Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

491 Pages·2012·6.142 MB·English

by  Antonio Napolitano(auth.)

GENERALIZATIONS OF
CYCLOSTATIONARY
SIGNAL PROCESSING
GENERALIZATIONS OF
CYCLOSTATIONARY
SIGNAL PROCESSING
SPECTRAL ANALYSIS AND
APPLICATIONS
AntonioNapolitano
UniversityofNapoliParthenope,Italy
Thiseditionfirstpublished2012
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LibraryofCongressCataloging-in-PublicationData
Napolitano,Antonio,1964–
Generalizationsofcyclostationarysignalprocessing:spectralanalysisand
applications/AntonioNapolitano.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-1-119-97335-5
1. Signalprocessing. 2. Spectrumanalysis. 3. Cyclostationarywaves. I. Title.
TK5102.9.N37 2012
621.382(cid:1)2–dc23
2012015740
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
PrintISBN:9781119973355
SetinTimesNewRoman10/12byThomsonDigital,Noida,India
ToUbaldaandAsha
Contents
AbouttheAuthor xiii
Acknowledgements xv
Preface xvii
ListofAbbreviations xxi
1 Background 1
1.1 Second-OrderCharacterizationofStochasticProcesses 1
1.1.1 Time-DomainCharacterization 1
1.1.2 Spectral-DomainCharacterization 3
1.1.3 Time-FrequencyCharacterization 4
1.1.4 Wide-SenseStationaryProcesses 5
1.1.5 EvolutionarySpectralAnalysis 6
1.1.6 Discrete-TimeProcesses 8
1.1.7 LinearTime-VariantTransformations 8
1.2 Almost-PeriodicFunctions 11
1.2.1 UniformlyAlmost-PeriodicFunctions 11
1.2.2 APFunctionsintheSenseofStepanov,Weyl,andBesicovitch 12
1.2.3 WeaklyAPFunctionsintheSenseofEberlein 14
1.2.4 PseudoAPFunctions 15
1.2.5 APFunctionsintheSenseofHartmanandRyll-Nardzewski 16
1.2.6 APFunctionsDefinedonGroupsandwithValuesinBanachand
HilbertSpaces 17
1.2.7 APFunctionsinProbability 17
1.2.8 APSequences 18
1.2.9 APSequencesinProbability 19
1.3 Almost-CyclostationaryProcesses 19
1.3.1 Second-OrderWide-SenseStatisticalCharacterization 20
1.3.2 JointlyACSSignals 23
1.3.3 LAPTVSystems 24
1.3.4 ProductsofACSSignals 28
1.3.5 CyclicStatisticsofCommunicationsSignals 30
1.3.6 Higher-OrderStatistics 31
1.3.7 CyclicStatisticEstimators 33
viii Contents
1.3.8 Discrete-TimeACSSignals 33
1.3.9 SamplingofACSSignals 35
1.3.10 MultirateProcessingofDiscrete-TimeACSSignals 38
1.3.11 Applications 39
1.4 SomePropertiesofCumulants 39
1.4.1 CumulantsandStatisticalIndependence 40
1.4.2 CumulantsofComplexRandomVariablesandJointComplex
Normality 41
2 GeneralizedAlmost-CyclostationaryProcesses 45
2.1 Introduction 46
2.2 CharacterizationofGACSStochasticProcesses 49
2.2.1 Strict-SenseStatisticalCharacterization 49
2.2.2 Second-OrderWide-SenseStatisticalCharacterization 51
2.2.3 Second-OrderSpectralCharacterization 61
2.2.4 Higher-OrderStatistics 64
2.2.5 ProcesseswithAlmost-PeriodicCovariance 67
2.2.6 MotivationsandExamples 68
2.3 LinearTime-VariantFilteringofGACSProcesses 72
2.4 EstimationoftheCyclicCross-CorrelationFunction 74
2.4.1 TheCyclicCross-Correlogram 74
2.4.2 Mean-SquareConsistencyoftheCyclicCross-Correlogram 78
2.4.3 AsymptoticNormalityoftheCyclicCross-Correlogram 83
2.5 SamplingofGACSProcesses 87
2.6 Discrete-TimeEstimatoroftheCyclicCross-CorrelationFunction 89
2.6.1 Discrete-TimeCyclicCross-Correlogram 90
2.6.2 AsymptoticResultsasN →∞ 93
2.6.3 AsymptoticResultsasN →∞andTs →0 97
2.6.4 ConcludingRemarks 104
2.7 NumericalResults 107
2.7.1 AliasinginCycle-FrequencyDomain 107
2.7.2 SimulationSetup 107
2.7.3 CyclicCorrelogramAnalysiswithVaryingN 108
2.7.4 CyclicCorrelogramAnalysiswithVaryingN andTs 112
2.7.5 Discussion 112
