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General Parallel Adaptor for Force Waves

In the more general case of $ N$ wave digital element ports being connected in parallel, we have the physical constraints

    $\displaystyle f_1(n) = f_2(n) = \cdots = f_N(n) \isdef f_J(n)$ (F.14)
    $\displaystyle v_1(n) + v_2(n) + \cdots + v_N(n) = 0$ (F.15)

The derivation for the two-port case extends to the $ N$-port case without modification:
0 $\displaystyle =$ $\displaystyle \sum_{i=1}^N v_i$  
  $\displaystyle =$ $\displaystyle \sum_{i=1}^N\frac{f^{{+}}_i-f^{{-}}_i}{R_i}$  
  $\displaystyle =$ $\displaystyle \sum_{i=1}^N\frac{2f^{{+}}_i-f_J}{R_i}$  
  $\displaystyle \isdef$ $\displaystyle \sum_{i=1}^N \left(2\Gamma _if^{{+}}_i-\Gamma _i f_J \right)$  
$\displaystyle \,\,\Rightarrow\,\,
\sum_{j=1}^N \Gamma _j f_J$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N 2\Gamma _i f^{{+}}_i$  
$\displaystyle \,\,\Rightarrow\,\,
f_J$ $\displaystyle =$ $\displaystyle \frac{\sum_{i=1}^N 2\Gamma _i f^{{+}}_i}{\sum_{j=1}^N \Gamma _j} .
\protect$ (F.16)

The outgoing wave variables are given by

$\displaystyle f^{{-}}_i(n) = f_J(n) - f^{{+}}_i(n)
$

Alpha Parameters

It is customary in the wave digital filter literature to define the alpha parameters as

$\displaystyle \fbox{$\displaystyle \alpha_i \isdef \frac{2\Gamma _i}{\sum_{j=1}^N \Gamma _j}$} \protect$ (F.17)

where $ \Gamma _i \isdef 1/R_i$ are the admittances of the wave digital element interfaces (or ``reference admittances,'' in WDF terminology). In terms of the alpha parameters, the force-wave parallel adaptor performs the following computations:
$\displaystyle f_J(n)$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N \alpha_i f^{{+}}_i(n)\protect$ (F.18)
$\displaystyle f^{{-}}_i(n)$ $\displaystyle =$ $\displaystyle f_J(n) - f^{{+}}_i(n)\protect$ (F.19)

We see that $ N$ multiplies and $ 2N-1$ additions are required. However, by observing from Eq.$ \,$(F.17) that

$\displaystyle \sum_{i=1}^N \alpha_i = 2,
$

we may implement $ \alpha_1$, say, as $ 2-\sum_{i=2}^N\alpha_i$ in order to eliminate one multiply. In WDF terminology, port 1 is then a dependent port.


Reflection Coefficient, Parallel Case

The reflection coefficient seen at port $ i$ is defined as

$\displaystyle \rho_i \isdef \left. \frac{f^{{-}}_i(n)}{f^{{+}}_i(n)} \right\vert _{f^{{+}}_j(n)=0, \forall j\neq i} \protect$ (F.20)

In other words, the reflection coefficient specifies what portion of the incoming wave $ f^{{+}}_i(n)$ is reflected back to port $ i$ as part of the outgoing wave $ f^{{-}}_i(n)$. The total outgoing wave on port $ i$ is the superposition of the reflected wave and the $ N-1$ transmitted waves from the other ports:

$\displaystyle f^{{-}}_i(n) = \rho_i f^{{+}}_i + \sum_{j\neq i} \tau_{ji} f^{{+}}_j \protect$ (F.21)

where $ \tau_{ji}$ denotes the transmission coefficient from port $ j$ to port $ i$. Starting with Eq.$ \,$(F.19) and substituting Eq.$ \,$(F.18) gives

\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*}

Equating like terms with Eq.$ \,$(F.21), we obtain

$\displaystyle \rho_i$ $\displaystyle =$ $\displaystyle \alpha_i - 1
\protect$ (F.22)
$\displaystyle \tau_{ji}$ $\displaystyle =$ $\displaystyle \alpha_j, \quad (i\neq j)
\protect$ (F.23)

Thus, the $ j$th alpha parameter is the force transmission coefficient from $ j$th port to any other port (besides the $ i$th). To convert the transmission coefficient from the $ i$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.


Physical Derivation of Reflection Coefficient

Physically, the reflection coefficient seen at port $ i$ is due to an impedance step from $ R_i$, that of the port interface, to a new impedance consisting of the parallel combination of all other port impedances meeting at the junction. Let

$\displaystyle \Gamma _J(i) \isdef \sum_{i\neq j} \Gamma _i \protect$ (F.24)

denote this parallel combination, in admittance form. Then we must have

$\displaystyle \rho_i = \frac{R_J(i)-R_i}{R_J(i)+R_i} = \frac{\Gamma _i-\Gamma _J(i)}{\Gamma _i+\Gamma _J(i)} \protect$ (F.25)

Let's check this ``physical'' derivation against the formal definition Eq.$ \,$(F.20) leading to $ \rho_i = \alpha_i - 1$ in Eq.$ \,$(F.22). Toward this goal, let

$\displaystyle \Gamma _J \isdef \sum_{j=1}^N \Gamma _j
$

denote the parallel combination of all admittances connected to the junction. Then by Eq.$ \,$(F.24), we have $ \Gamma _J = \Gamma _i + \Gamma _J(i)$ for all $ i$. Now, from Eq.$ \,$(F.17),

\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*}

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 $ i$ 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}
$

Thus, the port's impedance must equal the parallel combination of the other port impedances at the junction. In this case, the junction as a whole ``perfectly terminates'' the reflection free port, so no reflections come back from it.

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 $ A$, $ B$, and $ C$, we may proceed as follows: Let $ A$ be augmented with two unconstrained ports, having impedances $ R_1$ and $ R_2$. Add a reflection-free port to $ B$, and suppose its impedance has to be $ R_B$. Add a reflection-free port to $ C$, and suppose its impedance has to be $ R_C$. Now set $ R_1=R_B$ and connect $ B$ to $ A$ via the corresponding ports. Similarly, set $ R_2=R_C$ and connect $ C$ to $ A$ 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