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Wave Digital Filters

The idea of wave digital filters is to digitize RLC circuits (and certain more general systems) as follows:

  1. Determine the ODEs describing the system (PDEs also workable).

  2. Express all physical quantities (such as force and velocity) in terms of traveling-wave components. The traveling wave components are called wave variables. For example, the force $ f(n)$ on a mass is decomposed as $ f(n) = f^{{+}}(n)+f^{{-}}(n)$ , where $ f^{{+}}(n)$ is regarded as a traveling wave propagating toward the mass, while $ f^{{-}}(n)$ is seen as the traveling component propagating away from the mass. A ``traveling wave'' view of force mediation (at the speed of light) is actually much closer to underlying physical reality than any instantaneous model.

  3. Next, digitize the resulting traveling-wave system using the bilinear transform7.3.2,[452, p. 386]). The bilinear transform is equivalent in the time domain to the trapezoidal rule for numerical integration7.3.2).

  4. Connect $ N$ elementary units together by means of $ N$ -port scattering junctions. There are two basic types of scattering junction, one for parallel, and one for series connection. The theory of scattering junctions is introduced in the digital waveguide context (§C.8).
A more detailed introduction to WDFs is provided in Appendix F. In particular, the force-driven mass is considered in §F.3.4, and its wave digital model is shown in Fig.1.12.

Figure 1.12: Wave digital mass driven by external force $ f(n)$ .

We will not make much use of WDFs in this book, preferring instead more prosaic finite-difference models for simplicity. However, we will utilize closely related concepts in the digital waveguide modeling context (Chapter 6).

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University