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EEL 3135:
Dr. Fred J. Taylor, Professor
Lesson Title: IIR Systems
Lesson Number: 22 [Section 84 to 86]
Background:
In the previous lesson infinite impulse response or IIR system.
These systems are
characterized by the discretetime difference equation:

+

=
∑
∑
=
=
M
0
m
m
N
1
m
m
0
m]
x[k
b
m]
y[k
a
a
1
y[k]
1
where y[k] is the system’s output timeseries to an input x[k] time series. If a
0
=1, the system is
called
monic
.
An IIR with all
initial conditions
zero is said to be
atrest
.
The response of an LTI
to an arbitrary input time series x[k] is formally defined by the convolution theorem is stated
below.
[
]
[
]
[
]
[
]
[
]
m
k
x
m
h
m
x
m
k
h
k
y
m
m

=

=
∑
∑
∞
=
∞
=
0
0
2
where h[k] is the IIR’s impulse response.
A key tool in quantifying the IIR filter’s impulse
response is the
z
transform.
In Lesson 21 the convolution theorem was reintroduced and it
stated that:
[
]
[
]
[
]
(
29
(
29
(
29
z
X
z
H
z
Y
k
x
k
h
k
y
Z
=
→
←
=
3.
The mechanics and relationships developed in Chapter 7 can be directly applied to the studies
from Chapter 8.
Consider again the IIR described in Equation 1.
The ztransform of the monic
version of this Equation is:
(
29
(
29
(
29
+
=
→
←

+

=
∑
∑
∑
∑
=
=

=
=
M
0
m
m
m
N
1
m
m
m
M
0
m
m
N
1
m
m
z
X
z
b
z
a
z
Y
m]
x[k
b
m]
y[k
a
y[k]
z
Y
Z
4.
After a small amount of manipulation:
(
29
(
29
(
29
(
29
(
29
∑
∑
∑
∑
=
=

=
=

=

=
=
M
0
m
m
m
N
1
m
m
m
M
0
m
m
m
N
1
m
m
m
z
X
z
b
z
a
1
z
Y
z
X
z
b
z
a
z
Y
z
Y
5.
As a final step, the transfer function of the atrest system is given by:
1
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View Full DocumentEEL 3135:
Dr. Fred J. Taylor, Professor
(
29
(
29
(
29

=
=
∑
∑
=

=
N
1
m
m
m
M
0
m
m
m
z
a
1
z
b
z
X
z
Y
z
H
6.
The
inverse ztransform
of H(z), if we knew how to perform one, would produce the system’s
impulse response.
That is:
Z
1
(H(z) = h[k]
7.
For an IIR, the transfer function H(z) is the ratio of two polynomials in
z
.
As such, an IIRs
transfer function is also defined in terms of
poles
and
zeros
. The transfer function given in
Equation 6 can, therefore, also be expressed as:
(
29
(
29
(
29
∏
∏
∑
∑
=
=
=

=


=

=
=
n
m
m
M
m
m
p
z
z
z
k
0
0
N
1
m
m
m
M
0
m
m
m
)
(
)
(
z
a
1
z
b
z
X
z
Y
z
H
8.
where z
m
is a filter zero and p
m
is a filter pole.
One of an LTI system’s attributes that historically
been connect to the polezero locations is stability.
Stability
The authors motivate the concept of stability in the context of a suite of simple example
beginning with a difference equation:
y[k]=0.5y[k1] + 2x[k]
9.
which translates into a transfer function given by:
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