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Advanced Techniques for Synthetic Aperture Radar Image Reconstruction

Permanent Link: http://ufdc.ufl.edu/UFE0043913/00001

Material Information

Title: Advanced Techniques for Synthetic Aperture Radar Image Reconstruction
Physical Description: 1 online resource (131 p.)
Language: english
Creator: Vu, Duc H
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: aperture -- estimation -- image -- radar -- reconstruction -- sar -- sparse -- spectral -- synthetic
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation discusses advanced aspects of Synthetic Aperture Radar (SAR). SAR is a crucial capability for radar applications in both civilian as well as government sectors. This dissertation focuses on two actively researched aspects of SAR systems. First, a framework for reconstructing SAR images from a spectral analysis perspective is introduced. This approach is more robust and more accurate due to lower sidelobes level and can be used in situations that full data collection is not permissible. Second, a framework for identifying movers directly from SAR data is also discussed. The framework takes in account for SAR systems with multiple antennas and is shown to offer superior performance in such cases. The dissertation also provides a basic overview of the signal processing involve in constructing SAR images. This background knowledge lays the foundation to which further advanced concepts are built upon. We first focus on a new approach to reconstructing SAR images. This new approach uses spectral estimation. In particular we utilize the Bayesian framework as it results in more sparse images. We utilize an algorithm named SLIM, which can be thought of as a sparse signal recovery algorithm with excellent sidelobe suppression and high resolution properties. For a given sparsity promoting prior, SLIM cyclically minimizes a regularized least square cost function. We show how SLIM can be used for SAR image reconstruction as well as used for SAR image enhancement. We evaluate the performance of SLIM using realistically simulated complex-valued backscattered data from a backhoe vehicle. The numerical results show that SLIM satisfactorily suppress the sidelobes and yield higher resolution than the conventional matched filter or delay-and-sum (DAS) approach. SLIM outperform the widely used compressive sampling matching pursuit (CoSaMP) algorithm, which requires the delicate choice of user parameter. Compared with the recently developed iterative adaptive approach (IAA), which iteratively solves weighted least squares problem. Beside bringing SLIM to the field of SAR image reconstruction, we show how SLIM can be made more computationally efficient by utilizing the Fast Fourier Transform (FFT) and the conjugate gradient (CG) method. In the event that full data collection is not possible, we proposes several methods to reconstruct SAR images based on missing data samples. This is coined as interrupted SAR. For this scenario we consider nonparametric adaptive spectral analysis of complex-valued data sequences with missing samples occurring in arbitrary patterns. We once again utilize SLIM and IAA. However these algorithms were not adapted for the case of missing data samples. We consider how these algorithms can be adapted to the missing data sample case. Furthermore, we consider fast implementations of these algorithms using the Conjugate Gradient (CG) technique and the Gohberg-Semencul-type (GS) formula. Our proposed implementations fully exploit the structure of the steering matrices and maximize the usage of the Fast Fourier Transform (FFT), resulting in much lower computational complexities as well as much reduced memory requirements. The effectiveness of the adaptive spectral estimation algorithms is demonstrated via several numerical examples including both 1-D spectral estimation and 2-D interrupted synthetic aperture radar (SAR) imaging examples. We then shift gears to discuss how SAR data can be used to detect moving targets. Mover detections, whether in post-processing or the main focus of a surveillance system is increasingly more common in SAR systems. The detection of small movers is a challenging task, detection of small movers in high clutter environment is even more difficult. While mover detections algorithms exist, they are catered toward a 2-channel system. We explore how multi-channel can be used to detect smaller moving targets. We consider moving target detection and velocity estimation for multi-channel synthetic aperture radar (SAR) based ground moving target indication (GMTI). Via forming velocity versus cross-range images, we show that small moving targets can be detected even in the presence of strong stationary ground clutter. Furthermore, the velocities of the moving targets can be estimated, and the misplaced moving targets can be placed back to their original locations based on the estimated velocities. An iterative adaptive approach (IAA), which is robust and user parameter free, is used to form velocity versus cross-range images for each range bin of interest. Moreover, we discuss calibration techniques to estimate the relative antenna distances and antenna gains in practical systems. Furthermore, we present a simple algorithm for stationary clutter cancelation. We conclude by demonstrating the effectiveness of our approaches by using the Air Force Research Laboratory (AFRL) publicly-released Gotcha airborne SAR based GMTI data set. We then take it further and investigate a framework for the detection of ground moving targets using a Multiple-Input-Multiple-Output (MIMO) radar system. We look at how phase histories collected from a MIMO Synthetic Aperture Radar (SAR) system can be used for target detection in a Ground Moving Target Indication (GMTI) mode while SAR images are being formed. We propose the usage of the recently developed Iterative Adaptive Approach to detect movers and estimate their velocities in the post-Doppler domain. We show how the waveform diversity afforded by a MIMO radar system enables its superiority over its Single-Input-Multiple-Output (SIMO) counterpart.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Duc H Vu.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Li, Jian.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043913:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043913/00001

Material Information

Title: Advanced Techniques for Synthetic Aperture Radar Image Reconstruction
Physical Description: 1 online resource (131 p.)
Language: english
Creator: Vu, Duc H
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: aperture -- estimation -- image -- radar -- reconstruction -- sar -- sparse -- spectral -- synthetic
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation discusses advanced aspects of Synthetic Aperture Radar (SAR). SAR is a crucial capability for radar applications in both civilian as well as government sectors. This dissertation focuses on two actively researched aspects of SAR systems. First, a framework for reconstructing SAR images from a spectral analysis perspective is introduced. This approach is more robust and more accurate due to lower sidelobes level and can be used in situations that full data collection is not permissible. Second, a framework for identifying movers directly from SAR data is also discussed. The framework takes in account for SAR systems with multiple antennas and is shown to offer superior performance in such cases. The dissertation also provides a basic overview of the signal processing involve in constructing SAR images. This background knowledge lays the foundation to which further advanced concepts are built upon. We first focus on a new approach to reconstructing SAR images. This new approach uses spectral estimation. In particular we utilize the Bayesian framework as it results in more sparse images. We utilize an algorithm named SLIM, which can be thought of as a sparse signal recovery algorithm with excellent sidelobe suppression and high resolution properties. For a given sparsity promoting prior, SLIM cyclically minimizes a regularized least square cost function. We show how SLIM can be used for SAR image reconstruction as well as used for SAR image enhancement. We evaluate the performance of SLIM using realistically simulated complex-valued backscattered data from a backhoe vehicle. The numerical results show that SLIM satisfactorily suppress the sidelobes and yield higher resolution than the conventional matched filter or delay-and-sum (DAS) approach. SLIM outperform the widely used compressive sampling matching pursuit (CoSaMP) algorithm, which requires the delicate choice of user parameter. Compared with the recently developed iterative adaptive approach (IAA), which iteratively solves weighted least squares problem. Beside bringing SLIM to the field of SAR image reconstruction, we show how SLIM can be made more computationally efficient by utilizing the Fast Fourier Transform (FFT) and the conjugate gradient (CG) method. In the event that full data collection is not possible, we proposes several methods to reconstruct SAR images based on missing data samples. This is coined as interrupted SAR. For this scenario we consider nonparametric adaptive spectral analysis of complex-valued data sequences with missing samples occurring in arbitrary patterns. We once again utilize SLIM and IAA. However these algorithms were not adapted for the case of missing data samples. We consider how these algorithms can be adapted to the missing data sample case. Furthermore, we consider fast implementations of these algorithms using the Conjugate Gradient (CG) technique and the Gohberg-Semencul-type (GS) formula. Our proposed implementations fully exploit the structure of the steering matrices and maximize the usage of the Fast Fourier Transform (FFT), resulting in much lower computational complexities as well as much reduced memory requirements. The effectiveness of the adaptive spectral estimation algorithms is demonstrated via several numerical examples including both 1-D spectral estimation and 2-D interrupted synthetic aperture radar (SAR) imaging examples. We then shift gears to discuss how SAR data can be used to detect moving targets. Mover detections, whether in post-processing or the main focus of a surveillance system is increasingly more common in SAR systems. The detection of small movers is a challenging task, detection of small movers in high clutter environment is even more difficult. While mover detections algorithms exist, they are catered toward a 2-channel system. We explore how multi-channel can be used to detect smaller moving targets. We consider moving target detection and velocity estimation for multi-channel synthetic aperture radar (SAR) based ground moving target indication (GMTI). Via forming velocity versus cross-range images, we show that small moving targets can be detected even in the presence of strong stationary ground clutter. Furthermore, the velocities of the moving targets can be estimated, and the misplaced moving targets can be placed back to their original locations based on the estimated velocities. An iterative adaptive approach (IAA), which is robust and user parameter free, is used to form velocity versus cross-range images for each range bin of interest. Moreover, we discuss calibration techniques to estimate the relative antenna distances and antenna gains in practical systems. Furthermore, we present a simple algorithm for stationary clutter cancelation. We conclude by demonstrating the effectiveness of our approaches by using the Air Force Research Laboratory (AFRL) publicly-released Gotcha airborne SAR based GMTI data set. We then take it further and investigate a framework for the detection of ground moving targets using a Multiple-Input-Multiple-Output (MIMO) radar system. We look at how phase histories collected from a MIMO Synthetic Aperture Radar (SAR) system can be used for target detection in a Ground Moving Target Indication (GMTI) mode while SAR images are being formed. We propose the usage of the recently developed Iterative Adaptive Approach to detect movers and estimate their velocities in the post-Doppler domain. We show how the waveform diversity afforded by a MIMO radar system enables its superiority over its Single-Input-Multiple-Output (SIMO) counterpart.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Duc H Vu.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Li, Jian.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043913:00001


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ADVANCEDTECHNIQUESFORSYNTHETICAPERTURERADARIMAGERECONSTRUCTIONByDUCH.VUADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012DucH.Vu 2

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ToGodandfamily 3

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ACKNOWLEDGMENTS Thisdissertation,andthisacademicjourneywouldhavenotbeenpossiblewithoutthesupportofmanypeople.Iwouldliketotakethisspacetothankrstofallmyparents.Yourguidanceandinnitesacricesenabledmetobeatmyfullpotential.WithoutyourloveandsupportIwouldnotbeabletotakeonthisjourney.Iwouldliketothankmyadvisor,JianLi,herpatienceandworkethicsisinspiring.Iwillalwaysbegratefulforallthediscussionsthatwehad;eachonehavemademeabetterresearcherandabetterhumanbeing.Iwouldliketothankmymentor,coworkerandfriend,OliverAllen.YourselessnessmadepossiblethispursuitofknowledgeandyourencouragementenabledanachievementthatIneverthoughtcouldbepossible.IwouldliketothankmyfriendsandfellowlabmatesoftheSpectralAnalysisLaboratory:Hao,Jun,Bill,Xing,Zhaofu,Tarik,Ming,Lin,Enrique,Matteo,Xiang,Yubo,Arsen,Erik,Yuanxiang,Kexin,Qilin,Xianqi,Bin,Luzhou,Ode,WillandJohan.Forthesmallspanoftimeourlifehasoverlapped,ourtimeandfriendshipwillneverbeforgotten.Lastbutnotleast,Iwouldliketothankmycommitteemembers:Dr.HenryZmuda,Dr.JenshanLinandDr.MingzhouDing,fortheirtimeandguidance. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTIONTOSYNTHETICAPERTURERADAR ............. 13 1.1TheSignalProcessingAspectofSAR .................... 15 1.2ResolutioninRange .............................. 15 1.2.1MatchedFiltering ............................ 17 1.2.2PulseCompression ........................... 21 1.2.3LinearFrequencyModulatedSignal ................. 21 1.3ResolutioninCrossRange .......................... 25 1.3.1SyntheticAperture ........................... 26 1.3.2ProcessingoftheReturnSignals ................... 30 2SARRECONSTRUCTIONUSINGABAYESIANAPPROACH .......... 33 2.1PreliminariesandDataModel ......................... 36 2.2Data-AdaptiveAlgorithms ........................... 40 2.2.1CompressedSamplingMatchingPursuit(CoSaMP) ........ 40 2.2.2IterativeAdaptiveApproach(IAA) ................... 41 2.2.3SparseLearningviaIterativeMinimization(SLIM) .......... 41 2.3NumericalExamples .............................. 43 2.3.1AnExampleofIdealPointScatterers ................. 44 2.3.2ImagingoftheBackhoe ........................ 45 2.3.33-DSARImaging ............................ 45 3SARRECONSTRUCTIONOFINTERRUPTEDDATA .............. 50 3.1NonparametricSpectralAnalysis ....................... 53 3.1.1DataModel ............................... 53 3.1.2IterativeAdaptiveApproach(IAA) ................... 54 3.1.3SparseLearningviaIterativeMinimization(SLIM) .......... 55 3.1.4ComputationalComplexitiesandMemoryRequirements ...... 57 3.2MissingDataSpectralAnalysisandFastImplementations ......... 58 3.2.1DataModel ............................... 58 3.2.2FastIterativeAdaptiveApproach(Fast-IAA) ............. 61 3.2.2.1EfcientComputationoftheIAACovarianceMatrix ... 61 3.2.2.2EfcientComputationof^B ................. 62 5

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3.2.3FastSLIMUsingConjugateGradient(CG-SLIM) .......... 64 3.2.4FastSLIMUsingtheGohberg-Semencul-TypeFormula(GS-SLIM) 67 3.3NumericalExamples .............................. 72 3.3.11-DSpectralEstimation ........................ 72 3.3.22-DInterruptedSARImaging ..................... 78 4SARGROUNDMOVINGTARGETINDICATION ................. 84 4.1GeometryandDataModel .......................... 85 4.2ArrayCalibration ................................ 88 4.2.1DistancesAmongAntennas ...................... 89 4.2.2AntennaGains ............................. 90 4.3GroundMovingTargetIndication(GMTI) ................... 91 4.3.1GroundClutterCancelationUsingRELAX .............. 92 4.3.2MovingTargetDetectionUsingIAA .................. 94 4.4AnalysisoftheAFRLGOTCHADataSet .................. 96 4.4.1DescriptionoftheAFRLGotchaGMTIDataSet ........... 96 4.4.2SARImaging .............................. 98 4.4.3ArrayCalibration ............................ 98 4.4.4VelocityAmbiguityAnalysis ...................... 99 4.4.5AdaptiveGMTI ............................. 101 5MULTIPLE-INPUTMULTIPLE-OUTPUTSARGMTI ............... 107 5.1ExistingGMTIMethods ............................ 107 5.2MIMOGMTISystemModel .......................... 108 5.2.1SceneofInterest ............................ 109 5.2.2AntennaArrayandTransmissionWaveforms ............ 109 5.2.3ReceivedSignalModel ......................... 110 5.3GMTIAlgorithm ................................. 111 5.3.1RangeCompression .......................... 111 5.3.2DopplerProcessingandPhaseCompensation ........... 112 5.3.3MovingTargetDetectionintheRange-Doppler-VelocityDomain .. 113 5.4NumericalExamples .............................. 115 5.4.1Single-InputandMultiple-Output(SIMO)System .......... 116 5.4.2Multiple-InputandMultiple-Output(MIMO)System ......... 117 6CONCLUDINGREMARKSANDFUTUREWORKS ............... 121 REFERENCES ....................................... 123 BIOGRAPHICALSKETCH ................................ 131 6

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LISTOFTABLES Table page 3-1ComputationtimesneededbyIAA,SLIMandtheirfastimplementations. .... 77 3-2ComputationtimesneededbyIAA,SLIMandtheirfastimplementationsforinterruptedSARimagingundervariousinterruptionconditions. ......... 79 4-1Targetvelocityversusfv. ............................... 100 7

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LISTOFFIGURES Figure page 1-1Resolutionofapixel ................................. 16 1-2ScenewithImpulseTargets ............................. 16 1-3TransmittedSignal .................................. 18 1-4ReturnedSignal ................................... 18 1-5MatchedFilterofReturnSignals .......................... 19 1-6TwoCloseTargets .................................. 20 1-7ReturnedSignalsofTwoCloseTargets ...................... 20 1-8MatchedFilterReturnofTwoCloseTargets .................... 20 1-9ExampleofaChirpSignal .............................. 22 1-10SincFunction ..................................... 24 1-11ReturnFromSceneDuetoaChirp ......................... 24 1-12ReturnsfromSceneAfterMatchedFilteringforaChirp ............. 24 1-13CrossRangeWidthofaRealApertureAntenna ................. 25 1-14ConceptofDoppler ................................. 26 1-15DopplerResolution .................................. 27 1-16CrossRangeWidthofaRealApertureAntenna ................. 29 2-1SARimagingschematics. .............................. 36 2-2ImagingofScatterers ................................ 46 2-3VisualizingtheBackhoeDataset .......................... 47 2-4Backhoe3-DSARImageReconstruction ..................... 48 2-5Fused3-DimageusingSLIM ............................ 48 3-1Nonparametricspectralestimationwithoutandwithmissingsamples ...... 74 3-2Nonparametricspectralestimatesfordatasequenceswithmissingsamples .. 75 3-3Missingsamplesestimation ............................. 76 3-4NonparametricspectralestimatesfordatasequenceswithmissingsamplesusingSLIM-IAA .................................... 77 8

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3-5SlicyobjectandbenchmarkSARimage ...................... 78 3-6ModulusoftheSARimagesoftheSlicyobjectobtainedfroma4040completedatamatrix. ...................................... 80 3-7SARimagesoftheSlicyobjectunderrandom68%dataloss. .......... 81 3-8SARimagesoftheSlicyobjectunderrandom30%dataloss. .......... 82 4-1Geometryofanairboremulti-channelSARsystem. ................ 86 4-2TheowchartoftheentireprocessingchainfortheproposedadaptiveSARbasedGMTI. ..................................... 96 4-3AFRLGotchadatascenesetup ........................... 97 4-4SARimagesofthe46thsecond .......................... 99 4-5EstimateddistancebetweenAntennas. ...................... 100 4-6SpectralwindowsforthearraygeometryoftheAFRLGotchaGMTIsystem. .. 101 4-7GMTIresultsofDPCAatthe46thsecond. ..................... 103 4-8GMTIresultsofATIatthe46thsecond. ...................... 103 4-9GMTIresultsofIAAatthe46thsecond. ...................... 104 4-10GMTIresultsusingIAAwithallthreeantennasatthe46thsecond ....... 105 4-11GMTIresultsusingIAAwithallthreeantennasatthe51standthe68thsecond. 105 5-1Illustrationofgroundmovingtargetindication(GMTI)usingamultiple-inputandmultiple-output(MIMO)radarsystem. ..................... 108 5-2Groundclutter,andsimulatedtargetlocationsandvelocities. .......... 116 5-3DetectionviaDASfortheSIMOcase. ....................... 117 5-4DetectionviaIAAfortheSIMOcase. ........................ 118 5-5DetectionviaDASfortheMIMOcase. ....................... 119 5-6DetectionviaIAAfortheMIMOcase. ....................... 120 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyADVANCEDTECHNIQUESFORSYNTHETICAPERTURERADARIMAGERECONSTRUCTIONByDucH.VuMay2012Chair:JianLiMajor:ElectricalandComputerEngineering ThisdissertationdiscussesadvancedaspectsofSyntheticApertureRadar(SAR).SARisacrucialcapabilityforradarsystemsinbothcivilianaswellasgovernmentsectors.ThisdissertationfocusesontwoactivelyresearchedaspectsofSARsystems.First,aframeworkforreconstructingSARimagesfromaspectralanalysisperspectiveisintroduced.Thisapproachismorerobustandmoreaccurateduetolowersidelobeslevelandcanbeusedinsituationsthatfulldatacollectionisnotpermissible.Second,aframeworkforidentifyingmoversdirectlyfromSARdataisalsodiscussed.Furthermore,thisframeworktakesinaccountSARsystemswithmultipleantennasandisshowntooffersuperiorperformanceinsuchcases.ThedissertationalsoprovidesabasicoverviewofthesignalprocessinginvolveinconstructingSARimages.Thisbackgroundknowledgelaysthefoundationtowhichfurtheradvancedconceptsarebuiltupon. WerstfocusonanewapproachtoreconstructingSARimages.Thisnewapproachisfromthespectralestimationcommunity.InparticularweutilizetheBayesianframeworkasitresultsinasparserSARimagethatismoreamenableforidenticationandclassicationpurposes.WeutilizeanalgorithmnamedSLIM,whichcanbethoughtofasasparsesignalrecoveryalgorithmwithexcellentsidelobessuppressionandhighresolutionproperties.Foragivensparsitypromotingprior,SLIMcyclicallyminimizesaregularizedleastsquarecostfunction.WeshowhowSLIMcanbeusedforSARimagereconstructionaswellasusedforSARimageenhancement.Weevaluate 10

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theperformanceofSLIMusingrealisticallysimulatedcomplex-valuedbackscattereddatafromabackhoevehicle.ThenumericalresultsshowthatSLIMsatisfactorilysuppressthesidelobesandyieldhigherresolutionthantheconventionalmatchedlterordelay-and-sum(DAS)approach.SLIMoutperformthewidelyusedcompressivesamplingmatchingpursuit(CoSaMP)algorithm,whichrequiresthedelicatechoiceofuserparameter.BesidebringingSLIMtotheeldofSARimagereconstruction,weshowhowSLIMcanbemademorecomputationallyefcientbyutilizingtheFastFourierTransform(FFT)andtheconjugategradient(CG)method. Intheeventthatfulldatacollectionisnotpossible,weproposeseveralmethodstoreconstructSARimagesbasedonmissingdatasamples.ThisiscoinedasinterruptedSAR.Forthisscenarioweconsidernonparametricadaptivespectralanalysisofcomplex-valueddatasequenceswithmissingsamplesoccurringinarbitrarypatterns.WeonceagainutilizeSLIMandIAA.Howeverthesealgorithmswerenotadaptedforthecaseofmissingdatasamples.Weconsiderhowthesealgorithmscanbeadaptedtothemissingdatasamplecase.Furthermore,weconsiderfastimplementationsofthesealgorithmsusingtheConjugateGradient(CG)techniqueandtheGohberg-Semencul-type(GS)formula.OurproposedimplementationsfullyexploitthestructureofthesteeringmatricesandmaximizetheusageoftheFastFourierTransform(FFT),resultinginmuchlowercomputationalcomplexitiesaswellasmuchreducedmemoryrequirements.Theeffectivenessoftheadaptivespectralestimationalgorithmsisdemonstratedviaseveralnumericalexamplesincludingboth1-Dspectralestimationand2-Dinterruptedsyntheticapertureradar(SAR)imagingexamples. WethenshiftgearstodiscusshowSARdatacanbeusedtodetectmovingtargets.Moverdetections,whetherinpost-processingorthemainfocusofasurveillancesystemisincreasinglymorecommoninSARsystems.Thedetectionofsmallmoversisachallengingtask,detectionofsmallmoversinhighclutterenvironmentisevenmoredifcult.Whilemoverdetectionalgorithmsexist,theyarecateredtowarda2-channels 11

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system.Weexplorehowmulti-channelscanbeusedtodetectsmallermovingtargets.Weconsidermovingtargetsdetectionandvelocitiesestimationformulti-channelssyntheticapertureradar(SAR)basedgroundmovingtargetindication(GMTI).Viaformingvelocityversuscross-rangeimages,weshowthatsmallmovingtargetscanbedetectedeveninthepresenceofstrongstationarygroundclutter.Furthermore,thevelocitiesofthemovingtargetscanbeestimated,andthemisplacedmovingtargetscanbeplacedbacktotheiroriginallocationsbasedontheestimatedvelocities.Aniterativeadaptiveapproach(IAA),whichisrobustanduserparameterfree,isusedtoformvelocityversuscross-rangeimagesforeachrangebinofinterest.Moreover,wediscusscalibrationtechniquestoestimatetherelativeantennadistancesandantennagainsinpracticalsystems.Furthermore,wepresentasimplealgorithmforstationarycluttercancellation.WeconcludebydemonstratingtheeffectivenessofourapproachesbyusingtheAirForceResearchLaboratory(AFRL)publicly-releasedGotchaairborneSARbasedGMTIdataset.Finally,withtheemergenceofMultiple-Input-Multiple-Output(MIMO)radar,weinvestigateaframeworkforthedetectionofgroundmovingtargetsusingaMIMOradarsystem.WelookathowphasehistoriescollectedfromaMIMOSyntheticApertureRadar(SAR)systemcanbeusedfortargetdetectioninaGroundMovingTargetIndication(GMTI)modewhileSARimagesarebeingformed.WeproposetheusageoftherecentlydevelopedIterativeAdaptiveApproachtodetectmoversandestimatetheirvelocitiesinthepost-Dopplerdomain.WeshowhowthewaveformdiversityaffordedbyaMIMOradarsystemenablesitssuperiorityoveritsSingle-Input-Multiple-Output(SIMO)counterpart. 12

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CHAPTER1INTRODUCTIONTOSYNTHETICAPERTURERADAR ThehistoryofRADAR(RadioDetectionandRanging)isrichandisconsideredtobeoneofthehighpointsofhumaninnovations.Ameltingpotofmultitudesofdisciplinesresultedinasystemthatenablesusthepeaceandprosperitythatweenjoytoday.Fromweatherapplicationsthatenablesustobeonestepaheadofmothernature,toareasurveillancethatkeepsourtroopsandborderssafe.Whilethehistoryofradarisrootedintheprinciplesofelectromagneticsthatcanbetracedbacktooneofthegreatestmindofthe19thcentury,JamesClerkMaxwell,theprocessingofitsdatawhichwenowcalltheeldofradarsignalprocessing,isaveryrecenteldthatgrewduetotheoncomingofthedigitalage.Now,fastandcheapcomputinghardwareallowsustobetrulymajesticwiththeprocessingofitsreturnedsignals. Withintheeldofradarsignalprocessing,oneofthemostelegantdevelopmentisthatoftheconceptofSyntheticApertureRadar(SAR).TheinventionofSARcanbeconsideredoneofthemostingeniousresultofsignalprocessing,asitallowsasystemtoovercomeaphysicallimitationsbycarefulmanipulationsofthesignalsattheoutputoftheradar.ByusingSAR,aradarwhichuptothatpointismainlyusedinthe1-Ddomain,istransformedintotoasystemthatiscapableofformingboth2-Dand3-Dimages.Thisenablesunprecedentedachievementsinterrainmappingapplicationsandwideareasurveillancefortimeofpeaceandtimeofwar. ThisdissertationrevolvesaroundthedevelopmentofSARandproposesadvancedtechniquesonhowtobetterreconstructSARdataandhowwecanusenovelsignalprocessingtechniquestoovercomesomeoftheshortcomingofSAR.Itisaculminationofseveralyearsofresearchthatresultedinasetofacademicpublications[ 1 10 ].ThedissertationbeginswithanintroductorychaptertoSyntheticApertureRadarfromasignalprocessingperspective.IthighlightsthefoundingprinciplesbehindSARandidentiesdifferentaspectsthatcanbeexploitedforimprovements.Thenwemove 13

