The embodiments of the invention provide methods to deal with problems of cycle slips, angular skew, and residual phase noise for high-speed optical communications employing any arbitrary high-order multi-dimensional modulation formats. The embodiments use a slip process analyzer, a skew angle estimator, and a phase noise variance estimator to provide feedforward soft-decision information of a carrier phase recovery (CPE) for more accurate likelihood calculation based on a high-order hidden Markov model (HMM). The log-likelihood calculation can be done jointly in dual polarization with joint Markov state transition. Some embodiments use a kernel filter or a particle filter for log-likelihood calculation.
