Maximum Likelihood Phase Estimation Research Articles

Low-complexity two-stage carrier phase recovery using principal component reconstruction analysis and optimal decision maximum likelihood for PS-MQAM systems

A low-complexity two-stage carrier phase recovery (CPR) scheme based on principal component reconstruction analysis and optimal decision maximum likelihood (PCRA-OML) is proposed for probabilistic shaping (PS)-M quadrature amplitude modulation (QAM) coherent optical communication systems. In the first stage, symbols on the QPSK-shaped ring are selected, their amplitude noise is eliminated, and their amplitudes are increased, thus completing the reconstruction of the principal component. Lastly, the carrier phase is recovered using principal component phase estimation (PCPE). In the second stage, a maximum likelihood phase estimation algorithm based on optimal decision-making is employed to further enhance the stability and robustness of the scheme. The effectiveness of the proposed scheme is validated through 28 GBaud polarization division multiplexing PS-64QAM simulations and 28 GBaud PS-16QAM experiments. The experimental results indicate that in the PS-16QAM entropy of 3 bit/symbol system, when the threshold for normalized generalized mutual information is 0.9, the proposed scheme provides a 1.7 dB optical signal-to-noise ratio (OSNR) gain compared to blind phase search (BPS), and an additional 0.9 dB OSNR gain compared to the PCPE scheme. The proposed PCRA-OML scheme exhibits versatility across various shaping strengths and is less susceptible to probabilistic influences than the BPS and PCPE schemes. Additionally, under the premise of comparable performance in the PS-64QAM system, the proposed scheme reduces over 90% of the computational complexity associated with multiplication compared to BPS. In the PS-16QAM system, the proposed scheme's real multiplication complexity is only 17.8% of BPS, achieving an overall O(N) complexity.

Optimization of Partial Transmit Sequences for PAPR Reduction of OFDM Signals Without Side Information

This paper develops a novel optimization of partial transmit sequences (PTS) with phase quantization to reduce the peak-to-average-power ratio (PAPR) of orthogonal frequency-division multiplexing (OFDM) signals and enable data detection without side information. Since the formulated optimization problem is non-convex and has exponential time complexity, we employ a convex relaxation method to convert the original optimization problem into a series of convex programming whose solutions converge to a sub-optimal point satisfying the Karush-Kuhn-Tucker (KKT) conditions in polynomial time. Moreover, the obtained phase factors are quantized to enable the maximum likelihood (ML) phase estimation at the receiver, hence removing the need of sending side information. Analytical and numerical results are provided to show that our proposed PTS design achieves better PAPR reduction over existing PTS methods, while no performance degradation is incurred in data detection when the number of phase factors is properly chosen.

Characteristics of Attention Bias in Patients With Depression Disorder, and the Influence of High Frequency Transcranial Magnetic Stimulation Therapy on the Attention Bias

We propose a general model to entirely describe XPM effects induced by 16QAM channels in hybrid QPSK/16QAM wavelength division multiplexed (WDM) systems. A power spectral density (PSD) formula is presented to predict the statistical properties of XPM effects at the end of dispersion management (DM) fiber links. We derive the analytical expression of phase error variance for optimizing block length of QPSK channel coherent receiver with decision-aided (DA) maximum-likelihood (ML) phase estimation (PE). With our theoretical analysis, the optimum block length can be employed to improve the performance of coherent receiver. Bit error rate (BER) performance in QPSK channel is evaluated and compared through both theoretical derivation and Monte Carlo simulation. The results show that by using the DA-ML with optimum block length, bit signal-to-noise ratio (SNR) improvement over DA-ML with fixed block length of 10, 20 and 40 at BER of 10−3 is 0.18 dB, 0.46 dB and 0.65 dB, respectively, when in-line residual dispersion is 0 ps/nm.

Low Complexity Blind Carrier Phase Recovery for Probabilistically Shaped QAM

We propose a novel two-stage blind carrier phase recovery scheme tailored for probabilistically shaped QAM. Our algorithm is based on modifying the classic Viterbi&Viterbi (V&V) and maximum likelihood (ML) phase estimators by optimizing the blind decisions over the received symbols taking into account the probability distribution of the received signal. Our technique improves the standard V&V + ML in all conditions and outperforms the state-of-the-art blind phase search (BPS) algorithm for a constellation entropy ≤4. At the same time our implementation preserves the low complexity of standard V&V + ML. This results in a remarkable reduction in complexity of up to a factor thirty compared to BPS.