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ontotalkaboutmovingtargetsindicationsusingSARdatadirectly.Weextendtheapplicationsofmovingtargetindicationstomulti-channelsystems. InChapter 1 ,webeginsbybacktrackingthedevelopmentofSARandexploreindetailsthestandardtechniquesthatareemployedtodayinmodernSARsystems.Wediscusstheimportantconceptofrangeandcross-rangeresolutionswhicharetwoessentialscriteriaindevelopmentofaSARsystem.Wethenexplainhowthesetechniquesarecurrentlyemployedanddiscusseditsshortcoming.WealsomaketheconnectionbetweenthesetechniquestomoregeneralizedspectralestimationtechniquesandhowadvancedalgorithmsinspectralestimationmaybeusedtoimprovethemodernSARsystems. InChapters 2 and 3 ,weproposesseveraladaptivesignalprocessingtechniquestothereconstructionofSARimages.Inparticular,wefocusonthereconstructionofSARimagesinaBayesianframework,weshowthatthismethodcanproducesparseresultwithgreatsidelobesuppressioncapability.Inthechapterfollowing,wealsodiscusshowthisapproachcanbeusedintheapplicationofInterruptedSAR.Thisisaspecialcasewheremode-switchinginmodernSARsystemsresultsinmissingdata,ordataarethrownawayonpurposetoreducetheamountofdatatransfer. InChapters 4 and 5 ,wegiveathoroughtreatmentontheusageofSARdatatodetectmovingtargets.Currentlyaspecialcollectionmodeisrequiredformoversdetection.WeproposetouseSARdatatosimultaneouslyidentifymovingtargets,aswellastoestimatetheirparameterssuchasspeedanddirection.Weintroduceseveraladaptivetechniquesthatcanbeusedtoidentifythemoversandestimateitsparameters.WeextendthesemethodstothecaseofusinganantennaarraytoformSARimagesandhowhavingmultipleelementsincreasestheabilitytocorrectlyestimatethespeedofthetargets.WethenconcludethisdissertationwithatheoreticaltreatmentofusingMultiple-Input-Multiple-Output(MIMO)RadartoformSARimagesaswellastodetectmovers.ThewaveformdiversityaffordedbyMIMOradarleadstonewbreak 14

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throughinthesuppressingofgroundclutterwhichleadstobetteraccuracyinidentifyingourparametersofinterests. 1.1TheSignalProcessingAspectofSAR ThedevelopmentofSARwasmotivatedbythedesiretohaveconstantvisualsoftheterrainofinterestsduringwartime.Beforewehavesatellitesorbitingtheearthwithhighresolutionopticalimagingandthermalimagingequipment,theonlywaytoobtainvisualsofthegroundiswithanopticalimagingequipmentyingonboardanaircraft.Thisposestwoproblem,rst,highresolutionopticalcameraswerenotyetavailable,andsecond,opticalcamerasonlyworkunderidealweatherconditionsandwhenlightwasavailable.Thismeanscovertoperationsdoneatnightreliesondatathatmaybequiteobsolete.ThisimportantbarriergaverisetothedevelopmentofSAR.ForwithSARimaging,visionofthegroundispossibleinallweatherconditions,nightorday,andcanbeobtainwithratherlongstandoffrangessafefromanyenemythreat.Whendiscussingaphotograph,anaturalquestionistoinquireaboutitsresolution.Whatdoesapixelinaphotorepresentsinitsspatialdomain?Thisisaquestionofresolution,seegure 1-1 .Intermsofradarterminology,dyistheresolutioninrange,anddxistheresolutionincross-range.Wenowtalkaboutthetwocomponentindividuallyandhowtheresolutionisdetermined. 1.2ResolutioninRange Rangeresolutionrelatesbacktothemostfundamentalusageofaradar.Thegoalofaradaristorstdetectapresenceofatarget,andsecondlytodetermineitsdistance.Toachievethis,theradaremitsapulseofenergyinthedirectionofinterestandmeasurethetimedelayofthereturningecho.Sinceweknowthespeedoflightandwecanmeasurethetimedelay,wecaneffectivelydeterminetherangeoftheobjectassociatedwiththeecho.Iftherewasonlyoneobject,andonlyonepulsewassent,thereisnoproblem.However,iftherearemorethanoneobject,andmorethanone 15

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Figure1-1. Resolutionofapixel Figure1-2. ScenewithImpulseTargets pulseofenergyissent,thereexistsomeambiguitybetweendiscerningthereturnofonepulsefromtheother. Lets(t)beasimplesinusoidalpulsethatistransmittedfromthetransmitterofaradar.s(t)=8><>:Ae2if0tif0tT;0otherwise. IthasanamplitudeofAoperatingatacarrierfrequencyf0andhasadurationofT.Werethispulsefromourtransmittertoasceneofinterestthathasimpulsetargetsthatoccupiesonlyonerangebins.SeeFigure 1-2 Thereturnedsignalfromthefollowing 16

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scenewouldbeanattenuated,timedelayedversionofthesignalwesent.SeeFigure 1.2.1 .Ifourreturnsignalscontainsnonoise,itiseasytoseethatthetimedelayofthereturnedsignalcontainsthedistanceinformation,asweknowthatthewavetravelsatthespeedoflight.However,inthepresenceofnoise,thesignalisnolongercleanandinpracticewehavetoperformsomesignalprocessingonthereturnedsignal. 1.2.1MatchedFiltering Thestandardmethodtodetectthereturnedsignalsiscalledthematchedlter.Sinceweknowwhatwassentoutwecanmatcheditwiththereturnsignal.Fromalteringperspective,wecanrepresentthematchedlteras, h(t)=s()]TJ /F5 11.955 Tf 9.3 0 Td[(t)(1) wheres(t)isthetimereversedandconjugatedversionofthetransmittedsignal.GiventhattheFourierTransformofthetransmittedsignalandthelterisS(f)andH(f),respectively,thematchedlteroutputis, Y(f)=H(f)S(f)(1) inthetimedomain,theoutputis, y(t)=h(t)s(t)(1) wheretheistheconvolutionoperator,andthencanbewrittenoutas, y(t)=Z1h(t)]TJ /F14 11.955 Tf 11.96 0 Td[()s()d(1) substitutingthedenitionofh(t)wehavethedenitionoftheautocorrelationofasignal. y(t)=Z1s(t+)s()d(1) 17

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Figure1-3. TransmittedSignal Figure1-4. ReturnedSignal Letthematchedlterresultbedenedas, y[n]=k=1Xk=r[n)]TJ /F5 11.955 Tf 11.96 0 Td[(k]s[k](1) wherer[n]isthereturnedsignal,s[k]isthetransmittedsignalandkisthelag.Thematchedlterresultisgiveningure 1-5 ,ifweapplyathresholddetector,wecanpinpointwherethetargetlies.Fromthiswecandeterminetheexactdelaytimeofeachtarget.Nowsupposeifweputtwotargetsclosertogetherinrange,suchasthatgiveninFigure 1-6 .Firstwenotethattherawreturnedsignals(Figure 1-7 )showsanoverlapbetweenthereturnsofthetwotargets.Afterltering,(Figure 1-8 )weonlyseeonepeak.Athresholddetectorwouldthengivefalsedetectionsasthelocationofthetwopeaksareambiguous. 18

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Figure1-5. MatchedFilterofReturnSignals Fromthisexample,weseethatfortwotargetstobeseparable,thereturnsofthetwotargetsmustbeatleastseparatedbyT,thedurationofapulse.Sincethewavetravelatthespeedoflight,c,andthedurationofthewaveisT,thedistancetraversedbythewaveiscT.NotforgettingthatTisaround-triptimedelay,weconcludethattherangeresolutionofasinusoidalpulsewithnitedurationTis, 1 2cT(1) Sotogetnerresolution,weneedtodecreasethepulseduration,T.Anideally,wewouldliketohaveaninnitelyshortpulse.However,thedurationofthepulseTcontainsthetransmittedenergy.Remindedthatanenergyofasignalisdenedas: E=n=TXn=0js[n]j2=A2T(1) soifwemakeTsmall,wearereducingtheamountofenergyoutputoftheradar.Thisisimportantastheamountofenergyreturnedisanattenuatedamountoftheenergyimpingedonthetarget.Thereforeiftheamountofenergyistoolow,thereturnsignalcouldbebelowthenoiselevel,andnoamountofsignalprocessingcanrecoverthesignal.Atthisjuncture,withoutanyothertoolsavailabletous,itseemsthattogetthedesiredrangeresolution,acarefultradeoffbetweenpulsedurationandoutputenergymustbemade. 19

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Figure1-6. TwoCloseTargets Figure1-7. ReturnedSignalsofTwoCloseTargets Figure1-8. MatchedFilterReturnofTwoCloseTargets 20

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1.2.2PulseCompression Duringthistime,mostoftheeffortwithinthepracticingradarcommunityevolvearoundthedevelopmentofhigh-peakpowerradartubes.InthediscussionaboveitisnotedthatoneapproachistomakeAmuchbigger.IncreasingArequiresapowersourcewithhighpeakpower.Radartubeswithhighpoweroutputarecostly,andweightbecomesanissuesforairborneplatform.Withsuchahighpeakpower,safetywasalsoaconcernforitsoperator.Aroundthe1950sand1960s,amoreelegantapproachwasfoundanditiscalledpulsecompression.Thedevelopmentofpulsecompressionwasnotwidelyacknowledgeatthetime,becauseitwasrstleinpatentformbyR.HDickein1953,S.Darlingtonin1954,andlaterinpublishedformbyCharlesCook,in1960.Allconceptswerearrivedindependently.TherstradartousetheconceptofpulsecompressionwasbuiltbyMITLincolnLaboratoryalsointhe1960sbytheRadarTechniquesGroupnamedtheAN/FPS-17. Whiletheintuitionbehindpulsecompressionisabitcomplicatedduetoitbeingthoughtoffromaanaloglteringperspective,itisquitesimpleintermofsignalprocessing.Intheabovediscussion,atruncatedrectangularpulselteredreturngaveusatrianglefunction.Wenotethatthetrianglefunctionspans2T,andfortwotriangularfunctiontobeseparated,itneedstobeatseparatedbyT.Fromasignalprocessingperspective,aclearobjectiveishowtodesignapulsesuchthattheconvolutionofthatpulseanditsreturnsyieldanimpulselikefunction.Itturnsoutthattherstradarwaveformthathasthisdesirablepropertyisthechirp. 1.2.3LinearFrequencyModulatedSignal ThechirpisthemostfamousexampleofalinearFrequencyModulatedsignal,seegure 1-9 .Itisthemostwidelyusedsignalinpulsedradarsystem.AtimelimitedchirpwithadurationofTpandachirprateofcanbeexpressedasfollow,s(t)=8><>:Aej(2fct+t2)if)]TJ /F5 11.955 Tf 9.3 0 Td[(Tp=2tTp=2;0otherwise. 21

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Figure1-9. ExampleofaChirpSignal Nowsupposedwesendthisasthetransmittedsignalinsteadofapuresinusoidalpulse,thenaftermatchedltering,y(t)=ZTp0s(t+)s()d (1)=ZTp0ei(2fc+2)e)]TJ /F6 7.97 Tf 6.59 0 Td[(i(2fc(t+)+(t+)2)d=e)]TJ /F6 7.97 Tf 6.59 0 Td[(i(2fct+tTp+t2)Tpsin(tTp) tTp weseethatthemagnitudeofthisfunctionisasincfunction,refertogure 1-10 .Thereareseveralthingstonotehere.First,wenotethatforasincfunction,its3dBbeamwidthcorrespondstotherstzerocrossingintimedomain.Thishappenswhentheargumentisequalto1,i.etTp=1.This3dBwidthisalsorefertoastheeffectivecompressedpulsewidth,Te.Recallthatforauncompressedrectangularpulse,iftwotargetsareseparatedintimebyTpthenitisdistinguishable.Howeverforthechirp,theyareseparableifthetwotargetsareseparatedintimebyTe.Foranymodulatedsignal,wecandeneitseffectivecompressedpulseduration.Inthecaseofthechirpitis, Te=1 Tp(1) Goingbacktoourdiscussionontheseparabilityofthetargets,ifweapplyEquation 1 andinsteadofthepulsedurationtimeT,weusetheeffectivepulsedurationTe,thenfor 22

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achirpthetargetcanbeseparableifitisseparatedindistanceby,D=cTe 2 (1)=c 2Tp Thismeansifweincreasethetransmissiontimeofachirp,wecanincreasethesystem'sresolution,contrarytoarectangularpulse.Abettermetricforresolutionistospeakintermsofasignal'sbandwidth.Forachirp,itsbandwidthisdenedas, Bc=Tp(1) andtheeffectivepulsedurationis, Te=1 B(1) andourresolutionis, c 2B(1) Sobytransmittingachirplongerintime,wedecreaseitseffectiveduration,andthusforcedahigherbandwidthrequirement.Fromthebandwidthperspective,wecanseethatintheearliercaseofarectangularburst,Te=Tp,itsbandwidthisequaltotheinverseofitsdurationanditsexpressionforbandwidthisthengiveninEquation 1 .Toillustratewhatwehavediscussed,toresolvethetwoclosedtargets,insteadoftransmittingasinusoidalburst,weemployachirpsignalwithachirprateof10Hz/softhesameduration.Whatweseefromgure 1-11 isthatthereturnsfromachirpattheradarreceiverisnolongerrecognizable,soathresholddetectorwouldbeofnouseinthiscaseandfurtherpost-processingisneeded.Aftermatchedlteringhowever(gure 1-12 ),wewereabletoresolvethetwoclosedtargetswiththesametimeduration!Essentiallywetradedsystemperformanceinexchangeforincreasedsystemcomplexity. 23

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Figure1-10. SincFunction Figure1-11. ReturnFromSceneDuetoaChirp Figure1-12. ReturnsfromSceneAfterMatchedFilteringforaChirp 24

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Figure1-13. CrossRangeWidthofaRealApertureAntenna 1.3ResolutioninCrossRange Allthistimewewereonlyfocusingonhowtoimprovetheresolutionalongonedimensionofourdata,therange.Nowweneedtotalkabouttheotherdimension,thecross-range.ThewholedevelopmentofSARiscenteredaroundincreasingresolutioninthecrossrange.Theantennathatweusetotransmitoursignalshasabeamwidthassociatedwithit,whichisafunctionoftheantenna'seffectiveaperture.Whenwesendonepulse,andprocessitsreturns,wecanonlyresolvethetargetsintherangedirection.Fortargetsthatarenexttooneanotherincross-range,ortheazimuthaldirection,iftheyarewithinthemainbeamoftheantennathereisnowaytoseparatethem.InFigure 1-13 ,foranantennawitha2beamwidth,at5kmmetersinrange,theilluminatedpatchis175mwide,andat10000metersitis349mwide.Soanyobject,at10kminrange,cannotbedistinguishedifitisseparatedincross-rangeby349metersorless.Toincreaseresolutioninthecrossrangeweneedtouseanantennawithasmallerbeamwidth.Thebeamwidthisafunctionofanantenna'seffectiveapertureandtherelationshipisgivenasfollow, = D(1) whereisthebeamwidthinradians,andDisthephysicalsizeoftheantennaaperture,andisthewavelengthofourcarrierfrequency.Foragivenrange,R,thewidthoftheilluminatedswathis, W=R(1) 25

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Figure1-14. ConceptofDoppler andsuch, W=R D(1) soifwewanttoresolveacarthatis3mwide,atX-band(10Ghz)and10kmaway,wewouldneedtheairplanetocarryanantennathatis100meterlong.Thisiscertainlyimpracticalandthisbeamwidthrestrictionisahindrancetohighresolutionimagery. 1.3.1SyntheticAperture Intheearly1950s,CarlWileycameupwithasolutiontothisproblem.HeledapatenttitledDopplerBeamSharpening,itiswiththisworkthatprovidedthebasisforwhatwenowcallSyntheticApertureRadar.Inithepositsthatbytakingadvantageofthephaseinformationofthereturnsignals,wecanresolvetargetswithinthebeamofanilluminatedpatch.InthesamemannerthatamovingobjecthasaDopplershiftsassociatedwithitsvelocity,stationaryobjectswithinthesamerange,hasaDopplershiftsthatisassociatedwiththeaircraft'svelocity.Figure 1-14 ,takenfrom[ 11 ]illustratesthewellknownDopplereffect. Inordertoseewhataffectstheazimuthalresolution,considersthecaseinFigure 1-15 ,alsotakenfrom[ 11 ],whereislarge,andthus0issmall.Bythesmallangle 26

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Figure1-15. DopplerResolution approximationwehavefd=2vcos (1)=2vsin0 =2v0 0=fd 2v(1) 0=fd 2v(1) sotheresolutionof0dependsontheresolutionoftheDopplerfrequency,fd,andsincewesampletheazimuthalDoppleratthepulserepetitionfrequency,PRF,fdisrelatedtoPRFby, fd=PRF Npulses(1) onepulserepetitioninterval,PRI,isinverselyrelatedtothePRF, PRI=1 PRF(1) andsincewetransmittedNpulses,thetotaltimedurationthatweassumecoherency,is CPI=NPRI(1) 27

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whichisthecoherentprocessinginterval.Pluggingthesedenitionsinto 0= 2vCPI(1) andsinceweknowv,theaircraftvelocity,andtheCPI,thetotaldistancethattheaircrafttravelduringthistimeis, Ldistance=vCPI(1) pluggingitintothepreviousequationforangularresolution,wehave 0= 2Ldistance(1) intermsofthespatialresolutionthisis, azimuth=R 2Ldistance(1) intermsofthebeamwidthoftherealantennaapertureis, 0= 2(1) soourabilitytoresolveanobjectinazimuth,dependsonlyonthedistancetravel!ThisdistancetravelLdistanceisinfact,thesyntheticaperture.Tofurtherunderstandwhythisisso,welookatitintermsoftheDopplerbandwidththatisanalogoustohowwedenerangeresolutionpreviously.LookingatFigure 1-16 ,supposethattherearetwotargetsatthetopandbottomedgesoftheilluminatedbeam,thesetwotargetspossesstwodifferentDopplershifts. ftop=2vcos()]TJ /F10 7.97 Tf 13.15 5.26 Td[( 2) (1) fbottom=2vcos(+ 2) (1) 28

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Figure1-16. CrossRangeWidthofaRealApertureAntenna ThesetwoboundsformwhatisknownastheazimuthalDopplerbandwidth,BWazimuth=ftop)]TJ /F5 11.955 Tf 11.95 0 Td[(fbottom (1)=2vcos()]TJ /F14 11.955 Tf 13.15 8.08 Td[( 2) )]TJ /F3 11.955 Tf 13.15 12.87 Td[(2vcos(+ 2) =4vsin( 2)sin() =2vLsin() R (1) thisexpressionoftheDopplerbandwidthmatcheswhatweconcludedearlierregardingtheimprovedresolution,thatthelongertheaperturelengththehighertheresolutionduetotheincreaseinbandwidth.Anotherimportantaspectthatisnecessaryinordertoprocesstheazimuthreturnsistounderstandthenatureofthereturnsignal.Inrangecompression,weusedthematchedltertoextractthetargetsincethetransmittedsignalispulsedcompressed.Itturnsoutthatinazimuthwehavetodothesamething.Toseethis,wenotethatforagiventarget,asitmovesthroughtheilluminatedbeam,itsassociatedDopplershiftschangesataconstantrate.Thisrateisgivenas, 29

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=BWazimuth Ta (1)=4v2sin( 2)sin() L (1) whereTa=L vThereturnsinazimuthisactuallyachirp,withachirprate,.Toextractthetargetfromthereturnsignal,wehavetoalsoperformamatchedlterinazimuth,wherewematchedthereturntoachirpinslowtime,whichisthewaveformthatexistpulsetopulse,insteadoftheintra-pulseintherangecase. Inshort,toincreasetheresolutioninazimuth,weneedtoyalongerdistance.Therearehowever,certainlimitstothisdistance.First,theobjectofinterestmustremainintheilluminatedbeamoftheradiatingantennaduringtheCPI.Thismeansthatthebroadertheantennabeamwidththemoretimetheobjectspendswithinitsbeam,whichisoppositeoftherealantennaaperturecase.Second,becauseweeffectivelyaresamplingtheDopplerfrequencies,wemustsampleatNyquist,meaningourPRFmustbetwiceashighastheazimuthalDopplerbandwidth. 1.3.2ProcessingoftheReturnSignals Uptothispoint,westillhaven'tdiscusshowwecanformSARimagesfromthereturnsignals.Thereisaveryelegantrelationshipbetweenthereturnedsignals,andthephysicalreturnsofthesceneofinterest.Inourearliestdiscussionwherewespeakoftransmittingaconstantpulse,thereturnswereobviousandcansimplybedetectedusingathresholddetector.OrinthecaseofSARimages,thereturnsfromarectangularpulseistheactualscene,butonlyasliceinrange.Howeveraswehaveshown,whenweuseacompressedpulse,thereturnsarenolongerdirect,butsomeformofpulsecompressionmustbeused.Wewillnowshowhowthecompressedpulsedarerelatedtothephysicalscene.Again,letthetransmittedsignalbeachirp, s(t)=ej(!ot+t2)(1) 30

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andisequalto0everywhereoutsideoftheinterval)]TJ /F14 11.955 Tf 9.3 0 Td[(c=2tc=2.Thissignalhasacarrierfrequencyof!oandhasabandwidthof, Bc=c (1) byourearlierdiscussion,therealtimereturnattheradarreceiverisaconvolutionofthereectivityofthesceneandthetransmittedsignal, rc(t)=ReZ1)]TJ /F10 7.97 Tf 6.58 0 Td[(1g1()s(t)]TJ /F14 11.955 Tf 11.96 0 Td[()d(1) beforewecanmakeanysenseofthisreturnsweneedtoperformsomeformofpulsecompression.Wediscussedthematchedlter,whichisalsotheautocorrelator,herewewillperformsomethingcalladeramp.Forthechaseofthechirp,wecanmixthereturnsignalwithitsin-phase(I)andquadrature(Q)component.Thisisdoneinpracticeasitcanbeimplementedcheaplyinhardware.Wemixthereal,orthein-phasecomponentwith cI(t)=cos(!o(t)+t2)(1) andtheimaginary,orthequadraturewith cQ(t)=)]TJ /F3 11.955 Tf 11.29 0 Td[(sin(!o(t)+t2)(1) aftermixingandperformingalowpasslterandgettingridofsomeresidualterm,theoutputis, 1 2Zu1)]TJ /F6 7.97 Tf 6.59 0 Td[(u1g(u)ej[)]TJ /F11 5.978 Tf 7.78 3.26 Td[(2u c(!o+2t)]du(1) andifwedenethefollowing,U=2 c! (1)=2 c(!o+2t) 31

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wecanseethatpulsecompressedsignal, 1 2Zu1)]TJ /F6 7.97 Tf 6.58 0 Td[(u1g(u)e)]TJ /F6 7.97 Tf 6.59 0 Td[(juUdu(1) whichisnothingmorethanaFouriertransformofthereectivityofthesceneevaluatedofarangeoffrequencyspeciedbythetimesupportoftransmittedsignal.Thissignalwillhavethesupportof, 2 c(!o)]TJ /F14 11.955 Tf 11.95 0 Td[(c)U2 c(!o+c)(1) makeuseofthefactthat2Bc=2c,thesupportcanbewrittenintermofthebandwidthofthetransmittedsignal, 2 c(!o)]TJ /F14 11.955 Tf 11.96 0 Td[(Bc)U2 c(!o+Bc)(1) essentiallywhenwesendacompressedpulse,suchasachirp,wearesamplingtheFouriertransformofthereectivityfunctionforasetoffrequencythatspansUcenteredatanoffsetfrequency,!owhichisthecarrierfrequency.Togettheactualreectivityofthescene,wejustneedtodoaninverseFouriertransform!Sincetheazimuthreturnsisalsoachirp,afterazimuthcompression,wejusttaketheFouriertransformwithrespecttotheazimuth.Ora2DFouriertransformtotheentire2Dreturndatasetoffasttimeandslowtimesamples. InthischapterwehavethoroughlyexplaintheconceptofSAR,itssignalmodel,andhowtotransmitaswellastohowtohandlethereturnsignal.Wehavetalkedabouthowonecanachieveresolutioninrange,andalsoincross-range.Welearnthathigherresolutioninrangecanbeachievedifwesendamodulatedpulsewithahighbandwidth.Welearnthatresolutionincross-rangeisfullydependentonthelengthoftheightpath,whichislimitedtotherealaperturebeamwidth.Wealsomaketheimportantconnectionbetweenthereturnsignalsandthereectivityofthescene.Weseethatbytransmittingacompresspulsed,weareessentiallysamplingthespatialfrequenciesofthescene,andtogetourreectivity,wejustneedtoperformaninverseFouriertransform. 32

