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Wed Apr 17 20:58:19 EDT 2013

## Inverse Lattice

So, what is the inverse lattice in [1]? It's related, but not the
same as the normalized ladder. It appears it is not normalized, but
cosines do show up in one direction due to the multiple paths.
EDIT: Cosines show up because the junctions are still
energy-perserving (unit norm eigenvalues) but no longer orthogonal. A
non--normalized lattice allpass implements the same transfer frunction
as a normalized ladder, but will have undergone a coordinate
transformation for the state variables, i.e. the signals traveling
through the sections and delays.
It seems that the inverse lattice is really just the lattice filter
with the delay--free chain reversed. There is no 1/c normalization,
but this only has an effect on the allpole output, not the allpass, as
in the forward lattice, both forward and backward predictors are
scaled in the same way, and they serve as numerator and denominator of
the allpass.
The important thing to realize is that the equations harbor several
filter realizations: forward/backward FIR, reconstructing allpole,
allpass, both with and without the 1/c normalization.
% [1] ftp://129.132.148.131/hg/EMIS/journals/BMMSS/Bulletin/bul941/BULTHEEL.PDF

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