Finite-Wordlength Effects

We have just seen how the FDN can be seen as a simple DWN having a not-necessarily physical scattering matrix. In order to provide an easy control over the decay, the scattering matrix has to satisfy the condition of losslessness (27). A finite-precision implementation of the FDN might incur in limit cycles or overflow oscillations, due to departures from the infinite-precision lossless prototype. Departures can be of two kinds: the finite-precision scattering matrix does not satisfy the lossless condition (27), or the round-off noise in the matrix by vector multiplication ${\bf A}{\bf p}^+$ introduces signal amplitude modifications. By assuming that the scattering matrix satisfies (27) in a large extent even in finite precision, it is possible to apply the arguments used in [27,28,26,6] for the DWNs, in order to avoid limit cycles or overflow oscillations. If the matrix by vector multiplication is performed in the straightforward way as a collection of inner products, and the matrix coefficients have the same $n$ bits of precision as the signals, it is just sufficient to perform these order-$N$ inner products in the extended precision of $2 n + N - 1$ bits, and apply a passive truncation scheme on the output signal. In two's complement arithmetic, a simple passive truncation scheme is the following:

• If the N-1 most significant bits are not equal, replace the output value by the maximum-magnitude number in $n$-bit two's complement having the correct sign (saturation).
• Discard the $n$ least significant bits and add $2^{-n+1}$ to the result if it is negative.

As far as the condition on the losslessness of the scattering matrix is concerned, general requirements for the construction of ``structurally lossless,'' or at least ``structurally passive'' scattering matrices have to be worked out. This topic, previously touched by Gray [6] in the $N=2$ case, will be discussed in a forthcoming paper, since a complete treatment would enlarge the scope of this paper significantly.