Fourth-order Cumulant Research Articles

Cumulant-based DOA Estimation of Noncircular Signals Against Unknown Mutual Coupling.

To effectively find the direction of non-circular signals received by a uniform linear array (ULA) in the presence of non-negligible perturbations between array elements, i.e., mutual coupling, in colored noise, a direction of arrival (DOA) estimation approach in the context of high order statistics is proposed in this correspondence. Exploiting the non-circularity hidden behind a certain class of wireless communication signals to build up an augmented cumulant matrix, and carrying out a reformulation of the distorted steering vector to extract the angular information from the unknown mutual coupling, by exploiting the characteristic of mutual coupling, i.e., a limited operating range and an inverse relation of coupling effects to interspace, we develop a MUSIC-like estimator based on the rank-reduction (RARE) technique to directly determine directions of incident signals without mutual coupling compensation. Besides, we provide a solution to the problem of coherency between signals and mutual coupling between sensors co-existing, by selecting a middle sub-array to mitigate the undesirable effects and exploiting the rotation-invariant property to blindly separate the coherent signals into different groups to enhance the degrees of freedom. Compared with the existing robust DOA methods to the unknown mutual coupling under the framework of fourth-order cumulants (FOC), our work takes advantage of the larger virtual array and is able to resolve more signals due to greater degrees of freedom. Additionally, as the effective aperture is virtually extended, the developed estimator can achieve better performance under scenarios with high degree of mutual coupling between two sensors. Simulation results demonstrate the validity and efficiency of the proposed method.

Estimation in additive models and ANOVA-like applications

ABSTRACT A well-known property of cumulant generating function is used to estimate the first four order cumulants, using least-squares estimators. In the case of additive models, empirical best linear unbiased predictors are also obtained. Pairs of independent and identically distributed models associated with the treatments of a base design are used to obtain unbiased estimators for the fourth-order cumulants. An application to real data is presented, showing the good behaviour of the least-squares estimators and the great flexibility of our approach.

A Virtual Nested MIMO Array Exploiting Fourth Order Difference Coarray

In active direction of arrivals (DOA) sensing, the multiple-input multiple-output (MIMO) radar can lead to more degrees of freedom (DOFs). The DOFs can be further increased when the fourth order cumulants are used in MIMO. However, most array geometries are not designed for fourth order cumulants based MIMO and this array geometry optimization problem is still an open issue. To have a simple guideline for this optimization problem, we reformulate it into two separate problems, which are a fourth order sum coarray optimization problem and a second order difference coarray problem. The fourth order sum coarray optimization is realized by solving a postage stamp problem and a virtual nested array can be obtained. The number of elements in this virtual nested array can be significantly increased with only a few physical sensors. The corresponding inter-element spacing of virtual nested array is expanded simultaneously. Then the second order difference coarray is adopted to the virtual nested array to further increase the DOFs. Numerical results validate the high DOFs achieved by our proposition.

Near-Field Noncircular Sources Localization Based on Fourth-Order Cumulant

This paper presents a cumulant-based algorithm for near-field noncircular sources localization with a symmetric uniform linear array. It first constructs three extended matrices. Then, based on the rotational invariance property in extended cumulant-domain signal subspace, coarse direction-of-arrival (DOA) estimates are obtained. With the course results used as reference, the algorithm is able to disambiguate the cyclic phase ambiguities when the equivalent sub-array spacing exceeds a half wavelength. Thus, higher precision DOA estimates can be obtained. Finally, with the estimated DOAs, only one-dimensional searching is needed to obtain the range estimates via rank-reduction algorithm. The proposed algorithm avoids two-dimensional searching and parameters pairing. In addition, compared with the existing near-field noncircular sources localization algorithm, the significant characteristic of the proposed algorithm is that it exploits cumulant and the noncircularity of signal to achieve extended steering vector. Cumulant is insensitive to Gaussian (white or color) noise. Consequently, the proposed one provides DOA estimates with improved precision especially at low signal-to-noise ratios. Furthermore, it doubles the number of detectable sources. Total least square ESPRIT algorithm is exploited to yield search free estimates of near-field bearing parameters. Computer simulations are carried out to demonstrate the superiority of the proposed algorithm.

Фазовые переходы в низкоразмерных неупорядоченных моделях Поттса

Abstract The Monte Carlo method is used to study phase transitions in disordered two-dimensional Potts models in which disorder is realized as nonmagnetic impurities. The temperature dependences of the thermodynamic parameters have been calculated for specific heat C , susceptibility χ, and fourth-order Binder cumulants in the dependence on spin concentration p . The calculations have been performed for the system with periodic boundary conditions. The systems with linear sizes L × L = N , L = 20–160 are considered. The incorporation of nonmagnetic impurities to a spin system is shown to possibly lead to a change of the first-order phase transition to a second-order phase transition. The problem of the universality classes of the critical behavior of low-dimensional diluted systems is discussed.

