The velocity reflection coefficient seen at port
is defined as
![$\displaystyle \rho^v_i \isdef \left. \frac{v^{-}_i(n)}{v^{+}_i(n)} \right\vert _{v^{+}_j(n)=0, \forall j\neq i} \protect$](http://www.dsprelated.com/josimages_new/pasp/img4895.png) |
(F.31) |
Representing the outgoing velocity wave
![$ v^{-}_i(n)$](http://www.dsprelated.com/josimages_new/pasp/img4896.png)
as the
superposition of the reflected wave
![$ \rho^v_iv^{+}_i(n)$](http://www.dsprelated.com/josimages_new/pasp/img4897.png)
plus the
![$ N-1$](http://www.dsprelated.com/josimages_new/pasp/img1067.png)
transmitted waves from the other ports, we have
![$\displaystyle v^{-}_i(n) = \rho^v_i v^{+}_i + \sum_{j\neq i} \tau^v_{ji} v^{+}_j \protect$](http://www.dsprelated.com/josimages_new/pasp/img4898.png) |
(F.32) |
where
![$ \tau^v_{ji}$](http://www.dsprelated.com/josimages_new/pasp/img4899.png)
denotes the
velocity transmission
coefficientvelocity!transmission
coefficient from
port
![$ j$](http://www.dsprelated.com/josimages_new/pasp/img664.png)
to port
![$ i$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
. Substituting Eq.
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
(
F.29) into
Eq.
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
(
F.30) yields
Equating like terms with Eq.
(F.32) gives
Thus, the
![$ j$](http://www.dsprelated.com/josimages_new/pasp/img664.png)
th beta parameter is the velocity transmission
coefficient from
![$ 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$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
th port to the
reflection coefficient for that port, we simply subtract 1. These
relationships are specific to velocity waves at a series junction
(cf. Eq.
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
(
F.22)). They are exactly the dual of Equations (
F.22-
F.23)
for
force waves at a parallel junction.
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