Introductory Remarks

The last chapter was concerned with techniques for deriving a digital filter design from an analog network. It should be clear that such a digital filter structure can also be considered to be an explicit numerical solver for the system of of ordinary differential equations (ODEs) defined by the analog network which performs the filtering on continuous-time signals. This is perhaps an obvious point, but was apparently first noted in the literature in [65]. It is interesting that this link was not made immediately in the multidimensional case, which is the subject of this chapter.

A multidimensional generalization of wave digital filters (*MDWDFs*) first appeared rather early on [44], and most of the initial work involved applications to 2D filter design [118,119]. It is itself an outgrowth of earlier work in the area of multidimensional circuits and systems [20,105,135] where the emphasis was on the synthesis of so-called variable networks (i.e., lumped passive networks with variable elements). The procedure for deriving a wave digital filter is largely the same in multiple dimensions as for the lumped case: to a given reference circuit, made up of elements connected either in series or parallel, various transformations are applied, specifically a change to wave variables, and spectral mappings. The end product is a wave digital network which has nearly all of the same desirable properties as lumped WD networks, especially recursive computability, and insensitivity to signal and coefficient truncation. The difference in MD, however, is that the reference circuit, usually called a *multidimensional Kirchoff circuit* or *MDKC* is now far more of a mathematical abstraction than a lumped circuit; the circuit state is a function of several variables, which may or may not include time, and the circuit elements (as well as the connections between them) must be interpreted in a distributed sense. In particular, it is not a circuit which can be ``built'' (except in the case of the variable networks mentioned previously). As such, the major problem confronting the designer of a MDWDF is the construction of this reference circuit [45,44]. Various techniques were put forth, some involving MD circuits obtained by rotation of a known lumped reference network [46]. A good deal of work went into the related synthesis problem for general multidimensional reactance two-ports [8,9,52], which is much more difficult than in the lumped case (and not possible in general).

The first paper to consider MDWDFs from a simulation perspective appeared in 1990 [59], though it was foreshadowed much earlier in [178]. That is to say, in analogy with the lumped case, a closed MDWD network could be considered to be a simulator of a distributed system which is defined by a system of partial differential equations (PDEs) and *represented* by an MDKC. Here, unlike for filtering, there is a clear interpretation of the reference circuit, which is simply a symbolic restatement of the defining equations of a particular model system. The wave digital numerical integration approach is applicable to a wide variety of physical systems, including electromagnetics [50], coupled transmission lines [63,106] and elastic solid and beam dynamics [131]. Most surprisingly, the method can be applied to highly nonlinear systems [127] such as those of fluid dynamics [16,49,70], as well as even more complex hybrids, such as the magnetohydrodynamic system [191]. The method requires that the propagation speeds in the problem to be modeled be bounded; this is equivalent to saying that the system should be of *hyperbolic type*^{}[176]. This requirement is important because numerical methods derived in this way from this approach can be interpreted as *explicit finite difference schemes* [82]; as such, they must obey a requirement (the Courant-Friedrichs-Lewy criterion [176]) relating the physical region of dependence for the model problem to a similar region on a numerical grid. We would also like to note that a related approach to numerical integration, based on a transfer function formulation has been taken in [108,141,143,144].

Although this chapter is intended in part as an extended review and compendium of the work to date in the field of numerical integration through the use of wave digital filters, the subtext is certainly that these methods can and should be treated as a particular class of finite difference methods endowed with a special property, namely passivity. This point has not been explored in any depth in the literature, except in the lumped case [131]. Such a treatment will also make it easier to compare wave digital methods to digital waveguide networks (DWNs)[166,198,200], which can also be used for numerical integration purposes in a very similar way. Chapter 4, which is devoted to DWNs, will return to the subject of MDWDFs for such a comparison, and eventually, a unification of the two methods. In Chapter 5, we will apply the concepts discussed here to a variety of more complex systems, in particular those describing the vibration of beams, plates, shells and elastic solids.

We refer to §1.3 for a full technical summary of this chapter.