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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 [299]. 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 [436]. Appendix C develops this topic in more detail.

The scattering matrix for normalized pressure waves is given by

$\displaystyle \tilde{\mathbf{A}}= \left[ \begin{array}{llll} \frac{2 \Gamma_{1}}{\Gamma_J} - 1 & \frac{2 \sqrt{\Gamma_{1}\Gamma_{2}}}{\Gamma_J} & \dots & \frac{2 \sqrt{\Gamma_{1}\Gamma_{n}}}{\Gamma_J} \\ \\ \frac{2 \sqrt{\Gamma_{2}\Gamma_{1}}}{\Gamma_J} & \frac{2 \Gamma_{2}}{\Gamma_J}-1 & \dots & \frac{2 \sqrt{\Gamma_{2}\Gamma_{n}}}{\Gamma_J} \\ \dots & & \dots\\ \dots & & \dots\\ \frac{2 \sqrt{\Gamma_{n}\Gamma_{1}}}{\Gamma_J} & \frac{2 \sqrt{\Gamma_{n}\Gamma_{2}}}{\Gamma_J} & \dots & \frac{2 \Gamma_{n}}{\Gamma_J} -1 \end{array} \right]$ (C.119)

The normalized scattering matrix can be expressed as a negative Householder reflection

$\displaystyle \tilde{\mathbf{A}}= \frac{2}{ \vert\vert\,\tilde{{\bm \Gamma}}\,\vert\vert ^2}\tilde{{\bm \Gamma}}\tilde{{\bm \Gamma}}^T-\mathbf{I}$ (C.120)

where $ \tilde{{\bm \Gamma}}^T= [\sqrt{\Gamma_1},\ldots,\sqrt{\Gamma_N}]$ , and $ \Gamma_i$ is the wave admittance in the $ i$ th waveguide branch. To eliminate the sign inversion, the reflections at the far end of each waveguide can be chosen as -1 instead of 1. The geometric interpretation of (C.121) is that the incoming pressure waves are reflected about the vector $ \tilde{{\bm \Gamma}}$ . Unnormalized scattering junctions can be expressed in the form of an ``oblique'' Householder reflection $ \mathbf{A}= 2\mathbf{1}{\bm \Gamma}^T/\left<\mathbf{1},{{\bm \Gamma}}\right>-\mathbf{I}$ , where $ \mathbf{1}^T=[1,\ldots,1]$ and $ {\bm \Gamma}^T= [\Gamma_1,\ldots,\Gamma_N]$ .

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2022-07-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University