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

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Reflection Coefficient, Series Case

The velocity!reflection coefficient seen at port is defined as

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

(Q.31)

Representing the outgoing velocity wave $v^{-}_i(n)$ as the superposition of the reflected wave $\rho^v_iv^{+}_i(n)$ plus the

transmitted waves from the other ports, we have

$\displaystyle v^{-}_i(n) = \rho^v_i v^{+}_i + \sum_{j\neq i} \tau^v_{ji} v^{+}_j \protect$

(Q.32)

where $\tau^v_{ji}$ denotes the velocity!transmission coefficient from port

to port

. Substituting Eq. $\,$ (Q.29) into Eq. $\,$ (Q.30) yields

$\begin{eqnarray*} v^{-}_i(n) &=& v_J(n) - v^{+}_i(n)\\ &=& \left(\sum_{j=1}^N ... ... &=& (\beta_i - 1)v^{+}_i(n) + \sum_{j\neq i} \beta_j v^{+}_j(n) \end{eqnarray*}$

Equating like terms with Eq. $\,$ (Q.32) gives

$\displaystyle \rho^v_i$	$\displaystyle =$	$\displaystyle \beta_i - 1 \protect$	(Q.33)
$\displaystyle \tau^v_{ji}$	$\displaystyle =$	$\displaystyle \beta_j, \quad (i\neq j)$	(Q.34)

Thus, the

th beta parameter is the velocity transmission coefficient from

th port to any other port (besides the

th). To convert the transmission coefficient from the

th port to the reflection coefficient for that port, we simply subtract 1. These relationships are specific to velocity waves at a series junction (cf. Eq. $\,$ (Q.22)). They are exactly the dual of Equations (Q.22-Q.23) for force waves at a parallel junction.

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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