Wigner-Ville Spectrum Research Articles

Multi-taper S-transform method for estimating Wigner-Ville and Loève spectra of quasi-stationary harmonizable processes

Current non-stationary load models based on the evolutionary power spectral density (EPSD) may lead to overestimation and ambiguity of structural responses. The quasi-stationary harmonizable process with its Wigner-Ville spectrum (WVS) and Loève spectrum, which do not suffer from the deficiencies of EPSD, is suitable for modeling non-stationary loads and analyzing their induced structural responses. In this study, the multi-taper S-transform (MTST) method for estimating WVS and Loève spectrum of multi-variate quasi-stationary harmonizable processes is presented. The analytical biases and variances of the WVS, Loève spectrum, and time-invariant and time-varying coherence estimators from the MTST method are provided under the assumption that the target multi-variate harmonizable process is Gaussian. Using a numerical case of a bivariate harmonizable wind speed process, the superiority and reliability of the MTST method are demonstrated through comparisons with several existing methods for the WVS and Loève spectrum estimations. Finally, the MTST method is applied to two pieces of ground motion acceleration records measured during the Turkey earthquake in 2023.

Continuous-time locally stationary time series models

AbstractWe adapt the classical definition of locally stationary processes in discrete time (see e.g. Dahlhaus, ‘Locally stationary processes’, in Time Series Analysis: Methods and Applications (2012)) to the continuous-time setting and obtain equivalent representations in the time and frequency domains. From this, a unique time-varying spectral density is derived using the Wigner–Ville spectrum. As an example, we investigate time-varying Lévy-driven state space processes, including the class of time-varying Lévy-driven CARMA processes. First, the connection between these two classes of processes is examined. Considering a sequence of time-varying Lévy-driven state space processes, we then give sufficient conditions on the coefficient functions that ensure local stationarity with respect to the given definition.

Time-frequency feature extraction for classification of episodic memory

This paper investigates the extraction of time-frequency (TF) features for classification of electroencephalography (EEG) signals and episodic memory. We propose a model based on the definition of locally stationary processes (LSPs), estimate the model parameters, and derive a mean square error (MSE) optimal Wigner-Ville spectrum (WVS) estimator for the signals. The estimator is compared with state-of-the-art TF representations: the spectrogram, the Welch method, the classically estimated WVS, and the Morlet wavelet scalogram. First, we evaluate the MSE of each spectrum estimate with respect to the true WVS for simulated data, where it is shown that the LSP-inference MSE optimal estimator clearly outperforms other methods. Then, we use the different TF representations to extract the features which feed a neural network classifier and compare the classification accuracies for simulated datasets. Finally, we provide an example of real data application on EEG signals measured during a visual memory encoding task, where the classification accuracy is evaluated as in the simulation study. The results show consistent improvement in classification accuracy by using the features extracted from the proposed LSP-inference MSE optimal estimator, compared to the use of state-of-the-art methods, both for simulated datasets and for the real data example.

Passive advection of fractional Brownian motion by random layered flows

We study statistical properties of the process Y(t) of a passive advection by quenched random layered flows in situations when the inter-layer transfer is governed by a fractional Brownian motion X(t) with the Hurst index H ∈ (0,1). We show that the disorder-averaged mean-squared displacement of the passive advection grows in the large time t limit in proportion to , which defines a family of anomalous super-diffusions. We evaluate the disorder-averaged Wigner–Ville spectrum of the advection process Y(t) and demonstrate that it has a rather unusual power-law form with a characteristic exponent which exceed the value 2. Our results also suggest that sample-to-sample fluctuations of the spectrum can be very important.

