This document discusses the Z-transform and its inverse. It begins by defining the Z-transform as a mapping from a discrete-time signal to a power series. The region of convergence (ROC) is introduced as the set of values where the Z-transform has a finite value. Common properties of the ROC are described. Rational Z-transforms containing poles and zeros are covered. The inverse Z-transform is discussed using inspection and partial fraction expansion methods. The relationship between the Z-transform of a system's impulse response and its transfer function is explained. Finally, the four-step process for analyzing discrete-time linear time-invariant systems in the transform domain using the Z-transform is outlined.

1. Chapter Five
Z-Transform and Its Inverse
Addis Ababa Science and Technology University
College of Electrical & Mechanical Engineering
Electromechanical Engineering Department
Signals and Systems Analysis ( EME3262)

2. Z-transform and Its Inverse
Outline
 The Z-transform
 Properties of the Z-transform
 Transfer function of Discrete-time LTI Systems
 Transform Domain Analysis using the Z-transform
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3. The Z-transform
 The Z-transform of a discrete-time signal x(n), denoted by X(z),
is defined as:
 The Z-transform is a mapping (transformation) from a sequence
to a power series.
 We say that x(n) and X(z) are Z-transform pairs and denote this
relationship as:
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





n
n
z
n
x
z
X )
(
)
(
)
certain
(with
)
(
)
( ROC
z
X
n
x Z



4. The Z-Plane
 The variable z is complex and can be viewed in the z-plane.
 The Z-transform of a discrete-time signal x(n) is a function X(z)
defined on the z-plane.
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5. Region of Convergence (ROC)
 The region of convergence (ROC) is defined as the set of all
values of z for which X(z) has a finite values.
 Every time we cite a Z-transform, we should indicate its ROC.
Example:
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6. Region of Convergence (ROC)……
Exercise:
1. Find the Z-transform of the following discrete-time signals and
state the ROC.
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)
(
)
(
.
)
1
(
)
(
.
)
(
)
(
.
)
1
(
)
(
.
)
(
)
(
.
n
u
a
n
x
c
n
u
a
n
x
e
n
u
n
x
b
n
u
a
n
x
d
n
n
x
a
n
n
n









 

7. Region of Convergence (ROC)……
2. Find the Z-transform of the following discrete-time signals and
state the ROC.
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)
1
(
3
)
(
.
)
(
3
)
1
(
2
)
(
.
)
(
)
2
(
)
(
.
)
1
(
3
)
(
2
)
(
.
)
(
2
)
(
.
















n
u
n
x
c
n
u
n
u
n
x
e
n
u
n
x
b
n
u
n
u
n
x
d
n
u
n
x
a
n
n
n
n
n
n
n

8. Properties of the ROC
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9. Properties of the ROC……
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10. Properties of the ROC……
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11. Properties of the ROC……
 In general, the ROC has the following properties.
i. The ROC can not contain any poles inside it.
ii. If x(n) is left-sided signal, then:
iii.If x(n) is right-sided signal, then:
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pole
outermost
the
is
:
,
: 2
2 r
r
z
ROC 
pole
innermost
the
is
:
,
: 1
1 r
r
z
ROC 

12. Properties of the ROC……
iv. If x(n) is two-sided signal, then:
v. If x(n) is a finite length signal, then ROC is the entire z-plane
except possibly at
vi. The DTFT of x(n) exists if and only if the ROC of x(n)
includes the unit circle.
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: 1
2 r
z
r
ROC 

.
or
0 

 z
z

13. Properties of the ROC……
Exercise:
The Z-transform of a discrete-time signal x(n) is given by:
Determine:
a. all the possible ROCs
b. the corresponding discrete-time signal x(n) for each of the
above ROCs
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2
1
8
1
4
3
1
1
)
(





z
z
z
X

14. Summary of the ROC
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15. Summary of the ROC……..
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16. Summary of the ROC……..
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17. Some Common Z-transform Pairs
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18. Some Common Z-transform Pairs……
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19. Some Common Z-transform Pairs……
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20. Rational Z-transforms
 The most important and most commonly used Z-transforms are
those for which X(z) is a rational function of the form:
 The roots of the numerator N(z) are known as the zeros of X(z).
 The roots of the denominator D(z) are known as the poles of
X(z).
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N
M
M
M
N
k
k
k
M
k
k
k
z
a
z
a
a
z
b
z
b
b
z
a
z
b
z
D
z
N
z
X 


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
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


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





.....
.....
)
(
)
(
)
( 1
1
0
1
1
0
0
0

21. Rational Z-transforms……
 The above rational Z-transform contains:
 M zeros at z1, z2, ……, zM
 N poles at p1, p2, ……, pM
 If M<N, then there are N-M additional zeros at the origin z=0.
 If M>N, then there are M-N additional poles at the origin z=0.
 If M=N, then X(z) has exactly the same number of poles and
zeros.
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22. Rational Z-transforms……
Exercise:
Find the Z-transform and sketch the pole-zero plots of the
following discrete-time signals.
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)
1
(
)
5
.
1
(
)
(
)
5
.
0
(
)
(
.
)
1
(
5
.
0
)
(
.
)
(
5
.
0
)
(
.





