0 Td[(A)jj2+jjjj1 (4b) ^=argminjjUAT(y)]TJ /F12 11.955 Tf 11.96 0 Td[(A)jj2+jjjj1ConsiderthefollowingsingularvaluedecompositionofthesteeringmatrixA2RQL,withQ>Linpractice1: (4) A=UVT=264UA~UA375264A0375VT=UAAVTwherethecolumnsof[UA;~UA]aretheleftsingularvectorsofAandthecolumnsofVaretheirrightcounterparts,andthesingularvaluesofAareonthediagonalofthediagonalmatrixA.DuetothendgridusedfortheROI,someofthesingularvaluesofAinAarequitesmall.BydiscardingthesmallsingularvaluesofA,weapproximateAasAUssVsT.ThentheoptimizationprobleminEq.( 4b )becomes: ^=argminjjUsT(y)]TJ /F12 11.955 Tf 11.95 0 Td[(A)jj2+jjjj1(4)ThenusingUsfororthogonalprojectionyields: (4a) UsTy=UsTA+; (4b) ~y=~A+Viaaseriesofsimulations,wefoundthatthenumberofcolumnsinUsisnotsensitivetotheimageformationperformance.WechoosethedimensionofUsbyanalyzingthe 1NotethatsincetheradarilluminatesalargeareaM>Q,butwedonotconsiderMasthecontributionsoftheclutter(B)areeliminatedbasedonthedecomposition 93

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metricjjA)]TJ /F12 11.955 Tf 11.96 0 Td[(UssVsTjjF=jjAjjF.Noteherethat~y2Rs1and~A2Rs1wheres<
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Theupdateddatamodelin( 4b ),obtainedusingUscanbeobtainedusingthedecompositionin( 4 ).Thisdecompositioncanbeperformedofineaslongaspriorknowledgeoftheimaginggridisknown,whichistypicallythecase.Adifferentapproachfororthogonalprojectionthatimprovesoncomputationisdescribednext.ConsiderthesteeringmatrixGwhichcorrespondstotheimagingROIwithamuchcoarsergrid.ThematrixGcanthenbegeneratedbyselectingtheappropriatecolumnsofA.ThematrixGisusedtoapproximateUs.However,unlikeUsthismatrixisnotsemi-unitary.Theupdateddatavectoristhengivenas~y=(GTG))]TJ /F11 7.97 Tf 6.58 0 Td[(1GTyandtheupdatedsteeringmatrixisgivenas~A=(GTG))]TJ /F11 7.97 Tf 6.58 0 Td[(1GTA.Thisnewprojectionapproach,skipsthecomputationATAanditssubsequentdecomposition,signicantlyimprovingcomputationatcostoflessinterferencesuppression..Basedontheupdatedmodelin( 4b ),wepresentbelowtworecentlyproposed,data-adaptiveanditerativeapproachesforhighresolutionFLGPRSARimaging.TheyaretheSparseLearningviaIterativeMinimization(SLIM)[ 29 ]andtheSParseIterativeCovariance-basedEstimation(SPICE)[ 93 ],whichisequivalenttosquare-rootLASSOwith=1[ 99 ].Thesetwoapproachesareuser-parameterfreeandarecapableofproducingsparseandhighresolutionestimateswhenonlyasingledatavectorisavailableforimaging;theyaredescribednext. 4.5.2SLIMTheSLIMmethod[ 29 ]isamaximum-aposteriori(MAP)approachforsparsesignalrecovery.Thesparserecoveryproblemiscanbesolvedbyoptimizingthe`1optimizationcostfunctionin( 4 )basedonthelinearmodelin( 4 ): ^=argminjj~y)]TJ /F12 11.955 Tf 13.05 2.66 Td[(~Ajj2+jjjj1(4)TheSLIMalgorithmcanbeconsideredasan`qnormapproachnormfor0
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Table4-1. SLIMAlgorithm Initialization:Obtaininitialestimate^(0)withtheDASalgorithmand^(0)basedon^(0)and( 4 )SLIM(nth)Iteration:RepeatthefollowingstepsuntilconvergenceStep1:Compute:^R(n+1)=~Adiag(j(n)j2)]TJ /F8 7.97 Tf 6.58 0 Td[(q)~AT+(n)I=~AP(n)~AT+(n)I.Step2:Compute:^(n)=P(n)~AT^R)]TJ /F11 7.97 Tf 6.59 0 Td[(1(n)~y.Step3:Update:^(n)=1 Ljj~y)]TJ /F3 11.955 Tf 13.2 2.66 Td[(~A^(n)jj22. estimation[ 29 ]. (4a) ~yj;CN(~A;I) (4b) f()/Yie)]TJ /F18 5.978 Tf 7.93 3.25 Td[(2 q(jijq)]TJ /F11 7.97 Tf 6.59 0 Td[(1) (4c) f()/1SLIMestimatesthedesiredsparsevector,andthenoisevariance,iterativelybyminimizingthenegativelogarithmcostoftheposteriordensitygivenby: cq(;)=Llog+1 jj~y)]TJ /F3 11.955 Tf 13.2 2.65 Td[(~Ajj22+Xi2 q(jijq)]TJ /F3 11.955 Tf 9.3 0 Td[(1)(4)Thechoiceofq=1simpliesthiscostfunctiontothewell-known`1normconstraintforsparseestimation[ 29 ].Minimizingthecostfunctionin( 4 )yieldsthefollowingestimates: (4a) ^=PAT(~AP~AT+I))]TJ /F11 7.97 Tf 6.59 0 Td[(1~y=P~ATR)]TJ /F11 7.97 Tf 6.59 0 Td[(1~y (4b) ^=1 Ljj~y)]TJ /F3 11.955 Tf 13.2 2.66 Td[(~Ajj22whereP=diag(p)andp=jj2)]TJ /F8 7.97 Tf 6.58 0 Td[(q.TheseestimatesareobtainedinaniterativemannerbasedonthestepsinTab. 4-1 4.5.3SPICETheSPICEmethodforparameterestimation[ 30 93 ]modiesthemodelin( 4 )asfollows: (4) ~y=~A+=Dx 96

