离散时间信号处理（英文版）chap8-第1讲.pptVIP

Ch8. The Discrete Fourier Transform * Main contents The DFS (Discrete Fourier Series ) of periodic sequences The DFT (Discrete Fourier Transform) of finite-duration sequences Linear convolution using DFT * 8.1 Representation of periodic sequences: DFS : period is N Represent a periodic signal by a Fourier series 周期为N的序列，谐波成份仅N个独立 kth harmonica sequence 故求和只取N项，即 * Determine Fourier series coefficients So, —— period is N * Synthesis: Analysis: notation: DFS representation of a periodic sequence * 8.2 Properties of the DFS 1. Linearity Let , both with period N Then 2. Shift of a sequence 3. Shift of a Fourier coefficients (modulation property) If Then 4. Duality (对偶性) If Then 5. Symmetry Properties (p550: Table 8.1 9-17) Proof: That is, 6. Periodic Convolution (i) If Then Proof: (ii) If Then (iii) The difference between periodic convolution and aperiodic convolution 周期卷积的结果也是周期为N的周期序列 周期卷积的求和只在一个周期[0, N-1]上进行，将所得结果进行周期延拓，就得到整个周期序列。 (iv) The computation of periodic convolution * 8.3 The Fourier Transform of Periodic Signals FT * Example 8.5 The FT of a Periodic Impulse Train So, DFS * : finite-length (i) (ii) Relationship between a periodic sequence and a finite-length sequence FT FT * 周期序列 的DFS系数 可通过对有限长序列x[n]的傅里叶变换 进行等间隔采样而得到，采样间隔为 ，且有限长序列x[n]是 的一个周期，即： Example 8.3, 8.6 Let , FT then * 8.5. Fourier Representation of Finite-duration Sequence: DFT Relationship between periodic sequence and finite-duration sequence —— 周期延拓 —— 主值序列 * From DFS to DFT DFS 取主值区间 * DFT of finite-duration sequences notation: Synthesis: Analysis: * Relationship between DFT, ZT and FT FT: ZT: DFT: (ii) N点 DFT是ZT在单位圆上的N点等间隔采样 (i) N点 DFT是FT在区间 上的N点等间隔采样 * Example x[n]=R4[n], compute the DTFT, 4-point, 8-point and 16-point DFTs of x[n]. Solution: (ii) 4-point DFT (i) DTFT: (k=0,1,…, 7) (k=0,1,…,15) (iii) 8-point DFT (iv) 16-point DFT * 8.4 Sampling The Fourier Transform Let FT period: N samplin