DFT計(jì)算卷積教學(xué)課件

數(shù)字信號(hào)處理

(DigitalSignalProcessing)信號(hào)與系統(tǒng)系列課程組國(guó)家電工電子教學(xué)基地DFT計(jì)算卷積離散傅里葉變換(DFT)

問(wèn)題的提出有限長(zhǎng)序列的傅里葉分析離散傅里葉變換的性質(zhì)

利用DFT計(jì)算線性卷積利用DFT分析信號(hào)的頻譜DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積

利用DFT計(jì)算序列線性卷積的步驟長(zhǎng)序列和短序列的線性卷積利用DFT計(jì)算序列線性卷積DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積問(wèn)題提出：實(shí)際需要：LTI系統(tǒng)響應(yīng)y[k]=x[k]

h[k]可否利用DFT計(jì)算線性卷積？?jī)蓚€(gè)有限長(zhǎng)序列的線性卷積DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積兩個(gè)有限長(zhǎng)序列的線性卷積設(shè)x[k]的非零范圍是0

k

N-1g[k]的非零范圍是0

k

M-1y[k]=x[k]

h[k]非零范圍序列y[k]的長(zhǎng)度為0

k

N+M-2L=N+M-1至少要做L=N+M-1點(diǎn)的循環(huán)卷積DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積利用DFT計(jì)算序列線性卷積的步驟

若x[k]的長(zhǎng)度為N，h[k]的長(zhǎng)度為M，則L

N+M-1點(diǎn)循環(huán)卷積等于x[k]與h[k]的線性卷積。DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積例：利用MATLAB由DFT計(jì)算x[k]*h[k]。

x[k]={1,2,0,1},h[k]={2,2,1,1}%CalculateLinearConvolutionbyDFTx=[1201];h=[2211];%determinethelengthforzeropaddingL=length(x)+length(h)-1;%ComputetheDFTsbyzero-paddingXE=fft(x,L);HE=fft(h,L);%DeterminetheIDFToftheproducty1=ifft(XE.*HE);DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積直接計(jì)算與由DFT間接計(jì)算結(jié)果比較DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積長(zhǎng)序列和短序列的線性卷積

直接利用DFT計(jì)算的缺點(diǎn)：(1)信號(hào)要全部輸入后才能進(jìn)行計(jì)算，延遲太多(2)內(nèi)存要求大(3)算法效率不高

解決問(wèn)題方法：采用分段卷積分段卷積可采用重疊相加法和重疊保留法DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積長(zhǎng)序列和短序列的線性卷積1.重疊相加法（overlapadd）將長(zhǎng)序列x[k]分為若干段長(zhǎng)度為L(zhǎng)的序列定義(短序列h[k]的長(zhǎng)度為M)DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積長(zhǎng)序列和短序列的線性卷積1.重疊相加法（overlapadd）y0[k]的非零范圍y1[k-L]的非零范圍序列y0[k],y1[k]的重疊部分重疊的點(diǎn)數(shù)L+M-2-L+1=M-1

依次將相鄰兩段的M-1個(gè)重疊點(diǎn)相加，即得到最終的線性卷積結(jié)果。DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積重疊相加法分段卷積舉例DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積重疊相加法分段卷積舉例(L=7,M=4)DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積解:

重疊相加法(M=3)y[k]=x[k]*h[k]={2,7,12,16,20,24,28,32,36,40,44,48,52,41,14}x1[k]={2,3,4,5,6}x2[k]={7,8,9,10,11}x3[k]={12,13,14,0,0}y1[k]=x1[k]*h[k]={2,7,12,16,20,17,6}y2[k]=x2[k]*h[k]={7,22,32,36,40,32,11}y3[k]=x3[k]*h[k]={12,37,52,41,14,0,0}例:

已知序列x[k]=k+2,0

k

12,h[k]={1,2,1}試?yán)弥丿B相加法計(jì)算線性卷積,取L=5。DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積fftfilt(h,x,n)h:FIRfilterx:inputsequencen

為DFT點(diǎn)數(shù)，一般取2的整數(shù)次冪利用MATLAB實(shí)現(xiàn)分段卷積DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積利用MATLAB實(shí)現(xiàn)分段卷積%GeneratethenoisesequenceN=64;d=rand(1,N)-0.5;%Generatetheuncorruptedsequenceandaddnoisen=0:N-1;s=2*n.*((0.9).^n);x=s+d;%thelengthofmovingaveragefilterM=4;%Generatethemovingaveragefiltercoefficientsh=ones(1,M)/M;%Performtheoverlap-addfilteringoperationy=fftfilt(h,x,8);DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積4點(diǎn)滑動(dòng)平均系統(tǒng)去噪結(jié)果DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積若x1[k]為M

點(diǎn)序列,x2[k]為L(zhǎng)

點(diǎn)序列，L>Mx1[k]L

x2[k]中哪些點(diǎn)不是線性卷積的點(diǎn)?問(wèn)題討論DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積0

k

M-2不是線性卷積的結(jié)果，即前(M-1)個(gè)點(diǎn)與線性卷積不一樣。DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積對(duì)比圖DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積

x1[k]L

x2[k]k=0~M-2,前M-1個(gè)點(diǎn)不是線性卷積的點(diǎn)k=M-1~L-1,L-M+1個(gè)點(diǎn)與線性卷積的點(diǎn)對(duì)應(yīng)線性卷積L~L+M-2后M-1點(diǎn)沒有計(jì)算

則L點(diǎn)循環(huán)卷積結(jié)論若x1[k]為M

點(diǎn)序列,x2[k]為L(zhǎng)

點(diǎn)序列，L>MDFT計(jì)算卷積計(jì)算卷積計(jì)算卷積2.重疊保留法（overlapsave）方法：

(1)

將x[k]長(zhǎng)序列分段，每段長(zhǎng)度為L(zhǎng)。

(2)

各段序列xn[k]與M點(diǎn)短序列h[k]循環(huán)卷積。

(3)

從各段循環(huán)卷積中提取線性卷積結(jié)果。前M-1個(gè)點(diǎn)不是線性卷積的點(diǎn)，因yn[k]=xn[k]Lh[k]故分段時(shí)，每段與其前一段有M-1個(gè)點(diǎn)重疊。長(zhǎng)序列和短序列的線性卷積DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積長(zhǎng)序列和短序列的線性卷積2.重疊保留法（overlapsave）--x[k(M1)]M-1--L(M1)L-1x0[k]x1[k]2L-Mk第一段前需補(bǔ)M-1個(gè)零DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積2.重疊保留法（overlapsave）記：yn[k]=xn[k]Lh[k]長(zhǎng)序列和短序列的線性卷積DFT計(jì)算卷積計(jì)算卷積計(jì)算卷積解:

重疊保留法y[k]={2,7,12,16,20,24,28,32,36,40,44,48,52,41,14}x1[k]={0,0,2,3,4}x2[k]={3,4,5,6,7}x3[k]={6,7,8,9,10}y1[k]=x1[k]⑤h[k]={11,4,2,7,12}x4[k]={9,10,11,12,13}y2[k]=x2[k]⑤

h[k]={23,17,16,20,24}y3[k]=x3[k]⑤

h[k]={35,29,28,32,36}y4[k]=x4[k]⑤

h[k]={47,41,40,44,48}x5[k]