General Parallel Adaptor for Force Waves
In the more general case of wave digital element ports being
connected in parallel, we have the physical constraints
![]() |
(F.14) | ||
![]() |
(F.15) |
The derivation for the two-port case extends to the
![$ N$](http://www.dsprelated.com/josimages_new/pasp/img20.png)
The outgoing wave variables are given by
![$\displaystyle f^{{-}}_i(n) = f_J(n) - f^{{+}}_i(n)
$](http://www.dsprelated.com/josimages_new/pasp/img4842.png)
Alpha Parameters
It is customary in the wave digital filter literature to define the alpha parameters as
where
![$ \Gamma _i \isdef 1/R_i$](http://www.dsprelated.com/josimages_new/pasp/img4844.png)
We see that
![$ N$](http://www.dsprelated.com/josimages_new/pasp/img20.png)
![$ 2N-1$](http://www.dsprelated.com/josimages_new/pasp/img759.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
![$\displaystyle \sum_{i=1}^N \alpha_i = 2,
$](http://www.dsprelated.com/josimages_new/pasp/img4849.png)
![$ \alpha_1$](http://www.dsprelated.com/josimages_new/pasp/img4850.png)
![$ 2-\sum_{i=2}^N\alpha_i$](http://www.dsprelated.com/josimages_new/pasp/img4851.png)
Reflection Coefficient, Parallel Case
The reflection coefficient seen at port is defined as
In other words, the reflection coefficient specifies what portion of the incoming wave
![$ f^{{+}}_i(n)$](http://www.dsprelated.com/josimages_new/pasp/img4830.png)
![$ i$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
![$ f^{{-}}_i(n)$](http://www.dsprelated.com/josimages_new/pasp/img4831.png)
![$ i$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
![$ N-1$](http://www.dsprelated.com/josimages_new/pasp/img1067.png)
where
![$ \tau_{ji}$](http://www.dsprelated.com/josimages_new/pasp/img4854.png)
![$ j$](http://www.dsprelated.com/josimages_new/pasp/img664.png)
![$ i$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
![\begin{eqnarray*}
f^{{-}}_i(n) &=& f_J(n) - f^{{+}}_i(n)\\
&=& \left(\sum_{j=1...
...\alpha_i - 1)f^{{+}}_i(n) + \sum_{j\neq i} \alpha_j f^{{+}}_j(n)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4855.png)
Equating like terms with Eq.(F.21), we obtain
Thus, the
![$ j$](http://www.dsprelated.com/josimages_new/pasp/img664.png)
![$ j$](http://www.dsprelated.com/josimages_new/pasp/img664.png)
![$ i$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
![$ i$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
Physical Derivation of Reflection Coefficient
Physically, the reflection coefficient seen at port is due to an
impedance step from
, that of the port interface, to a new
impedance consisting of the parallel combination of all other
port impedances meeting at the junction. Let
denote this parallel combination, in admittance form. Then we must have
Let's check this ``physical'' derivation against the formal definition
Eq.(F.20) leading to
in Eq.
(F.22).
Toward this goal, let
![$\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$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
![\begin{eqnarray*}
\rho_i &\isdef & \alpha_i - 1
\;\isdef \; \frac{2\Gamma _i}{\...
..._i + \Gamma _J(i)}
\;=\; \frac{R_J(i) - R_i}{\Gamma _J(i)-R_i}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4865.png)
and the result is verified.
Reflection Free Port
It is useful in practice, such as when connecting two adaptors
together, to make one port reflection free. A
reflection-free port is defined to have a zero reflection coefficient. For port
of a parallel adaptor to be reflection free, we must have, from
Eq.
(F.25),
![$\displaystyle R_i = R_J(i) \isdef \frac{1}{\sum_{i\neq j} \Gamma _i}
$](http://www.dsprelated.com/josimages_new/pasp/img4866.png)
Connecting two adaptors at a reflection-free port prevents the formation of a delay-free loop which would otherwise occur [136]. As a result, multi-port junctions can be joined without having to insert unit elements (see §F.1.7) to avoid creating delay-free loops. Only one of the two ports participating in the connection needs to be reflection free.
We can always make a reflection-free port at the connection of two adaptors because the ports used for this connection (one on each adaptor) were created only for purposes of this connection. They can be set to any impedance, and only one of them needs to be reflection free.
To interconnect three adaptors, labeled ,
, and
, we may
proceed as follows: Let
be augmented with two unconstrained
ports, having impedances
and
. Add a reflection-free
port to
, and suppose its impedance has to be
. Add a
reflection-free port to
, and suppose its impedance has to be
. Now set
and connect
to
via the
corresponding ports. Similarly, set
and connect
to
accordingly. This adaptor-connection protocol clearly extends to any
number of adaptors.
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Two-Port Series Adaptor for Force Waves
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Two-Port Parallel Adaptor for Force Waves