Digital Signal Processing Reference
In-Depth Information
11.5.3 Design of an IIR Filter
Theorem
11.2
is used to check whether the poles of an IIR filter lie inside the disk
D(α,r)
. This stability check should be performed in each generation of an evolu-
tionary algorithm used to design an IIR filter. Before proceeding, we first define the
vector of the coefficients in the denominator of
H(z),a
=
(
1
a
1
a
2
... a
n
)
,asa
chromosome of GA. The following definition is helpful to describe the introduced
method.
Definition 11.4
A chromosome
a
=
(
1
a
1
a
2
...a
n
)
is stable if the corresponding
=
k
=
0
a
k
z
−
k
is
P
-stable. Moreover, it is
D(α,r)
-stable if the
corresponding polynomial is
PD(α,r)
-stable [
15
].
polynomial
a(z)
After the derivation of the stability criterion for the design of an IIR filter, based
on GA, a design strategy of a robust CSD coded stable IIR filter is then proposed as
follows. Moreover, the flowchart of the design procedure is depicted in Fig.
11.8
for
clarification.
Step 1. Initial generation.
Define the coefficients of the denominator of
H(z)
,
a(z)
=
k
=
0
a
k
z
−
k
, as a chromosome
a
(a
0
a
1
a
2
... a
n
)
. Generate
the initial generation of the chromosomes based on the CSD code format.
Step 2. Check the stability property.
Check the stability of the chromosome ac-
cording to Theorem
11.2
and Definition
11.4
. If a chromosome does not
satisfy the criterion in Theorem
11.2
, then regenerate a new chromosome
based on the CSD code format.
Step 3. Evaluate the fitness value of the chromosomes.
Evaluate the fitness value
of the chromosomes according to (
11.6
).
Step 4. Check whether the result is acceptable.
If the result is acceptable or the
number of iterations is larger than an assigned maximum number, go to the
end of this procedure (Step 7), otherwise go to the next step.
Step 5. Generate offspring.
Generate new chromosomes by the crossover and mu-
tation based on the CSD format which are proposed in Sect.
11.3
.
Step 6. Check the stability criterion.
Check the new chromosomes generated
from Step 5 to see whether they satisfy the stability criterion in Theo-
rem
11.2
. Go to Step 3 if they do, or go back to Step 5.
Step 7. End of this procedure.
=
11.5.4 Design Example of an IIR Filter [
5
]
Suppose that the transfer function of a CSD coded IIR filter is described as
H(z)
=
b
1
z
−
1
b
0
+
=
=
0
,
1, are all de-
signed in the CSD format. The target frequency response is
H
I
(Ω)
as depicted in
a
2
z
−
2
, in which the coefficients
a
i
,i
0
,
1
,
2, and
b
i
,i
a
1
z
−
1
a
0
+
+
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