Signal-Phase Estimation for Product Array Processors

In coprime arrays and intensity processing of vector sensor arrays, product array processing can offer advantages in detection or spatial-spectrum estimation over conventional processing of the full array (e.g., in heavy-tailed noise). Product array processing generally focuses on the incoherent response formed by the product of conventional beamforming of subarrays while ignoring the phase of the signal. However, estimating signal phase is important when postdetection estimation is performed coherently over multiple beamforming epochs. In this article, a weak-signal maximum-likelihood (ML) estimator is developed to estimate the phase of a narrowband signal in product array processors from the individual subarray phase estimates, which allows forming the complex envelope of the product beam response. When the subarray signal-to-noise ratios (SNRs) are equal, the estimator simplifies to the angle of a unit-circle average (UCA) of direction vectors, also known as the sample mean direction. A Fourier series representation of the probability density function (PDF) of the subarray phase estimate is used to simplify evaluation of the PDF of the UCA estimator and its circular variance. The UCA estimator is seen to suffer a performance loss relative to the full-array ML phase estimator of no more than 1-dB SNR in independent Gaussian noise. However, increasing SNR or subarray noise correlation reduces the disparity to the point where there is little practical difference. The models of circular variance, including a mixture-based approximation for correlated subarray noise, were seen to very accurately predict phase estimation performance measured from experimental data obtained on a vertical line array in shallow water.

Cold-start of coherent optical receivers with decision-aided maximum likelihood phase estimation scheme

Abstract In coherent optical communication systems adopting decision-aided maximum likelihood (DA-ML) phase estimation, receiver initialization requires transmission of a training sequence followed by data symbols. However, such a start-up scheme is infeasible under the scenarios like burst-mode transmission and multi-receiver scheme. We propose a novel cold-start concept for DA-ML receiver without using a training sequence. By adopting the M th-power scheme, initial data symbols are estimated and further utilized to start up the DA-ML receiver effectively. Extensive numerical simulations are conducted under different situations for comparison. Results verify the feasibility of the proposed cold-start scheme. Based on conventional DA-ML, adopting cold-start can effectively start up the DA-ML receiver as well as achieve comparative bit error rate (BER) performance under burst-mode transmission and multi-receiver scenarios, with good robustness and nearly the same system complexity.

Maximum-likelihood analysis of axial displacement in fluorescence phase-shifting interferometry.

Fluorescence phase-shifting interferometry (FPSI) is an optical technique that coherently combines the phase-shifted 4π steradian emission wavefronts of a single fluorescent emitter to obtain multiple interferograms from which the emitter axial displacement can be retrieved with high precision. Here, we study the axial displacement sensitivity in 4-step FPSI within the framework of maximum-likelihood (ML) phase estimation. Using Monte-Carlo simulations, we show that regardless of the method used to preprocess the measured interferograms, the variance of the ML estimate of the axial displacement approaches the Cramér-Rao lower bound and is closely limited from above by the variance of the classical 4-step phase shifting estimator. The difference between these lower and upper bounds depends on the interferogram visibility and signal-to-noise-ratio (SNR), with a percentage change of up to 29% that yields an absolute change of a few sub-nanometers to some nanometers for SNRs larger than ~5. Our results suggest that for these levels of SNR, the use of the computationally simpler classical 4-step phase shifting estimation can be adequate to accurately determine axial displacements in FPSI, as we also experimentally verified. FPSI interferograms with lower SNR but good visibility can benefit from the use of the ML estimator, provided that the spatiotemporal phase stability of the FPSI system is high.

Flexible DAML phase estimation algorithm in coherent optical M-PSK systems

Decision-aided maximum likelihood (DAML) phase estimation has been applied in coherent optical communication systems due to its high computational efficiency. However, conventional DAML algorithm only considers constant laser phase noise over the entire block length, thus causing block length effect. Given time-varying laser phase noise, a flexible DAML phase estimation algorithm is proposed in coherent optical M-PSK systems, which can eliminate the block length effect of conventional DAML. Weighted coefficients are introduced to estimate carrier phase more accurately than the conventional algorithm. Meanwhile, the phase estimation error is derived and simulations justify its validity. Simulation results also show that the flexible DAML algorithm can eliminate the block length effect of conventional DAML and relax the requirement of laser linewidth in coherent optical M-PSK systems.

