Digital Signal Processing Reference
In-Depth Information
passband ripple magnitude, maximum stopband ripple magnitude, and desirable
magnitude response of the prototype filter as that in [
23
], that is,
B
p
=[−
0
.
4
π,
0
.
4
π
]
,
(1.79)
B
s
=[
0
.
6
π,π
]∪[−
π,
−
0
.
6
π
]
,
(1.80)
N
=
36
,
(1.81)
ε
p
=−
50 dB
,
(1.82)
ε
s
=−
50 dB
,
(1.83)
D(ω)
=
1
for
ω
∈
B
p
,
(1.84)
and
D(ω)
=
0
for
ω
∈
B
s
.
(1.85)
In order to guarantee that the performance of the QMF bank designed via our pro-
posed method is better than that in [
23
], the specification on the maximum ampli-
tude distortion of the filter bank is chosen as
ε
a
=−
58 dB, which is better than that
in [
23
](
ε
a
=
50
.
4576 dB). In order not to have any bias among the max-
imum amplitude distortion of the filter bank, the maximum passband ripple mag-
nitude, and the maximum stopband ripple magnitude of the prototype filter, all the
weights in the objective function are chosen to be the same, that is,
α
=
β
=
γ
=
0
.
003
=−
1.
ε
=
10
−
6
In this chapter,
ε
=
are chosen, which is small enough for most applica-
tions.
¯
1
is chosen as the filter coefficients obtained via the Remez exchange algo-
˜
rithm, which is guaranteed to satisfy the specifications on the maximum passband
ripple magnitude and the maximum stopband ripple magnitude of the prototype fil-
ter.
¯
is chosen as the diagonal matrix with all diagonal elements equal to 10
−
3
,
which is small enough for most applications.
To compare the efficiency of the designed method, our proposed method only
takes three iterations to converge and the total time required for the computer nu-
merical simulations is 0.8 seconds. On the other hand, the method discussed in [
23
]
takes 68 iterations to converge and the total time required for the computer numeri-
cal simulations is 80 seconds. Hence, it can be concluded that the method discussed
in [
23
] requires more computational efforts than our proposed method and our pro-
posed method is more efficient than that discussed in [
23
]. The magnitude responses
of the filter banks as well as the magnitude responses of the prototype filters in both
the passband and the stopband designed via our proposed method are shown in
Figs.
1.1
,
1.2
,
1.3
. It can be seen from these figures that the prototype filter designed
by our proposed method can achieve
δ
p
=−
64
.
2416 dB and
δ
s
=−
50
.
3625 dB,
and the QMF bank could achieve
δ
a
=−
58
.
1557 dB. It can be checked easily that
the QMF bank designed via our proposed method achieves better performance with
respect to the maximum amplitude distortion of the filter bank, the maximum pass-
band ripple magnitude, and the maximum stopband ripple magnitude ripple of the
prototype filter than that designed by the method discussed in [
23
]. This is because
the QMF bank designed by the method discussed in [
23
] is not the global minimum,
while that designed by our proposed method is the global minimum.
Search WWH ::
Custom Search