Фазовые переходы в двумерной слабо разбавленной пятивершинной модели Поттса

Computer simulation was used to study phase transitions in the two-dimensional weakly diluted Potts model on a square lattice at q=5. Systems with linear dimensions L×L=N, L=10-120 are considered. Based on fourth-order Binder cumulants, it was shown that the introduction of nonmagnetic impurities into the spin system described by the two-dimensional Potts model with q=5 leads to a change in the first-order phase transition to the second-order phase transition.

Фазовые переходы и термодинамические свойства модели Поттса с числом состояний спина q=4 на решетке кагоме

Phase transitions and thermodynamic properties of the 2D ferromagnetic Potts model with the number of spin states q = 4 on a kagome lattice are investigated by the Mote Carlo method based on the Wang-Landau algorithm. The orders of the phase transitions are investigated using the Binder fourth-order cumulant method and histogram analysis of data. It is established that a first-order transition is observed in the model being investigated.

Исследование фазовых переходов и термодинамических свойств модели Поттса с C-=SUB=-q-=/SUB=-=4 на гексагональной решетке с взаимодействиями вторых ближайших соседей

Magnetic structures of the ground state, phase transitions and thermodynamic properties of the 2D ferromagnetic Potts model with the number of spin states q = 4 on a hexagonal lattice with next-nearest neighbors interactions are investigated by the Monte Carlo method based on the Wang-Landau algorithm. It is shown that taking into account the antiferromagnetic interactions of the next-nearest neighbors leads to the appearance of frustration and a violation of magnetic ordering. The orders of the phase transitions are investigated using the Binder fourth-order cumulant method and histogram analysis of data. It is established that a first-order transition is observed in the model being investigated.

High Accuracy Near-Field Localization Algorithm at Low SNR Using Fourth-Order Cumulant

Source localization is a key technology in array signal processing. However, it has a serious problem that the direction of arrival (DOA) estimation accuracy is reduced at low signal-to-noise ratio (SNR). Thus, this Letter proposes an improved near-field multiple signal classification (INF-MUSIC) algorithm to improve DOA estimation accuracy at low SNR. Firstly, the fourth-order cumulant is used to construct a Hermitian matrix with only DOA information. Secondly, the spatial spectrum has physical property that is a breakpoint and approaches a larger value at the DOA, thus first derivative of spatial spectrum is used to improve DOA estimation accuracy at low SNR. Finally, the corresponding ranges are estimated one by one by the one-dimensional MUSIC algorithm. The simulation results show that the INF-MUSIC algorithm makes the DOA estimation accuracy improved by 2° at low SNR.

Cumulant-Based Biquaternion Approach to Direction Finding of Wideband Signals

A direction-of-arrival (DOA) estimator is presented for statistically independent non-Gaussian and non-circular wideband signals in the presence of Gaussian noise (possibly spatially colored). With joint exploitation of non-Gaussianity and non-circularity of incident signals, enhanced aperture extension is achieved. The incorporation of the aligned biquaternion adjoint subspace augmentation and the aligned virtual array synthesis within the biquaternion framework, using both the standard and the complementary fourth-order cumulant matrices, enables wideband underdetermined DOA estimation: handling more signals than the number of sensors. The efficacy of the presented method has been demonstrated using simulated data.

Real‐valued off‐grid DOA estimation based on fourth‐order cumulants using sparse Bayesian learning in spatial coloured noise

In this study, the authors address the problem of off-grid sparsity-inducing direction-of-arrival (DOA) estimation in the context of real-valued fourth-order cumulants (FOC) in the presence of spatially coloured noise. Firstly, a selection matrix is constructed to eliminate the redundant data of FOC and rearrange the data in the de-redundant FOC matrix to facilitate real processing. Then a new virtual overcomplete dictionary is constructed with coupling symmetric property by linear transform with the selection matrix. Next, the FOC matrix is transformed into a real-valued matrix via a unitary transformation which can be sparsely represented by a real-valued virtual overcomplete dictionary. The real-valued sparse model is vectorised for transforming to a single measurement vector (SMV) model, and the redundant data in the vector model is further removed by another selection matrix. Finally, an off-grid sparse model based on the real-valued SMV is established and solved by utilising the SBL strategy. The proposed method not only reduces the computational complexity but also obtains an extended-aperture array with increased degrees of freedom which yields high resolution, and provides superiority in performance and robustness against coloured noise. The simulation results demonstrate the effectiveness of the proposed method.

Phase transitions in the Ising model on a triangular lattice with different values of interlayer exchange interaction

The Monte Carlo method was used to study the phase transitions and thermodynamic properties of the three-dimensional antiferromagnetic Ising model on a layered triangular lattice at various values of interlayer exchange interaction. A number of investigations have been carried out for the ratios of the intralayer (J1) and interlayer (J2) exchange interactions within the range of r = J2/J1 = 0.01–1.0. The nature of the phase transitions was studied using the fourth-order Binder cumulant method and histogram data analysis. It has been shown that in the considered range r, the transition from a disordered phase to an ordered phase is a second-order phase transition.

