Databases Reference
In-Depth Information
Example12.9.2:
In the previous example we found that
∞
z
a
n
z
−
n
=
a
,
|
z
|
>
|
a
|
(80)
z
−
n
=
0
If we take the derivative of both sides of the equation with respect to
a
, we get
∞
z
na
n
−
1
z
−
n
=
2
,
|
z
|
>
|
a
|
(81)
(
z
−
a
)
n
=
0
Thus,
z
na
n
−
1
u
Z
[
[
n
]] =
2
,
|
z
|
>
|
a
|
(
z
−
a
)
If we differentiate Equation (
80
)
m
times, we get
∞
m
!
z
a
n
−
m
z
−
n
n
(
n
−
1
)
···
(
n
−
m
+
1
)
=
m
+
1
(
z
−
a
)
n
=
0
In other words,
n
m
a
n
−
m
u
z
Z
[
n
]
=
(82)
m
+
1
(
z
−
a
)
In these examples the Z-transform is a ratio of polynomials in
z
. For sequences of interest
to us, this will generally be the case, and the Z-transform will be of the form
N
(
z
)
F
(
z
)
=
D
(
z
)
(
)
(
)
(
)
The values of
z
for which
F
z
is zero are called the
zeros
of
F
z
; the values for which
F
z
is infinity are called the
poles
of
F
(
z
)
. For finite values of
z
, the poles will occur at the roots
of the polynomial
D
.
The inverse Z-transform is formally given by the contour integral [?]
(
z
)
1
z
n
−
1
dz
F
(
z
)
2
π
j
C
where the integral is over the counterclockwise contour
C
, and
C
lies in the region of con-
vergence. This integral can be difficult to evaluate directly; therefore, in most cases we use
alternative methods for finding the inverse Z-transform.
