Databases Reference
In-Depth Information
Then we can write
F
(
z
)/
z
as
L
F
(
z
)
A
i
=
(84)
−
z
z
z
i
i
=
1
If we can find the coefficients
A
i
, then we can write
F
(
z
)
as
L
A
i
z
(
)
=
F
z
z
−
z
i
i
=
1
and the inverse Z-transform will be given by
L
A
i
z
i
u
f
n
=
[
n
]
i
=
1
The question then becomes one of finding the value of the coefficients
A
i
. This can be simply
done as follows: Suppose we want to find the coefficient
A
k
. Multiply both sides of Equation
(
84
)by
(
z
−
z
k
)
. Simplifying this we obtain
L
i
F
(
z
)(
z
−
z
k
)
A
i
(
z
−
z
k
)
=
(85)
z
z
−
z
i
=
1
L
i
A
i
(
z
−
z
k
)
=
A
k
+
(86)
=
1
z
−
z
i
i
=
k
Evaluating this equation at
z
=
z
k
, all the terms in the summation go to zero and
z
=
z
k
F
(
z
)(
z
−
z
k
)
A
k
=
(87)
z
Example12.9.4:
Let us use the partial fraction expansion method to find the inverse Z-transform of
6
z
2
−
9
z
F
(
z
)
=
z
2
−
2
.
5
z
+
1
Then
6
z
2
F
(
z
)
1
z
−
9
z
=
(88)
z
2
z
−
2
.
5
z
+
1
6
z
−
9
=
(89)
(
−
.
)(
−
)
z
0
5
z
2
We want to write
F
(
z
)/
z
in the form
(
)
F
z
A
1
A
2
=
5
+
z
z
−
0
.
z
−
2
