Image Processing Reference
In-Depth Information
1
x
(
n
)
z
n
X
(
z
) ¼
(
3
:
40
)
n
¼
0
Convergence of the z-transform requires that
1
j
x
(
n
)
r
n
j < 1
(
3
:
41
)
n
¼1
for some positive values of r. This means absolute summability of the exponentially
weighted sequence x
(
n
)
. For example, the following sequences do not have
z-transform, since neither of these sequences multiplied by r
n
would be absolutely
summable for any value of r. They are not satisfying the condition given above by
Equation 3.41.
x
1
(
n
) ¼
A
sin (v
0
n
)
v
0
n
(
3
:
42
)
x
2
(
n
) ¼
A
cos (v
0
n
þ u)
(
3
:
43
)
x
3
(
n
) ¼
a
n
1 <
n
< 1
(
3
:
44
)
Example 3.10
¼ a
n
u(n).
Find the z-transform of the one-sided signal x(n)
S
OLUTION
¼
1
n¼1
¼
1
n¼
¼
1
n¼
1
z
z a
x(n)z
n
a
n
z
n
(az
1
)
n
X(z)
¼
az
1
¼
1
0
0
jaz
1
The convergence of X(z) requires that
j <
1. Thus, the ROC is the set of
points in complex z-plane for which
, as shown in Figure 3.2. Notice that
the function X(z) has a single pole located at z ¼ a, which is outside ROC of X(z).
jzj > jaj
Im(
z
)
|
a
|
Re(
z
)
FIGURE 3.2
ROC for Example 3.10.
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