SPECTRAL ANALYSIS OF SIGNALS - Uppsala University

1 \sm2"2004/2/22pageiiiiiiiiiSPECTRALANALY SISOFSIGNALSP etreStoicaandRandolphMosesPRENTICEHALL,U pper Saddle River,NewJersey07458\sm2"2004/2/22pageii iiiiiiiiLibraryof CongressCataloging-in-PublicationDataSpe ctralAnalysisof SIGNALS /PetreStoicaandRandolphMosesp. Spectraltheory(Mathematics)I. Moses,RandolphII. Title512'{ :TomRobbinsEditor-in-Chief:?Assistant VicePresident of ProductionandManufacturing:?Executive ManagingEditor:?SeniorManagingEditor:?Pr oductionEditor:?ManufacturingBuyer:?Manu facturingManager:?MarketingManager:?Mark etingAssistant:?Directorof Marketing:?EditorialAssistant:?}

2 ArtDirector:?InteriorDesigner:?Cover Designer:?Cover Photo:?c 2005by PrenticeHall, SaddleRiver,NewJersey07458 All rightsreserved. Nopartof thisbook maybe reproduced, in anyformor by anymeans,withoutpermissionin theUnitedStatesof America10987654321 ISBN0-13-113956-8 PearsonEducationLTD.,LondonPearsonEducat ionAustraliaPTY,Limited,SydneyPearsonEdu cationSingapore, ,HongKongPearsonEducationCanada,Ltd.,Tor ontoPearsonEducaciondeMexico, Japan,TokyoPearsonEducationMalaysia, \sm2"2004/2/22pageiiiiiiiiiiiContents1 .. of DeterministicSignals.. SpectralDensity of RandomSignals.. nitionof Power SpectralDensity.

3 Nitionof Power SpectralDensity .. Power SpectralDensities.. 142 .. {2 FFT.. thePeriodogramMethod .. thePeriodogram.. thePeriodogram.. {Tukey Method .. {Tukey SpectralEstimate.. theBlackman{Tukey SpectralEstimate. DesignConsiderations.. {BandwidthProductandResolution{VarianceT rade-o sin Window Design.. DesignExample.. nedPeriodogramMethods.. Method.. {BasedComputationof Windowed Blackman{Tukey Pe-riodograms.. TemporalWindows:TheApodizationApproach .. 59iii\sm2"2004/2 Cross{SpectraandCoherencySpectra.. {BandwidthProductResults.}}}}}}}}}}

4 713 ParametricMethods .. ARMAP rocesses.. {Walker Method .. {Recursive Solutionsto theYule{Walker Equations.. {DurbinAlgorithm.. {GeninAlgorithm.. edYule{Walker Method.. {StageLeastSquaresMethod.. {SpaceEquations.. |TheoreticalAspects.. |ImplementationAspects.. CovarianceExtensions.. forARParameterEstimation.. {SemenculFormula.. PolynomialTime.. 1294 ParametricMethods .. SinusoidalSignalsin Noise.. {OrderYule{Walker Method .. andMUSICM ethods.. {NormMethod .. {BackwardApproach .. SampleCovariancesforLineSpectralProcesse s.}}}}}}}}}}}}}

5 EodoryParameterizationof a CovarianceMatrix. 172\sm2"2004/2 PeriodogramforSineWave Detectionin WhiteNoise.. SinusoidalSignalwithTime-VaryingAmplitud e.. ESPRIT-basedMethod .. UsefulResultforTwo-Dimensional(2D) .. 1985 .. thePeriodogram.. nedFilterBankMethod .. forHigh{ResolutionSpectralAnalysis.. forStatisticallyStableSpectralAnalysis.. theCaponMethod .. thePeriodogram.. Interpretationof DaniellandBlackman{TukeyPeriodograms.. (APES).. forGappedData(GAPES).. FilterBankApproachesto Two{DimensionalSignals.. 2576 .. Model.}}}

