The entry point, for any study of numerical methods based on wave and scattering ideas, must necessarily be a review of *wave digital filters* (WDFs) [41,46]. This filter design technique, proposed by Alfred Fettweis in the early 1970s, was an attempt at translating analog filters into the digital realm with a pointed emphasis on preserving as much of the underlying physics as possible. In particular, a digital filter structure arrived at through Fettweis's procedure has the same precise network topology and energetic properties as the lumped analog electrical circuit (called the *reference circuit*) from which it is derived.

The theory is straightforward; analog circuit components (-port devices or elements), usually defined by a voltage-current relation, are first given an equivalent characterization in terms of *wave variables*. While this is merely a change of variables, it has the advantage of allowing an alternate description of the dynamic behavior of the network: energy incident on a circuit element (incident from the rest of the network to which it is connected through a *port*) may be reflected back from the element through the same port, or transmitted through to another part of the network through a different port. The incident, reflected and transmitted energies are carried by *waves*^{}. The reflectances and transmittances themselves are determined by arbitrary positive constants called *port resistances* which are assigned to individual wave ports. An important result of using wave variables is that the entire network may then be parametrized by these reflection and transmission coefficients which are, at least for passive networks, bounded independently of the numerical circuit element values themselves (inductances, capacitances and resistances etc.), which may vary over a wide range.

The true advantage of using wave variables becomes much more tangible when we seek to obtain, from a given analog filter design, a digital filter structure. This is usually done in the WDF context at steady state via a particular type of *bilinear transformation* or *spectral mapping* [41] from continuous to discrete frequency variables. Unless wave variables are employed, the resulting filter structure will usually not be recursively computable, and hence not directly implementable as a computer program. In addition, because the reflectances and transmittances of the network (which become the *filter multiplier values*) are bounded in a simple way, a host of desirable filter properties result which are especially valuable in a fixed-point computer implementation: complete elimination of certain types of limit cycles or parasitic oscillations and very low sensitivity of the filter response to coefficient truncation are the most frequently mentioned [46]. A further advantage stems from the fact that because the network topology of the reference circuit has been inherited by the digital filter structure, we have convenient access to a simple energy measure for the discrete dynamical system; this energy, which is a direct analogue of the energy stored in the electrical and magnetic fields surrounding the reference circuit, may be used as a discrete-time Lyapunov function [37,42] in order to provide further rules for dealing with the inevitable truncation of the filter state in a fixed-point implementation.

Many of the underlying ideas, however, had existed for some time before they coalesced into Fettweis's digital filter design technique. In fact, it is perhaps best to describe wave digital filtering not as an unprecedented invention, but as the successful synthesis of two principal preexistent ideas. The crucial wave variable and scattering concepts were borrowed from microwave filter design [11,12], and digital structures based on the reflection and transmission of waves had appeared previously, especially in ``layer-peeling'' and ``layer-adjoining'' methods for solving inverse problems that arise in geophysics [22,23,213], and in models of the human vocal tract used in the analysis and synthesis of speech [104,145], as we saw in §1.1.1. Many other digital filter structures make use of similar ideas, and have similar useful properties--among these are digital ladder and lattice forms [79], normalized filters [80] and orthogonal filters. This last type of structure has been formally unified with WDFs in [192]. The other cornerstone of wave digital filtering, the concept of a continuous-to-discrete spectral mapping which is, in some-sense, energy preserving, was not new to circuit discretization approaches. It appeared in the 1960s in the numerical analysis community which was concerned with the stability of the discretization of sets of ordinary differential equations (ODEs); indeed, wave digital filtering can be thought of as an *A-stable* [32,65,75] numerical method which discretizes the defining differential equations of an analog electrical network.

Wave digital filtering has, since its inception, developed in many directions, and has become a large subfield of the vast expanse of digital filter design. Because this thesis is devoted to the use of wave digital filters for simulation purposes, and not for filtering, this introductory chapter is intended merely to motivate material in the sequel, and to provide enough basic information for the reader to understand the WDF symbology (which is, unfortunately, somewhat idiosyncratic and takes a bit of getting used to). Indeed, many filtering issues do not arise at all in a simulation setting, at least from the point of view of traditional numerical analysis^{}. The single best WDF review paper is certainly [46], which is filled with practical filter design information and references. We briefly mention that some of the recent lines of development have been in the areas of multi-rate systems and filter banks [54,117,186], cochlear modelling [76], vocal tract modelling [177], the modelling of nonlinear circuit components [39] such as transistors [36], switching elements [151] as well as applications to nonlinear transmission lines [40,126]. The concept of a generalized adaptor (see §2.3.5) with memory, as another means of approaching nonlinear circuit elements has been explored in [154]. Another important direction has been the generalization of WDFs to the multidimensional case [62], and we will discuss this in detail in the next chapter.