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Table4-2. CGSPICEAlgorithm Initialization:Obtaininitialestimate^xj(0)=dTj~y=dTjdj,andpj(0)=j^xj(0)j=!jj=1;:::;L+s.SPICE(nth)Iteration:RepeatthefollowingstepsuntilconvergenceStep1:Compute^R(n)=Ddiag(p(n))DT=DP(n)DT.Step2:Compute(usingCG):s(n)=R)]TJ /F11 7.97 Tf 6.59 0 Td[(1(n)~y.CGInitialization:s(n)(0)=0;r(0)=q(0)=~yCGIterations(mth):-(m)=(rT(m)r(m))=(qT(m)q(m))-s(n)(m+1)=s(n)(m)+(m)q(m)-r(m+1)=r(m))]TJ /F4 11.955 Tf 11.96 0 Td[((m)^R(n)q(m)-q(m+1)=q(m)+(rT(m+1)r(m+1))=(rT(m)r(m))Step3:Update^x(n+1)=P(n)DTR)]TJ /F11 7.97 Tf 6.58 0 Td[(1(n)~y:j=1;:::;L+s.Step4:Updatepj(n+1)=^xj(n+1)=!jj=1;:::;L+s. whereD=[~A;I]andx=[T;T]T.Thismethodisacovariancettingapproachtoparameterestimationthatminimizesthefollowingcovariancettingcostfunction: jjR1=2(~y~yT)]TJ /F12 11.955 Tf 11.95 0 Td[(R)jjF(4)whereR=Ef~y~yTg=DPDTisthecovariancematrixofthedataandthediagonalmatrixPisnowgivenas: P=264Ps00P375(4)withPs=diag(ps)andP=diag(p)beingthediagonalmatricescontainingthepowerestimatesofandthenoiseandinterferenceresidue,respectively.Thecriterionin( 4 )simpliestothefollowingminimizationproblemtoestimatebothxandp=[psT;pT]T. f^x;^pg=argminx;pxTP)]TJ /F11 7.97 Tf 6.58 0 Td[(1x+L+sXj=1!jpjs.t.Dx=~y(4)where!j=jjdjjj=jj~yjjandjjdjjjisthejthcolumnofD.Theestimatesaregivenas: 97