Stochastic Modeling and Optimal Time-Frequency Estimation of Task-Related HRV

In this paper, we propose a novel framework for the analysis of task-related heart rate variability (HRV). Respiration and HRV are measured from 92 test participants while performing a chirp-breathing task consisting of breathing at a slowly increasing frequency under metronome guidance. A non-stationary stochastic model, belonging to the class of Locally Stationary Chirp Processes, is used to model the task-related HRV data, and its parameters are estimated with a novel inference method. The corresponding optimal mean-square error (MSE) time-frequency spectrum is derived and evaluated both with the individually estimated model parameters and the common process parameters. The results from the optimal spectrum are compared to the standard spectrogram with different window lengths and the Wigner-Ville spectrum, showing that the MSE optimal spectral estimator may be preferable to the other spectral estimates because of its optimal bias and variance properties. The estimated model parameters are considered as response variables in a regression analysis involving several physiological factors describing the test participants’ state of health, finding a correlation with gender, age, stress, and fitness. The proposed novel approach consisting of measuring HRV during a chirp-breathing task, a corresponding time-varying stochastic model, inference method, and optimal spectral estimator gives a complete framework for the study of task-related HRV in relation to factors describing both mental and physical health and may highlight otherwise overlooked correlations. This approach may be applied in general for the analysis of non-stationary data and especially in the case of task-related HRV, and it may be useful to search for physiological factors that determine individual differences.

Inverse Acoustic Methodology for Continuous-Scan Phased Arrays

The paper presents a methodology for the direct estimation of the spatiospectral distribution of an acoustic source from microphone measurements that comprise fixed and continuously scanning sensors. The nonstationarity introduced by the sensor motion is quantified by means of the Wigner–Ville spectrum. Its strongest effect is on the correlations of the sensor signals. Suppression of the nonstationarity in the signal processing involves division of the signals into blocks and application of a frequency-dependent window within each block. The direct estimation approach entails the inversion of an integral that connects the source distribution to the measured coherence of the acoustic field. A Bayesian estimation approach is developed that allows for efficient inversion of the integral and performs similarly to the much costlier conjugate-gradient method. The methodology is applied to acoustic fields emitted by impinging jets approximating a point source and an overexpanded supersonic jet. The measurement setup comprises one continuously scanning microphone and a number of fixed microphones, which are all arranged on a linear array. Comparisons are made between array configurations with fixed microphones only and with the scanning microphone: all having the same sensor count. The noise source maps with the scanning microphone have improved spatial fidelity and suppressed side lobes. The ability of the continuous-scan paradigm to provide high-definition noise source maps with a low sensor count is demonstrated.

Channel Modeling for Satellite Communication Channels at Q-Band in High Latitude

This paper proposes a three-dimensional (3D) channel model for satellite communications at Q-band in a high latitude, including the path loss, shadowing, and small-scale fading. The shadowing effect is modelled by a Markov chain. The three states in the Markov chain are separated by the threshold of the received power level for the link budget and system optimization. The probability density function (PDF) of shadowing amplitude is modelled by a mixture of two Gaussian distributions with parameters obtained by the expectation-maximum (EM) algorithm. The small-scale fading is represented by a 3D geometry-based stochastic model (GBSM) where scatterers are located on the spherical surface of a hemisphere. The movement of the receiver and the Rician factor influenced by environment scattering are considered. Statistical properties including the local temporal autocorrelation function (ACF) and Wigner-Ville spectrum are derived. The satellite communication channel measurement at Q-band is conducted on the campus of Heriot-Watt University (HWU) in Edinburgh, UK. The parameters of our proposed channel model are estimated by the measurement data. Numerical and simulation results demonstrate that our proposed channel model has the ability to reproduce main statistical properties which are also consistent well with the corresponding theoretical and measurement results.

A Non-Stationary Mobile-to-Mobile Channel Model Allowing for Velocity and Trajectory Variations of the Mobile Stations

In mobile-to-mobile (M2M) communication systems, both the transmitter and the receiver are moving with a certain velocity, which is usually assumed to be constant over time. However, in realistic propagation scenarios, the velocity of the mobile stations (MSs) is subject to changes resulting in a non-stationary fading process. In this paper, we develop a non-stationary narrow-band M2M multipath fading channel model, where the transmitter and the receiver experience changes in their velocities and trajectories. For this model, we derive expressions for the local autocorrelation function (ACF), the Wigner–Ville spectrum, the local average Doppler shift, and the local Doppler spread under isotropic scattering conditions. In addition, we investigate the correlation properties of the proposed model assuming non-isotropic scattering around the MSs. By relaxing the standard assumption of constant velocities of the MSs, this paper shows that the local ACF and the Wigner–Ville spectrum differ completely from known expressions derived for wide-sense stationary M2M channel models. Furthermore, it is shown that our model provides consistent results with respect to the Doppler spread. The proposed channel model is useful for the performance analysis of M2M communication systems under non-stationary conditions caused by velocity variations of the MSs.

