The document discusses the z-transform, which converts discrete time-domain signals into a complex frequency-domain representation and serves as a discrete equivalent of the Laplace transform. It outlines the definition, properties, and region of convergence (ROC) of z-transforms, including stability conditions and examples for different types of sequences. The ROC determines where the z-transform converges and is crucial for analyzing the stability of systems.

1. Digital Signal Processing
Asst. Prof. Dr. Mohammed Ahmed Shakir
Lecture No. 6
Z-transform

2. Digital Signal Processing 2022-2023 Dr. Mohammed Ahmed
Shakir
Definition
•The Z-transform converts a discrete time-domain signal,
which is a sequence of real or complex numbers, into a
complex frequency-domain representation.
•It can be considered as a discrete equivalent of the Laplace
transform.

3. Digital Signal Processing 2022-2023 Dr. Mohammed Ahmed
Shakir
Definition
The z-transform of sequence x(n) is defined by






n
n
z
n
x
z
X )
(
)
(
 Let z = ej.
Fourier
Transform
( ) ( )
j j n
n
X e x n e
 


 
 

4. Digital Signal Processing 2022-2023 Dr. Mohammed Ahmed
Shakir
Why z-Transform?
• A generalization of Fourier transform
• Why generalize it?
• FT does not converge on all sequence
• Notation good for analysis
• Bring the power of complex variable theory deal with the
discrete-time signals and systems

5. Digital Signal Processing 2022-2023 Dr. Mohammed Ahmed
Shakir
Z-Plane
Re
Im






n
n
z
n
x
z
X )
(
)
(
( ) ( )
j j n
n
X e x n e
 


 
 
Fourier Transform is to evaluate z-transform
on a unit circle.
z = ej


6. Digital Signal Processing 2022-2023 Dr. Mohammed Ahmed
Shakir
Properties
• z-transforms are linear:
• The transform of a shifted sequence:
• Multiplication:
But multiplication will affect the region of
convergence and all the pole-zero locations will be
scaled by a factor of a.
  )
(
)
(
)
(
)
( z
bY
z
aX
n
by
n
ax 


Z
  )
(
)
( 0
0 z
X
z
n
n
x n


Z
  )
(
)
( 1
z
a
Z
n
x
an 

Z

7. Digital Signal Processing 2022-2023 Dr. Mohammed Ahmed
Shakir
Region of convergence
• Give a sequence, the set of values of z for which the
z-transform converges, i.e., |X(z)|<, is called the
region of convergence.



 









n
n
n
n
z
n
x
z
n
x
z
X |
||
)
(
|
)
(
|
)
(
|
ROC is centered on origin and
consists of a set of rings.

8. Digital Signal Processing 2022-2023 Dr. Mohammed Ahmed
Shakir
Properties of ROC
• A ring or disk in the z-plane centered at the origin.
• The Fourier Transform of x(n) is converge absolutely
iff the ROC includes the unit circle.
• The ROC cannot include any poles
• Finite Duration Sequences: The ROC is the entire z-
plane except possibly z=0 or z=.
• Right sided sequences: The ROC extends outward
from the outermost finite pole in X(z) to z=.
• Left sided sequences: The ROC extends inward from
the innermost nonzero pole in X(z) to z=0.

9. Digital Signal Processing 2022-2023 Dr. Mohammed Ahmed
Shakir
Stable Systems
• A stable system requires that its Fourier transform is
uniformly convergent.
 Fact: Fourier transform is to
evaluate z-transform on a unit
circle.
 A stable system requires the
ROC of z-transform to include
the unit circle.
Re
Im

10. Digital Signal Processing 2022-2023 Dr. Mohammed Ahmed
Shakir
Z-Transform Pairs
Sequence z-Transform ROC
)
(n
 1 All z
)
( m
n 
 m
z All z except 0 (if m>0)
or  (if m<0)
)
(n
u 1
1
1

 z
1
|
| 
z
)
1
( 

 n
u 1
1
1

 z
1
|
| 
z
)
(n
u
an 1
1
1

 az
|
|
|
| a
z 
)
1
( 

 n
u
an 1
1
1

 az
|
|
|
| a
z 

11. Digital Signal Processing 2022-2023 Dr. Mohammed Ahmed
Shakir
Example: A right sided Sequence
)
(
)
( n
u
a
n
x n
 |
|
|
|
,
)
( a
z
a
z
z
z
X 


Re
Im
a
ROC is bounded by the
pole and is the exterior
of a circle.

12. Digital Signal Processing 2022-2023 Dr. Mohammed Ahmed
Shakir
Example: A left sided Sequence
)
1
(
)
( 


 n
u
a
n
x n
|
|
|
|
,
)
( a
z
a
z
z
z
X 


Re
Im
a
ROC is bounded by the
pole and is the interior
of a circle.

13. Digital Signal Processing 2022-2023 Dr. Mohammed Ahmed
Shakir
Example: A Two Sided Sequence
)
1
(
)
(
)
(
)
(
)
( 2
1
3
1




 n
u
n
u
n
x n
n
2
1
3
1
)
(




z
z
z
z
z
X
Re
Im
1/2
)
)(
(
)
(
2
2
1
3
1
12
1




z
z
z
z
1/3
1/12
ROC is bounded by poles
and is a ring.
ROC does not include any pole.