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Discrete-Time Signal processing Chapter 3 the Z-transform

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1 Discrete-Time Signal processing Chapter 3 the Z-transform
Zhongguo Liu Biomedical Engineering School of Control Science and Engineering, Shandong University 山东省精品课程《生物医学信号处理(双语)》 2019/4/25 Zhongguo Liu_Biomedical Engineering_Shandong Univ.

2 Chapter 3 The z-Transform
2019年4月25日1时50分 Chapter 3 The z-Transform 3.0 Introduction 3.1 z-Transform 3.2 Properties of the Region of Convergence for the z-transform 3.3 The inverse z-Transform 3.4 z-Transform Properties 3.5 z-Transform and LTI Systems 3.6 the Unilateral z-Transform Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

3 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
3.0 Introduction Fourier transform plays a key role in analyzing and representing discrete-time signals and systems, but does not converge for all signals. Continuous systems: Laplace transform is a generalization of the Fourier transform. Discrete systems : z-transform, generalization of DTFT, converges for a broader class of signals. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

4 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
3.0 Introduction Motivation of z-transform: The Fourier transform does not converge for all sequences and it is useful to have a generalization of the Fourier transform. In analytical problems the z-Transform notation is more convenient than the Fourier transform notation. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

5 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
3.1 z-Transform If , Fourier transform z-transform z-Transform: two-sided, bilateral z-transform one-sided, unilateral z-transform Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

6 Relationship between z-transform and Fourier transform
Express the z in polar form as The Fourier transform of the product of and (the exponential sequence ). Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

7 Complex z plane Region Of Convergence (ROC)
Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

8 periodic sampling T:sampling period; fs=1/T: sampling rate;
Review periodic sampling T:sampling period; fs=1/T: sampling rate; Ωs=2π/T: sampling rate

9 Complex frequency domain:
Review Relation between Laplace Transform and Z-transform Laplace transform Continuous Time domain: Complex frequency domain: Region Of Convergence (ROC)

10 Laplace Transform and Fourier transform
Since So Fourier Transform frequency domain : Fourier Transform is the Laplace transform when s have the value only in imaginary axis, s=jΩ

11 z-transform of discrete-time signal
For sampling signal, the Laplace transform z-transform of discrete-time signal

12  let: so: 数字频率和模拟频率的关系 Laplace transform continuous time signal
z-transform discrete-time signal so: 数字频率和模拟频率的关系

13 DTFT : Discrete Time Fourier Transform
2019年4月25日1时50分 DTFT : Discrete Time Fourier Transform S plane Z plane - Go

14 plane

15 Region of convergence (ROC)
For any given sequence, the set of values of z for which the z-transform converges is called the Region Of Convergence (ROC). Absolute Summability the ROC consists of all values of z such that the inequality in the above holds Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

16 Region of convergence (ROC)
Convergence of the z-transform for a given sequence depends only on if some value of z, say, z =z1, is in the ROC, z1 then all values of z on the circle defined by |z|=|z1| will also be in the ROC. uniform convergence if ROC includes unit circle, then Fourier transform and all its derivatives with respect to w must be continuous functions of w.

17 Region of convergence (ROC)
The Fourier transforms are not continuous, infinitely differentiable functions, so they cannot result from evaluating a z-transform on the unit circle. it is not strictly correct to think of the Fourier transform as being the z-transform evaluated on the unit circle. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

18 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Zero and pole The z-transform is most useful when the infinite sum can be expressed in closed form, usually a ratio of polynomials in z (or z-1). Zero: The value of z for which Pole: The value of z for which Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

19 Example 3.1: Right-sided exponential sequence
Determine the z-transform, including the ROC in z-plane and a sketch of the pole-zero-plot, for sequence: DTFT Solution: ROC: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

20 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
: zeros  : poles Gray region: ROC Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

21 Ex. 3.2 Left-sided exponential sequence
Determine the z-transform, including the ROC, pole-zero-plot, for sequence: Solution: ROC: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

22 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
2019/4/25

23 Ex. 3.3 Sum of two exponential sequences
Determine the z-transform, including the ROC, pole-zero-plot, for sequence: Solution: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

24 Example 3.3: Sum of two exponential sequences
ROC: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

25 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
2019/4/25

26 Example 3.4: Sum of two exponential
Solution: ROC: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

27 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
2019/4/25

28 Example 3.5: Two-sided exponential sequence
Solution: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

29 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
ROC, pole-zero-plot Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

30 Finite-length sequence
Example : or entire z-plane, except z=0 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

31 Example 3.6: Finite-length sequence
Determine the z-transform, the ROC, pole-zero-plot(N=16, -1<a<1), for sequence: Solution: z-transform or , including z=a 2019/4/25 Zhongguo Liu_Biomedical Engineering_Shandong Univ.

