Digital Waveguide Networks as Multidimensional Wave Digital Filters

Multidimensional wave digital filters (MDWDFs) [1, 2] have recently been applied toward the numerical simulation of distributed systems. The basic procedure for deriving an algorithm is similar to that which was originally developed for deriving wave digital filter [3, 4] structures from lumped analog networks, though in that case, the application was to filter design, and not explicitly to simulation. In the lumped case, one begins from a given analog network structure, composed typically of RLC elements (and possibly more exotic devices such as transformers, gyrators, etc.). One then appl;ies a continuous-to-discrete spectral mapping to each reactive element, and, after the introduction of wave variables [3], ends up with a recursible filter structure. The spectral mapping (a particular type of bilinear transform, which corresponds, in the discrete-time domain, to the use of the trapezoid rule of numerical integration) is chosen so that the energetic properties of the analog network are mirrored by the discrete-time structure. It has the form