**6. Conclusions**

This work explored long-haul fiber-optic SDM coherent systems with PAM raisedcosine pulses. We investigated the overall system performance under different configurations of the optical channel. For that purpose, we modeled this channel with a MIMO multi-span structure that included the several dispersion terms and modal losses factors.

It was demonstrated that the linear MMSE MIMO receiver completely eliminated ISI and crosstalk when the number of taps was sufficiently high and the optical channel was free of MDL. Moreover, the generic structure of the linear MMSE MIMO receiver could be simplified to a continuous-time matched filter and still retained the same properties. This observation paves the way for analog receivers that simply implement the matched filter of the optical channel, eliminating all channel impairments when MDL is negligible.

We also defined performance metrics to assess the losses of a linear MIMO receiver implemented using a MIMO FSE with an oversampling factor of two for an optical channel that exhibited significant MDL. We have shown that such loss depends on the transmitter PAM pulses roll-off factor and the SNR level at the receiver input. We also determined that the performance degradation could be limited by processing the *D* output modes together by averaging the *D* output SNDR values at the receiver.

This fact opens a way to exploit this loss compensation between modes at the receiver output. The design of specific forward error correction codes taking into account this aspect could improve the final performance of the system in terms of bit error rate. Constructing the message to be encoded, including bits or signals belonging to all modes, could improve system performance w.r.t. constructing messages with bits or signals from only one mode.

**Author Contributions:** Conceptualization, methodology, validation, formal analysis, resources, investigation, data curation, writing—review and editing, writing—original draft preparation, and visualization, L.M.T., F.J.C. and L.D.; software, L.M.T. and L.D.; supervision, project administration, and funding acquisition F.J.C. and L.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported in part by Junta de Andalucía, under PAIDI program, TIC-102 research group, and by the Spanish Ministry of Ciencia e Innovación, under Project PID2019-109842RB-I00.

**Institutional Review Board Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
