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Beta Parameters

It is customary in the wave digital filter literature to define the beta parameters as

$\displaystyle \fbox{$\displaystyle \beta_i \isdef \frac{2R_i}{\sum_{j=1}^N R_j}$} \qquad\mbox{(Beta Parameters)} \protect$ (F.33)

where $ R_i$ are the port impedances (attached element reference impedances). In terms of the beta parameters, the force-wave series adaptor performs the following computations:
$\displaystyle v_J(n)$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N \beta_i v^{+}_i(n)$   (Common Junction Velocity)$\displaystyle \protect$ (F.34)
$\displaystyle v^{-}_i(n)$ $\displaystyle =$ $\displaystyle v_J(n) - v^{+}_i(n)$   (Outgoing Velocity Waves)$\displaystyle \protect$ (F.35)

However, we normally employ a mixture of parallel and series adaptors, while keeping a force-wave simulation. Since $ f^{{+}}_i(n) = R_i
v^{+}_i(n)$ , we obtain, after a small amount of algebra, the following recipe for the series force-wave adaptor:

$\displaystyle f^{{+}}_J(n)$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N f^{{+}}_i(n)$   (Total Incoming Force)$\displaystyle \protect$ (F.36)
$\displaystyle f^{{-}}_i(n)$ $\displaystyle =$ $\displaystyle f^{{+}}_i(n) - \beta_if^{{+}}_J(n)$   (Outgoing Force Waves)$\displaystyle \protect$ (F.37)

We see that we have $ N$ multiplies and $ 2N-1$ additions as in the parallel-adaptor case.

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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