Wigner Distribution Research Articles

A Protection Scheme for a Power System with Solar Energy Penetration

As renewable energy (RE) penetration has a continuously increasing trend, the protection of RE integrated power systems is a critical issue. Recently, power networks developed for grid integration of solar energy (SE) have been designed with the help of multi-tapped lines to integrate small- and medium-sized SE plants and simultaneously supplying power to the loads. These tapped lines create protection challenges. This paper introduces an algorithm for the recognition of faults in the grid to which a solar photovoltaic (PV) system is integrated. A fault index (FI) was introduced to identify faults. This FI was calculated by multiplying the Wigner distribution (WD) index and Alienation (ALN) index. The WD-index was based on the energy density of the current signal evaluated using Wigner distribution function. The ALN-index was evaluated using sample-based alienation coefficients of the current signal. The performance of the algorithm was validated for various scenarios with different fault types at various locations, different fault incident angles, fault impedances, sampling frequencies, hybrid line consisting of overhead (OH) line and underground (UG) cable sections, different types of transformer windings and the presence of noise. Two phase faults with and without the involvement of ground were differentiated using the ground fault index based on the zero sequence current. This study was performed on the IEEE-13 nodes test network to which a solar PV plant with a capacity of 1 MW was integrated. The performance of the algorithm was also tested on the western part of utility grid in the Rajasthan State in India where solar PV energy integration is high. The performance of the algorithm was effectively established by comparing it with the discrete Wavelet transform (DWT), Wavelet packet transform (WPT) and Stockwell transform-based methods.

Deformation quantization of constrained systems: a group averaging approach

Motivated by certain concepts introduced by the refined algebraic quantization formalism for constrained systems which has been successfully applied within the context of loop quantum gravity, in this paper we propose a phase space implementation of the Dirac quantization formalism to appropriately include systems with constraints. In particular, we propose a physically prescribed Wigner distribution which allows for the definition of a well-defined inner product by judiciously introducing a star version of the group averaging of the constraints. This star group averaging procedure is obtained by considering the star-exponential of the constraints and then integrating with respect to a suitable measure. In this manner, the proposed physical Wigner distribution explicitly solves the constraints by including generalized functions on phase space. Finally, in order to illustrate our approach, our proposal is tested in a couple of simple examples which recover standard results found in the literature.

Wigner distribution and Boltzmann-Shannon entropy for a diatomic molecule under polarized electric field interaction

We study the classical chaos appearing in a diatomic molecules BeO, CO and CN due to the interaction with a circularly polarized electric field, and its signature in Quantum Mechanics through the Wigner distribution function and the Boltzmann-Shannon entropy. We found a motion out of the center of the quantum phase space defined by Wigner function when the classical system becomes chaotic, and we found a jumping behavior of the average Boltzmann-Shannon entropy with respect the electric field strength when the classical system becomes chaotic, indicating a sudden increasing in the disorder (or sudden lost of information) in the quantum system.

On Integrals Over a Convex Set of the Wigner Distribution

We provide an example of a normalized $$L^{2}({\mathbb {R}})$$ function u such that its Wigner distribution $${\mathcal {W}}(u,u)$$ has an integral $$>1$$ on the square $$[0,a]\times [0,a]$$ for a suitable choice of a. This provides a negative answer to a question raised by Flandrin (Proc IEEE Int Conf Acoustics 4(1):2176–2179, 1988). Our arguments are based upon the study of the Weyl quantization of the characteristic function of $${{\mathbb {R}}_{+}\times {\mathbb {R}}_{+}}$$ along with a precise numerical analysis of its discretization.

QCD resummation in hard diffractive dijet production at the electron-ion collider

Diffractive dijet production at the electron-ion collider (EIC) has been proposed to study the gluon Wigner distribution at small-x. We investigate the soft gluon radiation associated with the final state jets and an all order resummation formula is derived. We show that the soft gluon resummation plays an important role to describe E791 data on π-induced diffractive dijet production at Fermilab. Predictions for the EIC are presented, and we emphasize that the soft gluon resummation is an important aspect to explore the nucleon/nucleus tomography through these processes.

