Fourth-order Cumulant Research Articles

Test of universality at first order phase transitions: The Lebwohl-Lasher model.

Finite size scaling for a first order phase transition, where a continuous symmetry is broken, is tested using an approximation of Gaussian probability distributions with a phenomenological "degeneracy" factor. Predictions are compared to the data from Monte Carlo simulations of the Lebwohl-Lasher model on L × L × L simple cubic lattices. The data show that the intersection of the fourth-order cumulant of the order parameter for different lattice sizes can be expressed in terms of the relative degeneracy q = 4π of the ordered and disordered phases. This result further supports the concept of universality at first order transitions developed recently.

Nonequilibrium tricritical behavior in anisotropic XY ferromagnet driven by elliptically polarized propagating magnetic field wave

The three-dimensional anisotropic XY ferromagnet, driven by an elliptically polarized propagating magnetic field wave, has been extensively investigated by Monte Carlo simulation with the Metropolis single spin flip algorithm. Both the effects of the bilinear exchange type and the single-site anisotropies are thoroughly investigated. The time-averaged magnetization (over the complete cycle of the elliptically polarized propagating magnetic field wave) components play the role of the dynamic order parameter. For a fixed set of values of the strength of anisotropy and the field amplitudes, the system has been found to get dynamically ordered at a pseudocritical temperature. The pseudocritical temperature of such a dynamic nonequilibrium phase transition has been found to depend both on the strength of anisotropy and the amplitudes of the elliptically polarized propagating magnetic field wave. A comprehensive phase diagram is represented here in the form of an image plot of the pseudocritical temperature in the plane formed by the strength of anisotropy and field amplitudes. Interestingly, this nonequilibrium phase transition has been found to be discontinuous (first order) for higher values of the field amplitude. On the other hand, the continuous (second order) transition has been noticed for lower values of the field amplitude. Such an interesting nonequilibrium tricritical behavior has been observed in driven XY ferromagnet. The order of such a nonequilibrium phase transition has been confirmed by the thermal variation (near the transition) of the statistical distribution of the order parameter and by the thermal variation of the fourth-order Binder cumulant. In the plane formed by field amplitude and anisotropy, a tricritical line has been shown as the accompanying (and complementary) phase diagram. The dependence of the pseudocritical temperature, on the frequency and wavelength of the elliptically polarized propagating magnetic field wave, has also been reported.

Open-loop quantum control of small-size networks for high-order cumulants and cross-correlations sensing

Quantum control techniques are one of the most efficient tools for attaining high-fidelity quantum operations and a convenient approach for quantum sensing and quantum noise spectroscopy. In this work, we investigate dynamical decoupling while processing an entangling two-qubit gate based on an Ising-xx interaction, each qubit affected by pure dephasing classical correlated 1/f-noises. To evaluate the gate error, we used the Magnus expansion introducing generalized filter functions that describe decoupling while processing and allow us to derive an approximate analytic expression as a hierarchy of nested integrals of noise cumulants. The error is separated in contributions of Gaussian and non-Gaussian noise, with the corresponding generalized filter functions calculated up to the fourth order. By exploiting the properties of selected pulse sequences, we show that it is possible to extract the second-order statistics (spectrum and cross-spectrum) and to highlight non-Gaussian features contained in the fourth-order cumulant. We discuss the applicability of these results to state-of-the-art small networks based on solid-state platforms.

Debiasing Welch’s method for spectral density estimation

Abstract Welch’s method provides an estimator of the power spectral density that is statistically consistent. This is achieved by averaging over periodograms calculated from overlapping segments of a time series. For a finite-length time series, while the variance of the estimator decreases as the number of segments increases, the magnitude of the estimator’s bias increases: a bias-variance trade-off ensues when setting the segment number. We address this issue by providing a novel method for debiasing Welch’s method that maintains the computational complexity and asymptotic consistency, and leads to improved finite-sample performance. Theoretical results are given for fourth-order stationary processes with finite fourth-order moments and an absolutely convergent fourth-order cumulant function. The significant bias reduction is demonstrated with numerical simulation and an application to real-world data. Our estimator also permits irregular spacing over frequency and we demonstrate how this may be employed for signal compression and further variance reduction. The code accompanying this work is available in R and python.

A novel variable tap-length adaptive decision feedback equalization in underwater wireless communication

Fix-parameter decision feedback equalization (DFE) is prone to the challenge posed by the time-varying underwater acoustic (UWA) channels. This paper proposes a new adaptively adjusting the DFE’s tap length (TL) algorithm to solve this problem. The number of feedforward filter (FFF) taps of DFE is changed by comparing the fourth-order cumulants and second-order moments related to receiving signals with a threshold value. The feasibility of this approach is verified via theoretical analysis and numerical simulations. Further, the new system has processed the massive experimental data from two lakes. In light of the outcomes, for achieving the same bit error ratio (BER), the signal noise ratio (SNR) that the new algorithm requires is approximately 4 dB less than the variable tap-length method based on the mean square error (MSE). Additionally, this method can improve the system’s flexibility in addressing time-varying channels in underwater wireless communication.

