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Module 2
The Z
The Ztransform and its Applications
transform and its Applications
0
Chen 9/9/201
L R
Module 2
The Z
The Ztransform and its applications
transform and its applications
1
Content
• Definition of bilateral Z transform
• Definition of bilateral Ztransform
• Region of convergence (ROC)
• Pole location and temporal behaviour of DT causal signals
• Properties of the Ztransform
• The inverse Z transform
• The inverse Ztransform
• Unilateral Ztransform; applications
Bibliography
• “The ZTransform” (Chapter 10) in
Signals and Systems
, 2
nd
Edition by
A V O
h i
A S W
i
l
lk
dS H N
b
Module 2
The Z
The Ztransform and its applications
transform and its applications
2
A. V. Oppenheim, A. S. Willsky, and S. H. Nawab.
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View Full Document Learning Objectives
• Understand the notion of
poles
and
zeros
• Understand the notion of
poles
and
•Beab
leto
compute the Ztransform
and
specify the ROC
for different DT signals
• Understand the relation between the
ROC
and the
time
domain properties of a DT signal
0
•Understand
pole location
and the
impact on the temporal
behaviour
of causal, DT signals
Chen 9/9/201
•Beable to
compute the inverse Ztransform
solve difference equations
and
analyze DTLTI
systems in the Zdomain
L R
Module 2
The Z
The Ztransform and its applications
transform and its applications
3
systems in the Z domain
Motivation for the Ztransform
• plays same role in analysis of DT signals and LTI systems as the
Laplace transform
does in CT signals and LTI systems
•provides means of characterizing an LTI system and its response to
signals by its polezero locations
Module 2
The Z
The Ztransform and its applications
transform and its applications
4
Definition of the direct (bilateral, 2sided) Ztransform
• the Ztransform of the DT signal
x
[
n
] is defined by the following
power series:
∑
∞
−∞
=
−
≡
n
n
z
n
x
z
X
]
[
)
(
• since the Ztransform comprises an infinite series, it
exists
only for
the values of
z
for which
the series converges
0
•
region of convergence (ROC)
:
set of all values of
z
for which
X
(
z
)
is
finite
Chen 9/9/201
• whenever we cite a Ztransform,
we also need to specify its ROC
L R
Module 2
The Z
The Ztransform and its applications
transform and its applications
5
Definition of the direct (bilateral, 2sided) Ztransform
• if we write
z
=
r
exp[j
θ
]
where
r
= 
z
 and
θ
=
∠
z
,
then
∑
∞
−
−
=
jn
n
e
r
n
x
z
X
θ
]
[
)
(
now,
−∞
=
n
∑
∞
−
−
=
jn
n
e
r
n
x
z
X
]
[
)
(
∑
∞
−
−
−∞
=
≤
jn
n
n
e
r
n
x
]
[
∑
∞
−
−∞
=
=
n
n
r
n
x
]
[
thus in the ROC of
X
(
z
), 
X
(
z
) <
∞⇒
need to ensure that
x
[
n
]
r
n
is
absolutely summable
⇒
the values of
r
which guarantee this
−∞
=
n
Module 2
The Z
The Ztransform and its applications
transform and its applications
6
the values of
condition form the ROC
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View Full Document Definition of the direct (bilateral, 2sided) Ztransform
•note
∑
∑
∑
∞
−
−
−
∞
−
+
=
=
1
]
[
]
[
]
[
)
(
n
n
n
r
n
x
r
n
x
r
n
x
z
X
∑
∑
∞
∞
=
−∞
=
−∞
=
+
−
=
0
]
[
]
[
n
n
n
n
n
n
x
r
n
x
=
=
0
1
n
n
r
converges
converges
0
converges
for
r
<
∞
converges
for
r
>
r
2
(
r
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This note was uploaded on 07/09/2011 for the course ECSE 304 taught by Professor Chenandbacsy during the Spring '11 term at McGill.
 Spring '11
 ChenandBacsy

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