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.

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