Bài giảng Digital Signal Processing - Chapter 5: z-Transform - Nguyen Thanh Tuan

In inverting a z-transform, it is convenient to break it into its partial
fraction (PF) expression form, i.e., into a sum of individual pole
terms whose inverse z transforms are known 
In general, the z-transform is of the form
 The poles are defined as the solutions of D(z)=0. There will be M
poles, say at p1, p2,…,pM . Then, we can write
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  1. Chapter 5 z-Transform Nguyen Thanh Tuan, ClickM.Eng. to edit Master subtitle style Department of Telecommunications (113B3) Ho Chi Minh City University of Technology Email: nttbk97@yahoo.com
  2. Content 1. z-transform 2. Properties of the z-transform 3. Causality and Stability 4. Inverse z-transform Digital Signal Processing 3 z-Transform
  3. Example 1  Determine the z-transform of the following finite-duration signals a) x1(n)=[1, 2, 5, 7, 0, 1] b) x2(n)=x1(n-2) c) x3(n)=x1(n+2) d) x4(n)=(n) e) x5(n)=(n-k), k>0 f) x6(n)=(n+k), k>0 Digital Signal Processing 5 z-Transform
  4. z-transform and ROC  It is possible for two different signal x(n) to have the same z- transform. Such signals can be distinguished in the z-domain by their region of convergence.  z-transforms: and their ROCs: ROC of a causal signal is the ROC of an anticausal signal exterior of a circle. is the interior of a circle. Digital Signal Processing 7 z-Transform
  5. 2. Properties of the z-transform  Linearity: x (n)z X (z) with ROC if 1 1 1 z and x2 (n) X 2 (z) with ROC2 then z x(n) x1(n) x2 (n) X (z) X1(z) X 2 (z) with ROC ROC1 ROC2  Example: Determine the z-transform and ROC of the signals a) x(n)=[3(2)n-4(3)n]u(n) b) x(n)=cos(w0 n)u(n) c) x(n)=sin(w0 n)u(n) Digital Signal Processing 9 z-Transform
  6. 2. Properties of the z-transform  Time reversal: z if x(n) X (z) ROC: r1 | z | r2 1 1 then x( n)z X (z 1) ROC: | z | r2 r1 Example: Determine the z-transform of the signal x(n)=u(-n).  Scaling in the z-domain: x(n)z X (z) ROC: r | z | r if 1 2 then n z 1 a x(n) X (a z) ROC: | a | r1 | z | | a | r2 for any constant a, real or complex n Example: Determine the z-transform of the signal x(n)=a cos(w0n)u(n). Digital Signal Processing 11 z-Transform
  7. 3. Causality and stability  Mixed signals have ROCs that are the annular region between two circles.  It can be shown that a necessary and sufficient condition for the stability of a signal x(n) is that its ROC contains the unit circle. Digital Signal Processing 13 z-Transform
  8. Partial fraction expression method  In general, the z-transform is of the form 1 N N(z) b0 b1z  bN z X (z) 1 M D(z) 1 a0 z aM z  The poles are defined as the solutions of D(z)=0. There will be M poles, say at p1, p2, ,pM . Then, we can write 1 1 1 D(z) (1 p1z )(1 p2 z )(1 pM z )  If N < M and all M poles are single poles. where Digital Signal Processing 15 z-Transform
  9. Example 5od Digital Signal Processing 17 z-Transform
  10. Example 6  Compute all possible inverse z-transform of Solution: -2 - Find the poles: 1-0.25z =0 p1=0.5, p2=-0.5 - We have N=2 and M=2, i.e., N = M. Thus, we can write where Digital Signal Processing 19 z-Transform
  11. Example 7 (cont.)  Determine the causal inverse z-transform of Solution: - We have N=5 and M=2, i.e., N > M. Thus, we have to divide the denominator into the numerator, giving Digital Signal Processing 21 z-Transform
  12. Example 8  Determine the causal inverse z-transform of Solution: Digital Signal Processing 23 z-Transform
  13. Some common z-transform pairs Digital Signal Processing 25 z-Transform
  14. Homework 1 Digital Signal Processing 27 z-Transform
  15. Homework 3 Digital Signal Processing 29 z-Transform
  16. Homework 5 Digital Signal Processing 31 z-Transform
  17. Homework 7  Tìm biến đổi z và miền hội tụ của các tín hiệu sau: 1) cos( n)u(n) 2) cos( n/2)u(n) 3) sin( n/2)u(n) 4) cos( n/3)u(n) 5) sin( n/3)u(n) 6) cos( n)u(n-1) 7) cos( n)u(1-n) 8) cos( n)u(-n-1) 9) 2ncos( n/2)u(n) 10) 2nsin( n/2)u(n) 11) 3ncos( n/3)u(n) 12) 3nsin( n/3)u(n) Digital Signal Processing 33 z-Transform