# Recent Progress on Novel DSP Techniques for Mode Division Multiplexing Systems: A Review
## 3 RLS MIMO Algorithms and Its Application on Equalization of TV Effects
So, at the beginning of the equalization, we set \(\lambda\) to be a lower value for faster convergence. When the filter matrix converges to near steady-state solution through enough iterations, \(\lambda\) reaches \(\lambda_{max}\) to ensure low misadjustment. Our proposed VFF-RLS-FDE algorithm was utilized to a 6 \(\times\) 6 MIMO-MDM transmission data. The experimental results showed that the algorithm achieve better performance than conventional LMS and the RLS has a fixed  
Figure 4: Block diagram of the RLS FDE. \(\lambda\) equaling 0.9. It can maintain faster convergence speed as shown in Figure 5a, which plots the mean square error (MSE) curves in a 600-km MDM transmission. The part of beginning convergence is zoomed in in the inset, where the green line of dashes stands for the target MSE level. For better comparison under the same standard, the desirable MSE is fixed to 1.9 \(\times\) 10\({}^{-5}\), which represents an MSE level that could be achieved by both the VFF-RLS/RLS and LMS algorithms even through 1000-km MDM transmission. In order to further highlight the improvement of the convergence speed of VFF-RLS/RLS, we defined the ratio of convergence speed (RCS) as the ratio of the number of training symbols needed by LMS algorithm to that by VFF-RLS or RLS algorithm for converging to achieve the same MSE. Figure 5b shows the RCS values at different transmission distances for conventional RLS and VFF-RLS, where the solid and dotted lines are cubic polynomial fits for RLS and VFF-RLS, respectively. Faster convergence speed was obtained for VFF-RLS/RLS at longer transmission distance. The advantage in convergence speed could be explained by that lower SNR caused by longer transmission distance requires more robust performance to reach the target MSE, which condition could be satisfied by VFF RLS or RLS. In particular, after transmission excessing 800-km, compared to conventional LMS algorithm, the convergence speed is improved over 90% by the VFF-RLS and RLS algorithm. Most notably, the RCS of VFF-RLS-FDE is 15.3% higher than conventional RLS-FDE at 600 km, which means that the VFF-RLS algorithm can also obtain a significant RCS improvement over the conventional RLS algorithm. In addition to the computational complexity, performance and convergence speed, the tracking speed of the equalization algorithms also attracts much attention for long-haul and high-dimension MDM transmission, due to the TV effects in FMF. On the one hand, the evolution rate of transmission matrix of FMFs scales with the number of spatial modes [18]. Compared with SMFs, the evolution speed of two-mode fiber is one order of magnitude faster. What's more, the rotation speed of Stokes vector of four-core coupled-core fiber on the Poincare sphere may exceed 100 krad/s [48]. On the other hand, the evolution rate of transmission matrix also scales with fiber length [18]. Therefore, for long-haul MDM transmission, which are easier to be disturbed by perturbations, dealing with TV MDM channels and developing faster-tracking MIMO equalizers are more necessary. In addition, there are no effective optical devices to control or compensate the time varying modal dynamics of MDM system. Therefore, people will rely more on DSP techniques to deal with TV MDM channels. Since RLS algorithm has been proposed theoretically for the MDM system to achieve a fast convergence speed and tracking capability, we had further extended the VFF-RLS algorithm to the equalization of TV channels in FMF links to  
Figure 5: Convergence speed comparison between VFF-RLS/RLS and LMS: (**a**) MSE convergence at 600-km vs. symbols; (**b**) Ratio of convergence speed of LMS and VFF-RLS/RLS vs. Distance. track the constantly changing transfer matrix [49], which is FD error-dependent VFF-RLS algorithm (ED-VFF-RLS). The forgetting factor of our previously proposed ED-VFF-RLS algorithm could be decided automatically by error signals.