Semi-Poisson nearest-neighbor distance statistics for a 2D dissipative map

Typical pseudotrajectories of 2D ergodic maps are known to possess Wignerian nearest-neighbor distance distributions (NNDDs). In the case of 2D chaotic dissipative maps, bounded aperiodic pseudotrajectories typically evolve on planar strange attractors. In this microarticle, the hypothesis that such pseudotrajectories should possess NNDDs that are intermediate between the Poisson and Wigner distributions is put forward and a rare example for which the intermediate distribution can be clearly identified is presented. In particular, it is demonstrated numerically that typical pseudotrajectories evolving in the strange attractor of the standard 2D Ikeda map possess semi-Poissionian NNDDs.

Transferring Einstein-Podolsky-Rosen State Cross Frequency Bands via Cavity Electro-Opto-Mechanical Converters

In order to transfer two-mode quantum state from optical input to microwave output, we propose a scheme based on double cavity electro-opto-mechanical converters. Principles and operation modes of cavity electro-opto-mechanical converter are introduced in detail, and theoretical model of transferring Einstein-Podolsky-Rosen state via our scheme is built. The dynamics and corresponding scattering matrixes of our scheme are analyzed carefully. Meanwhile, Ulhmann fidelity is utilized to quantify the similarity of initial state and final state, which also stands for the performance of our scheme. Influences of coupling rates and optical input bandwidth on fidelity are well discussed, and some significant factors are also taken into consideration, such as transmission spectrum, interaction rates and decoherence rates. Furthermore, Wigner distributions of initial state and final state as well as the von Neumann entropy are obtained to verify the performance of our scheme. The results show that our scheme could transfer the Einstein-Podolsky-Rosen state from optical input to microwave output with high fidelity (0.74), and the entropy ratio between initial and final states is about 0.78, which is quite close with the fidelity. Besides, Wigner distributions also show that initial state and final state are slightly different in phase structures. In case of strong coupling rates, ultralow temperature and narrow input bandwidth, the performance of our scheme could be further enhanced.

Decoherence of a Damped Anisotropic Harmonic Oscillator under Magnetic Field Effects in a Two-Dimensional Noncommutative Phase-Space

In this paper, decoherence of a damped anisotropic harmonic oscillator in the presence of a magnetic field is studied in the framework of the Lindblad theory of open quantum systems in noncommutative phase-space. General fundamental conditions that should follow our quantum mechanical diffusion coefficients appearing in the master equation are kindly derived. From the master equation, the expressions of density operator, the Wigner distribution function, the expectation and variance with respect to coordinates and momenta are obtained. Based on these quantities, the total energy of the system is evaluated and simulations show its dependency to phase-space structure and its improvement due to noncommutativity effects and the environmental temperature as well. In addition, we also evaluate the decoherence time scale and show that it increases with noncommutativity phase-space effects as compared to the commutative case. It turns out from simulations that this time scale is significantly improved under magnetic field effects.

Twisted elliptical multi-Gaussian Schell-model beams and their propagation properties.

We introduce a new kind of partially coherent source whose cross-spectral density (CSD) function is described as the incoherent superposition of elliptical twisted Gaussian Schell-model sources with different beam widths and transverse coherence widths, named twisted elliptical multi-Gaussian Schell-model (TEMGSM) beams. Analytical expression for the CSD function propagating through a paraxial ABCD optical system is derived with the help of the generalized Collins formula. Our results show that the TEMGSM beam is capable of generating a flat-topped elliptical beam profile in the far field, and the beam spot during propagation exhibits clockwise/anti-clockwise rotation with respect to its propagation axis. In addition, the analytical expressions for the orbital angular momentum (OAM) and the propagation factor are also derived by means of the Wigner distribution function. The influences of the twisted factor and the beam index on the OAM and the propagation factor are studied and discussed in detail.

