Digital Signal Processing Reference
In-Depth Information
where
C
is a closed contour traversed in the counterclockwise direction within
the ROC. Solving Eq. (13.5) involves the application of contour integration
techniques and is, therefore, seldom used directly. In Section 13.3, we will
consider alternative approaches based on the look-up table, partial fraction
expansion, and power series expansion to evaluate the inverse z-transform.
Collectively, Eqs. (13.4) and (13.5) form the bilateral z-transform pair, which
is denoted by
x
[
k
]
←→
X
(
z
)or
Z
x
[
k
]
=
X
(
z
)
.
(13.6)
To illustrate the steps involved in computing the z-transform, we consider the
following examples.
Example 13.1
Calculate the bilateral z-transform of the exponential sequence
x
[
k
]
= α
k
u
[
k
].
Solution
Substituting
x
[
k
]
= α
k
u
[
k
] in Eq. (13.4), we obtain
k
=−∞
α
k
u
[
k
]
z
∞
∞
−
k
−
1
)
k
X
(
z
)
=
=
(
α
z
k
=
0
1
1
− α
z
−
1
−
1
<
1
α
z
=
undefined
elsewhere
.
In the above expression, if
α
z
−
1
≥
1 the bilateral z-transform has an infinite
value. In such cases, we say that the z-transform is not defined. The set of values
of
z
over which the bilateral z-transform is defined is referred to as the region of
convergence (ROC) associated with the z-transform. In this example, the ROC
for the z-transform pair
1
1
− α
z
−
1
←→
α
k
u
[
k
]
is given by
−
1
ROC:
α
z
<
1or
z
> α.
Figure 13.1 highlights the ROC by shading the appropriate region in the complex
z-plane.
x
[
k
] =
a
k
u
[
k
]
1
Im{
z
}
a
a
2
a
3
a
4
Fig. 13.1. (a) DT exponential
sequence
x
[
k
] = α
k
u
[
k
]; (b) the
ROC,
z
> α, associated with
its bilateral z-transform. The ROC
is shown as the shaded area and
lies outside the circle of radius α.
Re{
z
}
k
(0, α)
−2
−1
0
1
3
(a)
(b)
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