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Two-Port Parallel Adaptor for Force Waves

Figure F.5a illustrates a generic parallel two-port connection in terms of forces and velocities.

Figure: a) Two-port description of the adaptor implementing a parallel connection between reference impedances $ R_1$ and $ R_2$ . b) Corresponding parallel force scattering junction (adaptor wave flow diagram) in Kelly-Lochbaum form. Compare with Fig.F.7.
\includegraphics[width=\twidth]{eps/lAdaptorParallel}

As discussed in §7.2, a parallel connection is characterized by a common force and velocities which sum to zero:

\begin{eqnarray*}
&& f_1(n) = f_2(n) \isdef f_J(n)\\
&& v_1(n) + v_2(n) = 0
\end{eqnarray*}

Following the same derivation leading to Eq.$ \,$ (F.2), and defining $ \Gamma _i=1/R_i$ for notational convenience, we obtain

\begin{eqnarray*}
0 &=& v_1+v_2 \\
&=& \frac{f^{{+}}_1-f^{{-}}_1}{R_1} + \frac{f^{{+}}_2-f^{{-}}_2}{R_2} \\
&=& \frac{2f^{{+}}_1-f_J}{R_1} + \frac{2f^{{+}}_2-f_J}{R_2} \\
&\isdef & 2\Gamma _1f^{{+}}_1-\Gamma _1 f_J + 2\Gamma _2f^{{+}}_2-\Gamma _2 f_J \\
\,\,\Rightarrow\,\,\quad
(\Gamma _1+\Gamma _2) f_J &=& 2\left[\Gamma _1 f^{{+}}_1 + \Gamma _2 f^{{+}}_2 \right] \\
\,\,\Rightarrow\,\,\quad
f_J &=& 2 \frac{\Gamma _1 f^{{+}}_1 + \Gamma _2 f^{{+}}_2 }{\Gamma _1+\Gamma _2} .
\end{eqnarray*}

The outgoing wave variables are given by

\begin{eqnarray*}
f^{{-}}_1(n) &=& f_J(n) - f^{{+}}_1(n) \\
f^{{-}}_2(n) &=& f_J(n) - f^{{+}}_2(n)
\end{eqnarray*}

Defining the reflection coefficient as

$\displaystyle \rho \isdef \frac{R_2-R_1}{R_2+R_1}
$

we have that the scattering relations for the two-port parallel adaptor are

\begin{eqnarray*}
f^{{-}}_1 &=& \rho f^{{+}}_1 + (1-\rho) f^{{+}}_2
\protect
\\
f^{{-}}_2 &=& (1+\rho)f^{{+}}_1 - \rho f^{{+}}_2
\protect
\end{eqnarray*}

as diagrammed in Fig.F.5b. This can be called the Kelly-Lochbaum implementation of the two-port force-wave adaptor.

Now that we have a proper scattering interface between two reference impedances, we may connect two wave digital elements together, setting $ R_1$ to the port impedance of element 1, and $ R_2$ to the port impedance of element 2. An example is shown in Fig.F.35.

The Kelly-Lochbaum adaptor in Fig.F.5b evidently requires four multiplies and two additions. Note that we can factor out the reflection coefficient in each equation to obtain

\begin{eqnarray*}
f^{{-}}_1 &=& f^{{+}}_2 + \rho(f^{{+}}_1 - f^{{+}}_2)\\
f^{{-}}_2 &=& f^{{+}}_1 + \rho(f^{{+}}_1 - f^{{+}}_2)
\end{eqnarray*}

which requires only one multiplication and three additions. This can be called the one-multiply form. The one-multiply form is most efficient in custom VLSI. The Kelly-Lochbaum form, on the other hand, may be more efficient in software, and slightly faster (by one addition) in parallel hardware.



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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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