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 , 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 . The inserted waveguide impedance is real and positive, but otherwise arbitrary. b) Expanded view of the interior of the infinitesimal waveguide section, also representing the termination impedance as an impedance-step within the waveguide.

   Points to note:

   • The infinitesimal waveguide is terminated by the element. The element reflects waves as if it were a new waveguide section at impedance , as depicted in Fig.F.1b.

   • The interface to the element is recast as traveling-wave components and at impedance . In terms of these components, the physical force on the element is obtained by adding them together: .

   • The waveguide impedance is arbitrary because it has been physically introduced. We will need to know it when we connect this element to other elements. The element's interface to other elements is now a waveguide (transmission line) at real impedance .

   • The junction is ``parallel'' (cf. §7.2):
     • Force (voltage) must be continuous across the junction, since otherwise there would be a finite force across a zero mass, producing infinite acceleration.

     • The sum of velocities (currents) into the junction must be zero by conservation of mass (charge).

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Subsections

• Reflectance of a General Lumped Waveguide Termination
• Reflectances of Elementary Impedances
• Capacitor Reflectance
• Inductor Reflectance
• Resistor Reflectance
• Choosing Impedance to Simplify Element Reflectance
• Digitizing Elementary Reflectances by Bilinear Transform

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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.