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Physical Audio Signal Processing
    Wave Digital Filters
       Adaptors for Wave Digital Elements
          General Series Adaptor for Force Waves
             Physical Derivation of Series Reflection Coefficient

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Physical Derivation of Series Reflection Coefficient

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

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

(Q.35)

denote this series combination. Then we must have, as in Eq. $\,$ (Q.25),

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

(Q.36)

Let's check this ``physical'' derivation against the formal definition Eq. $\,$ (Q.31) leading to $\rho^v_i = \beta_i - 1$ in Eq. $\,$ (Q.33). 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. $\,$ (Q.35),

for all

. From Eq. $\,$ (Q.26), the velocity reflection coefficient is given by

$\begin{eqnarray*} \rho^v_i &\isdef & \beta_i - 1 \;\isdef \; \frac{2R_i}{R_J} -... ..._J(i)}\\ &=& \frac{R_i - R_J(i)}{R_i + R_J(i)}\\ &=& -\rho_i \end{eqnarray*}$

Since

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

the result follows.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.