Digital Signal Processing Reference
In-Depth Information
transfer function. Because the poles of FIR filters are located at the origin and lie
within the unit circle, they are inherently stable. Also, FIR filters can be designed
as linear phase filters, which makes them a better choice in phase-sensitive applica-
tions. FIR filters whose phase response is linear with respect to frequency are said to
be linear phase. An FIR filter is linear phase if its coefficients are symmetric around
the center coefficient. Delay through such filters is constant at all frequencies; hence,
they do not cause phase distortion.
An FIR filter can be described by the transfer function:
N
a
i
z
−
i
.
H(z)
=
(7.7)
i
=
0
The parameters
a
0
,a
1
,a
2
,...,a
N
appearing in (
7.7
) are called filter coefficients,
and they determine the characteristics of a filter. Filter specifications, which are im-
portant in a filter design process, include the passband and stopband normalized
frequencies (
ω
p
,
ω
s
), passband and stopband ripple (
δ
p
) and (
δ
s
), stopband attenu-
ation and transition width. These specifications are satisfied by the filter coefficients
in (
7.7
). In any filter design problem, some of these specifications are fixed while
others are determined. In this chapter, swarm, evolutionary, quantum, and hybrid
optimization algorithms are applied in order to obtain an actual filter response that
comes as close as possible to the ideal response.
7.4 FIR Filter Design Using PSO-QI
From (
7.7
), the transfer function of the FIR filter can also be represented as:
a
1
z
−
1
a
2
z
−
2
a
N
z
−
N
.
H(z)
=
a
0
+
+
+···+
(7.8)
For (
7.8
), the numerator coefficient vector
a
0
,a
1
,a
2
,...,a
N
is represented in
N
dimensions. In PSO-like algorithms, each particle is distributed in a
D
-dimensional
search space, where
D
N
for the FIR filter. The position of each particle in this
D
-dimensional search space represents the coefficients of the FIR filter's transfer
function. During each iteration, these particles find a new position, which is the
new set of coefficients. Using the new values of the coefficients, the performance of
each particle is evaluated based on some predefined fitness function. The fitness is
then used to improve the search during each iteration, and the result obtained after a
certain number of iterations or after the error falls below a certain limit is considered
the final result. The error between the filter response of the desired and approximate
filters is given by (
7.9
):
=
G(ω)
H
d
e
jω
−
H
e
jω
=
E(ω)
(7.9)
where
G(ω)
is the weighting function used to provide different weights for the ap-
proximate errors in different frequency bands,
H
d
(e
jω
)
is the frequency response of
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