The Analysis of Spaceborne SAR Interferometry Based on Strapdown Inertial Navigation Mechanism and Maximum Likelihood Method

In this paper, a spaceborne SAR interferometry system operates on the strapdown inertial navigation mechanism, which is based on maximum likelihood phase estimation method. It has been shown using simulated data that phase estimation of cross-track multi-baseline synthetic aperture radar (SAR) interferometric data is most efficiently achieved through a maximum likelihood phase estimation method. With the help of strapdown inertial navigation mechanism and compared to simulated data, dealing with real data implies that several calibration steps be carried out to ensure that the data fits the model. It is well known that a nonlinear frequency- modulation (NLFM) chirp waveform can shape the signal’s power spectral density and provide a radar matched filter output with lower sidelobes without loss of the signal-to-noise ratio. The strapdown inertial navigation mechanism can measure the apex angle and azimuth angle, the motion attitude of the carrier can be calculated by the three-axis displacement and revolution which are measured by this system. Compares the value of spaceborne SAR interferometry with the value which is measured by strapdown inertial navigation mechanism, the measurement accuracy will be improved by the analysis of strapdown inertial navigation mechanism and maximum likelihood phase estimation method.

Analysis of a Maximum Likelihood Phase Estimation Method for Airborne Multibaseline SAR Interferometry

It has been shown using simulated data that phase estimation of cross-track multibaseline synthetic aperture radar (SAR) interferometric data was most efficiently achieved through a maximum likelihood (ML) method. In this paper, we apply and assess the ML approach on real data, acquired with an experimental Ka-band multibaseline system. Compared to simulated data, dealing with real data implies that several calibration steps be carried out to ensure that the data fit the model. A processing chain was, therefore, designed, including steps responsible for compensating for imperfections observed in the data, such as beam elevation angle dependent phase errors or phase errors caused by imperfect motion compensation. The performance of the ML phase estimation was evaluated by comparing it to results based on a coarse-to-fine (C2F) algorithm, where information from the shorter baselines was used only to unwrap the phase from the longest available baseline. For this purpose, flat areas with high coherence and homogeneous texture were selected in the acquired data. The results show that with only four looks, the noise level was marginally better with the C2F approach and contained fewer outliers. However, with more looks, the ML method consistently delivered better results: noise variance with the C2F approach was slightly but steadily larger than the variance obtained with ML method.

Performance of Subcarrier PSK Systems Using PSAM Maximum Likelihood Estimation in Turbulence Channels

A pilot-symbol assisted subcarrier phase-shift keying (PSK) free-space optical communication system is studied with maximum likelihood (ML) phase estimation. Using the phase error probability density function of ML estimation phase error, we study the error rate performance of the subcarrier PSK system with carrier phase error in lognormal and Gamma–Gamma turbulence channels. The closed-form expressions are derived for the asymptotic signal-to-noise ratio (SNR) performance loss for the imperfectly synchronized systems over turbulence channels, and the analytical expressions are verified with numerical results. These expressions reveal how the subcarrier PSK systems with ML estimated phase error are influenced by the pilot length and the modulation order in large SNR regimes.

Flexible decision-aided maximum likelihood phase estimation in coherent optical phase-shift-keying systems

Decision-aided maximum likelihood (DA-ML) phase estimation has been applied in coherent optical communication systems due to its high computational efficiency. However, conventional DA-ML scheme only assumes constant phase noise within each observation block, thus causing block length effect (BLE) which degrades system performance. In this paper, we take into account the time-varying laser phase noise and propose a flexible DA-ML phase estimation method for carrier phase recovery in coherent optical phase-shift-keying systems so as to eliminate BLE. Weighted coefficients based on ML criterion are introduced to strengthen the estimation accuracy. The statistical property of phase estimation error is derived, and the bit error rate (BER) performance is also evaluated. Numerical simulation results show that our flexible DA-ML scheme is very robust against time-varying phase noise. Compared with conventional DA-ML receiver, it can significantly reduce the phase estimation variance, improve the BER performance and increase the laser linewidth tolerance. By adopting the flexible DA-ML method with a relatively larger block length, BLE can be effectively eliminated. Thus, the BER performance can be significantly improved without carefully finding out the optimum block length or the optimum forgotten factor.