Statistical properties of Gaussian Schell-model beams propagating through anisotropic hypersonic turbulence

Hypersonic turbulence, causing the environment disturbance, is an important factor influencing optical propagation and optical communication. Based on the extended Huygens-Fresnel principle and second-order moments of the Wigner distribution function, we have derived the analytical expressions of the average intensity, the normalized propagation factor, and the normalized mean-squared beam width of Gaussian Schell-model beams in anisotropic hypersonic turbulence. The evolution properties demonstrate the severe beam expansion and beam quality degradation, and the influences of the source and turbulent parameters are discussed in detail. The comparison between anisotropic hypersonic turbulence and atmosphere turbulence is given, and it is proved that the influence of anisotropic hypersonic turbulence is much stronger. Our results play a guiding role in optical signal transmission in the turbulent environment induced by hypersonic aircraft.

Wigner Distribution of Accelerated TripartiteW-state

In this contribution, we introduce an analytical form of the Wigner distribution function for any tripartite quantum system. These forms are extended to the accelerated tripartite quantum systems of W-state type. The quantum and classical correlations are predicted by negative/ positive values, respectively of the Wigner function. It is shown that for the non-accelerated W-state, the quantum correlations are predicted at large values of the distribution angles. However, for the accelerated W-state, the quantum correlations are predicted at different intervals of distribution angle and different initial acceleration. Due to the acceleration the quantum correlation decays, where the decay rate increases as the number of the accelerated qubits increases. We show that, the local noisy channels; bit and phase bit channels curb the decoherence of the quantum correlation. The ability of the phase-bit channel to defend the quantum correlation is stronger than that shown for the bit-channel, where it decreases if all the three qubits are accelerated. Moreover, we quantify, the amount of the survival quantum correlation by using the volume of the negativity. Our results show that, the volume of negativity doesn’t vanish even at large acceleration, which means thatW-state keeps some of its quantum correlation during the acceleration process.

Zeros of the Wigner distribution and the short-time Fourier transform

We study the question under which conditions the zero set of a (cross-) Wigner distribution W(f, g) or a short-time Fourier transform is empty. This is the case when both f and g are generalized Gaussians, but we will construct less obvious examples consisting of exponential functions and their convolutions. The results require elements from the theory of totally positive functions, Bessel functions, and Hurwitz polynomials. The question of zero-free Wigner distributions is also related to Hudson’s theorem for the positivity of the Wigner distribution and to Hardy’s uncertainty principle. We then construct a class of step functions S so that the Wigner distribution W(f,mathbf {1}_{(0,1)}) always possesses a zero fin S cap L^p when p<infty , but may be zero-free for fin S cap L^infty . The examples show that the question of zeros of the Wigner distribution may be quite subtle and relate to several branches of analysis.

Regular and chaotic motion domains in the channeling electron's phase space and mean level density for its transverse motion energy

The motion of charged particles in a crystal in the axial channeling regime can be both regular and chaotic. The chaos in quantum case manifests itself in the statistical properties of the energy levels set. These properties have been studied previously for the electrons channeling along [110] direction of the silicon crystal, in the case when the classical motion was completely chaotic. The case of channeling along [100] direction is of special interest because the classical motion here can be both regular and chaotic for the same energy depending on the initial conditions. The semiclassical energy level density (as well as its part that corresponds to the regular motion domains in the phase space) is computed for the 10 GeV channeling electrons and positrons. It is demonstrated that the level spacing distribution for both electrons and positrons can be better described by Berry-Robnik distribution than by both Wigner (completely chaotic case) or Poisson (completely regular case) distributions.

