Databases Reference
In-Depth Information
Example12.9.1:
Given the sequence
a
n
u
f
n
=
[
n
]
where
u
[
n
]
is the unit step function
1
n
0
u
[
n
]=
(73)
0
n
<
0
the Z-transform is given by
n
=
0
a
n
z
−
n
F
(
z
)
=
(74)
n
=
0
(
az
−
1
n
=
)
(75)
This is simply the sum of a geometric series. As we confront this kind of sum quite often, let
us briefly digress and obtain the formula for the sum of a geometric series.
Suppose we have a sum
n
x
k
x
m
x
m
+
1
x
n
S
mn
=
=
+
+···+
(76)
k
=
m
Then
x
m
+
1
x
m
+
2
x
n
+
1
xS
mn
=
+
+···+
(77)
Subtracting Equation (
77
) from Equation (
76
), we get
x
m
x
n
+
1
(
1
−
x
)
S
mn
=
−
and
x
m
x
n
+
1
−
S
mn
=
−
x
If the upper limit of the sum is infinity, we take the limit as
n
goes to infinity. This limit exists
only when
1
|
|
<
1.
Using this formula, we get the Z-transform of the
x
{
f
n
}
sequence as
az
−
1
<
1
(
)
=
az
−
1
,
(78)
F
z
1
1
−
z
=
a
,
|
z
|
>
|
a
|
.
(79)
z
−
a
. For the Fourier transform
to exist, we need to include the unit circle in the region of convergence. In order for this to
happen,
a
has to be less than one.
Using this example, we can get some other Z-transforms that will be useful to us.
In this example the region of convergence is the region
|
z
|
>
