Signal and Coefficient Quantization

In a machine implementation of a wave digital filter, the signals and coefficients must necessarily be represented with a finite number of bits. As such, it is not immediately obvious that the passivity properties for a given WDF, which are framed in terms of real-valued signals (waves) and filter multipliers (related to the port resistances) will hold in a finite word-length computer implementation. All digital filter implementations are vulnerable to a host of undesirable effects which result from signal and coefficient quantization; among them are parasitic oscillations and high sensitivity of filter pole and zero locations (and thus the frequency response). WDFs, however, offer a number of means of combating these problems. The exploration of these means has produced a large body of literature [43,46,58,125,179,204]. We give only a brief outline here, for completeness sake.

From the discussion of wave digital elements, it is easy to see that in most cases, the only arithmetic operations in a WDF will occur as signals are scattered from adaptors^{}; the wave digital inductor, capacitor and unit element involve only shifts and possibly sign inversion, and the wave digital resistor, which behaves as a sink, can essentially be ignored by the programmer once its port resistance has been absorbed into the adaptor to which it is connected. Simple quantization procedures [56,201] were first proposed, and later the concept of incremental pseudopassivity [125] was developed for ensuring that a finite word-length implementation of a wave digital adaptor behaves passively under signal truncation. The most straightforward scheme appears in Figure 2.13, for the case of a three-port adaptor (either series or parallel).

are the input waves (assumed voltage waves) to the junction, for (we have in Figure 2.13), and are assumed to be of some finite word-length. Extended precision is used within the adaptor in order exactly calculate the output waves , from (2.37). We have assumed that the multiplier coefficients within the junction are of finite word-length as well--we will discuss this presently. The output waves thus satisfy (2.42), where is replaced by , with . Scattering is lossless. In general, however, the number of bits required to represent will now be greater than the number required for ; in order to reduce the size of the output word-length, we may apply

This implies, then, that

so that passivity is maintained even considering the finite word-length wave variables. In this way (by ensuring a decrease in the overall energy measure of the WD network), both large- and small-scale parasitic oscillations can be completely eliminated, at least the zero-input case [46]. Various types of overflow characteristics have been examined in [56,125]. Such a quantization rule has also appeared in other contexts [193], and applies equally well to digital waveguide networks [166], which are the subject of Chapter 4.

The quantization of coefficients in WDFs [42,43,46,111] as well as other similar filter structures [193] has been shown to have a minimal effect on the filter response. That is, in many lossless configurations [46], variations in the values of the multiplier coefficients (which are usually the reflection and transmission parameters or at an adaptor) can be shown to have a second-order effect on the filter response. In contrast, when such variations occur in direct-form filter structures, large changes in pole locations can result, and a stable filter may even become unstable [133]. This robustness property of scattering-based filter structures is sometimes called *structural passivity* [147,169,193]. As a simple example, consider the scattering equations (2.38) for a series adaptor; as mentioned above, the parameters in the vector are the filter multiplier coefficients, and recall also that the sum of the elements in is exactly 2, in infinite-precision arithmetic. Suppose that the elements of are truncated to some finite word-length values, which can be written as the vector
. If they are truncated such that all elements of
are positive, and their sum is still exactly 2, then it is easy to show that there must correspond a set of non-negative port resistances, and thus the quantized adaptor can still be considered as exactly lossless. More generally, it is possible to ensure passivity if the sum of the elements of
is less than or equal to 2; this has been discussed in the waveguide filter context in [169].

While most of the approaches to quantization have been concerned with fixed-point implementations, many of the same ideas can be applied in floating-point as well. Floating-point signal truncation rules were proposed in [34], and an early study of coefficient sensitivity and roundoff noise appeared in [111]. More recent developments include a generalized WDF which is simply realized using multiply/accumulate operations [53], and a description of passive coefficient-truncation rules [121] based on scattering matrix factorization.