Alpha Parameters

In the lossless (unloaded) case, , in the time domain:

$\displaystyle \alpha_i = \frac{2R_i}{R_1 + \cdots + R_N}$

These alpha parameters are analogous to those in the ``adaptors'' of wave digital filters (Fettweis)

In the lossles case

$\displaystyle 0\leq\alpha_i \leq 2$

and

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

Lossless series scattering in terms of alpha parameters:

$\begin{eqnarray*} v_J(t) &=& \sum_{i=1}^N \alpha_i v^+_i(t) \\ v^-_i(t) &=& v_J(t) - v^+_i(t) \end{eqnarray*}$

Alpha parameters conveniently parametrize lossless junctions:

Explicit coefficients of incoming traveling waves for computing junction velocity
Losslessness assured when alpha parameters are nonnegative and sum to
When alpha parameters sum to less than , there is conceptually a ``resistive loss'' at the junction

In the lossless, equal-impedance case, $R_i=R,\forall i$ , we have

$\displaystyle \alpha_i = \frac{2}{N}$

When

is a power of two, no multiplies are needed
(multiply-free reverberators, waveguide meshes, etc., are based on this)

2D mesh

3D mesh