成果2sjtu-dsp課件老師群

1、Chapter 8 the discrete Fourier transform8.1 represenion of periodic sequen:the discrete Fourier seriesthe Fourier transform of periodic signalsproperties of the discrete Fourier series8.4 Fourier represenion of finite-duration sequen:Definition of the discreteFourier transformsampling the Fourier tr

2、ansform (poproperties of the Fourier transformof sampling)8.7 linear convolution using the discrete Fourier transform8.8 the discrete cosine transform(DCT)18.1 represenion of periodic sequen:the discrete Fourier series（離散級數(shù)）xn：periodic sequence ，Nx% r % n nfor xanyeger values of n and rX k : Fourier

3、 series 1N 1 knx n XkWN(1 n),.Nk 0N 1 Xk xnknW(2 k), ,.Nn0Wknj e 2 kn/N NNotet the sequence X kis periodic with period N:N 1Xk rN x kn(rN W)nXk2Nn 0證明二者互逆見課堂筆記Period N=10Figure 8.14sin( k / 2 )sin( k / 10 )X k n 0e j 4 kkn10/ 10W5EXAMPLE.相位X表示：幅度為零，相位不確定Figure 8.26時域離散化導(dǎo)致頻域周期化時域周期化導(dǎo)致頻域離散化8.2 the Fou

4、rier transform of periodic signalsthe Fourier transform ofx%nis defined as2 2 N 1 22kNX kN0ke ) jk (X (X)NNotherk 0DFS是計算周期信號頻譜段7時域離散化導(dǎo)致頻域周期化時域周期化導(dǎo)致頻域離散化證明見課堂筆記relationship betn the Fourier series coefficientsand the Fourier transform.FIGURE 8.5不看曲線98.3 properties of the discrete Fourier seriesAme:

5、xDFS DFS DFS n Xxk,1n X1xk, 2 n X2klinea1rit.y:ax n a DFS kn X k12Xbx1b 2N=4，作12點DFS兩個序列周期不同N=6，作12點DFS均當(dāng)成周期12序列N=12，作12點DFS證明見課堂筆記11asequence: xn m DFSW km k2.shiftofXNnl DFS Wxnk l XNn DFSNx k 3.duality: X14證明見課堂筆記4. Symmetry properties（對稱性質(zhì)）x *n DFSDFS * *X k ,x nXk Re xn 1 (xn x * n) DFS 1 ( k X

6、 k ) X e k *X2122DFS12jImxn (xn x n)( X k X k ) Xo k *DFS1212x n (xn x n)( X k X k ) Re X k *ex n 1 (xn x *n) DFS 1 ( k *k ) j Im k XXXo22175. For a real sequence:xn x*nX k X *k X *N k Re k Re k XXIm k Im k XXk | Xk | X X k N)點取樣，可用時域補零求大點數(shù)的DFT的方法。38Discuss teral situation:If the length of sequence

7、is N (can be infinite)，and the sampling number in frequencyis M (can be greatern, equal to or lessn N)，then the reconstructed signals are periodic with period M (may beoverlap).t is , ifX (e) k x nW, M0k ,1. M1j| 2knM nThen the IDFT is：xn(x nrM M)Rn r40證明見課堂筆記結(jié)論：當(dāng)頻域取樣點數(shù)M序列長度N時，重構(gòu)的時域信號是原始信號的有混迭的周期性延拓

8、并取主周期。反之，若想用DFT求解FT的M(MN，xn補零到M點，再用M點DFTMN，xn以M為周期延拓混迭，取長M的主周期，再用M點DFT頻域采樣能否恢復(fù)原始信號的時域頻域采樣定理：頻域取樣點數(shù)大于等于信號長度的可以重構(gòu)時域，反之則不能。頻譜相等，則頻譜取樣相等；反之不成立。頻譜線性相位，則頻譜取樣線性相位；反之不成立。488.6 properties of the Fourier transformAxme:n DFT ,DFTDFTX Xk x nXkx,nk11221. linearityDFT n axbxaXnk bXk12122. circular shift （循環(huán)或圓周移位）

9、of a sequenceDFTkmm )x( n R nWX kNNNlnDFT l N)W xn X( kRNkN49Definition of circular shift of a sequencex1n x(n m) N RN n x(n ( N m) N RN n圖示循環(huán)移位Figure 8.1250EXAMPLE.|Hk|s D|FTHk|8po12（8.42）h1nh2njejes DFT |H()|H()1| 024 po21EXAMPLE.3. Duality（對偶性）X n DFT Nx(k )R k NxN k NNxn n cos(0.2n)| X k | DFTxn

