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)》

2. Chapter 3 The z-Transform
2018年12月7日5时57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

3. 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.

4. 3.0 Introduction
Motivation of z-transform:
The Fourier transform does not converge for all sequences;
It is useful to have a generalization of the Fourier transform→z-transform.
In analytical problems the z-Transform notation is more convenient than the Fourier transform notation.

5. 3.1 z-Transform Fourier transform z-transform If ,
z-Transform: two-sided, bilateral z-transform
one-sided, unilateral
z-transform Go

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 ).

7. Region Of Convergence (ROC)
Complex z plane
Region Of Convergence (ROC)

8. periodic sampling T: sampling period (单位s); fs=1/T: sampling rate(Hz);
Review
periodic sampling
T: sampling period (单位s);
fs=1/T: sampling rate(Hz);
Ωs=2π/T: sampling rate
模拟频率
模拟角频率
(rad/s) t n

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 令
z-transform of discrete-time signal

12.  let： so： 数字频率(单位rad)和模拟频率(rad/s)的关系
Laplace transform continuous time signal
z-transform discrete-time signal
so： 
数字频率(单位rad)和模拟频率(rad/s)的关系

13. DTFT : Discrete Time Fourier Transform
2018年12月7日5时57分
DTFT : Discrete Time Fourier Transform
S plane
Z plane - Go

14. 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

15. 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.
设|z|=|z1|=1在收敛域内
一致收敛uniform convergence
if ROC includes unit circle, then Fourier transform and all its derivatives with respect to w must be continuous functions of w.

16. 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.

17. 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

18. 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:
no DTFT
DTFT
Solution:
ROC:

19. : zeros
 : poles
Gray region: ROC

20. Ex. 3.2 Left-sided exponential sequence
Determine the z-transform, including the ROC, pole-zero-plot, for sequence:
Solution:
ROC:

21. 比较:
: zeros
 : poles
Gray region: ROC

22. Ex. 3.3 Sum of two exponential sequences
Determine the z-transform, including the ROC, pole-zero-plot, for sequence:
Solution:

23. Example 3.3: Sum of two exponential sequences
ROC:

25. Example 3.4: Sum of two exponential
another solution for Ex.3.3
Solution:
ROC:

27. Example 3.5: Two-sided exponential sequence
Solution:

28. ROC, pole-zero-plot

29. Finite-length sequence
Example :
or entire z-plane, except z=0

30. Example 3.6: Finite-length sequence
Determine the z-transform, the ROC, pole-zero-plot, for sequence:
(N=16, 0<a<1)
Solution: z-transform
or ,
including z=a

31. with N=16, a is real, and 0<a<1
pole-zero-plot
零极点图
N=16

32. z-transform pairs 1. 2. 3. 4.

33. z-transform pairs 5. 6. 7. 8.

34. z-transform pairs 9.
10.

35. z-transform pairs
11.
12.

36. z-transform pairs
13.

37. 3.2 Properties of the ROC for z-transform
2018年12月7日5时57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.,

38. 3.2 Properties of the ROC for z-transform
2018年12月7日5时57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:

39. 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.

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.

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 possibly or

42. 3.2 Properties of the ROC for the z-transform
Property 5: If is a right-sided sequence, i.e., for , then the ROC extends outward from the outermost finite pole in to (may including)
Proof: for
ROC
must
let
i.e.
ROC proved.

43. 3.2 Properties of the ROC for the z-transform
Property 6: If is a left-sided sequence, i.e., for , then the ROC extends inward from the innermost nonzero pole in to
Proof: for
ROC
must
let
i.e.
ROC proved.

44. 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.

45. 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.

46. Example 3.7: Non-Overlapping ROCs
Solution:
Since there is no overlap between
x[n] has no z-transform (nor Fourier transform).

47. Example: Different possibilities of the ROC define different sequences
A system with three poles

48. Different possibilities of the ROC.
(c) ROC to a
left-handed sequence
(b) ROC to a
right-sided sequence

49. (e) ROC to another
Unit-circle
included
Unit-circle
(d) ROC to a
two-sided sequence
two-sided sequence
not stable
stable

50. 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

51. 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

52. 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.

53. 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.

54. 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=∞.

55. 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.