Direct Position Determination of Non-Gaussian Sources for Multiple Nested Arrays: Discrete Fourier Transform and Taylor Compensation Algorithm.

This paper delves into the problem of direct position determination (DPD) for non-Gaussian sources. Existing DPD algorithms are hindered by their high computational complexity from exhaustive grid searches and a disregard for the received signal characteristics by multiple nested arrays (MNAs). To address these issues, the paper proposes a novel DPD algorithm for non-Gaussian sources with MNAs: the Discrete Fourier Transform (DFT) and Taylor compensation algorithm. Initially, the fourth-order cumulant matrix of the received signal is computed, and the vectorizing method is applied. Subsequently, a computationally efficient DPD cost function is proposed by leveraging a normalized DFT matrix to reduce complexity. Finally, first-order Taylor compensation is utilized to enhance the accuracy of the localization results. The superiority of the proposed algorithm is demonstrated through numerical simulation results.

Three-dimensional Heisenberg model with Dzyaloshinskii-Moriya interaction: A Monte Carlo study.

The three-dimensional classical Heisenberg model on a simple cubic lattice with Dzyaloshinskii-Moriya (DM) interactions between nearest-neighbors in all directions has been studied using Monte Carlo simulations. The Metropolis algorithm, combined with single histogram reweighting techniques and finite-size scaling analyses, has been used to obtain the thermodynamic behavior of the system in the thermodynamic limit. Simulations were performed with the same set of interaction parameters for both shifted boundary conditions (SBC) and fluctuating boundary conditions (FBC). Because of an incommensurability caused by the DM interaction, the SBC incorporated a fixed shift angle at the boundary which varies as a function of the DM interaction and lattice size. This SBC method decreases the simulation time significantly, but the distribution of states is somewhat different than that obtained with FBC. The ground state for nonzero DM interaction is a spiral configuration where the spins are restricted to lie in planes perpendicular to the DM vector. We found that this spiral configuration undergoes a conventional second-order phase transition into a disordered, paramagnetic state with the transition temperature being a function of the magnitude of the DM interaction. The limiting case with only DM interaction in the model has also been considered. The critical exponent ν, the critical exponent ratios α/ν, β/ν, γ/ν, as well as the critical temperature T_{c} and fourth-order cumulant of the order parameter U_{4}^{*} at T_{c} have been estimated for different magnitudes of DM interaction. The critical exponents and cumulants at the transition are different from those for the three-dimensional Heisenberg model, but the ratios α/ν, β/ν, γ/ν, U_{4}^{*}/ν are the same, implying that weak universality is valid for all values of DM interaction. Structure factor calculations for particular cases have been performed considering SBC and FBC in the simulations with different lattice sizes at the critical temperatures.

ИСПОЛЬЗОВАНИЕ КУМУЛЯНТНОГО АНАЛИЗА ДЛЯ РАСПОЗНАВАНИЯ QAM-МОДУЛЯЦИИ В УСЛОВИЯХ АПРИОРНОЙ НЕОПРЕДЕЛЕННОСТИ

The exponential growth in the volume of transmitted data necessitates the development of efficient methods for signal processing and analysis, particularly under conditions of a priori uncertainty. This paper presents a novel method for estimating carrier frequency and initial phase offsets in quadrature amplitude modulation (QAM) signals based on cumulant analysis. The method leverages the properties of fourth-order cumulants, which are invariant to phase shifts. An algorithm is developed that enables accurate determination of carrier frequency and initial phase values for QAM signals in the absence of a priori information about signal parameters. Experimental results demonstrate the high accuracy and efficiency of the proposed method in radio monitoring and radio frequency spectrum analysis applications.

Fazovye perekhody v chasovoy modeli s chislom sostoyaniy spina <i>q </i>= 5 na treugol'noy reshetke

The phase transitions and thermodynamic properties of the clock model with q = 5 spin states on a triangular lattice have been investigated using the Wang–Landau Monte Carlo algorithm. The phase transitions have been analyzed with the histogram method and the fourth-order Binder cumulant method. Two Berezinskii–Kosterlitz–Thouless phase transitions are shown to be observed in the ferromagnetic clock model, while a second-order phase transition has been detected in the antiferromagnetic clock model.

