An Improved Second-Order Multisynchrosqueezing Transform for the Analysis of Nonstationary Signals

Second-order multisynchrosqueezing transform(SMSST),an effective tool for the analysis of nonstationary signals,can significantly improve the time-frequency resolution of a nonstationary signal.Though the noise energy in the signal can also be enhanced in the transform which can largely affect the characteristic frequency component identification for an accurate fault diagnostic.An improved algorithm termed as an improved second-order multisynchrosqueezing transform(ISMSST)is then proposed in this study to alleviate the problem of noise interference in the analysis of nonstationary signals.In the study,the time-frequency(TF)distribution of a nonstationary signal is calculated first using SMSST,and then aδfunction is constructed based on a newly proposed time-frequency operator(TFO)which is then substituted back into SMSST to produce a noisefree time frequency result.The effectiveness of the technique is validated by comparing the TF results obtained using the proposed algorithm and those using other TFA techniques in the analysis of a simulated signal and an experimental data.The result shows that the current technique can render the most accurate TFA result within the TFA techniques employed in this study.

Anti-aliasing nonstationary signals detecion algorithm based on interpolation in the frequency domain using the short time Fourier transform

To eliminate the aliasing that appeared during the measurement of multi-components nonstationary signals, a novel kind of anti-aliasing algorithm based on the short time Fourier transform （STFT） is brought forward. First the physical essence of aliasing that occurs is analyzed; second the interpolation algorithm model is setup based on the Hamming window; then the fast implementation of the algorithm using the Newton iteration method is given. Using the numerical simulation the feasibility of algorithm is validated. Finally, the electrical circuit experiment shows the practicality of the algorithm in the electrical engineering.

Seismic spatial effects on long-span bridge response in nonstationary inhomogeneous random fields

The long-span bridge response to nonstationary multiple seismic random excitations is investigated using the PEM (pseudo excitation method). This method transforms the nonstationary random response analysis into ordinary direct dynamic analysis, and therefore, the analysis can be solved conveniently using the Newmark, Wilson-9 schemes or the precise integration method. Numerical results of the seismic response for an actual long-span bridge using the proposed PEM are given and compared with the results based on the conventional stationary analysis. From the numerical comparisons, it was found that both the seismic spatial effect and the nonstationary effect are quite important, and that both stationary and nonstationary seismic analysis should pay special attention to the wave passage effect.

Generalized Demodulation Transform for Bearing Fault Diagnosis Under Nonstationary Conditions and Gear Noise Interferences

It is a challenging issue to detect bearing fault under nonstationary conditions and gear noise interferences. Meanwhile, the application of the traditional methods is limited by their deficiencies in the aspect of computational accuracy and e ciency, or dependence on the tachometer. Hence, a new fault diagnosis strategy is proposed to remove gear interferences and spectrum smearing phenomenon without the tachometer and angular resampling technique. In this method, the instantaneous dominant meshing multiple(IDMM) is firstly extracted from the time-frequency representation(TFR) of the raw signal, which can be used to calculate the phase functions(PF) and the frequency points(FP). Next, the resonance frequency band excited by the faulty bearing is obtained by the band-pass filter. Furthermore, based on the PFs, the generalized demodulation transform(GDT) is applied to the envelope of the filtered signal. Finally, the target bearing is diagnosed by matching the peaks in the spectra of demodulated signals with the theoretical FPs. The analysis results of simulated and experimental signal demonstrate that the proposed method is an e ective and reliable tool for bearing fault diagnosis without the tachometer and the angular resampling.

A Recent Approach Incorporating External Forces To Predict Nonstationary Processes

Most real-world time series have some degree of nonstationarity due to external perturbations of the observed system; external driving forces are the essential reason that leads to the nonstationarity of dynamics system. In this paper, the authors present a novel technique in which the authors incorporate external forces to predict nonstationary time series. To test the effect, the authors also examined two prediction experiments with an ideal time series from a logistic map and a proxy climate dataset for the past millennium. The preliminary results show that the resulting algorithm has better predictive ability than the one that does not consider the external forces.

Nonstationary Time Series Prediction by Incorporating External Forces

Almost all climate time series have some degree of nonstationarity due to external forces of the observed system. Therefore, these external forces should be taken into account when reconstructing the climate dy- namics. This paper presents a novel technique in predicting nonstationary time series. The main difference of this new technique from some previous methods is that it incorporates the driving forces in the pre- diction model. To appraise its effectiveness, three prediction experiments were carried out using the data generated from some known classical dynamical models and a climate model with multiple external forces. Experimental results indicate that this technique is able to improve the prediction skill effectively.

Low order nonconforming mixed finite element method for nonstationary incompressible Navier-Stokes equations

This paper studies a low order mixed finite element method （FEM） for nonstationary incompressible Navier-Stokes equations. The velocity and pressure are approximated by the nonconforming constrained Q1^4ot element and the piecewise constant, respectively. The superconvergent error estimates of the velocity in the broken H^1-norm and the pressure in the L^2-norm are obtained respectively when the exact solutions are reasonably smooth. A numerical experiment is carried out to confirm the theoretical results.

