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).

Figure 2.13: Signal truncation at a three-port adaptor.

$\tilde{a}_{j}$ are the input waves (assumed voltage waves) to the junction, for $j=1,\hdots,M$ (we have $M=3$ 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 $b_{j}$, 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 $b_{j}$ thus satisfy (2.42), where ${\bf a}$ is replaced by $\tilde{{\bf a}}$, with $\tilde{{\bf a}}=[\tilde{a}_{1},\hdots, \tilde{a}_{M}]^{T}$. Scattering is lossless. In general, however, the number of bits required to represent $b_{j}$ will now be greater than the number required for $\tilde{a}_{j}$; in order to reduce the size of the output word-length, we may apply magnitude truncation (represented graphically in Figure 2.13 by boxes labelled ``Q'', which are not wave digital one-ports. Magnitude truncation may be incorporated formally into the scattering picture through the use of circulators [125]). A reduced word-length wave $\tilde{b}_{j}$ is obtained from $b_{j}$ by truncating it in any way as long as magnitude is decreased. In other words, for any port $j$,

$$\tilde{b}_{j}^{2}\leq b_{j}^{2}$$

This implies, then, that

$$\Vert\tilde{{\bf b}}\Vert _{{\bf P},2}\leq\Vert{\bf b}\Vert _{{\bf P},2}=\Vert\tilde{{\bf a}}\Vert _{{\bf P},2}$$

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 $\alpha$$_{s}$ or $\alpha$$_{p}$ 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 $\alpha$$_{s}$ are the filter multiplier coefficients, and recall also that the sum of the elements in $\alpha$$_{s}$ is exactly 2, in infinite-precision arithmetic. Suppose that the elements of $\alpha$$_{s}$ are truncated to some finite word-length values, which can be written as the vector $\tilde{\mbox{\boldmath $\alpha$}}$$_{s}$. If they are truncated such that all elements of $\tilde{\mbox{\boldmath $\alpha$}}$$_{s}$ are positive, and their sum is still exactly 2, then it is easy to show that there must correspond a set of non-negative port resistances, and thus the quantized adaptor can still be considered as exactly lossless. More generally, it is possible to ensure passivity if the sum of the elements of $\tilde{\mbox{\boldmath $\alpha$}}$$_{s}$ 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.