Класифікація моделей двокомпонентних сумішей симетричних розподілів з нульовим коефіцієнтом ексцесу

On the basis of a family of two-component mixtures of distributions, a class K of symmetric non-Gaussian distributions with a zero kurtosis coefficient is defined, which is divided into two groups and five types. The dependence of the fourth-order cumulant on the weight coefficient of the mixture is studied, as a result of which the conditions are determined under which the kurtosis coefficient of the mixture is equal to zero. The use of a two-component mixture of Subbotin distributions for modeling single-vertex symmetric distributions with a zero kurtosis coefficient is justified. Examples of symmetric non-Gaussian distributions with zero kurtosis coefficient are given. The use of class K models gives a practical opportunity at the design stage to compare the effectiveness of the developed methods and systems for non-Gaussian signals with zero coefficients of asymmetry and kurtosis processing.

Study of Interference Detection of Rail Transit Wireless Communication System Based on Fourth-Order Cyclic Cumulant.

The wireless communication system is used to provide dispatching, control, communication and other services for rail transit operations. In practice, interference from other wireless communication systems will affect the normal operation of trains, so it is an urgent problem to study the interference detection algorithms for rail transit applications. In this paper, the fourth-order cyclic cumulant (FOCC) of signals with different modulation modes is analyzed for the narrow-band wireless communications system of rail transit. Based on the analysis results, an adjacent-frequency interference detection algorithm is proposed according to the FOCC of the received signal within the predetermined cyclic frequency range. To detect interference with the same carrier frequency, a same-frequency interference detection algorithm using the relationship between the FOCC and the received power is proposed. The performance of the proposed detection algorithms in terms of correct rate and computational complexity is analyzed and compared with the traditional second-order statistical methods. Simulation results show that when an interference signal coexists with the expected signal, the correct rates of the adjacent-frequency and the same-frequency interference detection algorithms are greater than 90% when the signal-to-noise ratio (SNR) is higher than 2 dB and -4 dB, respectively. Under the practical rail transit wireless channel with multipath propagation and the Doppler effect, the correct rates of the adjacent-frequency and the same-frequency interference detection algorithms are greater than 90% when the SNR is higher than 3 dB and 7 dB, respectively. Compared with the existing second-order statistical methods, the proposed method can detect both the adjacent-frequency and the same-frequency interference when the interference signals coexist with the expected signal. Although the computational complexity of the proposed method is increased, it is acceptable in the application of rail transit wireless communication interference detection.

Retracted: Estimation Method for Mixed Near-Field and Far-Field Sources Based on Four-Order Cumulant and Polarization Sensitive Array

Retracted: Estimation Method for Mixed Near-Field and Far-Field Sources Based on Four-Order Cumulant and Polarization Sensitive Array

Critical behavior in a chiral molecular model.

Understanding the condensed-phase behavior of chiral molecules is important in biology as well as in a range of technological applications, such as the manufacture of pharmaceuticals. Here, we use molecular dynamics simulations to study a chiral four-site molecular model that exhibits a second-order symmetry-breaking phase transition from a supercritical racemic liquid into subcritical D-rich and L-rich liquids. We determine the infinite-size critical temperature using the fourth-order Binder cumulant, and we show that the finite-size scaling behavior of the order parameter is compatible with the 3D Ising universality class. We also study the spontaneous D-rich to L-rich transition at a slightly subcritical temperature of T = 0.985Tc, and our findings indicate that the free energy barrier for this transformation increases with system size as N2/3, where N is the number of molecules, consistent with a surface-dominated phenomenon. The critical behavior observed herein suggests a mechanism for chirality selection in which a liquid of chiral molecules spontaneously forms a phase enriched in one of the two enantiomers as the temperature is lowered below the critical point. Furthermore, the increasing free energy barrier with system size indicates that fluctuations between the L-rich and D-rich phases are suppressed as the size of the system increases, trapping it in one of the two enantiomerically enriched phases. Such a process could provide the basis for an alternative explanation for the origin of biological homochirality. We also conjecture the possibility of observing nucleation at subcritical temperatures under the action of a suitable chiral external field.

Sparse Non-Uniform Linear Array-Based Propagator Method for Direction of Arrival Estimation

A novel approach that does not require the number of sources as a priori is proposed to estimate the direction of arrival (DOA) based on a sparse non-uniform linear antenna array. To ensure the identifiability of the DOA, a specific configuration scheme of sparse array is designed. Based on this specific sparse array, firstly the fourth-order cumulant (FOC) is adopted to eliminate the impact imposed by Gaussian noise. Secondly, to circumvent eigenvalue decomposition or singular value decomposition, a propagator is constructed by using a Hermitian FOC matrix and a hyperparameter. Finally, a projection onto an irregular Toeplitz set is proposed to further improve estimation accuracy.

