&b_{jk}^d \equiv \frac{V_{jk}^d-I_{jk}^d Z_{jk}^d}{2 \sqrt{Z_{jk}^d}} \notag\\
&a_{jk}^c \equiv \frac{V_{jk}^c+I_{jk}^c Z_{jk}^c}{2 \sqrt{Z_{jk}^c}}
&b_{jk}^c \equiv \frac{V_{jk}^c-I_{jk}^c Z_{jk}^c}{2 \sqrt{Z_{jk}^c}}
\label{eq:pseudowaves_mixed_mode}\end{aligned}$$ A pseudowave state vector of a single mixed-mode port $jk$ can be defined as: $$\begin{aligned}
\bm{\dot{p_{jk}}} \equiv (a_{jk}^d,b_{jk}^d,a_{jk}^c,b_{jk}^c)^T
\label{eq:mixed-mode_ab__jk_state_vector}\end{aligned}$$ Using \[eq:from\_single-ended\_to\_mix\_mode\_vi,eq:pseudowaves\_signle\_ended,eq:pseudowaves\_mixed\_mode\], the relation between single-ended and mixed-mode pseudowaves state vectors becomes $$\begin{aligned}
\bm{\dot{p_{jk}}} = \Theta \bm{p_{jk}},\end{aligned}$$ where $\Theta$ is reported in .

$$\begin{aligned}
    &\bm{\Theta}
    = K
    \begin{bmatrix}
    \sqrt{Z_j}+\sqrt{Z_k}  & -(\sqrt{Z_j}+\sqrt{Z_k}) & \sqrt{Z_j}-\sqrt{Z_k} & -(\sqrt{Z_k}-\sqrt{Z_j})\\
    \sqrt{Z_j}-\sqrt{Z_k} & -(\sqrt{Z_k}-\sqrt{Z_j}) & \sqrt{Z_j}+\sqrt{Z_k}  & -(\sqrt{Z_j}+\sqrt{Z_k})\\
    \sqrt{Z_j}+\sqrt{Z_k}  & \sqrt{Z_j}+\sqrt{Z_k} & \sqrt{Z_j}-\sqrt{Z_k} & \sqrt{Z_k}-\sqrt{Z_j}\\
    \sqrt{Z_j}-\sqrt{Z_k}  & \sqrt{Z_k}-\sqrt{Z_j} & \sqrt{Z_j}+\sqrt{Z_k} & \sqrt{Z_j}+\sqrt{Z_k}\\
    \end{bmatrix}
    &K  = \frac{1}{2 \sqrt{2\sqrt{Z_j Z_k}}}
    \label{eq:from_single-ended_to_mixed-mode_pseudowaves}
    \end{aligned}$$

If $Z_j=Z_k$, then the matrix $\bm{\Theta}$ becomes [@1573844]:

$$\begin{aligned}
&\bm{\Theta}
= K
\begin{bmatrix}
1  & -1 & 0  & 0 \\
0  & 0  & 1  & -1\\
1  & 1  & 0  & 0 \\
0  & 0  & 1  & 1 \\
\end{bmatrix}
&K  = \frac{1}{\sqrt{2}}.
\label{eq:from_single-ended_to_mixed-mode_pseudowaves_simpified}\end{aligned}$$

The scattering matrix $\bm{S}_{jk}$ for single-ended ports and $\dot{\bm{S}_{jk}}$ for mixed-mode ports obey the following relations $$\begin{aligned}
&\bm{b}_{jk}\equiv \bm{S}_{jk}\bm{a}_{jk},  &\dot{\bm{b}_{jk}}\equiv \dot{\bm{S}_{jk}} \dot{\bm{a}_{jk}}.
\label{eq:scattering_mx_se}\end{aligned}$$ where $\bm{b}_{jk} =(b_j, b_k)$, $\bm{a}_{jk} =(a_j, a_k)$, $\dot{\bm{b}}_{jk} =(b^d_{jk}, b^c_{jk})$, $\dot{\bm{a}}_{jk} =(a^d_{jk}, a^c_{jk})$.

Reordering the rows of $\bm{\Theta}$, it is possible to compute the relationship between the mixed-mode 