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CHAPTER2SARRECONSTRUCTIONUSINGABAYESIANAPPROACH WeintroduceanewapproachusingtheBayesianframeworkforthereconstructionofsparseSyntheticApertureRadar(SAR)images.Thealgorithm,namedSLIM,canbethoughtofasasparsesignalrecoveryalgorithmwithexcellentsidelobesuppressionandhighresolutionproperties.Foragivensparsitypromotingprior,SLIMcyclicallyminimizesaregularizedleastsquarecostfunction.WeshowhowSLIMcanbeusedforSARimagereconstructionaswellasusedforSARimageenhancement.WeevaluatetheperformanceofSLIMusingrealisticallysimulatedcomplex-valuedbackscattereddatafromabackhoevehicle.ThenumericalresultsshowthatSLIMsatisfactorilysuppressthesidelobesandyieldhigherresolutionthantheconventionalmatchedlterordelay-and-sum(DAS)approach.SLIMoutperformthewidelyusedcompressivesamplingmatchingpursuit(CoSaMP)algorithm,whichrequiresthedelicatechoiceofuserparameter.Comparedwiththerecentlydevelopediterativeadaptiveapproach(IAA),whichiterativelysolvesweightedleastsquaresproblem.DuetocomputationalcomplexityinvolvedwithSARimaging,weshowhowSLIMcanbemademorecomputationallyefcientbyutilizingtheFouriertransform(FFT)andconjugategradient(CG)method.Furthermore,sincethetwoalgorithmswerederivedundertheBayesianmodel,theaposterioridistributiongivenbythealgorithmsprovidesuswithacondentmeasureregardingthestatisticalpropertiesoftheSARimagepixels. Theusageofasyntheticapertureinradarsystemstoovercometheazimuthalresolutionlimitationhasplayedaubiquitousroleinbothgovernmentandcommercialindustries.Theseclassofradars,convenientlynamed,SyntheticApertureRadar(SAR),haswiderangingapplications.Whilemostapplicationsarefoundinthesecurityanddefenseindustrieswhichincludes,reconnaisance,surveillance,targetidentications,foliagepenetrations,andmovingtargetindications,thecommercialindustriesalsomakeusesofSARsystemsfornavigations,mappings,andenvironmentalmonitoring. 33

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Atpresent,satellitesimaging(optical)areoftenusedforrealtimeresponsetoenvironmentaldisasters,SARanditsabilitytoimageinallweatherconditionsoffersacomplimentaryviewofthesituations.UsingvariousSARmodesandSARimagereconstructiontechniquescertaindetailsandobservationscanbemadethatotherwisewouldbemissedwhenusingopticalimaging. WhileasignicantpartofSARimagequalitydependsonthehardwareparametersoftheradarsystems,theSARimageconstructionalgorithmsplaysapivotalroleinmakingthemostofthecollecteddata.TheSARcommunityhasacceptedvariousmethodsforSARimagereconstruction,allofwhicharedata-independentapproaches,suchastheFourierTransforms,andbackprojection[ 12 ].Therearevariantsimprovementtothesebasicalgorithmsviapreandpostprocesses,buttheareahasbeenwelldeveloped.Theattractivenessofthesealgorithmsarethattheyaresimpletoimplementandcomputationallyefcientbuttheysufferfromhighsidelobelevelsandthusyieldlowresolutions.Tomitigatethisproblemwetradeitslowcomplexityforhigherresolutions.Inthischapterwewillfocusondata-adaptiveapproachesforSARimagereconstruction,withemphasisonalgorithmsthatpromotessparsityintheresultingimage.Assparserimagesaremorebenecialforfeatureextractionsandtargetsrecognitionsapplications. Variousdata-adaptiveapproacheshasbeenproposedforSARimaging[ 13 18 ].However,someofthesemethods,suchasCapon[ 18 19 ]andamplitudeandphaseestimation(APES)[ 15 16 ]requiremultiplesnapshotstoformthesamplecovariancematrix.ThisrequirementishardtosatisfyinSARsystems,duetothemovingnatureoftheplatform.Anothercategoryofalgorithm,thesparsesignalrecoveryalgorithms[ 20 22 ],includingthepopularregularizedl1-normbasedmethodsandthegreedy-typeapproaches,suchascompressivesamplingmatchingpursuit(CoSaMP)(see,e.g.,[ 23 ]andthereferencestherein),canbeusedtorecoversparseradarimageswithhighresolutionandtheydonotrequiremultiplesnapshots;nevertheless,theirperformanceissensitivetouserparameterswhicharehardtochooseinpractice[ 24 ].Inaddition,these 34

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approacheslackrobustnesstonoiselevels,whichwewillshowlaterinthenumericalexamples.Onepossibledata-adaptivealgorithm,theiterativeadaptiveapproach(IAA)[ 25 28 ]hasbeenintroducedfortheSARimagingapplication.IAAisrobustanduserparameterfreeandworkswithfeworevenasinglesnapshot.Comparedwiththedata-independentapproaches,IAAisshowntoproduceimageswithsignicantlyreducedsidelobesandmuchenhancedresolution[ 29 ].However,IAAiscomputationallyintensive,whichtosomeextentlimitsitsapplicability. WeconsideraBayesian-basedsparsesignalrecoveryalgorithms,namedSLIM(sparselearningviaiterativeminimization).SLIMisacyclicalgorithmthatiterativelytsthesignaltotheobservationwhilesimultaneouslysatisfyingthesparsity-promotingconstraint.Inanotherperspecive,SLIMisessentiallyalb-norm(0
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Figure2-1. SARimagingschematics. algorithms,namelyCoSaMPandIAA,usingidealscatterersaswellastheBackhoeDataDomeprovidedbyDARPAIXOandAFRL/SNAS(realisticallysimulatedusingcomputationalelectromagneticssoftware).Finally,weconcludethechapterinSectionV. 2.1PreliminariesandDataModel Inthissection,weintroducetheprinciplesofSARimaging,withspecialemphasisplacedonthegeometryofthephysicalsceneanddataacquisitionmechanism.Basedonthephysicalinterpretationofthedataacquisitionprocess,wewillapplysomereasonableapproximationstoderiveanappropriatedatamodel,withwhichvariousmethodscanbeappliedtoformtheSARimage. Thedataacquisitiongeometryforspot-lightmodeSARisillustratedinFig. 2-1 .A3-DCartesiancoordinatesystemisestablished,centeredatthescenepatchilluminatedbytheradarbeam.Thenon-shadedcirclerepresentsasensoratelevationangleandazimuthangle.Ascattererinthescenepatchlocatedat(x0,y0,z0)isdenotedbythelledcircle. 36

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InSARradarsystem,theantennatransmitsanelectromagneticpulsetowardthesceneofinterest(SOI),andthereectedsignalbearstheinformationofthepotentialscatterersintheSOI.Thetimedelay(x0,y0,z0)ofthereturnedsignalassociatedwithascattererlocatedat(x0,y0,z0)isdirectlydeterminedbythedistanceR(x0,y0,z0),whichcanbecalculatedasfollows: R(x0,y0,z0)=p (Rccoscos)]TJ /F5 11.955 Tf 11.95 0 Td[(x0)2+(Rccossin)]TJ /F5 11.955 Tf 11.96 0 Td[(y0)2+ (Rcsin)]TJ /F5 11.955 Tf 11.96 0 Td[(z0)2=Rcq 1+(x0 Rc)2+(y0 Rc)2+(z0 Rc)2)]TJ /F3 11.955 Tf 11.96 0 Td[(2x0 Rccoscos)-166( 2y0 Rccossin)]TJ /F3 11.955 Tf 11.96 0 Td[(2z0 Rcsin.(2) Sincethesensorisusuallyair-borneandthedistanceRcismuchlargerthanthedimensionofthescenepatch,theexpressionabovecanbesimpliedusingthefollowingapproximations: R(x0,y0,z0)Rcq 1)]TJ /F3 11.955 Tf 11.96 0 Td[(2x0 Rccoscos)]TJ /F3 11.955 Tf 11.95 0 Td[(2y0 Rccossin)]TJ /F3 11.955 Tf 11.96 0 Td[(2z0 RcsinRc(1)]TJ /F6 7.97 Tf 14.13 4.88 Td[(x0 Rccoscos)]TJ /F6 7.97 Tf 14.13 5.03 Td[(y0 Rccossin)]TJ /F6 7.97 Tf 14.18 4.88 Td[(z0 Rcsin)=Rc)]TJ /F3 11.955 Tf 11.95 0 Td[((x0coscos+y0cossin+z0sin).(2) Theexpressionin( 2 )givesthedistancebetweenthescattererandthesensorrelativetoRc. Letp(t)=a(t)ej2fctdenotethepulse(e.g.,achirpwaveform)transmittedtowardtheSOI,wherea(t)isthecomplexenvelopeandfcisthecarrierfrequency.Thereectedsignalr(t,,)canbeinterpretedasthesumofdelayedversionsofp(t)withtheamplitudesandphasesmodiedproportionallytothereectivityg(x,y,z)ofthescatterers.Mathematically,thisrelationcanbeexpressedas: r(t,,)=Xx,y,zg(x,y,z)p(t)]TJ /F14 11.955 Tf 11.96 0 Td[((x,y,z)),(2) 37

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where(x,y,z)=2R(x,y,z) candcisthespeedoflight.Withthismodel,thematchedlteroutputy(t,,)canbeexpressedas y(t,,)=Xx,y,zg(x,y,z)s(t)]TJ /F14 11.955 Tf 11.96 0 Td[((x,y,z))+e(t),(2) wheres(t)istheauto-correlationfunctionofthetransmittedsignalp(t)ande(t)istheadditivenoise.BytakingtheFouriertransformof( 2 )andusingtherelationsin( 2 )above,thephasehistorydataarerelatedtothescattererreectivityfunctionby: Y(f,,)=Xx,y,zg(x,y,z)S(f)e)]TJ /F6 7.97 Tf 6.59 0 Td[(j2f2R(x,y,z) c+E(f) (2a) =C(f)Xx,y,zg(x,y,z)ej4 c(xfcoscos+yfcossin+zfsin)+E(f) (2b) whereE(f)istheFouriertransformofe(t)andC(f)=S(f)e)]TJ /F6 7.97 Tf 6.58 0 Td[(j2f2Rc ciscommontoallscatterersinthescenepatchsothatwewilldropitfromtheexpressioninthefollowingdiscussionforthesakeofnotationalsimplicity.Substitutingthetransformationpairsu(f,,)=fcoscos,v(f,,)=fcossinandw(f,,)=fsininto( 2b ),theoutputsignalcanbefurtherwrittenas Y(f,,)=Px,y,zg(x,y,z)ej4 c(xu(f,,)+yv(f,,)+zw(f,,))+E(f).(2) With( 2 ),thephasehistorydatacanthusbeinterpretedasthe3-DFouriertransformsofthereectivityfunctionofthescenepatch.Inotherwords,thecollectedphasehistorydataformapolarrasterinthetransformedspatialfrequencyspace. Consequently,wehaveasetofsamplesnonuniformlyplacedinthespatialfrequencydomain.SincethereisnofastalgorithmforcomputingFouriertransformsonanon-Cartesianspace,wenextseektoreformatthedatasamples.Basically,theapproachesforthispurposefallintothefollowingtwocategories:approximationandinterpolation.Theapproximation-basedapproachisappropriatewhenthefrequency 38

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spanf,theelevationapertureandtheazimuthaperturearerelativelysmall.Insuchacase,wecanviewthepolargridpointsasapproximatelyuniforminaCartesiancube,whichfacilitatestheapplicationoffastFouriertransformsdenedonaCartesiancoordinatesystem. However,forwide-apertureSARphasehistorydata,theapproximationaboveisinaccurate.Inaddition,thismethodcreatestechnicalproblemsforthe3-DSARimagefusionapplication,asisdiscussedinSectionIV.Inthissituation,wecanresorttotheinterpolation-basedapproachtotransformthedatafromthepolarcoordinateintotheCartesiancoordinate.Oneparticularinterpolation-typemethodinvolvesusingnon-uniformfastFouriertransforms(NUFFT)[ 33 ].Inthiscase,thenon-uniformdatasamplesinthe3-Dspatialfrequencyspacearerstbackprojectedtotheintermediateimagespace.Sincetheimagesamplesareuniforminthe3-DCartesiancoordinatesystem,theycanbeconvenientlyprojectedagaintothespatialfrequencyspaceusing3-DFFT.Afterthistwo-stepprocedure,thedatasamplesinthespatialfrequencyspaceareuniforminthedimensionsofu,v,andw. Basedonthescenegeometryandthedatareformationabove,wecanderivethedatamodelasfollows.LetM1,M2andM3denotethenumbersofimagegridpointsinthex,yandzdirections,respectively.LetN1,N2andN3denotethenumberoffrequencygridpoints,thenumberofazimuthanglegridpointsandthenumberofelevationanglegridpoints,respectively.ThetotalnumberofunknownsisM=M1M2M3andthetotalnumberofobserveddatapointsisN=N1N2N3.Forascattererlocatedat(xj,yk,zl),theN-elementsteeringvectorcanbewrittenas: aj,k,l=[bj,k,l(1,1,1),,bj,k,l(N1,N2,1),,bj,k,l(1,1,N3),,bj,k,l(N1,N2,N3)]T,1jM1,1kM2,1lM3,(2) 39

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wherebj,k,l(p,q,r)=ej4 c(upxj+vqyk+wrzl)for1pN1,1qN2and1rN3.Byusingthesedenitions,thesteeringmatrixcanbewrittenas: A=[a1,1,1,,aM1,M2,1,,a1,1,M3,,aM1,M2,M3].(2) Wedenes=[g(x1,y1,z1),,g(xM1,yM2,z1),,g(x1,y1,zM3),,g(xM1,yM2,zM3)]TtobeanM1vectorcontainingthereectivitycoefcients,andwestacktheobserveddataintothevectory.Thedatamodelcanbecompactlyexpressedinmatrixformas: y=As+e,(2) whereeaccountsfortheadditivenoise. ToformSARimages,weessentiallyneedtoestimatesfromyandA.ThemoststraightforwardmethodistoperformtheinverseFouriertransform,whichgivestheconventionaldelay-and-sum(DAS)ormatchedlteringapproach[ 34 ].Inspiteofthecomputationalefciency,thequalityoftheDASimageisusuallyunsatisfactoryduetothelowresolutionandhighsidelobesintheimage.Comparedwiththedata-independentDASmethod,thedata-adaptiveapproaches,whichutilizetheinformationcontainedinthedataacquired,usuallyyieldbetterimagingresults.Inthefollowingsections,wediscussseveraldata-adaptivealgorithmsthatcanbeusedtoenhancetheimagingquality. 2.2Data-AdaptiveAlgorithms Inthissection,werstreviewtwodata-adaptivealgorithms,namelyCoSaMPandIAA,astheyhavetheclosesttiestoSLIMandcommentontheirperformanceandapplicability.ThenweintroduceSLIManddiscusshowitcanbeadaptedtotheSARimagingproblem. 2.2.1CompressedSamplingMatchingPursuit(CoSaMP) CoSaMP[ 23 ]isasparsesignalrecoveryalgorithm,whichiterativelyidentiesthesupportofthesparsesignalvectorandestimatesthecorrespondingentryvalues. 40

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CoSaMPhastwouserparameters,whichdeterminethesparsityoftheestimationresultandthedimensionofthepotentialsupport.Usually,thesignalsparsityissettobehalfofthesupportdimension,hencewewillonlygivevaluesofthesignalsparsityinthenumericalexamples.CoSaMPisshowntogenerateSARimageswithhigherresolutionthantheconventionalDASalgorithm[ 35 ].However,theperformanceofCoSaMPishighlydependentontheuserparameters,whosechoice,unfortunatelylacksaclearguidance.Therefore,theperformanceofCoSaMPcanbepoorwithanimproperselectionoftheuserparameters,whichisillustratedlateroninthenumericalexamples.Furthermore,theperformanceofCoSaMPdegradesasthenoiselevelincreases,asillustratedbythenumericalexamples. 2.2.2IterativeAdaptiveApproach(IAA) IAA[ 25 ]isanonparametricadaptivealgorithmoriginallyproposedforthearraysignalprocessingapplication.IAAhasbeenshowntoprovidemuchimprovedperformanceovermanyotherdata-adaptivealgorithmsaswellasDASinapplicationssuchasMIMOradar,activesonar,andmedicalimaging.Inarecentwork[ 29 ],IAAisintroducedtotheSARimagingproblemanditisshowntherethatIAAcansignicantlyreducethesidelobelevelsandproducemuchcleanerimages.ThoughtheperformanceofIAAissatisfactory,itshighcomputationalcomplexitytosomeextentlimitsitsapplicabilitytolargedimensionalproblems.Sofar,reducingthecomputationalcomplexitiesofIAAisstillanopenproblem.WewilldiscussfurtherabouttheperformanceofIAAwiththenumericalexampleslateron. 2.2.3SparseLearningviaIterativeMinimization(SLIM) Inthissection,weintroduceourmainalgorithmthatisbasedonaBayesianmodel.WenameditSLIM(SparseLearningviaIterativeMinimization)sinceittsitselftoagivensparsepromotingprioriteratively.TheSLIMalgorithmcanbecastintothe 41

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BayesianframeworkbyconsideringthefollowinghierarchicalBayesianmodel: yjs,CN(As,I)f(s)/Mm=1e)]TJ /F11 5.978 Tf 7.83 3.26 Td[(2 b(jsmjb)]TJ /F9 7.97 Tf 6.59 0 Td[(1)f()/1,(2) whereisthenoisepowerandf(s)isasparsitypromotingpriorfor0
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SLIMisusuallyinitializedbyDASandterminatedwhenthenormofthedifferencebetweentwoconsecutiveestimatesofsfallsbelowapre-speciedthreshold. Theaposterioridistributionofscanbeapproximatedbythefollowingconditionalprobabilitydistribution: f(sjy,op,o)/f(s,y,opjo)=f(yjs,o)f(sjop)f(op)/expf)]TJ /F9 7.97 Tf 18.74 4.71 Td[(1 2ojjy)]TJ /F4 11.955 Tf 11.95 0 Td[(Asjj2gexpf)]TJ /F9 7.97 Tf 16.47 4.71 Td[(1 2sH(oP))]TJ /F9 7.97 Tf 6.58 0 Td[(1sg=expf)]TJ /F9 7.97 Tf 18.74 4.71 Td[(1 2oyHy+1 2osHAHy+1 2oyHAs)]TJ /F9 7.97 Tf -186.86 -19.9 Td[(1 2sH(1 oAHA+(oP))]TJ /F9 7.97 Tf 6.58 0 Td[(1)s)g,(2) whereopandoaretheestimatedvaluesatconvergence.From( 2 ),weobservethattheconditionaldistributionofsgiveny,opandoisaGaussiandistribution.LetanddenotethemeanandcovariancematrixforthisconditionalGaussianrandomvector,i.e.,sjy,op,oCN(,).Expandingtheexpressionaboveandequatingthecorrespondingterms,wecannd: =oP)]TJ /F18 7.97 Tf 14.03 10.7 Td[(oPAH(oI+AoPAH))]TJ /F9 7.97 Tf 6.59 0 Td[(1AoP,and=oPAH(AoPAH+oI))]TJ /F9 7.97 Tf 6.59 0 Td[(1y.(2) Thesearetheaposteriorimeanandcovariancematrixofthesignalvectorsandtheestimateofsistakentobetheaposteriorimean.Withthisaposterioridistribution,thestatisticalpropertiesoftheestimatescanbeutilizedforperformancepredictionpurposesforapplicationssuchasautomatictargetrecognition. 2.3NumericalExamples Inthissection,weshowtheSARimagingcapabilityofSLIMandcompareitsperformanceswiththoseofCoSaMPandIAAaswellasthedata-independentDASalgorithmwhichwewillimplementviaFFT.Werstconsiderasimulatedexampleofidealpointscatterers,withwhichimagingpropertiesofthevariousadaptivealgorithmsintroducedabovearestudied.WethenexaminetheimagingcapabilityofthevariousalgorithmsusingtherealisticallysimulatedBackhoeDataDome,Version1.0dataset. 43

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2.3.1AnExampleofIdealPointScatterers Inthisexample,theSOIconsistsof12idealpoint-likescattererswithuniformamplitudes1,denotedbyblackcirclesinFigure 2-2 (a).Basedonthedatamodelin( 2 ),wedene SNR=jjAsjj2 jjejj2=jjAsjj2 N2,(2) where2istheaveragenoisepower.Throughoutthisexample,theSNRissettobe10dB.IAAissettorunfor10iterations,afterwhichnosignicantchangeinimagequalityisobserved.SLIMandCoSaMParesettorunfor20iterations. Figure 2-2 (b)showstheSARimageobtainedusingDAS.Thoughcomputationallyefcient,DASyieldsanimagewithhighsidelobelevelsandpoorresolution.NotethatthefourcloselyspacedscatterersinthecenteroftheSOIsmearwitheachotherandappearasonesinglescatterer.Figures 2-2 (d)and(f)showtheSARimagesgeneratedbyCoSaMP,whichhasoneuserparameterthatcontrolstheSOIsparsity.Sincethisuserparameterdirectlydeterminesthenumberofnon-zeropixelspresentintheimageandcansignicantlyaffecttheimagequality,thelackofproperguidanceforthechoiceofthisparameterrestrictstheapplicabilityofthisalgorithm.AsisseenfromFigures 2-2 (d)and(g),differentvaluesofthisuserparametercanleadtoverydifferentresults,whichisundesirableinpractice.Figures 2-2 (c),(e)and(g)showtheSARimagesproducedbyIAAandSLIM,respectively.Asevidentfromthegures,IAAandSLIMallpossessexcellentsidelobesuppressioncapability,producingcleanimageswherethepresenceofthescatterersisclearlyindicatedandthecloselyspacedscatterersaredistinguished. Next,weconsidertherobustnessofthevariousalgorithmsagainstadditivenoise.AgainbylookingatFigure 2-2 ,withSNRat10dB,CoSaMPismoresensitivetothenoiselevelthanSLIMandIAA.Whentheuserparameterdoesnotmatchthescatterersparsity,theperformanceofCoSaMPcanberatherpoor,asisseenfromFigure 2-2 (d).Incontrast,SLIMandIAAareallmuchmorerobusttothenoiselevel. 44

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Regardingcomputationalcomplexity,DASisthemostefcientalgorithmsinceitcanpurelybeimplementedviaFFTs.SLIMcanbemadecomputationallyefcientvialeveragingFFTsandtheConjugateGradientmethod,detailsofwhicharetoolongandsidetracksthetopicofthispaper.OurimplementationsofSLIMincorporatesthesespeed-upmethods,andthecomputationtimeisasfollow.Foratypicalrun(ComputerSpecications:Xeon2.33GHzCPU,16.0GBRAM,MatlabR2009a64bit),ittakesSLIMlessthan4secondstogenerateanimageshowninFigure 2-2 .CoSaMPisslightlyfasterthanSLIM,andthecomputationaltimeisaround3seconds.IAAiscomputationallythemostintensiveamongallthealgorithmsconsideredherein,andthetypicalrunningtimeisabout2hours. 2.3.2ImagingoftheBackhoe WenowusetheBackhoeDataDome,Version1.0toevaluatetheSARimagingcapabilityoftheSLIMalgorithm.FromthispointonwedropthecomparisonwithDASandIAA.FirstwedropDASbecausethehighsidelobesresultedinsignicantsmearingwhichmakestheobjectunrecognizable.Secondly,wedropIAAbecauseofitscomputationtimes.Bothofthesereasonsarewellfoundedbytheidealpointscatterersexampleabove.InterestedreaderwhowishtoseetheperfromanceofIAAcanreferto[ 29 ].WewillwillsolelyfocusonSLIManditsnearestbenchmark,CoSaMP.Thedatasetconsistsofrealisticallysimulated,fullypolarized,complexbackscattereddatafromabackhoevehicleinfreespace.ThedatasetwasgeneratedbyX-Patch,aleadingCEM(ComputationalElectromagnetic)code.The3-DCADmodelofthebackhoeandanillustrationofthebackhoedatadomeareshowninFigure 2-3 .Wewillnowlookatthe3-D(range-azimuth-elevation)imagingofthebackhoeobject. 2.3.33-DSARImaging Inthissection,weconsiderthe3-DSARimagingproblem.ThephasehistorydatausedhereconsistoftheportionwithabandwidthB=0.5GHzcenteredatf0=12GHz.Theelevationapertureof25centeredat0=30isequallydividedinto5 45

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(a)(b)(c) (d)(e)(f) (g) Figure2-2. ImagingofScatterersA)ThetruescattererdistributionB)DAS.C)IAA.D)CoSaMP(300).E)SLIM(b=1).F)CoSaMP(12).G)SLIM(b!0). non-overlappingsubapertures,andtheazimuthapertureof45centeredat0=90isequallydividedinto9non-overlappingsubapertures. WewillfocusoncomparingSLIMwithCoSaMP.ForCoSaMP,fourdifferentvaluesoftheuserparameterareconsidered.Theimagesshownhereareprojectionsontoa2-Dplanefromacertainazimuthangle.AscanbeseenfromFigure 2-4 ,SLIMwithb=1canproducecleanimagesrepresentingthestructureofthebackhoevehicle. 46

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(a)(b) Figure2-3. VisualizingtheBackhoeDataSetA)3-DCADmodelofthebackhoe.B)Thebackhoedatadome. CoSaMP,ontheotherhand,tendstodistorttheimagewithsometargetfeatures(suchasthelinearoundthe-1elevation)thatappearandthendisappearastheuserparameterincreases. InFigure 2-5 weshowthefused3-DSARimagesofthebackhoevehiclegeneratedusingtheSLIMalgorithm.WeapplyNUFFTtointerpolatethephasehistorydatasamplesontoauniform3-DCartesiangrid,whichisusedforallsubapertures.Thefusedimageisobtainedbycombiningthesubimageswiththemaximum-magnitudeoperatorasfollows: Image(x,y,z)=max1q9,1r5jSubimage(x,y,z;q,r)j.(2) Ascanbeseenfromthegure,theimagesgeneratedarecleanandfocused,givingminimumartifactsandnicelyshowingtheimagedvehicle.Moreover,SLIMwitheitherb!0orb=1workswellinthis3-Dfusedimagingapplication. Withtheaforementionednumericalexamples,theadvantagesofSLIMoverDAS,IAAandCoSaMPbecomeapparent.FirstSLIMgiveresultswithhighdelity,nicelyindicatingtheunderlyingstructureofthetargets.Withtheaddedbenetsofauserparameterthatiseasytochoose.Furthermore,SLIMisrobusttoadditivenoiselevels.Inaddition,SLIMiscomputationallyefcientandpreferableforproblemswithlargedata 47

