常用Fourier变换表

Function, f(t)Definition of Inverse Fourier TransformFourier Transform, F(w)Definition of Fourier Transform1f(t)=2pf(t-t0)¥-¥jwtF(w)edwòF(w)=¥-¥òf(t)e-jwtdtF(w)e-jwt0F(w-w0)f(t)ejw0tf(at)1wF()aa2pf(-w)(jw)nF(w)F(t)dnf(t)dtn(-jt)nf(t)dnF(w)dwn-¥òf(t)dttF(w)+pF(0)d(w)jw12pd(w-w0)2jwd(t)ejw0tsgn(t)Signals & Systems - Reference Tables

1

1jptsgn(w)u(t)1pd(w)+jwjnw0tn=-¥åFne¥2pn=-¥åFnd(w-nw0)wt)2¥trect()ttSa(BBtSa()2p2tri(t)wrect()BwSa()22Acos(ptt)rect()2t2tApcos(wt)t(p)2-w22tp[d(w-w0)+d(w+w0)]cos(w0t)sin(w0t)p[d(w-w0)-d(w+w0)]jp[d(w-w0)+d(w+w0)]+2jw22w0-wu(t)cos(w0t)u(t)sin(w0t)pw2[d(w-w0)-d(w+w0)]+222jw0-wu(t)e-atcos(w0t)(a+jw)2w0+(a+jw)2Signals & Systems - Reference Tables

2

u(t)e-atsin(w0t)w02w0+(a+jw)2e-at2aa2+w2e-t2/(2s2)s2p e-s2w2/2u(t)e-at1a+jw1(a+jw)2u(t)te-atØ Trigonometric Fourier Series

f(t)=a0+å(ancos(w0nt)+bnsin(w0nt))n=1¥

where

1a0=

T

ò0

T

2T

f(t)dt , an=òf(t)cos(w0nt)dt ,and

T0

2T

bn=òf(t)sin(w0nt)dt

T0

Ø Complex Exponential Fourier Series

f(t)=

n=-¥

åFne

¥

jwnt

1T

, where Fn=òf(t)e-jw0ntdt

T0

Signals & Systems - Reference Tables

3

Some Useful Mathematical Relationships

ejx+e-jxcos(x)=2ejx-e-jxsin(x)=2jcos(x±y)=cos(x)cos(y)msin(x)sin(y)sin(x±y)=sin(x)cos(y)±cos(x)sin(y)cos(2x)=cos2(x)-sin2(x)sin(2x)=2sin(x)cos(x)2cos2(x)=1+cos(2x)2sin2(x)=1-cos(2x)cos2(x)+sin2(x)=12cos(x)cos(y)=cos(x-y)+cos(x+y)2sin(x)sin(y)=cos(x-y)-cos(x+y)2sin(x)cos(y)=sin(x-y)+sin(x+y)Signals & Systems - Reference Tables

4

Useful Integrals

òcos(x)dxòsin(x)dxòxcos(x)dxòxsin(x)dx2xòcos(x)dx2xòsin(x)dxaxeòdxsin(x)-cos(x)cos(x)+xsin(x)sin(x)-xcos(x)2xcos(x)+(x2-2)sin(x)2xsin(x)-(x2-2)cos(x)eaxaaxxeòdxéx1ùeaxê-2úëaaûéx22x2ùeê-2-3úaûëaaax2axxòedxdxòa+bx1lna+bxbòa2+b2x2dxbx1tan-1()abaSignals & Systems - Reference Tables

5