FOURIER ANALYSIS傅立叶分析解析.pdf

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1、FOURIERANALYSISLucasIlling2008Contents1FourierSeries21.1GeneralIntroduction........................21.2DiscontinuousFunctions......................51.3ComplexFourierSeries.......................72FourierTransform82.1Denition..............................82.2Theissueofconvention......

2、................112.3ConvolutionTheorem.......................122.4SpectralLeakage..........................133DiscreteTime173.1DiscreteTimeFourierTransform.................173.2DiscreteFourierTransform(andFFT)..............194ExecutiveSummary2011.FourierSeries1FourierSeries1.1General

3、IntroductionConsiderafunctionf()thatisperiodicwithperiodT.f(+T)=f()(1)Wemayalwaysrescaletomakethefunction2periodic.Todoso,deneanewindependentvariablet=2,sothatTf(t+2)=f(t)(2)Soletusconsiderthesetofallsucientlynicefunctionsf(t)ofarealvariabletthatareperiodic,withperiod2.Sinc

4、ethefunctionisperiodicweonlyneedtoconsideritsbehaviorononeintervaloflength2,e.g.ontheinterval(;).Theideaistodecomposeanysuchfunctionf(t)intoaninnitesum,orseries,ofsimplerfunctions.FollowingJosephFourier(1768-1830)considertheinnitesumofsineandcosinefunctionsX1a0f(t)=+[ancos(nt)+b

5、nsin(nt)](3)2n=1wheretheconstantcoecientsanandbnarecalledtheFouriercoecientsoff.Therstquestiononewouldliketoanswerishowtondthosecoecients.Todosoweutilizetheorthogonalityofsineandcosinefunctions:ZZ1cos(nt)cos(mt)dt=[cos((mn)t)+cos((m+n)t)]dt28><2;m=n=0=;m=n6=0>:0;m6=n(2;

6、m=n=0=(4)mn;m6=02GeneralIntroductionSimilarly,ZZ1sin(nt)sin(mt)dt=[cos((mn)t)cos((m+n)t)]dt2(0m=0=(5)mnm6=0andZZ1sin(nt)cos(mt)dt=[sin((mn)t)+sin((m+n)t)]dt2=0(6)UsingtheorthogonalityandtheassumedexpressionfortheinniteseriesgiveninEq.(3),itfollowsthattheFouriercoe

7、cientsareZ1an=f(t)cos(nt)dt(7)Z1bn=f(t)sin(nt)dt(8)ThisinitialinsightbyFourierwasfollowedbycenturiesofaworkonthesecondobviousquestion:AretheRHSandLHSinEq.(3)actuallythesame?ClearlyoneneedstodetermineforwhichclassoffunctionsftheinniteseriesontherighthandsideofEq.(3)willconverg

8、e.Tha