# Matched Filtering for MIMO Coherent Optical Communications with Mode-Dependent Loss Channels
## 5 Numerical Simulation of Linear MIMO FSE Receiver for MDL-Impaired Optical Channel
### Numerical Simulation Results  
The first result is focused on the impact of PAM pulses roll-off factor \(\alpha\) and MDL level, represented by \(\sigma_{g}\), on the SDM optical system performance. Figure 6 shows that for systems with transmitters using a higher \(\alpha\), the degradation is a bit lower. The effect is higher with increasing \(\sigma_{g}\) for systems working at a high regime of \(\overline{SNR}_{in}\), as seen in Figure 6b. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value and Reference** \\ \hline Span length & \(\ell_{span}\) & 50 km \\ Number of spans & \(K_{amp}\) & 100 \\ Number of spatial and polarization modes & \(D\) & 6 \\ \hline Center wavelength & \(\lambda_{c}\) & 1469 nm [41] \\ Modal dispersion & \(\sigma_{\tau}/\sqrt{\ell_{span}}\) & 3.1 ps\(/\sqrt{\text{km}}\)[41] \\ Dispersion coefficient & \(D_{CD}=-\frac{\delta\pi\sigma}{\lambda_{c}^{2}}\delta_{2}\) & 20.1 ps\(/\)(nm\(\cdot\)km) [41] \\ Underestimation dispersion factor & \(U_{CD}\) & 2\% [9] \\ Amplifier gain STD & \(\sigma_{g}\) & 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 dB \\ \hline Symbol rate & \(R_{s}=1/T_{s}\) & 64 GBaud \\ Oversampling factor & \(r_{\text{ov}}\) & 2 \\ Roll off factor & \(\alpha\) & 0.1, 0.5, 0.7, 0.9 \\ Number of channel realizations & \(N_{ch}\) & 10000 \\ Signal to noise ratio at the receiver input & \(\overline{SNR}_{in}\) & 60 30 15 10 6.2 5 dB [27] \\ Number of taps & \(N_{taps}\) & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 6: _Cont._  
In practical systems, the allowable loss of \(\underline{SNR}_{out}\) in a channel with elements introducing MDL w.r.t. an ideal channel without MDL is around 1-2 dB. We can observe that, assuming a maximum degradation of 2 dB in the system with a 95% confidence, \(\sigma_{g}\) values of the amplifiers should not exceed 0.2 dB. These results are in agreement with the capacity limits of a MIMO MMSE receiver and MDL channel calculated in [27]. A second result is presented in Figure 7, where the probability distribution of \(AL\) and \(\overline{SNR}_{in}\) estimated from the analysis of all channel realizations is plotted. We are comparing different levels of SNR regimes, with \(\overline{SNR}_{in}\) = 5, 6.2, and 10 dB (Figure 6(a)) and \(\overline{SNR}_{in}\) = 15, 30, and 60 dB (Figure 6(b)), for a roll-off factor of \(\alpha\) = 0.9. The upper and lower limits of the blue boxes represent the 25 th and 75 th percentiles respectively. The red line inside the box indicates the median of the metric. In case the distribution of values was Gaussian, the whisker bounds correspond to 2.7 times the STD of the metric or, in other words, the number of values between the upper and lower bounds of the whiskers contains 99.3% of the values. Values outside these limits are considered outliers and are individually represented by red crosses. We make the following observations from Figure 7:  
* The distribution of \(AL\) is not Gaussian, as we can observe by comparing the difference between upper and lower outliers for higher \(\sigma_{g}\) values and their asymmetry;
* There are no negative values of \(AL\), since the FSE MIMO receiver cannot improve on average the \(\overline{SNR}_{in}\). However, by taking all values of \(L(i)\) for any received mode \(i\), we can find that, for certain channels and modes, the FSE MIMO receiver can locally improve the \(SNR_{in}(i)\) of a particular mode \(i\), but always at the cost of another mode of the receiver;
* The performance degradation depends on the \(\overline{SNR}_{in}\). In a low \(\overline{SNR}_{in}\) regime (5 dB, Figure 6(a)), the degradation is measured lower in absolute values when compared to the high \(\overline{SNR}_{in}\) regime (60 dB, Figure 6(b));
* The STD of the performance degradation also depends on the \(\overline{SNR}_{in}\).