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Digital Waveguide Theory

In this appendix, the basic principles of digital waveguide acoustic modeling are derived from a mathematical point of view. For this, the reader is expected to have some background in linear systems and elementary physics. In particular, facility with Laplace transforms [284], Newtonian mechanics [180], and basic differential equations is assumed.

We begin with the partial differential equation (PDE) describing the ideal vibrating string, which we first digitize by converting partial derivatives to finite differences. This yields a discrete-time recursion which approximately simulates the ideal string. Next, we go back and solve the original PDE, obtaining continuous traveling waves as the solution. These traveling waves are then digitized by ordinary sampling, resulting in the digital waveguide model for the ideal string. The digital waveguide simulation is then shown to be equivalent to a particular finite-difference recursion. (This only happens for the lossless ideal vibrating string with a particular choice of sampling intervals, so it is an interesting case.) Next digital waveguides simulating lossy and dispersive vibrating strings are derived, and alternative choices of wave variables (displacement, velocity, slope, force, power, etc.) are derived. Finally, an introduction to scattering theory for digital waveguides is presented.



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Next: The Ideal Vibrating String

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About the Author: Julius Orion Smith III
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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