scattering matrix $\dot{\bm{S_{jk}}}$ and the single-ended scattering matrix $\bm{S}$. To this aim, $\bm{\Theta}$ is split in two rectangular matrices $ \bm{\Theta_1,\Theta_2} $. $\bm{\Theta_1}$ is composed by the first two rows of $\bm{\Theta}$ and $\bm{\Theta_2}$ the last two, respectively. A new matrix $\bm{\Theta_\alpha}$ is made by exchanging the position of $\bm{\Theta_1}$ and $\bm{\Theta_2}$ so that $$\begin{aligned}
&\bm{\Theta}
=
\begin{bmatrix}
\bm{\Theta_1} \\
\bm{\Theta_2} \\
\end{bmatrix}
&\bm{\Theta_\alpha}
=
\begin{bmatrix}
\bm{\Theta_2} \\
\bm{\Theta_1} \\
\end{bmatrix}.
\label{eq:theta_m}\end{aligned}$$

Finally, from ,, the direct and inverse transformations linking $\dot{\bm{S}}$ and $\bm{S}$ are: $$\begin{aligned}
&\hat{\bm{S_{jk}}} = {\bm{\Theta_\alpha}}^{-1} \hat{\dot{\bm{S_{jk}}}} \bm{\Theta} \\
&\hat{\dot{\bm{S_{jk}}}} = \bm{\Theta_\alpha} \hat{\bm{S_{jk}}} \bm{\Theta^{-1}},
\label{eq:transform_s}\end{aligned}$$ where $\hat{\bm{S_{jk}}}$ and $\hat{\dot{\bm{S_{jk}}}}$ are defined as

$$\begin{aligned}
&\bm{\hat{{S_{jk}}}}
=
\begin{bmatrix}
\bm{S_{jk}} & \bm{I}_2 \\
\bm{I}_2  & \bm{{S_{jk}}^{-1}}\\
\end{bmatrix}
&\dot{\hat{\bm{S_{jk}}}}
=
\begin{bmatrix}
\bm{\dot{S_{jk}}} & \bm{I}_2 \\
\bm{I}_2  & \bm{\dot{{S_{jk}}^{-1}}}\\
\end{bmatrix}
\label{eq:S_explanation}\end{aligned}$$

and $\bm{I}_2$ is the 2x2 identity matrix.

The $\bm{\dot{S_{jk}}}$ matrix is a 2x2 matrix, with the following parameters: $S^{dd}_{jk}$ denoting the differential mode, $S^{cc}_{jk}$ denoting the common mode and $S^{dc}_{jk}$ as well as $S^{cd}_{jk}$ denoting the conversion between both modes.

$$\begin{aligned}
&\bm{\dot{S_{jk}}}
=
\begin{bmatrix}
S^{dd}_{jk} & S^{dc}_{jk} \\
S^{cd}_{jk} & S^{cc}_{jk} \\
\end{bmatrix}
\label{eq:S_mix-mode_explanation}\end{aligned}$$

The final scattering matrix respectively for the single-ended and mixed-mode ports are $$\begin{aligned}
&\bm{b}\equiv \bm{S}\bm{a},  &\dot{\bm{b}}\equiv \dot{\bm{S}} 