Image Processing Reference
In-Depth Information
the intersection of the individual regions of convergence. The proof
is straightforward and is based on the fact that sum of two sequences is
equal to sum of individual sequences.
Example 3.13
Find the z-transform of the following sequence:
2
3
n
u(n)
3
2
n
u(n)
x(n)
¼
þ
S
OLUTION
To
nd
X(z), we use the linearity property and the
z-transform pair
a
n
u(n)
z
za
,
!
jzj > jaj
. Then
z
z
z
z
z(5z
2)
z
X(z)
¼
2
3
þ
3
2
¼
1
1
1
3
1
2
z
and the ROC is
\ zj >
¼jzj >
1
3
1
2
1
2
R ¼
ROC
¼ zj >
(b)
Delay Property of the
z
-Transform
:If
x
(
n
)
u
(
n
) !
X
(
z
)
,
ROC ¼
R
(
3
:
51
)
then
x
(
n
k
)
u
(
n
k
) !
z
k
X
(
z
)
,
ROC ¼
R
(
3
:
52
)
The delay k is an integer.
Proof:
1
1
1
x
(
n
k
)
u
(
n
k
)
z
n
x
(
n
k
)
z
n
x
(
l
)
z
l
k
¼
¼
n
¼1
n
¼
k
l
¼
0
¼
z
k
1
l
¼
0
x
(
l
)
z
l
¼
z
k
X
(
z
)
(
:
)
3
53
Example 3.14
Find the z-transform of the following sequence:
0
n
3
x(n)
¼
6)
n
(0
:
n
4
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