Error performance of optically preamplified hybrid BPSK-PPM systems with transmitter and receiver imperfections

In this paper, we investigate the impact of the transmitter finite extinction ratio and the receiver carrier recovery phase offset on the error performance of two optically preamplified hybrid M-ary pulse position modulation (PPM) systems with coherent detection. The first system, referred to as PB-mPPM, combines polarization division multiplexing (PDM) with binary phase-shift keying and M-ary PPM, and the other system, referred to as PQ-mPPM, combines PDM with quadrature phase-shift keying and M-ary PPM. We provide new expressions for the probability of bit error for PB-mPPM and PQ-mPPM under finite extinction ratios and phase offset. The extinction ratio study indicates that the coherent systems PB-mPPM and PQ-mPPM outperform the direct-detection ones. It also shows that at Pb=10-9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_b=10^{-9}$$\end{document} PB-mPPM has a slight advantage over PQ-mPPM. For example, for a symbol size M=16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=16$$\end{document} and extinction ratio r=30\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=30$$\end{document} dB, PB-mPPM requires 0.6 dB less SNR per bit than PQ-mPPM to achieve Pb=10-9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_b=10^{-9}$$\end{document}. This investigation demonstrates that PB-mPPM is less complex and less sensitive to the variations of the offset angle θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} than PQ-mPPM. For instance, for M=16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=16$$\end{document}, r=30\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=30$$\end{document} dB, and θ=10∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =10^{\circ }$$\end{document} PB-mPPM requires 1.6 dB less than PQ-mPPM to achieve Pb=10-9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_b=10^{-9}$$\end{document}. However, PB-mPPM enhanced robustness to phase offset comes at the expense of a reduced bandwidth efficiency when compared to PQ-mPPM. For example, for M=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=2$$\end{document} its bandwidth efficiency is 60 % that of PQ-mPPM and ≈86%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx 86\,\%$$\end{document} for M=1024\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=1024$$\end{document}. For these reasons, PB-mPPM can be considered a reasonable design trade-off for M-ary PPM systems.