Null space method for tap value initialization in constant modulus algorithm for mode division multiplexing systems

Abstract

In this paper, we propose a constant modulus algorithm (CMA) for mode division multiplexing (MDM) systems with improved convergence performance. In order to adapt to sparse channels with large differential mode group delay (DMGD) in MDM systems, the CMA adopts a variable step size similar to the improved proportionate normalized least-mean-square (IPNLMS) algorithm. In additional to that, when a singularity problem is encountered or the tap values fail to converge, it reinitializes the tap coefficients according to the tap vectors of the successfully de-multiplexed data tributaries. The proposed initialization approach is based on the fact that the channel matrix is unitary in the frequency domain in the absence of mode dependent loss (MDL), which means the channel coefficient vectors for each data tributary should be orthogonal to each other. By limiting the initial values of the taps within the null space of the complex conjugate vectors of the successfully de-multiplexed channels, singularity can be effectively avoided and the convergence of the taps is guaranteed. When the number of modes is two, the proposed algorithm becomes the constrained CMA, which has been commonly implemented in polarization division multiplexing (PDM) systems. Although the algorithm has been derived under zero MDL assumption, it is found that the proposed CMA can be quite resilient to MDL. No singularity/tap convergence failure problem occurs when the MDL is below 4 dB at both the input and the output ports.