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 [298]. 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 G develops this topic in more detail.

The scattering matrix for normalized pressure waves is given by

$$\tilde{\mathbf{A}}= \left[ \begin{array}{llll} \frac{2 \Gamma_{1}... ..._{2}}}{\Gamma_J} & \dots & \frac{2 \Gamma_{n}}{\Gamma_J} -1 \end{array} \right]$$ (G.108)

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

$$\tilde{\mathbf{A}}= \frac{2}{ \vert\vert\,\tilde{{\bm \Gamma}}\,\vert\vert ^2}\tilde{{\bm \Gamma}}\tilde{{\bm \Gamma}}^T-\mathbf{I}$$ (G.109)

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 (G.110) 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 ${\bm \Gamma}^T= [\Gamma_1,\ldots,\Gamma_N]$.