# Recent Progress on Novel DSP Techniques for Mode Division Multiplexing Systems: A Review
## 1 Introduction  
SDM has gained great attention over the last decades as a potential technology to accommodate the demand for future long-haul and power-efficient optical fiber communication systems with extremely high capacity [1, 2]. In SDM transmission systems, fibers with parallel spatial transmission channels are used to carry independent data-streams in the same frequency band, which can be either uncoupled multi-core fibers (MCFs) or few- and multi-mode fibers (FMFs and MMFs) [3]. Each type of fiber for SDM systems has its own special advantages. Uncoupled MCFs are easy to be expanded from current single mode fiber (SMF) schemes [4] since each core is used as an independent spatial channel where crosstalk from other cores is well suppressed [5]. While strongly coupled MCFs have been proved with better nonlinear tolerance compared to SMFs, which makes longer transmission distance or higher transmitting power possible [6]. FMFs or MMFs on the other hand can support much high spatial channel-count in a fixed cross-section area compared with other typeof fibers, using additional DSP to undo the linear, nonlinear and other impairments that recoverable after several kilometers of transmission. The impairments over long-distance FMF systems mainly include modal dispersion (MD), linear coupling or crosstalk (XT) among different fiber modes, time-varying (TV) effect in transmission channels, mode dependent loss (MDL) and nonlinear effects. A combination of MD and linear coupling causes visible inter-symbol interference between different modes and different time slots. Accumulated MD is also called differential mode group delay (DMGD), which means different spatial modes propagate along the fiber with different speed. While linear coupling either appears at discrete points like fiber splices and connectors or is distributed along the fiber due to bends or fabrication imperfections. Generally, linear coupling and MD are mitigated by multiple-input-multiple-output (MIMO) equalizers [7]. To decrease the high computational complexity of MIMO equalizers, frequency-domain equalization (FDE) algorithms were utilized to recover the signals that have experienced impairments in fibers [8; 9; 10]. The FDE methods can reduce the computational complexity dramatically, holding the same performance compared with the time-domain equalization (TDE). What's more, when it refers to the specific updating methods of weighting matrix, the least mean squares (LMS) and recursive least squares (RLS) algorithms are two most widely adopted and studied methods [11; 12; 13]. Compared with LMS algorithm, faster convergence and better performance can be accomplished by applying RLS at the cost of higher complexity [14; 15; 16]. It should also be noticed that linear coupling between different modes can be easily influenced by a random temperature fluctuation or mechanical perturbation [17], so the transmission matrix of fiber channels is actually time-varying with a certain speed related to the perturbation rate. To achieve timely signal recovery, the equalizer on the receiver needs to re-estimate the transmission matrix more frequently at the maximum speed of the fiber channel evolution [18]. Moreover, if the speed of the fiber channel fluctuation is too fast to track for the equalizer, the quality of the recovery would be degraded, leading to decrease in transmission capacity [19]. So, the TV effect will not only increase the burden on a receiver but also degrade the quality of channel equalization. Thus, aside from the computational complexity and performance, the tracking capability or adaptation time of the equalizers also becomes important for SDM systems, which naturally means RLS-based MIMO algorithm with faster convergence is much valuable as a possible solution for TV effects. Long-haul SDM systems are also impaired by MDL arising from the imperfections or mode-dependent difference of inline components such as fibers, amplifiers and multiplexers, who are inducing modal loss/gain disparities [20; 21; 22; 23].