Waterfilling Theorems for Linear Time-Varying Channels and Related Nonstationary Sources

The capacity of the linear time-varying (LTV) channel, a continuous-time LTV filter with additive white Gaussian noise, is characterized by waterfilling in the time-frequency plane. Similarly, the rate distortion function for a related nonstationary source is characterized by reverse waterfilling in the time–frequency plane. Constraints on the average energy or on the squared-error distortion, respectively, are used. The source is formed by the white Gaussian noise response of the same LTV filter as before. The proofs of both waterfilling theorems rely on a Szegő theorem for a class of operators associated with the filter. A self-contained proof of the Szegő theorem is given. The waterfilling theorems compare well with the classical results of Gallager and Berger. In the case of a nonstationary source, it is observed that the part of the classical power spectral density is taken by the Wigner–Ville spectrum. The present approach is based on the spread Weyl symbol of the LTV filter, and is asymptotic in nature. For the spreading factor, a lower bound is suggested by means of an uncertainty inequality.

Bearing Fault Diagnosis Method Based on Local Mean Decomposition and Wigner Higher Moment Spectrum

Combining local mean decomposition (LMD) and Wigner higher moment spectrum (WHOS), a bearing fault diagnosis method was proposed, called LMD-WHOS method. Firstly, LMD decomposed fault diagnosis signal into a series of production function (PF), and then the “real” components were found out from the decomposed components by calculating correlation coefficient between the component and the original signal. Secondly, the WHOS of the selected components was estimated. These estimated spectrums were added up to obtain the WHOS of original signal. Finally diagnosis conclusion can be drawn from the WHOS and its corresponding marginal spectrum. The algorithms of LMD and WHOS were described, and the major steps of proposed method were provided. Simulated signal and some measured rolling bearing fault signal were analyzed based on the presented method, and the results were compared with that of Wigner-Ville distribution (WVD) method. Results show that the proposed method reserves the advantages of LMD and WHOS, and can effectively inhibit the cross-term effect, which arises in Wigner-Ville spectrum. With the new method, the nature of the bearing fault signal is kept exactly and its dynamic changing characteristics of energy distribution with time and frequency can be clearly exhibited in the WHOS spectrum. LMD-WHOS method provides a new way for the accurate judgment of bearing fault state.

Research on Cutting Tool Wear Based on Fractional Brownian Motion

Cutting tool wear is a very complex process. Various factors have a direct or indirect effect on cutting tool wear, resulting in uncertainty, so it is difficult for experimental data and result to have good stability. However, Vibration analysis is a very important means for condition monitoring and fault diagnosis. This paper aims to study the methods of tool vibration signal processing, pattern recognition and trend prediction. Collected on tool vibration signal at different times, wavelet noise reduction is used to pretreat the vibration signals. Then, for the self-similar vibration signals, we propose the fractional Brownian motion (FBM) theory with long-range dependence (LRD). Combined with Wigner-Ville spectrum, characteristic parameter can be extracted, so the cutting tool wear state can be determined according to fractal dimension and average slope of the fitting curve of the logarithm power spectrum. Finally, we use FBM model to predict the trend of tool vibration signals. Experiments show that the methods have a good effect on tool wear state recognition and trend prediction.

Vibration Signal of Generator Features and its Fault Diagnosis Application

This paper study the wind turbine bearing vibration signal characterization and nonlinear transient characteristics, the transient signal characteristics were established based on Wigner-Ville spectrum entropy and support vector machine (SVM) extraction and fault diagnosis model and quantitative description of nonlinear characteristics of multi-scale method based on LMD sort of entropy, and the effectiveness of dynamic signal feature extraction and fault diagnosis method of the study, a typical experiment platform bearing bearing fault diagnosis experiment verified.

