## 2 Principle
### MIMO Equalizers: State of the Art  
Although the SDM conception was present for quite a long time, the real SDM transmission systems were not pursued in industry until recent years [22]. This is because, in MDM systems, it is assumed that signals could be immaculately conveyed under a particular mode without interfering with other spatial channels within an ideal FMF; however, in reality, that might not be the case, for the signals in numerous modes are often cross-coupled to each other due to bending, twisting, and/or fiber fabrication imperfections; henceforward, the orthogonality of modes might only be preserved for a very short distance in industrial SDM applications. In order to resolve such issue, MIMO signal processing was developed to undo channel cross-talk and recover the transmitted data quality. The coherent receiver structure for a single-carrier system with a 6 \(\times\) 6 MIMO FDE is exemplified in Figure 3, where the coefficient adaptation can be accomplished with LMS or RLS algorithms along with data-aided training. The conventional time-domain equalization (TDE) uses a finite impulse response (FIR) filter matrix adapted by the LMS algorithm. The complexity for the TDE C\({}_{TDE}\) is given by [23]  
\[C_{TDE}=3\cdot m\cdot\Delta\beta_{1}\cdot L\cdot R_{S}, \tag{1}\]  
where \(\Delta\beta_{1}\) indicates the DMGD of the fiber, \(m\) signifies the number of modes within the fiber, \(L\) represents the length of the fiber, and \(R_{S}\) denotes the symbol rate. Subsequently, the complexity of TDE scales linearly with the mode group delay of the optical link, the number of modes, and the symbol rate. Meanwhile, the complexity for FDE C\({}_{\text{FDE}}\) can be written in the form of  
\[\text{C}_{FDE}=4\cdot(m+1)\cdot\log_{2}(2\cdot\Delta\beta_{1}\cdot L\cdot R_{S} )+8\cdot m. \tag{2}\]  
It can be resolved from the equations above that FDE might substantially reduce the complexity compared with TDE. Characteristically, a block length equal to twice the equalizer length is applied for the adaptive FDE. By dint of even and odd sub-equalizers, an equivalent half-symbol period delay spacing finite impulse response (FIR) filter can be realized in the frequency domain. The required MIMO equalizer duration to compensate for DMGD in the weak-coupling regime can be transcribed as  
\[\text{N}_{DMGD}=\Delta\beta_{1}\cdot L\cdot R_{OS}\cdot R_{S}, \tag{3}\]  
where... means the ceiling function, \(\Delta\beta_{1}\) signifies the DMGD in the fiber, and R\({}_{\text{OS}}\) represents the oversampling rate. Furthermore, the MDL can be expressed as [24]  
Figure 3: The typical coherent receiver configuration for a single-carrier transmission system with a 6 \(\times\) 6 multi-input multi-output (MIMO) frequency-domain equalization (FDE) [7]. \[\text{MDL}=\underset{k=1\sim N}{\text{max}}\left({\lambda_{k}}^{2}\right)/\underset{k= 1\sim N}{\text{min}}\left({\lambda_{k}}^{2}\right), \tag{4}\]  
where \(\lambda_{k}\) stands for singular values of the coupler transfer matrix, which can be calculated by employing the singular value decomposition (SVD). To achieve the MIMO equalization, the impulse-response matrix measurement needs to be performed first. An example of the squared magnitude of the polarization-division multiplexing (PDM) SDM 6 \(\times\) 6 impulse responses for 96 km of a six-mode FMF is depicted in Figure 4, where the columns correspond to the transmitted ports and the rows correspond to the received ports [25]. In Figure 4, the sharp peaks in the subplots identify the main coupling points evidently, which are divided into four regions designated with A (the coupling between the polarization modes LP\({}_{01\text{x}}\) and LP\({}_{01\text{y}}\)), B (the coupling between the spatial and polarization modes LP\({}_{11\text{ax}}\), LP\({}_{11\text{ay}}\), LP\({}_{11\text{bx}}\), and LP\({}_{11\text{by}}\)), and C and D (the cross-talk between LP\({}_{01}\) and LP\({}_{11}\) modes).