Time-Varying Normalized Waveguide Networks

A time-varying waveguide network can be created by changing one or more impedances over time. This induces changes in the scattering junctions connecting the various impedances. Defining time variation this way also preserves the physical interpretation of the time-varying network which is critical for keeping a handle on passivity and hence on stability and numerical robustness.

Since the signal power associated with, say, a single traveling pressure sample $p(n)$ stored within a delay element in a scalar digital waveguide is ${\cal P}_{p(n)} = p^2(n)\Gamma$, where $\Gamma$ is the (scalar) wave admittance of the associated waveguide, modulating a waveguide impedance also modulates the stored signal power in that waveguide. That is, the stored power associated with the sample $p(n)$ varies proportional to $\Gamma(n)$. This is not the case with normalized waves. The normalized counterpart of pressure sample $p(n)$ is ${\tilde p}(n) = p(n)\sqrt{\Gamma(n)}$, and the associated power is always just the square of the sample value: ${\cal P}_{{\tilde p}(n)} = {\tilde p}^2(n) = p^2(n)\Gamma(n) = {\cal P}_{p(n)}$. The scattering junctions connecting normalized waveguides are modulated by the time-varying impedances, but the stored signal power in the normalized waveguides is not. For the case of lossless scattering junctions, signal energy is constant throughout the time-varying network. This gives a general class of energy conserving time-varying digital filters which, along with passive rounding rules, are also free of limit cycles and overflow oscillations [92]. Thus, normalized waveguide networks decouple signal power in the network from time variations in the scattering coefficients: A change in the wave impedance changes the scattering properties of the junctions but does not alter the instantaneous signal power in the network.