\dot{\bm{a}}.
\label{eq:scattering_mx_tot}\end{aligned}$$ where $\bm{b} =(b_1, \dots ,b_M)$, $\bm{a} =(a_1, \dots ,a_M)$, $\dot{\bm{b}} =(b^d_{jk}, \dots ,b^c_{jk})$ and $\dot{\bm{a}} =(a^d_{jk}, \dots ,a^c_{jk})$. The conversion between $\bm{S}$ and $\dot{\bm{S}}$ is computed by applying and to all the possible $j,k$ couples.

Mixed-mode transmission matrix
------------------------------

The description of MTL blocks with the mixed-mode scattering matrix helps to understand the conversion between DM and CM and vice versa. A drowback of the scattering parameters is that they do not allow to easily manipulate the concatenation of multiple scattering matrices. However, MMSMs can be converted to mixed-mode transmission matrices (MMTMs) [@1573844]. The system resulting from the connection of $N$ devices can be represented by a MMTM that is given by the multiplication of the MMTMs of the single subsystems.

The conversion procedure comprises: repositioning of the matrix parameters and a linear transformation. The ports should be repositioned in order to reach this form: $\bm{b}=(\bm{b_I},\bm{b_O})^T$ where $\bm{b_I} = (b^d_1,b^c_1,b^d_3,b^c_3)^T$, $\bm{b_O} = (b^d_2,b^c_2,b^d_4,b^c_4)^T$; $\bm{a}=(\bm{a_I},\bm{a_O})^T$ where $\bm{a_I} = (a^d_1,a^c_1,a^d_3,a^c_3)^T$, $a_O = (a^d_2,a^c_2,a^d_4,a^c_4)^T$, where subscripts $I$ and $O$ refer to the input and output side of the network (see ).

The following linear transformation then applies:

$$\begin{aligned}
\begin{bmatrix}
\bm{b_R} \\
\bm{a_R} \\
\end{bmatrix}
=
\begin{bmatrix}
\bm{T_{11}} & \bm{T_{12}} \\
\bm{T_{21}} 