Compressive Spectral Estimation for Nonstationary Random Processes

Estimating the spectral characteristics of a nonstationary random process is an important but challenging task, which can be facilitated by exploiting structural properties of the process. In certain applications, the observed processes are underspread, i.e., their time and frequency correlations exhibit a reasonably fast decay, and approximately time-frequency sparse, i.e., a reasonably large percentage of the spectral values are small. For this class of processes, we propose a compressive estimator of the discrete Rihaczek spectrum (RS). This estimator combines a minimum variance unbiased estimator of the RS (which is a smoothed Rihaczek distribution using an appropriately designed smoothing kernel) with a compressed sensing technique that exploits the approximate time-frequency sparsity. As a result of the compression stage, the number of measurements required for good estimation performance can be significantly reduced. The measurements are values of the ambiguity function of the observed signal at randomly chosen time and frequency lag positions. We provide bounds on the mean-square estimation error of both the minimum variance unbiased RS estimator and the compressive RS estimator, and we demonstrate the performance of the compressive estimator by means of simulation results. The proposed compressive RS estimator can also be used for estimating other time-dependent spectra (e.g., the Wigner-Ville spectrum), since for an underspread process most spectra are almost equal.

A NewFault Detection and Diagnosis Method Based on Wigner-Ville Spectrum Entropy for the Rolling Bearing

The rolling bearings are one of the most critical components in rotary machinery. To prevent unexpected bearing failure, it is crucial to develop the effective fault detection and diagnosis techniques to realize equipment’s near-zero downtime and maximum productivity. In this paper, a new fault detection and diagnosis method based on Wigner-Ville spectrum entropy (WVSE) is proposed. First, the local mean decomposition (LMD) and the Wigner-Ville distribution (WVD) are combined to develop a new feature extraction approach to extract the fault features in time-frequency domain of the bearing vibration signals. Second, the concept of the Shannon entropy is integrated into the WVD to define the Wigner-Ville spectrum entropy to quantify the energy variation in time-frequency distribution under different work conditions. The research results from the bearing vibration signals demonstrate that the proposed method based on WVSE can identify different fault patterns more accurately and effectively comparing with other methods based on singular spectrum entropy (SSE) or power spectrum entropy (PSE).

Noise resistant time frequency analysis and application in fault diagnosis of rolling element bearings

Rolling element bearings are frequently used in rotary machinery, but they are also fragile mechanical parts. Hence, exact condition monitoring and fault diagnosis for them plays an important role in ensuring machinery's reliable running. Timely diagnosis of early bearing faults is desirable, but the early fault signatures are easily submerged in noise.In this paper, Wigner–Ville spectrum based on cyclic spectral density (CSWVS for a brief notation) is studied, which is able to represent the cyclostationary signals while reducing the masking effect of additive stationary noise. Both simulations and experiments show that CSWVS is a noise resistant time frequency analysis technique for extracting bearing fault patterns, when bearing signals are under influences of random noise and gear vibrations. The 3-D feature of the CSWVS is proved useful in extracting bearing fault pattern from gearbox vibration signals, where bearing signals are affected by gear meshing vibration and noise. Besides, CSWVS utilizes the second order cyclostationary property of the vibration signals produced by bearing distributed fault, and clearly extracts its fault features, which cannot be extracted by envelope analysis.To quantitatively describe the extent of bearing fault, Renyi information encoded in the time frequency diagram of CSWVS is studied. It is shown to be a more sensitive index to reflect bearing performance degradation, compared with the spectral entropy (SE), squared envelope spectrum entropy (SESE) and Renyi informations for WVD, PWVD, especially when SNR is low.

Anechoic Blind Source Separation Using Wigner Marginals

Blind source separation problems emerge in many applications, where signals can be modeled as superpositions of multiple sources. Many popular applications of blind source separation are based on linear instantaneous mixture models. If specific invariance properties are known about the sources, for example, translation or rotation invariance, the simple linear model can be extended by inclusion of the corresponding transformations. When the sources are invariant against translations (spatial displacements or time shifts) the resulting model is called an anechoic mixing model. We present a new algorithmic framework for the solution of anechoic problems in arbitrary dimensions. This framework is derived from stochastic time-frequency analysis in general, and the marginal properties of the Wigner-Ville spectrum in particular. The method reduces the general anechoic problem to a set of anechoic problems with non-negativity constraints and a phase retrieval problem. The first type of subproblem can be solved by existing algorithms, for example by an appropriate modification of non-negative matrix factorization (NMF). The second subproblem is solved by established phase retrieval methods. We discuss and compare implementations of this new algorithmic framework for several example problems with synthetic and real-world data, including music streams, natural 2D images, human motion trajectories and two-dimensional shapes.

