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In-Depth Information
10
5
0
−
5
−
10
−
15
−
20
−2
5
−
30
0
0.5
1
1.5
2
2.5
3
Frequency
F I GU R E 14 . 18
Magnitude characteristics of the two-tap PR filters.
Thus, for perfect reconstruction with no aliasing and no amplitude or phase distortion, the
mirror condition does not seem like such a good idea. However, if we slightly relax these rather
strict conditions, we can obtain some very nice designs. For example, instead of attempting
to eliminate all phase and amplitude distortion, we could elect to eliminate only the phase
distortion and
minimize
the amplitude distortion. We can optimize the coefficients of
H
1
(
z
)
such that
T
e
j
ω
)
is made as close to a constant as possible, while minimizing the stopband
energy of
H
1
(
(
in order to have a good low-pass characteristic. Such an optimization has
been suggested by Johnston [
204
] and Jain and Crochiere [
208
]. They construct the objective
function
z
)
π
π
0
(
2
2
e
j
ω
)
e
j
ω
)
J
=
α
H
1
(
d
ω
+
(
1
−
α)
1
−
T
(
)
d
ω
(74)
ω
s
which has to be minimized to obtain
H
1
(
z
)
and
T
1
(
z
)
, where
ω
s
is the cutoff frequency of the
filter.
We can also go the other way and eliminate the amplitude distortion, then attempt to
minimize the phase distortion. A review of these approaches can be found in [
207
,
206
].
14.6.2 Power Symmetric FIR Filters
Another approach, independently discovered by Smith and Barnwell [
205
] and Mintzer [
209
],
can be used to design a two-channel filter bank in which aliasing, amplitude distortion, and
