傅里葉變換特性（常用的一些，至于微分積分性質(zhì)感覺不是很常用，這里沒寫）設(shè)基本傅里葉變換F[f(t)]=F(r),F[g(x)]=G(r)線性性F[af(W+bg(x)]=aF(目)+bG(目)對稱性F[F(x)]=f(-r)縮放性質(zhì)1rF[f(at)]=a^F(a)平移特性F[f(x土x0)]=exp(±j2叫x0)F(r)相移特性F[exp(±j2冗rox)f(x)]=F(rr0)面積對應(yīng)關(guān)系（沒啥用，對應(yīng)的就是變換后的基頻分量）8Sjf(x)dx=F(0),jF(r)dx=f(0)-8-8復(fù)共軛函數(shù)傅里葉變換（）F[f*(x)]=F*(-r),F[f*(-x)]=F*(r)微分特性「d“，、】fF^^f(x)=j2兀uf(u)，F(xiàn)dn]f(x)=(j2兀u)nf(u)dxnJ積分特性F「jxf(t)dt〕=F(u)+兀F(0)5(u)L-8」j2兀u

傅里葉變換對空間域空間頻域8(x)1cos(2兀hx)—[8(h-h)+8(h+H)]sin(2兀hx)2-[5(h-H0)-8(H+H0)]rect(x)sinc(h)tri(x)sinc2(h)sgn(x)j^Hstep(x)8(H)+2j2兀Hcomb(x)comb(H)exp(一兀x2)exp(一兀H2)

卷積及相關(guān)性質(zhì)交換性f(x)⑤g(x)=g(x)⑤f(x)平移不變性f(x-x)⑤h(x-x)=Jsf(a-x)h(x-a-x)da=g(x-x-x)121212多重卷積不變性f(x)⑤h(x)⑤h(x)=[f(x)⑤h(x)]⑤h(x)=f(x)⑤[h(x)⑤h(x)]121212縮放性質(zhì)f(ax)⑤h(ax)=~rg(ax)lal微積分性質(zhì)微分性質(zhì)—[f(x)⑤f(x)]=f(x)⑤df(x)=df(x)⑤f(x)dx121dx2dx12積分性質(zhì)Jx[f(X)⑤f(X也X=f(x)?Jtf(X)dX=f(x)?Jtf(X)dX121O21綜合性質(zhì)s(x)=[f1(x)⑤f2(x)]s(i)(x)=f(j)(x)?f(i-j)(x)其中i，j為求導(dǎo)的階次微積分性質(zhì)對求解廣義函數(shù)非常有用對于課上習(xí)題：計算rect(x/a)與rect(x/b)的卷積。采用卷積微積分性質(zhì)的話xxdxxtrect(―)⑥r(nóng)ect(一)=(—rect(一))⑥Jrect(—)dtabdxa_?b=—(step(x+~)-step(x-~))?Jxrect(~)dtdx22-sb=(8(x+—)-8(x一—))?Jxrect(-)dt22-sb其中rectU)的積分為bfxtxbbJrect()dt=rect()(x+)+step(x一)-sbb22這樣，可以在不討論a，b大小的情況下解出卷積axt。a。a2abab原式=Jrect(_)dt⑥(8(x+~)-8(x-~))=(rect()(x+—+—)+step(x+—+—)}-sax—2一(r