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A Physical Derivation of Wave Digital Elements

This section provides a ``physical'' derivation of Wave Digital Filters (WDF), which contrasts somewhat with the more formal derivation common in the literature. The derivation is presented as a numbered series of steps (some with rather long discussions):

  1. To each element, such as a capacitor or inductor, attach a length of waveguide (electrical transmission line) having wave impedance $ R_0$ , and make it infinitesimally long. (Take the limit as its length goes to zero.) A schematic depiction of this is shown in Fig.F.1a. For consistency, all signals are Laplace transforms of their respective time-domain signals. The length must approach zero in order not to introduce propagation delays into the signal path.

    Figure F.1: a) Physical schematic for the derivation of a wave digital model of driving-point impedance $ R(s)$ . The inserted waveguide impedance $ R_0$ is real and positive, but otherwise arbitrary. b) Expanded view of the interior of the infinitesimal waveguide section, also representing the termination impedance $ R(s)$ as an impedance-step within the waveguide.

    Points to note:

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2017-05-16 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University