# Matched Filtering for MIMO Coherent Optical Communications with Mode-Dependent Loss Channels
## 1 Introduction  
The increasing demand of higher bit rates, combined with the environmental requirement of energy-efficient communication systems, is driving the development of ultra-high-capacity fiber optic communications. In this context, recent advances in spatial division multiplexing (SDM) using multimode or multicore fibers in long- and short-distance links [1, 2] cannot be possible without the extensive use of multiple-input multiple-output (MIMO) signal processing. Since the initial proposal to use polarization-division multiplexing (PDM) in a single mode fiber (SMF) [3] to double the capacity of a coherent optical communication system, MIMO signal processing [4, 5] has become necessary to process and recover the parallel transmitted data streams even before the signal processing used was identified as a MIMO equalizer [6]. The channel model for PDM in SMF and its relation with the non-linear Schrodinger equation [7], its representation by means of the \(2\times 2\) Jones matrix [8] and as a multi-section system [9] has been extensively discussed in previous works. Multiple contributions to adaptive MIMO equalizers using the flexibility of digital signal processors (DSP) have been developed [7, 10, 11], where normally the equalization is divided into two parts: The first one, with an invariant chromatic dispersion (CD) compensation for eachof the polarizations; and a second one, with an adaptive 2 \(\times\) 2 MIMO linear equalizer to resolve the crosstalk between the modes [12]. SDM [13] using multimode fibers (MMF) [14] or few mode fibers (FMF) [15; 16], appeared as a solution for communication systems reaching speeds well above 100 Tb/s when combined with wavelength multiplexing techniques [17]. Therefore, the optical channel model, based on the Jones matrix, was extended to represent the multiple fiber modes [18; 19; 20], and adaptive linear MIMO equalization [21] was studied and updated as an extension of the PDM case [2]. There are works that study the complexity of direct time and frequency domain implementations of the adaptive MIMO equalization, both for a linear design [12; 22; 23; 24] and a nonlinear one [25], and also in the optical domain [26]. An important difference of SDM systems w.r.t. PDM systems is that the modal dispersion (MD) in SDM systems is higher than the equivalent polarization dispersion in PDM systems, reaching the same order of magnitude of the CD [20]. This boosts looking for simpler DSP schemes that avoid the enormous complexity required from the classical equalizers proposed for a PDM system and initially adapted to SDM systems [24]. In particular, linear MIMO receiver designs have been proposed for SDM systems [27; 28] by expanding PDM systems [12; 29; 21], where a fractional-spaced equalizer (FSE) with an oversampling rate \(r_{ov}\) of two is used. A review of different combinations of fiber types and DSP schemes reported in the literature with their associated complexity is summarized in [1]. The impact of mode-dependent loss (MDL) in long-haul optical links has more recently been studied, especially in the associated loss in the channel capacity when using minimum mean square error (MMSE) MIMO receivers [27; 30]. This fact has initiated a race towards nonlinear receivers that can improve performance in the presence of MDL, increasing the receiver complexity notably [31; 32]. However, performance evaluation of SDM systems that incorporate MIMO FSE receivers in the presence of MDL and the impact that pulse-amplitude modulation (PAM) pulses roll-off factor have on this performance, still deserve attention. This paper provides a framework for the analysis of linear MIMO receivers for SDM that includes a continuous-time MIMO matched filter followed by a MIMO linear filter, without making prior assumptions about oversampling or the continuous-time optical front-end. This approach provides, for example, a theoretical basis for possible silicon photonics optical front-ends capable of SDM equalization.