离散时间信号处理（英文版）chap8-第2讲.pptVIP

* Homework 8.10, 8.12，8.27, 8.58 , 8.56 (selective) * 8.6 Properties of the DFT 1. Linearity If x1[n] has length N1, and x2[n] has length N2 Then the maximum length of x3[n] is So, both DFTs must be computed with the same length 2. Circular Shift of a sequence (i) definition x(n) e.g. N-point circular shift in one direction by m is the same as a circular shift in the opposite direction by (N-m) (ii) The theorem of circular shift in time-domain (iii) The theorem of circular shift in frequency-domain 3. Duality If Then (i) Proof 4. Symmetry Properties (ii) To any finite-length sequence x[n] —— periodic conjugate-symmetric component —— periodic conjugate-antisymmetric component Then Example: x1[n] and x2[n] are two N-point real sequence, compute their DFTs by using a N-point DFT only once. Solution: Let So, Then, 5. Circular Convolution (ii) Circular convolution Theorem (i) Definition N N Note: The circular convolution of x1[n] and x2[n] is the principle sequence of the periodic convolution of and . N N 圆周卷积的长度: (iii) The procedure of circular convolution 5 7. Summary of Properties of DFT (p576: Table 8.2) 6. Pasvel theorem in DFT form Prove: * 8.7 Linear Convolution Using the DFT 两个序列时域圆周卷积在频域上相当于它们的DFT相乘，因而可以采用DFT的快速算法——FFT，与线性卷积相比，计算速度可大大加快。 一般实际问题都是线性卷积运算。 如果信号以及系统的单位冲激响应都是有限长序列，那么能否用圆周卷积运算来代替线性卷积运算呢？ * (i) Linear convolution Let (ii) Circular convolution length: (L+P-1) N length: Relationship between linear convolution and circular convolution * Relationship between linear convolution and circular convolution N点的圆周卷积 是线性卷积 以N为周期进行周期延拓所得周期序列的主值序列 * N (i) Discussion N (ii) L (iii) N L+P-1 L Let L P, N = L (iv) Conclusion When , N-point circular convolution can substitute for linear convolution. N * Eg (L=5) (P=3) (L+P-1=7) 5 (L+P-1=7) 6 (N=6) (L+P-1=7) 7 (N=L+P-1=7) (L+P-1=7) 8 (L+P-1=7) (N=8) * Implementing LTI Systems Using the DFT Let N Then N So, zero-padding