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傅傅立葉變換的幾種形連續(xù)的時間非周期信號x。xt(xt連續(xù)的時間非周期信號x。xt(xt-xWt1|x(jW( xtpW級傅立xt周期為 的周期性連續(xù)時間函數(shù)x級傅立xt周期為 的周期性連續(xù)時間函數(shù)xt)展成立葉級數(shù)，其系數(shù)x(jkWx(jkW)是離散頻率12t0tdt0Tp2T-p￥t0-|x(jkW0WtW0=2pW0=2p|x(jkW0WtW0=2pW0=2pF=k序列的Xejw序列的Xejwne-pXjwejwnx |w=WTW=2p,那,fssTTn=-Xj=2p,那,fssTTn=-XjWnTeT2WXsjWjnWTd W2-Wss|Xejwxn)nW=-WWWsT|Xejwxn)nW=-WWWsTW0=2pW0=2pTWs=離散的周期序列的傅離散的周期序列的傅立葉變立葉奇數(shù)設(shè)nn￥-|xn||z-|=jj2pW0=2p2pj2pNw0=2p2pj2n neNjj2n neNjNnknj2pk+rNj2pe=,rNN：erNn=n所以對DFS而言，只能取k=0到kN-1個獨所以對DFS而言，只能取k=0到kN-1個獨-j2p1kxn=NN這里xk是k次的諧波系數(shù)，下一步求解系數(shù)xkrmN為任意整1=Nnk1nk1xkn=0kxnN=Nnk-rjNN-N-=xk=xrk=n=-j2pxk)=nN這就是k=0,",N-1的N個諧波系數(shù)，x這就是k=0,",N-1的N個諧波系數(shù)，xk的公式nkxk+mN)=xnNn-j2p=xn=xkNn即：xk亦是以N為周期2Nj-we=N-j2pxk)=DFSx-j2pxk)=DFSx=xn=xnwNNxn)=IDFSxkxkxk-=NNNk周期序列xkxn一個周期xn作Zx)xz)=x)xz)=)x ￥則xn的Z變換為xk)=xz)z=wN-kNRe(zDFSDFS的性nxk)若：x1nx2n是周期為Nx1k)=DFSkDFS1n+bx2=ax1k+bx2kDFSxn+mDFSxn+m=kxkNDFSn+m)=n+mwxiwikw-=NNN和wNik都是以Nx)DFSwNnlxn=DFSwNnlxn=xk+lnnwwNNn)k)n=xklN如果：yk)x如果：yk)x1kx2kynxnxn12=12k11NN-1N-k11NN-1N-yn)=xmxknmk12Nk=0k)w=N2k=x1m)x2n-k同樣通過變量代換，有同樣通過變量代換，有2m)1n1）結(jié)果yn)的周期為N；2）求和只在一個周ynnlx)llk12離散傅立葉變化有限長列的離離散傅立葉變化有限長列的離散頻率表xn0￡n￡N-￥xnrN)xn=()=x通常把x通常把xn)的第一周n0到nN-1“主值區(qū)間xn)xnN)xnn)xn)用nN表示n對N取余數(shù)或稱n對N取模值。xn是周期為9\9=7∵n=\9=7∵n=25=∵n=-51·9+4\9=25=x259x7)x5)=)=x4)9利用矩形序列RNn，即：0￡n￡N-R N0所以xn)x所以xn)xnRNn)對于頻率的周期序列xk)xk)=xk)xkkRNkNn0~N-1及k0~N-主值序xn)Xk)k=k=NXxn=DFT=(kNx2x21-x2x21-x22-mx2x22-mx2DFT變DFT變換的性kk.DFTax1n+bx2=aX1k)+bX2kn),nn),n周期為kk②若x1n的周期為N12n)的周期為,aXkk的周期?且N=max(N1,N2的周的周期，將x延拓成周期序xn)位，然后取主值區(qū)間n0~N-1n)=xnNnn+m)fixn的周期延拓序列xn)的移Nxn+mRNn表示對此延拓xn+mRNn表示對此延拓移位后的周期序 n knmmR=wNNNDFSxn+m)=DFSxn+=-mkxkNNn+mN=wN-mk = -NN-j2pwNN同樣對于頻域有限長序列Xk同樣對于頻域有限長序列XkXk-j2pk) k=n)=xn)NNNNDFTxnsin2pnl=1DFTxnsin2pnl=1Xk-l-Xk+l)k2NNNNk+Xk+l)k2NNNNm0m0m0m0m0m0m0m01212(((((1212(((((( nxn+ * , nxn- *neo軛對稱分量及反對稱共軛分量的長度都是2N-n和xonn和xon入手因此須從fiNn)n：圓周共軛反對稱分nnNxo(n)x(n)=1[x(n)+xo(n)x(n)=1[x(n)+x*(-n)]=1+x*((N-]eNN22x(n)=1[x(n)-x*(-n)]=1-x*((N-]oNN22x(n)=x*(-eex(n)=x*(-oo 的圓周共軛對稱分量xep(n)和圓xop(n)=x(n)R(n)=1(n)=x(n)R(n)=1x+x*((N-eNNNN21 (n)=x(n)R(n)= -x*((N-n))N]RN2∵x(n)=xe(n)+xo\x(n)=x(n)RN(n)=[xe(n)+xo(n)]RN=xep(n)+xopRe[x(n)]=1[x(n)+2jIm[x(n)]=1[x(n)-2DFT[x*(nDFT{Re[x(nj DFT[x*(n)]=X*((-k))DFT[x*(nDFT{Re[x(nj DFT[x*(n)]=X*((-k))NRN(k)=X*((N-k))RN(kx*(n)表DFT[x*(n)]**xR(k=x(k= NN*(NNk =X*((-kR(k=x](k N=X*((N-k))NRN(k-j2p=e-j2p=-j2p=e-j2p==NNw-=w(N-kNNDFT[x*((-n))X*(k(2R (n)]N證DFT[x*((-n))NRN=x*((-n))RNNw-nkx*((-= =[w-nkx*((-= =[wnk wnkx*(n)= =X*(kDFT{Re[x(n)]}=Xep(k+X*((N-k(kNNN2Re[x(n)]=1[x(n)+2\DFT{Re[x(n)]}=1{DFT[x(n\DFT{Re[x(n)]}=1{DFT[x(n)]+2=1[X(k)+X*((N-kR 2=1[X((k))+X*((N-k))NN2=XepDFT{jIm[x(n)]}=Xop(k)-DFT{jIm[x(n)]}=Xop(k)-X*((N-k(kNNN2jIm[x(n)]=1[x(n)-x*(n)]2DFT{jIm[x(n)]}=1{DFT[x(n)]-2故=1[X(k)-X*((N-R 2=1[X((k))-X*((N-kNN2=Xop*X(k)= ((N-kR(k*X(k)= ((N-kR(k) Xep((N-k))NRN(k)=RN(k*X((N-k R(k)(k)=-NNXop(kRe[Xop(Xop(kRe[Xop(k)]=-Re[Xop((N-k))NRN(kIm[ (k)]=Im[R(k((N-k （5）.若x(n)是實序列，則顯X(k)=X*((N-k))RN(k（6）.若x(（6）.若x(n)是純序數(shù)序列，那X(k)=-X*((N-kN (k).N上述兩種情況，只要知道一半數(shù)目的X(k)可以了，另一半可以利用對稱性求出x(n)X(k)的對稱特性，同樣可以找的實部、虛部與X(n)的圓周共軛對稱部分及(7(8)DFT[Xep(7(8)DFT[Xep(n)]=Re[x(kDFT[Xop(n)]jIm[x(kx1(nx2(n)DFT[x1(n)],DFT[x2w(n)=x1(n)+jx2則DFT[w(n)]=W(k)=DFT[x1(n)+jx2則DFT[w(n)]=W(k)=DFT[x1(n)+jx2=DFT[x1(n)]+jDFT[x2=X1(k)+jX2(k又x1(nRe[w(n)]x1(k)=DFT{Re[w(n)]}=Wep(k1*x(k)=[W(k)+W((N-k(k1NN2x2(n)=故x(k)=11[W(k)-W*((N-k(k)(k2NNj24.DFT形式下的帕斯瓦爾定N**x(n)y(n)X(k4.DFT形式下的帕斯瓦爾定N**x(n)y(n)X(k)Y(kkN11Y(k)W-nk證：x(n(n)*NN Y*(k =NN*X(k)Y(k=N若y(n)=N**x(n)x(若y(n)=N**x(n)x(n)X(k)X(k=122X(k=即N5.圓周卷設(shè)x5.