## 3 Design and Optimization of Pre- and Post-Equalization  
According to the literature, a few-tap pre-equalizer is preferred to compensate for the channel influence without any noise enhancement [16; 17; 20; 22]. The zero-forcing design of a pre-equalizer is chosen predominantly due to the simple implementation. Moreover, the post-equalizer filter taps are separately calculated. By contrast, in the proposed transceiver-joint equalization, the pre-equalizer and the post-equalizer are calculated and jointly optimized to yield performance benefits. The pre-equalizer is characterized as a finite impulse response (FIR) filter, and the role of pre-equalizer taps \(p_{i,\mu}\) is shown as  
\[b_{\mu}(k)=\sum_{i=1}^{L_{\text{p}}}p_{i,\mu}\cdot s_{\mu}[k-(i-1)],\qquad\mu =1,\ldots,n_{\text{T}}\enspace, \tag{15}\]  
where \(L_{\text{p}}\) is the length of the pre-equalizer at the \(\mu\)-th MIMO layer. The structure of a linear FIR pre-equalizer present on each MIMO layer is shown in Figure 3. The main goal is to design and optimize the PPE filter taps to enhance the system performance while maintaining power and cost budgets. In this work, a numerical convex optimization is applied to calculate and optimize the PPE coefficients. The primal-dual interior-point constrained algorithm is tailored with  
Figure 3: The structure of a linear FIR pre-equalizer filter, where the delay by one symbol is denoted by \(z^{-1}\). respect to the analytical solution of the given system model [28]. Hence, the objective function subjected to the constraints is stated as follows  
\[\underset{p_{\mu}(k),f_{\
u\mu}(k)}{\text{minimize}} \theta_{\mu}(h_{\
u\mu}(k))\hskip 56.905512 pt,\
u=1,\ldots,n_{ \text{R}}\] (16) subject to \[\mathbf{F}\cdot\mathbf{H}\cdot\mathbf{P}-\mathbf{I}=0\hskip 5.690551 pt,\] and \[s_{\mu,M_{\mu}-1}-\text{q}\cdot P_{\text{s},\mu}^{(\text{max})}=0 \hskip 5.690551 pt.\]  
The objective function is defined to minimize the NWF on each MIMO layer in accordance with the channel state information. The values of \(p_{\mu}(k)\) and \(f_{\
u\mu}(k)\) will converge to their optimal value with each iteration. In (16), the first constraint determines the complete removal of the ISI and ICI with the PPE filter. Additionally, the second constraint assures that the symbol with the highest amplitude is limited to the maximum transmit power of the laser source. In (16), the optimization problem is called the primal problem. In order to solve this problem, the duality optimization is utilized, where the constraints are incorporated with the objective function. The modified objective function of the optimization statement is  
\[\underset{p_{\mu}(k),f_{\
u\mu}(k)}{\text{minimize}}\hskip 5.690551 pt\theta_{ \mu}(h_{\
u\mu}(k))+\underset{\alpha,\beta}{\text{max}}\left((\mathbf{F}\cdot\mathbf{ H}\cdot\mathbf{P}-\mathbf{I})\cdot\alpha+\left(s_{\mu,M_{\mu}-1}-\text{q}\cdot P_{\text{s}, \mu}^{(\text{max})}\right)\cdot\beta\right)\hskip 5.690551 pt, \tag{17}\]  
where \(\alpha\) and \(\beta\) represent the scalar slopes of the penalty function. The purpose of the penalty function is to penalize the objective function when the constraints are not followed. Furthermore, the objective function is rewarded when the constraints are obeyed. The objective function (17) is termed as the primal problem, which is converted into the duality problem, as the min and max operators are in the function simultaneously. Thus, the dual problem can be written as  
\[\underset{\alpha,\beta}{\text{max}}\hskip 5.690551 pt\underset{p_{\mu}(k),f_{ \
u\mu}(k)}{\text{minimize}}\left(\theta_{\mu}(h_{\
u\mu}(k))+(\mathbf{F}\cdot\mathbf{ H}\cdot\mathbf{P}-\mathbf{I})\cdot\alpha+\left(s_{\mu,M_{\mu}-1}-\text{q}\cdot P_{\text{s}, \mu}^{(\text{max})}\right)\cdot\beta\right)\hskip 5.690551 pt. \tag{18}\]  
The objective function in (18) can be easily solved with the Lagrangian solver.