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(a)(b)(c) (d)(e)(f) Figure2-4. Backhoe3-DSARimagereconstruction.A)SLIM(b!0)B)SLIM(b=1)C)CoSaMP(128)D)CoSaMP(256)E)CoSaMP(512)F)CoSaMP(1024) (a)(b) Figure2-5. Fused3-DImageUsingSLIM.A)SLIM(q!0)B)SLIM(q=1) sets.Themaincompetingalgorithm,CoSaMP,isagoodalgorithmbutsinceitsuserparameterdirectlytieswiththenumberofnon-zerosinthesignalspace,itrequiressomeclairvoyantonthesideoftheuser. 48

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WehavediscussedthedataacquisitionprocessanddatamodelofSARimagingproblem.WehaveintroducedasparsesignalrecoveryalgorithmnamedSLIMtotheSARimagingapplicationforthepurposeofsidelobesuppressionandresolutionenhancement.SLIMsimplycyclicallymaximizingtheaposterioriprobabilityoftheunderlyingparametersgiventheobservations.SLIMcanalsobeviewedastheiterativeminimizationoftheregularizednegativelog-likelihoodfunction.SLIMiseasytouse(doesnotrequirethedelicatetuningofuserparameters),providesaccurateandsparsesignalestimatesinacomputationallyefcientmanner,andisrobusttoadditivenoiselevels.WehavetestedtheperformancesofSLIMusingvariousnumericalexamples.ItprovidesbetterresolutionthanthetraditionalDAS/FFTapproach,itiscomputationallymorefeasiblethanIAA,anditsuserparameter,unlikeCoSaMP,doesnotgreatlyinuenceitsperformance. 49

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CHAPTER3SARRECONSTRUCTIONOFINTERRUPTEDDATA Weconsidernonparametricadaptivespectralanalysisofcomplex-valueddatasequenceswithmissingsamplesoccurringinarbitrarypatterns.Werstpresenttwohigh-resolutionmissing-dataspectralestimationalgorithms:theIterativeAdaptiveApproach(IAA)andtheSparseLearningviaIterativeMinimization(SLIM)method.Bothalgorithmscansignicantlyimprovethespectralestimationperformance,includingenhancedresolutionandreducedsidelobelevels.Moreover,weconsiderfastimplementationsofthesealgorithmsusingtheConjugateGradient(CG)techniqueandtheGohberg-Semencul-type(GS)formula.OurproposedimplementationsfullyexploitthestructureofthesteeringmatricesandmaximizetheusageoftheFastFourierTransform(FFT),resultinginmuchlowercomputationalcomplexitiesaswellasmuchreducedmemoryrequirements.Theeffectivenessoftheadaptivespectralestimationalgorithmsisdemonstratedviaseveralnumericalexamplesincludingboth1-Dspectralestimationand2-Dinterruptedsyntheticapertureradar(SAR)imagingexamples. Spectralestimationisimportantinmanyeldsincludingastronomy,communications,medicalimaging,radar,andunderwateracoustics.Mostexistingspectralestimationalgorithmsaredevisedforuniformlysampledcomplete-datasequences.However,inmanypracticalapplications,themeasureddatamaybeincompletedue,forexample,tosensorfailures,outliers,thedatacompressionneeds,etc.Missingdataproblemsarealsoencounteredinmodernradarsystems.Thesesystemshavemultipledutiesincludingsearching,trackingandtheautomaticclassicationoftargets.Switchinginandoutofthesemodesleadstoanincompletephasehistorydataforsyntheticapertureradar(SAR)imaging[ 36 38 ].ThislossofthephasehistorydataisalsoprevalentduringSARmissionsencounteringhighradiofrequency(RF)interferences[ 39 40 ]. Manymissing-dataspectralestimationtechniqueshavebeendevelopedpreviously.Aconceptuallyandcomputationallysimplemethodisthematchedlter(MF),implemented 50

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viausingthefastFouriertransform(FFT),withthemissingsamplessettozero.However,asadata-independentapproach,theFFTapproach,aswellasitswindowedFFT(WFFT)variations,suffersfrompoorresolutionandhighsidelobelevelproblems.Thishighsidelobelevelproblemisespeciallysevereforthemissingdatacase.TheCLEANalgorithm[ 41 ]isusedtoestimatethespectrumbydeconvolvingthemissing-dataFFTspectrumintothetruesignalspectrumandtheFouriertransformofthemissingpatternwindowingfunctionviaaniterativeapproach.AlthoughtheCLEANalgorithmworksforbothmissingandirregularlysampleddatasequences,itcannotresolvecloselyspacedspectrallines,andhenceitmaynotbeasuitabletoolforhigh-resolutionspectralestimation.Themulti-tapermethods[ 42 43 ]computespectralestimatesbyassumingcertainquadraticfunctionsoftheavailabledatasamples.Thecoefcientsinthecorrespondingquadraticfunctionsareoptimizedaccordingtocertaincriteria,butitappearsthatthistypeofapproachescannotovercometheresolutionlimitofFFT.Toachievehighresolution,severalparametricalgorithms,e.g.,thosebasedonautoregressive(AR)orautoregressivemoving-average(ARMA)models,wereusedtohandlethemissing-dataproblem[ 44 47 ].Althoughtheseparametricmethodscanprovideimprovedspectralestimates,theyaresensitivetomodelerrors. Severalnonparametricadaptivelteringbasedtechniqueshavebeendevelopedforthemissing-dataspectralestimationproblem[ 38 ].Forexample,in[ 48 ]and[ 38 ],twononparametricmissing-dataamplitudeandphaseestimation(MAPES)algorithmsarederivedbyusingamaximumlikelihood(ML)ttingcriterion.Thewell-knownexpectationmaximization(EM)technique[ 49 ]isusedtosolvetheestimationproblemiteratively.MAPESworksforthegeneralspectralanalysisproblemwithmissingsamplesoccurringinarbitrarypatterns.However,MAPESisconceptuallycomplicatedanditiscomputationallyintensive.Furthermore,itiswell-knownthatEMmayconvergeratherslowly,and,itmayevenfailtoconvergeglobally.Aswewillshowvianumerical 51

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examples,thespectralestimationperformanceofMAPESdegradesseverelyinhighdatamissingratiocases. Recently,sparsesignalrecovery(SSR)techniques(see,e.g.,[ 23 ]andthereferencestherein)haveattractedmuchattentioninthesignalprocessingcommunity.Differentfromtheconventionaldata-adaptiveapproaches,suchasCapon[ 50 ]andAPES[ 51 ],SSRcanworkforveryfeworevenonesnapshot,andhencecanbeusedforspectralanalysisofirregularlysampleddata.However,mostoftheexistingSSRalgorithms,includingthewell-knownCoSaMP(CompressiveSamplingMatchingPursuit)[ 23 ],requirethebasesofthesignalcomponentstoobeyarestrictedisometryproperty[ 52 ],whichhardlyholdsinhigh-resolutionspectralestimationproblems.Aswewillshowvianumericalexamples,CoSaMPfailstoworkproperlyinboth1-Dspectralestimationand2-DinterruptedSARimagingproblems. Inthischapter,wewillrstpresenttworecently-developedhigh-resolutionspectralestimationalgorithms,i.e.,theiterativeadaptiveapproach(IAA)[ 25 ]andthesparselearningviaiterativeminimization(SLIM)method[ 32 53 ].Bothalgorithmscanworkforbothcompleteandincompletedatasamples,andareabletoachieveexcellentspectralestimationperformanceundervariousconditions.However,bothIAAandSLIMarecomputationallyprohibitive,especiallyforhigh-dimensionalspectralestimationproblemssuchasinterruptedSARimaging.WewillconsiderthefastimplementationofIAAforthespectralestimationwithmissingsamples.Moreover,wewillproposetwofastimplementationsofSLIMwithinthemissing-dataspectralestimationframework.Wewillapplytheproposedalgorithmstoboth1-Dspectralestimationand2-DinterruptedSARimagingproblems.Viaseveralnumericalexamples,weshowthatIAAandSLIMoutperformtheconventionalFFTandMAPESapproaches,aswellasCoSaMP,signicantly.ComparedtotheoriginalIAAandSLIMalgorithms,thefastimplementationapproachescandramaticallyreducethecomputationalcomplexitiesandmemoryrequirements. 52

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Theremainderofthischapterisorganizedasfollows.Section 3.1 introducestheIAAandSLIMalgorithms.InSection 3.2 ,weproposethreefastmissing-dataspectralestimationmethods.Several1-Dspectralestimationand2-DinterruptedSARimagingexamplesarepresentedinSection 3.3 .Finally,Section 3.3.2 containsourconclusions. 3.1NonparametricSpectralAnalysis Inthissection,werstpresenttworecently-developedadaptivespectralestimationalgorithms,i.e.,IAAandSLIM,withinthemissingdataframework.Bothalgorithmscanworkfor1-Dandhigherdimensionalspectralestimationproblemswithcompleteorincompletedatasamples. 3.1.1DataModel WeconsideraspectralestimationproblemwithMgivencomplex-valueddatasamples,andstacktheMdatasamplesintoanM1columnvectorx.Letl(l=0,1,,L)]TJ /F3 11.955 Tf 13.18 0 Td[(1)bethecomplexamplitudeatthelthfrequencygridpointofthespectrum,andal(l=0,1,,L)]TJ /F3 11.955 Tf 13.08 0 Td[(1)bethenormalizedcontributionofthelthfrequencygridpointtotheMavailabledatasamples.Then,thegivendatasamplescanbemodeledasfollows: x=A+n,(3) wherendenotesthenoisevector, A=[a0,a2,,aL)]TJ /F9 7.97 Tf 6.58 0 Td[(1],(3) and =[0,1,,L)]TJ /F9 7.97 Tf 6.59 0 Td[(1]T,(3) with()Tdenotingthetranspose.In( 3 ),Aisreferredtoasthesteeringmatrix,andthecolumnvectorrepresentsthespectrumtobeestimated.Notethatthedatamodelin( 3 )canbeusedforboth1-Dandhigherdimensionalspectralestimationproblems.Inparticular,intheSARimagingapplication,containsthecomplex-valuedreectioncoefcientsofscattererswithinanimagingarea(orvolume)ofinterest.Notealsothat 53

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inhigh-resolutionspectralestimationproblems,thegridpointnumberLisusuallymuchlargerthanthenumberofdatasamples.Hence,theconventionalapproachesbasedon( 3 ),suchasthemaximumlikelihood(ML)andleast-square(LS)approaches,failtoworkcompletely. 3.1.2IterativeAdaptiveApproach(IAA) IAAisaniterativeadaptivealgorithmbasedonanonparametricweightedleastsquares(WLS)approach.Letpl=j^lj2(l=0,1,,L)]TJ /F3 11.955 Tf 12.99 0 Td[(1)with^kbeinganintermediateestimateofl.Thenoiseandinterferencecovariancematrixlforthelthfrequencygridpointcanbeestimatedas: l=Xk6=lpkakaHk=R)]TJ /F5 11.955 Tf 11.96 0 Td[(plalaHl,(3) where()Hdenotestheconjugatetranspose, R=L)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xk=0pkakaHk=Adiag(p)AH,(3) and p=p0p1pL)]TJ /F9 7.97 Tf 6.59 0 Td[(1T(3) withdiag()beingadiagonalmatrixformedbytheelementsofagivenvector. Giventhenoiseandinterferencecovariancematrixl,theestimateoflcanberenedusingtheWLScriterionasfollows: ^l=argminkx)]TJ /F4 11.955 Tf 11.95 0 Td[(allk2)]TJ /F11 5.978 Tf 5.76 0 Td[(1l,(3) wherekxk2)]TJ /F11 5.978 Tf 5.75 0 Td[(1l=xH)]TJ /F9 7.97 Tf 6.59 0 Td[(1lx.Solvingtheoptimizationproblemof( 3 )yields ^l=aHl)]TJ /F9 7.97 Tf 6.58 0 Td[(1lx aHl)]TJ /F9 7.97 Tf 6.58 0 Td[(1lal=aHlR)]TJ /F9 7.97 Tf 6.59 0 Td[(1x aHlR)]TJ /F9 7.97 Tf 6.58 0 Td[(1al,(3) wherewehaveused( 3 )andthematrixinversionlemma[ 54 ]. 54

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Thenalspectrumestimateisobtainedviaiterating( 3 )and( 3 ).TheinitializationofflgL)]TJ /F9 7.97 Tf 6.59 0 Td[(1l=0canbedonebyastandardMF: ^l,MF=aHlx kalk2,forl=1,2,,L,(3) withkkdenotingtheEuclideannormofavector. Weremarkthatintheabovederivationwehaveimplicitlytakenintoaccounttheeffectofnoise.AregularizedversionofIAA,referredtoasIAA-R[ 55 ],canbeusedtotakeintoaccountthenoiseeffectexplicitly.ThefastimplementationofIAAproposedinSection 3.2 canbeextendedtoIAA-Rstraightforwardly.Sowewillfocusontheformerherein.WeformallyoutlinethestepsintheIAAalgorithmbelow. TheIAAAlgorithm: Initialization:EstimatefkgusingMF. Iteration:Repeatthefollowingsteps: Step1-Computep=[j0j2,j1j2,,jL)]TJ /F9 7.97 Tf 6.59 0 Td[(1j2]T. Step2-UpdatethesignalcovariancematrixR=Adiag(p)AH. Step3-Updatethespectrumestimate^l=aHlR)]TJ /F11 5.978 Tf 5.76 0 Td[(1x aHlR)]TJ /F11 5.978 Tf 5.75 0 Td[(1alforl=0,1,,L)]TJ /F3 11.955 Tf 11.96 0 Td[(1. 3.1.3SparseLearningviaIterativeMinimization(SLIM) TheSLIMalgorithm[ 32 53 ]considersthefollowingoptimizationproblem: (^,^)=min,g(,),(3) where g(,),Mlog+1 kx)]TJ /F4 11.955 Tf 11.95 0 Td[(Ak2+L)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xl=02 q(jlj2)q=2)]TJ /F3 11.955 Tf 11.96 0 Td[(1,(3) withkkdenotingtheEuclideannormofavector.In( 3 ),qisauserparameterthatrangesbetween0and1.InournumericalexamplesinSection 3.3 ,weuseq=1forboth1-Dspectralestimationand2-DinterruptedSARimagingproblems.TheSLIM 55

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algorithmcanbeviewedasamaximumaposteriori(MAP)approachwithbeingthenoisevarianceandthepriordistributionofflgbeingf(l)/e)]TJ /F11 5.978 Tf 7.9 3.25 Td[(2 qjljq)]TJ /F9 7.97 Tf 6.59 0 Td[(1. Theoptimizationproblemcanbesolvediterativelybyusingthecyclicminimization(CM)andmajorization-minimization(MM)techniques[ 56 ].Let^(t)and^(t)beintermediateestimatesofand,respectively,forthetthiteration.For0q1,2 q(jlj2)q=2)]TJ /F3 11.955 Tf 11.95 0 Td[(1isaconcavefunctionwithrespecttojlj2.Hence,thefunction2 q(jlj2)q=2)]TJ /F3 11.955 Tf 11.95 0 Td[(1ismajorizedbyitstangentlinearfunctionatj^(t)lj2,i.e., 2 q(jlj2)q=2)]TJ /F3 11.955 Tf 11.95 0 Td[(12 qhj^(t)ljq)]TJ /F3 11.955 Tf 11.95 0 Td[(1i+1 p(t)lhjlj2)-222(j^(t)lj2i,(3) where p(t)l=j^(t)lj2)]TJ /F6 7.97 Tf 6.59 0 Td[(q,(3) andtheequationholdswhenjlj=j^(t)lj.Therefore,wehave: g(,)Mlog+1 kx)]TJ /F4 11.955 Tf 11.95 0 Td[(Ak2+L)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xl=0p(t)lhjlj2)-221(j^(t)lj2i+L)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xl=02 qhj^(t)ljq)]TJ /F3 11.955 Tf 11.95 0 Td[(1i,h(,j^(t)), (3) where,again,theequationholdswhenjlj=j^(t)ljforl=0,1,,L)]TJ /F3 11.955 Tf 11.96 0 Td[(1. Given=^(t),minimizingthemajorizationfunctionofg(,),i.e.,h(,j^(t)),withrespecttoyields: ^(t+1)=AHA+(t)diag)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(p(t))]TJ /F9 7.97 Tf 6.59 0 Td[(1AHx=diag)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(p(t)AH)]TJ /F9 7.97 Tf 6.58 0 Td[(1x, (3) where =Adiag)]TJ /F4 11.955 Tf 5.47 -9.68 Td[(p(t)AH+^(t)I,(3) andp(t)isdenedsimilarlytopin( 3 ),i.e., p(t)=^(t)2)]TJ /F6 7.97 Tf 6.58 0 Td[(q,^(t)02)]TJ /F6 7.97 Tf 6.59 0 Td[(q^(t)12)]TJ /F6 7.97 Tf 6.58 0 Td[(q^(t)L)]TJ /F9 7.97 Tf 6.59 0 Td[(12)]TJ /F6 7.97 Tf 6.59 0 Td[(qT.(3) 56

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Furthermore,given=^(t+1),minimizingh(,j^(t))withrespecttoyields: ^(t+1)=1 Mx)]TJ /F4 11.955 Tf 11.95 0 Td[(A^(t+1)2.(3) TheSLIMalgorithmestimatesviaiterating( 3 ),( 3 ),and( 3 ). ByusingtheCMandMMproperties,wereadilyhave: g^(t+1),^(t+1)h^(t+1),^(t+1)j^(t)h^(t+1),^(t)j^(t)h^(t),^(t)j^(t)=g^(t),^(t), (3) i.e.,thecostfunctiondecreasesmonotonically. Again,weinitializeflgL)]TJ /F9 7.97 Tf 6.58 0 Td[(1l=0usingMFin( 3 ).Thevariableisinitializedas: ^=1 Lx)]TJ /F4 11.955 Tf 11.95 0 Td[(A^MF2,(3) where^MFisacolumnvectorformedbyf^l,MFgL)]TJ /F9 7.97 Tf 6.58 0 Td[(1l=0similarlyto( 3 ),andischosentobe10inournumericalexamplesinSection 3.3 .TheSLIMalgorithmissummarizedbelow. TheSLIMAlgorithm: Initialization:EstimateusingMFin( 3 ),andsetaninitialvalueof^using( 3 ). Iteration:Repeatthefollowingsteps: Step1-Computep=j^j2)]TJ /F6 7.97 Tf 6.59 0 Td[(q. Step2-Compute=Adiag(p)AH+^I. Step3-Update^=diag(p)AH)]TJ /F9 7.97 Tf 6.58 0 Td[(1x. Step4-Update^=1 Mkx)]TJ /F4 11.955 Tf 11.96 0 Td[(A^k2. 3.1.4ComputationalComplexitiesandMemoryRequirements FortheIAAalgorithm,itiscomputationallyintensivetocomputethecovariancematrixRinStep2andaHlR)]TJ /F9 7.97 Tf 6.59 0 Td[(1al(forl=0,1,,L)]TJ /F3 11.955 Tf 9.51 0 Td[(1)inStep3,withthecomputationalcomplexitiesbeingO(M2L)periteration.Similarly,SLIMrequiresO(M2L)opsper 57

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iterationtocomputeinStep2.Furthermore,bothIAAandSLIMneedtopre-computeandstorethesteeringmatrixAofsizeML.Boththecomputationalcomplexityandmemoryrequirementofthealgorithms,ifimplementeddirectly,wouldbeprohibitive,especiallyforhigh-dimensionalspectralestimationproblemssuchasSARimaging,whereMandLcouldbeupto105and107,respectively.WepresentbelowseveralfastimplementationsofIAAandSLIMtodramaticallyreducetheircomputationalcomplexitiesandmemoryrequirements. 3.2MissingDataSpectralAnalysisandFastImplementations Hereafter,wefocusourdiscussionona2-Dspectralestimationproblem,sincetheinterruptedSARimagingapplicationisofparticularinteresttous.The1-Dspectralestimationproblemisaspecialcaseofits2-Dcounterpart.Moreover,ourresultscanbereadilyextendedto3-orhigher-dimensionalspectralestimationproblems. 3.2.1DataModel Weconsiderthefollowing2-Dspectralestimationproblem: x(m1,m2)=L1)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xl1=0L2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xl2=0l1,l2zm1l11zm2l22+n(m1,m2),(3) where z1=ej2 L1andz2=ej2 L2.(3) In( 3 ),fx(m1,m2)g(m1=0,1,,M1)]TJ /F3 11.955 Tf 12.91 0 Td[(1,andm2=0,1,,M2)]TJ /F3 11.955 Tf 12.91 0 Td[(1)arethecomplex-valueddatasamples,fl1,l2g(l1=0,1,,L1)]TJ /F3 11.955 Tf 13.11 0 Td[(1,andl2=0,1,,L2)]TJ /F3 11.955 Tf 12.68 0 Td[(1)arethecomplexamplitudesofsignalspectrumatfrequencygridf(l1,l2)g,andfn(m1,m2)gdenotethenoise. Fornotionalconvenience,weorganizethecompletedatasamplesfx(m1,m2)gandthecomplex-valuedamplitudesfl1,l2gintoanM1M2matrixXcandanL1L2matrix 58

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B,respectively,asfollows: Xc=266666664x(0,0)x(0,1)x(0,M2)]TJ /F3 11.955 Tf 11.95 0 Td[(1)x(1,0)x(1,1)x(1,M2)]TJ /F3 11.955 Tf 11.95 0 Td[(1)............x(M1)]TJ /F3 11.955 Tf 11.95 0 Td[(1,0)x(M1)]TJ /F3 11.955 Tf 11.96 0 Td[(1,1)x(M1)]TJ /F3 11.955 Tf 11.96 0 Td[(1,M2)]TJ /F3 11.955 Tf 11.96 0 Td[(1)377777775,(3) and B=2666666640,00,10,L2)]TJ /F9 7.97 Tf 6.59 0 Td[(11,01,11,L2)]TJ /F9 7.97 Tf 6.59 0 Td[(1............L1)]TJ /F9 7.97 Tf 6.58 0 Td[(1,0L1)]TJ /F9 7.97 Tf 6.58 0 Td[(1,1L1)]TJ /F9 7.97 Tf 6.59 0 Td[(1,L2)]TJ /F9 7.97 Tf 6.58 0 Td[(1377777775.(3) Then,thecompletedatamodelin( 3 )canberewrittenas: Xc=F1BFT2+Nc,(3) whereNcisdenedsimilarlytoXcin( 3 )forthenoise,andF1andF2areM1L1andM2L2inverseFouriertransformmatrices,respectively,i.e., Fk=26666666666411111zkz2kzLk)]TJ /F9 7.97 Tf 6.59 0 Td[(1k1z2kz4kz2(Lk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)k...............1zMk)]TJ /F9 7.97 Tf 6.59 0 Td[(1kz2(Mk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)kz(Mk)]TJ /F9 7.97 Tf 6.58 0 Td[(1)(Lk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)k377777777775,fork=1,2.(3) Byapplyingthematrixvectorizationoperatortobothsidesof( 3 ),thecompletedatamodelcanberewrittenas: xc=Fvec(B)+vec(Nc),withF=F2F1andxc=vec(Xc),(3) whereFisthe2-DinverseFouriertransformmatrix,vec()denotesthevectorizationoperator(i.e.,stackingthecolumnsofamatrixontopofeachother),anddenotes 59

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theKroneckerproductofmatrices.In( 3 ),wehaveusedthefactthatvec(ABC)=(CTA)vec(B)(see,e.g.,[ 54 ]). NowweassumethatonlyMoutofM1M2samplesareavailable.LetxbeanM1columnvectorformedbythegivendatasamples,andJgbethecorrespondingM(M1M2)selectionmatrix,i.e., x=Jgxc.(3) TheselectionmatrixJghaselements0and1.Eachrowoftheselectionmatrixcontainsoneandonlyoneelementequalto1,whosepositionindicatestheselectedrow/cloumnindex.Asanexample,let Jg=264010000000010375.(3) ItcanbeeasilyveriedthatJgCisa2-rowmatrixformedbythesecondandfthrowsofa6-rowmatrixC,whileDJTgisa2-columnmatrixformedbythesecondandfthcolumnsofa6-columnmatrixD.NotealsothatJTgCexpandsa2-rowmatrixCtoa6-rowmatrixwhosesecondandfthrowsareequaltotherstandsecondrowsofC,respectively,withtheremainingrowsbeingzero.Also,DJgexpandsa2-columnmatrixDtoa6-columnmatrixsimilarly. From( 3 )and( 3 ),wegetthedatamodelforthe2-Dmissing-dataspectralestimationproblem: x=JgFvec(B)+n.(3) Comparing( 3 )with( 3 ),wecanseethattheaforementionedIAAandSLIMalgorithmsinSection 3.1 canbeappliedto( 3 )directlywiththesteeringmatrixAbeingJgF.Ifimplementeddirectly,thecomputationalcomplexityandmemoryrequirementofIAAandSLIMareO(M2L1L2)andO(ML1L2),respectively,whichareprohibitiveforpracticalapplications.WepresentbelowseveralcomputationallyefcientimplementationsofIAAandSLIMthatexploitthestructureofJgF. 60

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3.2.2FastIterativeAdaptiveApproach(Fast-IAA) WepresentbelowanefcientimplementationofIAAviafullyexploitingthestructureofthesteeringmatrixJgF.WerefertothisfastimplementationofIAAasFast-IAA. 3.2.2.1EfcientComputationoftheIAACovarianceMatrix WerstconsiderhowtocomputetheIAAcovariancematrixR(step2oftheIAAalgorithmoutline)efciently. Let^BbeanintermediateestimateofB,andP=j^Bj2bethecorrespondingpowerestimate,wherejj2denotestheelement-wisemodulus-squareoperation.Then,theIAAcovariancematrixRin( 3 )canbewrittenas: R=JgFdiag(vec(P))JgFH=JgRcJTg, (3) where Rc=Fdiag(vec(P))FH(3) representstheIAAcovariancematrixofthecompletedatasamplescorrespondingtoP. Byusingthe2-DinverseFouriertransformpropertyofF,itcanbereadilyveriedthatRcisaToeplitz-Block-Toeplitzmatrix[ 57 ],whichcanbewrittenasfollows: Rc=266666666664R0RH1RH2RH(M2)]TJ /F9 7.97 Tf 6.59 0 Td[(1)R1R0RH1RH(M2)]TJ /F9 7.97 Tf 6.59 0 Td[(2)R2R1R0RH(M2)]TJ /F9 7.97 Tf 6.59 0 Td[(3)...............RM2)]TJ /F9 7.97 Tf 6.59 0 Td[(1RM2)]TJ /F9 7.97 Tf 6.59 0 Td[(2RM2)]TJ /F9 7.97 Tf 6.59 0 Td[(3R0377777777775,(3) 61

