One of the greatest benefits of a scattering formulation is its guaranteed stability even under the highly nonlinear quantization operations that must be applied to both signals and multipliers in a computer implementation (see §2.3.6). To date, however, there have been no published comparisons of quantization effects in scattering structures vs. standard finite difference schemes.

This is, of course, a huge research problem, and certainly worth a dissertation or two by itself. The time is, however, ripe for such work since (a), there is large body of work devoted to quantization strategies in wave digital and other related filter designs, and (b), it is a necessary first step towards building special-purpose simulation hardware, which is the ultimate goal of all this work (such hardware has in fact already been built [202], but as mentioned above, there has been no attempt at any comparison with the performance of standard difference methods). The principal question is of how much there is to gain, in terms of memory savings, using a scattering implementation which employs small word lengths.

We would recommend a comparison of signal quantization effects in a simple (2+1)D structure such as the rectangular mesh for the (2+1)D wave equation, and its finite difference counterpart (to be discussed in Appendix A), subject to various boundary conditions. Because coefficient truncation effects will probably be most noticeable in a problem with material variation, it would be worthwhile to examine such effects in the (1+1)D transmission line problem under very simple conditions (losslessness, and spatial variation of a very simple form in one of the line parameters or ). By ``comparison,'' we mean that the error between the exact solution to the problem and a numerical solution should be computed for various signal and coefficient word lengths.