10、 |DFTX n52EXAMPLE.X (k ) N RN k X ( N k ) N RN k X N k 近似寫法X k X k 544. Symmetry properties:x*n DFT X *(k)R k X *(N k)R k X *N kNNNNRn x*N n DFT X *k x*(n)NNRexn 1 (xn x *n) DFT 1 ( X k X *N k ) Xk ep22j Imxn 1 (xn x *n) DFT 1 ( X k X *N k ) Xk op22xn 1 (xn x *N n) DFT 1 ( X k X *k ) ReX k ep22xn 1

11、(xn x *N n) DFT 1 ( X k X *k ) j ImX k op2255Here, we define:X epk : the perioonjugate-symmetric components（圓周共軛對稱分量）X opk : the perioonjugate-antisymmetric components（圓周共軛稱分量）Any finiength sequence can beed as:X k X epkop X k where,X Xk(k 1 X*Nk) *XN kep2epX Xk(k 1 X*N k)*XN kop2opThe length of the

12、 three sequenare all N.565. for a real sequence:X k X *N k xn x *nReX k ReX N k ImX k ImX N k | X k | X N k | X k =N1+N2-1,then xn*hn=xn(N)hn74EXAMPLE.Figure 8.18線性卷積線性卷積右移線性卷積6點循環(huán)卷積=線性卷積混迭加12點循環(huán)卷積=線性卷75pNpNbNc:calculate N pocircular convolution by linear convolution(a) y n xn*h nb(x)nN(h )n ynN rNR

13、 n rcalculainear convolution by circular convolutionaddinga () zxero nandhn toleng1th21ofb(x)nh n xn (N)hncalculainear convolution by DFTadding a () zexro nandhn toleng1th 21 ofTofxn andhnx(n) N (h) n IDFTXkHkx n *hnx n (N )hn76.(exercise 8.15)x1n x2n x1n(4)x2n77EXAMPLE.實際應(yīng)用中常用FIR對無限長或不定長序列濾波impleme

14、ntinglinear time-invariant FIR systems using the DFTFigure 8.22需要實時處理（即邊輸入邊處理，并且速度快）。可采用以下兩種方法。也可用時域直接實現(xiàn)（速度較慢）7。9overlap-add methodhn length is P=P 的段，斷間相連（2）分別對 h(n)點FFT和本段的x(n)補零作L+P-1y(n) IFFTH (k) X (k)，n 0,.L P 2求取y(n)的n=0P-2點與前段yn的后P-1點相加，n=0L-1點作為該段的輸出81overlap-save method線性卷積結(jié)果hn length is P

15、=L最前P-1 點與前段相同L點的循環(huán)卷積是線性卷積以L為周期延拓，能保證中間L-P+1點沒有混迭，可作為當(dāng)前段的輸出Figure 8.2482步驟:hn長度為P（1）將x (P-1點n分)成長度為L的段，斷間（2）分別對h (n)和本段的x (n補)k零作L點DFTIFFT)，X k0 y（(n3）) 求（4）取y () H(n,.L1n)中的L-P+1點作為該段的輸出n=P-1,.L-1如果L+P-1點DFT，則將結(jié)果的前后各P-1點去除，輸出中間L-P+1點，浪費。保證輸出的L-P+1是線性卷積結(jié)果的最小DFT點數(shù)是L，不是L+P-1。83總結(jié)DFT的用途：計算信號的頻譜的取樣計算系統(tǒng)的

16、頻響的取樣（FIR和IIR）FIR系統(tǒng)的頻域?qū)崿F(xiàn)848.8 the discrete cosine transform(DCT)N 1kn2DCT 1：X k ),0 k N 1cc xncos(knN 1N 1n0N 1kn2xn ),0 n N 1cc X kcos(nkN1N1n0N 1k (2n 1)2NDCT 2：X k ),0 k N 1cxncos(k2Nn0N 1k (2n 1)n02Nxn ),0 n N 1c X k cos(k2N85N 1n(2k 1)2NDCT 3：X k ),0 k N 1c xncos(n2Nn0N 1n(2k 1)2Nxn ),0 n N 1cX

17、 k cos(k2Nn0N 1 (2n 1)(2k 1)2NDCT 4：X k ),0 k N 1xncos(4Nn0N 1 (2n 1)(2k 1)n02Nxn ),0 n N 1X k cos(4N1/ 2, k 0ck 1,1 k N 186對信號作對稱延拓和周期延拓，作DFS再取主周期得到DCT。DCT-1-2 1. 0. 1. 2. 3.4. 5DCT-2-4 3 -2-10. 1. 2. 3. 4. 5. 6. 787延拓后序列的2N點DFT與原序列的 N點DCT的關(guān)系DCT 2xn, n 0,.N 1yn X 2N 1 n, n N ,.2N 12N 1N 12N 1n NknknknYk ynWxnW 1 nWx2N2 N2N2Nn0n0N 1N 1k (2N 1n)knxnWxnW2N2 Nn0n0 xncos(2n 1)k )N 1 2W k / 22N2Nn02 c 2W k / 2 X k /()2NkN89Comparewith DFT