The reflection coefficient seen at port is defined as
In other words, the reflection coefficient specifies what portion of the incoming wave is reflected back to port as part of the outgoing wave . The total outgoing wave on port is the superposition of the reflected wave and the transmitted waves from the other ports:
where denotes the transmission coefficient from port to port . Starting with Eq.(F.19) and substituting Eq.(F.18) gives
Equating like terms with Eq.(F.21), we obtain
Thus, the th alpha parameter is the force transmission coefficient from th port to any other port (besides the th). To convert the transmission coefficient from the 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