Image Processing Reference
In-Depth Information
TABLE 3.3
z-Transform of Elementary Sequences
x(n)
X(z)
ROC
d(
n
)
1
Entire z-plane
z
k
d(
n
k
)
Entire z-plane
z
z
1
u
(
n
)
j
z
j >
1
z
z
a
a
n
u
(
n
)
j
z
j > j
a
j
z
z
a
a
n
u
(
n
1
)
j
z
j < j
a
j
)
(
z
a
)(
z
a
1
z
(
z
þ
a
a
1
1
a
a
j
n
j
a
< j
z
j <
0
<
a
<
1
)
z
(
z
1
)
nu
(
n
)
j
z
j >
1
2
az
(
z
a
)
na
n
u
(
n
)
j
z
j > j
a
j
2
az
(
z
þ
a
)
(
z
a
)
n
2
a
n
u
(
n
)
j
z
j > j
a
j
3
n
(
n
1
) (
n
m
þ
1
)
m
!
z
(
z
a
)
a
n
m
u
(
n
m
þ
1
)
j
z
j > j
a
j
m
þ
1
z
(
z
cos v
0
)
cos (v
0
n
)
u
(
n
)
j
z
j >
1
z
2
2z
cos v
0
þ
1
z
sin v
0
sin (v
0
n
)
u
(
n
)
j
z
j >
1
z
2
2z
cos v
0
þ
1
z
(
z
a
cos v
0
)
a
n
cos (v
0
n
)
u
(
n
)
j
z
j > j
a
j
z
2
2az
cos v
0
þ
a
2
za
sin v
0
a
n
sin (v
0
n
)
u
(
n
)
j
z
j > j
a
j
z
2
2za
cos v
0
þ
a
2
z
(
z
þ
a
a
1
)
(
z
a
)(
z
a
1
1
a
a
j
n
j
a
< j
z
j <
0
<
a
<
1
)
and
Z
x
2
(
n
) !
X
2
(
z
)
,
ROC ¼
R
2
(
3
:
49
)
then
Z
a
x
1
(
n
) þ b
x
2
(
n
) !
a
X
1
(
z
) þ b
X
2
(
z
)
,
ROC ¼
R
¼
R
1
\
R
2
(
3
:
50
)
This means that the z-transform of linear combination of two sequences
is the linear combination of their respective z-transforms and the ROC is
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