Fourth-order direction finding in antenna arrays with partial channel gain/phase calibration

In this paper, we consider the problem of direction finding in partly calibrated uniform linear array (ULA) system with channel gain/phase mismatch, by taking advantage of the high-order statistics (HOS) of the array observations. In particular, we devise a (2M−1)×(2M−1) fourth-order cumulant (FOC) matrix for an M-element ULA and exploit it to estimate the directions-of-arrival (DOAs) with subspace techniques. The proposed FOC direction finding method is computationally attractive without spatial spectrum search and robust against spatially correlated noise. It is also able to achieve better DOA estimation performance than the existing method, as illustrated by numerical simulations.

Consensus formation in continuous opinion dynamics on quasiperiodic lattices

We studied the Biswas–Chatterjee–Sen (BCS) consensus formation model, also known as the Kinetic Continuous Opinion Dynamics (KCOD) model on quasiperiodic lattices by using Kinetic Monte Carlo simulations and Finite Size Scaling technique. Our results are consistent with a continuous phase transition, controlled by an external noise. We obtained the order parameter [Formula: see text], defined as the averaged opinion, the fourth-order Binder cumulant [Formula: see text] and susceptibility [Formula: see text] as functions of the noise parameter. We estimated the critical noises for Penrose and Ammann–Beenker lattices. We also considered seven-fold and nine-fold quasiperiodic lattices and estimated the respective critical noises as well. Irrespective of rotational and translational long-range order of the lattice, the system falls in the same universality class of the two-dimensional Ising model. Quasiperiodic order is irrelevant and it does not change any critical exponents for BCS model.

Some remarks on Koziol’s kurtosis

James Koziol, in his 1987 and 1989 papers, proposed to use the sums of either the squared fourth-order cumulants or moments as test statistics for multivariate normality. His proposals are by far less popular than Mardia’s measure of multivariate kurtosis, that is the fourth moment of the Mahalanobis distance of a random vector from its mean. We investigate some properties of Koziol’s measures of multivariate kurtosis which motivate their use in statistical practice. Firstly, we show some of their connections with Mahalanobis angles. Secondly, we use inequalities to highlight their connections with other measures of multivariate skewness and kurtosis. Thirdly, we obtain their analytical formulae for some well-known multivariate statistical models. Simple examples illustrate the interpretation of Koziol’s measures of multivariate kurtosis and detect a wrong statement about them which appeared in the statistical literature. We suggest that Mardia’s and Koziol’s measures of kurtosis should be used together to detect interesting data structures.

The QCD topological charge and its thermal dependence: the role of the η′

We analyze the contribution of the η′ (958) meson in the first two non-trivial moments of the QCD topological charge distribution, namely, the topological susceptibility and the fourth-order cumulant of the vacuum energy density. We perform our study within U(3) Chiral Perturbation Theory up to next-to-next-to-leading order in the combined chiral and large-Nc expansion. We also describe the temperature dependence of these two quantities and compare them with previous analyses in the literature. In particular, we discuss the validity of the thermal scaling of the topological susceptibility with the quark condensate, which is intimately connected with a Ward Identity relating both quantities. We also consider isospin breaking corrections from the vacuum misalignment at leading order in the U(3) framework.

ICA Model o f Densities f or Images ( ICA DGGDD

In data analysis we use ICA is a basic tool, the aim is that find out a co-ordinate system where are the components of the data are independent.Mostly ICA method such as fastICA and Jointapproximation and diogonalization of eigen matrix (JADE), uses kurtosis as a metric of non gaussianity. But the assumption of kurtosis (fourth order cumulant) may not always satisfies practically. So there are one possible solution is to use skewness (third order cumulant) instead of kurtosis. In this paper we are going to introduce ICA based method, that approach is good for heavy-tailed (fourth order kurtosis) as well as asymmetric data (third order skewness).

Dimension degradation of fractionally spaced super‐exponential algorithm for sparse channel equalisation

Fractionally spaced super-exponential (FSSE) algorithm has the disadvantage of computational complexity since it exploits high-order statistics explicitly. The authors propose a dimension degradation technique for FSSE when it is applied to the equalisation of a sparse channel in accordance with the relationship between the coefficients of the cross fourth-order cumulant (CFOC) in FSSE and the channel impulse response. They implement partial updating on the prominent coefficients of the CFOC with all the small coefficients remaining unchanged. The computational complexity of the modified FSSE reduces significantly with an acceptable performance loss, and its performance is validated via numerical simulations.

Three-dimensional Baxter-Wu model.

A classic three-dimensional spin model, based upon the Baxter-Wu scheme, is presented. It is found, by entropic sampling simulations, that the behavior of the energy and magnetization fourth-order cumulants points to a first-order phase transition. A finite-size procedure was performed, confirming that the system scales with the dimensionality d=3, yielding a high-resolution estimate of the transition temperature as T_{c}=11.377485(29).

Phase Transitions and Thermodynamic Properties of the Potts Model with Spin States Number q = 4 on a Hexagonal Lattice

Phase transitions and thermodynamic properties of the 2D ferromagnetic Potts model with the number of spin states q = 4 on a hexagonal lattice are investigated by the Mote Carlo method based on the Wang–Landau algorithm. The orders of the phase transitions are investigated using the Binder fourth-order cumulant method and histogram analysis of data. It is established that a first-order transition is observed in the model being investigated.