Alpha Parameters

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

$$\fbox{$\displaystyle \alpha_i \isdef \frac{2\Gamma _i}{\sum_{j=1}^N \Gamma _j}$} \protect$$ (P.17)

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:

$$f_J(n) = \sum_{i=1}^N \alpha_i f^{{+}}_i(n)\protect$$ (P.18)

$$f^{{-}}_i(n) = f_J(n) - f^{{+}}_i(n)\protect$$ (P.19)

We see that $N$ multiplies and $2N-1$ additions are required. However, by observing from Eq.$\,$(P.17) that

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

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.