dsp2纯课件数字信号处理第1讲

1、Ch8. The Discrete Fourier Transform2Main contentsThe DFS (Discrete Fourier Series ) of periodic sequencesThe DFT (Discrete Fourier Transform) of finite-duration sequencesLinear convolution using DFT38.1 Representation of periodic sequences: DFS: period is N Represent a periodic signal by a Fourier s

2、eries 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4 Determine Fourier series coefficientsSo, period is N5Synthesis:Analysis:notation: DFS representation of a periodic sequence68.2 Properties of the DFS1. LinearityLet, both with period NThen2. Shift of a sequence3. Shift of a Fourier coefficients

3、 (modulation property)IfThen4. Duality (对偶性)IfThen 5. Symmetry Properties(p550: Table 8.1 9-17)Proof: That is,6. Periodic Convolution(i)If ThenProof: (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

4、。(iv) The computation of periodic convolution138.3 The Fourier Transform of Periodic SignalsFT14Example 8.5 The FT of a Periodic Impulse TrainSo,DFS15: finite-length(i)(ii)Relationship between a periodic sequence and a finite-length sequenceFTFT16周期序列 的DFS系数 可通过对有限长序列xn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

5、列xn是 的一个周期，即：Example 8.3, 8.6Let , FTthen188.5. Fourier Representation of Finite-duration Sequence: DFTRelationship between periodic sequence and finite-duration sequence 周期延拓 主值序列19 From DFS to DFTDFS取主值区间20 DFT of finite-duration sequencesnotation:Synthesis:Analysis:21Relationship between DFT, ZT

6、and FTFT: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22Example xn=R4n, compute the DTFT, 4-point, 8-point and 16-point DFTs of xn. Solution: (ii) 4-point DFT(i) DTFT:(k=0,1, 7) (k=0,1,15) (iii) 8-point DFT(iv) 16-point DFT258.4 Sampling The Fourier TransformLet FTperiod: NsamplingLet

7、 be the IDFS of Conclusion: Sampling in F-domainPeriodic continuation in T-domain Sampling in T-domainPeriodic continuation in F-domain Compared with sampling in time-domainThen 27 Discuss: (i) xn: infinite-duration (ii) aliasing in no aliasing in aliasing in 8.4 Sampling The Fourier Transform28Frequency Sampling TheoremLet N: the number of frequency samples Compared with Nyquist Sampling Th