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(4a) ^x=PDTR)]TJ /F11 7.97 Tf 6.59 0 Td[(1~y (4b) pj=jjj !j;j=1;:::;L+swhicharesolvediterativelytillconvergence[ 30 ];withx=fxjgL+sj=1andp=fpjgL+sj=1.ToimproveonthecomputationallyefciencyoftheSPICEalgorithmforFLGPRSARimaging,aconjugategradientbasedSPICEalgorithm(CG-SPICE)ispresentedinthispaper.ThisapproachissimilartotheconjugategradientSLIMalgorithmdescribedin[ 29 ],[ 100 ].ThestepsofthisCG-SPICEalgorithmaredescribedinTab 4-2 .Basedon[ 101 ],theSPICEoptimizationproblemcanbere-writtenas: argmin^xjjxjj1=L+sXj=1jxjjs.t.Dx=~y(4)Forreal-valueddata,let~hj,max(xj;0)andhj,)]TJ /F1 11.955 Tf 9.3 0 Td[(min(xj;0).Notethatxj=~hj)]TJ /F3 11.955 Tf 12.24 2.66 Td[(hjandjxjj=~hj+hj.TheoptimizationprobleminEq.( 4 )canthenbeaugmentedto[ 102 ]: argmin^huThs.t.Dh=~yandh0(4)whereh=[~h1;:::~hL+s;h1:::hL+s],D=[D;)]TJ /F12 11.955 Tf 9.29 0 Td[(D]andu=[1;1:::;1]T.Thelinearprogramin( 4 )canbesolvedefcientlytoprovidesparseestimatesforFLGPRSARimaging.Thisapproachisonthesameorder(computationally)asthecyclicoptimizationapproachinTab 4-2 implementedusingtheconjugategradient,andfaster(approximately2times)withoutCGbasedonnumericalsimulations.NotethattheestimatesinSLIMandSPICEhavethesameformwiththedifferencelyingintheestimationofthenoiseandinterferenceresidue.SPICEestimatesthereectioncoefcientsandthenoiseandinterferenceresiduesimultaneouslywiththenoiseandinterferenceresiduevarianceofeachelementsnotnecessarilybeingequalunlikethecaseinSLIM.However,simulationresultsshowsimilarperformanceofthetwoalgorithmswithSPICEbeinglesssusceptibletothenoiseandinterferenceresidue.Thisalgorithmsarerobustandeffectiveforgeneratingsparseresults. 98

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Table4-3. Subspaceapproximation No.ofsingularvectors-sjjA)]TJ /F12 11.955 Tf 11.96 0 Td[(UssVsTjjF=jjAjjF 150.9021000.5123000.182 Unlikemostwell-knownadaptivealgorithms,SLIMcandynamicallyestimatetheuserparameter(whichinthiscase,correspondstothenoisepowerestimate)oftheoriginalLASSOcostfunction(forsparseparameterestimation)[ 103 ]whichissensitivetothechoiceofthisparameter().TheSPICEcriteriacanalsobereducedtothecriteriain( 4 )with=1[ 99 ](aspecialcaseofthesquare-rootlasso[ 98 ]whichisinsensitivetothechoiceoftheuser-parameter).TheserobustuserparameterfreealgorithmsareappliedheretoproblemofFLGPRSARimaging.AnalysisisperformedonbothsimulatedandrealexperimentallymeasuredSIREFLGPRdataforimaging,andtheresultsarepresentedinthenextsection. 4.6NumericalandExperimentalResultsInthissection,weperformsparsehighresolutionimagingforFLGPRusingorthogonalprojection(usingUs)forclutteranddatareduction.TheSPICEandSLIMalgorithmsareconsideredforhighresolutionimaging.Wealsoanalyzethewell-knownCLEANapproachforimagingbasedontheproposeddatamodelandshowtheabilityofthiswell-knownalgorithmtoyieldsparseandaccurateresults.TheCLEANapproach,however,islimitedinimagingresolutionanddoesnotimproveresolutionoverthestandardBPalgorithm.ThecoarsegridapproachusingGisalsoanalyzedforprojection.ForFLGPRSARimaginganalysis,weusetheSIREradardesignedbyARLforimagingbasedonthesetupinFigs. 4-1 and 4-2 .Forsimulations,weconsideranimagingareawhichhasarangeswathof4mandacross-rangeswathof5m.A 99

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A B C DFigure4-4. Subspacedimension(s)forhighresolutionimaging(A)EigenvaluesofATA,(B)s=15,(C)s=100,and(D)s=300 minimumstandoffdistanceof8misusedforsimulations,withamaximumstandoffdistanceof14m.Simulationwasrunwiththreetargetsplacedatvarious[x;y;z]locationsinmetersmarkedbythesymbol'X'.TheimagingareaconsistsofL=10000pixels.Ananalysisofthesubspacedimension,(i.e.,thenumberofdominantsingularvaluesofA)isperformedrst.Targetsatlocations[0,0,0],[-0.3,1,0],and[1.5,1.5,0]aresimulated.Fig. 4-4 showstheSPICEalgorithmappliedwithvariousthresholds.FromTab. 4-3 ,thresholdsaslarge0.5basedonthecriterionjjA)]TJ /F12 11.955 Tf 11.95 0 Td[(UssVsTjjF=jjAjjF 100

