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

Z-plane Unit circle xIn==a ul-n x(z) for z<a 19e X(z) r|2>a 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ

23 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. ( ) z X z z a = − for z a  xn= −a u− n −1 n for z a  xn a un n = ( ) z X z z a = −

EX, 3.3 Sum of two exponential sequences Determine the z-transform, including the roc pole-zero-plot, for sequence XIn 小+- Solution n x(2)=∑巛uln n|+ uln>z n=(2/un|2”+ ∑(-= n=-00 n ∑ n=0 n=0 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ

24 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Ex. 3.3 Sum of two exponential sequences       1 1 2 3 n n x n u n u n     = + −         ( )     1 1 2 3 n n n n X z u n u n z  − =−         = + −                0 0 1 1 1 1 2 3 n n n n z z   − = − =     = + −               1 1 2 3 n n n n n n u n z u n z   − − =− =−     = + −           ◆Determine the z-transform, including the ROC, pole-zero-plot, for sequence: Solution:

Example 3.3: Sum of two exponential sequences X(z)=∑ n=0 2z z 12 1+-z t-Z 2 ROC: z> and z>3=ROC: 2> 25 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ

25 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Example 3.3: Sum of two exponential sequences 1 1 2 3 1 2 z  and R z   OC : z  ( ) 1 0 0 1 1 1 2 3 n n n n X z z z   − = − =     = + −           1 1 1 2 12 1 1 1 1 2 3 z z z z − −     −   =         − +     1 1 1 1 1 1 1 1 2 3 z z − − = + − + ROC:

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

26 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 3.4: Sum of two exponential x[n 2 Solution X(2)= az 2 2 or z> +-z ROC. 小+(-小 2 2 +-z vi99w-MMi vn. g UniV

27 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Example 3.4: Sum of two exponential   2 1 2 1 1 1 2 1 1  −        − , z z u n Z n   3 1 3 1 1 1 3 1 1  +        − − , z z u n Z n     2 1 3 1 1 1 2 1 1 1 3 1 2 1 1 1  + + −         + −      − − , z z z u n u n Z n n       1 1 2 3 n n x n u n u n     = + −         xn a un n = ( ) 1 1 1 X z az − = − for z a  Solution: ROC: