## 5 Results
### Simulation Results  
The discrete-time Monte Carlo simulations are incorporated with a \((2\times 2)\) MIMO system using a 1.4 km long MMF channel. The simulations are implemented using three sequential steps. In the first step, the MMSE-based PE-only is applied to the provided system model while considering the pre-equalizer as a Dirac delta function. Additionally, the channel state information is computed from the received data using the pilot-based least-square algorithm. The second step involves the design and optimization of the PPE coefficients with the help of the given constrained numerical optimization. In the last step, the optimal joint-PPE coefficients are integrated in the system model, and the BER performance is calculated. Finally, the performance of the joint-PPE, the ZF-PPE and the PE-MMSE (also known as PE-only) are compared. Please note that the ZF-PPE incorporates a ZF-based pre-equalizer with an MMSE-based post-equalizer [29; 30]. The primary features of an IM/DD system are simple implementation and cost-effectiveness. Therefore, limiting the complexity of the transmitter is a factor, which should  
Figure 4: Optical MIMO channel utilizing the SVD technique. Figure 5: Simplified layer-specific MIMO system model with joint-PPE scheme. be considered while evaluating the system performance. In order to determine the optimal number of pre-equalizer filter taps, the relationship of the NWF with the pre-equalizer taps is calculated. The results are illustrated in Figure 6, where the optimized pre-equalizer coefficients achieve a significant reduction in the NWF on both the MIMO layers as compared to the PE-MMSE NWF of 7 on the first layer and 58.48 on the second layer. The extent of NWF decrement is directly related to the improvements in the system performance. Clearly, the optimally designed pre-equalizer using the joint-PPE scheme surpasses the ZF-PPE scheme. Hence, the joint-PPE is expected to yield better BER performance. It is noteworthy that the maximum reduction in the NWF is observed with the joint-PPE using \(L_{\mathrm{p}}=4\) at the first layer. However, the ZF-PPE scheme on the first MIMO layer has attained the maximum reduction in the NWF with \(L_{\mathrm{p}}=30\). Similarly, the maximum reduction in the NWF on the second MIMO layer is observed at \(L_{\mathrm{p}}=4\) and \(L_{\mathrm{p}}=30\) with the joint-PPE and the ZF-PPE schemes, respectively. On the basis of the reduction of the NWF and minimal transmitter's complexity, a pre-equalizer with at most four-taps on each layer is chosen for further performance analysis. Evidently, the joint-PPE requires a considerably smaller number of pre-equalizer taps. Thus, the prerequisites of the IM/DD systems are followed by a few pre-equalizer taps in conjunction with a post-equalizer. Prior to the system performance evaluation, the measured channel impulse response of a 1.4 km MMF is illustrated in Figure 7. The measured channel state information is obtained using an SDM-based optical MIMO experimental setup with offset launching conditions into an MMF. Afterwards, the estimated channel contains dispersion impairments and mode-dependent losses in accordance with the excited mode groups. This channel information is incorporated in the computer simulation to imitate the SDM technique. From Figure 7, it is clear that both MIMO layers are adversely affected due to crosstalk \(h_{vp}(k)\), when \(\
u\
eq\mu\). Additionally, a significant amount of ISI is also observed due to differential mode group delays. The proposed joint-transceiver equalization system performance is compared with the conventional ZF-PPE and PE-MMSE. The BER performances using PAM-2 and PAM-4 transmissions are demonstrated in Figures 8 and 9. The enhancements in the average received power \(P_{\mathrm{r}}\) compared to the PE-MMSE are presented in Table 2 at a BER of \(10^{-4}\). The BER results prove that the joint-PPE outperforms the ZF-PPE and PE-MMSE. Additionally, the BER performance of the joint-PPE can be advanced using the SVD-based joint-PPE.