Advanced Signal Processing Tools for the Vibratory Surveillance of Assembly Faults in Diesel Engine Cold Tests

This paper addresses the use of several signal processing tools for monitoring and diagnosis of assembly faults in diesel engines through the cold test technology. One specific fault is considered here as an example: connecting rod with incorrectly tightened screws. First, the experimental apparatus concerning the vibration tests is introduced. Subsequently, the dynamic analysis of the engine has been carried out in order to calculate the connecting rod forces against the crankpin for predicting the position where mechanical impacts are expected. Then, a vibration signal model for this type of faults is introduced. It deals with the cyclostationary model in which the signal is subdivided into two main parts: deterministic and nondeterministic. Finally, the acceleration signals acquired from the engine block during a cold test cycle at the end of the assembly line are analyzed. For quality control purposes in order to obtain reliable thresholds for the pass/fail decision, a method based on the image correlation of symmetrized dot patterns is proposed. This method visualizes vibration signals in a diagrammatic representation in order to quickly detect the faulty engines in cold tests. Moreover, the fault identification is discussed on the basis of the cyclostationary model of the signals. The first-order cyclostationarity is exploited by the analysis of the time synchronous average (TSA). In addition, the residual signal is evaluated by subtracting the TSA from the raw synchronized signal, and thus, the second-order cyclostationarity analysis is developed by means of the Wigner–Ville distribution (WVD), Wigner–Ville spectrum (WVS), and mean instantaneous power. Moreover, continuous wavelet transform is presented and compared with the WVD and WVS.

Kernels and Multiple Windows for Estimation of the Wigner-Ville Spectrum of Gaussian Locally Stationary Processes

This paper treats estimation of the Wigner-Ville spectrum (WVS) of Gaussian continuous-time stochastic processes using Cohen's class of time-frequency representations of random signals. We study the minimum mean square error estimation kernel for locally stationary processes in Silverman's sense, and two modifications where we first allow chirp multiplication and then allow nonnegative linear combinations of covariances of the first kind. We also treat the equivalent multitaper estimation formulation and the associated problem of eigenvalue-eigenfunction decomposition of a certain Hermitian function. For a certain family of locally stationary processes which parametrizes the transition from stationarity to nonstationarity, the optimal windows are approximately dilated Hermite functions. We determine the optimal coefficients and the dilation factor for these functions as a function of the process family parameter

Cyclic spectral analysis in practice

This paper addresses the spectral analysis of cyclostationary (CS) signals from a generic point of view, with the aim to provide the practical conditions of success in a wide range of applications, such as in mechanical vibrations and acoustics. Specifically, it points out the similarities, differences and potential pitfalls associated with cyclic spectral analysis as opposed to classical spectral analysis. It is shown that non-parametric cyclic spectral estimators can all be derived from a general quadratic form, which yields as particular cases “cyclic” versions of the smoothed, averaged, and multitaper periodograms. The performance of these estimators is investigated in detail on the basis of their frequency resolution, cyclic leakage, systematic and stochastic estimation errors. The results are then extended to more advanced spectral quantities such as the cyclic coherence function and the Wigner–Ville spectrum of CS signals. In particular an optimal estimator of the Wigner–Ville spectrum is found, with remarkable properties. Several examples of cyclic spectral analyses, with an emphasis on mechanical systems, are finally presented in order to illustrate the value of such a general treatment for practical applications.

CYCLIC SPECTRAL ANALYSIS FROM THE AVERAGED CYCLIC PERIODOGRAM

The theory of cyclostationarity has emerged as a new approach to characterizing a certain type of nonstationary signals. Many aspects of the spectral analysis of cyclostationary signals have been investigated but were essentially based on the use of the smoothed cyclic periodogram. This paper proposes a cyclic spectral estimator based on the averaged cyclic periodogram which benefits from better implementation properties. It shows that an unexpected but important condition for this estimator to be valid is to set enough overlap between adjacent segments in order to prevent cyclic leakage. It proves that setting the percentage of overlap to 75% with a hanning window, or 50% with a half-sine window fixes the problem. It also shows that in certain situations the cyclic leakage associated with the averaged cyclic periodogram can be made exactly zero, in contrast with the smoothed cyclic periodogram. Illustrative examples finally confirm the obtained results, where it is also demonstrated how to use them for efficiently estimating the Wigner-Ville spectrum.