Image Processing Reference
In-Depth Information
It can be shown that
(
P
Residues of
X
(
z
)
z
n
1
at poles of
X
(
z
) inside
Cn
0
x
(
n
) ¼
P
Residues of
X
(
z
)
z
n
1
at poles of
X
(
z
) outside
Cn
1
(
3
:
73
)
Example 3.19
Find the inverse z-transform of the following function:
z
2
X(z)
¼
3)
, ROC
: jzj >
0
:
3
(z
0
:
2)(z
0
:
S
OLUTION
The ROC and the contour C for this example are shown in Figure 3.6.
Since there are no poles outside C, we have
x(n)
¼
0 for n <
0
For n
0, we have
¼
X
Residues of
z
2
3)
z
n1
at poles of X(z) inside C ¼A
1
þA
2
x(n)
(z
0
:
2)(z
0
:
The residue A
1
corresponding to pole p
1
¼
0
:
2is
2)
n1
z
2
2
2
(0
0
:
:
3)
z
n1
2)
n
A
1
¼
(z
0
:
2)
j
z¼0:2
¼
3)
¼
2(0
:
(z
0
:
2)(z
0
:
(0
:
2
0
:
Im(
z
)
C
0.2
Re(
z
)
0.3
ROC of
X
(
z
)
FIGURE 3.6
ROC and contour C for Example 3.19.
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