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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 [137]. 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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``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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