Viterbi-Viterbi Carrier Phase Estimation for Multi-Ring $M$-APSK With Wiener Carrier Phase Noise and Its Performance

High-order modulations are currently widely used in optical communications to increase the data rate. However, due to the existence of laser phase noise, the performance of high-order modulated systems will be greatly degraded. Therefore, phase estimation (PE) methods and laser linewidth tolerance analysis are particularly important in practice. In this paper, we first propose a common refined Viterbi-Viterbi carrier phase estimator for multi-ring $M$ -ary amplitude phase-shift keying ( $M$ -APSK) with Wiener phase noise. The design is based on the linear minimum mean square error (LMMSE) criterion to optimize the weight coefficients with respect to the statistics of Wiener phase noise and additive, white, Gaussian noise. For simple implementation, approximate ring-detection-based and average-energy-based LMMSE estimators are further derived for multi-ring modulations. The laser linewidth tolerance in Wiener phase noise is analyzed thoroughly compared to the popular PE methods, e.g., decision-aided maximum likelihood and QPSK partitioning. Numerical results show in detail the receiver sensitivity penalties as a function of the estimation memory length, and verify that the proposed estimators do not suffer from the block length effect and perform much better at high signal-to-noise ratio (SNR). They are suitable for coherent demodulation of a multi-ring signal constellation in high SNR or fast-varying phase noise.