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Chapters

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

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Reflection Coefficient, Parallel Case

The reflection coefficient seen at port $ i$ is defined as

$\displaystyle \rho_i \isdef \left. \frac{f^{{-}}_i(n)}{f^{{+}}_i(n)} \right\vert _{f^{{+}}_j(n)=0, \forall j\neq i} \protect$ (Q.20)

In other words, the reflection coefficient specifies what portion of the incoming wave $ f^{{+}}_i(n)$ is reflected back to port $ i$ as part of the outgoing wave $ f^{{-}}_i(n)$. The total outgoing wave on port $ i$ is the superposition of the reflected wave and the $ N-1$ transmitted waves from the other ports:

$\displaystyle f^{{-}}_i(n) = \rho_i f^{{+}}_i + \sum_{j\neq i} \tau_{ji} f^{{+}}_j \protect$ (Q.21)

where $ \tau_{ji}$ denotes the transmission coefficient from port $ j$ to port $ i$. Starting with Eq.$ \,$(Q.19) and substituting Eq.$ \,$(Q.18) gives

\begin{eqnarray*} f^{{-}}_i(n) &=& f_J(n) - f^{{+}}_i(n)\\ &=& \left(\sum_{j=1... ...\alpha_i - 1)f^{{+}}_i(n) + \sum_{j\neq i} \alpha_j f^{{+}}_j(n) \end{eqnarray*}

Equating like terms with Eq.$ \,$(Q.21), we obtain

$\displaystyle \rho_i$ $\displaystyle =$