the output side of MTL are

$$\begin{aligned}
\bm{I}_O =& \bm{Z_C^{-1}}(e^{-\bm{\Gamma }l}\bm{V^+}_m - e^{\bm{\Gamma }l}\bm{V^-}_m) \\
\bm{V}_O =& (e^{-\bm{\Gamma }l}\bm{V^+}_m + e^{\bm{\Gamma }l}\bm{V^-}_m).
\label{eq:V_I_forw_back_right}\end{aligned}$$

The subscripts $I$ and $O$ denote the input and output ports respectively. In order to solve these equations, it is necessary to introduce the impedance matrices $Z_I$ and $Z_O$ at the input and the output of the line respectively. These impedance matrices provide a boundary condition through to the system and allow it to be solved.

The resulting scattering matrix $\bm{S}$ is a 2N x 2N matrix, partitioned in 4 NxN matrices as $$\begin{aligned}
&\bm{S} =
\begin{bmatrix}
\bm{S_{II}} & \bm{S_{IO}}\\
\bm{S_{OI}} & \bm{S_{OO}}\\ 
\end{bmatrix} ,
\label{eq:S_2nx2n}\end{aligned}$$ where $S_{II}$ is the auto-scattering matrix of the inputs, $S_{OO}$ is the auto-scattering matrix of the outputs, and $S_{IO}$ and $S_{OI}$ are the transition scattering matrices from the inputs to the outputs and vice versa. The complete derivation of the scattering matrix is presented in [@kajfez].

Mixed-mode scattering matrix
----------------------------

In order to improve the understanding of the signal propagation through a transmission line, $\bm{S}$ can be redefined to explicitly show the relation between differential and common modes, using the theory of the mixed-mode scattering matrix (MMSM) [@392911].

The single-ended generic 