信号处理导论（英）9.4 Matrix Form of DFT.pptVIP

9.4 Matrix Form of DFT 9.5 Modulo-N Reduction 9.6 Inverse DFT 9.7 Sampling of Periodic Signals and the DFT * * The N-point DFT ( 9.2.8: ) can be thought of as a linear matrix transformation of the L-dimensional vector of time data into an N-dimensional vector of frequency data . (9.4.6) , (9.4.7) P.497中间的矩阵A的化简 * modulo-N reduction 定义： It is defined by dividing the signal x into contiguous non-overlapping blocks of length N ,wrapping the blocks around to be time-aligned with the first block , and adding them up . (9.5.1) , Fig.9.5.1 Example 9.5.1: * periodic extension of the signal x(n) with period N : (9.5.5) (9.5.2) ： xm(n) 是x(n) 的第m 段长度为N的子块 (9.5.3) ：用xm(n) 表示的modulo-N reduction 或者说Fig.9.5.1的结 果，注意此时 n=0,1,….,N-1。 (9.5.4) ： (9.5.3) 的 compact form ，注意此时 n=0,1,….,N-1。 (9.5.5) ： periodic extension of the signal x(n) with period N 注意此时 Fig.9.5.2 The connection of the mod-N reduction to the DFT : The connection is that the length-N wrapped signal has the same N-point DFT as the original unwrapped signal 即： Example 9.5.2: Example 9.5.3: 结论：Thus, even ….are the same. (Example 9.5.3 结束一段） * The two signals x and are not the only ones that have a common DFT . Any other signal that has the same mod-N reduction as x will have the same DFT as x 例： p.504 最上面四个序列都有相同的 ，所以有相同的DFT * 用 z 变换从另一个角度理解上述结果： Suppose the length-L signals y and x have equal mod-N reduction , p.504—p.505（了解一下） 注意：IDFT的结果是 ，而不是 。 因为 In order for the IDFT to generate the original unwrapped signal , it is necessary to have .This happens only if the DFT length N is at least L , so that there will be only one length-N sub-block in and there will be nothing to wrap around. (p.507最后一段) Fig.9.6.1 , (9.6.12) , (9.6.13) Example 9.6.1: * 用DFT计算I