Image Processing Reference
In-Depth Information
(d)
Multiplication by n or Differentiation of X(z)
: The z-transform of nx
(
n
)
is
z
d
X
(
z
)
d
z
, where X
(
z
)
is the z-transform of x
(
n
)
.
Proof: The z-transform of x
(
n
)
is
1
x
(
n
)
z
n
X
(
z
) ¼
(
3
:
58
)
n
¼
0
Differentiating both sides of Equation 3.58 with respect to the complex
variable z results in
1
¼
z
1
1
n
¼1
d
X
(
z
)
d
z
¼
x
(
n
)(
n
)
z
n
1
nx
(
n
)
z
n
(
3
:
59
)
n
¼1
Multiplying both sides of Equation 3.59 by
z yields
1
z
d
X
(
z
)
nx
(
n
)
z
n
d
z
¼
(
3
:
60
)
n
¼1
The following example uses the differentiation property.
Example 3.17
Find the z-transform of the sequence
¼ na
n
u(n)
x(n)
(3
:
61)
S
OLUTION
Since the z-transform of a
n
u(n)is
z
¼ na
n
u(n)is
za
, then the z-transform of x(n)
¼z
z
a
z
d
dz
z
z a
az
(z a)
2
X(z)
¼z
(z a)
2
¼
(3
:
62)
(e)
Initial Value Theorem
:Ifx
(
n
) ¼
0 for n
<
0, then
x
(
0
) ¼ lim
z
!1
X
(
z
)
(
3
:
63
)
Proof: To derive this property, take the limit on both sides of Equation
3.64 as z approaches in
nity:
1
x
(
n
)
z
n
)
z
1
)
z
2
X
(
z
) ¼
¼
x
(
0
) þ
x
(
1
þ
x
(
2
þ
(
3
:
64
)
n
¼
0
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