Image Processing Reference
In-Depth Information
Therefore,
x(0)
¼
1, x(1)
¼
0
:
9, x(2)
¼
1
:
05, x(3)
¼
0
:
813, x(4)
¼
0
:
5577,
...
In general, it is not possible to obtain closed-form solution using long-
division technique.
Example 3.22
Find the inverse z-transform of the irrational function
þ az
1
) ROC
X(z)
¼
log (1
: jzj > a
S
OLUTION
Using Taylor series expansion, we have
1
1)
nþ1
a
n
z
n
n
þ az
1
)
log (1
¼
(
n¼
1
Therefore,
1)
nþ1
a
n
x(n)
¼
(
n
u(n
1)
(c)
Inversion by Partial Fraction Expansion
:IfX
(
z
)
is a rational function,
that is,
P
k
¼
0
b
k
z
k
P
k
¼
0
a
k
z
k
¼
P
k
¼
0
b
k
z
k
Q
i
¼
1
(
z
p
i
)
X
(
z
) ¼
(
3
:
74
)
m
i
where
p
i
, i
¼
or roots of polynomial
P
k
¼
1
a
k
z
k
1, 2,
...
, Q are the poles of X
(
z
)
m
i
is the multiplicity of the ith pole. Note that m
1
þ
m
2
þþ
m
Q
¼
M
Partial fraction of
X
(
z
)
z
yields
X
(
z
z
¼
A
0
z
þ
A
1
z
p
1
þ
A
2
(
z
p
1
)
A
m
1
(
z
p
1
)
2
þþ
m
1
B
1
z
p
2
þ
B
2
(
z
p
2
)
B
m
2
(
z
p
2
)
þ
2
þþ
þ
(
3
:
75
)
m
2
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