This module introduces the ideas behind and design issues of Orthogonal Perfect Reconstruction of FIR filterbanks.
Orthogonal pr filterbanks
The
FIR
perfect-reconstruction (PR) conditions leave some
freedom in the choice of
and
.
Orthogonal PR filterbanks are defined
by causal real-coefficienteven-length-
analysis filters
that satisfy the following two equations:
To verify that these design choices satisfy the FIR-PR
requirements for
and
, we evaluate
under the second condition above. This yields
which corresponds to
and
in the FIR-PR determinant condition
. The remaining FIR-PR conditions then imply that
the synthesis filters are given by
The orthogonal PR design rules imply that
is "power symmetric" and that
form a "power complementary" pair. To see the power
symmetry, we rewrite the first design rule using
and
, which gives
The last two steps leveraged the fact that the
DTFT of a
real-coefficient filter is conjugate-symmetric. Thepower-symmetry property is illustrated in
:
Power complementarity follows from the second orthogonal PR
design rule, which implies
. Plugging this into the previous equation, we find
The power-complimentary property is illustrated in
:
