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Tue Apr 16 02:02:46 EDT 2013

## Ladder vs. Lattice?

EDIT: See later this week.

Lattice has to do with AR modeling.  What is a (normalized) ladder
filter?  It shows up here.

Just staring at some pictures, something tells me the lattice vs. ladder
distinction has to do with hyperbolic vs. orthogonal rotations...

EDIT: Yes +-.  A ladder filter can be derived directly from a
waveguide, while a lattice filter is derived from the Levinson-Durbin
algorithm for linear prediction.  Both arrive at +- the same point,
with the former being engergy-normalized over the sections and the
other one not.  They have the same transfer function for the allpass.

Moonen's course notes might have some more info.  From Bultheel's
linear prediction article it is clear that the hyperbolic form
comes from moving from an energy equation for traveling waves moving
into opposite directions specified at different points in space, to a
form that is parameterized in space only.  Basically, these are
equations that can be interpreted in two directions.

Ladder is phase shifter: Given an allpass filter $h_n(z)$, adding an
orthogonal ladder junction (wihtout delay) parameterized by angle
$\alpha$ will give a new allpass filter

$$h_{n+1}(z) = z^{-1} \frac{s + h_n}{1 + s h_n(z)}.$$ with $s = \sin \alpha.$

I.e. given a phase shaper, it will shape it some more.  Without the
delay this doesn't do much: such operations form a monoid (actually a
group).  However, adding the delay will add a full phase wrap on each
section.

?? Now the relation between ladder and lattice is to observe that the
recursion relation above can also be implemented more directly ??

 http://thesounddesign.com/MIO/EQ-Coefficients.pdf
 http://homes.esat.kuleuven.be/~moonen/asp_course.html

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