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with Rm2=266666666664r0,m2r)]TJ /F9 7.97 Tf 6.59 0 Td[(1,m2r)]TJ /F9 7.97 Tf 6.58 0 Td[(2,m2r)]TJ /F9 7.97 Tf 6.59 0 Td[((M1)]TJ /F9 7.97 Tf 6.59 0 Td[(1),m2r1,m2r0,m2r)]TJ /F9 7.97 Tf 6.58 0 Td[(1,m2r)]TJ /F9 7.97 Tf 6.59 0 Td[((M1)]TJ /F9 7.97 Tf 6.59 0 Td[(2),m2r2,m2r1,m2r0,m2r)]TJ /F9 7.97 Tf 6.59 0 Td[((M1)]TJ /F9 7.97 Tf 6.59 0 Td[(3),m2...............rM1)]TJ /F9 7.97 Tf 6.59 0 Td[(1,m2rM1)]TJ /F9 7.97 Tf 6.58 0 Td[(2,m2rM1)]TJ /F9 7.97 Tf 6.59 0 Td[(3,m2r0,m2377777777775.(3) In( 3 ),rm1,m2(form1=)]TJ /F5 11.955 Tf 9.3 0 Td[(M1+1,)]TJ /F5 11.955 Tf 9.3 0 Td[(M1+2,,M1)]TJ /F3 11.955 Tf 11.96 0 Td[(1;andm2=0,1,,M2)]TJ /F3 11.955 Tf 11.97 0 Td[(1)isthe2-DinverseFouriertransformofP,whichcanbecomputedefcientlybyusingthe2-DinverseFFT(IFFT)technique.Furthermore,from( 3 ),bytheselectionmatrixproperty,RisasubmatrixofRcformedbytherowsandcolumnsspeciedbytheselectionmatrixJg.Byapplying2-DIFFTtoPtoobtainRcrstandthengetR,wecanreducethecomputationalcomplexityfromO(M2L1L2)toO(L1L2log(L1L2)). 3.2.2.2EfcientComputationof^B Nowweconsideranefcientmethodtocompute^BgivenR.ApplyingtheWLSmethod(i.e.,( 3 ))tothe2-Dspectralestimationdatamodelin( 3 ),aftersomestraightforwardmanipulations,wereadilyhave: [^B]l1,l2=cl1,l2 dl1,l2,forl1=0,1,,L1)]TJ /F3 11.955 Tf 11.96 0 Td[(1andl2=0,1,,L2)]TJ /F3 11.955 Tf 11.96 0 Td[(1,(3) where cl1,l2=hJgFiH.,l1+M1l2R)]TJ /F9 7.97 Tf 6.59 0 Td[(1x=[F]H.,l1+M1l2JTgR)]TJ /F9 7.97 Tf 6.59 0 Td[(1x, (3) and dl1,l2=hJgFiH.,l1+M1l2R)]TJ /F9 7.97 Tf 6.59 0 Td[(1hJgFi.,l1+M1l2=[F]H.,l1+M1l2JTgR)]TJ /F9 7.97 Tf 6.59 0 Td[(1Jg[F].,l1+M1l2, (3) 62

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with[]l1,l2and[].,ldenotingthe(l1,l2)thelementandthelthcolumnofagivenmatrix,respectively. GivenR)]TJ /F9 7.97 Tf 6.59 0 Td[(1,cl1,l2in( 3 )canbecomputedefcientlyasfollows:(1)computeR)]TJ /F9 7.97 Tf 6.58 0 Td[(1x;(2)expandtheso-obtainedM1vectortoanM1M21vectorbyusingtheselectionmatrixproperty;(3)organizethevectorintoanM1M2matrix,and(4)apply2-DFFTtotheso-obtainedM1M2matrix. Wenowconsiderhowtocomputedl1,l2in( 3 )efciently.LetQ=JTgR)]TJ /F9 7.97 Tf 6.58 0 Td[(1Jg,which,obviously,canbeobtainedfromR)]TJ /F9 7.97 Tf 6.58 0 Td[(1byinsertingsomezerorowsandcolumnsspeciedbytheselectionmatrixJTg.Forconvenience,werewriteQasanM2M2blockmatrixformedbyM1M1sub-matrices,i.e., Q=266666664Q0,0Q0,1Q0,M2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Q1,0Q1,1Q1,M2)]TJ /F9 7.97 Tf 6.59 0 Td[(1............QM1)]TJ /F9 7.97 Tf 6.58 0 Td[(1,0QM1)]TJ /F9 7.97 Tf 6.58 0 Td[(1,1QM1)]TJ /F9 7.97 Tf 6.59 0 Td[(1,M2)]TJ /F9 7.97 Tf 6.59 0 Td[(1377777775.(3) Then,dl1,l2canbeexpressedasapolynomialfunctionofzl11andzl22asfollows: dl1,l2=M1)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xm1=)]TJ /F9 7.97 Tf 6.59 0 Td[((M1)]TJ /F9 7.97 Tf 6.59 0 Td[(1)M2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xm2=)]TJ /F9 7.97 Tf 6.59 0 Td[((M2)]TJ /F9 7.97 Tf 6.59 0 Td[(1)em1,m2z)]TJ /F6 7.97 Tf 6.58 0 Td[(l1m11z)]TJ /F6 7.97 Tf 6.59 0 Td[(l2m22,(3) where em1,m2=8>>>>>>><>>>>>>>:PM1)]TJ /F9 7.97 Tf 6.59 0 Td[(1k1=m1PM2)]TJ /F9 7.97 Tf 6.58 0 Td[(1k2=m2[Qk2,k2)]TJ /F6 7.97 Tf 6.59 0 Td[(m2]k1,k1)]TJ /F6 7.97 Tf 6.58 0 Td[(m1,form10andm20,PM1)]TJ /F9 7.97 Tf 6.59 0 Td[(1+m1k1=0PM2)]TJ /F9 7.97 Tf 6.59 0 Td[(1k2=m2[Qk2,k2)]TJ /F6 7.97 Tf 6.59 0 Td[(m2]k1,k1)]TJ /F6 7.97 Tf 6.58 0 Td[(m1,form1<0andm20,PM1)]TJ /F9 7.97 Tf 6.59 0 Td[(1k1=m1PM2)]TJ /F9 7.97 Tf 6.58 0 Td[(1+m2k2=0[Qk2,k2)]TJ /F6 7.97 Tf 6.59 0 Td[(m2]k1,k1)]TJ /F6 7.97 Tf 6.58 0 Td[(m1,form10andm2<0,PM1)]TJ /F9 7.97 Tf 6.59 0 Td[(1+m1k1=0PM2)]TJ /F9 7.97 Tf 6.58 0 Td[(1+m2k2=0[Qk2,k2)]TJ /F6 7.97 Tf 6.59 0 Td[(m2]k1,k1)]TJ /F6 7.97 Tf 6.59 0 Td[(m1,form1<0andm2<0.(3) From( 3 ),dl1,l2(forl1=0,1,,L1)]TJ /F3 11.955 Tf 13.29 0 Td[(1andl2=0,1,,L2)]TJ /F3 11.955 Tf -411.8 -23.91 Td[(1)canbecomputedefcientlyviaapplying2-DFFTtothepolynomialcoefcientsfem1,m2gin( 3 ),whichreducesthecomputationalcomplexityfromO(M2L1L2)toO(L1L2log(L1L2)). 63

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TheFast-IAAalgorithmissummarizedbeow. TheFast-IAAAlgorithm: Initialization:Apply2-DFFTtoxtogetaninitialestimateofB. Iteration:Repeatthefollowingsteps: Step1-ComputeP=j^Bj2. Step2-Apply2-DIFFTtoP,andconstructthematrixRusing( 3 )and 3 ). Step3-ComputethematrixinverseofR. Step4-ComputeR)]TJ /F9 7.97 Tf 6.58 0 Td[(1x,andthenapply2-DFFTtoJTR)]TJ /F9 7.97 Tf 6.59 0 Td[(1xtogetfcl1,l2g. Step5-Computefel1,l2gfromR)]TJ /F9 7.97 Tf 6.59 0 Td[(1using( 3 ). Step6-Apply2-DFFTtofel1,l2gtocomputefdl1,l2g. Step7-Update^Bbyusing( 3 )andtheresultsofSteps4and6. NotethatthecomputationallymostintensivestepsoftheFast-IAAalgorithmarethematrixinverseofRinStep3and2-DFFT/IFFT,withcomputationalcomplexitiesO(M3)andO(L1L2log(L1L2)),respectively.Hence,theoverallcomputationalcomplexityoftheIAAalgorithmisreducedtoO(M3)+O(L1L2log(L1L2)),whichismuchsmallerthanO(M2L1L2)oftheoriginalIAAalgorithm.Weremarkthatinthecompletedatacase,Rin( 3 )isaToeplitz-Block-Toeplitzmatrix[ 57 ],whoseinversecanbecomputedefcientlybyusingtheGohberg-Semencul(GS)typeformula[ 58 59 ].WerefertothisGS-typeformulabasedFast-IAAalgorithmasFast-IAA-GS[ 60 ].ThecomputationalcomplexityofFast-IAA-GSisO(L1L2log(L1L2)).Furthermore,wenotethatFast-IAA,aswellasFast-IAA-GS,doesnotusethesteeringmatrixexplicitly.Hence,itdoesnotneedtopre-computeandstorethesteeringmatrix,whichsignicantlyreducesthememoryrequirement. 3.2.3FastSLIMUsingConjugateGradient(CG-SLIM) Next,weconsiderafastimplementationoftheSLIMalgorithmbyusingtheconjugategradient(CG)technique[ 61 ].WerefertothisalgorithmasCG-SLIM. 64

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CGisanefcientmethodtosolvealinearsystemiteratively.ThenumberofiterationsneededbyCGisusuallymuchsmallerthantheproblemsize,whichcansignicantlyreducethecomputationalcomplexity.Inthissection,weemploytheCGtechniquetocomputethemostintensivestepofSLIM,i.e.,)]TJ /F9 7.97 Tf 6.59 0 Td[(1x.Let y=)]TJ /F9 7.97 Tf 6.58 0 Td[(1x.(3) Then,theunknownvectoryin( 3 )canbecomputedefcientlybyusingtheCGapproachwhichisoutlinebelow. TheConjugateGradientAlgorithm: Initialization:Setaninitialvectory,andr0=r1=g=x)]TJ /F4 11.955 Tf 11.96 0 Td[(Ry. Iteration:Repeatthefollowingstepsuntilconvergence: Step1-Computeh=g. Step2-Compute=kr1k2 gHh. Step3-Updatey=y+g. Step4-Updater0=r1andr1=r1)]TJ /F14 11.955 Tf 11.95 0 Td[(h. Step5-Updateg=r1+kr1k2 kr0k2g. FortheCGalgorithmabove,Step1iscomputationallythemostintensive,whichinvolvesamatrix-vectorproduct.However,inthe2-Dspectralestimationproblemin( 3 ),thecomputationalcomplexityofthisstepcanbereducedbyusingthestructureofthesteeringvector.Let^BbeanintermediateestimateofB,andP=j^Bj2)]TJ /F6 7.97 Tf 6.58 0 Td[(q.ReplacingAandpin( 3 )byJgFandvec(P),respectively,wehave: =JgFdiag(vec(P))JgFH+^I=Jg[Fdiag(vec(P))F]HJTg+^I. (3) 65

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Then,Step1oftheCGalgorithmcanbewrittenas: h=JgFdiag(vec(P))JgFHg+^g=JgFhvec(P)FHJTggi+^g=F)]TJ /F9 7.97 Tf 6.59 0 Td[(12d[vec(P)F2d(g)]+^(t)g, (3) wheredenotestheHadamardproductofmatrices,andthefunctionsF)]TJ /F9 7.97 Tf 6.59 0 Td[(12d()andF2d()aredened,respectively,asfollows: F)]TJ /F9 7.97 Tf 6.58 0 Td[(12d(b)=JgFb,(3) and F2d(g)=FHJTgg.(3) Bythepropertiesofthe2-DinverseFouriertransformmatrixFandtheselectionmatrixJg,( 3 )and( 3 )canbecomputedefcientlybyusingthe2-DFFT/IFFTtechnique.FortheF)]TJ /F9 7.97 Tf 6.58 0 Td[(12d()functionin( 3 ),wecanre-organizetheL1L21vectorbintoanL1L2matrix,apply2-DIFFTtotheso-obtainedmatrix,andthenpickuptheelementsofthe2-DIFFToutputspeciedbytheselectionmatrixtoformacolumnvector.FortheF2d()functionin( 3 ),wecanrstexpandtheM1columnvectortoanM1M21vectorviainsertingzerosatthepositionsspeciedbytheselectionmatrixJTg,re-organizethevectorintoanM1M2matrix,andthenapply2-DFFTtotheso-obtainedmatrix.Byusingthese2-DFFTandIFFTtechniques,thecomputationalcomplexityofStep1oftheConjugateGradientalgorithmisreducedtoO(L1L2log(L1L2)). BycombiningtheSLIMalgorithmandtheCGalgorithm,theCG-SLIMalgorithmissummarizedbelow. TheCG-SLIMAlgorithm: Initialization:Apply2-DFFTtoxtogetaninitialestimateofB,initialize^,andy=0. Iteration:Repeatthefollowingsteps: 66

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Step1-ComputeP=j^Bj(2)]TJ /F6 7.97 Tf 6.58 0 Td[(q). Step2-InitializeCGusingr1=x)-222(F)]TJ /F9 7.97 Tf 6.59 0 Td[(12d(vec(P).F2d(y))andg=r1. Step3-Repeatthefollowingstepsuntilkr1k2 kxk2CG: Step3.a-Computeh=F)]TJ /F9 7.97 Tf 6.58 0 Td[(12d(vec(P).F2d(g)). Step3.b-Compute=kr1k2 gHh. Step3.c-Updatey=y+g. Step3.d-Updater0=r1andr1=r1)]TJ /F14 11.955 Tf 11.95 0 Td[(h. Step3.e-Updateg=r1+kr1k2 kr0k2g. Step4-Updatevec(^B)=vec(P).(F2d(y)). Step5-Update^=1 Mx)-222(F)]TJ /F9 7.97 Tf 6.58 0 Td[(12dvec(^B)2. WeterminatetheCGiterationswhentheratiooftheEuclideannormoftheresiduevectorr1totheEuclideannormofthedatavectorxbecomeslessthanapredenedthresholdCG.InthenumericalexamplesinSection 3.3 ,wechooseCG=10)]TJ /F9 7.97 Tf 6.58 0 Td[(6.WenotethatcomputationallythemostintensivestepintheCG-SLIMalgorithmisnow2-DFFT/IFFT,withacomputationalcomplexityO(L1L2log(L1L2)).Hence,theoverallcomputationalcomplexityofCG-SLIMisO(KL1L2log(L1L2))perSLIMiteration,withKbeingtheaverageCGiterationnumber,whichisusuallymuchsmallerthanM.Forinstance,intheinterruptedSARexampleinSection 3.3 ,thedatasamplenumberMisupto1600,buttheCGalgorithmusuallyconvergesinlessthanK=30iterations. 3.2.4FastSLIMUsingtheGohberg-Semencul-TypeFormula(GS-SLIM) Foracomplete-dataspectralestimationproblem,i.e.,whenJg=I,similarlytoRcin( 3 ),thematrixin( 3 )willbeaToeplitz-Block-Toeplitzmatrix[ 57 60 ].Inthiscase,thevectory=)]TJ /F9 7.97 Tf 6.58 0 Td[(1xin( 3 )canbecomputedefcientlybyusingtheGS-typeformula[ 58 59 ],whosecomplexityisaroundmin(M31M22,M21M32),whichismuchlowerthanO(M21M22L1L2)oftheoriginalSLIMalgorithm.Inspiredbythis,weintroducebelowanewfastimplementationofSLIM,referredtoasGS-SLIM,forthemissingdata 67

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spectralestimationproblem.Thisalgorithmincludestwosteps:(1)estimatemissingsamplesfromanintermediatespectralestimate,and(2)estimatethesignalspectrumfromthegivendatasamples,aswellastheestimatedmissingsamples.NotethatthesecondstepoftheGS-SLIMalgorithmisessentiallyacomplete-dataspectralestimationproblem,wheretheaforementionedGStechniquecanbeutilized. Forthe2-Dspectralestimationproblemin( 3 ),theSLIMformulationin( 3 )and( 3 )canbewrittenasfollows: (^B,^)=minB,g(B,),(3) where g(B,),Mlog+1 kx)]TJ /F4 11.955 Tf 11.95 0 Td[(JgFvec(B)k2+L1)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xl1=0L2)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xl2=02 q(jl1,l2j2)q=2)]TJ /F3 11.955 Tf 11.95 0 Td[(1.(3) Below,weemploythemaximization-minimization(MM)techniquetosolvetheoptimizationproblemiteratively. LetxmandJmbethemissingsamplesandtheassociatedselectionmatrix,respectively.Let^B(t)and^(t)betheintermediateestimatesofBandatthetthiteration,respectively.Naturally,themissingdatacanbeestimatedasfollows: ^x(t)m=JmFvec(^B(t)).(3) WeconsiderbelowanefcientmethodtoupdatetheBandestimatessuchthatthecostfunctionin( 3 )decreasesmonotonically. 68

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From( 3 ),wereadilyhave: g(B,)=Mlog+1 kx)]TJ /F4 11.955 Tf 11.95 0 Td[(JgFvec(B)k2+L1)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xl1=0L2)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xl2=02 q(jl1,l2j2)q=2)]TJ /F3 11.955 Tf 11.95 0 Td[(1 (3) Mlog+1 kx)]TJ /F4 11.955 Tf 11.95 0 Td[(JgFvec(B)k2+1 k^x(t)m)]TJ /F4 11.955 Tf 11.95 0 Td[(JmFvec(B)k2+L1)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xl1=0L2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xl2=02 q(jl1,l2j2)q=2)]TJ /F3 11.955 Tf 11.95 0 Td[(1 (3) =Mlog+1 k^x(t)c)]TJ /F4 11.955 Tf 11.95 0 Td[(Fvec(B)k2+L1)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xl1=0L2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xl2=02 q(jl1,l2j2)q=2)]TJ /F3 11.955 Tf 11.96 0 Td[(1 (3) Mlog+1 k^x(t)c)]TJ /F4 11.955 Tf 11.95 0 Td[(Fvec(B)k2+L1)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xl1=0L2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xl2=0p(t)l1,l2hjl1,l2j2)-222(j^tl1,l2j2i+L1)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xl1=0L2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xl2=02 qhj^(t)l1,l2jq)]TJ /F3 11.955 Tf 11.95 0 Td[(1i (3) ,h(B,j^B(t),^x(t)c), (3) where^x(t)cisthecompletedatacomposedbytheestimatedmissingsamplevector^x(t)m,aswellasthegivensamplevectorx.Theinequalityin( 3 )isobtainedsimilarlyto( 3 ),and p(t)l1,l2=j^(t)l1,l2j2)]TJ /F6 7.97 Tf 6.59 0 Td[(q.(3) Obviously,theequalitying(B,)h(B,j^B(t),^x(t)c)holdswhenB=^B(t).Inotherwords,thefunctiong(B,)ismajorizedbyh(B,j^B(t),^x(t)c)atthepointB=^B(t).Therefore,wecanobtaintheestimatesofBandviaminimizingh(B,j^B(t),^x(t)c)withrespecttoBand. Similarlyto( 3 ),given=^(t),minimizingh(B,j^B(t),^x(t)c)withrespecttoB,yields: vec(^B(t+1))=vec(^P(t))FH)]TJ /F9 7.97 Tf 6.59 0 Td[(1c^x(t)c,(3) where c=Fdiagvec(^P(t))FH+^(t)I.(3) 69

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Asdiscussedabove,cisaToeplitze-Block-Toeplitzmatrix[ 57 60 ]havingtheform: c=2666666666640H1H2H(M2)]TJ /F9 7.97 Tf 6.59 0 Td[(1)10H1H(M2)]TJ /F9 7.97 Tf 6.59 0 Td[(2)210(M2)]TJ /F3 11.955 Tf 11.96 0 Td[(3)H...............M2)]TJ /F9 7.97 Tf 6.59 0 Td[(1M2)]TJ /F9 7.97 Tf 6.58 0 Td[(2M2)]TJ /F9 7.97 Tf 6.59 0 Td[(30377777777775,(3) and m2=2666666666640,m2)]TJ /F9 7.97 Tf 6.59 0 Td[(1,m2)]TJ /F9 7.97 Tf 6.59 0 Td[(2,m2)]TJ /F9 7.97 Tf 6.59 0 Td[((M1)]TJ /F9 7.97 Tf 6.58 0 Td[(1),m21,m20,m2)]TJ /F9 7.97 Tf 6.59 0 Td[(1,m2)]TJ /F9 7.97 Tf 6.59 0 Td[((M1)]TJ /F9 7.97 Tf 6.58 0 Td[(2),m22,m21,m20,m2)]TJ /F9 7.97 Tf 6.59 0 Td[((M1)]TJ /F9 7.97 Tf 6.58 0 Td[(3),m2...............M1)]TJ /F9 7.97 Tf 6.59 0 Td[(1,m2M1)]TJ /F9 7.97 Tf 6.58 0 Td[(2,m2M1)]TJ /F9 7.97 Tf 6.58 0 Td[(3,m20,m2377777777775+^(m2)I,(3) where(m2)isaKroneckerdeltafunction,andfm1,m2gisthe2-DFouriertransformofPandcanbecomputedefcientlybyusing2-DFFT.Furthermore,viaexploitingtheToeplitze-Block-Toeplitzstructureofc,)]TJ /F9 7.97 Tf 6.59 0 Td[(1^xcin( 3 )canbecomputedefcientlybyusingthegeneralizedGSalgorithm[ 57 60 ]. Ontheotherhand,givenB=^B(t+1),minimizingh(B,j^B(t),^x(t)c)withrespecttoyields: ~=1 M^x(t)c)]TJ /F4 11.955 Tf 11.95 0 Td[(Fvec^B(t+1)2.(3) Equation( 3 )isanaturalestimateofgivenB=^B(t+1).However,wendempiricallythattheGS-SLIMalgorithmsuffersfromaslowconvergenceproblemusing( 3 ).Instead,weproposetoupdate^asfollows: ^(t+1)=~+(1)]TJ /F14 11.955 Tf 11.96 0 Td[()^(t)= M^x(t)c)]TJ /F4 11.955 Tf 11.96 0 Td[(Fvec^B(t+1)2+(1)]TJ /F14 11.955 Tf 11.95 0 Td[()^(t),(3) whereischoosetobethegiven-to-completedataratio,i.e.,=M M1M2,inthenumericalexamplesinSection 3.3 70

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The2-DspectrumBisestimatedviaiterating( 3 ),( 3 ),( 3 )and( 3 ).Weprovebelowthatthecostfunctiong(B,)decreasesmonotonically.First,wereadilyhavetherst-orderderivativeofh(^B(t+1),j^B(t),^x(t)c)asfollows: @h(^B(t+1),j^B(t),^x(t)c) @=M )]TJ /F8 11.955 Tf 13.15 26.19 Td[(^x(t)c)]TJ /F4 11.955 Tf 11.95 0 Td[(Fvec^B(t+1)2 2=M 2()]TJ /F3 11.955 Tf 12.57 0 Td[(~), (3) where~isdenedin( 3 ).From( 3 ),wecanseethat@h(^B(t+1),j^B(t),^x(t)m,^(t)) @<0for0<<~and@h(^B(t+1),j^B(t),^x(t)m,^(t)) @>0for>~.Inotherwords,thefunctionh(^B(t+1),j^B(t),^x(t)m,^(t))monotonicallydecreaseswithrespecttofor0<<~,andmonotonicallyincreasesfor>~.Ontheotherhand,obviouslywehave^(t)^(t+1)~when^(t)~,and~^(t+1)^(t)when(t)>~.Thatimmediatelyimplies: h(^B(t+1),^(t+1)j^B(t),^x(t)c)h(^B(t+1),^(t)j^B(t),^x(t)c).(3) Using( 3 )andthefactthath(B,)j^B(t),^x(t)c)majorizesg(B,)(see( 3 )),weimmediatelyhave: g(B(t+1),t+1)h(^B(t+1),^(t+1)j^B(t),^x(t)c)h(^B(t+1),^(t)j^B(t),^x(t)c)h(^B(t),^(t)j^B(t),^x(t)c)=g(B(t),^(t)), (3) i.e.,thecostfunctiong(B,)ismonotonicallydecreasing.TheGS-SLIMalgorithmissummarizedbelow. TheGS-SLIMAlgorithm: Initialization:Setaninitialvaluefor,let^xm=0,andapply2-DFFTtoxtogetaninitialestimateofB. Iteration:Repeatthefollowingsteps: Step1-ComputeP=j^Bj(2)]TJ /F6 7.97 Tf 6.58 0 Td[(q). 71