Linear canonical Stockwell transform

The linear canonical transform (LCT) has long been recognized as a competent tool for optics and signal processing. However, the LCT cannot reveal the local LCT-frequency contents due to its global kernel and as such makes the LCT incompetent in situations demanding a joint information of time and linear canonical domain-frequency. In this paper, we propose the linear canonical Stockwell transform (LCST) in order to rectify the limitations of the LCT and the Stockwell transform (ST). Firstly, we examine the resolution of the novel transform in the time and LCT domains and then derive some of its basic properties, such as Moyal's formula, inversion formula and provide a characterization of the range. Furthermore, we derive a direct relationship between the well known Wigner distribution and the proposed LCST. In sequel, a discrete version of the linear canonical Stockwell transform is also presented. Finally, we extend the scope of the proposed work by studying the LCST in the realm of almost periodic functions.

Automatic Detection and Suppression of Parasitic Chirp Signals in the Intermediate Time–Frequency Domain

The problem of automatic parasitic chirp components detection and suppression in a digital signal is considered. The detection is based on the calculation of the discrete Wigner distribution accompanied by the Hough transform. The filtration is achieved by a series of discrete fractional Fourier transforms converting the signal to special coordinates, where the entire energy of the parasitic signals is accumulated at a few samples and then reset. The main novelty is the utilization of new construction of discrete fractional Fourier transform coordinated with the discrete Wigner distribution. The advantages of the proposed approach are illustrated by the simulations.

Linear Canonical Wigner Distribution Based Noisy LFM Signals Detection Through the Output SNR Improvement Analysis

In our previous work, we addressed a unified representation problem on Wigner distribution (WD) in linear canonical transform (LCT) domains by introducing a kind of closed-form instantaneous cross-correlation function type of Wigner distribution (CICFWD) that can be regarded as the WD's closed-form representation in the linear canonical domain. We then discussed the application of CICFWD in the detection of linear frequency-modulated (LFM) signals through the numerical simulation analysis approach, and it turns out that there is a causation between LCT free parameters and the detection performance. The main contribution of this paper is to provide a mathematical model for this causality in order to disclose the intrinsic mechanism of non-stationary signal detection performance improvement triggered by LCT free parameters. We first revisit the definition of CICFWD, and establish some related theories including its essential properties and discretization. We then formulate a mathematical theory foundation of the output signal-to-noise ratio (SNR) improvement, an output SNR inequality relation between the traditional WD and the CICFWD for a general noisy signal, and discuss how to solve this inequality to come out LCT free parameters. Within the well-established output SNR improvement analysis framework, we also explore LCT free parameters selection results on LFM signals added with white noise. Finally, numerical experiments are carried out to validate the correctness of strategies on determination LCT free parameters and the feasibility of output SNR improvement analysis approach.

Solvability of the Caldeira–Leggett model

In this paper, we illustrate the use of the method of the characteristics in various dissipative models of a single harmonic oscillator. The master equation governing the process can be transformed to a partial differential equation on the Wigner distribution, which in turn can be split to a system of coupled differential equations. We present a useful technique that can be used to separate the system without increasing the order and then the solutions can be obtained. The obtained solutions are used to calculate the average of energy observable of the system. This procedure can be extended to solve some other complex similar problems.

