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

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

The normalized scattering matrix can be expressed as a Householder reflection

$${\tilde {\bf A}} = {2\over \vert\vert{\bf {{\tilde \Gamma}}}... ...rt ^2}{\bf {{\tilde \Gamma}}}{\bf {{\tilde \Gamma}}}^T-{\bf I}$$ (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 ${{\bf\Gamma}}^T= [\Gamma_1,\ldots,\Gamma_N]$.