A flexible decision-aided maximum likelihood phase estimation in hybrid QPSK/OOK coherent optical WDM systems

Although decision-aided (DA) maximum likelihood (ML) phase estimation (PE) algorithm has been investigated intensively, block length effect impacts system performance and leads to the increasing of hardware complexity. In this paper, a flexible DA-ML algorithm is proposed in hybrid QPSK/OOK coherent optical wavelength division multiplexed (WDM) systems. We present a general cross phase modulation (XPM) model based on Volterra series transfer function (VSTF) method to describe XPM effects induced by OOK channels at the end of dispersion management (DM) fiber links. Based on our model, the weighted factors obtained from maximum likelihood method are introduced to eliminate the block length effect. We derive the analytical expression of phase error variance for the performance prediction of coherent receiver with the flexible DA-ML algorithm. Bit error ratio (BER) performance is evaluated and compared through both theoretical derivation and Monte Carlo (MC) simulation. The results show that our flexible DA-ML algorithm has significant improvement in performance compared with the conventional DA-ML algorithm as block length is a fixed value. Compared with the conventional DA-ML with optimum block length, our flexible DA-ML can obtain better system performance. It means our flexible DA-ML algorithm is more effective for mitigating phase noise than conventional DA-ML algorithm.

Decision-aided maximum likelihood phase estimation with optimum block length in hybrid QPSK/16QAM coherent optical WDM systems

We propose a general model to entirely describe XPM effects induced by 16QAM channels in hybrid QPSK/16QAM wavelength division multiplexed (WDM) systems. A power spectral density (PSD) formula is presented to predict the statistical properties of XPM effects at the end of dispersion management (DM) fiber links. We derive the analytical expression of phase error variance for optimizing block length of QPSK channel coherent receiver with decision-aided (DA) maximum-likelihood (ML) phase estimation (PE). With our theoretical analysis, the optimum block length can be employed to improve the performance of coherent receiver. Bit error rate (BER) performance in QPSK channel is evaluated and compared through both theoretical derivation and Monte Carlo simulation. The results show that by using the DA-ML with optimum block length, bit signal-to-noise ratio (SNR) improvement over DA-ML with fixed block length of 10, 20 and 40 at BER of 10−3 is 0.18dB, 0.46dB and 0.65dB, respectively, when in-line residual dispersion is 0ps/nm.

Mathematical Framework for Phase-Triangulation Algorithms in Distributed-Scatterer Interferometry

To improve the spatial density of measurement points of persistent-scatterer interferometry, distributed scatterer (DS) should be considered and processed. An important procedure in DS interferometry is the phase triangulation (PT). This letter introduces two modified PT algorithms (i.e., equal-weighted PT and coherence-weighted PT) and analyzes the mathematical relations between different published PT methods (i.e., the maximum-likelihood phase estimator, least squares estimator, and eigendecomposition-based phase estimators). The analysis shows that the above five PT methods share very similar mathematical forms with different weight values in the estimation procedure.

Validation of Pulsar Phase Tracking for Spacecraft Navigation

Previous theoretical work has shown that pulsars have fascinating potential as aids for autonomous, deep space navigation. In this paper, phase tracking is investigated as a technique to measure the distance traveled by a spacecraft over an observation interval. Phase tracking was experimentally validated using photons emitted by a modulated x-ray source and detected by a silicon-drift detector using the NASA Goddard X-Ray Navigation Laboratory Testbed. Models of the Crab pulsar and PSR B1821-24 were used. Three simulated, one-dimensional trajectories were considered: one stationary, one with constant velocity, and one with constant acceleration. For constant signal frequency, a maximum-likelihood phase estimator was used. Otherwise, the observation window was broken into smaller blocks over which maximum-likelihood phase estimates were looped through a digital phase-locked loop to reduce errors and estimate the Doppler frequency. The maximum-likelihood phase estimator tracked the initial phase within 0.15% for the simulations and 2.5% in the experiments. The maximum-likelihood phase estimator–digital phase-locked loop cascade tracked the phase and frequency in both the simulations and experiments. The phase error and Doppler frequency error tended toward zero in under 250 and 100 s, respectively, with the Crab pulsar.