圓周卷設(shè)x1(n)和x2(n)都是長度為N的有限長序(0￡n￡N-DFT[x1(n)]=X1(k),DFT[x2(n)]=X2(kY(k)=X1(k)X2(ky(n)=IDFT[Y(k=[x1(m)Nx2((n-m))N]RN=[x2(m)Nx1((n-m))N]RN先將先將Y(k)Y(k)=X1(kX2(ky(n)=x1(m)x2(n-m)=x1((m))Nx2((n-由于0￡m￡N-1=x1\y(n)=y(n)RN(n)=[x1((m))Nx2((n-m))N]RNm=x2((n-到N-m=x2((n-到N-x2(n)=[x1(m)x2((n-m))N]RNN=[x2(m)x1((n-m))N]RNy(n)=x2若1則Y(k)DFTy(n)]X(kX(kN12N………x2(N-xxx(Nx2………x2(N-xxx(Nx22x2#2#21x2(3)=####…若y(n)=x2x1(n),x2NN]RN(kY(k)=DFT[y(n)][X(l)X2((k-l＝1[(l)X((k-(k21NNN1l=X1(kX2(kNNx2((0-m))NRNx2((1-m))NRNx2((2-m))x2((0-m))NRNx2((1-m))NRNx2((2-m))NRN有限長序列的線性卷積與圓周卷x1(n)長度為(n)長度為有限長序列的線性卷積與圓周卷x1(n)長度為(n)長度為￥x1(m)x2(n-m)=x1(m)x2(n-l(nx2(m):0￡n-m￡N2-對x1(m):0￡m￡N1-有：0￡n￡N1N22長度為N2-1Nx0￡n￡N1-N1￡n￡L-xN2￡n￡L-1x(n)2x(n)12yc(n)=[x1(m)xyc(n)=[x1(m)x2((n-]RLL￥x2(n)=x2((n))L=x2(n+yc(n)=[x1(m)x2((n-m))L]RL￥=[x1(m)x2(n+rL-m)]RL￥L-￥=[x1(m)x2(n+rL-m)]RL(n)=[ylCircularConvolutionasConvolutionCircularConvolutionasConvolutionwithX(ejw)canbeconsideredassampledatwk=2pk/N,thentheresultingsequencecorrespondstotheDFScoefficientsoftheperiodicsequence￥x[n]=x[n-rNX(ej(2pk/N)0￡k￡N-X[k]=[n]=x[n],0￡n￡N[n]=x[n],0￡n￡N-xp0, X3 )=X1(e )X2 j(2pk/NX3(k)=X3 X3(k)=X1(k)X2(k0￡k￡N-￥x[n-rN0￡n￡N1 [n]=33x3p[n]=x1[n]:ExampleExample8.12CircularasLinearConvolutionwithConsidertwofinite-durationsequencesx1[n]andxConsidertwofinite-durationsequencesx1[n]andx2[n],withx1[n]oflengthLandx2[n]oflengthP,P<LLetusconsidertheL-pointcircularconvolutionx1[n]andx2[n].Thelinearconvolutionofx1[n]andx2[n]willhavethelengthofL+P-1x￥:x[n]x[n-rL],0￡n￡L1 [n]=1L233r=-r=-??isinfluencedonlyx3??isinfluencedonlyx3[n]andx3[n+x3ponlythex3[n+P<willaliasintothe?Block￥x[n]=x[Block￥x[n]=x[n-rL],rx[n]=x[n+rL],0￡n￡L-r￥y[n]=x[n]*h[n]=yr[n-ryr[n]=xr[n]*Overlap-addOverlap-addOverlap-saveOverlap-save§抽樣Z變換§抽樣Z變換—頻域抽樣理Xk)X￥x(z)=x(n)z-→→￥=X(k)=X(Z)∴N→→￥=X(k)=X(Z)∴N1N1jiiXkNN-[￥￥N=W(m-] =NNN1N1k(m-n)k1,m=1N1k(m-n)k1,m=n+W=N￥XN(n)=x(n+rNXkN→→x(n)有限長序列M,x(n)N≥→x(n)有限長序列M,x(n)N≥M￥X(n)= (n)=x(n),N?NNNX(ejW)W=X(k),=0,1,...,N-NkX(kN0≤n≤N-1。X(Z)x(n)Z-N-1-nkx(n)X(kX(kN0≤n≤N-1。X(Z)x(n)Z-N-1-nkx(n)X(k)W∵Nk=011---X(Z)[x(kx(kW =NNNNkkx(k-NkZ-1-1-Z-1kx(kkN==-kZ-N1-N1-NN1-Z-1X(Z)1-Z-1X(Z)=x(k)jk(Z),jk(Z).-k 1-k=Nj2pZ=,r=0,1,2,...,k,...,N-NZ