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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Wave Digital Filters
       Adaptors for Wave Digital Elements
          General Series Adaptor for Force Waves
             Beta Parameters

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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}$} \protect$ (F.26)

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:
$\displaystyle v_J(n)$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N \beta_i v^{+}_i(n)\protect$ (F.27)
$\displaystyle v^{-}_i(n)$ $\displaystyle =$ $\displaystyle v_J(n) - v^{+}_i(n)\protect$ (F.28)

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:

$\displaystyle f^{{+}}_J(n)$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N f^{{+}}_i(n)\protect$ (F.29)
$\displaystyle f^{{-}}_i(n)$ $\displaystyle =$ $\displaystyle f^{{+}}_i(n) - \beta_if^{{+}}_J(n)\protect$ (F.30)

We see that we have $ N$ multiplies and $ 2N-1$ additions as in the parallel-adaptor case. However, we again have from Eq.$ \,$(F.26) that

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

so that we may implement one beta parameter as 2 minus the sum of the rest, thus eliminating a multiplication by creating a dependent port.

Previous: General Series Adaptor for Force Waves
Next: Reflection Coefficient, Series Case

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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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