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One-Multiply Series Reflection-Free Three-Port Adaptor

As we saw for the parallel case in §F.2.3, the series three-port adaptor (scattering junction) with reflection-free port can be implemented using only one multiply.

The scattering relations in terms of the beta parameters introduced in §F.2.5 for a series junction of any number of ports:

$\displaystyle v_J$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N \beta_i \, v^{+}_i,$ (F.45)
$\displaystyle v^{-}_i$ $\displaystyle =$ $\displaystyle v_J - v^{+}_i$ (F.46)

where $ v_J$ denotes the junction force (or voltage), the $ \beta$ parameters are defined for series junctions by

$\displaystyle \beta_i \isdefs \frac{2R_i}{\sum_{j=1}^N R_j},

and $ R_i=1/\Gamma _i$ is the wave impedance on port $ i$ .

As for the parallel case, we set $ N=3$ , $ R_1=R_A$ , $ R_2=R_B$ , and $ R_3=R_C$ . With port A reflection-free, we have $ R_A=R_B+R_B$ . Dividing all impedances by $ R_A$ gives $ R_1=1$ , and $ R_2+R_3=1$ . Our one degree of freedom may be chosen as $ R_2\in[0,1]$ , and then $ R_3=1-R_2\in[0,1]$ .

The beta parameters become $ \beta_1=1$ , $ \beta_2=R_2\in[0,1]$ , and $ \beta_3=R_3=1-R_2\in[0,1]$ , so that the scattering formulas are

$\displaystyle v_J$ $\displaystyle =$ $\displaystyle v^{+}_1 + R_2v^{+}_2 + (1-R_2)v^{+}_3 \eqsp
v^{+}_1 + v^{+}_3 + R_2(v^{+}_2 - v^{+}_3),$ (F.47)
$\displaystyle v^{-}_i$ $\displaystyle =$ $\displaystyle v_J - v^{+}_i.$ (F.48)

Thus, we can compute the junction velocity (or current) using one multiply and three additions. Computing the outgoing waves requires three more additions, giving six total. However, as before, we can reuse intermediate results to get this down to four additions for all scattering:
$\displaystyle v_\delta$ $\displaystyle \isdef$ $\displaystyle v^{+}_2 - v^{+}_3$ (F.49)
$\displaystyle g_\delta$ $\displaystyle \isdef$ $\displaystyle R_2 v_\delta$ (F.50)
$\displaystyle v^{-}_1$ $\displaystyle =$ $\displaystyle v^{+}_3 + g_\delta$ (F.51)
$\displaystyle v^{-}_3$ $\displaystyle =$ $\displaystyle v^{+}_1 + g_\delta$ (F.52)
$\displaystyle v^{-}_2$ $\displaystyle =$ $\displaystyle v^{+}_1 + v^{+}_3 + g_\delta - v^{+}_2
\eqsp v^{+}_1 + g_\delta - v_\delta$ (F.53)
  $\displaystyle =$ $\displaystyle v^{-}_3 - v_\delta$ (F.54)

Thus, one multiply and four additions suffice for the three-port series junction with reflection-free port.

The series adaptor has now been derived in a way which emphasizes its duality with respect to the parallel adaptor.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2022-07-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University