STFT

Short-TimeFourierTransformContentsIntroductionDefinitionCharacteristicsTime&FrequencyResolutionWindowtypeSpectrogramSignalsaredifferent,spectrumsaresimilar!FourierTransformFourierTransformnonstationarysignalsstationarysignalsnonstationarysignalsFourierTransformnonstationarysignalsnonstationarysignalsFourierTransformIntroductionIntroductionWhatiswrongwiththeFourierTransform?Let’sconsidertwobasisfunctions:isnotlocalizedintime.isnotlocalizedinfrequency.Fouriertransformcan’texplicitlyindicatehowasignal’sfrequencycontentsevolveintime.tsinttsintIntroductionThesenonstationarysignalshavestatisticalcharacteristicsthatchangeappreciablyovertime,sothattheirFouriertransformswouldnotmakesense.nonstationarysignalsFourierTransformIntroductionintroducetechniquesforprocessingnon-stationarysignals:Short-TimeFourierTransform(STFT)Thebasicideaistodividethesignalintoshorttimesegmentsor“frames”overwhichthesignalisapproximatelystationary,thenmakeasetofmeasurementsforeachframe.Suchtime-dependentprocessingisapplicablewheneverthesignalisquasi-stationary,i.e.whenitsstatisticalcharacteristicschangeslowlyrelativetotheframelength.DefinitionTheSTFTisfunctionoftwovariables,timeandfrequency,whichdescribeshowthespectrumofrestrictedsegmentsofasignalevolveswithtime.Formally,itisdefinedby:detgxtjx,STFTmjmxemngnxn,STFTGiventimeseriesx[n]:Givenasignalx(t):DefinitionCharacteristicsA:LinearityB:FrequencyDomainShiftingC:TimeDomainShifting22112211STFTSTFTxxSTFTaatfatfa,STFT00tttfxSTFT0,STFT0xSTFTjetfTime&FrequencyResolutionHightimeandfrequencyresolutioncan’tbeachievedsimultaneouslyTime&FrequencyResolutionPrincipleofuncertainty若f(t)与F(ω)构成傅里叶变换对，且已经由其幅度的平方归一化(即f*(t)f*(t)相当于t的概率密度；F*(ω)F(ω)/2π相当于ω的概率密度，*表示共轭)，则无论f(t)的形式如何，t和ω的标准差的乘积σtσω不会小于某一常数(该常数的具体形式与f(t)的形式有关)Inparticular,thismeansthatthereisnoinstantaneousfrequencyanalysisforfiniteenergysignalsTime&FrequencyResolutionAfundamentalproblemofSTFTistheselectionofthewindowstoachieveagoodtradeoffbetweentimeandfrequencyresolution.STFTwithmediumwindowSTFTwithnarrowwindowSTFTwithwidewindowWindowtypeTheJointTime-FrequencyResolutionofSTFTisdeterminedbythetypeoftheWindowfunction,everytypeofwindowhaveaconstantJointTime-FrequencyResolution.HanningorBlackmanwindowaremostcommonlyusedGaussianwindowhasthebestjointT-FresolutionSpectrogramSpectrogramisgraphicalrepresentationofSTFTWhitenoiseSpectrogramChirpsignalSpectrogramMusicscaleSpectrogramHumanvoice“Bat”