The reflection coefficient seen at port
is defined as
![$\displaystyle \rho_i \isdef \left. \frac{f^{{-}}_i(n)}{f^{{+}}_i(n)} \right\vert _{f^{{+}}_j(n)=0, \forall j\neq i} \protect$](http://www.dsprelated.com/josimages_new/pasp/img4852.png) |
(F.20) |
In other words, the
reflection coefficient specifies what portion of
the incoming wave
![$ f^{{+}}_i(n)$](http://www.dsprelated.com/josimages_new/pasp/img4830.png)
is
reflected back to port
![$ i$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
as
part of the outgoing wave
![$ f^{{-}}_i(n)$](http://www.dsprelated.com/josimages_new/pasp/img4831.png)
. The total outgoing wave on port
![$ i$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
is the
superposition of the reflected wave and the
![$ N-1$](http://www.dsprelated.com/josimages_new/pasp/img1067.png)
transmitted waves from the other ports:
![$\displaystyle f^{{-}}_i(n) = \rho_i f^{{+}}_i + \sum_{j\neq i} \tau_{ji} f^{{+}}_j \protect$](http://www.dsprelated.com/josimages_new/pasp/img4853.png) |
(F.21) |
where
![$ \tau_{ji}$](http://www.dsprelated.com/josimages_new/pasp/img4854.png)
denotes the
transmission coefficient from
port
![$ j$](http://www.dsprelated.com/josimages_new/pasp/img664.png)
to port
![$ i$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
. Starting with Eq.
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
(
F.19) and substituting
Eq.
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
(
F.18) gives
Equating like terms with Eq.
(F.21), we obtain
Thus, the
![$ j$](http://www.dsprelated.com/josimages_new/pasp/img664.png)
th alpha parameter is the
force transmission coefficient
from
![$ j$](http://www.dsprelated.com/josimages_new/pasp/img664.png)
th port to any other port (besides the
![$ i$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
th). To convert the
transmission coefficient from the
![$ i$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
th port to the reflection
coefficient for that port, we simply subtract 1. This general
relationship is specific to
force waves at a parallel junction, as we
will soon see.
Next Section: Physical Derivation of Reflection CoefficientPrevious Section: Alpha Parameters