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1、FILTERINGGG313Lecture27December1,2005AFILTERisadeviceorfunctionthatallowscertainmaterialtopassthroughitwhilenotothers.Inelectronicsandsignalanalysis,afilterisadeviceorfunctionthatchangestheshapeofthephaseand/oramplitudespectruminthefrequencydomain,i.e.itattenuatesenergyatsomefrequenciesrelativet
2、oothers,anditmayalsomakeconsiderablechangesinphase,causingdelaysoradvancesinthetimedomain.ALINEARTIME-INVARIENT(LTI)filteralwaysbehavesthesamewayregardlessoftheinputorthetime.Forexampleafilterthatattenuates60Hzenergyby48dB,ALWAYSattenuatesenergyat60Hzby48dB.ALTIfilter’sresponseiscompletelyspecif
3、iedbyitsIMPULSERESPONSE-theoutputwhentheinputisanimpulse.AfilterisCAUSALiftheoutputdoesnotanticipatetheinput.Azero-phasefilter(onethatdoesnotcauseanydelay)cannotbephysicallyreliasable(exceptforaunitimpulse).Forexample,considerafilterwiththefollowingimpulseresponse:h=[.2.2.2.2.2]Thisfilterissymme
4、tricaboutitscenter(zerolag)anditwillspreadanysignalattimet(i)overtimest(i-2)tot(i+2).ItthusseesAHEAD,anditcannotbebuiltwithphysicalcomponents.Soitisnoncausal.Wecanmakethisfiltercausalbyaddingzerosatthefront,addinga2-stepdelay:h=[00.2.2.2.2.2]Nowitspreadsoutthesignalatt(i)fromt(i)tot(i+4),anditdo
5、esnotseeahead.Smoothingfilters.Oftenyouwillhaveapowerspectrumthatisverynoisywithlargechangesfromonespectralestimatetoanother.Inthecasebelowitjustlookslikespikes.First,wetakethelogtoseethe“real”spectrum,andnotethatthereisanunderlyingstructure:IFwewanttheunderlyingstructure,wecanconvolvethepowersp
6、ectrumwitha25-pointboxcar(eachpoint=1/25)toobtaintheheavylinebelow:andthenexponentiatebacktolinearamplitude:Thistypeofboxcarfilterisamovingaveragesmoother.Itisthesameastakinga25-pointmean,movingonepoint,andtakinganother.Whichdomainyouarein-timeorfrequency-isnotimportant,smoothingfiltersworkinbot
7、hdomains.Notethatthetransferfunctionofthissmoothingfilterisasincfunction-notaparticularlygoodfilterforseparatingsignalswithdifferentfrequencies.Forthat,wewantaboxcarintheFREQUENCYdomain,sotheimpulseresponseintheTIMEdomainwil
