Discrete-Time Signal processing Chapter 3 the Z-transform

1
Discrete-Time Signal processingChapter 3 the
Z-transform

• Zhongguo Liu
• Biomedical Engineering
• School of Control Science and Engineering,
  Shandong University

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

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
  FT, converges for a broader class of signals.

4
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.

5
3.1 z-Transform

• z-Transform two-sided, bilateral z-transform

• one-sided, unilateral z-transform

6
Relationship between z-transform and Fourier
transform

• Express the complex variable z in polar form as

7
Complex z plane
8
Condition for convergence of the z-transform
9
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).

if some value of z, say, z z1, is in the ROC,
then all values of z on the circle defined by
zz1 will also be in the ROC.
if ROC includes unit circle, then Fourier
transform and all its derivatives with respect to
w must be continuous functions of w.
10
Region of convergence (ROC)
-8ltnlt8

• Neither of them is absolutely summable, neither
  of them multiplied by r-n (-8ltnlt8) would be
  absolutely summable for any value of r. Thus,
  neither them has a z-transform that converges
  absolutely.

11
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.

12
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).

13
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
14
(No Transcript)
15
Ex. 3.2 Left-sided exponential sequence

• Determine the z-transform, including the ROC,
  pole-zero-plot, for sequence

Solution
ROC
16
(No Transcript)
17
Ex. 3.3 Sum of two exponential sequences

• Determine the z-transform, including the ROC,
  pole-zero-plot, for sequence

Solution
18
Example 3.3 Sum of two exponential sequences
ROC
19
(No Transcript)
20
Example 3.4 Sum of two exponential
Solution
ROC
21
Example 3.5 Two-sided exponential sequence
Solution
22
ROC, pole-zero-plot
23
Finite-length sequence
Example
24
Example 3.6 Finite-length sequence

• Determine the z-transform, including the ROC,
  pole-zero-plot, for sequence

Solution
25
N16, a is real
pole-zero-plot
26
z-transform pairs
27
z-transform pairs
28
z-transform pairs
29
z-transform pairs
30
z-transform pairs
31
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.

32
3.2 Properties of the ROC for the z-transform
33
3.2 Properties of the ROC for the z-transform

• Property 3 The ROC cannot contain any poles.

34
3.2 Properties of the ROC for the z-transform
35
3.2 Properties of the ROC for the z-transform
36
3.2 Properties of the ROC for the z-transform
37
3.2 Properties of the ROC for the z-transform
Proof
38
Property 5 right-sided sequence
For other terms
ROC
39
3.2 Properties of the ROC for the z-transform
Proof
40
Property 6 left-sided sequence
41
3.2 Properties of the ROC for the z-transform
42
3.2 Properties of the ROC for the z-transform

• Property 8 ROC must be a connected region.

43
Example Different possibilities of the ROC
define different sequences
A system with three poles
44
Different possibilities of the ROC.
(b) ROC to a right-sided sequence
45
(d) ROC to a two-sided sequence.
46
LTI system Stability, Causality, and ROC

• A z-transform does not uniquely determine a
  sequence without specifying the ROC

• Its convenient to specify the ROC implicitly
  through time-domain property of a sequence

• Consider a LTI system with impulse response hn.
  The z-transform of hn is called the system
  function H (z) of the LTI system.

• stable system(hn is absolutely summable and
  therefore has a Fourier transform) ROC include
  unit-circle.
• causal system (hn0,for nlt0) right sided

47
Ex. 3.7 Stability, Causality, and the ROC

• Consider a LTI system with impulse response hn.
  The z-transform of hn i.e. the system function
  H (z) has the pole-zero plot shown in Figure.
  Determine the ROC, if the system is

• (1) stable system (ROC include unit-circle)

• (2) causal system (right sided sequence)

48
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.

49
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.

50
Ex. 3.7 Stability, Causality, and the ROC

• ROC , the impulse response is
  left-sided, system is non-causal, unstable since
  the ROC does not include unit circle.

51
3.3 The Inverse Z-Transform

• Formal inverse z-transform is based on a Cauchy
  integral theorem.

(??residue?)
Zi?X(z)zn-1???C?????
??cX(z)????????????????????????
52
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

53
3.3 The inverse z-Transform

• 3.3.1 Inspection Method

54
3.3 The inverse z-Transform

• 3.3.1 Inspection Method

55
3.3 The inverse z-Transform

• 3.3.2 Partial Fraction Expansion

56
Example 3.8Second-Order z-Transform
57
Example 3.8Second-Order z-Transform
58
Br is obtained by long division
59
Inverse Z-Transform by Partial Fraction Expansion
Br is obtained by long division
60
Example 3.9Inverse by Partial Fractions
61
(No Transcript)
62
(No Transcript)
63
(No Transcript)
64
3.3 The inverse z-Transform

• 3.3.3 Power Series Expansion

65
Example 3.10Finite-Length Sequence
66
Ex. 3.11 Inverse Transform by power series
expansion
67
Example 3.12 Power Series Expansion by Long
Division
68
Example 3.13 Power Series Expansion for a
Left-sided Sequence
69
3.4 z-Transform Properties
3.4.1 Linearity
70
Example of Linearity
71
3.4.2 Time Shifting
72
Time Shifting Proof
73
Example 3.14Shifted Exponential Sequence
74
3.4.3 Multiplication by an Exponential sequence
75
Example 3.15Exponential Multiplication
76
(No Transcript)
77
3.4.4 Differentiation of X(z)
78
Example 3.16Inverse of Non-Rational z-Transform
79
Example 3.17 Second-Order Pole
80
3.4.5 Conjugation of a complex Sequence
81
3.4. 6 Time Reversal
82
Example 3.18 Time-Reverse Exponential Sequence
83
3.4. 7 Convolution of Sequences
84
Ex. 3.19 Evaluating a Convolution Using the
z-transform
Solution
85
Example 3.19 Evaluating a Convolution Using the
z-transform
86
3.4. 8 Initial Value Theorem
87
Region of convergence (ROC)
-8ltnlt8,
88
Example

• For

89
Chapter 3 HW

• 3.3, 3.4, 3.8, 3.9, 3.11, 3.16

• 3.2, 3.20

Zhongguo Liu_Biomedical Engineering_Shandong Univ.
89
2014-11-17
???
???
? ?