This module will drop the restrictive QMF conditions and focus on using FIR filters to achieve perfect reconstruction from filterbanks.
Fir perfect-reconstruction conditions
The
QMF design
choices prevented the design of a useful(
i.e. , frequency selective)
perfect-reconstruction (PR) FIR filterbank. This motivates usto re-examine PR filterbank design without the
overly-restrictive QMF conditions. However, we will stillrequire causal FIR filters with real coefficients.
Recalling that the
two-channel
filterbank (
),
has the input/output relation:
we see that the delay-
perfect
reconstruction requires
where
or, equivalently, that
where
For FIR
and
, we require
Since we cannot assume that FIR
and
share a common root.
that
for
and
. Under this determinant condition, we find that
Assuming that
and
are causal with non-zero initial coefficient, we
choose
to keep
and
causal and free of unnecessary delay.
