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Updated: May 5, 2026

Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator
Published on: January 28, 2019
Improving the generalized mutual information approximation in the presence of residual phase noise for general 4D
Abstract:
Shaping modulation formats in four-dimensional (4D) signal space has been shown to be an effective approach for enhancing spectral efficiency in both additive white Gaussian noise (AWGN) channels and coherent dual-polarization (DP) optical communication systems. As residual phase noise (RPN) normally exists after imperfect phase estimation, the most commonly used mismatched 2D Gaussian receiver for two independent polarizations is suboptimal for generalized mutual information (GMI) analysis in the presence of phase noise. Existing approaches either fail to apply to full dual-polarization 4D formats or neglect phase noise contributions. In this paper, we propose a modified log-likelihood ratio (LLR) calculation to improve the GMI approximation for general 4D modulation formats in the presence of RPN. The enhanced GMI is derived by separating amplitude and phase in polar coordinates, thereby fully accounting for phase noise arising from both RPN and AWGN. As an application of the proposed LLR approximation, we investigate the performance improvement for soft-decision forward error correction (FEC). Compared to conventional methods, our modified GMI yields more accurate predictions of the post-FEC bit error rate (BER) in RPN-impaired systems. Moreover, numerical results are given in detail to show the comparison of different GMI approximations in the presence of phase noise and the phase noise tolerance for various 4D modulation formats. Finally, we demonstrate that under severe RPN, re-optimizing the entire 4D modulation format can enhance the tolerance to phase noise and mitigate GMI loss.
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