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A B C D EFigure4-5. FLGPRSARImaging-detectionofweaktarget(A)Back-projection,(B)RSM,(C)CLEAN,(D)SLIM,and(E)SPICE (normalizedscaleof0to1)yielddesirableresultswithallthetargetsdetectedascanbeseeninFig. 4-4 .Thenextanalysisinvolvesdetectingweaktargetsburiedbythesidelobesofmuchstrongertargets.Threetargetsareagainsimulatedatlocations[0,0,0],[-0.1,0.8,0], 101

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A B C D EFigure4-6. FLGPRImaging-resolutionimprovement(A)Back-projection,(B)RSM,(C)CLEAN,(D)SLIM,and(E)SPICE and[1.5,1.5,0],withthestrongtargets10timesstrongerthantheweaktargetasshowninFig. 4-5 .Fromthisgurewecanseethattheweaktargetisburiedbythesidelobesofthestrongertargetusingthestandardbackprojectionalgorithm.TheCLEANapproach,whichiterativelysubtractsoutthecontributionsofthestrongesttarget 102

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A B C DFigure4-7. Orthogonalprojectionusing(A)Us(HighSNR),(B)G(HighSNR),(C)Us(LowSNR),and(D)G(LowSNR) fromthereceivemeasurements,caneffectivelyandaccuratelydetectthisweaktarget.TheSPICEandSLIMalgorithmsarealsoappliedpostorthogonalprojection(withthresholdjjA)]TJ /F12 11.955 Tf 11.96 0 Td[(UssVsTjjF=jjAjjF=0:2)andtheweaktargetisrevealed.TheCLEANapproach,althougheffectiveinprovidingsparseandaccurateresults,islimitedinresolutionandhasnoimprovementoverthestandardbackprojectionalgorithm.ThehighresolutionimagingmethodsforFLGPR,provideimprovementinresolutionoverthebackprojection-basedalgorithms(BP,RSMandCLEAN).Fig. 4-6 showsthreetargetsatlocations[0,0,0],[0,0.75,0],and[1.5,1.5,0].Twoofthesetargetsare'closely'spacedandareclearlyresolvedbySLIMandSPICE.TheSPICEandSLIMalgorithmsprovidealmostafactorof2improvementinimagingresolution. 103

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A B CFigure4-8. Realdata-SIREFLGPRSARImaging:(A)Back-projection,(B)RSM,and(C)SPICE 104

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Figure4-9. ReceiverOperatingCurve(ROC)comparison:FLGPRSARImaging ThisexperimentisrepeatedusingGfororthogonalprojection.Theresultsarecomparedtousingthesemi-unitarymatrixUsforprojectionandareshowninFig. 4-9 .Againincomputationisachievedatacostoflessinterferencesuppression.Theproposedapproachisveriedusingrealexperimentallymeasureddata.ResultsbasedonrealSIREdataprovidedbytheArmyresearchlabcanbeseeninFig. 4-8 .Inthisgureasubimages2minrangearecontinuouslyformedbasedonoverlapping2Daperturestogeneratetheentireimage[ 54 ].Basedonthisgure,signicantinterferencereductioncanbeseenbythehighresolutionSPICEcomparedtothestandardbackprojectionalgorithm,withsomeofthetargetsofinterestmarkedinredovalcircles.Aquantitativenumericalanalysisisperformedtoshowtheeffectivenessofthehigh-resolutionSPICEapproachforFLGPRSARimaging.Severaltargetswithvaryingstrengthsaresimulatedandthereceiveroperatingcharacteristicscurve(ROC)isshowninFig. 4-9 .Thiscurvewhichshowstheprobabilityofdetectionversusthenumberoffalsealarmsisgeneratedbyusingasimplethresholddetector.Theimageunderanalysisissegmentedintoregions,andthemaximumpixelvalueineachregionisretained.Thethresholdisincrementedinstepsandforeachthreshold,thenumber 105

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ofalarmsarerecorded.Thealarmsthatfalloutsidetheregionswithtargetspresent(basedonpriorknowledge)areconsideredfalsealarms.AsshowninFig. 4-9 ,whenthereisfulldetection,theSPICEapproachhaslessfalsealarmsthetheBPalgorithms. 4.7ConclusionsInthischapter,wehaveconsiderednewapproachestoimagingforforwardlookinggroundpenetratingradar.Thepre-processinginvolvesapropositionofadatamodelinthetimedomain,whichtakesintoaccountthecontributionsofclutteroutsidetheimagingarea.AnorthogonalprojectionofthemeasureddatatoasubspacespannedbythesteeringmatrixcorrespondingtotheimagingROIisthenusedforclutterreductionaswellassignicantdatareduction,makingitfeasibleforpracticalapplicationsofhighresolutionmethods.Thesteeringmatrixdecompositionisperformedefcientlyanddependsonlyonthepriorknowledgeofthedesiredimagingareaandhencecanbeperformedofine.Tworecentlyproposed,data-adaptiveapproaches,SPICEandSLIMareusedforFLGPRSARimaging.Theyareuserparameterfreealgorithmsandhavetheabilitytoprovidesparseandhighresolutionimagesusingasingledatavector,unlikeotherwell-knownhighresolutionmethods.TheresultsusingsimulateddatashowthatSLIMandSPICEprovideimprovementinresolutionclosetoafactoroftwocomparedtothebackprojectionbasedalgorithmsincludingBP,RSM,CLEAN.AnewconjugategradientbasedSPICEalgorithmisalsointroducedinthispaperformoreefcientcomputationsoftheestimates. 106