Nonstationary inversion-based directional deconvolution of airgun array signature

Airgun arrays are widely used in marine seismic exploration because signatures excited by airgun arrays have high energy and high-peak bubble ratio, whereas the considerable length and width of the array and ghost reflections make the airgun array signature directional. As a result, the relation of the reflection amplitude with the incident and azimuth angles is variable. This means that the directivity of the airgun array results in a nonstationary wavelet and distorts the relation of the amplitude variation with the incident and azimuth angles. To remove the directivity effect, we propose a nonstationary inversion-based directional deconvolution. At fi rst, the signature as a function of take-off angle and azimuth angle is calculated using the spatial configuration of the airgun array and the near-field signatures. Then, based on the velocity model, the time-variant take-off angles are estimated and directional fi lters are designed using the take-off angles. Finally, the directivity-dependent signatures are shaped to the signature right below the airgun array using nonstationary inversion in the directional deconvolution.

Nonstationary signal inversion based on shaping regularization for random noise attenuation

Prediction filtering is one of the most commonly used random noise attenuation methods in the industry;however,it has two drawbacks.First,it assumes that the seismic signals are piecewise stationary and linear.However,the seismic signal exhibits nonstationary due to the complexity of the underground structure.Second,the method predicts noise from seismic data by convolving with a prediction error filter(PEF),which applies inconsistent noise models before and after denoising.Therefore,the assumptions and model inconsistencies weaken conventional prediction filtering's performance in noise attenuation and signal preservation.In this paper,we propose a nonstationary signal inversion based on shaping regularization for random noise attenuation.The main idea of the method is to use the nonstationary prediction operator(NPO)to describe the complex structure and obtain seismic signals using nonstationary signal inversion instead of convolution.Different from the convolutional predicting filtering,the proposed method uses NPO as the regularization constraint to directly invert the eff ective signal from the noisy seismic data.The NPO varies in time and space,enabling the inversion system to describe complex(nonstationary and nonlinear)underground geological structures in detail.Processing synthetic and field data results demonstrate that the method eff ectively suppresses random noise and preserves seismic refl ection signals for nonstationary seismic data.

Wavelet-Based Nonstationary Wind Speed Model in Dongting Lake Cable-Stayed Bridge

The wind-rain induced vibration phenomena in the Dongting Lake Bridge (DLB) can be observed every year, and the field measurements of wind speed data of the bridge are usually nonstationary. Nonstationary wind speed can be decomposed into a deterministic time-varying mean wind speed and a zero-mean stationary fluctuating wind speed component. By using wavelet transform (WT), the time-varying mean wind speed is extracted and a nonstationary wind speed model is proposed in this paper. The wind characteristics of turbulence intensity, integral scale and probability distribution of the bridge are calculated from the typical wind samples recorded by the two anemometers installed on the DLB using the proposed nonstationary wind speed model based on WT. The calculated results are compared with those calculated by the empirical mode decomposition (EMD) and traditional approaches. The compared results indicate that the wavelet-based nonstationary wind speed model is more reasonable and appropriate than the EMD-based nonstationary and traditional stationary models for characterizing wind speed in analysis of wind-rain-induced vibration of cables.

Nonstationary probability densities of system response of strongly nonlinear single-degree-of-freedom system subject to modulated white noise excitation

The nonstationary probability densities of system response of a single-degree- of-freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to modulated white noise excitation are studied. Using the stochastic averaging method based on the generalized harmonic functions, the averaged Fokl^er-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude is derived. The solution of the equation is approximated by the series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. According to the Galerkin method, the time-dependent coefficients can be solved from a set of first-order linear differential equations. Then, the semi-analytical formulae of the nonstationary probability density of the amplitude response as well as the nonstationary probability density of the state response and the statistic moments of the amplitude response can be obtained. A van der Pol-Duffing oscillator subject to modulated white noise is given as an example to illustrate the proposed procedures. The effects of the system parameters, such as the linear damping coefficient and the nonlinear stiffness coefficient, on the system response are discussed.

THE COUPLING OF BOUNDARY ELEMENT AND FINITE ELEMENT METHODS FOR THE EXTERIOR NONSTATIONARY NAVIER-STOKES EQUATIONS

In this paper, we represent a new numerical method for solving the nonstationary Stokes equations in an unbounded domain. The technique consists in coupling the boundary integral and finite element methods. The variational formulation and well posedness of the coupling method are obtained. The convergence and optimal estimates for the approximation solution are provided.

Nonstationary Wavelets Related to the Walsh Functions

Using the Walsh-Fourier transform, we give a construction of compactly supported nonstationary dyadic wavelets on the positive half-line. The masks of these wavelets are the Walsh polynomials defined by finite sets of parameters. Application to compression of fractal functions are also discussed.

