Digital Signal Processing Reference
In-Depth Information
and a Temporal Difference Q-Learning (TDQL) [
27
,
28
] for local refinement. A con-
striction factor has been included in the expression for standard deviation for dis-
persal of seeds. It is important to mention here that the constriction factors for all
members of the population should not be equal for the best performance. A mem-
ber with a good fitness should search in the local neighborhood, whereas a poor
performing member should participate in the global search. A good member thus
should have small constriction factors, while worse members should have relatively
large constriction factors. The selection of constriction factors from the meme pool
is governed by the TDQL learning policy.
8.2 The Design Problem Formulation
The general prototype transfer function of an
N
dimensional recursive IIR digital
filter is represented by the following expression:
i
=
0
j
=
0
p
ij
z
i
1
z
j
2
H(z
1
,z
2
)
=
H
0
,p
00
=
1
.
(8.1)
k
=
1
(
1
+
q
k
z
1
+
r
k
z
2
+
s
k
z
1
z
2
)
The variables
z
1
and
z
2
represent the complex indeterminants in the discrete
Laplace Transform and are related to the Fourier domain frequency terms
ω
1
and
ω
2
by the relationship
z
1
=
e
−
jω
1
and
z
2
=
e
−
jω
2
(where
ω
1
,ω
2
∈[−
).
Let us assume that the user-specified amplitude response of the filter is desig-
nated by
M
d
(ω
1
,ω
2
)
. The design task reduces to finding an appropriate transfer
function
H(z
1
,z
2
)
such that
M(ω
1
,ω
2
)
π,π
]
H(e
−
jω
1
,e
−
jω
2
)
follows the desired re-
sponse
M
d
(ω
1
,ω
2
)
as closely as possible. The approximation can be achieved by
minimizing [
4
,
8
,
16
,
18
],
=
N
1
N
2
M(ω
1
,ω
2
)
−
M
d
(ω
1
,ω
2
)
b
J(p
ij
,q
k
,r
k
,s
k
,H
0
)
=
(8.2)
n
1
=
0
n
2
=
0
where
ω
1
=
(π/N
1
)n
1
,
ω
2
=
(π/N
2
)n
2
, and
b
is a positive integer (usually
b
=
1
,
2
,
4, or 8).
Here the prime objective is to reduce the error between the desired and actual
amplitude responses of the filter at
N
1
N
2
points. Since the denominator consists of
only first degree factors, we assert the stability conditions following [
14
,
26
]as:
|
q
k
+
r
k
|−
1
<s
k
<
1
−|
q
k
−
r
k
|
,
(8.3)
where
k
1
,
2
,...,N
.
Thus the design of a 2D recursive filter is equivalent to the following constrained
optimization problem:
=
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