Digital Waveguide Theory

Digital Waveguide Filters

Conventional Ladder Filters

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Given a reflecting termination on the right, the half-rate DWF chain of Fig.C.25 can be reduced further to the conventional ladder/lattice filter structure shown in Fig.C.26.

To make a standard ladder/lattice filter, the sampling rate is cut in
half (*i.e.*, replace by ), and the scattering junctions are
typically implemented in one-multiply form (§C.8.5) or
normalized form (§C.8.6), etc. Conventionally, if the
graph of the scattering junction is nonplanar, as it is for the
one-multiply junction, the filter is called a *lattice filter*;
it is called a *ladder filter* when the graph is planar, as it is
for normalized and Kelly-Lochbaum scattering junctions. For all-pole
transfer functions
, the *Durbin
recursion* can be used to compute the reflection coefficients
from the desired transfer-function denominator polynomial coefficients
[449]. To implement arbitrary transfer-function zeros, a
linear combination of delay-element outputs is formed using weights
that are called ``tap parameters'' [173,297].

To create Fig.C.26 from Fig.C.24, all delays along the top rail are pushed to the right until they have all been worked around to the bottom rail. In the end, each bottom-rail delay becomes seconds instead of seconds. Such an operation is possible because of the termination at the right by an infinite (or zero) wave impedance. Note that there is a progressive one-sample time advance from section to section. The time skews for the right-going (or left-going) traveling waves can be determined simply by considering how many missing (or extra) delays there are between that signal and the unshifted signals at the far left.

Due to the reflecting termination, conventional lattice filters cannot be extended to the right in any physically meaningful way. Also, creating network topologies more complex than a simple linear cascade (or acyclic tree) of waveguide sections is not immediately possible because of the delay-free path along the top rail. In particular, the output cannot be fed back to the input . Nevertheless, as we have derived, there is an exact physical interpretation (with time skew) for the conventional ladder/lattice digital filter.

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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