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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:

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

$\displaystyle \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)


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Download Scattering.pdf
Download Scattering_2up.pdf
Download Scattering_4up.pdf

``Scattering at an Impedance Discontinuity'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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