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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[1].

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[2].  From Bultheel's
linear prediction article[4] 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 ??


[1] http://thesounddesign.com/MIO/EQ-Coefficients.pdf
[2] http://homes.esat.kuleuven.be/~moonen/asp_course.html
[3] https://ccrma.stanford.edu/~jos/pasp/Conventional_Ladder_Filters.html
[4] ftp://129.132.148.131/hg/EMIS/journals/BMMSS/Bulletin/bul941/BULTHEEL.PDF




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