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Step2-Apply2-DFFTtoPtocomputecin( 3 )and( 3 ). Step3-Computey=)]TJ /F9 7.97 Tf 6.59 0 Td[(1c^xcusingthegeneralizedGSalgorithm. Step4-Updatevec(^B)=vec(P)F2d(y). Step5-Compute~xc=F)]TJ /F9 7.97 Tf 6.58 0 Td[(12d(vec(B)). Step6-Update^=~+(1)]TJ /F14 11.955 Tf 11.95 0 Td[()^with~=1 Mk^xc)]TJ /F3 11.955 Tf 11.8 0 Td[(~xck2. Step7-Updatemissingdataestimate^xm=Jm~xcandtheassociatedcompletedata^xc. 3.3NumericalExamples Wepresentbelowseveral1-Dspectralestimationand2-DinterruptedSARimagingexamplestodemonstratetheperformanceofthemissingdataspectralestimationalgorithmsandtheirfastimplementations. 3.3.11-DSpectralEstimation Todemonstratetheperformanceimprovementsofthenewmissingdataspectralanalysisapproachesoversomeexistingones,werstconsiderasimpleexampleinFigure 3-1 .Thecompletedatahas128samples,andcontains8complexsinusoids.ThefrequenciesandamplitudesofthesinusoidsareindicatedbycirclesinFigure 3-1 .Thedataiscorruptedbyazero-meancircularlysymmetriccomplexwhiteGaussiannoisewithvariance20=0.001.InFigures 3-1 (a)3-1 (e),weprovidethespectraestimatesobtainedbywindowedFFT(WFFT),APES/MAPES,CoSaMP,IAAandSLIM,respectively.ATaylorwindowwitha)]TJ /F3 11.955 Tf 9.3 0 Td[(30dBsidelobelevelisusedforWFFT.Theiterationnumbersforalliterativeapproaches,includingMAPES,CoSaMP,IAA,andSLIM,arexedto20.(Inourexamples,wedidnotndsignicantperformanceimprovementafter20iterationsforallalgorithms.)WechoosetheuserparameterofCoSaMPtobe32.ForSLIM,wechooseq=1.Thesubplotsattheleft,middleandrightcolumnsinFigure 3-1 showtheresultsobtainedfromthecompletedata,50%and30%data,respectively.Theavailabledatasamplesareselectedrandomly.ForFigure 3-1 (b),weuseAPESforthecompletedatacase,andMAPESforthetwomissingdatacases. 72

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FromFigure 3-1 (a),itisclearthatWFFTsuffersfromthelowresolutionandhighsidelobelevelproblems.Thesidelobelevelincreasessignicantlyinthemissingdatacase.Duetothehighsidelobelevels,severaltargetsfailtobedetected.InFigure 3-1 (b),APESprovidesagoodspectralestimationperformanceforthecompletedatacase.Itsmissingdatavariant,i.e.,MAPES,behaveswellwhen50%ofthedatasamplesareavailable.However,itsperformancedegradessignicantlywhentheavailabledatapercentagedecreasesto30%.Moreover,MAPESiscomputationallyveryintensive.ThecomputationtimesneededbyMAPES,whenimplementedonanordinarypersonalcomputer(IntelXeonCPUat2.67GHz;12GBRAM),forthetwomissing-dataexamplesinFigure 3-1 (b)are969and847seconds,respectively,whilethecomputationtimesneededbyalltheotheralgorithmsarelessthan1second.WecanalsoseeinFigure 3-1 (c)thatCoSaMPproducesmanyfalsepeaksinthemissingdatacases.FromFigures 3-1 (d)and 3-1 (e),wenotethatbothIAAandSLIMprovidegoodspectralestimationperformanceundervariousconditions.Allofthe8sinusoidscanbeidentiedclearlyevenwithonly30%datasamplesavailable.WeremarkthattheamplitudeestimatesofSLIMaresignicantlybiaseddownward.However,withthecorrectlyidentiedsinusoidalnumberandfrequencies,theamplitudeestimatescanberenedreadilyusing,forexample,thestandardleast-squares(LS)methodortheRELAXalgorithm[ 62 ]. Wenextconsideramorechallengingproblemofanalyzingalongdatasequencewhere800outof1000datasamplesareavailableforspectralestimation.Thedatasequencecontains100complex-valuedsinusoids,andiscorruptedbyazero-meancircularlysymmetriccomplexwhiteGaussiannoisewithvariance20=0.001.Figures 3-2 (a)3-2 (f)showthespectralestimates(zoomed-intoshowdetailswithinthe[0.21,0.24]Hzband)obtainedbyWFFT,IAA,SLIM,Fast-IAA,CG-SLIM,andGS-SLIM,respectively.Again,thedata-adaptivealgorithmsoutperformtheconventionaldata-independentWFFTapproachsignicantly.Theperformanceofthefastimplementations 73

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(a) (b) (c) (d) (e) Figure3-1. Nonparametricspectralestimationwithoutandwithmissingsamples,wherethecompletedatalengthis128.Thesubplotsintheleft,middleandrightcolumnsareobtainedwiththecompletedata,50%data,and30%data,respectively.A)WFFTB)MAPESC)CoSaMPD)IAAE)SLIM 74

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(a)(d) (b)(e) (c)(f) Figure3-2. Nonparametricspectralestimatesfordatasequenceswithmissingsampleswhen800outof1000datasamplesareavailableforspectralanalysis.A)WFFTB)IAAC)Fast-IAAD)SLIME)CG-SLIMF)GS-SLIM ofIAAandSLIMareveryclosetothoseoftheiroriginalalgorithms.FromFigure 3-2 ,weseeagainthattheamplitudeestimatesofSLIManditsvariantsarebiaseddownward,whileIAAanditsvariantprovideratheraccurateamplitudeestimates. ThemissingdatasamplescanberecoveredbysimplyapplyingFFT(seeEquation( 3 ))tothespectralestimatesofSLIMoritsfastimplementations.Themissing-dataIAA(MIAA)algorithmin[ 63 ]canalsobeusedtorecoverthemissingsamplesfromtheIAAorFast-IAAspectraestimates,which,however,ismorecomplicatedbothconceptuallyandcomputationallythanitsSLIMcounterpart.Figures 3-3 (a)and 3-3 (b) 75

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(a) (b) Figure3-3. MissingsamplesestimationbySLIM,thesubplotsintheleftandrightcolumnsrepresenttherealandimaginarypartsofthedatasamples,respectively.A)CG-SLIMB)GS-SLIM showthemissingsamplesestimatedbyCG-SLIMandGS-SLIM,respectively.Aswecansee,themissingsamplescanbeestimatedquiteaccurately.TomitigatethebiasedamplitudeestimationproblemofCG-SLIMandGS-SLIM,wecanapplytheefcientcomplete-dataIAAalgorithmdiscussedinSection 3.2.2 ,i.e.,Fast-IAA-GS,totheestimatedmissingsamplesobtainedusingtheSLIMvariants,aswellasthegivensamples.Werefertothesetwonewspectralestimationprocedures(i.e.,CG-SLIMfollowedbyFast-IAA-GS,andGS-SLIMfollowedbyFast-IAA-GS)asCG-SLIM-IAAandGS-SLIM-IAA,respectively.NotethatduetotheusageoftheGS-typeformula,Fast-IAA-GSiscomputationallyquiteefcient.Furthermore,theFast-IAA-GSstepcanbeinitializedbyusingtheCG-SLIMorGS-SLIMspectralestimatetoreducetheneedednumberofiterations.Figures 3-4 (a)and 3-4 (b)showthespectralestimatesobtainedbyCG-SLIM-IAAandGS-SLIM-IAA,respectively.TheFast-IAA-GSiterationnumberissettobe5forbothalgorithms.FromFigures 3-4 (a)and 3-4 (b),wecanseethatCG-SLIM-IAAandGS-SLIM-IAAprovidemoreaccurateamplitudeestimatesthanCG-SLIMandGS-SLIM.Inparticular,comparingFigures 3-2 (c)and 3-4 (a),wesee 76

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thatCG-SLIM-IAAprovidesasimilarspectralestimateasFast-IAA.However,aswewillshowbelow,CG-SLIM-IAAiscomputationallymuchmoreefcientthanFast-IAA. (a)(d) Figure3-4. Nonparametricspectralestimatesfordatasequenceswithmissingsampleswhen800outof1000datasamplesareavailableforspectralanalysis.A)CG-SLIM-IAAB)GS-SLIM-IAA Thecomputationtimesneededbythe1-DspectralestimationmethodstoobtainFigures 3-2 and 3-4 areshowninTable 3-1 .Clearly,theproposedfastimplementationsofIAAandSLIMarecomputationallymuchmoreefcientthantheiroriginalcounterparts.Inparticular,theproposedCG-SLIMandGS-SLIMalgorithmsreducethecomputationtimesneededbySLIMfrom44.34secondsto0.4and1.4seconds,respectively.ThecomputationalefcienciesofCG-SLIM-IAAandGS-SLIM-IAAareclosetothoseofCG-SLIMandGS-SLIM,respectively.Inparticular,CG-SLIM-IAAneedsonly0.78seconds,whichamounttoonly9.3%ofthecomputationtimeneededbyFast-IAA(8.39seconds,whileprovidingasimilarspectralestimationperformancetoFast-IAA.Althoughforthis1-Dspectralestimationexample,CG-SLIMiscomputationallymoreefcientthanGS-SLIM,aswewillshowinSection 3.3.2 ,GS-SLIMcanbecomputationallymoreefcientthanCG-SLIMfor2-DinterruptedSARimaging. Table3-1. ComputationtimesneededbyIAA,SLIMandtheirfastimplementations. Alg.WFFTIAASLIM Fast-IAACG-SLIMGS-SLIMCG-SLIM-IAAGS-SLIM-IAA Time(s)0.0484.9844.34 8.390.41.40.781.75 77

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3.3.22-DInterruptedSARImaging Inthefollowingexamples,weillustratetheapplicationsofthemissing-dataspectralanalysisalgorithmstointerruptedSARimagingusingincompletephasehistorydata. (a)(b) Figure3-5. SlicyobjectandbenchmarkSARimageA)Photographoftheobject(takenat45azimuthangle).B)benchmarkSARimageformedwitha288288(not4040)datamatrix. Weusethetwo-dimensionalphase-historydataofanobjectcalledSlicygeneratedat0azimuthangleusingXPATCH[ 64 ],ahighfrequencyelectromagneticscatteringpredictioncodeforcomplex3-Dobjects.AphotooftheSlicyobjecttakenat45azimuthangleandaSARimagebenchmarkobtainedfromacomplete288288datamatrixareshowninFigures 3-5 (a)and 3-5 (b),respectively.Below,weuseonlya4040centerblockofthephase-historydataforSARimaging. Figures 3-6 3-8 showtheSARimagesobtainedfromthe4040completedata,andtworandomlyinterrupteddata.TheavailabledatapatternsforthetwointerruptioncasesareshowninFigures 3-7 (a)and 3-8 (a),wheretheavailabledataratiosare68%and30%,respectively.WeprovidetheSARimagesobtainedbyIAA,SLIM,andtheirfastvariants,i.e.,Fast-IAA/Fast-IAA-GS,CG-SLIMandGS-SLIM,aswellasWFFTandCoSaMP.NotethatweuseFast-IAA-GSforthecompletedatacaseinFigure 3-6 ,andFast-IAAforthetwomissingdatacasesinFigures 3-7 and 3-8 .Again,theiterationnumbersforalliterativeapproaches,includingCoSaMP,IAA,andSLIM,areallxedto 78

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10.WesettheCoSaMPuserparametertobe3000,theSLIMparameterq=1,andCG=10)]TJ /F9 7.97 Tf 6.59 0 Td[(6forCG-SLIM.ThenoisevarianceestimateofSLIMisinitializedusing( 3 ). Figures 3-6 (b), 3-7 (b)and 3-8 (b)showtheSARimagesformedbyWFFT,wherea2-DTaylorwindowwitha)]TJ /F3 11.955 Tf 9.3 0 Td[(30dBsidelobelevelisapplied.Again,WFFThaslowresolutionandhighsidelobeproblems,especiallyfortheinterruptedscenarios.Theobjectfeaturesaresmearedundertheinterrupteddataconditions.WecanalsoseethattheCoSaMPapproachfailstoworkproperlyunderallconditions.IAA,SLIM,andtheirfastvariants,i.e.,Fast-IAA,CG-SLIMandGS-SLIM,signicantlyimprovetheinterruptedSARimagingquality,intermsofbothhigherimagingresolutionandlowersidelobelevels,asevidencedbysharperSARimageswithlessghostartifacts.Inthisexample,SLIManditsvariantsprovidethebestimagingperformance,ascomparedtothebenchmarkinFigure 3-5 forallinterruptionconditions.Notethatonly0.56%ofthedatasampleswasusedtoformtheSARimagesinFigure 3-8 ,comparedtotheentire288288datasamplesinFigure 3-5 Table 3-2 providesthecomputationtimesneededbytheaforementionedalgorithmsundervariousinterruptionconditions.Again,theproposedfastimplementationapproacheshavesignicantlyenhancedcomputationalefciencies.Inparticular,fortheinterruptionexampleinFigure 3-7 ,GS-SLIMneedsonly2.4seconds,whichamounttoonly0.4%ofthecomputationtimeneededbySLIM(584.1seconds). Table3-2. ComputationtimesneededbyIAA,SLIMandtheirfastimplementationsforinterruptedSARimagingundervariousinterruptionconditions. Alg.WFFTCoSaMPIAASLIMFast-IAA-GSCG-SLIMGS-SLIM Fig. 3-6 (s)0.01173.01193392602.637.081.87Fig. 3-7 (s)0.32192.5111958412.906.502.40Fig. 3-8 (s)0.03163.72401313.804.101.80 Wehaveconsideredusingdata-adaptivemethods,namelyIAAandSLIM,forspectralestimationofdatasequenceswithmissingsamples.Bothmethodscanprovidesignicantlyimprovedspectralestimates,intermsofspectralresolutionandlower 79

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(a)(e) (b)(f) (c)(g) (d)(h) Figure3-6. ModulusoftheSARimagesoftheSlicyobjectobtainedfroma4040completedatamatrix.A)Availabledatapattern.B)WFFTC)IAAD)Fast-IAA-GSE)CoSaMPF)SLIMG)CG-SLIMH)GS-SLIM 80

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(a)(e) (b)(f) (c)(g) (d)(h) Figure3-7. ModulusoftheinterruptedSARimagesoftheSlicyobjectunderarandominterruptioncondition(68%dataavailablecomparedtoFigure 3-6 ).A)Availabledatapattern.B)WFFTC)IAAD)Fast-IAAE)CoSaMPF)SLIMG)CG-SLIMH)GS-SLIM 81

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(a)(e) (b)(f) (c)(g) (d)(h) Figure3-8. ModulusoftheinterruptedSARimagesoftheSlicyobjectunderarandominterruptioncondition(30%dataavailablecomparedtoFigure 3-6 ).A)Availabledatapattern.B)WFFTC)IAAD)Fast-IAAE)CoSaMPF)SLIMG)CG-SLIMH)GS-SLIM 82

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sidelobelevels,ascomparedtotheconventionaldata-independentapproaches.Wehaveintroducedseveralfastimplementationsofthesealgorithms,namelyFast-IAA,CG-SLIMandGS-SLIM,byexploitingthestructuresofthesteeringmatricesandmaximizingtheutilityofFFTs.Viaseveralnumericalexamplesofboth1-Dspectralanalysisand2-DinterruptedSARimaging,wehavedemonstratetheeffectivenessofIAAandSLIMforspectralestimationandthecomputationalefcienciesoftheirfastimplementations. 83

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CHAPTER4SARGROUNDMOVINGTARGETINDICATION Syntheticapertureradar(SAR)imaginghasbeenwidelyusedinmanycivilianandmilitaryapplications[ 12 65 ].SinceSARwasoriginallydevelopedforimagingthestationaryscene,amovingtargetisusuallymisplacedduetothemotion-inducedDoppler-shiftintheSARimage[ 66 69 ].ThegoalofSAR-basedgroundmovingtargetindication(GMTI)istwofold:detectionofmovingtargetsintheSARimages,andparameterestimation,includingvelocityandlocationdeterminationofthemovingtargets.Thisproblemisratherchallengingwhenthemovingtargetsareslowandsmallandareburiedinstrongstationarygroundclutter. GMTItechniqueshavebeendevelopedformostoftherecenthistoryofradarsystems.GMTIcanbeperformedincoherentlywithasinglechannelSAR[ 70 71 ],orcoherentlywithatwo-ormulti-channelSAR[ 72 75 ].Techniquessuchasdisplacedphasecenterantenna(DPCA)andalong-trackinterferometry(ATI)havebeenwidelyusedinthesurveillancecommunity[ 74 76 78 ].Thetraditionaldata-independentDPCAandATIinvestigationsaremostlylimitedtotwo-channelSAR.Multi-channelDPCAandATIhavebeenpresentedrecently[ 79 81 ].However,theymayfailtoworkproperlyforchallengingGMTIproblems,suchasforsmallmovingtargetdetectioninthepresenceofstrongstationaryclutter.Withtheavailabilityofincreasedcomputationalpower,adaptiveSARbasedGMTIcanmakeupfortheshortcomingsofconventionalSARbasedGMTImethods.Viaadaptivelyformingangle-Dopplerimages,movingtargetscanbedetectedeffectivelyfromgroundclutterviaspace-timeadaptiveprocessing(STAP)[ 82 83 ].STAPcansignicantlyimprovethelow-velocityandsmalltargetdetectioncapabilityinthepresenceofstrongclutterandjamminginterferences.However,itiswell-knownthatconventionalSTAPhasahighcomputationalrequirement[ 82 83 ].Moreover,itneedsalargeamountoftarget-freehomogeneoussecondarydata,whichishardlyavailableinpractice[ 82 83 ].Anl1-normbasedmethod,whichbelongstothecategory 84

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ofcompressivesensing(CS)algorithms,isusedforSTAPin[ 84 ].However,theCSalgorithmsusuallyrequireuser-parameters,whicharenoteasytotune,andhaveratherhighcomputationalcost. Inthischapter,newmovingtargetindicationtechniquesformulti-channelairborneSARarepresented.Viaadaptivelyformingthevelocityversuscross-rangeimagesforeachrangebinofinterest,smallmovingtargetscanbedetectedeffectivelyeveninthepresenceofstrongstationarygroundclutter.Aniterativeadaptiveapproach(IAA)[ 25 55 ]isusedtoformvelocityversuscross-rangeimagesforeachrangebinofinterest.ThisweightedleastsquaresbasedIAAalgorithmisrobustanduserparameterfree.IAAcanprovidebetterperformancethanDPCAandATI,whilehavingmuchlowercomputationalrequirementthanconventionalSTAPandobviatingtheneedofhomogeneoustarget-freesecondarydata.TheperformancesoftheproposedalgorithmsaredemonstratedusingtheAirForceResearchLaboratory(AFRL)GotchaairborneSARbasedGMTIdataset[ 85 ]. Theremainderofthischapterisorganizedasfollows.InSection 4.1 ,weintroducetheSARimaginggeometryandthedatamodelofthisSARbasedGMTIproblem.WethendiscusstheneedforarraycalibrationsinSection 4.2 .Wepresentourapproachtobothcalibratetheantennagainsanddeterminethearrayinter-elementspacings.Stationarycluttercancelation,targetdetectionandvelocityestimationalgorithmsarepresentedinSection 4.3 .InSection 4.4 ,wediscusstheAFRLGotchaGMTIdatasetandpresentseveralexamplestodemonstratetheperformanceoftheproposedalgorithms.Finally,concludingremarksaregiven. 4.1GeometryandDataModel Weconsiderthe3-DgeometryofanairborneSARsystemasshowninFigure 4-1 .Theairplanemoveswithavelocityvpalongtheydirection.ThereareMreceivechannelsplacedalongtheightpath,andthedistancebetweenthemthantennaandtherstantennaisdenotedasLm.HenceL1=0.Assumethatamovingtargetinthekth 85

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Figure4-1. Geometryofanairboremulti-channelSARsystem. rangebinwithradialvelocityvtisattheanglerelativetothenormalofthesyntheticaperture,asshowninFigure 4-1 .TheanglebetweentheradialvelocityofthetargetandtheslantrangelineBCinFigure 4-1 isdenotedas.Then,afterrangecompression,thereceiveddataduetothemovingtargetforthemthantennaandthenthprobingpulsecanbemodeledas:ym(k,n)=gmbej2h2vp fPRFsin()+2vt fPRFcos()inej2Lm sin(), (4)m=1,2,...,M,k=1,2,...,K,n=1,2,...,N, (4) wheregmistheantennagainofthemthantennarelativetotherstantenna(andhenceg1=1),bisthecomplex-valuedresponsefromthemovingtarget,isthewavelengthoftheradar,andfPRFisthepulserepetitionfrequency(PRF).In( 4 ),ej22vp fPRFnsin()denotesthephasedelayduetothemotionoftheplatform,ej22vt fPRFncos()denotesthephasedelayduetothemotionofthemovingtarget,andej2Lm sin()denotesthephasedelayduetotheantennadistanceLm.Byletting fd=2vp fPRF,(4) 86

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fv=2vt fPRFcos(),(4) and lm=Lm ,(4) wehave ym(k,n)=gmbej2[fdsin()+fv]nej2lmsin().(4) Let ft=fdsin()+fv(4) denotethetotalDopplercausedbybothplatformandtargetmotions,andlet m=lm fd.(4) Thenwecanrewriteym(k,n)as ym(k,n)=gmbej2ft(n+m)e)]TJ /F6 7.97 Tf 6.59 0 Td[(j2fvm.(4) Notethatwehavefv=0forthestationarygroundclutter,mcanbeuniformlyornon-uniformlyspaceddependingonthearraygeometry,and1=L1=fd=0. Foreachm,applyingthediscrete-timeFouriertransformalongnto( 4 ),wegettheSARimageforthemthantenna: zm(k,s)=gmbe)]TJ /F6 7.97 Tf 6.59 0 Td[(j2fvmej2fsm(fs)]TJ /F5 11.955 Tf 11.96 0 Td[(ft),s=1,2,...,N,(4) wherefs=(s)]TJ /F3 11.955 Tf 12.61 0 Td[(1)=NisthenormalizedDopplerfrequencyforthesthDopplerbin(orcross-rangebin),and (fs)]TJ /F5 11.955 Tf 11.95 0 Td[(ft)=sin[N(fs)]TJ /F5 11.955 Tf 11.96 0 Td[(ft)] sin[(fs)]TJ /F5 11.955 Tf 11.95 0 Td[(ft)]e)]TJ /F6 7.97 Tf 6.58 0 Td[(j(N)]TJ /F9 7.97 Tf 6.59 0 Td[(1)(fs)]TJ /F6 7.97 Tf 6.59 0 Td[(ft).(4) 87

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Whenthedistancesandrelativegainsamongtheantennasareknown,thegainsandthesecondphaseshiftterminEquation( 4 )canbecompensatedoutasfollows: ~zm(k,s)=e)]TJ /F6 7.97 Tf 6.59 0 Td[(j2fsm gmzm(k,s)=be)]TJ /F6 7.97 Tf 6.58 0 Td[(j2fvm(fs)]TJ /F5 11.955 Tf 11.96 0 Td[(ft).(4) Becausefv=0forthestationarygroundclutter,thedatamodelinthepresenceofthestationarygroundclutterandnoisecanbewrittenas: ~zm(k,s)=be)]TJ /F6 7.97 Tf 6.59 0 Td[(j2fvm(fs)]TJ /F5 11.955 Tf 11.96 0 Td[(ft)+Xs0cs0(fs)]TJ /F5 11.955 Tf 11.95 0 Td[(fcs0)+em(fs),(4) wherefcs0istheDopplershiftofthestationaryorgroundclutter,cs0isthecomplex-valuedresponseoftheclutterinthes0thDopplerbin,andem(f)isthenoiseterm.Stackingf~zm(k,s)gMm=1intoavector~z(k,s),wehave: ~z(k,s)=a(fv)b(fs)]TJ /F5 11.955 Tf 11.95 0 Td[(ft)+a(0)Xs0cs0(fs)]TJ /F5 11.955 Tf 11.96 0 Td[(fcs0)+e(fs),(4) where ~z(k,s)=[~z1(k,s)~z2(k,s)~zM(k,s)]T,(4) and a(fv)=[1e)]TJ /F6 7.97 Tf 6.59 0 Td[(j2fv2e)]TJ /F6 7.97 Tf 6.58 0 Td[(j2fvM]T,(4) isthesteeringvector. Ourproblemofinterestistodetectthemovingtargetandestimateitsvelocityfrom~z(k,s). 4.2ArrayCalibration Forapplicationsthatrelyonaccuratephasehistories,arraycalibrationiscrucial[ 86 87 ].Forourdatamodel,thismeanshavingcorrectvaluesfortheantennainter-elementspacing,Lm,andtheindividualantennagain,gm.Evenwithpriorknowledgeofthearraygeometryandtheantennaspecications,arraycalibrationisstillnecessary.Depending 88

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onthephysicalarraycongurationsandplatformgeometry,in-situantennaperformanceisneverthesameasitsfree-spacecounterparts. 4.2.1DistancesAmongAntennas Theantennainter-elementspacingmaybeunknowninpractice.WeconsiderestimatingthespacingsusingtheRank-1method[ 86 ]inthissubsection.ConsidertheestimationofthedistancebetweenAntennas1and2.ForagivenDopplerbins,wehave(intheabsenceofnoise):z1,2(k,s)=264z1(k,s)z2(k,s)375=2641g2ejs)]TJ /F11 5.978 Tf 5.75 0 Td[(1 NL2 D375z1(k,s),k=1,,K, (4) whereD=vp=fPRFisthedistancetheradartraveledwithinapulserepetitioninterval(PRI).Inthederivationof( 4 ),wedroppedthetermej2fvmin( 4 )byassumingfv=0,sincethetargetpowerisnegligiblecomparedtothestationarygroundclutterpower.ForeachDopplerbins,theideal(noiseless)covariancematrixforz1,2(k,s)is R1,2(s)=Efz1,2(k,s)zH1,2(k,s)g=p(s)a1,2(s)aH1,2(s),s=1,,N, (4) wherep(s)=PKk=1jz1(k,s)j2isthepowerofz1(k,s)(k=1,,K),and a1,2(s)=h1g2ejs)]TJ /F11 5.978 Tf 5.75 0 Td[(1 NL2 DiT(4) isthesteeringvector.Notethat,a1,2(s)aH1,2(s)isa22Rank-1matrix.NotealsothatthesteeringvectorcontainsthevariableL2.Thesteeringvectorcanbeestimatedfromtheeigen-decompositionofthecovariancematrixR1,2(s),whichisusuallyreplacedbythesamplecovariancematrix[ 88 ]inpractice.Thentheantennaseparationcanbedeterminedfromtheestimatedsteeringvector.Moredetailscanbefoundinreferences 89

