Wave Digital Filter (WDF) Construction

Wave digital elements may be derived from their describing differential equations (in continuous time) as follows:

1. Express forces and velocities as sums of traveling-wave components (``wave variables''):
   $$\begin{eqnarray*} f(t) &=& f^{{+}}(t)+f^{{-}}(t)\\ v(t) &=& v^{+}(t)+v^{-}(t) \end{eqnarray*}$$
   

   The actual ``travel time'' is always zero.
   (For historical reasons, WDFs typically use traveling-wave components scaled by 2.)

2. Digitize via the bilinear transform (trapezoid rule)

3. Use scattering junctions (``adaptors'') to connect elements together in
   • series (common velocity, summing forces), or
   • parallel (common force, summing velocities).