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:?ArtDirect or:?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.}

2 ,LondonPearsonEducationAustraliaPTY,Limi ted,SydneyPearsonEducationSingapore, ,HongKongPearsonEducationCanada,Ltd.,Tor ontoPearsonEducaciondeMexico, Japan,TokyoPearsonEducationMalaysia, \sm2"2004/2/22pageiiiiiiiiiiiContents1 .. of DeterministicSignals.. SpectralDensity of RandomSignals.. nitionof Power SpectralDensity .. 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.}}}}}}}}

3 TemporalWindows:TheApodizationApproach .. 59iii\sm2"2004/2 Cross{SpectraandCoherencySpectra.. {BandwidthProductResults.. 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.. eodoryParameterizationof a CovarianceMatrix. 172\sm2"2004/2 PeriodogramforSineWave Detectionin WhiteNoise.}}}}}}}}}}}}}}}

4 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.. {Transmission{DemodulationProcess.. theModelEquation.. {Walker Method .. andMUSICM ethods .. {NormMethod .. 285\sm2"2004/2 .. Estimationfora Constant-ModulusSignal.. : FurtherInsights andDerivations.. forUncertainDirectionVectors.. withNoiseGainConstraint .. (APES).. theCovari-anceMatrix.}}}}}}}

5 319 APPENDICESA .. ,NullSpace,andMatrixRank.. (Semi)De niteMatrices.. LinearEquations.. Systems.. Systems.. 353B Cram er{ .. 367C .. ParameterEstimation.. Posteriori(MAP)SelectionRule.. :TheoreticalandPracticalPerspectives.. (KL)Approach: No-NameRule.. 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(!)}}}}}

6 WhiteNoisethePeriodogramis Estimationof (!) from^ p(!) {SampleVariance/CovarianceAnalysisof {WeightedACSE stimateInterpretationof BartlettandWelch FurtherLookat theTime{ Blackman{Tukey Window Property of nedMethods: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.)}}}}}}}}}}}}}}

7 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).Thiswordappearsto be a variant of theLatinword\specter"whichmeans\ghostlya pparition".

8 ThecontemporaryEnglishwordthathasthesame meaningas 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,andalsoin anintroductorycourse,thereadercanskipthe complements andreferto resultsin theappendiceswithouthavingto understandin themoreadvancedreader.}

9 We have includedthreeappendicesanda num-ber of complement sectionsin each summaryof themaintechniquesandresultsin linearalgebra,statisticalaccuracybounds, andmodelorderselection,respectively. Thecomplements present a broadrangeofadvancedtopicsin of thesearecurrent or recent researchtopicsin theendof each chapterwe have {or{lessorderedfromleastto mostdi cult;thisorderingalsoapproximatelyfollow sthechronologicalpresen-tationof materialin cultexercisesexploreadvancedtopicsin spectralanalysisandprovideresultswhich arenotavailablein selectedexercisesarefoundin illustratethemainpoints of thetextandto providethereaderwith rst{handinformationon thebehaviorandperformanceof \sm2"2004/2/22pagexiviiiiiiiixivperforma nceof themethods andexploreothertopicssuch as statisticalaccuracy,resolutionproperties ,andthelike, thatarenotanalyticallydeveloped in have usedMatlab1tominimizetheprogrammingchore andtoencouragethereaderto \play" providea setofMatlabfunctionsfordatagenerationand spectralestimationthatforma basisfora comprehensivesetof spectralestimationtools.}}}

10 Thesefunctionsareavailableat alsobe prepareda setof overheadtransparencieswhich canbe usedas a teachingaidfora believe thatthesetransparenciesareusefulnotonlyt o courseinstructorsbutalsoto otherreaders,becausetheysummarizetheprin cipalmethods andresultsin readerswhostudythetopicontheirown,it shouldbe a usefulexerciseto referto themainpoints addressedinthetransparenciesaftercomplet ingthereadingof each mentionedearlier,thistextis a revisedandexpandedversionofIn-troduction to SPECTRAL ANALYSIS (PrenticeHall,1997).We have maintainedtheconcisenessandaccessability of themaintext;therevisionhasprimarilyfocus edonexpandingthecomplements,appendices,a ndbibliography. Speci cally, we haveexpandedAppendixB to includea detaileddiscussionof Cram er-Raoboundsfordirection-of-arrival have addedAppendixC,which coversmodelorderselection,andhave morethandoubledthenumber of complements fromthepreviousbook to 32,mostof which present recent resultsin have alsoexpandedthebibliography to includenewtopicsalongwithrecent resultson organizedas introducesthespectralanalysisproblem,mot ivatesthede nitionof power spectraldensity functions,andreviewssomeimportant propertiesof autocorrelationsequencesandspectraldensi ty and5 presents classicaltechniques,includingtheperiodog ram,thecorrelogram,andtheirmodi includeananalysisof biasandvarianceof thesetechniques.