Coordinate Descent Detection Algorithms Using Adaptive Moment Estimation for MDL-Impaired Mode Division Multiplexing Systems

The performance of mode division multiplexing (MDM) systems is limited by mode-dependent loss (MDL), which seriously deteriorates the unitarity of the transmission matrix, making the traditional detection algorithms ineffective. In order to mitigate the effect of MDL in MDM transmission and improve the efficiency of digital signal processing at the receiver, a novel coordinate descent (CD)-based box-constrained detection algorithm using adaptive moment estimate (CDbox-Adam) is proposed. In the process of each coordinate update for the CDbox detection algorithm, the adaptive moment estimation (Adam) method adaptively updates the step size for different parameters from the first moment estimate and the second moment estimate of the gradients. To simplify the process of each coordinate updating of the CDbox-Adam algorithm, the CDbox-Adamax detection algorithm is also proposed. The CDbox-Adamax detection algorithm generalizes the 2-norm update rule of weights to the infinite norm. At the same time, an early stop criterion is proposed to efficiently avoid unnecessary iterations. Simulation results show that the early stop criterion reduces the number of iterations of the proposed detection algorithms. For a 12-mode MDM system impaired by MDL, the average number of iterations of the proposed detection algorithms is less than the CDbox detection algorithm. Thus, the proposed detection algorithms present lower computational complexity than the CDbox detection algorithm. Furthermore, compared with the conventional minimum mean square error (MMSE) detection algorithm, the proposed detection algorithms, CDbox-Adam and CDbox-Adamax, gain signal to noise ratio (SNR) improvement of 2.3 dB and 1.6 dB at BER = $10^{-5}$ when MDL = 5 dB with QPSK and 16QAM, respectively. In the presence of MDL = 5 dB and 10 dB, when compared with the proposed detection algorithms with perfect CSI, the proposed detection algorithms with LS channel estimation requires 5.1 dB and 2.5 dB more of SNR to get the same BER performance, respectively. Moreover, the results suggest that at high SNR the proposed detection algorithms outperform the least mean squares (LMS) and recursive least squares (RLS) algorithms for modal dispersion (MD) and MDL-impaired MDM systems by using least squares (LS) channel estimation, which indicate their strong potential for MDM transmissions.