Line spectrum extraction method of underwater targets based on the polarization characteristic

In order to suppress the environmental noise interference of underwater targets, a line spectrum extraction method is proposed based on the polarization characteristic of underwater acoustic signals received by a single vector sensor. In this paper, the polarization filtering function of a fourth-order cumulant diagonal slice is developed, and the fourth-order cumulant diagonal slice characteristic spectrum corresponding to the maximum singular value decomposition is used to extract the line spectrum. The simulation and experimental results show that the proposed method can extract the line spectrum signal adaptively, which has a good abatement effect on environmental noise interference.

High-resolution frequency-difference beamforming for a short linear array.

Conventional beamforming (CBF) is a commonly employed approach for detecting and estimating the direction-of-arrival (DOA) of acoustic signals in underwater environments. However, CBF becomes ambiguous due to spatial aliasing when the received signal's half wavelength is smaller than the array spacing. Frequency-difference beamforming (FDB) allows for processing data in the lower frequency Δf without encountering spatial aliasing by utilizing the product of array data at frequency f with its complex conjugate at frequency f+Δf. However, lower frequency results in a wider mainlobe, which can lead to poorer DOA performance for short arrays. In this paper, a fourth-order cumulants FDB method and a conjugate augmented FDB method are proposed to extend an M-element uniform linear array to 2M-1 and 4M-3 elements. The proposed methods generate narrower beams and lower sidelobe levels than the original FDB for short arrays with large spacing. And by setting the signal subspace dimension reasonably, the proposed methods can improve the weak target detection ability under strong interference compared with FDB. Last, we verify the excellent performance of the proposed methods through simulations and experimental data.

Fourth-order cumulants based-least squares methods for fractional Multiple-Input-Single-Output Errors-In-Variables system identification

This paper presents new consistent methods for continuous-time Multiple-Input-Single-Output (MISO) Errors-In-Variables (EIV) systems by fractional models. The proposed idea is to use Higher-Order Statistics (HOS), such as fourth-order cumulants (foc), instead of noisy input and output measurements to obtain unbiased estimates. Firstly, all differentiation orders are assumed to be known a priori and linear coefficients are estimated. The developed estimator is based on minimizing the equation error and it is called fractional fourth-order based-least squares estimator ( $$frac-foc-ls$$ ). Secondly, the global commensurability of the fractional MISO system is considered. The $$frac-foc-ls$$ is combined with a non linear technique to estimate the global commensurate order along with linear coefficients. The developed algorithm is based on minimizing the output error and called fractional fourth-order cumulants based-least squares combined with global commensurate order optimization ( $$frac-foc-gcools$$ ). The consistency of the developed estimators, in presence of high levels of noise corrupting both the input and output measurements, is assessed through a numerical example with the help of Monte Carlo simulations.

DOA estimation for uniform circular array without the source number based on beamspace transform and higher-order cumulant

Fourth-order cumulant is one of most widely used high-order cumulant for direction of arrival (DOA) estimation due to its ability of expanding the virtual array aperture as well as suppressing Gaussian noise. To address the two-dimensional (2D) DOA estimation problem, we propose a modified MUSIC scheme for uniform circular array (UCA) in this paper. Firstly, the fourth-order cumulant of UCA is considered to construct a new propagator, resulting in the elimination of a priori knowledge of the number of signals. Secondly, the UCA is transformed by beamspace transformation, reducing the time computational complexity of the algorithm since the two-dimensional grid search and singular value decomposition are avoided. And finally a low-rank recovery algorithm is adopted to improve the accuracy regarding the limited snapshots scenario. The numerical simulations validate the superiority of the proposed method.

Adaptive Time Delay Estimation Based on Signal Preprocessing and Fourth-Order Cumulant

Adaptive Time Delay Estimation Based on Signal Preprocessing and Fourth-Order Cumulant

Ultra-short baseline positioning delay estimation based on signal preprocessing and fourth-order cumulant

To address the problem that the traditional generalized cross correlation (GCC) method in ultra-short baseline (USBL) positioning systems has a poor delay estimation accuracy in a low signal-to-noise ratio environment or complex noise background, a generalized quadratic cross correlation (GQCC) time delay estimation algorithm based on signal preprocessing and fourth-order cumulants is proposed. The noisy signal was first preprocessed using singular value decomposition and wavelet denoising. Then, the delay was calculated using an algorithm that combined the GQCC and the fourth-order cumulant. The results of the simulation and sea trial demonstrate that the proposed method is superior to the GCC method and the GQCC method and may significantly increase the positioning accuracy of the USBL system. This technique can offer a fresh technological perspective for weak signal detection and passive positioning of small targets in ocean detection.