Bài giảng Xử lý tín hiệu số - Z-Transform

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1. Xử lý tín hiệu số Z - transform Ngô Quốc Cường ngoquoccuong175@gmail.com sites.google.com/a/hcmute.edu.vn/Ngô Quốc Cường ngoquoccuong
2. Z - transform • Z- transform • Properties of Z-transform • Inversion of Z- transform • Analysis of LTI systems in Z domain 2
3. 4.1. Z - transform • Given a discrete-time signal x(n), its z-transform is defined as the following series: where z is a complex variable. • Writing explicitly a few of the terms: • Z-transform is an infinite power series, it exists only for those values of z for this series converges. • The region of convergence (ROC) of X(z) is the set of all values of z for which X(z) attains a finite value. 3
4. 4.1. Z - transform • Example: Determines the z-transform of the following finite duration signals 4
5. 4.1. Z - transform • Solution 5
6. 4.1. Z - transform • Example 6
7. 4.1. Z - transform Recall that 7
8. 4.1. Z - transform • Example • Solution 8
9. 4.1. Z - transform • Example • We have (l = -n), • Using the formula (when A<1) 9
10. 4.1. Z - transform • We have identical closed-form expressions for the z transform • A closed-form expressions for the z transform does not uniquely specify the signal in time domain. • The ambiguity can be resolved if the ROC is specified. • Z – transform = closed-form expressions + ROC 11
11. 4.1. Z - transform • Example • Solution – The first power series converges if |z| > |a| – The second power series converges if |z| < |b| 12
12. • Case 1 13
13. • Case 2 14
14. • Characteristics families of signals with their corresponding ROC 15
15. 4.1. Z - transform • The z-transform of the impulse response h(n) is called the transfer function of a digital filter: • Determine the transfer function H(z) of the two causal filters 17
16. 4.2. Properties of Z-transform 18
17. • Example 19
18. • Example 21
19. Exercise 24
20. 4.2. Properties of Z-transform • The ROC of z-k X(z) is the same as that of X(z) except for z=0 if k>0 and z=∞ if k<0. 26
21. • Solution 27
22. 4.2. Properties of Z-transform 28
23. • Example 29
24. 4.2. Properties of Z-transform 31
25. • Example 32
26. 4.2. Properties of Z-transform 33
27. 4.2. Properties of Z-transform • Example 34
28. 4.2. Properties of Z-transform 35
29. 4.2. Properties of Z-transform • Example 36
30. 4.2. Properties of Z-transform • Convolution in Z domain 38
31. 4.3. RATIONAL Z-TRANSFORM 39
32. 4.3.1. Poles and Zeros • An important family of z-transforms are those for which X(z) is a rational function. • Poles and Zeros – Zeros: value of z for which X(z) = 0; – Poles: value of z for which X(z) = ∞ 40
33. 4.3.1. Poles and Zeros • Example • Solution 41
34. 4.3.2. Causality and Stability • A causal signal of the form will have z-transform • the common ROC of all the terms will be 42
35. 4.3.2. Causality and Stability • if the signal is completely anticausal • the ROC is in this case 43
36. 4.3.2. Causality and Stability • Causal signals are characterized by ROCs that are outside the maximum pole circle. • Anticausal signals have ROCs that are inside the minimum pole circle. • Mixed signals have ROCs that are the annular region between two circles—with the poles that lie inside the inner circle contributing causally and the poles that lie outside the outer circle contributing anticausally. 44
37. 4.3.2. Causality and Stability • Stability can also be characterized in the z-domain in terms of the choice of the ROC. • A necessary and sufficient condition for the stability of a signal x(n) is that the ROC of the corresponding z-transform contain the unit circle. • A signal or system to be simultaneously stable and causal, it is necessary that all its poles lie strictly inside the unit circle in the z-plane. 45
38. 4.3.2. Causality and Stability 46
39. 4.3.3. System function of LTI • System function • From a linear constant coefficient equation • We have, 47
40. 4.3.3. System function of LTI • Or equivalently 48
41. 4.3.3. System function of LTI • Example 49
42. 4.3.3. System function of LTI • Solution • The unit sample response 50
43. 4.4. Inverse Z- transform • By contour integration. • By power series expansion. • By partial fraction expansion. 51
44. • The partial fraction expansion method can be applied to z- transforms that are ratios of two polynomials • The partial fraction expansion of X(z) is given by 52
45. • Example • The two coefficients are obtained as follows: 53
46. • If the degree of the numerator polynomial N(z) is exactly equal to the degree M of the denominator D(z), then the PF expansion must be modified 54
47. • Example • Compute all possible inverse z-transforms of 55
48. • Solution • Where • |z| > 0.5: • |z|<0.5: 56
49. • Example • Determine all inverse z-transforms of 57
50. • Solution 58
51. • there are only two ROCs I and II: 59
52. MORE ABOUT INVERSE Z-TRANSFORM 60
53. • Distinct poles 61
54. • Multiple order poles • Solution • In such a case, the partial fraction expansion is: 62
55. Exercise 64
56. • a) • b) • c) 65