This module examines a type of Quadrature Mirror Filterbank (QMF) in regards to its reconstruction properties and the ideas presented by Johnston.
Two-channel perfect-reconstruction
QMF banks are not very useful because the
analysis filters have poor frequency selectivity. Theselectivity characteristics can be improved, however, if we
allow the system response
to have magnitude-response ripples while keeping its
linear phase.
Say that
is causal, linear-phase, and has
impulse response length
. Then it is possible to write
in terms of a real-valued zero-phase response
, so that
Note that if
is odd,
,
A null in the system response would be very undesirable, and so
we restrict
to be an even
number. In that case,
The system response is linear phase, but will have amplitude
distortion if
is not equal to a constant.
Johnston's idea was to assign a cost function that penalizesdeviation from perfect reconstruction as well as deviation from
an ideal lowpass filter with cutoff
. Specifically, real symmetric coefficients
are chosen to minimize
where
balances between the two conflicting objectives.
Numerical optimization techniques can be used to determine thecoefficients, and a number of popular coefficient sets have been
tabulated. (See
Crochiere and
Rabiner ,
Johnston , and
Ansari and Liu )
"12b" filter
As an example, consider the "12B" filter from
Johnston :
which gives the following DTFT magnitudes (
).
The top plot shows the analysis filters and the bottom one
shows the system response.
