现代数字信号处理AdvancedDigitalSignalProcessing-ch4PSE

AdvancedDigitalSignalProcessing(ModernDigitalSignalProcessing)Chapter4PowerSpectrumEstimation,AmplitudeSpectrum(2)Thepowerleakagebysidelobe(inter-spectruminterference).Inconsistentestimation,ThePSEofHarmonicProcessWithPeriodogram,ThePSEofWhiteNoiseWithPeriodogram,ModificationsofClassicalPSEAchievinglowvarianceattheexpenseofbiasandfrequencyresolutionAveragingPeriodogram(Bartletmethod),Averaging,N:datalength,N=LM,M,M,M,M,Periodogram,Periodogram,Periodogram,Periodogram,AveragingPeriodogram,ItsbiasislargerthantheperiodogramwhileitsvarianceislessthanthePeriodogram:,ModifiedPeriodogram,ThewindowwillsmooththePSDacquiredbyperiodogram.Itsfunctionissimilartoalowpassfilter.,Averagingmodifiedperiodogram(Welchmethod),Averaging,N:datalength,N=LM,M,M,M,M,ModifiedPeriodogram,AveragingPeriodogram,ModifiedPeriodogram,ModifiedPeriodogram,ModifiedPeriodogram,ThePSEofHarmonicProcessWithWelchThesequenceisdividedintoeightsectionswith50%overlap,eachsectioniswindowedwithaHammingwindow,ThePSEofWhiteNoiseWithWelchThesequenceisdividedintoeightsectionswith50%overlap,eachsectioniswindowedwithaHammingwindow,4.3ParameterModelMethodsforPSE,BasicPrinciples,Autocorrelationfunction,ClassicalPSE,PSD,Fouriertransform,ObservationsxN(n),EstimationofsignalmodelH(z),Parametermodelmethods,PSD,ObservationsxN(n),TheTimeSeriesModelofStationaryRandomSignal,LinearsystemwithtransferfunctionH(z),Whitenoisew(n),Stationaryrandomsequencex(n),MA(q)model(all-zeromodel)SuitableforsignalswhosepowerspectrahavevalesbutnopeaksAR(p)model(all-polemodel,mostwidelyused)Suitableforsignalswhosepowerspectrahavepeaksbutnovales,butbewidelyusedsincethelinearrelationbetweenitsparametersandthesignalautocorrelationfunction,ARMA(p,q)model(zero-polemodel)Suitableforsignalswhosepowerspectrahavevalesandpeaks,Modelparameterstobeestimated,AR(p):,MA(q):,ARMA(p,q):,TheRelationbetweentheAutocorrelationFunction&theModelParametersARMA(p,q)model,Inverseztransform,GeneralizedYule-Walkerequations:anonlinearequationset,buttheequationsarelinearwhenmq.,MA(q)model,AR(p)model,Yule-Walkerequation:alinearequationset,Initialvaluetheorem,ARmodelpowerspectrumestimation(ARPSE),Estimationofautocorrelationfunction,ObservationsxN(n),EstimationofARmodelparameters,ARmodelestimation,TheARmodelparameterestimationisobtainedbysolvingtheYule-Walkerequation(m=0,1,p):,PropertiesofARPSETheimpliedautocorrelationfunctionextensionWiththep+1samplesofautocorrelationfunction(ACF)estimation,Formp,thecanbeextrapolatedfromthoseACFestimationsofmpby,i.e.,extrapolatingfrom,to,Continuousvaluerandomvariable,Maximumentropyspectralestimation(MESE)&ARPSEEntropy,Discretevaluerandomvariable,EntropyofrandomvariableX,UncertaintyofrandomvariableX,Maximumentropy,PDFwithmaximumuncertainty(leastrestriction),KnownACFestimation,MESE,Maximumentropyextrapolation,UnknownACFestimation,ACFswithmaximumuncertainty,MESEofzero-meanGaussianrandomsequencePDFofN-dimensionGaussianrandomsequence,and,andsoon.,TheequivalencebetweentheARPSEandtheMESEofGaussianrandomsequenceARPSEforp=N:,MESEofGaussianrandomsequence,ForGaussianrandomsequence,Maximumentropyextrapolation,ARPSEimpliedextrapolation,=,MESE,ARPSE,=,TherearenopolesofitsARmodeloutsidetheunitcircle,else,ThestabilityofARmodel,Stationaryrandomsequencex(n),TheARmodelofstationaryrandomsequenceisstable(minimumphasemodel),TherelationshipbetweentheARPSEandthelinearpredictionOne-steppurelinearoptimalpredictionfilter,Aone-steppurelinearoptimalpredictionfilteristhesolutionoftheYule-Walkerequation:,AR(p)model,TheAR(p)parameterscouldbeobtainedasthecoefficientsthatminimizedthepredictionerrorpowerofap-thorderlinearpredictor.,MethodsofARPSE(SolutionsofY-KEquation)Levinson-Durbinrecursivealgorithm,porderARmodelequation,p+1orderARmodelequation,Let,Let,Expandedequation,Preparativeequation,If,Then,i.e.,Thepredictiveerrorisreducedgraduallyaspincreases.,Levinson-Durbinrecursivealgorithm,Burgsrecursivealgorithm,ComputingARmodelparametersfromobservationdatadirectly,ThePSEofHarmonicProcessWithBurgsMethod,ThePSEofWhiteNoiseWithBurgsMethod,SelectionofARModelsOrderBasicprinciples,Higherorderhaslowerpredictionerrororhigherapproximationaccuracy,CriteriatoselectARmodelorderFinalpre