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



n
u
n
u
n
x
c
n
u
n
x
b
n
u
n
x
a
n
n
n
n

23. Rational Z-transforms……
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24. Rational Z-transforms……
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25. Properties of Z-Transform
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26. Inverse Z-transform
Inverting by Inspection:
 The simplest inversion method is by inspection, or by comparing
with the table of common Z-transform pairs.
Exercise:
Find the inverse of the following Z-transforms by inspection.
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5
.
0
:
,
5
.
0
1
1
)
(
.
5
.
0
:
,
5
.
0
1
1
)
(
.
1
1








z
ROC
z
z
X
b
z
ROC
z
z
X
a

27. Inverse Z-transform……
Inverting by Partial Fractional Expansion:
 This is a method of writing complex rational Z-transforms as a
sum of simple terms.
 After expressing the complex rational Z-transform as a sum of
simple terms, each term can be inverted by inspection.
Exercise:
Find the inverse Z-transform by partial fractional expansion method.
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   5
.
0
:
,
5
.
0
1
25
.
0
1
1
)
( 1
1



 

z
ROC
z
z
z
X

28. Transfer Function of Discrete-time LTI Systems
 The Z-transform of the impulse response h(n) is known as the
transfer function of the system.
 Mathematically:
 We say that h(n) and H(z) are Z-transform pairs and denote this
relationship as:
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





n
n
z
n
h
z
H )
(
)
(
)
(
)
( z
H
n
h Z



29. Transfer Function of Discrete-time LTI Systems……
 The output y(n) of a discrete-time LTI system equals the
convolution of the input x(n) with the impulse response h(n),
i.e.,
 Taking the Z-transform of both sides of the above equation by
applying the convolution property, we obtain:
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)
(
*
)
(
)
( n
h
n
x
n
y 
)
(
)
(
)
(
)
(
)
(
)
(
z
X
z
Y
z
H
z
H
z
X
z
Y 



30. Transfer Function of Discrete-time LTI Systems……
i. Causal LTI Systems
 A discrete-time LTI system is causal if h(n)=0, n<0. In other
words, h(n) is right-sided signal.
 Therefore, ROC of H(z) is an exterior region starting from the
outermost pole.
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31. Transfer Function of Discrete-time LTI Systems……
ii. Anti-causal LTI Systems
 A discrete-time LTI system is anti-causal if h(n)=0, n>0. In
other words, h(n) is left-sided signal.
 Therefore, ROC of H(z) is an interior region starting from the
innermost pole.
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32. Transfer Function of Discrete-time LTI Systems……
iii. BIBO Stable LTI Systems
 A discrete-time LTI system is BIBO stable if h(n) is absolutely
summable, i.e. ,
 Therefore, ROC of H(z) always contains the unit circle.
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





n
n
h )
(

33. Transfer Function of Discrete-time LTI Systems……
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34. Transfer Function of Discrete-time LTI Systems……
iv. Causal & BIBO stable LTI Systems
 The ROC of H(z) must be an exterior region starting from the
outermost pole and contains the unit circle.
 In other words, all poles must be inside the unit circle.
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35. Transfer Function of Discrete-time LTI Systems……
Exercise:
1. The transfer function of a discrete-time LTI system is given by:
a. Find the poles and zeros of H(z).
b. Sketch the pole-zero plot.
c. Find the impulse response h(n) if the system is known to be:
i. causal iii. BIBO stable
ii. anti-causal
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2
1
1
5
.
2
1
3
3
)
( 






z
z
z
z
H

36. Transfer Function of Discrete-time LTI Systems……
2. Plot the ROC of H(z) for discrete-time LTI systems that are:
a. causal & BIBO stable
b. causal & unstable
c. anti-causal & BIBO stable
d. anti-causal & unstable
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37. Transform Domain Analysis using the Z-transform
 The procedure for evaluating the output y(n) of a discrete-time
LTI system using the Z-transform consists of the following four
steps.
1. Calculate the Z-transform X(z) of the input signal x(n).
2. Calculate the Z-transform H(z) of the impulse response h(n) of
the discrete-time LTI system.
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38. Transform Domain Analysis using the Z-transform….
3. Based on the convolution property, the Z-transform of the
output y(n) is given by Y(z) = H(z)X(z).
4. The output y(n) in the time domain is obtained by calculating
the inverse Z-transform of Y(z) obtained in step (3).
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39. Exercise
1. Find the Z-transform of the following discrete-time signals.
2. Find the inverse Z-transform of:
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)
1
(
2
1
)
(
3
1
)
(
.
)
(
3
1
)
(
2
1
)
(
.















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


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

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
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





n
u
n
u
n
x
b
n
u
n
u
n
x
a
n
n
n
n
2
1
:
,
2
1
1
4
1
1
1
)
(
1
1


















z
ROC
z
z
z
X

40. Exercise……
3. The input to a causal discrete-time LTI system is given by:
The Z-transform of the output of this system is:
a. Determine the impulse response h(n) of the system.
b. Find the output y(n) of the system.
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)
(
2
1
)
1
(
)
( n
u
n
u
n
x
n





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

 
1
1
1
1
2
1
1
2
1
)
(
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
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z
z
z
z
Y