Image Processing Reference
In-Depth Information
|
z
2
|
ROC
|
z
1
|
FIGURE 3.11
ROC of Example 3.24.
Example 3.28
Find the z-transform of the 2-D sequence
n
1
n
1
¼ n
2
0
x(n
1
, n
2
)
¼
(3
:
98)
0
otherwise
S
OLUTION
The 2-D z-transform of x(n
1
, n
2
)is
1
1
1
x(n
1
, n
2
)z
n
1
z
n
2
(z
1
z
2
)
n
1
X(z
1
, z
2
)
¼
¼
(3
:
99)
2
n
1
¼1
n
2
¼1
n
1
¼
0
If
jz
1
z
1
j <
1or
jz
1
jjz
2
j >
1, the sum in Equation 3.99 convergences to
2
1
z
1
z
2
z
1
z
2
X(z
1
, z
2
)
¼
z
1
z
2
¼
(3
:
100)
1
1
The ROC is shown in Figure 3.11
3.8 TWO-DIMENSIONAL DISCRETE-SPACE FOURIER TRANSFORM
The 2-D discrete-space Fourier transform (DSFT) of 2-D discrete signal x
(
n
1
, n
2
)
is
de
ned as
1
1
x
(
n
1
, n
2
)
e
jn
1
v
1
e
jn
2
v
2
X
(
j
v
1
, j
v
2
) ¼
(
3
:
101
)
n
2
¼1
n
1
¼1
evaluated on the boundary of unit circles z
1
¼
e
j
v
1
This is the z-transform X
(
z
1
, z
2
)
and z
2
¼
e
j
v
2
in the 4-D complex space
(
z
1
, z
2
)
, that is,
X
(
j
v
1
, j
v
2
) ¼
X
(
z
1
, z
2
)j
z
1
¼
e
j
v
1
,z
2
¼
e
j
v
2
(
3
:
102
)
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