port $j$ is defined by the single-ended voltage and current state vector: $$\begin{aligned}
\bm{s_j} \equiv (V_j,I_j)^T
\label{eq:sing_ended_vi_state_vector}\end{aligned}$$ where the symbol $T$ indicates the transpose operator.

On the other hand, the mixed-mode generic port $jk$ with $(j,k)\in (1\dots M)$, $j\neq k$, where M is the total number of ports, is defined based on the single-ended voltages and currents as: $$\begin{aligned}
V_{jk}^d &\equiv V_j-V_k, \notag\\
I_{jk}^d &\equiv \frac{(I_j-I_k)}{2}, \notag\\
V_{jk}^c &\equiv \frac{(V_j+V_k)}{2},\notag\\
I_{jk}^c &\equiv I_j+I_k  ,
\label{eq:from_single-ended_to_mix_mode_vi}\end{aligned}$$ where the $\{\cdot\}^d$ indicates the differential voltages and currents and the symbol $\{\cdot\}^c$ indicates the common mode ones. The corresponding mixed-mode voltage and current state vector is: $$\begin{aligned}
\bm{\dot{r_{jk}}} \equiv (V_{jk}^d, I_{jk}^d, V_{jk}^c, I_{jk}^c)^T
\label{eq:mixed-mode_vi_state_vector}\end{aligned}$$ After defining the complex reference impedance as $Z_j$, the forward and reverse pseudowaves for the single-ended ports are defined as: $$\begin{aligned}
&a_j  \equiv \frac{V_j+Ij Z_j}{2 \sqrt{Z_j}}
&b_j  \equiv \frac{V_j-Ij Z_j}{2 \sqrt{Z_j}}
\label{eq:pseudowaves_signle_ended}\end{aligned}$$ A pseudowave state vector of two single-ended ports $j,k$ can be defined as: $$\begin{aligned}
\bm{p_{jk}} \equiv (a_j,b_j,a_k,b_k)^T
\label{eq:sing_ended_ab__jk_state_vector}\end{aligned}$$ For the mixed-mode ports, the reference impedances are defined as $Z{jk}^d \equiv 2\sqrt{Z_j Z_k}$ and $Z{jk}^c \equiv \frac{\sqrt{Z_j Z_k}}{2}$ for the DM and CM respectively. The mixed-mode pseudo waves are therefore defined as: $$\begin{aligned}
&a_{jk}^d \equiv \frac{V_{jk}^d+I_{jk}^d Z_{jk}^d}{2 \sqrt{Z_{jk}^d}} \notag
