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Alpha Parameters

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

$\displaystyle \fbox{$\displaystyle \alpha_i \isdef \frac{2\Gamma _i}{\sum_{j=1}^N \Gamma _j}$} \qquad\mbox{(Alpha Parameters)} \protect$ (F.14)

where $ \Gamma _i \isdef 1/R_i$ are the admittances of the wave digital element interfaces (or ``reference admittances,'' in WDF terminology). In terms of the alpha parameters, the force-wave parallel adaptor performs the following computations:
$\displaystyle f_J(n)$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N \alpha_i f^{{+}}_i(n)$   (Common Junction Force)$\displaystyle \protect$ (F.15)
$\displaystyle f^{{-}}_i(n)$ $\displaystyle =$ $\displaystyle f_J(n) - f^{{+}}_i(n)$   (Outgoing Force Waves)$\displaystyle \protect$ (F.16)

We see that $ N$ multiplies and $ 2N-1$ additions are required.


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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
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