Digital Signal Processing Reference
In-Depth Information
Note that if the sequence
x
[
n
]
is a geometrically convergent series, then the
z
-transform will also
converge provided that
x
[
n
+
1
]
<
|
z
|
x
[
n
]
In terms of numbers, if
=
x
[
n
+
1
]
0
.
9
x
[
n
]
for example, then it is required that
|
z
|
>
0
.
9 for convergence to occur.
is a geometrically convergent series, and
z
is properly chosen, the sum of the infinite
series of numbers consisting of
x
[
n
]
z
−
n
may conveniently be written in a simple algebraically closed form.
If in fact
x
[
n
]
Example 2.1.
Determine the z-transform for a single pole IIR with a real pole
p
having a magnitude
less than 1.0.
The impulse response of such a filter may be written as
p
0
(
1
), p
1
,p
2
,p
3
, ... p
n
[
=
]
etc., or to pick a concrete example with the pole at 0.9,
0
.
9
n
=[
1
,
0
.
9
,
0
.
81
,
0
.
729
, ...
]
and the
z
-transform would therefore be:
0
.
9
z
−
1
0
.
81
z
−
2
0
.
729
z
−
3
...p
n
z
−
n
A(z)
=
1
+
+
+
+
or in generic terms as
...p
n
z
−
n
The summation of an infinite number of terms of the form
c
n
where
pz
−
1
p
2
z
−
2
p
3
z
−
3
A(z)
=
1
+
+
+
+
|
c
|
<
1 with
0
≤
n<
∞
is
1
(2.2)
1
−
c
pz
−
1
(note that
p
0
=
=
For the single pole IIR with a pole at
p
, and by letting
c
1) in Eq. (2.2),
we get the closed-form
z
-transform as
1
z
A(z)
=
=
(2.3)
1
−
pz
−
1
z
−
p
The
z
-transform in this case is defined or has a finite value for all
z
with
|
z
|
>
|
p
|
or
p
z
<
1
