Time-Domain 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 scattering junctions:

• Explicit coefficients of incoming traveling waves for computing junction velocity
• Losslessness assured when alpha parameters are nonnegative and sum to $2$
• When alpha parameters sum to less than $2$ , there is conceptually a ``resistive loss'' at the junction

In the lossless, equal-impedance case, $R_i=R,\forall i\;\Rightarrow$

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

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