2.7.6 ConjecturingtheNonstationarityTypeofthe
Continuous-TimeSignal 115
2.7.7 LTIFilteringofGACSSignals 119
2.8 Summary 119
3 ComplementsandProofsonGeneralizedAlmost-CyclostationaryProcesses 123
3.1 ProofsforSection2.2.2“Second-OrderWide-SenseStatistical
Characterization” 123
3.2 ProofsforSection2.2.3“Second-OrderSpectralCharacterization” 125
3.3 ProofsforSection2.3“LinearTime-VariantFilteringofGACSProcesses” 129
3.4 ProofsforSection2.4.1“TheCyclicCross-Correlogram” 131
Contents ix
3.5 ProofsforSection2.4.2“Mean-SquareConsistencyoftheCyclic
Cross-Correlogram” 136
3.6 ProofsforSection2.4.3“AsymptoticNormalityoftheCyclic
Cross-Correlogram” 147
3.7 ConjugateCovariance 149
3.8 ProofsforSection2.5“SamplingofGACSProcesses” 151
3.9 ProofsforSection2.6.1“Discrete-TimeCyclicCross-Correlogram” 151
3.10 ProofsforSection2.6.2“AsymptoticResultsasN →∞” 155
3.11 ProofsforSection2.6.3“AsymptoticResultsasN →∞andTs →0” 168
3.12 ProofsforSection2.6.4“ConcludingRemarks” 176
3.13 Discrete-TimeandHybridConjugateCovariance 177
4 SpectrallyCorrelatedProcesses 181
4.1 Introduction 182
4.2 CharacterizationofSCStochasticProcesses 186
4.2.1 Second-OrderCharacterization 186
4.2.2 RelationshipamongACS,GACS,andSCProcesses 194
4.2.3 Higher-OrderStatistics 195
4.2.4 MotivatingExamples 200
4.3 LinearTime-VariantFilteringofSCProcesses 205
4.3.1 FOT-DeterministicLinearSystems 205
4.3.2 SCSignalsandFOT-DeterministicSystems 208
4.4 TheBifrequencyCross-Periodogram 209
4.5 MeasurementofSpectralCorrelation–UnknownSupportCurves 215
4.6 TheFrequency-SmoothedCross-Periodogram 223
4.7 MeasurementofSpectralCorrelation–KnownSupportCurves 226
4.7.1 Mean-SquareConsistencyoftheFrequency-Smoothed
Cross-Periodogram 226
4.7.2 AsymptoticNormalityoftheFrequency-Smoothed
Cross-Periodogram 230
4.7.3 FinalRemarks 232
4.8 Discrete-TimeSCProcesses 234
4.9 SamplingofSCProcesses 238
4.9.1 Band-LimitednessProperty 238
4.9.2 SamplingTheorems 241
4.9.3 IllustrativeExamples 245
4.10 MultirateProcessingofDiscrete-TimeJointlySCProcesses 258
4.10.1 Expansion 259
4.10.2 Sampling 263
4.10.3 Decimation 264
4.10.4 ExpansionandDecimation 268
4.10.5 StrictlyBand-LimitedSCProcesses 269
4.10.6 InterpolationFilters 270
4.10.7 DecimationFilters 271
4.10.8 FractionalSamplingRateConverters 273
4.11 Discrete-TimeEstimatorsoftheSpectralCross-CorrelationDensity 273
x Contents
4.12 NumericalResults 274
4.12.1 SimulationSetup 274
4.12.2 UnknownSupportCurves 275
4.12.3 KnownSupportCurves 278
4.13 SpectralAnalysiswithNonuniformFrequencySpacing 280
4.14 Summary 286
5 ComplementsandProofsonSpectrallyCorrelatedProcesses 291
5.1 ProofsforSection4.2“CharacterizationofSCStochasticProcesses” 291
5.2 ProofsforSection4.4“TheBifrequencyCross-Periodogram” 293
5.3 ProofsforSection4.5“MeasurementofSpectralCorrelation–Unknown
SupportCurves” 298
5.4 ProofsforSection4.6“TheFrequency-SmoothedCross-Periodogram” 306
5.5 ProofsforSection4.7.1“Mean-SquareConsistencyofthe
Frequency-SmoothedCross-Periodogram” 310
5.6 ProofsforSection4.7.2“AsymptoticNormalityoftheFrequency-Smoothed
Cross-Periodogram” 325
5.7 AlternativeBounds 334
5.8 ConjugateCovariance 335
5.9 ProofsforSection4.8“Discrete-TimeSCProcesses” 338
5.10 ProofsforSection4.9“SamplingofSCProcesses” 340
5.11 ProofsforSection4.10“MultirateProcessingofDiscrete-Time
JointlySCProcesses” 343
6 FunctionalApproachforSignalAnalysis 355
6.1 Introduction 355
6.2 RelativeMeasurability 356
6.2.1 RelativeMeasureofSets 356
6.2.2 RelativelyMeasurableFunctions 357
