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)》
2018/12/7
Zhongguo Liu_Biomedical Engineering_Shandong Univ.

2. 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
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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.
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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.
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5. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
3.1 z-Transform
z-Transform: two-sided, bilateral z-transform
one-sided, unilateral z-transform
If , z-transform is Fourier transform.
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6. Relationship between z-transform and Fourier transform
Express the complex variable z in polar form as
The Fourier transform of the product of and the exponential sequence
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7. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Complex z plane
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8. periodic sampling Review
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
Continuous
Time domain：
Laplace transform
Complex frequency domain：

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 令
z-transform of discrete-time signal

12.  let： so： Laplace transform continuous time signal
z-transform discrete-time signal
so： 

13. 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
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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.
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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
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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:
Solution:
ROC:
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20. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
: zeros
 : poles
Gray region: ROC
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21. Ex. 3.2 Left-sided exponential sequence
Determine the z-transform, including the ROC, pole-zero-plot, for sequence:
Solution:
ROC:
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22. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
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23. Ex. 3.3 Sum of two exponential sequences
Determine the z-transform, including the ROC, pole-zero-plot, for sequence:
Solution:
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24. Example 3.3: Sum of two exponential sequences
ROC:
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25. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
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26. Example 3.4: Sum of two exponential
Solution:
ROC:
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27. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
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28. Example 3.5: Two-sided exponential sequence
Solution:
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29. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
ROC, pole-zero-plot
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30. Finite-length sequence
Example :
or entire z-plane, except z=0
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31. Example 3.6: Finite-length sequence
Determine the z-transformthe ROC, pole-zero-plot, for sequence:
Solution:
including z=a
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32. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
N=16, a is real
pole-zero-plot
with N = 16
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33. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
z-transform pairs
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34. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
z-transform pairs
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35. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
z-transform pairs
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36. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
z-transform pairs
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37. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
z-transform pairs
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38. 3.2 Properties of the ROC for the z-transform
2018年12月7日5时57分
3.2 Properties of the ROC for the z-transform
Property 1: The ROC is a ring or disk in the z-plane centered at the origin.
For a given x[n], ROC is dependent only on
(not on the angle of z).

39. 3.2 Properties of the ROC for the z-transform
Property 2: The Fourier transform of converges absolutely if the ROC of the z-transform of includes the unit circle.
The z-transform reduces to the Fourier transform when ie.
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40. 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.
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41. 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 possible or
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42. 3.2 Properties of the ROC for the z-transform
2018年12月7日5时57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:

43. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Property 5: right-sided sequence
for
the z-transform:
For other terms:
ROC
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44. 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:
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45. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Property 6: left-sided sequence
for
the z-transform:
For other terms:
ROC
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46. 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 not containing any poles.
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47. 3.2 Properties of the ROC for the z-transform
Property 8: ROC must be a connected region.
for finite-duration sequence
possible
ROC:
for right-sided sequence
possible
ROC:
for left-sided sequence
possible
ROC:
for two-sided sequence
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48. Example: Different possibilities of the ROC define different sequences
A system with three poles
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49. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Different possibilities of the ROC.
(c) ROC to a
left-handed sequence
(b) ROC to a
right-sided sequence
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50. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
(e) ROC to another
two-sided sequence
Unit-circle
included
(d) ROC to a
two-sided sequence.
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51. 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
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52. Ex. 3.7 Stability, Causality, and the ROC
Consider a LTI system with impulse response h[n]. The z-transform of h[n] i.e. the system function H (z) has the pole-zero plot shown in Figure. Determine the Stability, Causality,ROC, if the system is:
(1) stable system: (ROC include unit-circle)
(2) causal system: (right sided sequence)
(3) non-causal, unstable system
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53. Ex. 3.7 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.
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54. Ex. 3.7 Stability, Causality, and the ROC
(2) causal system: (right sided sequence)
ROC: ,the impulse response is right-sided. system is causal but unstable.
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=∞.
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55. Ex. 3.7 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.
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56. 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)?
If the poles of H (z) are all in the unit circle, is the system stable?
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57. 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 ?
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58. 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?
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59. 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。
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60. 3.3 The Inverse Z-Transform
Less formal ways are sufficient and preferable in finding the inverse z-transform. :
Inspection method
Partial fraction expansion
Power series expansion
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61. 3.3 The inverse z-Transform
3.3.1 Inspection Method
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62. 3.3 The inverse z-Transform
3.3.1 Inspection Method
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63. 3.3 The inverse z-Transform
3.3.2 Partial Fraction Expansion
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64. Ex. 3.8 find inverse z-Transform of 2nd-Order z-Transform
Solution:
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65. Example 3.8 Second-Order z-Transform
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66. Inverse Z-Transform by Partial Fraction Expansion
Br is obtained by long division
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67. Inverse Z-Transform by Partial Fraction Expansion
if M>N, and has a pole of order s at d=di
Br is obtained by long division
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68. Example 3.9: Inverse by Partial Fractions
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69. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
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70. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
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71. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
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72. 3.3 The inverse z-Transform
3.3.3 Power Series Expansion
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.
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73. Example 3.10: find Finite-Length Sequence x[n] of X(z):
Solution:
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74. Ex. 3.11: find Inverse Transform by power series expansion
Solution:
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75. Example 3.12: Power Series Expansion by Long Division
Solution:
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76. Example 3.13: Power Series Expansion for a Left-sided Sequence
Solution:
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77. 3.4 z-Transform Properties
3.4.1 Linearity
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78. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Example of Linearity
Find X(z) of
Solution:
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79. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
3.4.2 Time Shifting
is an integer
is positive, is shifted right
is negative, is shifted left
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80. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Time Shifting: Proof
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81. Example 3.14: Shifted Exponential Sequence
, Determine x[n].
Solution 2:
Solution 1:
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82. 3.4.3 Multiplication by an Exponential sequence
that is the frequency shifting of Fourier Transform
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83. Example 3.15: Exponential Multiplication
Use
, determine
Z-transform of
Solution :
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84. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
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85. 3.4.4 Differentiation of X(z)
Proof:
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86. Example 3.16: Inverse of Non-Rational z-Transform
Solution :
Check
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87. Example 3.17: Second-Order Pole
determine X(z) of
Solution :
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88. 3.4.5 Conjugation of a complex Sequence
Proof:
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89. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Time Reversal
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90. Example 3.18: Find X(z) of Time-Reverse Exponential Sequence
Solution:
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91. 3.4. 7 Convolution of Sequences
Proof:
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92. Ex. 3.19: Evaluating a Convolution Using the z-transform
Solution:
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93. Example 3.19: Evaluating a Convolution Using the z-transform
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94. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Initial Value Theorem
Proof:
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95. Chapter 3 HW
3.3, 3.4, 3.9, 3.16
3.2 , 3.8, 3.11, 3.20
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