## 1 Introduction  
The space division multiplexing (SDM) systems were proposed to improve the capacity of fiber-optic transmission links, either by means of core multiplexing exploiting multicore fibers (MCF), or with mode-division multiplexing (MDM) using few-mode fibers (FMF) [1; 2; 3]. Device integration is the most likely approach to accomplish power consumption and cost reduction, through a variety of proposals for realizing parallel spatial channels, including integrating parallel transmitters, receivers, and amplifiers in the same device, as well as utilizing various multimode or multicore fibers, as shown in Figure 1[4]. Amongst these proposals, the simplest solution is to solely apply fiber bundles to have the same behavior as \(N\) single-mode fibers, which can consist of a fiber ribbon or a multi-element fiber as displayed in Figure 1a. Such a multi-element fiber provides a smooth upgrade path where all existing components can be reused, and the power consumption and cost go up with the scale of the fiber bundles. In contrast, MCFs epitomize the next step in parallel fiber integration, where the adjacent fiber cores are enclosed in the same glass cladding, and it allows novel system architectures such as local oscillator sharing with less temperature-dependent fluctuation, as revealed in Figure 1b. It is worth mentioning that MCFs are usually designed with low inter-core cross-talk; thus, multiple-input multiple-output (MIMO) digital signal processing (DSP) is not required at the receiver. On the other hand, mode coupling is difficult to avoid in FMFs and, hence, necessitates the usage of MIMO DSP in the SDM systems, where orthogonal spatial modes within the same fiber core form the parallel channels, as exposed in Figure 1c. Furthermore, strongly coupled MCFs that behave like FMFs were also proposed with certain combined advantages, which shows there is great potential to combine MCF with few-mode cores in next-generation optical-fiber communication systems [5; 6]. Principally, the adaptive multi-input multi-output (MIMO) frequency-domain equalization (FDE) techniques were recognized to de-multiplex the signals upon diverse modes, as well as compensate for the differential mode group delay (DMGD) dynamically through digital signal processing (DSP) [7; 8; 9]. The FDE techniques can reduce the algorithmic complexity evocatively, in comparison with the time-domain equalization (TDE), while holding equivalent performance, amongst which the least mean squares (LMS) and recursive least squares (RLS) algorithms are most widely held and, hence, broadly studied [10; 11; 12]. Although the LMS approach has lower complexity, it suffers from severe performance deprivation and exhibits slower convergence, while encountering a large number of parameters to be simultaneously adapted in future SDM channels [13]. Alternatively, faster convergence can be accomplished by applying RLS with a more sophisticated algorithmic conformation at the price of higher complexity [14; 15]. Hereafter, it is significant to develop a MIMO-DSP FDE algorithm with the intention of achieving an RLS-level performance at an LMS-adapted complexity for an SDM-based optical communication system. Another bewildering deficiency of FMF-based optical fiber systems is the mode-dependent loss (MDL), which soars from inline component defectiveness and disorders the modal orthogonality, consequently degrading the overall capacity of MIMO channels [16]. With the aim of lessening the impact of MDL, old-fashioned tactics take account of mode scrambler and specialty fiber designs. These approaches were primarily disadvantaged with high cost, yet they cannot utterly eradicate the accumulated MDL in optical links [17]. Furthermore, the space-time trellis codes (STTC) were proposed to reduce the MDL, but they suffered from great complexity as well [18]. Henceforward, MDL is categorically another key challenge that needs to be triumphed for next-generation optical transmission systems.