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Isolating Time-Varying Junctions

Since waveguide sections are typically terminated on both ends by scattering junctions, a change in one waveguide impedance modulates both scattering junctions rather than only one as may be desired. It is possible to vary the coefficients of only a single junction when the network is an acyclic graph (which can be thought of as a generalization of the cascade chain as used in ladder/lattice filters). Consider the simple case of the cascade chain: Suppose there are $N$ waveguides abutted end to end and numbered $1$ through $N$ from left to right. Suppose further that we want to modulate only the scattering junction between sections $m$ and $m+1$. We can accomplish this by modulating the impedance $R_{m+1}$ of section $m+1$. Let the modulation signal be denoted by $g(n) =
R_{m+1}(n)/R_{m+1}(0)$, assuming the modulation begins after time $0$. Then we can cancel the modulation at the right endpoint of section $m+1$ by modulating the impedance of section $m+2$ by $g(n)$ (since the scattering coefficients at the junction of two waveguides depends only on the impedance ratio). However, now we have to also modulate the impedance of section $m+3$ by $g(n)$ in order to prevent modulation at the junction of sections $m+2$ and $m+3$. Continuing in this way, we must modulate the impedances of all waveguides to the right of section $m$ by $g(n)$ in order to obtain an isolated modulated junction between sections $m$ and $m+1$. This argument extends readily to an acyclic graph, and breaks down whenever a ``downstream'' branch is connected to an ``upstream'' branch, i.e., whenever there is a cycle in the waveguide network graph.

The simultaneous variation of many wave impedances determines an instantaneous variation of the stored signal power. When the waveguides are normalized, the signal power remains fixed. As a result, it is possible to vary isolated junctions in the normalized case without worrying about energy modulation consequences in other parts of the network.

Another way to isolate impedance variations in a time-varying network is by means of ideal transformers which can step the wave impedance up or down by an arbitrary factor without inducing reflections. The basic theory of the digital waveguide transformer is discussed in Appendix A, and Section 9 discusses further applications.

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``Aspects of Digital Waveguide Networks for Acoustic Modeling Applications'', by Julius O. Smith III and Davide Rocchesso , December 19, 1997, Web published at
Copyright © 2007-02-07 by Julius O. Smith III and Davide Rocchesso
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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