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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