Hilbert Method Research Articles

An improved phase retrieval method based on Hilbert transform in interferometric microscopy

An improved phase retrieval method based on Hilbert transform is introduced to quantitatively calculate the phase distribution from a micro-interferometric pattern. The property of Hilbert transform, losing DC offsets and owning π/2 phase shifting, is used to remove the DC component and obtain the phase distribution from an off-axis interferogram. Compared with traditional Hilbert method, the Fourier transform is omitted that can avoid the questions of frequency leakage and improve the computational efficiency. Based on the micro-interferometric measurement, the improved method is used to retrieve the phase distribution of rabbit red blood cells. And the corresponding results show that it owns high calculation efficiency and the ability of reserving details of samples.

Gearbox Fault Diagnosis Based on Empirical Mode Decomposition and Hilbert Transform

Gearbox is vital components in a wide range of industrial and transport applications. It is very important how to monitor operating state of automobile gearbox and detect incipient faults. This paper applies the empirical mode decomposition (EMD) and Hilbert spectrum methods to gearbox vibration signal analysis capture from vibrating acceleration sensor for gearbox fault diagnosis. The original modulation fault vibration signals are firstly decomposed into a number of intrinsic mode function (IMF) by the EMD method. Then Hilbert spectrum of intrinsic mode function at different fault characteristic frequencies is obtained by Hilbert transform. Finally, the time-frequency fault characteristics of gearbox are analyzed by the Hilbert spectrum value of intrinsic mode function. Experiment result has shown the feasibility and efficiency of the EMD algorithms and Hilbert spectrum characteristic method in fault diagnosis and fault message abstraction.

Single frame digital fringe projection profilometry for 3-D surface shape measurement

Multiple frame digital fringe projection technique is widely used for measuring the 3-D surface shape. In dynamic situations single frame analysis techniques are desirable. In this paper we discuss a Hilbert transform based single-frame analysis. Hilbert transformation method requires only one fringe pattern for the extraction of phase reducing the calculation time. The method is easy to implement, and it is capable of conducting automated measurements at video frame rate. The application of the proposed method for curved surfaces is emphasized. A few experimental results are presented.

An Applied Research on Reactive Current Detection Methods Based on Instantaneous Reactive Power Theory

The reactive current detection methods in reactive power compensation device based on instantaneous reactive power theory were discussed in the paper. By analyzing the quality of various kinds of harmonious and reactive current detection methods in power system, and according to different compensation requirement, two reactive current detection methods ip-iq method and Hilbert method were adopted. Then it gives simulations on these two models to analyze their comprehensive performance of both stable state and dynamic state. Finally, gives a verification under a IC hardware platform.The result shows that these two methods can well inspected the three-phase reactive current, and then to complete the system compensation.

Transient Power Quality Signal Generator and Detector Platform

a transient power quality detective platform which includes the disturbance signal's generator and detector is designed based on virtual instrument technology. The platform is developed under LabVIEW environment. It's generator can give the transient disturbance signals according to the standard EIEE1159-1995, such as voltage swells, voltage sags, voltage fluctuation and transient oscillation. The detector uses Hilbert phase-shifting Transform method to realize the real-time detection to the disturbance signals from the generator. The test example shows that the platform for transient disturbance detection is visual and convenient, and the phase-shifting method is effective for transient power quality disturbance signal's detection.

THE RIEMANN–HILBERT APPROACH TO OBTAIN CRITICAL ASYMPTOTICS FOR HAMILTONIAN PERTURBATIONS OF HYPERBOLIC AND ELLIPTIC SYSTEMS

The present paper gives an overview of the recent developments in the description of critical behavior for Hamiltonian perturbations of hyperbolic and elliptic systems of partial differential equations. It was conjectured that this behavior can be described in terms of distinguished Painlevé transcendents, which are universal in the sense that they are, to some extent, independent of the equation and the initial data. We will consider several examples of well-known integrable equations that are expected to show this type of Painlevé behavior near critical points. The Riemann–Hilbert method is a useful tool to obtain rigorous results for such equations. We will explain the main lines of this method and we will discuss the universality conjecture from a Riemann–Hilbert point of view.

The Hilbert basis method for D-flat directions and the superpotential

We discuss, using the Hilbert basis method, how to efficiently construct a complete basis for D-flat directions in supersymmetric Abelian and non-Abelian gauge theories. We extend the method to discrete (R and non-R) symmetries. This facilitates the construction of a basis of all superpotential terms in a theory with given symmetries.

