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Net Signal Power at a Two-Port Scattering Junction

A junction is passive if the power flowing away from it does not exceed the power flowing into it

\begin{eqnarray*}
\underbrace{\frac{[f^{{+}}_i(t)]^2}{R_i(t)}
+ \frac{[f^{{-}}_{i-1}(t+T)]^2}{R_{i-1}(t)}}_{\hbox{outgoing power}}
\leq
\underbrace{\frac{[f^{{+}}_{i-1}(t-T)]^2}{R_{i-1}(t)}
+ \frac{[f^{{-}}_i(t)]^2}{R_i(t)}}_{\hbox{incoming power}}
\end{eqnarray*}

Let $ {\hat f}$ denote the finite-precision version of $ f$ . Then a sufficient condition for junction passivity is

\begin{eqnarray*}
\left\vert{\hat f}^{+}_i(t)\right\vert&\leq&\left\vert f^{{+}}_i(t)\right\vert
\\
\left\vert{\hat f}^{-}_{i-1}(t+T)\right\vert&\leq&\left\vert f^{{-}}_{i-1}(t+T)\right\vert
\end{eqnarray*}


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``Scattering at an Impedance Discontinuity'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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