Time-Frequency（scale） Analysis and Diagnosis for Nonstationary Dynamic Signals of Machinery

When operating speed or load is changed, or mechanical faults appear in machinery, many dynamic signals coming from running machinery are nonstationary. It is difficult to analyse these kind of signals. Timefrequency(scale) analysis methods of WignerVille distribution (WVD), short time Fourier transform (STFT), and wavelet transform (WT) provide powerful new tools to analyse and diagnose nonstationary signals of machinery. This paper adopts these new methods to analyse vibration signals of a metallurgical rolling mill and a mining electric excavator, and to diagnose their operating conditions and mechanical faults. Mechanical shock, friction, wear and additive impulse are revealed successfully from nonstationary operating conditions.

A Novel Approach for Evaluating Nonstationary Response of Dynamic Systems to Stochastic Excitation

The transient state of a dynamic system,such as offshore structures,to random excitation is always nonstationary.Many studies have contributed to evaluating response covariances at the transient state of a linear multi-degree-of-freedom(MDOF)system to random excitations,but a closed-form solution was not available unless the excitation was assumed to be a physically unrealizable white noise process.This study derives explicit,closed-form solutions for the response covariances at the transient state by using a pole-residue(PR)approach operated in the Laplace domain when the excitations are assumed to be stationary random processes described by physically realizable spectral density functions.By using the PR method,we can analytically solve the triple integral in evaluating the nonstationary response covariance.As this approach uses the poles and residues of system transfer functions,rather than the conventional mode superposition technique,the method is applicable to MDOF systems with non-classical damping models.Particular application of the proposed method is demonstrated for multi-story shear buildings to stochastic ground acceleration characterized by the Kanai–Tajimi spectral density function model,and a numerical example is provided to illustrate the detailed steps.No numerical integrations are required for computing the response covariances as the exact closed-form solution has been derived.The correctness of the proposed method is numerically verified by Monte Carlo simulations.

An approximation of the first passage probability of systems under nonstationary random excitation

An approximate method is presented for obtaining analytical solutions for the conditional first passage probability of systems under modulated white noise excitation. As the method is based on VanMarcke＇s approximation, with normalization of the response introduced, the expected decay rates can be evaluated from the second-moment statistics instead of the correlation functions or spectrum density functions of the response of considered structures. Explicit solutions to the second-moment statistics of the response are given. Accuracy, efficiency and usage of the proposed method are demonstrated by the first passage analysis of single-degree-of-freedom （SDOF） linear systems under two special types of modulated white noise excitations.

The dynamics of nonstationary solutions in one-dimensional two-component Bose-Einstein condensates

This paper investigates the dynamical properties of nonstationary solutions in one-dimensional two-component Bose-Einstein condensates. It gives three kinds of stationary solutions to this model and develops a general method of constructing nonstationary solutions. It obtains the unique features about general evolution and soliton evolution of nonstationary solutions in this model.

THE WAVELET TRANSFORM OF PERIODIC FUNCTION AND NONSTATIONARY PERIODIC FUNCTION

Some properties of the wavelet transform of trigonometric Junction, periodic function and nonstationary periodic function have been investigated. The results show that the peak height and width in wavelet energy spectrum of a periodic function are in proportion to its period. At the same time, a new equation, which can truly reconstruct a trigonometric function with only one scale wavelet coefficient, is presented. The reconstructed wave shape of a periodic function with the equation is better than any term of its Fourier series. And the reconstructed wave shape of a class of nonstationary periodic function with this equation agrees well with the function.

Bayesian track-before-detect algorithm for nonstationary sea clutter

Radar detection of small targets in sea clutter is a particularly demanding task because of the nonstationary characteristic of sea clutter.The track-before-detect(TBD)filter is an effective way to increase the signal-to-clutter ratio(SCR),thus improving the detection performance of small targets in sea clutter.To cope with the nonstationary characteristic of sea clutter,an easily-implemented Bayesian TBD filter with adaptive detection threshold is proposed and a new parameter estimation method is devised which is integrated into the detection process.The detection threshold is set according to the parameter estimation result under the framework of information theory.For detection of closely spaced targets,those within the same range cell as the one under test are treated as contribution to sea clutter,and a successive elimination method is adopted to detect them.Simulation results prove the effectiveness of the proposed algorithm in detecting small targets in nonstationary sea clutter,especially closely spaced ones.

A Random Nonstationary Pulse Train Model

A simple and mathematically tractable model of a nonstationary process is developed.The process is the sum of waves where the parameters of the waves are random.Explicit expres-sions for the mean and autocorrelation function at each position as a function of time are obtained.In the case of infinite time,the model evolves into a stationary process.The time-frequency distri-bution at each position is also obtained.An explicit example is given where the initial waves are Gaussian.The case where there is dispersion in the propagation is also discussed.