Graphics Programs Reference
In-Depth Information
∞
∑
z
∫
x
()
na
x
()
z
a
k
a
1
------------
Z
=
u
d
(13.110)
+
n
=
0
0
Initial Value:
8.
xn
()
z
n
0
X
()
z
=
(13.111)
→
∞
Final Value:
9.
1
lim
x
()
=
lim
(
1
z
)
X
()
(13.112)
n
→
∞
z
→
1
10.
Convolution:
∞
∑
Z
h n
(
k
)
x
()
=
H
()
X
()
(13.113)
k
=
0
11.
Bilateral Convolution:
∞
∑
Z
h n
(
k
)
x
()
=
H
()
X
()
(13.114)
k
=
∞
Example:
Prove Eq. (13.109).
Solution:
Starting with the definition of the Z-transform,
∞
∑
n
X
()
=
x
()
z
n
=
∞
Taking the derivative, with respect to z, of the above equation yields
∞
∑
d
X
()
n
1
=
x
()
n
()
z
d
z
n
=
∞
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