Junction Passivity

In fixed-point implementations, the round-off error and other nonlinear operations should be confined when possible to physically meaningful wave variables. When this is done, it is easy to ensure that signal power is not increased by the nonlinear operations. In other words, nonlinear operations such as rounding can be made passive. Since signal power is proportional to the square of the wave variables, all we need to do is make sure amplitude is never increased by the nonlinearity. In the case of rounding, magnitude truncation, sometimes called ``rounding toward zero,'' is one way to achieve passive rounding. However, magnitude truncation can attenuate the signal excessively in low-precision implementations and in scattering-intensive applications such as the digital waveguide mesh [526]. Another option is error power feedback in which case the cumulative round-off error power averages to zero over time.

A valuable byproduct of passive arithmetic is the suppression of limit cycles and overflow oscillations [441]. Formally, the signal power of a conceptually infinite-precision implementation can be taken as a Lyapunov function bounding the squared amplitude of the finite-precision implementation.

The Kelly-Lochbaum and one-multiply scattering junctions are structurally lossless [513,468] (see also Appendix J) because they have only one parameter (or ), and all quantizations of the parameter within the allowed interval (or ) correspond to lossless scattering.^H.6

In the Kelly-Lochbaum and one-multiply scattering junctions, because they are structurally lossless, we need only double the number of bits at the output of each multiplier, and add one bit of extended dynamic range at the output of each two-input adder. The final outgoing waves are thereby exactly computed before they are finally rounded to the working precision and/or clipped to the maximum representable magnitude.

For the Kelly-Lochbaum scattering junction, given -bit signal samples and -bit reflection coefficients, the reflection and transmission multipliers produce and bits, respectively, and each of the two additions adds one more bit. Thus, the intermediate word length required is bits, and this must be rounded without amplification down to bits for the final outgoing samples. A similar analysis gives also that the one-multiply scattering junction needs bits for the extended precision intermediate results before final rounding and/or clipping.

To formally show that magnitude truncation is sufficient to suppress overflow oscillations and limit cycles in waveguide networks built using structurally lossless scattering junctions, we can look at the signal power entering and leaving the junction. A junction is passive if the power flowing away from it does not exceed the power flowing into it. The total power flowing away from the th junction is bounded by the incoming power if