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Reduction to Standard Forms

Digital waveguide filters (DWF) are obtained (conceptually) by sampling the unidirectional traveling waves which occur in a system of ideal, lossless waveguides. Sampling is across time and space. Thus, variables in a DWF structure are equal exactly (at the sampling times and positions, to within numerical precision) to variables propagating in the physical medium in an interconnection of uniform transmission-lines.

Figure 1: Waveguide digital filter structure.

A cascade chain of DWF sections, terminated by a pure reflection on the right, is shown in Fig.1. Each box enclosing the symbol $ k_i(t)$ denotes a scattering junction characterized by that reflection coefficient. While we have mentioned only the Kelly-Lochbaum and one-multiply junction, any type of lossless scattering junction will do [4]. The DWF employs delays between each scattering junction along both the top and bottom signal paths, unlike conventional ladder and lattice filters. As a result, it has a direct physical interpretation as a sampled acoustic tube.

The delays preceding the two inputs to a junction can be ``pushed'' into the junction so that they emerge on the outputs and combine with the delays there. (Show this using the Kelly-Lochbaum scattering junction.) By performing this operation on every other section in the DWF chain, the filter structure of Fig.2 is obtained. This structure has some advantages worth considering: (1) it consolidates delays to length $ 2T$ as do conventional lattice/ladder structures, (2) it does not require a termination by an infinite wave impedance, allowing it to be extended to networks of arbitrary topology (e.g., multiport branching, intersection, and looping), and (3) there is no long delay-free signal path along the upper rail as in conventional lattice/ladder structures--a pipeline segment is only two sections long. This structure appears to have better overall characteristics than any other digital filter structure for many applications. Advantage (2) makes it especially valuable for modeling physical systems.

Figure 2: Pipelineable, physically extendible, consolidated-delay, waveguide filter.

Given a reflecting termination on the right, the half-rate DWF chain of Fig.2 can be reduced further to the conventional ladder/lattice structure of Fig.3. Every delay on the upper rail is pushed to the right until they have all been worked around to the bottom rail. In the end, each bottom-rail delay becomes $ 2T$ seconds instead of $ T$ seconds. Such an operation is possible because of the termination at the right by an infinite (or zero) wave impedance. In the time-varying case, pushing a delay through a multiply results in a corresponding time advance of the multiplier coefficient. The time arguments of the reflection coefficients in the figure indicate the amount of the time shift for each section. Note that because of 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 series (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.

Figure 3: Conventional ladder/lattice filter structure.

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``Waveguide Filter Tutorial'', by Julius O. Smith III, Proceedings of the International Computer Music Conference (ICMC-87, Champaign-Urbana), pp. 9-16, Computer Music Association, 1987..
Copyright © 2010-11-22 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University