Maximum Likelihood Estimation of Wiener Phase Noise Variance in $M$PSK Modulated Systems

Phase noise is one of the fundamental impairments in radar, communications, and even the integration of sensing and communications, which is necessary to be suppressed to guarantee the system performance for high-order modulations. In order to obtain precise phase estimation or effectively track phase noise, many estimation algorithms rooted in digital signal processing operate under the premise that the variance of the phase noise is known. However, in practical applications, the receiver side can hardly get the premise knowledge of the phase noise variance. Thus, accurate estimation of the phase noise variance is significantly important for not only carrier recovery, but also performance monitoring. This paper proposes a maximum likelihood (ML)-based Wiener phase noise variance estimation scheme, based on the amplitude and phase-form of the noisy received signal model for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M$</tex>-ary phase-shift keying (<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M$</tex>PSK) modulated systems. Specifically, by making full use of the explicit statistics of the received phase after raising to the Mth power, the closed-form expressions for ML estimation of the incremental phase noise and the Wiener phase noise variance are derived. The estimated mean square error is both theoretically and numerically analyzed to validate the unbiased ML estimator. Numerical results are given to verify the estimation accuracy in terms of varing signal-to-noise ratio and memory length. The proposed ML estimator is demonstrated to have precise estimation performance with low computational complexity.