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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 $ i$ is defined as

$\displaystyle \rho^v_i \isdef \left. \frac{v^{-}_i(n)}{v^{+}_i(n)} \right\vert _{v^{+}_j(n)=0, \forall j\neq i} \protect$ (F.31)

Representing the outgoing velocity wave $ v^{-}_i(n)$ as the superposition of the reflected wave $ \rho^v_iv^{+}_i(n)$ plus the $ N-1$ transmitted waves from the other ports, we have

$\displaystyle v^{-}_i(n) = \rho^v_i v^{+}_i + \sum_{j\neq i} \tau^v_{ji} v^{+}_j \protect$ (F.32)

where $ \tau^v_{ji}$ denotes the velocity transmission coefficientvelocity!transmission coefficient from port $ j$ to port $ i$. Substituting Eq.$ \,$(F.29) into Eq.$ \,$(F.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.$ \,$(F.32) gives

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

Thus, the $ j$th beta parameter is the velocity transmission coefficient from $ j$th port to any other port (besides the $ i$th). To convert the transmission coefficient from the $ i$th port to the reflection coefficient for that port, we simply subtract 1. These relationships are specific to velocity waves at a series junction (cf. Eq.$ \,$(F.22)). They are exactly the dual of Equations (F.22-F.23) for force waves at a parallel junction.

Previous: Beta Parameters
Next: Physical Derivation of Series Reflection Coefficient

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About the Author: 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.


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