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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}$}$   (Alpha Parameters)$\displaystyle \protect$ (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. However, by observing from Eq.$ \,$ (F.14) that

$\displaystyle \sum_{i=1}^N \alpha_i = 2,

we may implement $ \alpha_1$ , say, as $ 2-\sum_{i=2}^N\alpha_i$ in order to eliminate one multiply. In WDF terminology, port 1 is then a dependent port.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University