6 {Transmission{DemodulationProcess.. theModelEquation.. {Walker Method .. andMUSICM ethods .. {NormMethod .. 285\sm2"2004/2 .. Estimationfora Constant-ModulusSignal.. : FurtherInsights andDerivations.. forUncertainDirectionVectors.. withNoiseGainConstraint .. (APES).. theCovari-anceMatrix.. 319 APPENDICESA .. ,NullSpace,andMatrixRank.. (Semi)De niteMatrices.. LinearEquations.. Systems.. Systems.. 353B Cram er{ .. 367C .. ParameterEstimation.. Posteriori(MAP)SelectionRule.. :TheoreticalandPracticalPerspectives.. (KL)Approach: No-NameRule.}}}}}

7 386\sm2"2004/2 : TheAICRule.. : theGICRule.. : TheBICRule.. 397D Answersto SelectedExercises399 Bibliography401 ReferencesGroupedby Subject413 Index420\sm2"2004/2/22pageviiiiiiiiiiivi ii\sm2"2004/2/22pageixiiiiiiiiListof { UsefulZ{ Proof thatjr(k)j r(0) a TruncatedAutocovarianceSequence(ACS)a ValidACS? a SequenceanAutocovarianceSequence? of theSumof Two { { (cont'd) leadto Negative of theEquality^ p(!) =^ c(!) AnotherProof of theEquality^ p(!) =^ c(!) WhiteNoisethePeriodogramis Estimationof (!) from^ p(!) {SampleVariance/CovarianceAnalysisof {WeightedACSE stimateInterpretationof BartlettandWelch FurtherLookat theTime{ Blackman{Tukey Window Property of nedMethods.}}}}}}}}

8 Variance{ResolutionTradeo {BasedEstimatorsappliedto MeasuredDataix\sm2"2004/2 of Yule{Walker Proof of theStability Property of Re ectionCoe Re ectionCoe cient NumeratorEstimatorsin ExpressionforARMAP ower e (Non)Uniquenessof AR,ARMAandPeriodogramMethods by a DopplerRadarasa SinusoidswithRandomAmplitudesor { {BasedDerivationof thePisarenko CombinedHOYW-ESPRITM ethod andtheConvergenceof { { AnotherRelationshipbetweenESPRITandMin{ SubspaceMethods forEstimationof appliedto Interpretationof BartlettandWelch a RelationshipbetweentheCaponMethod andMUSIC(Pseudo) Capon{like Implementationof MUSIC\sm2"2004/2 theParametersof a Derivationof Re Sensorin { Element (cont'd) MUSIC(cont'd.)}}}}}}}}}}

9 MUSIC(cont'd.) edMUSICforCoherent SpatialSpectralEstimatorsforCoherent MeasuredData\sm2"2004/2/22pagexiiiiiiiii ixii\sm2"2004/2/22pagexiiiiiiiiiiiPrefac eSpectralanalysisconsiderstheproblemof determiningthespectralcontent( , thedistributionof power over frequency)of a timeseriesfroma nitesetofmeasurements,by meansof eithernonparametricor spectralanalysisas anestablisheddisciplinestartedmorethana centuryagowiththeworkby Schusterondetectingcyclicbehaviorin onthedevelopments in this eldcanbe foundin[Marple1987]. Thisreferencenotesthattheword\spectrum"w as apparentlyintroducedby Newtonin relationto hisstudiesof thedecompositionof whitelightinto a bandof light colors,whenpassedthrougha glassprism(asillustratedonthefront cover).

10 Thiswordappearsto be a variant of theLatinword\specter"whichmeans\ghostlya pparition".ThecontemporaryEnglishwordtha thasthesamemeaningas theoriginalLatinwordis \spectre".Despitetheserootsof theword\spectrum",we hope thestudent willbe a \vividpresence"in thecoursethathasjuststarted!Thistext,whi ch is a revisedandexpandedversionofIntroductiont o SpectralAnalysis(PrenticeHall,1997),is designedto be usedwitha rstcoursein spec-tralanalysisthatwouldtypicallybe o eredto seniorundergraduateor rst{ usefulforself-study, as it is conciseby design,so thatit getsto themainpointsquicklyandshouldhencebe appealingto thosewhowouldlike a fastappraisalontheclassicalandmodernappr oachesof orderto keepthebookas conciseas possiblewithoutsacri cingtherigorof presentationor skippingover essentialaspects,we donotcover someadvancedtopicsof spectralestimationin themainpartof ,severaladvancedtopicsareconsideredin thecomplements thatappearat theendof each chapter.}