56. 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

57. 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)

58. 3.3 The inverse z-Transform
3.3.1 Inspection Method(观察法)

59. 3.3 The inverse z-Transform
3.3.1 Inspection Method

60. 3.3 The inverse z-Transform
3.3.2 Partial Fraction Expansion部分分式展开法

61. Ex. 3.9 find inverse z-Transform of 2nd-Order z-Transform
Solution:

62. Example 3.9 Second-Order z-Transform

63. Inverse Z-Transform by Partial Fraction Expansion
Br is obtained by long division

64. 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;

65. Example 3.10: Inverse by Partial Fractions

69. 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.

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

71. Ex. 3.12: find Inverse Transform by power series expansion
Solution:
Taylor series:

72. Example 3.13: Power Series Expansion by Long Division
Solution:

73. Example 3.13(第二版): Power Series Expansion for a Left-sided Sequence
Solution:

74. 3.4 z-Transform Properties
3.4.1 Linearity

75. Example of Linearity
Find X(z) of
Solution:

76. 3.4.2 Time Shifting is an integer is positive, is shifted right
is negative, is shifted left

77. Time Shifting: Proof

78. Example 3.14: Shifted Exponential Sequence
, Determine x[n].
Solution 2:
Solution 1:

79. 3.4.3 Multiplication by an Exponential sequence
that is the frequency shifting of Fourier Transform

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

81. Example 3.15: Exponential Multiplication

82. 3.4.4 Differentiation of X(z)
Proof:

83. Example 3.16: Inverse of Non-Rational z-Transform
Check
Solution :

84. Example 3.17: Second-Order Pole
2018年12月7日5时57分
Example 3.17: Second-Order Pole
determine X(z) of
Solution :

85. 3.4.5 Conjugation of a complex Sequence
Proof:

86. Time Reversal

87. Example 3.18: Find X(z) of Time-Reverse Exponential Sequence
Solution:

88. 3.4. 7 Convolution of Sequences
Proof:

89. Ex. 3.19: Convolution of Finite-Length Sequences
Given
Solution:

90. 3.4. 8(第二版) Initial Value Theorem
Proof:

91. 3.4. 8 Summary of Some z-Transform Properties
TABLE 3.2

92. 3.5 z-transforms and LTI systems
2018年12月7日5时57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)

93. 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:

94. Ex.3.20 Convolution of Infinite-Length Sequences
Solution:

95. Example 3.20: Convolution of Infinite-Length Sequences Using the z-transform

96. 3.5 z-transforms and LTI systems
h[n]
x[n]
y[n] Z
z-plane 
unit circle
for causal LTI System,
and if stable
all poles are in the unit circle.

97. 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).

98. 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..

99. Ex. 3.22 Unilateral Transform of an Impulse
2018年12月7日5时57分
Ex Unilateral Transform of an Impulse
compare bilateral and Unilateral z-transform of
Solution:
bilateral z-transform:
Unilateral z-transform
identical to
different

100. 3.6 the Unilateral Z-transform
h[n]
x[n]
usually initial rest conditions. in noninitial rest conditions, the linearity and time-shifting properties of the unilateral z-transform are particularly useful tools.
the linearity prop­erty is identical to that of the bilateral z-transform, time-shifting property is different:
for

101. 3.6 the Unilateral Z-transform
By a similar analysis,
Proof:

102. Ex. 3.23 Effect of Nonzero Initial Conditions
2018年12月7日5时57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

103. Ex. 3.23 Effect of Nonzero Initial Conditions
2018年12月7日5时57分
Ex Effect of Nonzero Initial Conditions
initial condition at n=-1: y[-1], find y[n]= ?
2) if x[n]=Au[n].
Solution:
T{‧}
齐次性
not linear, ∵ 0x[n]=0, T{0x[n]}≠0y[n]=0.

104. Ex. 3.23 Effect of Nonzero Initial Conditions
2018年12月7日5时57分
Ex Effect of Nonzero Initial Conditions
find y[n]= ?
2) if x[n]=Au[n].
Solution:

105. Ex. 3.23 Effect of Nonzero Initial Conditions
2018年12月7日5时57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)

106. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Chapter 3 HW
3.3, 3.4, 3.7, 3.16
3.2 , 3.8, 3.11, 3.20
Zhongguo Liu_Biomedical Engineering_Shandong Univ.
122
2018/12/7
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