Beta Parameters

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

$$\fbox{$\displaystyle \beta_i \isdef \frac{2R_i}{\sum_{j=1}^N R_j}$} \protect$$ (P.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:

$$v_J(n) = \sum_{i=1}^N \beta_i v^{+}_i(n)\protect$$ (P.27)

$$v^{-}_i(n) = v_J(n) - v^{+}_i(n)\protect$$ (P.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:

$$f^{{+}}_J(n) = \sum_{i=1}^N f^{{+}}_i(n)\protect$$ (P.29)

$$f^{{-}}_i(n) = f^{{+}}_i(n) - \beta_if^{{+}}_J(n)\protect$$ (P.30)

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

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

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.