32 with N=16, a is real, and 0<|a|<1
pole-zero-plot 零极点图 N=16 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

33 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
z-transform pairs 1. 2. 3. 4. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

34 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
z-transform pairs 5. 6. 7. 8. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

35 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
z-transform pairs 9. 10. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

36 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
z-transform pairs 11. 12. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

37 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
z-transform pairs 13. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

38 3.2 Properties of the ROC for z-transform
2019年4月25日1时50分 3.2 Properties of the ROC for z-transform Property 1: The ROC will either be of the form or, in general the annulus, i.e.,

39 3.2 Properties of the ROC for z-transform
2019年4月25日1时50分 3.2 Properties of the ROC for z-transform Property 1: The ROC : For a given x[n], ROC is dependent only on (not on the angle of z). Two-Sided Sequence:

40 3.2 Properties of the ROC for the z-transform
Property 2: The Fourier transform of converges absolutely if and only if the ROC of the z-transform of includes the unit circle. The z-transform reduces to the Fourier transform when ie. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

41 3.2 Properties of the ROC for the z-transform
Property 3: The ROC cannot contain any poles. is infinite at a pole and therefore does not converge. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

42 3.2 Properties of the ROC for the z-transform
Property 4: If is a finite-duration sequence, i.e., a sequence that is zero except in a finite interval : then the ROC is the entire z-plane, except possibly or Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

43 3.2 Properties of the ROC for the z-transform
2019年4月25日1时50分 3.2 Properties of the ROC for the z-transform Property 5: If is a right-sided sequence, i.e., a sequence that is zero for , the ROC extends outward from the outermost finite pole in to (possibly including) Proof:

44 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Property 5: right-sided sequence for the z-transform: For other terms, ROC: ROC of X(z): Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

45 3.2 Properties of the ROC for the z-transform
Property 6: If is a left-sided sequence, i.e., a sequence that is zero for , the ROC extends inward from the innermost nonzero pole in to Proof: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

46 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Property 6: left-sided sequence for the z-transform: For other terms, ROC: ROC of X(z): Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

47 3.2 Properties of the ROC for the z-transform
Property 7: A two-sided sequence is an infinite-duration sequence that is neither right-sided nor left-sided. If is a two-sided sequence, the ROC will consist of a ring in the z-plane, bounded on the interior and exterior by a pole and consistent with Property 3, not containing any poles. 2019/4/25

48 3.2 Properties of ROC for z-transform
Property 8: ROC must be a connected region. for finite-duration sequence possibly ROC: for right-sided sequence possibly ROC: for left-sided sequence possibly ROC: for two-sided sequence If no overlap between ROCs of right- and left-sided parts; i.e., rL<rR, no z-transform exists.

49 Example 3.7: Non-Overlapping ROCs
Solution: Since there is no overlap between x[n] has no z-transform (nor Fourier transform). Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

50 Example: Different possibilities of the ROC define different sequences
A system with three poles Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

51 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Different possibilities of the ROC. (c) ROC to a (b) ROC to a right-sided sequence left-handed sequence Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

52 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
(e) ROC to another Unit-circle included Unit-circle (d) ROC to a two-sided sequence two-sided sequence not stable stable Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

53 LTI system Stability, Causality, and ROC
A z-transform does not uniquely determine a sequence without specifying the ROC It’s convenient to specify the ROC implicitly through time-domain property of a sequence Consider a LTI system with impulse response h[n]. The z-transform of h[n] is called the system function H (z) of the LTI system. stable system(h[n] is absolutely summable and therefore has a Fourier transform): ROC include unit-circle. causal system: h[n]=0,for n<0, right sided Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