Optical domain scheme of pilot-tone-aided carrier phase recovery for Nyquist single-carrier optical communication system

We propose an optical domain pilot-aided carrier phase recovery (PA-CPR) scheme used in a single-carrier Nyquist system. A comprehensive mathematical description of the proposed PA-CPR scheme is presented. Optimal parameters for the system are obtained through optimizations. Furthermore, we propose the two-stage PA-CPR algorithm combinations which use the maximum likelihood or constellation transformation phase estimation as the second stage just after the pilot phase compensation. We also make the evaluation and comparison of the linewidth tolerance performances among the proposed two-stage PA-CPR algorithm combinations and other typical CPR algorithms in a 28-Gbaud Nyquist polarization-division multiplexed (PDM)-16QAM optical communication system. The calculation results show that the proposed two-stage PA-CPR algorithm combinations, which have lower complexities, show better performances than others.

A new maximum-likelihood phase estimation method for X-ray pulsar signals

X-ray pulsar navigation (XPNAV) is an attractive method for autonomous navigation of deep space in the future. Currently, techniques for estimating the phase of X-ray pulsar radiation involve the maximization of the general non-convex object functions based on the average profile from the epoch folding method. This results in the suppression of useful information and highly complex computation. In this paper, a new maximum likelihood (ML) phase estimation method that directly utilizes the measured time of arrivals (TOAs) is presented. The X-ray pulsar radiation will be treated as a cyclo-stationary process and the TOAs of the photons in a period will be redefined as a new process, whose probability distribution function is the normalized standard profile of the pulsar. We demonstrate that the new process is equivalent to the generally used Poisson model. Then, the phase estimation problem is recast as a cyclic shift parameter estimation under the ML estimation, and we also put forward a parallel ML estimation method to improve the ML solution. Numerical simulation results show that the estimator described here presents a higher precision and reduces the computational complexity compared with currently used estimators.

Relative Phase Noise-Induced Phase Error and System Impairment in Pump Depletion/Nondepletion Regime

Although the pump relative intensity noise transfer phenomenon in traditional Raman amplified amplitude-shift keying fiber communication systems has been investigated intensively, the stochastic intensity fluctuation of Raman pump laser will interplay with fiber nonlinearity and leads to additional phase noise of signal in the multilevel modulated coherent optical communication system. Such phase noise is defined as relative phase noise (RPN). In this paper, we present comprehensive analysis regarding the characteristics and impairments of RPN through both theoretical derivation and Monte Carlo simulation. We derive the analytical expressions of phase error variance for M-ary PSK and 16QAM modulation formats in consideration of RPN using decision-aided maximum-likelihood phase estimation algorithm. The analysis can be used to predict RPN induced impairment theoretically. Bit error rate (BER) performance is evaluated and compared for QPSK, 8PSK, 16PSK, and 16QAM modulation formats, and the result shows that in the context of RPN 16QAM signal outperforms 16PSK which has the same spectral efficiency. Finally, we extend the analytical result obtained in pump nondepletion regime to depletion regime via numerical analysis. It is observed by comparison that Q-factor penalty will increase in depletion regime.

Phase Noise Jitter Synchronization for Coherent Optical OFDM via Pilot-Data-Aided and Wiener Filter

modulation (M-QAM) coherent optical orthogonal frequency division multiplexing (CO-OFDM) signal employing a unique pilot’s system design, Feed forward maximum likelihood phase estimator as well as Wiener filter-type Minimum Mean square error (MMSE) interpolator. The wiener filter relies upon Kolmogorov type to interpolate the estimated phase noise with M taps. A 20 Gb/s CO-OFDM via 4-QAM, 16-QAM, 64-QAM then 256-QAM modulation is applied as simulation model in Optisystem. System efficiency is evaluated throughout phase root mean square error (RMSE) calculated in degree. A comparative investigation of four different modulation techniques found that 4-QAM performs with good RMSE versus the rest of square M-QAM. A free-noise receiver, a pilot aided feed forward maximum likelihood (PA-FF-ML) receiver and a PA-FF-ML with MMSE (PA-FF-ML-MMSE) are compared. PA-FF-ML-MMSE exhibited superior performance rather than receiver using just PA-FF-ML.