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Normalized Scattering

For ideal numerical scaling in the $L_2$ sense, we may choose to propagate normalized waves which lead to normalized scattering junctions analogous to those encountered in normalized ladder filters [13]. Normalized waves may be either normalized pressure ${\tilde p}_j^+ =
p_j^+\sqrt{\Gamma_i}$ or normalized velocity ${\tilde v}_j^+ =
v_j^+/\sqrt{\Gamma_i}$. Since the signal power associated with a traveling wave is simply ${\cal P_j^+} = ({\tilde p}_j^+)^2 = ({\tilde v}_j^+)^2$, they may also be called root-power waves [27].

The scattering matrix for normalized pressure waves is given by

\begin{displaymath}
{\tilde {\bf A}} =
\left[
\begin{array}{llll}
\frac{2 \Ga...
... \dots
& \frac{2 \Gamma_{n}}{\Gamma_J} -1
\end{array} \right]
\end{displaymath} (25)

The normalized scattering matrix can be expressed as a Householder reflection

\begin{displaymath}
{\tilde {\bf A}} = {2\over \vert\vert{\bf {{\tilde \Gamma}}}...
...rt ^2}{\bf {{\tilde \Gamma}}}{\bf {{\tilde \Gamma}}}^T-{\bf I}
\end{displaymath} (26)

where ${\bf {{\tilde \Gamma}}}^T= [\sqrt{\Gamma_1},\ldots,\sqrt{\Gamma_N}]$, and $\Gamma_i$ is the wave admittance in the $i$th waveguide branch. The geometric interpretation of (26) is that the incoming pressure waves are reflected about the vector ${\bf {{\tilde \Gamma}}}$. Unnormalized scattering junctions can be expressed in the form of an ``oblique'' Householder reflection ${\bf A}= 2{\bf 1}{{\bf\Gamma}}^T/\left<{\bf 1},{{\bf\Gamma}}\right>-{\bf I}$, where ${\bf 1}^T=[1,\ldots,1]$ and ${{\bf\Gamma}}^T= [\Gamma_1,\ldots,\Gamma_N]$.


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``Circulant and Elliptic Feedback Delay Networks for Artificial Reverberation'', by Davide Rocchesso and Julius O. Smith III, preprint of version in IEEE Transactions on Speech and Audio, vol. 5, no. 1, pp. 51-60, Jan. 1996.

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Copyright © 2005-03-10 by Davide Rocchesso and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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