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$M$-ary phase-shift keying ($M$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.

Polarization-division multiplexing (PDM) in coherent optical communications enhances system capacity but is vulnerable to various channel distortions including transmitter and receiver in-phase/quadrature (IQ) mismatch, rotation of state of polarization (RSOP), frequency offset (FO) and phase noise (PN), which significantly degrade the system performance. To address these challenges, we propose a novel approach using sequential number-theoretic optimization (SNTO) for the joint estimation of these distortions. We further introduce a decision-aided scheme with a window-split structure to accurately track and compensate for time-varying RSOP and PN, thereby implementing signal detection. Through comprehensive mean squared error (MSE) and bit error rate (BER) analysis under different signal-to-noise ratio (SNR) conditions and varying RSOP speeds, our method demonstrates high precision and effectiveness. The SNTO-based algorithm maintains robust performance with superior estimation accuracy and resilience against ultra-fast RSOP. This work introduces an innovative solution for high-dimensional channel equalization in coherent optical PDM systems with not only effectiveness but also robustness.

Coherent optical polarization-division multiplexing (PDM) is a promising technique to enhance spectral efficiency by transmitting data streams independently using orthogonal polarizations of light. However, in challenging conditions like lightning strikes, rotation of state of polarization (RSOP) due to Kerr and Faraday effects can cause significant signal distortions at speeds up to Mrad/s. In this paper, we propose a maximum likelihood (ML) and an expectation maximization (EM) estimator for the estimation of RSOP and laser phase noise (PN). The ML estimator utilizes pilot symbols for deriving explicit RSOP estimates, while the EM estimator iteratively refines estimates by incorporating unknown transmitted data as latent variables, which approximates the ML estimator’s performance while enhancing spectral efficiency. A decision-aided (DA) scheme is subsequently proposed for dynamic tracking of RSOP and PN, as well as signal detection. The Cramér-Rao lower bounds (CRLBs) are derived for performance evaluation, and simulation results demonstrate the superior RSOP tracking performance of our proposed estimators over conventional methods, with capable tracking speed beyond 100 Mrad/s. The proposed methods also exhibit robustness to impairments such as laser PN, polarization dependent loss (PDL), fiber nonlinearity and residual chromatic dispersion (CD).

The performance of modern communication and radar systems can suffer severe performance degradation from oscillator phase noise. Existing receivers commonly assume that the carrier phase is a constant over a window of a few symbol intervals and average the received signals over these intervals to obtain an estimate of the carrier phase for data demodulation. For mmWave/THz wireless and optical communications with fast time-varying phase noise, this quasi-static carrier phase assumption will no longer be applicable to ensure optimum estimation and compensation for the unknown carrier phase. We present here an Mth-power receiver for$M$-ary phase-shift keying ($M\text{PSK}$) modulation and a Wiener process carrier phase model that is applicable in many situations, especially in optical communications. The receiver can eliminate the$M$-ary phase modulation by raising the received signal samples (one sample per symbol interval) with noise to the power of$M$. The resulting unmodulated phase samples then enable the receiver to perform joint maximum likelihood estimation and maximum a posteriori probability estimation of the unknown initial carrier phase and Wiener carrier phase noise process. The estimation performance improves with a relatively small data block length for any given signal-to-noise ratio and Wiener phase noise variance, and this leads to better error probability performance in data detection. Simulation results are obtained for the estimation mean square error of the noisy carrier phase and the error probability of the detected$M\text{PSK}$symbols.

Based on globally supported radial basis functions, this paper presents a hierarchical approach to 3D scattered data interpolation. Given a scattered data set distributed along a surface, we first obtain a sparse approximation of the scattered data set by filtering the scattered data with two different steps, and identify the characteristic points by the measurements of bounding cube and the distance between the points and the interpolating surface. Then using multi-level hierarchy of point sets, the accuracy of interpolating surface can be gradually improved. On the basis of effective selection of the off-surface normal points, the number of constraint conditions is reduced, and trivial solutions of the interpolating equations can be avoided. Above all, we can identify the characteristic points in the model with less data points, and obtain interpolating surface with higher fitting accuracy by solving equations with lower order. The numerical experiments show that the method is more efficient and easy to implement.