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Reflection Coefficient, Parallel Case

The reflection coefficient seen at port $ i$ is defined as

$\displaystyle \rho_i \isdef \left. \frac{f^{{-}}_i(n)}{f^{{+}}_i(n)} \right\vert _{f^{{+}}_j(n)=0, \forall j\neq i} \protect$ (F.20)

In other words, the reflection coefficient specifies what portion of the incoming wave $ f^{{+}}_i(n)$ is reflected back to port $ i$ as part of the outgoing wave $ f^{{-}}_i(n)$ . The total outgoing wave on port $ i$ is the superposition of the reflected wave and the $ N-1$ transmitted waves from the other ports:

$\displaystyle f^{{-}}_i(n) = \rho_i f^{{+}}_i + \sum_{j\neq i} \tau_{ji} f^{{+}}_j \protect$ (F.21)

where $ \tau_{ji}$ denotes the transmission coefficient from port $ j$ to port $ i$ . Starting with Eq.$ \,$ (F.19) and substituting Eq.$ \,$ (F.18) gives

\begin{eqnarray*}
f^{{-}}_i(n) &=& f_J(n) - f^{{+}}_i(n)\\
&=& \left(\sum_{j=1}^N \alpha_j f^{{+}}_j(n)\right) - f^{{+}}_i(n)\\
&=& (\alpha_i - 1)f^{{+}}_i(n) + \sum_{j\neq i} \alpha_j f^{{+}}_j(n)
\end{eqnarray*}

Equating like terms with Eq.$ \,$ (F.21), we obtain

$\displaystyle \rho_i$ $\displaystyle =$ $\displaystyle \alpha_i - 1
\protect$ (F.22)
$\displaystyle \tau_{ji}$ $\displaystyle =$ $\displaystyle \alpha_j, \quad (i\neq j)
\protect$ (F.23)

Thus, the $ j$ th alpha parameter is the force transmission coefficient from $ j$ th port to any other port (besides the $ i$ th). To convert the transmission coefficient from the $ i$ th port to the reflection coefficient for that port, we simply subtract 1. This general relationship is specific to force waves at a parallel junction, as we will soon see.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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