# Recent Progress on Novel DSP Techniques for Mode Division Multiplexing Systems: A Review
## 5 Novel DSP Techniques for MIMO Systems
### Digital Back-Propagation Algorithm (DBP)  
It is well known to us that the DBP algorithm is the most common DSP method to compensate nonlinearities in single mode fiber. However, as mode coupling exists in FMFs, traditional DBP cannot be successfully used. The extension of single-mode split-step Fourier method (SSFM) based DBP has recently been proposed to solve the multimode propagation problem over FMFs [30; 31]. As illustrated in Figure 14, a symmetric SSFM is implemented, where the nonlinear effects are considered in the middle of the segment \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Classification** & \multicolumn{2}{c}{**Receiver-Side**} & \multicolumn{2}{c}{**Transmitter-Side**} \\ \hline
**Name** & **ML** & **IC** & **STC** \\ \hline Complexity & High & Low & Moderate (excluding equalizer) \\ \hline Advantages & Optimal equalizer & Suboptimal equalizer & \
\begin{tabular}{c} Easily compatible with other solutions; \\ Coder technique \\ \end{tabular} \\ \hline Disadvantages & High complexity & Worse performance &
\begin{tabular}{c} Loss of multiplexing \\ gain; Required for \\ extra coder and \\ equalizer \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison between different digital solutions for mitigating MDL effect. instead of the segment boundary. By utilizing this structure, the generalized nonlinear Schrodinger equation (GNLSE) can be simplified according to the strength of the linear mode coupling. Following the strategy above, two original Manakov-like equations with nonlinear coefficients are derived for FMFs in two extreme coupling regimes. In the weak coupling (WC) regime, we only need to consider the average value over birefringence fluctuations, which leads to a reduced intramodal degeneracy factor of 8/9 and a reduced intermodal degeneracy factor of 4/3 [75]. In the strong coupling (SC) regime, all propagation modes are included on average. A 19 channels wavelength-division multiplexed (WDM) MDM system is simulated to study the performance as shown in Figure 15[31]. There are six LP modes with dual-polarization in each channel, as 14 Gbaud polarization-multiplexed 16 QAM, the overall data rate of 12.8 Tb/s is transmitted over 12 spans of 20 km 6-mode FMF. In the receiver side, DBP is implemented by utilizing a fixed step size of 100 m, subsequently, chromatic dispersion (CD) compensation, time synchronization, channel estimation is performed, and finally Q-factor is calculated. Figure 16 shows the Q-factor performance with launched power in three different case. In Figure 16a, both WC-DBP and SC-DBP have a Q-factor improvement at low XT, which means that the Manakov approximations are reasonable. But it can be found that only SC-DBP can perform well at high XT shown in Figure 16b, which can be explained by the nonlinear coefficients in WC-DBP is larger than the actual channel leading to overcompensation. As shown in Figure 16c, when GD management with high DMD is utilized, WC-DBP can still perform well in low XT. Recently, DBP is realized in a deep neural network architecture [32]. The architecture of this deep neural network (DNN) is shown in Figure 17, where the inputs are the received signal samples and the outputs are the estimated symbols. The weight \(\mathbf{W}_{k}\) represent linear effects in transmission, such as mode dependent loss and chromatic dispersion. And \(\sigma_{k}\) is the nonlinear operator which represents the nonlinear phase derotation. A polarization rotation matrix \(\mathbf{R}_{k}\) is appended at the \(k\)-th stage of the DBP operation and can be optimized from training data without any knowledge of the principle states of polarization (PSP). Thus, all the linear and nonlinear parameters in traditional DBP can be optimized in the DNN. Figure 14: The symmetric SSFM for FMFs with linear coupling. Figure 15: Simulated WDM-MDM system based on new DBP. The PDM-WDM transmission system is experimentally constructed to investigate the performance of this DNN [32].