Reflection Coefficient, Parallel Case

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

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

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

where $\tau_{ji}$ denotes the transmission coefficient from port $j$ to port $i$. Starting with Eq. (N.19) and substituting Eq. (N.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. (N.21), we obtain

$$\rho_i = \alpha_i - 1 \protect$$ (N.22)

$$\tau_{ji} = \alpha_j, \quad (i\neq j) \protect$$ (N.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.