Shaping Distribution Identification of Phase Rotated Probabilistically Shaped Signals With Radius-Based Expectation Maximization

A radius-based expectation maximization (R-EM) algorithm is proposed for shaping distribution identification (SDI) with phase rotated probabilistically shaped (PS) signals in a flexible transceiver. Since the proposed SDI algorithm is independent of carrier phase, shaping distribution (SD)-aware algorithms for frequency offset estimation and carrier phase recovery can be implemented after SDI to fully leverage the shaping gain. To remove the impact of the carrier phase on the calculation of posterior probability, the channel is modeled as a function of the radius rather than the complex symbol as in the conventional EM SDI. An approximation method is employed to analytically derive the parameter updating equation. A low-complexity initialization step is also proposed for the R-EM algorithm. The performance of the proposed algorithm is investigated numerically via Monte-Carlo simulation of a polarization division multiplexing PS-64QAM system. The optimization of the R-EM algorithm and the impacts of the shaping granularity, the block length and the signal-to-noise ratio (SNR) estimation accuracy on the success rates are analyzed. In the case of ideal SNR estimation, high success rates (≥99.6%) are achieved at operation points (i.e., SNR threshold at the normalized generalized mutual information of 0.9) with an entropy granularity of 0.21 bit/symbol. For a coarser entropy granularity (0.42 bit/symbol), even under a SNR estimation error up to ±1 dB, the proposed algorithm provides robust performance (≥99.6% success rate) at operation points. The results indicate that the R-EM algorithm is a promising SDI method for elastic optical networks with PS signals.