A class of causal, FIR, two-channel, criticially sampled, exact perfect-reconstruction filter-banks is the set of so-called Conjugate Quadrature Filters (CQF).
In the z-domain, the CQF relationships are
In the time domain, the analysis and synthesis filters are given by
That is,
for the lowpass channel, and the highpass
channel filters are a modulation of their lowpass counterparts by
.
or
where
With the CQF constraints, () reduces to
Let
, such that
is a spectral factor of
the half-band filter
(i.e.,
is a nonnegative power
response which is lowpass, cutting off near
). Then,
(
) reduces to
A simple design of an FIR half-band filter would be to window a sinc function:
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(9) |
In this design procedure, we have chosen to satisfy the first two and give up the third.