Wave digital filter networks are derived from lumped analog circuits, and we have seen that they can be interpreted as numerical ODE integrators. Most importantly, we saw that a given analog circuit immediately implies a corresponding WDF structure; if the original circuit is lossless, then the WDF network, which is its discretetime image, will be lossless as well. It is easy to extend the maintenance of losslessness to the more general case of passivity (i.e., we allow our networks to dissipate energy, as well as recirculate it).
Fettweis and Nitsche [62] found a way of directly extending this simulation technique to distributed systems. First, it is necessary to generalize the definition of a circuit element to multiple dimensions, in which case it is called an MD circuit element; an MD oneport is shown at left in Figure 1.12. The oneport is still defined in terms of a voltage and current across its terminals, but these quantities are now more generally functions of an dimensional spatial coordinate as well as time . In particular, and will be in general related by partial differential operators. Though the representation is the same as in the lumped case, this circuit element is itself a distributed object, occupying physical space.

It is also possible to extend the notions of passivity and losslessness to multiple dimensions, and to introduce wave variables, which, like the voltages and currents, will also be distributed quantities. Finally, it is also possible to discretize these elements in such a way that this passivity is retained in the discrete time and space domain (through the use of the trapezoid rule in multiple dimensions). the result is the multidimensional wave digital (MDWD) element shown at right in Figure 1.12. Just as for the lumped case, where differential operators are mapped to delays, here partial differential operators are mapped to shifts in the discrete multidimensional problem space. We again have an input wave and an output wave , which take on values at a discrete set of locations; these are to be interpreted as grid functions over a set of points, indexed by an integervalued vector .

Though we will discuss the MDWD discretization procedure in much more detail in Chapter 3, we outline the basic steps in Figure 1.13. Beginning from a given passive physical system, we first model it with a suitable system of PDEs. It may be possible, then, to interpret the individual equations as loop equations in a closed multidimensional Kirchoff circuit (MDKC) made up of elements of the form shown at left in Figure 1.12. Typically, the currents flowing through the ``wires'' in such a network will be the dependent, or state variables describing the physical system; all the partial differential operators are then consolidated in the various elements. For a firstorder system of PDEs, it will usually be true that the number of equations is equal to the number of loops in the circuit. It is important, at this stage, to ensure that such a network representation is composed of multidimensional circuit elements which are individually passivethis can generally be determined by a cursory examination of the circuit element values (such as inductances, capacitances, etc., which may be functions of several variables). Once the work of manipulating the system into a suitable circuit form is complete, the discretization step is immediate, and a multidimensional wave digital network results; if the MDKC is made up of passive elements, then the discrete network will be as well. It can then be interpreted as a stable explicit numerical integration scheme for the original defining system of PDEs. The basic operations will be, just as for DWNs, the scattering and shifting of wave variables through a numerical grid of nodes. The resulting structures, however, differ markedly from DWNs in many ways, though they can still be viewed as finite difference schemes.
We note that each of the various steps (i.e., the arrows in Figure 1.13) involves a good deal of choice (and experience) on the part of the algorithm designer. For a given system, there is almost always a variety of PDE systems which could serve as adequate models; not all are suitable for circuitbased discretization. It is also true that for a given system of PDEs, there is not a unique network representation (though they should all be related by equivalence transformations from classical network theory). Finally, though Fettweis et al. make use of the MD trapezoid rule as a means of arriving at a passive discrete network, this is by no means the only way of proceedingmany integration rules possess the desired passivitypreserving properties. We will explore the consequences of these choices extensively throughout the rest of this thesis.