## 2 System Model
After the channel, the signal suffers from  
Figure 1: Electrical (\(2\times 2\)) MIMO system model with jointly designed pre- and post-equalization filter. Figure 2: Illustrating the multi-level signals after pre-equalization using PAM-2 configuration, where: (a) \(L_{\mathrm{P}}=2\); and (b) \(L_{\mathrm{P}}=3\). At the transmitter, the light launching conditions into the optical fiber core are included in (c), which represents the centric \(0\,\upmu\)m and eccentric \(15\,\upmu\)m light launching conditions for two MIMO inputs \(\mathrm{Tx}_{1}\) and \(\mathrm{Tx}_{2}\), respectively. additive, white and Gaussian noise (AWGN) [15]. The mathematical description of the provided system model in matrix notation is derived as  
\[\mathbf{r}=\mathbf{H}\cdot\mathbf{P}\cdot\mathbf{s}+\mathbf{w}\enspace, \tag{2}\]  
where \(\mathbf{r}\) is the received signal matrix, and \(\mathbf{H}\) is the channel matrix including direct and cross-talk components. The pre-equalizer and transmitted symbols are denoted by \(\mathbf{P}\) and \(\mathbf{s}\), respectively. The AWG noise is considered to be \(\mathbf{w}\) in (2). Finally, the received signal \(\mathbf{r}\) undergoes the post-equalization \(\mathbf{F}\). The output signal matrix \(\mathbf{y}\) is defined as  
\[ \begin{split}\mathbf{y}=&\mathbf{F}\cdot\mathbf{r}\\ \mathbf{y}=&\underbrace{\mathbf{F}\cdot\mathbf{H}\cdot\mathbf{P}}_ {\text{Term 1}}\cdot\mathbf{s}+\underbrace{\mathbf{F}\cdot\mathbf{w}}_{\text{Term 2}}\enspace.\end{split}  \tag{3}\]  
From (3), Term 1 emphases that the pre- and post-equalizers can jointly contribute to recover the transmitted data by mitigating the effect of the channel. Secondly, Term 2 highlights the contribution of the post-equalization in the noise amplification, which will result in a performance degradation. While considering that the ISI and ICI are completely mitigated with the help of pre-equalizer and post-equalizer, (3) is condensed further as follows  
\[ \begin{split}\mathbf{F}\cdot\mathbf{H}\cdot\mathbf{P}=\mathbf{I}\\ \mathbf{y}=\mathbf{s}+\mathbf{F}\cdot\mathbf{w}\enspace,\end{split}  \tag{4}\]  
with \(\mathbf{I}\) denoting the identity matrix. It is clear that the post-equalization is responsible for noise enhancement at the receiver. Therefore, the hypothesis implies that by shifting the equalization efforts to the transmitter will reduce the noise amplification and will compensate the channel interferences. Consequently, the system performance can be improved. The noise amplification due to the presence of the post-equalizer is quantified using the noise power \(\widetilde{P}_{\text{n},\mu}\). The expression is provided as follows  
\[\widetilde{P}_{\text{n},\mu}=\theta_{\mu}\cdot P_{\text{n}},\qquad\theta_{\mu }=\sum_{\
u=1}^{n_{\text{R}}}\sum_{k=0}^{L_{\text{f}}-1}|f_{\mu\
u}[k]|^{2}\enspace, \tag{5}\]  
where the noise weighting factor \(\theta_{\mu}\) is derived from the amplitude of the post-equalization coefficients \(f_{\mu\
u}[k]\). Prior to the post-equalization, the noise power is denoted as \(P_{\text{n}}\). Furthermore, \(L_{\text{f}}\) defines the length of the post-equalization filter. From (5), a direct relation between the post-equalization coefficients and the noise enhancement is established. In order to evaluate the performance of the provided \((2\times 2)\) optical MIMO system, the key performance indicator is considered as BER \(P_{\text{BER}}^{(\mu)}\) of \(\mu\)-th layer, and it is defined as  
\[ \begin{split} P_{\text{BER}}^{(\mu)}&=\frac{1}{ \log_{2}(M_{\mu})}\left(\frac{M_{\mu}-1}{M_{\mu}}\right)\text{erfc}\left(\sqrt {\frac{\theta_{\mu}}{2}}\right),\\ \text{where},\quad\varrho_{\mu}&=\frac{U_{\text{A}, \mu}^{2}}{\widetilde{P}_{\text{n},\mu}}=\frac{U_{\text{A},\mu}^{2}}{\theta_{ \mu}\,P_{\text{n}}}\enspace,\end{split}  \tag{6}\]  
with erfc(\(\cdot\)) denoting the complementary error function. In (6), at the \(\mu\)-th input, \(M_{\mu}\) defines the size of the constellation, the signal-to-noise-ratio (SNR) is \(\varrho_{\mu}\) and the half-vertical eye-opening is \(U_{\text{A},\mu}\).