山东大学：《生物医学信号处理 Biomedical Signal Processing》精品课程教学资源（PPT课件讲稿）Chapter 03 The z-Transform

手 m z-plane n z-plane 2 e ORe 1+-z 2z 12 +-Z Re 2 n xIn un+ 2 Zhongguo Liu_ Biomedical Engineering_shandong Univ

28 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 1 1 1 1 2 z − − 1 1 1 1 3 z − + 1 1 1 2 12 1 1 1 1 2 3 z z z z − −     −           − +           1 1 2 3 n n x n u n u n     = + −        

Example 3.5 wo-sided exponential sequence n n x[小]=(-5)2 [m-,x -au-n Solution X(2) az uln<> 1+-z O|2<a -n-1 2 22 X( ROC: -<z< +z-11 1+-z 2 2 29 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ

29 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Example 3.5: Two-sided exponential sequence   ( )   ( )   1 1 1 3 2 n n x n u n u n = − − − − ，   1 1 1 , 1 3 1 3 1 3 n Z u n z z −     −     +   1 1 1 1 , 2 1 1 1 2 2 n Z u z n z −   − − −    −    ( ) 1 1 1 1 1 2 1 1 12 : 1 1 1 1 1 1 3 3 3 2 2 2 z z X z ROC z z z z z − − − −     −   = + =     + −     + −       ( ) 1 1 1 X z az − = − for z a     1 n x n a u n = − − − Solution:

2z1 x(=)= ROC:-<z< 1+z1-x1|1+1z Z-plane ROC pole-zero-plot 9 30 2021/2/6 n

30 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. ROC, pole-zero-plot      1 2 1 3 1  − −       −      x n = − u n u n n n ( ) 1 1 1 1 1 2 1 1 12 : 1 1 1 1 1 1 3 3 3 2 2 2 z z X z ROC z z z z z − − − −     −   = + =     + −     + −      

Finite-length sequence X(z)=∑xn]z n=N Example x]=olz]+6[n-5 X(z)=1+ ROC or entire z-plane, except z=0 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ

31 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Finite-length sequence xn= n+ n−5 ( ) 5 X z z 1 − = + ， ( )   2 1 N n N n X z x n z = − =  Example : or entire z-plane, except z=0 ROC z : 0 

Example 3.6: Finite-length sequence Determine the z-transform the roc pole zero-plot(N=16 ,-1<a<1) for sequence x an,0<n<N一1 0. otherwise <0 Solution z-transform X(2)=∑a"z=∑(z) z<∞ =0 0 N ROC:|2≠0 az or 1-az N-1 Le including z=a 32 2021/2/6 Zhongguo Liu_ Biomedical Engineering_ Shandong Univ

32 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Example 3.6: Finite-length sequence        − = , otherwise a , n N x n n 0 0 1 ( ) ( ) 1 1 1 0 0 N N n n n n n X z a z az − − − − = = = =   ROC : 1 1 . N N N z a z z a − − = − ( ) 1 1 1 1 N az az − − − = − ◆Determine the z-transform, the ROC, pole￾zero-plot(N=16, -1<a<1), for sequence: Solution: z-transform including z=a , a   1 , az −   z  0 z  0 or