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[ 86 ].Thisprocesscanberepeatedforeachantennaelementtodeterminealloftheinter-elementdistancesofthearray. 4.2.2AntennaGains Toestimatethegainsoftheindividualantennaelement(relativetotherstantenna),wefollowasimilarregime.Withtheestimatedantennadistances,thephasedelayduetothisdistancecanbecompensatedout.WeassumeaconstantgainacrossallDopplerbins,butavaryinggainasafunctionofrangebin.WefoundthatthisassumptionhelpsimprovetheGMTIperformanceforourapplications.Foraxedrangebink(k=1,,K),weconsiderthefollowingdatamodel(intheabsenceofnoise): z(k,s)=266666664z1(k,s)z2(k,s)e)]TJ /F6 7.97 Tf 6.59 0 Td[(js)]TJ /F11 5.978 Tf 5.75 0 Td[(1 NL2 D...zM(k,s)e)]TJ /F6 7.97 Tf 6.58 0 Td[(js)]TJ /F11 5.978 Tf 5.76 0 Td[(1 NLM D377777775=2666666641g2(k)...gM(k)377777775z1(k,s),s=1,,N, (4) whereMisthenumberofarrayelements.Theideal(noiseless)covariancematrixis R(k)=Efz(k,s)zH(k,s)g=p(k)a(k)aH(k),k=1,,K, (4) wherep(k)=PNs=1jz1(k,s)j2isthepowerofz1(k,s)(s=1,,N),and a(k)=[1g2(k)gM(k)]T(4) 90

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isthesteeringvectoratthekthrangebin.Notethat,a(k)aH(k)isalsoaRank-1matrix.Again,thesamplecovariancematrix^R(k)isusedtoreplacethecovariancematrixR(k)inpractice,where ^R(k)=1 NNXs=1z(k,s)zH(k,s),k=1,,K.(4) Let ^R(k)=MXi=1i(k)ei(k)eHi(k)(4) denotetheeigen-decompositionof^R(k),andlet e1(k)=[e11(k),e12(k),,e1M(k)]T(4) betheeigenvectorcorrespondingtothelargesteigenvalue.Thentheestimationofthesteeringvectora(k)canbewrittenas ^a(k)=1e12(k) e11(k)e1M(k) e11(k)T,(4) andthemthantennagainattherangebinkisestimatedas: ^gm(k)=e1m(k) e11(k),m=2,,M.(4) 4.3GroundMovingTargetIndication(GMTI) Afterantennacalibration,ourgoalistodetectthemovingtargetandtoestimatefv(theDopplershiftinducedbythemotionofthemovingtarget)andft(thetotalDopplershiftwhichincludesboththelocationinformationinthecross-rangedimensionandthevelocity)ofthemovingtarget.Oncewehavefvandft,thetruelocationfdofthemovingtargetintheDopplerdimensioncanbecalculatedbyusingEquation( 4 ).WeestimatethevalueofbatallpossiblefvvaluesforeachpixelintheSARimagedomain.Thepeakvaluesandthecorrespondingpeaklocationsoftheseimagescanbeusedtoindicate 91

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thepresenceofthemovingtargetandtheassociatedtargetparameterestimates,^fvand^ft,offvandft. 4.3.1GroundClutterCancelationUsingRELAX Thedetectionofweakmovingtargetsinurbanenvironmentscanbechallengingduetothestrongresponseofthestationarygroundcluttersuchasbuildings,trees,andotherreectivestructures.ThisproblemisexacerbatedbytheDopplerinduceddisplacementofthemovingtargetsontoexistingstrongstationarygroundreturn,suchasreturnfromabuilding.Theremovalofstationarygroundclutterwouldthenimprovetheprobabilityofcorrectlyseparatingthemovingtargetsfromthebackground. WeconsiderthefollowingstationaryclutterrejectionapproachtoaddressthischallengingproblembyusingtheRELAXalgorithm[ 62 89 ].Bytakingthestationarygroundclutterandamovingtargetintoconsideration,thedatamodelinEquation( 4 )canberewrittenas: ~z=a(fv)~b+a(0)~c+~e,(4) where~z=~z(k,s),~b=b(fs)]TJ /F5 11.955 Tf 12.79 0 Td[(ft)and~c=Ps0cs0(fs)]TJ /F5 11.955 Tf 12.79 0 Td[(fcs0)are,respectively,thecomplex-valuedresponsesofthemovingtargetsandstationaryclutteratthegivennormalizedDopplerfrequencyf,and~e=e(fs)isthenoiseterm.Fornotationalconvenience,wehaveomittedthedependenceof~y,~b,~c,and~eonkands. Byapplyingthenonlinearleast-squares(NLS)criterion,~caswellasthe~bandfvcanbeestimatedbyminimizingthefollowingcostfunction: F[~c,~b,fv]=~z)]TJ /F4 11.955 Tf 11.95 0 Td[(a(fv)~b)]TJ /F4 11.955 Tf 11.95 0 Td[(a(0)~c2,(4) wherekkdenotestheFrobeniusnorm.TheoptimizationproblemcanbesolvedbyusingtheRELAXalgorithm[ 62 89 ]asfollows. Foragivenestimated^~c(themeanvalueof~zcanbeusedastheinitialvalueof^~cinpractice),let ~zb=~z)]TJ /F4 11.955 Tf 11.95 0 Td[(a(0)^~c.(4) 92

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Then,thecostfunctionof( 4 )canbesimpliedas Fb[~b,fv]=~zHb)]TJ /F4 11.955 Tf 11.95 0 Td[(a(fv)~b2.(4) Minimizing( 4 )withrespectto~bandfvyields: ^~b=aH(fv)~zb Mfv=^fv,(4) whereMistheantennanumber,and ^fv=argminfvI)]TJ /F4 11.955 Tf 13.15 8.09 Td[(a(fv)aH(fv) N~zb2=argmaxfvjaH(fv)~zbj2. (4) Thecostfunctionin( 4 )canbecomputedbyapplyingDTFT(discrete-timeFouriertransform)to~zb.Ontheotherhand,giventheestimated^fvand^~b,~ccanbere-estimatedbyminimizingthecostfunction Fc[~c]=k~yc)]TJ /F4 11.955 Tf 11.96 0 Td[(a(0)~ck2,(4) where ~zc=~z)]TJ /F4 11.955 Tf 11.96 0 Td[(a(^fv)^~b.(4) Solvingthisoptimizationproblemyields: ^~c=aH(0)~zc M.(4) Therefore,theoriginalNLSproblemcanbesolvedviaiterating( 4 ),( 4 )and( 4 )untilconvergence.ThestepsoftheRELAXalgorithmareillustratedbelow. TheRELAXAlgorithm: Initialize^~c=aH(0)~z M Iterate: Step1: ~zb=~z)]TJ /F4 11.955 Tf 11.96 0 Td[(a(0)^~c 93

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^fv=argmaxfvjaH(fv)~zbj2 ^~b=aH(fv)~zb Mfv=^fv Step2: ~zc=~z)]TJ /F4 11.955 Tf 11.95 0 Td[(a(^fv)^~b ^~c=aH(0)~zc M untilconvergenceoracertainnumberofiterationsisreached After^~cisestimated,thestationarycluttercanberemovedusingEquation( 4 ). InthisRELAXalgorithm,wehavexedthesteeringvectorofthestationaryclutterasa(0).Inpractice,becauseofvariouserrorsandperturbations,thetruesteeringvectorofthegroundcluttermaynotbeexactlya(0).RELAXcanstillbeusedtoremoveasignicantportionofthegroundclutteraslongasitssteeringvectorisclosetoa(0).AfterusingRELAXtoremoveamajorportionofthestationarygroundclutter,weusetherobustnon-parametricIAAalgorithmformovingtargetdetectionandtargetvelocityestimation. 4.3.2MovingTargetDetectionUsingIAA SincewehaveusedRELAXtoestimatetheintensityofthestationaryclutterreturn,wecansubtractitoutfrom~z,asshowninEquation( 4 ),toget ~zb=a(fv)~b+e,(4) whereeincludestheresidualofthegroundclutterandnoise. Thegoalnowistodetectmovingtargets.TherecentlyproposedIAAalgorithm[ 25 55 ]isconsideredhereintomakeanestimateof~b=[~b1,,~bN]T,where~bi=~b(fvi)(i=1,,N)aretheresponsesatdifferentpotentialvelocities,andfvi(i=1,,N)arescanningpointsoffv.IAAhasalsobeenpresentedinChapter2,interestedreadersshouldfollowthereferenceforadetaileddevelopmentofIAA.Let A=[a1,,aN](4) 94

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denotethesteeringmatrix,whereai=a(fvi),wehavethefollowingcompactdatamodel: ~zb=A~b+e.(4) IAAsolvesfor~bbyusingaweightedleastsquaresapproach,whichinvolvestheminimizationofthefollowingcostfunction, min~bik~zb)]TJ /F3 11.955 Tf 12.05 2.65 Td[(~biaik2~Q)]TJ /F11 5.978 Tf 5.76 0 Td[(1i,i=1,,N,(4) wherekuk2~Q)]TJ /F11 5.978 Tf 5.76 0 Td[(1,uH~Q)]TJ /F9 7.97 Tf 6.58 0 Td[(1u.InEquation( 5 ),theinterferenceandnoisecovariancematrix~Qiisgivenby: ~Qi=~R)-221(j~bij2aiaHi,(4) where ~R=NXi=1j~bij2aiaHi(4) istheIAAcovariancematrix.Byletting~PdenoteanNNdiagonalmatrixwhosediagonalcontainsthesignalpoweras~pi=j~bij2,wecanrewriteEquation( 5 )inthefollowingmatrixform: ~R=A~PAH.(4) TheminimizationofEquation( 5 )withrespectto~bigives: ~bi=aHi~Q)]TJ /F9 7.97 Tf 6.59 0 Td[(1i~zb aHi~Q)]TJ /F9 7.97 Tf 6.58 0 Td[(1iai.(4) Usingthedenitionof~Qiandthematrixinversionlemma,Equation( 5 )canbewrittenas[ 25 ]: ~bi=aHi~R)]TJ /F9 7.97 Tf 6.59 0 Td[(1~zb aHi~R)]TJ /F9 7.97 Tf 6.58 0 Td[(1ai,(4) whichavoidsthecomputationof~Q)]TJ /F9 7.97 Tf 6.59 0 Td[(1iforeachi=1,,N,andsignicantlysavesthecomputationtime.Sincetheaprioriknowledgeof~PisrequiredinEquation( 5 )tocalculatethecovariancematrix~R,IAAneedstobeimplementediteratively.Usually,IAAisinitializedwiththeperiodogram. 95

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Once~bisestimatedbyIAA,themovingtargetwillbedetectedbasedonthepeakvalueof^~b,theestimateof~b.Theelements,whosefvarenearzero,inside^~bwillberemovedfrom^~btofurtherreducetheeffectoftheresiduestationarygroundclutter.Thenthepeakvalueof^~biscomparedwithathreshold,andamovingtargetwillbeindicatedifthepeakvalueisgreaterthanthethreshold.Moreover,the^fvcorrespondingtothepeakvalueof^~bcanbeusedtoestimatetheradialvelocityofthetarget.Theradialvelocityofthemovingtargetcanbecalculatedfrom( 4 )as: vt=fPRF 2cos()^fv(meters=second).(4) Notethatcanbedeterminedfromtheightinformationoftheplatformwhencollectingthedata. TheowchartoftheentireprocessingchainfortheproposedadaptiveSARbasedGMTIisgiveninFigure 4-2 Figure4-2. TheowchartoftheentireprocessingchainfortheproposedadaptiveSARbasedGMTI. 4.4AnalysisoftheAFRLGOTCHADataSet 4.4.1DescriptionoftheAFRLGotchaGMTIDataSet Todemonstratetheeffectivenessofourapproach,weapplyittotheAFRLGotchaopendataset[ 85 ].ThisdatasetisanairborneX-bandSARdatacollectedinanurbanenvironment.Threephase-historysets,allwithHHpolarizationcorrespondingtothreedifferentchannels,arecontainedinthisdataset.Thecarrierfrequencyusedinthis 96

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systemis9.6GHz,theprobingsignalisalinearfrequencymodulated(LFM)signalwithabandwidthof640MHz,andthePRFis2171.6Hz. (a) (b) Figure4-3. AFRLGotchadatascenesetup.A)3Dview.B)2Dview. Thedatacollectiondurationisabout71seconds.Therearemultiplevehiclesmovingontheroadsnearbuildingsinthesceneunderinterrogation.Thegroundtruthofoneofthemovingvehiclesisprovidedinthedataset.Thelocationsoftheradarplatformandthevehicleofinterest,whichisreferredtoasthetargetvehicle,aregivenintheauxiliarydataoftheAFRLdataset.Thetracksoftheradarplatformandthetargetvehicleduringthe71-seconddurationareplottedinFigure 4-3 .Thevehicleismovingawayfromtheradarplatform,whichinducesanegativeDoppler-shift.Thereareatotalof5400originalrangebinsinthedataset,butonly384rangebins,range-gated 97

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tocenteraroundthetargetvehicle,areprovidedinthedataset.Thediagramoftherange-gatingisshowninFigure2ofthereference[ 85 ]. 4.4.2SARImaging WebeginthedataprocessingbyformingtheSARimages.Duetothetruncationofthedatasetfordistributionpurposes,the384rangebinsgivenbythisdatasetarecenteredaroundthemovingtargetvehicle.Theserangebinsarenotaligned,asshowninFigure3ofthereference[ 85 ].Fortunately,theoffsetvalueforeachprobingpulsewasgiveninthedataset.Sotherststepafterweloadthedataistoshifttherangebinsandalignthem.Asimple2DFFTofthealigneddatacangiveareasonableSARimage,asshowninFigure 4-4 (a)atthe46thsecondforaCPIof1secondforoneofthe3channels.However,theimageissmearedbecauseofthehighsidelobeproblemofFFT.Toreducethesidelobesandhencesmearing,beforethe2DFFT,a2DTaylorwindowwithparameter4and-80dBsidelobelevelisappliedtothedata,followedbyzeropadding.TheresultingSARimagewithTaylorwindowingandzeropaddingisshowninFigure 4-4 (b).However,therangemigrationinducedbythetargetmotionwillcausecouplingbetweentherangeandcross-rangedimensionsandthissmearstheimageofthemovingtarget.TheKeystonetransformmethod[ 90 91 ]isusedtoreformattheSARdata,correcttherangemigration,andimagethemovingtarget.TheresultingSARimageisshowninFigure 4-4 (c),withtheimagingqualityvisiblyenhanced. 4.4.3ArrayCalibration TheGotchaSARbasedGMTIsystemhasthreeantennas,butunfortunately,theinformationabouttheantennaarrayisnotprovidedinthedataset.Sowecalibratedthearraybyestimatingthedistancesamongtheantennasandtheircorrespondinggains(relativetotherstantenna)fromthedatasetusingtheapproachpresentedinSection 4.2 .Theestimateddistancesbetweenantennasfortheentire71secondsareshowninFigure 4-5 .Itisshowninthisgurethatthedistanceestimatesarestableandalmost 98

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(a) (b) (c) Figure4-4. SARimagesusingdataatthe46thsecond.A)2DFFTonly.B)2DFFTwith2DTaylorwindowandzeropadding.C)2DFFTwith2DTaylorwindow,zeropaddingandKeystonereformatting. constantovertime.FromFigure 4-5 ,theestimateddistancebetweenAntennas1and2isaround0.238meters,andbetweenAntennas1and3is0.476meters. 4.4.4VelocityAmbiguityAnalysis Equations( 4 )showsthatforagivenfv,theestimatedtargetvelocityisdependentonthewavelengthandthePRF.Table 4-1 showsvariousfvvaluesandthecorrespondingvelocitiesfortheAFRLGotchadataset,wherethewavelengthis=0.0312meters,andthePRFisfPRF=2171.6Hz. However,becauseofthelargeinter-antennaspacing,velocityambiguityproblemsexist.Tobetterunderstandtheambiguityproblem,weconsiderthespectralwindow 99

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Figure4-5. EstimateddistancebetweenAntennas. fvVelocity(meters/second)Velocity(miles/hour) 0.13.39257.63320.26.785015.26640.310.177622.89950.413.570130.53270.516.962638.16590.620.355145.79910.723.747753.43230.827.140261.06540.930.532768.69861.033.925276.3318 Table4-1. Targetvelocityversusfv. denedas[ 92 ] W(f)=MXm=1ej2fm2.(4) FortheAFRLGotchaGMTIdatasetatthe46thsecond,wehave1=0,2=4.952,and3=9.904.Thespectralwindowwithf2[)]TJ /F3 11.955 Tf 9.3 0 Td[(0.5,0.5]isplottedinFigure 4-6 .Itisclearlyshownthatthepeakrepetitionoccursataperiodofaround0.2.Wedenethemaximalfrequencywithoutambiguityasfmax,whichmeansthattherearenoambiguitywithinf2[)]TJ /F5 11.955 Tf 9.3 0 Td[(fmax,fmax].FromFigure 4-6 ,wendthatfmax=0.101,whichcorrespondstothevehiclevelocityofabout3.425meters/second(basedonTable 4-1 ),whichisfarbelowanormalvehiclespeedontheroad.Foraxednumberofreceivechannels,theinter-antennaspacingisconstrainedbytwoopposingfactors.Ontheonehand,large 100

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Figure4-6. SpectralwindowsforthearraygeometryoftheAFRLGotchaGMTIsystem. inter-antennaspacingincreasesvehiclevelocityestimationaccuracyandresolutionasaresultoflongeraperture.Ontheotherhand,toavoidvelocityambiguityproblems,theinter-antennaspacingshouldbesmall. TheaboveanalysisshowsthatbecauseofthelimitationsoftheAFRLGotchaGMTIdataset,theunambiguousvelocityrangeismuchsmallerthanthevelocityrangeofanormalmovingvehicle.Wewillsuggestlateronsomemethodstoresolvetheambiguityproblembasedonscenecontext. 4.4.5AdaptiveGMTI ThecenterrangebinoftheAFRLGotchaGMTIdatasetatthe46thsecondisusedtoevaluateourapproachrst,becausethegroundtruthofthetargetvehicleisknown.Forcomparisonpurposes,theresultsofDPCAandATIarealsopresented.BecauseDPCAandATIinvestigationssofararelimitedtotwo-channelSAR,onlythedataofAntennas1and2areusedforallmethods.Atthe46thsecond,thevehicleofinterest,aDurango,islocatedatthecenterrangebin,andthegroundtruthofthevehicleisknown.Thisvehicleismovingawayfromtheradarplatform,whichinducesanegativeDoppler-shift.ForalltheSARimagesinthissection,weusecirclestoindicatetheuncorrectedlocationsofthedetectedmovingtargets.Themultiplepotentialcorrectedlocations(causedbytheambiguityproblemdiscussedabove)ofthemovingtargetsaremarkedbytriangles4andr,withthedirectionsofthetrianglesindicatingtheleavingorclosingofthetargets. 101

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WeplottheDPCAoutputinthecomplexplaneinFigure 4-7 (a),andtheSARimagewithdetectedtargetsinFigure 4-7 (b).WeseethatthetraditionalDPCAfailstoidentifythemoverandfalselyidentifytwonon-movers.TheoutputofATIisgiveninFigure 4-8 (a),andthecorrespondingSARimagewithdetectedtargetsinFigure 4-8 (b).TheATItechniqueisabletondthemover,butalsogivestwofalsealarms.TheGMTIresultsoftheproposedadaptiveapproacharegiveninFigure 4-9 .Figure 4-9 (a)showsthevelocityversusnormalizedcross-rangeimageofIAA.Inthisimage,thelocationsandvaluesofthepeaksindicatethepresenceofthemovingtargetandtheassociatedtargetparameterestimates.TheSARimagewiththedetectedmovingtargetsisshowninFigure 4-9 (b).Itcanbeseenthatthemoverwasidentiedcorrectlywithoutanyfalsealarmbyusingtheadaptiveapproach.However,themovervelocitycannotbeuniquelydeterminedduetothevelocityambiguityproblemcausedbythelargeinter-antennaspacing.AsshowninFigure 4-9 (a),therearemorethanonepeakinthisvelocityversusnormalizedcross-rangeimage.Wenextattempttondthecorrectpeaklocationofthetargetvehiclebyshiftingthetargetvehicletoitspotentialoriginallocationsbasedontheestimatedpossiblevelocities.Thepotentialoriginallocationsarediscardediftheyareoutsidetherange[)]TJ /F3 11.955 Tf 9.3 0 Td[(0.25,0.25],whichistherangeofthenormalizedcross-rangeoftheSARimage.TheresultsareshowninFigure 4-9 (c),wheretwopotentialoriginallocationsofthevehiclearedenotedbythetriangle,andthetruelocationofthevehicleisalsogiveninthisSARimage.Therighttrianglematchesthetruelocationandisclosetoaroad.BasedonthescenecontextoftheSARimage,wecandiscardthelefttrianglebecauseitisnotclosetoaroad[ 93 ]. Inthesecondexample,weconsiderthescenarioatthe46thsecondagain,butdatafromallthreeantennasatallrangebinsareprocessedbyIAA.TheSARimagewithIAAdetectedmovingtargetsisshowninFigure 4-10 .Multipletargets,includingthevehicleofinterest,aredetectedinthisscenario.Wenextattempttondthecorrectpeaklocationsofthemovingtargetsbyshiftingthemovingtargetstotheirpotentialoriginal 102

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(a)(b) Figure4-7. GMTIresultsofDPCAatthe46thsecond.indicatesthedetectedmovingtargetsbyDPCA,withgroundtruthoverlay.A)DPCAdetection.B)DetectedmovingtargetsintheSARimage. (a)(b) Figure4-8. GMTIresultsofATIatthe46thsecond.indicatesthedetectedmovingtargetsbyATI,withgroundtruthoverlay.A)ATIdetection.B)DetectedmovingtargetsintheSARimage. locationsbasedontheestimatedpossiblevelocities.Thetargetvelocityandlocationcannotbedetermineduniquelyduetotheambiguitiescausedbythelargeinter-antennaspacing[ 92 93 ].Again,thepotentialestimatedlocationsarediscardediftheyareoutsidetherange[)]TJ /F3 11.955 Tf 9.3 0 Td[(0.25,0.25],whichistherangeofthenormalizedcross-rangeoftheSARimage.Thepotentialestimatedlocationsofthemovingtargetsandthegroundtruthofthemovingvehicleofinterest,theDurango,arealsoindicatedinFigure 4-10 .TheleftboxindicatestheDurango'sDoppler-shiftedlocationintheimage,andtherightboxindicatestheDurango'sactuallocationintheimage.Thegureshowsthatthemovingvehicleofinterestisdetectedcorrectly.FromtheauxiliarydataoftheGotchadataset,weknowthattheDurangoismovingawayfromtheairplaneatarateof13.95meters/second,thistranslatestoaradialvelocityof9.772meters/second.Thevelocity 103

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(a) (b) (c) Figure4-9. GMTIresultsofIAAatthe46thsecond.indicatesthedetectedmovingtargetsbyIAA,and4andrindicatetheestimatedoriginallocationsofthemovingtarget,withgroundtruthoverlay.A)Velocityversusnormalizedcross-rangeimageofIAA.B)DetectedmovingtargetsintheSARimage.C)Estimatedpotentiallocationsofthemovingtarget. estimatedbyIAAis9.824meters/second,whichmatchesthetruevelocityverywell.Wecommentinpassingthat,withdatafromthreechannels,wecanalsoperformSTAPforGMTI.However,thecomputationofSTAPinvolvesestimatingthecovariancematrixforthedatavectoroflengthMN(32172=6516inthisexample)andcomputingitsinversion.Incontrast,theproposedalgorithmhereonlyrequiressolvingNtimestheproblemwithdatadimensionM,whichiscomputationallymuchmoreefcient. TheIAAresultsatthe51stsecondandthe68thsecondareshowninFigures 4-11 (a)and 4-11 (b),respectively.Thesetwocasesaremorechallenging.TheDoppler-shifted 104

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Figure4-10. GMTIresultsusingIAAwithallthreeantennasatthe46thsecond.indicatesthedetectedmovingtarget,and4andrindicatetheestimatedoriginallocationsofthemovingtarget.Right(blue)andleft(red)boxesindicatetheactualtargetlocationandtheDopplershiftedlocationofthevehicleDurango,respectively. (a) (b) Figure4-11. GMTIresultsusingIAAwithallthreeantennas.indicatesthedetectedmovingtarget,and4andrindicatetheestimatedoriginallocationsofthemovingtarget.Right(blue)andleft(red)boxesindicatetheactualtargetlocationandtheDopplershiftedlocationofthevehicleDurango,respectively.A)Resultatthe51stsecond.B)Resultatthe68thsecond. imageofthemovingtargetofinterestismixedwithastrongstationaryclutteratthe51stsecond,andtheroadisdownhillatthe68thsecond.TheguresshowthatourIAAalgorithmcandetectthemovingtargetandestimateitstruelocationaccuratelyevenforthesechallengingscenarios. 105

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WehavepresentedaframeworkforadaptiveSARGMTIforanairborneplatform.BycombiningacluttercancelingtechniqueusingRELAX,andanadaptivetechniqueforDopplerspectralestimationusingIAA,weareabletooutperformexistingdata-independenttechniquesincludingDPCAandATI.ThisframeworkworksbeyondthecustomizedDPCAorATIradararchitecture,sinceitprovidesanaccuratelyestimatedtargetvelocity,workswithmorethantwochannels,andisabletodetectmovingtargetsinthepresenceofstrongstationaryurbanclutter.WehavedemonstratedtheeffectivenessofthisframeworkusingthepubliclyavailableAFRLGotchaGMTIdataset,andourGMTIresultsmatchthegroundtruthprovidedinthedataset.Wehavealsopresentedtechniquesforarraycalibration,whichworkwellevenwhenthearrayhardwarespecicationsarecompletelyunknown. 106

