Simple, Unified, and Accurate Prediction of Error Probability for Higher Order MPSK/MDPSK With Phase Noise in Optical Communications

Abstract

Higher order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math>$M$</tex-math></inline-formula> -ary phase shift keying (MPSK) and differential phase shift keying (MDPSK) ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math>$M\ge 4$</tex-math></inline-formula> ) are used in optical communications for increased spectral efficiency. This paper proposes a unified and systematic approach to predicting the error probability of MPSK with phase reference error (PRE) and MDPSK with residual phase noise (RPN) in additive white Gaussian noise (AWGN) channel. It is shown that AWGN leads to an equivalent additive observation phase noise (AOPN), whose probability density function (pdf) conditioned on knowing the received signal magnitude is Tikhonov. For high signal-to-noise ratio (SNR), the Tikhonov pdf can be accurately approximated by a Gaussian pdf. This AOPN can be combined with the PRE/RPN, and the distribution of the combined phase noise (AOPN + PRE/RPN) facilitates the computation of the probability of the received signal phasor falling in any sector in the complex plane. It thus enables us to express the symbol error probability of MPSK/MDPSK with phase noise as one Gaussian <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math>$Q$</tex-math></inline-formula> -function. Moreover, it facilitates the analysis of bit error probability (BEP) with Gray code mapping for MPSK/MDPSK. All the new BEP expressions obtained are linear combinations of single Gaussian <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math>$Q$</tex-math></inline-formula> -functions, and do not involve the product of Gaussian <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math>$Q$</tex-math></inline-formula> -functions. It is shown that our Gaussian AOPN <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math>$+$</tex-math></inline-formula> PRE/RPN model provides a simpler and quicker way to accurately estimate the error performance as a function of the phase error variance. Our unified approach is increasingly more accurate as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math>$M$</tex-math></inline-formula> increases.