Normalized Scattering
For ideal numerical scaling in the sense, we may choose to propagate
normalized waves which lead to normalized scattering junctions
analogous to those encountered in normalized ladder filters [297].
Normalized waves may be either normalized pressure
or normalized velocity
. Since the signal power associated with a traveling
wave is simply
,
they may also be called root-power waves [432].
Appendix C develops this topic in more detail.
The scattering matrix for normalized pressure waves is given by
The normalized scattering matrix can be expressed as a negative Householder reflection
where
![$ \tilde{{\bm \Gamma}}^T= [\sqrt{\Gamma_1},\ldots,\sqrt{\Gamma_N}]$](http://www.dsprelated.com/josimages_new/pasp/img4068.png)
![$ \Gamma_i$](http://www.dsprelated.com/josimages_new/pasp/img4069.png)
![$ i$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
![$ \tilde{{\bm \Gamma}}$](http://www.dsprelated.com/josimages_new/pasp/img4070.png)
![$ \mathbf{A}= 2\mathbf{1}{\bm \Gamma}^T/\left<\mathbf{1},{{\bm \Gamma}}\right>-\mathbf{I}$](http://www.dsprelated.com/josimages_new/pasp/img4071.png)
![$ \mathbf{1}^T=[1,\ldots,1]$](http://www.dsprelated.com/josimages_new/pasp/img4072.png)
![$ {\bm \Gamma}^T= [\Gamma_1,\ldots,\Gamma_N]$](http://www.dsprelated.com/josimages_new/pasp/img4073.png)
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General Conditions for Losslessness
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Lossless Scattering