短时傅里叶变换.ppt

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1、短时傅里叶变换贾会宾华中师范大学心理学院LimitationsoftheFourierTransformFTdoesnothavetimeresolutionForcalculatingtheFT,weassumethatthesignalisstationaryand,consequently,thattheactivityatdifferentfrequenciesisconstantthroughoutthewholesignal.Inmanyoccasions,however,signalshavetime-v

2、aryingfeaturesthatcannotberesolvedwiththeFouriertransform------event-relatedEEGdataisanexample!x(t)=cos(2*pi*5*t)+cos(2*pi*10*t)+cos(2*pi*20*t)+cos(2*pi*50*t)FTisquitemisleadingbecausethechirpsignalgivesthesamepowerspectrumastheoneofabroadbandrandomsignalwecanov

3、ercomethelackoftimeresolutionoftheFTbychoppingthedataintopiecesandthencalculatingthepowerspectrumforeachpieceor,evenbetter,byusingatime-evolvingwindowtofocusatdifferentsegmentsofdata.IfthewindowusedisaGaussian,itiscalledGabortransformShort-TimeFourierTransformTh

4、eSTFTofasignalx(t)isdefinedasItisthesameastheFTbutwiththeintroductionofthewindowgD*(t′−t)ofwidthDandcenterintTheSTFTquantifiestheactivityofthesignalaroundtimetandfrequencyωWithrespecttotheparticularwindowgDtobeused,Gabor(1946)proposedtouseaGaussianfunction:TheST

5、FTcanbeexpressedastheinnerproductbetweenthesignalx(t)andthecomplexsinusoidalfunctionse−jwt′modulatedbytheGaussianwindowgα:itcanbeseenasthe“matching”betweentheoriginalsignalx(t)andtheoscillatoryfunctionsItispossibletodefineanestimationofthetime-varyingpowerspectr

6、um,thespectrogram,asLimitationsoftheSTFTthewindowlengthisfixedanditmaygiveanoptimalcompromisebetweentimeandfrequencyresolutionforagivenfrequencybutnotforothers