54 Ex. 3.8 Stability, Causality, and the ROC
A LTI system with impulse response h[n]. The z-transform of h[n] i.e. system function H(z) has pole-zero plot shown in Figure 3.9. Determine the Stability, Causality,ROC, for: (1) stable system: (ROC include unit-circle) (2) causal system: (right sided sequence) (3) non-causal, unstable system Figure 3.9 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

55 Ex. 3.8 Stability, Causality, and the ROC
Solution: (1) stable system (ROC include unit-circle), ROC: , the impulse response is two-sided, system is non-causal. stable. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

56 Ex. 3.8 Stability, Causality, and the ROC
(2) causal system: (right sided sequence) ROC: ,the impulse response is right-sided. system is causal but unstable. Unit-circle A system is causal and stable if all the poles are inside the unit circle, and the ROC extends outward from the outermost pole to z=∞. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

57 Ex. 3.8 Stability, Causality, and the ROC
(3) ROC: , the impulse response is left-sided, system is non-causal, unstable since the ROC does not include unit circle. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

58 LTI system Stability, Causality, and ROC
Review For a LTI system with impulse response h[n], if it is causal, what do we know about h[n]? (Is h[n] one-sided or two-sided sequence? Left-sided or right-sided?) Then what do we know about the ROC of the system function H (z)? And more if the poles of H (z) are all in the unit circle, is the system stable? Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

59 LTI system Stability, Causality, and ROC
For H (z) with the poles as shown in figure , Review can we uniquely determine h[n] ? Unit-circle included If ROC of H(z) is as shown in figure, can we uniquely determine h[n] ? is the system stable ? Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

60 LTI system Stability, Causality, and ROC
For H (z) with the poles as shown in figure , Review If the system is causal (h[n]=0,for n<0,right-sided ), What’s the ROC like? If ROC is as shown in figure, is h[n] one-sided or two-sided? Is the system causal or stable? Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

61 3.3 The Inverse Z-Transform
Formal inverse z-transform is based on a Cauchy integral theorem. (留数residue法) Zi是X(z)zn-1在围线C内的极点。 围线c:X(z)的环状收敛域内环绕原点的一条逆时针的闭合单围线。 Zi Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

62 3.3 The Inverse Z-Transform
Less formal ways are sufficient and preferable in finding the inverse z-transform : Inspection method (观察法) Partial fraction expansion (部分分式展开法) Power series expansion (幂级数展开法) Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

63 3.3 The inverse z-Transform
3.3.1 Inspection Method(观察法) Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

64 3.3 The inverse z-Transform
3.3.1 Inspection Method Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

65 3.3 The inverse z-Transform
3.3.2 Partial Fraction Expansion部分分式展开法 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

66 Ex. 3.9 find inverse z-Transform of 2nd-Order z-Transform
Solution: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

67 Example 3.9 Second-Order z-Transform
Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

68 Inverse Z-Transform by Partial Fraction Expansion
Br is obtained by long division Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

69 Inverse Z-Transform by Partial Fraction Expansion
s重极点di if M>N, and has a pole of order s at d=di Br is obtained by long division

70 3.3 The Inverse Z-Transform
Review Inspection method Partial fraction expansion Power series expansion Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

71 Partial Fraction Expansion
Review Partial Fraction Expansion if M>N, and has a pole of order s at d=di Br is obtained by long division Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

72 Example 3.10: Inverse by Partial Fractions
Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

73 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
2019/4/25

74

75 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
2019/4/25

76 3.3 The inverse z-Transform
3.3.3 Power Series Expansion幂级数展开法 If the z-transform is given as a power series: Laurent series where x[n] are the coefficients of z-n we can determine any particular value of the sequence x[n] by finding the coefficient of the appropriate power of z-1. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

77 Example 3.11: find Finite-Length Sequence x[n] of X(z):
Solution:

78 Ex. 3.12: find Inverse Transform by power series expansion
Solution: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

79 Example 3.13: Power Series Expansion by Long Division
Solution:

80 Example 3.13(第二版): Power Series Expansion for a Left-sided Sequence
Solution: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

81 3.4 z-Transform Properties
3.4.1 Linearity Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

82 Example of Linearity Find X(z) of Solution:

83 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
3.4.2 Time Shifting is an integer is positive, is shifted right is negative, is shifted left Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

84 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Time Shifting: Proof Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

85 Example 3.14: Shifted Exponential Sequence
, Determine x[n]. Solution 2: Solution 1: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

