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

$$\tilde{\mathbf{A}}= \left[ \begin{array}{llll} \frac{2 \Gamma_{1}}{\Gamma_J} - 1 & \frac{2 \sqrt{\Gamma_{1}\Gamma_{2}}}{\Gamma_J} & \dots & \frac{2 \sqrt{\Gamma_{1}\Gamma_{n}}}{\Gamma_J} \\ \\ \frac{2 \sqrt{\Gamma_{2}\Gamma_{1}}}{\Gamma_J} & \frac{2 \Gamma_{2}}{\Gamma_J}-1 & \dots & \frac{2 \sqrt{\Gamma_{2}\Gamma_{n}}}{\Gamma_J} \\ \dots & & \dots\\ \dots & & \dots\\ \frac{2 \sqrt{\Gamma_{n}\Gamma_{1}}}{\Gamma_J} & \frac{2 \sqrt{\Gamma_{n}\Gamma_{2}}}{\Gamma_J} & \dots & \frac{2 \Gamma_{n}}{\Gamma_J} -1 \end{array} \right]$$ (C.143)

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

$$\tilde{\mathbf{A}}= \frac{2}{ \vert\vert\,\tilde{{\bm \Gamma}}\,\vert\vert ^2}\tilde{{\bm \Gamma}}\tilde{{\bm \Gamma}}^T-\mathbf{I}$$ (C.144)

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.145) 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 ${\bm \Gamma}^T= [\Gamma_1,\ldots,\Gamma_N]$ .