Addis Ababa Science and Technology University
College of Electrical & Mechanical Engineering
Electromechanical Engineering Department
Signals and Systems Analysis ( EEEg-2121)
Chapter Five
Z-Transform and Its Inverse
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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The Z-transform
 The Z-transform of a discrete-time signal x(n), denoted by X(z),
is defined as:
X ( z) 

n
x
(
n
)
z

n  
 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:
Z
x(n) 
X ( z ) (with certain ROC )
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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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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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Region of Convergence (ROC)……
Exercise:
1. Find the Z-transform of the following discrete-time signals and
state the ROC.
a. x(n)   (n)
d . x(n)  a nu (n  1)
b. x(n)  u (n)
e. x(n)  a nu (n  1)
c. x(n)  a nu (n)
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Region of Convergence (ROC)……
2. Find the Z-transform of the following discrete-time signals and
state the ROC.
a. x(n)  2 n u (n)
d . x(n)  2 n u (n)  3n u (n  1)
b. x(n)  (2) n u (n)
e. x(n)  2 n u (n  1)  3n u (n)
c. x(n)  3n u (n  1)
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Properties of the ROC
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Properties of the ROC……
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Properties of the ROC……
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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:
ROC : z  r1 ,
r1 : is the innermost pole
iii.If x(n) is right-sided signal, then:
ROC : z  r2 ,
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r2 : is the outermost pole
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Properties of the ROC……
iv. If x(n) is two-sided signal, then:
ROC : r2  z  r1
v. If x(n) is a finite length signal, then ROC is the entire z-plane
except possibly at z  0 or z  .
vi. The DTFT of x(n) exists if and only if the ROC of x(n)
includes the unit circle.
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Properties of the ROC……
Exercise:
The Z-transform of a discrete-time signal x(n) is given by:
X ( z) 
1
3 1 1  2
1 z  z
4
8
Determine:
a. all the possible ROCs
b. the corresponding discrete-time signal x(n) for each of the
above ROCs
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Summary of the ROC
April 2015
Prepared by: WelelawY.
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Summary of the ROC……..
April 2015
Prepared by: WelelawY.
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Summary of the ROC……..
April 2015
Prepared by: WelelawY.
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Some Common Z-transform Pairs
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Some Common Z-transform Pairs……
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Some Common Z-transform Pairs……
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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:
M
N ( z)
X ( z) 

D( z )
k
b
z
k
k 0
N
k
a
z
 k
b0  b1 z 1  .....  bM z  M

a0  a1 z 1  .....  aM z  N
k 0
 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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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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Rational Z-transforms……
Exercise:
Find the Z-transform and sketch the pole-zero plots of the
following discrete-time signals.
a. x(n)  0.5n u (n)
b. x(n)  0.5n u ( n  1)
c. x(n)  (0.5) n u (n)  (1.5) n u (n  1)
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Rational Z-transforms……
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Rational Z-transforms……
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Properties of Z-Transform
April 2015
Prepared by: Welelaw Y.
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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.
1
a. X ( z ) 
,
1
1  0.5 z
ROC : z  0.5
1
b. X ( z ) 
,
1
1  0. 5 z
ROC : z  0.5
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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.
X ( z) 
1
1  0.25z 1  0.5z 
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1
1
,
ROC : z  0.5
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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:
H ( z) 

n
h
(
n
)
z

n  
 We say that h(n) and H(z) are Z-transform pairs and denote this
relationship as:
Z
h(n) 
H ( z)
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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.,
y ( n )  x ( n ) * h ( n)
 Taking the Z-transform of both sides of the above equation by
applying the convolution property, we obtain:
Y ( z)
Y ( z)  X ( z)H ( z)  H ( z) 
X ( z)
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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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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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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. ,

 h( n)  
n  

Therefore, ROC of H(z) always contains the unit circle.
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Transfer Function of Discrete-time LTI Systems……
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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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Transfer Function of Discrete-time LTI Systems……
Exercise:
1. The transfer function of a discrete-time LTI system is given by:
3  3z 1
H ( z) 
1  2.5 z 1  z 2
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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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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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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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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Exercise
1. Find the Z-transform of the following discrete-time signals.
n
n
1
 1
a. x(n)    u (n)     u (n)
2
 3
n
n
 1
1
b. x(n)     u (n)    u ( n  1)
 3
2
2. Find the inverse Z-transform of:
1
X ( z) 
,
 1 1  1 1 
1  z 1  z 
 4  2 
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ROC : z 
2
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Exercise……
3. The input to a causal discrete-time LTI system is given by:
n
1
x(n)  u (n  1)    u (n)
2
The Z-transform of the output of this system is:
1 1
 z
2
Y ( z) 
 1 1 
1
1

z
1

z


 2 


a. Determine the impulse response h(n) of the system.
b. Find the output y(n) of the system.
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