A physical component, such as a mass or spring, or an electrical component, such as an inductor or capacitor, is considered to be represented by an impedance .
Attached to this impedance is a waveguide of infinitesimal length having wave impedance , as diagrammed in Fig. 15.
Figure 15:
The impedance representation of a wave digital filter element.
The interface to the element is described in terms of a scattering junction, as illustrated in Fig. 16, where physical variables of force (and velocity) are decomposed into traveling-wave components.
Figure 16:
Traveling-wave scattering view at the junction of a wave digital filter element.
The waveguide impedance is arbitrary because it has been introduced only to facilitate the interconnection of fundamental WDF elements.
At the junction, incoming velocities (or currents) must sum to zero. Forces (or voltages) on either side of the junction must be equal. These two properties define a parallel junction.
The WDF element can now be represented in terms of a reflectance as seen by the infinitesimal waveguide section:
This relationship can be found by expressing the junction properties above in terms of traveling-wave components,
and
and solving for in terms of and .
As an example, the driving-point impedance of a mass is given by , which implies a reflectance of
The arbitrary waveguide impedance is then determined so as to simplify the reflectance expression. Setting in results in
Finally, the reflectance expression is digitized using the bilinear transform by making the substitution
For the case of the WDF mass element, this reduces to