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Physical Derivation of Series Reflection Coefficient

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

$\displaystyle R_J(i) \isdef \sum_{i\neq j} R_i \protect$ (F.42)

denote this series combination. Then we must have, as in Eq.(F.22),

$\displaystyle \rho_i = \frac{R_J(i)-R_i}{R_J(i)+R_i}$ (F.43)

Let's check this ``physical'' derivation against the formal definition Eq.(F.38) leading to $ \rho^v_i = \beta_i - 1$ in Eq.(F.40). Define the total junction impedance as

$\displaystyle R_J \isdef \sum_{j=1}^N R_j

This is the series combination of all impedances connected to the junction. Then by Eq.(F.42), $ R_J = R_i + R_J(i)$ for all $ i$ . From Eq.(F.33), the velocity reflection coefficient is given by

\rho^v_i &\isdef & \beta_i - 1
\;\isdef \; \frac{2R_i}{R_J} - 1\\
&\isdef & \frac{2R_i - R_J}{R_J}
\;=\; \frac{2R_i - \left[R_i + R_J(i)\right]}{R_i + R_J(i)}\\
&=& \frac{R_i - R_J(i)}{R_i + R_J(i)}\\
&=& -\rho_i


$\displaystyle \rho^v_i\isdef \frac{v^{-}_i(n)}{v^{+}_i(n)} = \frac{-f^{{-}}_i(n)/R_i}{f^{{+}}_i(n)/R_i}
= - \frac{f^{{-}}_i(n)}{f^{{+}}_i(n)} \isdef -\rho_i

the result follows.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University