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CHAPTER5CONCLUDINGREMARKSANDFUTUREWORKDuetolimitationsofdataindependentmethodsforspectralestimation,data-adaptivemethodsarecurrentlybeinginvestigatedforimprovedperformance.Inthisdissertation,wefocusedonefcientandeffectiveapplicationsofdata-adaptivemethodstorealworldsensingproblems.InChapter2,thebasicproblemofharmonicretrievalisinvestigated.Theproblempertainstodigitalaudioforensics.Thecontributionwemaketothisprobleminvolvescomingupwithamorereliableandaccuratewayofestimatingthenetworkfrequencyburiedinanaudiorecordingusingdataadaptivetechniques.Theproposedapproachinvolvesspectralanalysisusingarobusthighresolutionalgorithmandtrackingthenetworkfrequencyviaadynamicprogrammingapproach.Theapproachyieldssignicantimprovementintheestimationoftheembeddednetworkfrequencywhenthissignalisweakcomparedtotheaudiorecording(amajorchallengeforthisproblem).Chapters3and4arethefocusofthisdissertation.Inthesechapters,theSynchronousImpulseReconstruction(SIRE)radar,whichisaremotesensingtoolforlandminedetectionisanalyzedandstudied.InChapter3,weproposeanewapproachofRadioFrequencyInterferencesuppressionforthisradar.Thisnewapproachcanprovideanimprovementofcloseto7dBinRFIsuppressionwithoutdistortingthedesiredtargetsignatures.Thisapproachisimplementedinanefcientwaybyexploitingtheequivalentsamplingtechniqueofthisradar.Chapter4focusesonsparsehighresolutionimagingforthisSIREForwardLookingGroundPenetratingradar(FLGPR).Inthischapter,weestablishasignalmodelinthetimedomainsincethetransmittedimpulseiswelllocalizedintime.Thisdatamodeltakesintoaccountthecontributionsofclutteroutsidetheimagingregionofinterest(ROI).Weproposeapre-processingstepoforthogonalprojectiontomitigatetheeffectsofclutteroutsidetheROIwhichispresentinthecollecteddata.Recentlyproposed 107

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robust,sparsehighresolutionalgorithmsforimagingarethenappliedtoprovideimprovedimagingresolution.Weachieveclosetoafactorof2improvementinimagingresolutioncomparedtothestandardmethodscurrentlyusedforSARimagingforthisradar.Ourcurrentandfutureworkfocusesonmoreefcientwaystoimplementthepre-processingstepoforthogonalprojection.InlieuofadecompositionofthesteeringmatrixcorrespondingtoimagingROIandthenprojectionofthedata,weproposeadirectprojectionofthedatatothesteeringmatrixcorrespondingtotheROIwithamuchcoarsergridtoimprovecomputation.ThismatrixapproximatesthesetoforthogonalvectorsthatspansthesubspaceofthematrixcorrespondingtotheROI.Resultsshowsignicantimprovementincomputationatcostoflessinterferencesuppression. 108

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BIOGRAPHICALSKETCH OdeOjowuJr.wasborninZaria,Nigeria.HecametotheUnitedStatesin2001topursueanacademiccareer.HereceivedaBachelorofArtsinphysicsfromGrinnellCollegein2005,aswellasaBachelorofScienceandMasterofScienceinelectricalengineeringfromWashingtonUniversityin2007.HeiscurrentlywiththeSpectralAnalysisLab(SAL)supervisedbyProf.JianLiattheUniversityofFlorida.HewillreceiveaDoctorofPhilosophyinelectricalengineeringfromtheUniversityofFloridaintheFallof2013.Hisgeneralresearchinterestliesintheeldofsignalsandsystemswithafocusondata-adaptivespectralestimationtechniques,arraysignalprocessingandradarsignalprocessing. 117


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