& \bm{T_{22}} \\
\end{bmatrix}
\begin{bmatrix}
\bm{a_L} \\
\bm{b_L} \\
\end{bmatrix}
\label{eq:T_mix-mode_matrix}\end{aligned}$$

Where, $$\begin{aligned}
\bm{T_{11}} =& S_{OI}-S_{OO}S_{IO}^{-1}S_{II}\\
\bm{T_{12}} =& S_{OO}S_{IO}^{-1}\\
\bm{T_{21}} =& -S_{IO}^{-1}S_{II}\\
\bm{T_{22}} =& S_{IO}^{-1}
\label{eq:T_mix-mode_matrix_explaniation}\end{aligned}$$

Application to power line networks {#sec:application}
==================================

![Example of an in-home power line network.[]{data-label="fig:inhome_topology"}](inhome_topology.eps){width="40.00000%"}

Power line networks as the one depicted in are composed by a mixture of different appliances connected by cables that often have different characteristics in different branches [@tonello2011bottomup]. Moreover, cable interconnections include junction boxes, power strips, plugs and sockets that increase the network complexity. In order to account for all of these elements when simulating the communication between two PLMs, an equivalent MMTM representation for each element has to be provided. All these matrices are combined using the chain rule [@1458855] to compute the total channel frequency response.

As for the cables, we represent them using the results reported in . We decided to use four input and four output single-ended ports to describe each MTRL block. This configuration is converted in two input and two output mixed-mode ports (see ). The defined number of ports has been chosen to obtain a squared shape for the final mixed-mode matrix. The square matrix is easily manipulated for the final conversion to 