МОДАЛЬНОЕ УПРАВЛЕНИЕ ПРОЦЕССАМИ ДОЗИРОВАНИЯ В СРЕДЕ ПРОСТРАНСТВА СОСТОЯНИЙ И ВЕЙВЛЕТ-ПРЕОБРАЗОВАНИЙ

The article presents the essential aspects of designing and operational functioning of computer-aided modal control systems, using the example of a control system for feeding processes in а mixture-producing set. The object of study is the structure of feeding devices unit and transient processes during feeding. The subjects of research are the issues of designing and creating a computer-aided modal control system ( CMCS ) which operates in the mode of tunable parametrically non-stationary in frequency transient processes ( TP ). In this case, the scalar signals of the feeding devices ( FD ) material flows are formed in terms of the state space method (in the format of the internal models “input-state-output”), and monitoring of such processes is carried out in the time-frequency wavelet medium. The latter makes it possible to analyze the processes occurring in the CMCS at a semantically transparent and information-saturated level. The parameters (pulse signal off-duty ratios) of the Fourier model for the signals of a discrete-type feeder are presented, on which the nature of the dosing process depends. A model of the feeders unit is formed in terms of the state space. The 1 D -flow rate signals and their representations in the form of time-frequency distributions ( W -maps) are considered. A new concept was developed for flow rate modal control of the feeding devices unit in the frame of the mixture-producing set by introducing the definition of parametric non-stationary state of the feeding devices internal environment that takes into account the real-time variation of the multidimensional state feedback coefficient during transients in the system and for the purpose of experimental estimation of transients parameters ( here: time constants), they are determined by flow signal distributions on W -maps. The wavelet medium used in the system deals with Cohen’s class time-frequency distributions (Wigner-Ville and Choi-Williams transformations). The combination of TP -models in terms of the state space and their representation in the Wigner distribution format allows one to evaluate, set and reassign the parameters of the internal structure of the object controlled (a separate feeder or feeding devices unit) in real time in order to achieve the effect of stabilizing the ingredients flow rate in a rated mode at pre-mixing stage. The transient modes are presented in the form of processes restoring the rated modes (the so-called reverse TP) . A procedure has been developed and described for the implementation of a modal control algorithm for the feeding process, including the technology of forced localization and relocalization of the object poles under non-stationary processes in the “Executive mechanism- FD ” system, using the example of a continuous-type feeder in a transient mode on load decrease (with a downward reverse transitional process). For this mode, the calculation of the non-stationary matrix of the modal controller is presented.

Analytic Wigner distribution function for a split potential well

The analytic Wigner function for a single well with infinite walls is extended to the configuration where half of the well is raised and the weighting of the wave functions is in accordance with a thermal Fermi-Dirac distribution. This requires a method for determining the energy eigenstates particularly near the height of the higher half-well. The methods provide a means for constructing analytic and accurate trajectories that can be used to model tunneling times. The fidelity of the trajectory model is assessed with respect to absorption of quanta related to changes within the allowable eigenstates.

Study of kaon structure using the light-cone quark model

We investigate the various distributions explaining multi-dimensional structure of kaon at the level of its constituents ($u$ and $\bar{s}$) using the light-cone quark model. The overlap form of wavefunctions associated with the light-cone quark model is adopted for the calculations. The generalized parton distributions(GPDs)of $u$ and $\bar{s}$ quarks are presented for the case when the momentum transfer in the longitudinal direction is non-zero. The dependence of kaon GPDs is studied in terms of variation of quark longitudinal momentum fraction, momentum transfer in longitudinal direction and total momentum transfer to the final state of hadron. The transverse impact-parameter dependent GPDs are also studied by taking the Fourier transformation of general GPDs. Further, the quantum phase-space distributions; Wigner distributions are studied for the case of unpolarized, longitudinally-polarized and transversely-polarized parton in an unpolarized kaon. The Wigner distributions are analysed in the transverse impact-parameter plane, the transverse momentum plane and the mixed plane. Further, to get a complete picture of kaon in terms of its valence quarks, the variation of longitudinal momentum fraction carried by quark and antiquark in the generalized transverse momentum-dependent parton distributions (GTMDs) is studied for different values of transverse quark and antiquark momentum $({\bf k}_\perp)$ as well as for different values of momentum transferred to the kaon in transverse direction $({\bf \Delta}_\perp)$. This has been done for zero as well as non-zero skewedness representing respectively the absence and presence of momentum transfer to the final state of kaon in longitudinal direction. Furthermore, the possible spin-orbit correlation for $u$ and $\bar{s}$ in kaon is elaborated in context of Wigner distributions and GTMDs.