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Chapter 3: Multidimensional Wave Digital Filters

Chapter 3 begins with a review of some of the basics of symmetric hyperbolic systems of partial differential equations, and then proceeds to the generalization of electrical network passivity into multiple dimensions, where it has been called *MD-passivity* [48,85]; this can be done in a straightforward way through the application of coordinate transformations [62,122]. Next, we introduce *multidimensional circuit elements* [45], which are similar to their lumped counterparts, except that they are distributed objects and may have particular directions associated with them. The transformation to wave variables and discretization proceeds as in the lumped case, but now the trapezoid rule must be interpreted in a directional sense, as must be the associated MD *spectral bilinear transformations*. We then proceed through some treatments of typical model problems, namely the *advection equation* [176], the *transmission line system* [107], and its extension to two spatial dimensions, in which case it is called the *parallel-plate system* [60,61]. We write down *multidimensional Kirchoff circuit* representations (MDKCs) and show the discrete time and space counterparts (MDWDFs) for all these systems. We then spend some time in **§3.9.1** examining MDWD structures as finite difference schemes and make some comments about modal behavior, paying particular attention in **§3.9.2** to *parasitic modes*. In **§3.10** we present a new treatment of the initialization of MDWD methods, and then give a brief overview of methods for setting boundary conditions. *Balanced forms* are introduced in **§3.12** as a means of increasing the computational efficiency of MDWD methods; to date they have been notoriously sub-optimal in that the maximum allowable time step can be a great deal smaller than that of conventional finite difference methods (such as, for example, the *finite-difference time domain* (FDTD) method [184,214] and, by extension, DWNs). Finally, in **§3.13** we turn to a means of incorporating higher-order spatially accurate [176] methods into a circuit framework; this is surprising, because it had long been assumed that MDWD methods, traditionally based on the use of the trapezoid rule could be no better than second-order accurate [130]. We circumvent this problem by applying an alternative integration rule, which is also passivity-preserving (and which will also serve as a ``back-door'' into the realm of digital waveguide networks).

** Next:** Chapter 4: Digital Waveguide
** Up:** Summary and Results
** Previous:** Chapter 2: Wave Digital
Stefan Bilbao
2002-01-22