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Physical Derivation of Reflection Coefficient

Physically, the reflection coefficient seen at port $ i$ is due to an impedance step from $ R_i$ , that of the port interface, to a new impedance consisting of the parallel combination of all other port impedances meeting at the junction. Let

$\displaystyle \Gamma _J(i) \isdef \sum_{i\neq j} \Gamma _i \protect$ (F.24)

denote this parallel combination, in admittance form. Then we must have

$\displaystyle \rho_i = \frac{R_J(i)-R_i}{R_J(i)+R_i} = \frac{\Gamma _i-\Gamma _J(i)}{\Gamma _i+\Gamma _J(i)} \protect$ (F.25)

Let's check this ``physical'' derivation against the formal definition Eq.$ \,$ (F.20) leading to $ \rho_i = \alpha_i - 1$ in Eq.$ \,$ (F.22). Toward this goal, let

$\displaystyle \Gamma _J \isdef \sum_{j=1}^N \Gamma _j
$

denote the parallel combination of all admittances connected to the junction. Then by Eq.$ \,$ (F.24), we have $ \Gamma _J = \Gamma _i + \Gamma _J(i)$ for all $ i$ . Now, from Eq.$ \,$ (F.17),

\begin{eqnarray*}
\rho_i &\isdef & \alpha_i - 1
\;\isdef \; \frac{2\Gamma _i}{\Gamma _J} - 1\\
&\isdef & \frac{2\Gamma _i - \Gamma _J}{\Gamma _J}
\;=\; \frac{2\Gamma _i - \left[\Gamma _i + \Gamma _J(i)\right]}{\Gamma _i + \Gamma _J(i)}\\
&=& \frac{\Gamma _i - \Gamma _J(i)}{\Gamma _i + \Gamma _J(i)}
\;=\; \frac{R_J(i) - R_i}{\Gamma _J(i)-R_i}
\end{eqnarray*}

and the result is verified.


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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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