## 4 Simulation Results  
The benefits of the joint-PPE schemes are estimated by using a Monte Carlo simulation. A \((2\times 2)\) MIMO system, as illustrated in Figure 1, with a \(100\,\mathrm{m}\) long MMF is considered for the simulation. There are three sequential steps, which are conducted to evaluate the performance of the proposed joint-PPE scheme. In the first step, an MMSE post-equalization is implemented while considering the pre-equalizer as a Dirac delta function. The channel state information is computed in the second step by using a pilot-based least square estimator algorithm. In the last step, the pre-and post-equalization filter taps are designed and optimized using the optimization algorithm in (8). Finally, the joint-PPE filter is incorporated with the provided system model and the calculated channel state information. The measured channel impulse response of a \(100\,\mathrm{m}\) long MMF is demonstrated in Figure 3. The cross-talk \(h_{12}[k]\) from the first layer to the second layer, as shown in Figure 3, is significant compared to \(h_{21}[k]\). Hence, the performance of the first layer is worse than the second layer due to the substantial modal dispersion in MMF. Moreover, the proposedhypothesis of this work targets the effective mitigation of this ICI and ISI by joint-PPE filter with minimal increase in the noise amplification. The simulation results are analyzed with two different scenarios, depending upon the utilized modulation format on both MIMO inputs. The first scenario employs a PAM-2 scheme on both MIMO inputs. Additionally, the second scenario uses the PAM-4 format on both MIMO inputs. While using the parameters mentioned above, the Monte Carlo simulations are conducted for both scenarios, and the BER is considered the key performance indicator. For the first scenario, the layer-wise and overall BERs dependent upon SNR are illustrated in Figure 4. It is evident that the joint-PPE scheme with four pre-equalizer filter taps with PAM-2 yields a better BER performance by 2.25 dB at \(10^{-4}\) BER compared to the PE-only with an MMSE principle. Moreover, the joint-PPE schemes significantly improve the performance of the first layer, which implies that the ICI and ISI are mitigated. Figure 5 shows the BER results for the second scenario, using PAM-4 on both MIMO inputs. In the second scenario, the joint-PPE with two-tap and four-tap pre-equalizer performs better than the MMSE-based PE-only. The noise amplification using PE-only is reduced by shifting the equalization taps from the receiver to the transmitter and jointly designing the PPE filter. The SNR gain of 1.90 dB at BER of \(10^{-4}\) is confirmed by using the joint-PPE with \(L_{\mathrm{P}}=4\) in the second scenario. When a comparison between the pre-equalizer with two and four taps is explored with PAM-4 format, an SNR gain of 0.60 dB is noted. It implies a slight improvement in the system performance. Therefore, in the analyzed case, when a trade-off between complexity and system performance is considered, a pre-equalizer with two taps should be preferred as it displays nearly similar performance benefits compared to a pre-equalizer with four taps. A detailed description of the SNR gains is depicted in Table 1.  
Figure 3: Measured channel impulse response of a 100 m long MMF with respect to the symbol rate \(f_{\mathrm{T}}=1/T_{\mathrm{s}}=2.5\,\mathrm{GHz}\) at an operating wavelength of 1550 nm. Figure 4: BER in dependent on SNR for the first scenario, where PAM-2 is utilized on both the MIMO inputs. The BER associated with PE-only and the proposed joint-PPE schemes are represented with solid and dashed lines, respectively. Figure 5: BER in dependent on SNR for the second scenario, where PAM-4 is utilized on both the MIMO inputs. The BER associated with PE-only and the proposed joint-PPE schemes are represented with solid and dashed lines, respectively. In summary, the simulated results demonstrate the usefulness of the proposed joint-PPE filter.