86 3.4.3 Multiplication by an Exponential sequence
that is the frequency shifting of Fourier Transform Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

87 Ex. 3.15 Exponential Multiplication
Use , determine Z-transform of Solution :

88 Example 3.15: Exponential Multiplication
Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

89 3.4.4 Differentiation of X(z)
Proof: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

90 Example 3.16: Inverse of Non-Rational z-Transform
Check Solution : Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

91 Example 3.17: Second-Order Pole
2019年4月25日1时50分 Example 3.17: Second-Order Pole determine X(z) of Solution : Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

92 3.4.5 Conjugation of a complex Sequence
Proof: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

93 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Time Reversal Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

94 Example 3.18: Find X(z) of Time-Reverse Exponential Sequence
Solution: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

95 3.4. 7 Convolution of Sequences
Proof: 2019/4/25 Zhongguo Liu_Biomedical Engineering_Shandong Univ.

96 Ex. 3.19: Convolution of Finite-Length Sequences
Given Solution: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

97 3.4. 8(第二版) Initial Value Theorem
Proof: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

98 3.4. 8 Summary of Some z-Transform Properties
TABLE 3.2 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

99 3.5 z-transforms and LTI systems
2019年4月25日1时50分 3.5 z-transforms and LTI systems z-transform can be used in the representation and analysis of LTI systems. h[n] x[n] y[n] X(z) Y(z) H(z) System Function: y[n] = h[n] *x[n] Y(z) = H(z) X(z) Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

100 3.5 z-transforms and LTI systems
analysis of LTI system in difference equation: h[n] x[n] initial rest conditions → causal LTI system for LTI System:

101 Ex.3.20 Convolution of Infinite-Length Sequences
Solution: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

102 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Example 3.20: Convolution of Infinite-Length Sequences Using the z-transform Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

103 3.5 z-transforms and LTI systems
h[n] x[n] y[n] Z z-plane unit circle for causal LTI System, and stable

104 Ex st-order System for causal LTI System, Solution: 1). Iterate the difference equation, not easy to get a closed form solution; 2). convolution techniques, y[n] = h[n] *x[n] 3).

105 3.6 the Unilateral Z-transform
two-sided, bilateral z-transform one-sided, unilateral z-transform ROC: If x[n]=0 for n< 0, the unilateral and bilateral z-transforms are identical, if x[n] is not zero for all n< 0, they will be different.. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

106 Ex. 3.22 Unilateral Transform of an Impulse
2019年4月25日1时50分 Ex Unilateral Transform of an Impulse compare bilateral and Unilateral z-transform of Solution: bilateral z-transform: Unilateral z-transform identical to different Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019/4/25

107 3.6 the Unilateral Z-transform
h[n] x[n] If noninitial rest conditions occur, the linearity prop­erty is identical to that of the bilateral z-transform, time-shifting property is different.

108 3.6 the Unilateral Z-transform
By a similar analysis,

109 Ex. 3.23 Effect of Nonzero Initial Conditions
2019年4月25日1时50分 Ex Effect of Nonzero Initial Conditions initial condition at n=-1: y[-1], find y[n]= ? 2) if x[n]=Au[n]. Solution: Apply the Unilateral z-transform, using the linearity property and the time-shift property: if initial rest condition: system function LTI system

110 Ex. 3.23 Effect of Nonzero Initial Conditions
2019年4月25日1时50分 Ex Effect of Nonzero Initial Conditions initial condition at n=-1: y[-1], find y[n]= ? 2) if x[n]=Au[n]. Solution: not linear, x[n]=0, y[n]≠0.

111 Ex. 3.23 Effect of Nonzero Initial Conditions
2019年4月25日1时50分 Ex Effect of Nonzero Initial Conditions find y[n]= ? 2) if x[n]=Au[n]. Solution:

112 Ex. 3.23 Effect of Nonzero Initial Conditions
2019年4月25日1时50分 Ex Effect of Nonzero Initial Conditions 2) if x[n]=Au[n]. find y[n]= ? Solution: ZIR ZICR not linear, not time-shifting zero input response (ZIR) The zero initial conditions response (ZICR)

113 Chapter 3 HW 3.2, 3.4, 3.7, 3.9, 3.16 3.20 3.2 , 3.8, 3.11, 3.20 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 121 2019/4/25 2019/4/25 返 回 上一页 下一页


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