Physical Derivation of Reflection Coefficient
Physically, the reflection coefficient seen at port is due to an
impedance step from
, that of the port interface, to a new
impedance consisting of the parallel combination of all other
port impedances meeting at the junction. Let
denote this parallel combination, in admittance form. Then we must have
Let's check this ``physical'' derivation against the formal definition
Eq.(F.20) leading to
in Eq.
(F.22).
Toward this goal, let
![$\displaystyle \Gamma _J \isdef \sum_{j=1}^N \Gamma _j
$](http://www.dsprelated.com/josimages_new/pasp/img4863.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
![$ \Gamma _J = \Gamma _i + \Gamma _J(i)$](http://www.dsprelated.com/josimages_new/pasp/img4864.png)
![$ i$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
![\begin{eqnarray*}
\rho_i &\isdef & \alpha_i - 1
\;\isdef \; \frac{2\Gamma _i}{\...
..._i + \Gamma _J(i)}
\;=\; \frac{R_J(i) - R_i}{\Gamma _J(i)-R_i}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4865.png)
and the result is verified.
Next Section:
Reflection Free Port
Previous Section:
Reflection Coefficient, Parallel Case