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CHAPTER5MULTIPLE-INPUTMULTIPLE-OUTPUTSARGMTI GroundMovingTargetIndication(GMTI)modeisanessentialpartofanyreal-timesurveillanceradarsystem,sinceitprovidestargetmovementinformationtotheusers,givinghintsastotheoverallintentofthegroundtargets.Detectionofgroundmovingtargetsisatrickyendeavour,unlikethegeneralMTImode,whichisusedmainlytodetectairborneobjects.GMTIhastodealwiththechallengingeffectsofgroundclutter.Inurbanenvironments,groundreturnsfrombuildingsandotherstationarystructurescansignicantlydrownoutreturnsfromsmallmovingtargets.Furthermore,sincethesurveillanceplatformisairborneandtravelsatagivenvelocitythiscausesthestationarygroundstructurestoexhibitDopplershiftsthatarefunctionsoftheplatformvelocityandtheirrespectivecross-rangelocations.TheseDopplershiftscomplicatemovingtargetdetectionsastheymayoverlapwiththeDopplershiftsofthetargetsofinterests. InthischapterweproposeanewGMTIframeworkfortheMIMOradarsystemarchitecturebyleveragingtheadvantagesofaMIMOsystem[ 94 97 ]andtheadvancesinMIMOwaveformdesign[ 98 ]forenhancedgroundcluttersuppression.WedemonstratetheMIMOradarpotentialviaseveralnumericalexamples,comparingitsperformancetoitsSIMOcounterpart.WealsodiscussexistingGMTImethods,andmotivateastowhytheynecessitateanalternative. 5.1ExistingGMTIMethods CurrentmethodsindetectionofgroundmovingtargetsincludeDisplacedPhaseCenterAntenna(DPCA)andAlong-TrackInterferometry(ATI).Thesesystemsrequiretwodedicatedantennasplacedinapre-denedconguration.InDPCA,theintensityiscomparedbetweentherstandsecondchannels,whileinATIthephaseisbeingcomparedbetweenthem.Sincethesamesceneissampledbythetwochannelsattwodifferenttimes,stationarytargetswillyieldthesamereturnsacrossbothchannels,whilemovingtargetswillhavedifferentreturns.DPCAandATIarecomputationallyefcient, 107

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buttheyarenotdataadaptiveandcannoteffectivelysuppressstrongclutter.Moreimportantly,forradarsystemswithmorethan2channels,theydonotmakethemostofthecollecteddata. Forsystemswithmorethan2channels,onesolutionistousetheSpace-TimeAdaptiveProcessing(STAP)[ 99 100 ]technique.STAPissuitableforadaptivegroundcluttersuppression,butduetoitshighcomputationalcomplexityanditsrequirementofaclutter-and-noisecovariancematrix,itsusageinarealisticGMTIscenario. WeproposeanewGMTImethodthatutilizesmorethan2channelseffectively,andsinceitoperatesonSARphasehistorydata,itcanbeusedsimulatneouslyforbothSARimagingandGMTI.OurmethodalsotakesadvantageoftheMIMOradarwaveformdiversityforenhancedGMTIperformance.InordertoillustratetheeffectivenessoftheproposedGMTImethod,werstdevelopareturnsignalmodelofourMIMOradarsystemandthenweexplainfurtherthestepsnecessarytodetectthemoversfromthesetofreturnsignals. 5.2MIMOGMTISystemModel Figure5-1. Illustrationofgroundmovingtargetindication(GMTI)usingamultiple-inputandmultiple-output(MIMO)radarsystem. Tofacilitateourdiscussions,wedevelopanairborneMIMOradarsystemmodel.Inthissectionwedescribethevariouscomponentsofthismodel,andthephysicsinvolved 108

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ingeneratingandreceivingthetransmittedsignals.Therearevariouscomponentstothismodelandwewillbreakthemdownasfollows. 5.2.1SceneofInterest AsshowninFigure 5-1 ,weconsideranairborneradarsystemattachedtoayingplatform.Letvpdenotethemovingvelocityoftheradarplatform.Weassumethattheimagingareaisfarawayfromtheradarplatform,andhencethereturnsignalsareplanewaves.Letandbethetargetangleandtheround-triptimedelaybetweentheradarandthetarget,respectively.Themovingspeedofthetargetisvt,andtheanglebetweenthetargetmovingdirectionandsignalarrivaldirectionis. 5.2.2AntennaArrayandTransmissionWaveforms Weconsideranairborneradarsystemequippedwithtwouniformlineararraysfortransmissionandreception.Thecongurationoftheantennasarespecial,byhavingasparsetransmitarray,andadensereceivearray,intheMIMOcase,alargerdensevirtualarraycanbeachievedwithgoodwaveforms.ThetransmittingarrayhasNantennaelementswiththeinter-elementspacingbeingdt,andthereceivingarrayhasMantennaelementswiththeinter-elementspacingbeingdr. Letsn(t),withn=0,1,,N)]TJ /F3 11.955 Tf 12.34 0 Td[(1andt=0,1,,T)]TJ /F3 11.955 Tf 12.34 0 Td[(1,bethetransmittedpulsewaveformfromthenthtransmittingantenna.Forsimplicity,weassumethatthetransmittedwaveformsfsn(t)ghaveperfectauto-andcross-correlationpropertieswithinacertaintimelagT0,i.e., T)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xt=0sn1(t)sn2(t)]TJ /F14 11.955 Tf 11.95 0 Td[()=8><>:1,when=0andn1=n2,0,jjT0..(5) Thesignalsarebeingtransmittedwithapulserepetitioninterval(PRI)ofTPRIandmodulatedtoacarrierfrequencycorrespondingtothewavelength.Toachievetherequirementofhavingagoodwaveformsetwithgoodauto-andcross-correlationproperties,weusedthePeCANalgorithm[ 101 ]todesigntheinitialsequencethen 109

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useshiftedcopiesofthesequencetoformasetofNsequenceswiththeprescribedzero-correlationzone(ZCZ). 5.2.3ReceivedSignalModel Letxm,p(t)(m=0,1,,M)]TJ /F3 11.955 Tf 13.31 0 Td[(1,andp=0,1,,P)]TJ /F3 11.955 Tf 13.31 0 Td[(1)bethereceiveddataatthemthreceiveantennaoutputduetothepthprobingpulsewithPbeingthenumberofpulseswithinacoherentprocessinginterval(CPI).Ignoringtheintra-pulseDopplereffectaswellasanyquantizationandarraycallibrationerrors,thereceiveddatasamplesforagivenpointcontainsaDopplershiftassociatedwiththeplatformmovement,2vpsin() ,andaDopplershiftduetothetargetmovement,2vtcos() .Bytransmittingfromthenthtransmitantennaandthenreceivingatthemthreceiveantenna,wealsohavephaseshiftsof2dr msin()and2dt nsin()relativetothereferencepoint.Thisresultsinthefollowingsignalmodel, xm,p(t)=,N)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xn=0ej22vpsin() +2vtcos() pTPRIej2dr msin()ej2dt nsin()sn(t)]TJ /F14 11.955 Tf 11.96 0 Td[(),(5) where,denotesthecomplex-valuedreectioncoefcientofthetarget.Fornotationalsimplicity,let b=2vp TPRI,(5) and v=2vtcos() TPRI.(5) Then,thedatamodelin( 5 )canberewrittenas: xm,p(t)=,N)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xn=0ej2fd(p+rm+tn)e)]TJ /F6 7.97 Tf 6.58 0 Td[(j2v(rm+tn)sn(t)]TJ /F14 11.955 Tf 11.96 0 Td[(),(5) where fd=bsin()+v(5) 110

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istheoverallDopplerfrequencycausedbybothplatformandtargetmovementsand r=dr bandt=dt b(5) arethenormalizedinter-elementspacingsofthereceivingandthetransmittingarrays,respectively.Notethatforstationarygroundclutter,wehavev=0andfd=bsin(),thenthesecondtermin( 5 ),i.e.,e)]TJ /F6 7.97 Tf 6.59 0 Td[(j2v(rm+tn),isequalto1. Takingthestationarygroundclutter,noiseandinterferencefromothertargetsintoaccount,theobserveddatasamplescanbemodeledas: xm,p(t)=,N)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xn=0ej2fd(p+rm+tn)e)]TJ /F6 7.97 Tf 6.59 0 Td[(j2v(rm+tn)sn(t)]TJ /F14 11.955 Tf 11.96 0 Td[()+zm,p(t),(5) wherezm,p(t)containsthenoise,groundclutter,andinterferencesfromothermovingtargets.Theproblemofinterestistodetectmovingtargetsfromtheobserveddatasamplesfxm,p(t)g.Thisproblemischallengingconsideringthefactthatthestationarygroundclutterismuchstrongerthanthemovingtargets. 5.3GMTIAlgorithm Inthissection,wedetailourGMTIalgorithmforaMIMOradarsystemarchitecture.Itinvolvesthreetasks:rangecompression,Dopplerprocessingandphasecompensation,andmovingtargetsdetection.OurmovingtargetdetectorisbasedontheIterativeAdaptiveApproach(IAA),ahighresolutionspectralestimationtechnique,thatwewillapplytoaspecialrange-Doppler-velocitydomain.WewillusetheDelay-and-Sumapproachasourbenchmark.Eachtaskisdetailedasfollows. 5.3.1RangeCompression Thereturnsignalsmustrstberangecompressedvia,forexample,thematchedltering.Let, ym,n(,p),T+)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xt=xm,p(t)sn(t)]TJ /F14 11.955 Tf 11.95 0 Td[()(5) 111

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betherangecompressedoutputatthemreceivingantennaforthenthtransmittingwaveformatthepthpulse,where()denotesthecomplexconjugate. From( 5 ),( 5 )and( 5 ),therange-compressedoutputcanbemodeledas, ym,n(,p)=,ej2fd(p+rm+tn)e)]TJ /F6 7.97 Tf 6.59 0 Td[(j2v(rm+tn)+~zm,n(,p).(5) 5.3.2DopplerProcessingandPhaseCompensation Toworkinthepost-Dopplerdomain,weapplytheFouriertransformtobothsidesof( 5 ): ~Ym,n(,f)=,e)]TJ /F6 7.97 Tf 6.59 0 Td[(j2v(rm+tn)ej2f(rm+tn)(f)]TJ /F5 11.955 Tf 11.95 0 Td[(fd)+~Zm,(,f),(5) where~Ym,n(,f)and~Zm,(,f)aretheFouriertransformsoffym,n(,p)gandf~zm,n(,p)gwithrespecttop,respectively,and()istheKroneckerdelta,i.e.,(0)=1and(f)=0forf6=0. Inordertolookatthereturnsofthesamesceneforallchannels,wemustundothephaseshiftsthatwascausedbytheinter-elementspacing.Weapplyaphasecompensationto~Ym,n(,f),i.e., Ym,n(,f)=e)]TJ /F6 7.97 Tf 6.58 0 Td[(j2f(rm+tn)~Ym,n(,f).(5) Then,( 5 )becomes: Ym,n(,f)=,e)]TJ /F6 7.97 Tf 6.59 0 Td[(j2v(rm+tn)(f)]TJ /F5 11.955 Tf 11.96 0 Td[(fd)+Zm,n(,f),(5) where Zm,n(,f)=e)]TJ /F6 7.97 Tf 6.58 0 Td[(j2f(rm+tn)~Zm,n(,f).(5) Let Y(,f)=Y0,0(,f)Y1,0(,f)YM)]TJ /F9 7.97 Tf 6.59 0 Td[(1,0(,f)Y0,1(,f)YM)]TJ /F9 7.97 Tf 6.59 0 Td[(1,N)]TJ /F9 7.97 Tf 6.58 0 Td[(1(,f)T,(5) 112

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and a(v)=1e)]TJ /F6 7.97 Tf 6.58 0 Td[(j2vre)]TJ /F6 7.97 Tf 6.58 0 Td[(j2vr(M)]TJ /F9 7.97 Tf 6.58 0 Td[(1)e)]TJ /F6 7.97 Tf 6.59 0 Td[(j2vte)]TJ /F6 7.97 Tf 6.59 0 Td[(j2v(t+t)e)]TJ /F6 7.97 Tf 6.59 0 Td[(j2v(t(N)]TJ /F9 7.97 Tf 6.59 0 Td[(1)+t(M)]TJ /F9 7.97 Tf 6.58 0 Td[(1))T,(5) where()Tdenotesthetranspose. From( 5 ),wehave Y(,f)=a(v),(f)]TJ /F5 11.955 Tf 11.96 0 Td[(fd)+Z(,f),(5) whereZ(,f)isdenedsimilarlytoX(,f)in( 5 ). Notethatthesteeringvectorofthestationarygroundclutteris a(0)=111T.(5) Thisimpliesthatafterphasecompensation,allthegroundclutterliesinthesubspacespannedbya(0),andhencecanbesuppressedeffectivelybyusingadaptivetechniques.Meanwhile,Pm,nYm,n(,f)providesaSAR(syntheticapertureradar)imageofthestationaryscene. 5.3.3MovingTargetDetectionintheRange-Doppler-VelocityDomain Todetectthemovers,welookattherange-Doppler-velocitydomain.Foragivenrange-Dopplercell,welookatthevirtualreturnsforeverypossiblevelocities.Toestimatethevirtualreturnsatthesevelocities,weproposetousearecentlydevelopedadaptivealgorithmtomakeanestimateofthesereturns.TherecentlyproposedIAAalgorithm[ 25 ]isconsideredhereintomakeanestimateofthevector(,)=[(,)1,,(,)K]T,where(,)i=,(vi)(i=1,,K)aretheresponsesatdifferentpotentialvelocitiesofthemover,andvi(i=1,,K)arethedifferentvelocities.Kisthenumberscanningpointsofthetarget'svelocity.WeproposetouseIAAduetoitshighresolutionandlowsidelobepropertiesandthusitsabilitytoresolveslowlymovingtargetsfromstationarygroundclutter.SimilartothestepsinChapter4,wenowshowhowIAAcanbeappliedourMIMOreceivedmodel. 113

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Let A=[a1,,aK](5) denotethesteeringmatrix,whereai=a(vi),wehavethefollowingcompactmodel,wherefornotationalexpediency,wehavedropthedependencyonand: y=A+e.(5) IAAsolvesforbyusingaweightedleastsquaresapproach,whichinvolvestheminimizationofthefollowingcostfunction, miniky)]TJ /F14 11.955 Tf 11.95 0 Td[(iaik2~Q)]TJ /F11 5.978 Tf 5.75 0 Td[(1i,i=1,,K,(5) wherekuk2~Q)]TJ /F11 5.978 Tf 5.76 0 Td[(1,uH~Q)]TJ /F9 7.97 Tf 6.58 0 Td[(1uand()Hdenotesthecojugatetranspose.InEquation( 5 ),theinterferenceandnoisecovariancematrixQiisgivenby: Qi=R)-222(jij2aiaHi,(5) where R=NXi=1jij2aiaHi(5) istheIAAcovariancematrix.BylettingPdenoteanMNMNdiagonalmatrixwhosediagonalcontainsthesignalpoweraspi=jij2,wecanre-writeEquation( 5 )inthefollowingmatrixform: R=APAH.(5) TheminimizationofEquation( 5 )withrespecttoigives: i=aHiQ)]TJ /F9 7.97 Tf 6.58 0 Td[(1iy aHiQ)]TJ /F9 7.97 Tf 6.59 0 Td[(1iai.(5) UsingthedenitionofQiandthematrixinversionlemma,Equation( 5 )canbewrittenas[ 25 ]: i=aHiR)]TJ /F9 7.97 Tf 6.59 0 Td[(1y aHiR)]TJ /F9 7.97 Tf 6.59 0 Td[(1ai,(5) 114

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whichavoidsthecomputationofQ)]TJ /F9 7.97 Tf 6.59 0 Td[(1iforeachi=1,,K.SincetheaprioriknowledgeofPisrequiredinEquation( 5 )tocalculatethecovariancematrixR,IAAhastobeimplementediteratively.Usually,IAAisinitializedwiththeperiodogramanditisiteratedforasetnumberofiterations(usually10iterations). OnceisestimatedbyIAA,weobtaina3DcubedenotedbyP(,f,v),whosevoxelvaluerepresentsthereectioncoefcientofthevirtualtarget.Todetectthemoversfromthiscubeofpossiblemovingtargetswecomputethefollowingmetric: (,f,v)=P(,f,v) 2v,(5) where2vrepresentstheaveragepoweroftheclutterthathasvelocityv.Inthiswaywemaketheassumptionthattheclutterpowerishomogenousovervariousrange-Dopplercellsforagivenvelocity.Hence2vcanbeestimatedusingthemeanvalueofP(,,v)whichisjustEquation( 5 )withJ(suspectednumberofmovers,sayJ=10)largestvaluesremoved.Thenwecanlookatthe3DcubeP(,,v)whereitspeaksindicatestheexistenceofthemovingtargets.Theassociatedrange^,Dopplerfrequency^fdandvelocity^vcanthenberelatedbacktothetruetargetangle,^bythefollowingequation, ^=sin)]TJ /F9 7.97 Tf 6.59 0 Td[(1 ^fd)]TJ /F3 11.955 Tf 12.25 0 Td[(^v b!.(5) 5.4NumericalExamples WepresentbelowseveralnumericalexamplestodemonstratetheadvantagesoftheproposedMIMOGMTIapproachoveritsSIMOcounterpart.TherealisticgroundclutterandtheassociatedmovingtargetsareshowninFigure 5-2 .ThisgroundclutterwascollectedviatheGOTCHAprogramsponsoredbytheAirForceResearchLab.Wemodiedtheclutterreturnsbyembeddingvemovingtargetswithvelocitiesvtcos()being)]TJ /F3 11.955 Tf 9.29 0 Td[(8.5,)]TJ /F3 11.955 Tf 9.3 0 Td[(5.1,5.1,6.8and8.1meterspersecond.Thecorrespondingnormalized 115

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velocitiesvare)]TJ /F3 11.955 Tf 9.3 0 Td[(0.25,)]TJ /F3 11.955 Tf 9.3 0 Td[(0.15,0.15,0.20,and.24respectively.Aswecansee,thechallengeistolocatethetwotargetsthatareveryclosetoeachother. Figure5-2. Groundclutter,andsimulatedtargetlocationsandvelocities. Wesimulatearadarsystemwithacarrierfrequencyof9.6GHz,withthebandwidthbeing640MHz,whichcorrespondstoa0.23meterrangeresolution.ThePulseRepetitionFrequency(PRF)isaround2.17KHz.Weassumethatthevelocityoftheradarplatformis100meterspersecond,withthecorrespondingnormalizedvelocitybbeing2.95.Theunimodularsequencedesignalgorithmin[ 101 ],PeCAN,isutilizedtodesignthetransmittedpulsewaveformssn(t).FortheSIMOcase,wedesigna4096-chipunimodularsequence,thencyclicallyshiftitwithvariousdelaysforeachtransmitantennasintheMIMOcase. 5.4.1Single-InputandMultiple-Output(SIMO)System First,weconsidertheperformanceofaSIMOradarsystem.Wecompareitsperformanceusingtwodetectionalgorithms.FirstwediscusstheperformancewhentheDelay-and-Sum(DAS)methodisusedformoverdetection,theDASisalsoknownasthematchedlter,andiswidelypopularinpractice.WerstconsidertheconventionalSIMOradarsystem.Figures 5-3 (a)-(c)showtheDoppler-VelocityimagesofthemoverusingDASforagivenrangebin.Thecirclesindicatethetruecross-rangelocationsandnormalizedvelocitiesofthemovingtargets.Aswecansee,theso-obtainedDoppler-Velocityimagessufferfromhighsidelobeproblemwhichmaycausefalsedetectionsandpoorvelocityestimation.ThisisexascerbatedfurtherviathepoorresolutionofDAS,andaswecanseeDASwasonlyabletodetectonetarget 116

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successfully.Withrespecttothetwocloselyspacedtargets,DASfailedtoresolvethetwobutdidindicatethepresentsofamover.ForclaritythedetectionresultistransposeontotheSARimageinFigure 5-3 (d).AswecanseethecombinationofalowresolutionmethodlikeDAScombinedwiththesmallapertureproblemofSIMOyieldsmanyfalseresults. Figures 5-4 (a)-(c)showsthesamesetupasabove,butinsteadofusingDASweusedIAAtodetectthemovers.Asmentionedearlier,IAAisahigh-resolutionapproachandshouldbeabletoperformbetterthanDAS,butduetothesmallaperturelimitationsofSIMO,weseethatwhileitcorrectlypicksupsometarget,therearealsosomefalsedetectionaswell. (a)(b) (c)(d) Figure5-3. DetectionviaDAS.Velocity-DopplerimagefortheSIMOcaseatvariousranges.A)10meters.B)15meters.C)-10meters.(D)Range-DopplerimageviaDASforSIMO. 5.4.2Multiple-InputandMultiple-Output(MIMO)System FortheMIMOcase,weusea4-elementUniformLinearArray(ULA)withaninter-elementspacingof0.1metersforreceiving,anda3-elementULAwithinter-elementspacingof0.5metersfortransmitting.Thecorrespondingrandtare1.09and5.43,respectively.WeprovidebelowtheGMTIresultsbyusingtheproposedMIMO 117

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(a)(b) (c)(d) Figure5-4. DetectionviaIAA.Velocity-DopplerimagefortheSIMOcaseatvariousranges.A)10meters.B)15meters.C)-10meters.D)Range-DopplerimageviaIAAforSIMO. radarsystem.WerstlookatusingDASinthisMIMOsetup,Figures 5-5 (a)-(c)showtheDoppler-velocityimagesatagivenrange.Aswecansee,theso-obtainedDoppler-VelocityimageshavemuchhigherresolutionthantheirSIMOcounterparts.However,duetothehighsidelobenatureofDAS,manyfalsealarmsexust,andDASisnotabletoresolvethetwocloselyspacedtargets. Figure 5-6 showstheperformanceofIAAwithaMIMOsetup.DuetothehighresolutionandlowsidelobenatureofIAAinadditiontotheincreasedapertureoftheMIMOconguration,IAAissuccessfulindetectingall5targetswithnofalsealarm(inthegivendynamicrange),aswellascorrectlyestimatingtheirvelocities. WehaveconsideredusingMIMOradarformovingtargetdetectioninthepresenceofstrongstationarygroundclutter.AMIMOSARbasedGMTIsystemhasbeenintroduced.Withthisframework,targetdetectioncanbemadeinparallelwithSARimaging.WehaveshowndetectionperformanceusingDASaswellasahighresolutiondetectionalgorithm,IAA.Withtherightwaveforms,theMIMOcongurationanditslargervirtualapertureallowformoreadvancedreceiversidealgorithmssuchasIAAtonot 118

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(a)(b) (c)(d) Figure5-5. DetectionviaDAS.Velocity-DopplerimagefortheMIMOcaseatvariousranges.A)10meters.B)15meters.C)-10meters.(D)Range-DopplerimageviaDASforMIMO. onlydetectthemoversbutalsocorrectlyestimatetheirvelocities,somethingthatwasnotpossibleusingaSIMOconguration. 119

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(a)(b) (c)(d) Figure5-6. DetectionviaIAA.Velocity-DopplerimagefortheSIMOcaseatvariousranges.A)10meters.B)15meters.C)-10meters.D)Range-DopplerimageviaIAAforMIMO. 120

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CHAPTER6CONCLUDINGREMARKSANDFUTUREWORKS ThisdissertationhaspresentedanewapproachtoformingSARimages,analternativetothetraditionallyusedFourierTransform.ThenewmethodbasedonaBayesianframework,ismorerobustandmoreaccurateandusesspectralestimationasitsfoundingprinciples.WeshowedhowitcanbeusedonSARdatathatareincompleteandweshowedhowitscomputationalcomplexitycanbemademorecompetitiveviaexploitingthestructureofthesensingmatrices,aswellasvariousnumericaltechniquessuchastheFastFourierTransformsandtheConjugateGradientMethod.ThesignicanceofthiscontributionliesintheamountofdatathatisneededtoformacceptableSARimagesthatcanbeusednotonlyforreconnaissancebutaremoreamenabletoautomatictargetrecognitionsalgorithms.Thislowerdatarequirementsmeanslesstimeisspentsensing,whichreducesriskstosensingsystems.Thelowertimerequirementalsoresultsinanincreaseintimeallottedforanalysisandresponse. ThesecondbodyofcontributionthatthisdissertationpresentsisaframeworkonhowtodetectmoversdirectlyfromSARdata.MoversdetectionasdiscussedinChapter 4 arespecictocertainSARsystems.ThedissertationpresentsaframeworkonhowmoverscanbedetectedsolelybasedonthemeasuredSARdata.ThisremovetheneedforadedicationGMTImodeaswellastheabilitytooperateonreconnaisancedatadirectly.ThedisserationalsoopensupavenueofusingMIMOsystemsformoverdetections.Fortargetsthataresmallandisclosetotheendo-clutter,aMIMOsystemsincreasestheminimumdetectablevelocityofasystem.Italsoallowsforincreasedparameteridentiability.WithaMIMOSARsystem,wecannotonlydetectthestateofthetargetsbutalsoitsvelocity. SARresearchisinsomesenseasmallniche.ThisislargelyinpartbecauseoftheapplicationsofSAR,whicharemoregearedtowardthedefensecommunity.HoweveranotherreasonisthelackofmeasuredSARdata.SARdatainitscomplex 121

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phasehistoriesformatisalmostnon-existent.CommunitiesthatworkswithSARdataforenvironmentalresearchworksonlywithcomplexSARdata.ComplexSARdataarepost-processeddataandmanyofitsusefuldatacharacteristicswhichareexploitedusingspectralestimationhasbeenlost.CompoudingthiswiththedifcultyofgatheringSARdatamakesthisalmostaclosedeld.Tospearheadresearchinthisarea,morephase-historySARdatashouldbemadeavailabletoresearchers.Atthetimeofthiswriting,weareintheprocessofbuildingaprototypeMIMORadar.Thisinitial2x2systemcanbeusedtogatherboth1-Dand2-Ddata.WiththisdatawecanmorereliablyverifytheworkingsoftheproposedalgorithmsaswellasmakemorecontributiontotheeldofMIMOSAR. 122

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BIOGRAPHICALSKETCH DucVuwasborninBienHoa,VietNam.HegrewupinDavenport,IowaandreceivedhisBachelorofScienceinelectricalengineeringfromIowaStateUniversityinAmes,Iowain2005.HethenworkedasaRadar&AntennaEngineerfortheNavalAirWarfareAircraftDivisionatPatuxentRiver,Maryland.DuringhistenurewiththeNavy,heattendedtheUniversityofSouthernCalifornia,LosAngeles,CaliforniaandreceivedhisMasterofScienceinelectricalengineeringin2008.HewillreceivehisDoctorofPhilosophyinelectricalengineeringfromtheUniversityofFloridaintheSpringof2012.HisresearchinterestsareinRadarsignalprocessingandspectralestimationwithemphasisonSyntheticApertureRadarsystems.CurrentlyheiswiththeSpectralAnalysisgroupattheUniversityofFlorida. 131