Arbitrary-order Hilbert spectral analysis for time series possessing scaling statistics: Comparison study with detrended fluctuation analysis and wavelet leaders

In this paper we present an extended version of Hilbert-Huang transform, namely arbitrary-order Hilbert spectral analysis, to characterize the scale-invariant properties of a time series directly in an amplitude-frequency space. We first show numerically that due to a nonlinear distortion, traditional methods require high-order harmonic components to represent nonlinear processes, except for the Hilbert-based method. This will lead to an artificial energy flux from the low-frequency (large scale) to the high-frequency (small scale) part. Thus the power law, if it exists, is contaminated. We then compare the Hilbert method with structure functions (SF), detrended fluctuation analysis (DFA), and wavelet leader (WL) by analyzing fractional Brownian motion and synthesized multifractal time series. For the former simulation, we find that all methods provide comparable results. For the latter simulation, we perform simulations with an intermittent parameter μ=0.15. We find that the SF underestimates scaling exponent when q>3. The Hilbert method provides a slight underestimation when q>5. However, both DFA and WL overestimate the scaling exponents when q>5. It seems that Hilbert and DFA methods provide better singularity spectra than SF and WL. We finally apply all methods to a passive scalar (temperature) data obtained from a jet experiment with a Taylor's microscale Reynolds number Re(λ)≃250. Due to the presence of strong ramp-cliff structures, the SF fails to detect the power law behavior. For the traditional method, the ramp-cliff structure causes a serious artificial energy flux from the low-frequency (large scale) to the high-frequency (small scale) part. Thus DFA and WL underestimate the scaling exponents. However, the Hilbert method provides scaling exponents ξ(θ)(q) quite close to the one for longitudinal velocity, indicating a less intermittent passive scalar field than what was believed before.

Vocal Fold Vibratory Characteristics in Normal Female Speakers From High-Speed Digital Imaging

The purpose of the study was to investigate relationships between vocal fold vibrations and voice quality. Laryngeal images obtained from high-speed digital imaging (HSDI) were examined for their open-closed timing characteristics and perturbation values. A customized software delineated the glottal edges and used the Hilbert transform-based method of analysis to provide objective quantification of glottal perturbation. Overlay tracings of the transformed glottal cycles provided visual patterns on the overall vibratory dynamics. In this paper, we described the use of this method in looking at vibratory characteristics of a group of young female speakers (N=23). We found that, females with no voice complaints and who had been perceived to have normal voices were not a homogeneous group in terms of their glottal vibratory patterns during phonation. Their vibratory patterns showed characteristics similar to exemplar voices targeted to be clear (50%), pressed (27%), breathy (15%), or a mixed quality (8%). Perturbation range in terms of cycle-to-cycle frequency and amplitude was small and did not discriminate patterns. All these patterns yielded perceptually normal voices suggesting that in normal young speakers, the level of perturbation may be more important to the judgment than the actual pattern of closure.

String-derived MSSM vacua with residual R symmetries

Recently it was shown that there is a unique Z 4 R symmetry for the MSSM which allows for the Yukawa couplings and dimension five neutrino mass operator, forbids the μ term and commutes with SO(10). This Z 4 R symmetry contains matter parity as a subgroup and forbids dimension four and five proton decay operators. We show how to construct string vacua with discrete R symmetries in general and this symmetry in particular, and present an explicit example which exhibits the exact MSSM spectrum, the Z 4 R symmetry as well as other desired features such as gauge-top unification. We introduce the Hilbert basis method for determining all D-flat configurations and efficient algorithms for identifying field configurations with a desired residual symmetry. These methods are used in an explicit example, in which we describe in detail how to construct a supersymmetric vacuum configuration with the phenomenologically attractive Z 4 R symmetry. At the perturbative level, this is a supersymmetric Minkowski vacuum in which almost all singlet fields (moduli) are fixed.

Scattering on two Aharonov–Bohm vortices with opposite fluxes

The scattering of an incident plane wave on two Aharonov–Bohm vortices with opposite fluxes is considered in detail. The presence of the vortices imposes non-trivial boundary conditions for the partial waves on a cut joining the two vortices. These conditions result in an infinite system of equations for scattering amplitudes between incoming and outgoing partial waves, which can be solved numerically. The main focus of the paper is the analytic determination of the scattering amplitude in two limits, the small flux limit and the limit of small vortex separation. In the latter limit the dominant contribution comes from the S-wave amplitude. Calculating it, however, still requires solving an infinite system of equations, which is achieved by the Riemann–Hilbert method. The results agree well with the numerical calculations.

Integrable properties of the general coupled nonlinear Schrödinger equations

In this paper, a general integrable coupled nonlinear Schrödinger system is investigated. In this system, the coefficients of the self-phase modulation, cross-phase modulation, and four-wave mixing terms are more general while still maintaining integrability. The N-soliton solutions in this system are obtained by the Riemann–Hilbert method. The collision dynamics between two solitons is also analyzed. It is shown that this collision exhibits some new phenomena (such as soliton reflection) which have not been seen before in integrable systems. In addition, the recursion operator and conservation laws for this system are also derived.

A method of integration for wave equation and some applications to wave physics

We present a Hilbert space analytical rigorous method to solve globally a class of important abstract wave equations in mathematical physics of continuum mechanics. We show its usefulness by proposing some sort of Trotter–Kato–Feynman method to analyze the usual wave motion in . We besides produce a proof for the exponential decay of the total energy associated to a planar magnetoelastic wave motion in an electric dispersive medium. As a last result of our work, we present the existence and uniqueness of an abstract Klein–Gordon functional wave equation by means of Hilbert space techniques.