6.2.3 JointlyRelativelyMeasurableFunctions 358
6.2.4 ConditionalRelativeMeasurabilityandIndependence 360
6.2.5 Examples 361
6.3 Almost-PeriodicallyTime-VariantModel 361
6.3.1 Almost-PeriodicComponentExtractionOperator 361
6.3.2 Second-OrderStatisticalCharacterization 362
6.3.3 SpectralLineRegeneration 365
6.3.4 SpectralCorrelation 366
6.3.5 StatisticalFunctionEstimators 367
6.3.6 Sampling,Aliasing,andCyclicLeakage 369
6.3.7 FOT-DeterministicSystems 371
6.3.8 FOT-DeterministicLinearSystems 372
6.4 NonstationarityClassificationintheFunctionalApproach 374
6.5 ProofsofFOTCounterpartsofSomeResultsonACSandGACSSignals 375
7 ApplicationstoMobileCommunicationsandRadar/Sonar 381
7.1 PhysicalModelfortheWirelessChannel 381
7.1.1 AssumptionsonthePropagationChannel 381
Contents xi
7.1.2 StationaryTX,StationaryRX 382
7.1.3 MovingTX,MovingRX 384
7.1.4 StationaryTX,MovingRX 387
7.1.5 MovingTX,StationaryRX 388
7.1.6 ReflectiononPointScatterer 388
7.1.7 StationaryTX,ReflectiononPointMovingScatterer,StationaryRX
(StationaryBistaticRadar) 390
7.1.8 (Stationary)MonostaticRadar 391
7.1.9 MovingTX,ReflectiononaStationaryScatterer,MovingRX 392
7.2 ConstantVelocityVector 393
7.2.1 StationaryTX,MovingRX 393
7.2.2 MovingTX,StationaryRX 394
7.3 ConstantRelativeRadialSpeed 395
7.3.1 MovingTX,MovingRX 395
7.3.2 StationaryTX,MovingRX 399
7.3.3 MovingTX,StationaryRX 401
7.3.4 StationaryTX,ReflectiononaMovingScatterer,StationaryRX
(StationaryBistaticRadar) 405
7.3.5 (Stationary)MonostaticRadar 407
7.3.6 MovingTX,ReflectiononaStationaryScatterer,MovingRX 407
7.3.7 Non-synchronizedTXandRXoscillators 408
7.4 ConstantRelativeRadialAcceleration 408
7.4.1 StationaryTX,MovingRX 409
7.4.2 MovingTX,StationaryRX 409
7.5 TransmittedSignal:Narrow-BandCondition 410
7.5.1 ConstantRelativeRadialSpeed 412
7.5.2 ConstantRelativeRadialAcceleration 415
7.6 MultipathDopplerChannel 417
7.6.1 ConstantRelativeRadialSpeeds–DiscreteScatterers 417
7.6.2 ContinuousScatterer 417
7.7 SpectralAnalysisofDoppler-StretchedSignals–ConstantRadialSpeed 418
7.7.1 Second-OrderStatistics(Continuous-Time) 419
7.7.2 MultipathDopplerChannel 423
7.7.3 Doppler-StretchedSignal(Discrete-Time) 428
7.7.4 SimulationofDiscrete-TimeDoppler-StretchedSignals 431
7.7.5 Second-OrderStatistics(Discrete-Time) 433
7.7.6 IllustrativeExamples 439
7.7.7 ConcludingRemarks 444
7.8 Spectral Analysis of Doppler-Stretched Signals – Constant Relative Radial
Acceleration 449
7.8.1 Second-OrderStatistics(Continuous-Time) 450
7.9 OtherModelsofTime-VaryingDelays 455
7.9.1 TaylorSeriesExpansionofRangeandDelay 455
7.9.2 PeriodicallyTime-VariantDelay 455
7.9.3 PeriodicallyTime-VariantCarrierFrequency 456
7.10 Proofs 457
xii Contents
8 BibliographicNotes 465
8.1 Almost-PeriodicFunctions 465
8.1.1 FundamentalTreatments 465
8.1.2 Generalizations 465
8.1.3 Reviews 465
8.2 CyclostationarySignals 465
8.2.1 FundamentalTreatments 465
8.2.2 DevelopmentsandApplications 465
8.2.3 Higher-orderCyclostationarity 466
8.3 GeneralizationsofCyclostationarity 466
8.3.1 GeneralizedAlmost-CyclostationarySignals 466
8.3.2 SpectrallyCorrelatedSignals 466
8.4 OtherNonstationarySignals 466
8.5 FunctionalApproachandGeneralizedHarmonicAnalysis 466
8.6 LinearTime-VariantProcessing 467
8.6.1 Channels 467
8.6.2 DopplerEffect 467
8.6.3 TimeWarping 467
8.6.4 FrequencyWarping 467
8.7 Sampling 467
8.8 ComplexRandomVariables,Signals,andSystems 467
8.9 StochasticProcesses 467
8.10 Mathematics 468
8.11 SignalProcessingandCommunications 468
References 469
Index 479