Two-channel perfect-reconstruction
QMF banks are not very useful because the
analysis filters have poor frequency selectivity. The
selectivity characteristics can be improved, however, if we
allow the system response
Tⅇⅈω
T
ω
to have magnitude-response ripples while keeping its
linear phase.
Say that
H
0
z
H
0
z
is causal, linear-phase, and has
impulse response length
NN. Then it is possible to write
H
0
ⅇⅈω
H
0
ω
in terms of a real-valued zero-phase response
H
~
0
ⅇⅈω
H
~
0
ω
, so that
H
0
ⅇⅈω=ⅇ-ⅈωN-12
H
~
0
ⅇⅈω
H
0
ω
ω
N
1
2
H
~
0
ω
(1)
Tⅇⅈω=
H
0
2ⅇⅈω-
H
0
2ⅇⅈω-π=ⅇ-ⅈωN-1
H
~
0
2ⅇⅈω-ⅇ-ⅈω-πN-1
H
~
0
2ⅇⅈω-π=ⅇ-ⅈωN-1
H
~
0
2ⅇⅈω-ⅇⅈπN-1
H
~
0
2ⅇⅈω-π
T
ω
H
0
ω
2
H
0
ω
2
ω
N
1
H
~
0
ω
2
ω
N
1
H
~
0
ω
2
ω
N
1
H
~
0
ω
2
N
1
H
~
0
ω
2
(2)
Note that if
NN is odd,
ⅇⅈωN-1=1
ω
N
1
1
,
Tⅇⅈω|ω=π2=0
ω
2
T
ω
0
(3)
A null in the system response would be very undesirable, and so
we restrict
NN to be an even
number. In that case,
Tⅇⅈω=ⅇ-ⅈωN-1
H
~
0
2ⅇⅈω+
H
~
0
2ⅇⅈω-π=ⅇ-ⅈωN-1|
H
0
ⅇⅈω|2+|
H
0
ⅇⅈω-π|2
T
ω
ω
N
1
H
~
0
ω
2
H
~
0
ω
2
ω
N
1
H
0
ω
2
H
0
ω
2
(4)
note:
The system response is linear phase, but will have amplitude
distortion if
|
H
0
ⅇⅈω|2+|
H
0
ⅇⅈω-π|2
H
0
ω
2
H
0
ω
2
is not equal to a constant.
Johnston's idea was to assign a cost function that penalizes
deviation from perfect reconstruction as well as deviation from
an ideal lowpass filter with cutoff
ω
0
ω
0
. Specifically, real symmetric coefficients
h
0
n
h
0
n
are chosen to minimize
J=λ∫
ω
0
π|
H
0
ⅇⅈω|2dω+1-λ∫0∞1-|
H
0
ⅇⅈω|2-|
H
0
ⅇⅈπ-ω|2dω
J
λ
ω
ω
0
H
0
ω
2
1
λ
ω
0
1
H
0
ω
2
H
0
ω
2
(5)
where
0<λ<1
0
λ
1
balances between the two conflicting objectives.
Numerical optimization techniques can be used to determine the
coefficients, and a number of popular coefficient sets have been
tabulated. (See
Crochiere and
Rabiner,
Johnston, and
Ansari and Liu)
Example 1: "12B" Filter
As an example, consider the "12B" filter from
Johnston:
h
0
0=-0.006443977=
h
0
11
h
0
0
-0.006443977
h
0
11
h
0
1=0.02745539=
h
0
10
h
0
1
0.02745539
h
0
10
h
0
2=-0.00758164=
h
0
9
h
0
2
-0.00758164
h
0
9
h
0
3=-0.0913825=
h
0
8
h
0
3
-0.0913825
h
0
8
h
0
4=0.09808522=
h
0
7
h
0
4
0.09808522
h
0
7
h
0
5=0.4807962=
h
0
6
h
0
5
0.4807962
h
0
6
which gives the following DTFT magnitudes (
Figure 1).
References-
R.E.Crochiere and L.B. Rabiner. (1983). Multirate Digital Signal Processing. Englewood Cliffs, NJ: Prentice Hall.
-
J.D. Johnston. (1980, April). A filter family designed for use in Quadrature mirror filterbanks. In Proc IEEE Internat. Conf. on Acoustics, Speech, and Signal Processing. (pp. 291-294).
-
R. Ansari and B. Liu. Multirate Signal Processing. In S.K. Mitra and J.F. Kaiser (Eds.), Handbook for Digital Signal Processing. [chp. 14 pp. 981-1084]. New York: Wiley Interscience.