1
Department of
Electrical and Computer Engineering
Yimin Zhang
Department of Electrical & Computer Engineering
Villanova University
ECE8231
Digital Signal Processing
http://www.ece.villanova.edu/~zhang/ECE8231/
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Z-Transform and its Properties, Region of Convergence, and Inverse Transform, Study notes of Digital Signal Processing
An in-depth explanation of the z-transform, its properties, region of convergence, and inverse transform. It includes examples of right-sided, left-sided, and two-sided sequences, as well as the relationship between stability, causality, and the region of convergence. The document also covers the inverse z-transform using inspection method and partial fraction expansion.
Typology: Study notes
Pre 2010
1 / 41
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Z-Transform and its Properties, Region of Convergence, and Inverse Transform and more Study notes Digital Signal Processing in PDF only on Docsity!
Department of
Electrical and Computer Engineering
Yimin Zhang
Department of Electrical & Computer Engineering
Villanova University
ECE
Digital Signal Processing
http://www.ece.villanova.edu/~zhang/ECE8231/
Chapter 3
The
z
-Transform
In general, we have
Let
z=re
j^ ω
Then, the
z
-transform of a sequence
x
[
n
] is
defined aswith
z
being a complex variable.
z
-Transform (2)
∞ −∞ =
−
n
n z n x z X
]
[
]
[
z
X
n
x
z
∞ −∞ =
−
−
∞ −∞ =
−
n
n
j
n
n
n
j
j^
e
r
n
x
re
n
x
re
X
ω
ω
ω
]
[
](
[
Because
z
is a complex number, we often use the
z
-plane.
When |
z
|=1, that is,
z
takes value from the unit circle, the
z
-transform reduces to the Fourier transform.
z
-Transform (3)
Unit circle
Im
Re
1
z
-plane
ω j
e
z
ω
Absolute summability of
z
-transform
It is possible for the
z
-transform to converge even if the
Fourier transform does not.
Convergence
depends
only
on
z
The
region
of
convergence (ROC) consists of all values of
z
such that
the last inequality holds.
If the ROC includes the unit circle, the Fourier transformof the sequence converges.
Region of Convergence
∑
∑
∞
−∞
−
∞ −∞ =
−
n
n
n
n
z n x z n x
]
[
]
[
Consider
x
[
n
]=
a
n
u
[
n
]. Because it is nonzero only for
n
this is an example of a
right-sided
sequence.
For convergence of
X
z
), we require
Thus, the ROC is the range of values of
z
for which |
az
or equivalently, |
z
a
|. Inside the ROC, the infinite series
converges to
Example – Right-Sided
Exponential Sequence (1)
∑
∑
∞ =
−
∞ −∞ =
−
0
1
) ( ] [ ) (
n
n
n
n
n
az
z n u a z X
∑
∞ −∞ =
−
n
n
az
1
a
z
z
az
z
X
−
1
a
z
Consider
x
[
n
]= –
a
n
u
[–
n –
1]. Because it is nonzero only for
n
≤
1, this is an example of a
left-sided
sequence.
ROC and
Example – Left-SidedExponential Sequence
∑
∑
∑
∑
∞ =
−
∞ =
−
−
−∞
−
∞
−∞
−
0
1
1
1
]
[
n
n
n
n
n
n
n
n
n
n
n
z
a
z
a
z a z n u a z X
1
−
z
a
a
z
z
az
z
a
z
X
−
−
1
1
a
z
Unit circle
Im
Re
1
x
z
-plane
ROC
a
a
z
As can be seen from the two examples,
the algebraic
expression
or
pole-zero
pattern
does
not
completely
specify the z-transform of a sequence; i.e., the ROC mustalso be specified.
Notes on ROC
a
z
z
az
z
X
−
1
a
z
a
z
z
az
z
X
−
1
a
z
]
[
]
[
n
u
a
n
x
n
]
[
]
[
−
n
u
a
n
x
n
Consider the sequenceThen The ROC is determined bywhich requires
Example – Finite-Length Sequence
a
z
a
z
z
az
az
az
z
a
z
X
N
N
N
N
N n
n
N n
n
n
−
− −
− =
−
− =
−
∑
∑
1
1 1
(^10)
1
(^10)
otherwise
]
[
N
n
a
n
x
n
∑
− =
−
(^10)
1
N n
n
az
and
z
a
Unit circle
Im
Re
1
x
z
-plane
11
th
order pole
a
ROC
In this example N
= 12 and 0<
a
<1.
Some Common
z
-Transform Pairs (1)
(
)
(
)
|a|
|z|
az az
n
u
na
|a|
|z|
az az
n
u
na
|a|
|z|
az
n
u
a
|a|
|z|
az
n
u
a
m
m
z
z
m
n
z
z
n
u
z
z
n
u
z
n
z
n
n n
n
m
<
−
−
−
−
>
−
<
−
−
−
−
>
−
<
∞
>
−
<
−
−
−
−
>
− −
− −
− − − −
−
− −
1
] 1
[
1
]
[
1
1
] 1
[
1
1
]
[
(if
or
(if 0
except
all
]
[
1
|
|
1
1
] 1
[
1
|
|
1
1
]
[
all
1
]
[
ROC
Transform
Sequence
2 1 1
2 1 1
(^11) 1 1
δ δ
Property 1:
The ROC is a ring or disk in the
z
-plane
centered at the origin; i.e., 0
r
R
z
r
L
Property 2:
The Fourier transform of
x
[
n
] converges
absolutely if and only if the ROC of the
z
-transform of
x
[
n
] includes the unit circle.
Property 3:
The ROC cannot contain any pole.
Property 4:
If
x
[
n
] is a finite-duration sequence, i.e., a
sequence that is zero except in a finite interval –
N
1
n
N
2
, then the ROC is the entire
z
-plane, except
possibly
z
= 0 or
z
Properties of the ROC (1)
Property 5:
If
x
[
n
] is a
right-sided sequence
, i.e., a
sequence that is zero for
n < N
1
, the ROC extends
outward from the outmost (i.e., largest magnitude) finitepole in
X
z
) to (and possibly include)
z
Property
If
x
[
n
]
is
a
left-sided
sequence
i.e.,
a
sequence that is zero for
n > N
2
, the ROC extends
inward from the innermost (smallest magnitude) nonzeropole in
X
z
) to (and possibly include)
z
Properties of the ROC (2)
z-Transform with Different ROC (1)
Im
Re
x
z
-plane
Unit circle
x
x
For
a
system
whose
poles
are
shown
in
the
figure,
consider the stability and causality of the system.
z-Transform with Different ROC (2)
Im
Re
x
z
-plane
ROC
x
x
Right-sided sequence
Im
Re
x
z
-plane
ROC
x
x
Left-sided sequence