Twochannel perfectreconstruction 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
Tejω
T
ω
to have magnituderesponse ripples while keeping its
linear phase.
Say that
H
0
z
H
0
z
is causal, linearphase, and has impulse response length
NN. Then it is possible to write
H
0
ejω
H
0
ω
in terms of a realvalued zerophase response
H
~
0
ejω
H
~
0
ω
, so that
H
0
ejω=e(−j)ωN−12
H
~
0
ejω
H
0
ω
ω
N
1
2
H
~
0
ω
(1)
Tejω=
H
0
2ejω−
H
0
2ej(ω−π)=e(−j)ω(N−1)
H
~
0
2ejω−e(−j)(ω−π)(N−1)
H
~
0
2ej(ω−π)=e(−j)ω(N−1)(
H
~
0
2ejω−ejπ(N−1)
H
~
0
2ej(ω−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,
ejω(N−1)=1
ω
N
1
1
,
Tejω
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,
Tejω=e(−j)ω(N−1)(
H
~
0
2ejω+
H
~
0
2ej(ω−π))=e(−j)ω(N−1)(
H
0
ejω2+
H
0
ej(ω−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)
The system response is linear phase, but will have amplitude
distortion if

H
0
ejω2+
H
0
ej(ω−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
ejω2dω−1∫0∞1−
H
0
ejω2−
H
0
ej(π−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)
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).

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. 291294).

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. 9811084]. New York: Wiley Interscience.