Squared eigenfunctions for the Sasa–Satsuma equation

Squared eigenfunctions are quadratic combinations of Jost functions and adjoint Jost functions which satisfy the linearized equation of an integrable equation. They are needed for various studies related to integrable equations, such as the development of its soliton perturbation theory. In this article, squared eigenfunctions are derived for the Sasa–Satsuma equation whose spectral operator is a 3×3 system, while its linearized operator is a 2×2 system. It is shown that these squared eigenfunctions are sums of two terms, where each term is a product of a Jost function and an adjoint Jost function. The procedure of this derivation consists of two steps: First is to calculate the variations of the potentials via variations of the scattering data by the Riemann–Hilbert method. The second one is to calculate the variations of the scattering data via the variations of the potentials through elementary calculations. While this procedure has been used before on other integrable equations, it is shown here, for the first time, that for a general integrable equation, the functions appearing in these variation relations are precisely the squared eigenfunctions and adjoint squared eigenfunctions satisfying, respectively, the linearized equation and the adjoint linearized equation of the integrable system. This proof clarifies this procedure and provides a unified explanation for previous results of squared eigenfunctions on individual integrable equations. This procedure uses primarily the spectral operator of the Lax pair. Thus two equations in the same integrable hierarchy will share the same squared eigenfunctions (except for a time-dependent factor). In the Appendix, the squared eigenfunctions are presented for the Manakov equations whose spectral operator is closely related to that of the Sasa–Satsuma equation.

Numerical optimization for the calculus of variations by gradients on non-Hilbert Sobolev spaces using conjugate gradients and normalized differential equations of steepest descent

Numerical optimization for the calculus of variations by gradients on non-Hilbert Sobolev spaces using conjugate gradients and normalized differential equations of steepest descent

Integrated Approach for Fetal QRS Detection

Fetal magnetocardiography provides reliable signals of the fetal heart dynamics with high temporal resolution that can be used in a clinical setting. We present a robust Hilbert transform method for extraction of the fetal heart rate. Our method may be applied to signals derived from a single channel or an array of channels. In the case of multichannel data, the channels can be combined to improve signal-to-noise ratio for the extraction of fetal heart data. The method is inherently insensitive to fetal position or movement and, in addition, can be automated. We demonstrate that the determination of R-wave timing is relatively insensitive to waveform morphology. The method can also be applied if the data were preprocessed by independent component analysis (ICA). We compared the Hilbert method, ICA, ICA + Hilbert, and raw signals and found that the Hilbert method gave the best overall performance. We demonstrated that there were approximately 171 errors in 46,789 fetal heart beats.

A Comparative Assessment of Two Techniques for Modal Identification From Power System Measurements

Power system oscillatory behavior can be analyzed in terms of modes, expressed as exponentially modulated sinusoids, exhibited in signals measured on the system. These signals are driven by the behavior of a large, nonlinear, time-variant system. In this paper, two modal identification methods are examined comparatively: Prony analysis and a method based on the Hilbert transform. Considerable structural differences exist between the two methods. Prony analysis yields modes which are directly expressed as exponentially modulated sinusoids, whereas the Hilbert method provides a more general solution. Synthetic and measured signals are used in the comparison. Some general conclusions are drawn from the analysis of several signals, including two sets of measured field data.

Projection operator techniques and Hilbert space averaging in the quantum theory of nonequilibrium systems

The development of efficient strategies for the treatment of the dynamics of relevant observables in complex quantum systems plays a decisive role in the theory of quantum relaxation and transport behavior. Here we discuss the most important tools that are based on the projection operator techniques of nonequilibrium statistical mechanics. For both the Nakajima-Zwanzig and the time-convolutionless projection operator technique we derive the equations of motion for a set of relevant observables and develop explicit expressions in second and fourth order of the corresponding perturbation expansions. We also discuss the Hilbert space average method which is based on the idea of a best guess of conditional quantum expectations determined by an average over a suitable region in the underlying Hilbert space, and relate this method to the projection operator technique.

Finite quantum environments as thermostats: an analysis based on the Hilbert space average method

We consider discrete quantum systems coupled to finite environments which may possibly consist of only one particle in contrast to the standard baths which usually consist of continua of oscillators, spins, etc. We find that such finite environments may, nevertheless, act as thermostats, i.e., equilibrate the system though not necessarily in the way predicted by standard open system techniques. Thus, we apply a novel technique called the Hilbert space Average Method (HAM) and verify its results numerically.

Modelling of Power Components in Non-sinusoidal Power Systems Using Hilbert Transform

AbstractThis article presents a Hilbert transform-based approach of power components definition in non-sinusoidal power systems. Instantaneous phasors of current and voltage, apparent power, and power factor are also defined. The proposed Hilbert transform-based method is applicable to sinusoidal or non-sinusoidal, single-phase systems as well as balanced and unbalanced three-phase systems. The method is also applicable in the presence of interharmonics. The properties and physical meanings of defined power-related quantities are discussed in detail. The presented method leads to the formulas used in the conventional RMS concept method. Study cases of power systems at different loads are examined to test the validity and practicability of the proposed approach.