Wigner-Ville Distribution Research Articles

Investigation and evaluation of cross-term reduction in masked Wigner-Ville distributions using S-transforms.

Non-linear and non-stationary signals are analyzed and processed in the time-frequency (TF) domain due to interpretation simplicity. Wigner-Ville distribution (WVD) delivers a very sharp resolution of non-stationary signals in the TF domain. However, cross-terms occur between true frequency modes due to their bilinear nature. Masked WVD reduces cross-terms by multiplying the time-frequency representation (TFR) obtained from the WVD with the TFR of the same signal obtained from another method, while S-transform (ST) is a linear signal analysis method that combines the advantages of short-time Fourier transform (STFT) and wavelet transform (WT). This paper investigated WVD masking with both original and modified STs to compare their cross-term reduction results. Moreover, additional parameters are integrated into the ST to deliver the better resolution of the ST and, consequently, more satisfactory cross-term reduction. However, these parameters must be carefully optimized by expert users in their respective application fields.

Dynamic mode decomposition-based technique for cross-term suppression in the Wigner-Ville distribution

This paper presents a new method for time-frequency representation (TFR) using dynamic mode decomposition (DMD) and Wigner-Ville distribution (WVD), which is termed as DMD-WVD. The proposed method helps in removing cross-term in WVD-based TFR. In the suggested method, the DMD decomposes the multi-component signal into a set of modes where each mode is considered as mono-component signal. The analytic modes of these obtained mono-component signals are computed using the Hilbert transform. The WVD is computed for each analytic mode and added together to obtain cross-term free TFR based on the WVD. The effectiveness of the proposed method for TFR is evaluated using Rényi entropy (RE). Experimental results for synthetic signals namely, multi-component amplitude modulated signal, multi-component linear frequency modulated (LFM) signal, multi-component nonlinear frequency modulated (NLFM) signal, multi-component signal consisting of LFM and NLFM mono-component signal, multi-component signal consisting of sinusoidal and quadratic frequency modulated mono-component signals, and synthetic mechanical bearing fault signal and natural signals namely, electroencephalogram (EEG) and bat echolocation signals are presented in order to show the effectiveness of the proposed method for TFR. It is clear from the results that the proposed method suppresses cross-term effectively as compared to the other existing methods namely, smoothed pseudo WVD (SPWVD), empirical mode decomposition (EMD)-WVD, EMD-SPWVD, variational mode decomposition (VMD)-WVD, VMD-SPWVD, and DMD-SPWVD.

Transient Analysis of Flow Unsteadiness And Machine Instabilities In a Centrifugal Compressor at Surge and Second Quadrant Operation

Abstract Instabilities in turbomachinery operation originate from intrinsic flow unsteadiness, sudden variations of operating conditions, and interactions with the surrounding system, affecting both system stability and machine reliability. The flow unsteadiness, presenting a characteristic behaviour, requires a deep understanding of the underlying flow fundamentals, a careful analysis using advanced experimental methods and a clear distinction on how the system might lead to instabilities. The identified approach addresses deep surge and second quadrant characterization. The analysis presented here benefits from the responsiveness and flexibility of the experimental test rig, fine tuning of the system layout, the instrumentation installation, and the transient tests procedure. The main focus area is the time-frequency analysis of flow and pressure signals at transients, thus providing a proper reconstruction of flow mechanisms and instability evolution. Valuable techniques include: short-time fast Fourier transform, wavelet analysis, and Wigner-Ville distribution; coherence between flow and pressure, to trace waves propagation, is considered too. These, presented outlining their strengths and weaknesses, are evaluated considering real time field applicability with regards to computational resources too. The experimental campaigns and data processing are presented, with a test matrix covering a wide range of parameters, including system layout and surge volume size, rotational speeds, flow rates ranging from nominal point to full second quadrant operation. These data are vital for system modeling validation for a full compressor-system digital twin. Results indicate the importance of ensuring reliable data acquisition and analysis, with adequate instrumentation range, accuracy, synchronization, responsiveness and positioning.

Adaptive stochastic resonance enhanced weak linear frequency modulated signal perception in low signal-to-noise ratio environments

Abstract The linear frequency modulated (LFM) signal has gained extensive utilization in radar, sonar and covert communication system due to its large time-bandwidth product, high-precision distance measurement and low detection probability. The perception of weak LMF signals holds significant importance yet encounters substantial challenges within the increasing complexity of the electromagnetic environments. Consequently, this paper proposes an adaptive stochastic resonance (ASR) enhanced approach for perceiving weak LFM signals in low signal-to-noise ratio (SNR) conditions. This approach initially commences a pioneering study on the quantitative synergistic resonance mechanism among LFM signals, random noise, and nonlinear stochastic resonance (SR) systems. Subsequently, the ASR system’s implementation becomes straightforward through the adaptive adjustment of SR system parameters according to the LFM signal and noise characteristics. This implementation leverages the inherent property of noise energy transfer to signal energy, facilitating the enhancement of ordered weak signals via random noise. Following the enhancement of the output signal by the ASR system, the Wigner–Ville distribution (WVD) transform’s time-frequency concentration property is employed to extract the time-frequency characteristics. Building on the WVD transform, the Radon transform with linear integral projection is applied to further harness the time-frequency domain energy concentration, thereby achieving the effective perception of weak LFM signals. Finally, the effectiveness of the proposed method in improving the perception performance of weak LFM signal in very low SNR conditions is verified through numerical simulations. It is anticipated that this method will have potential application value in fields such as radar, sonar, and communication under complex electromagnetic environments.

Radar-Based Elderly Fall Detection Using Smoothed Pseudo Wigner Ville Distribution and XGBoost Learning.

Falls among the elderly population pose significant health risks, often leading to morbidity and decreased quality of life. Traditional fall detection methods, namely wearable devices and cameras, have limitations such as lighting conditions and privacy concerns. Radar-based fall detection has emerged as a promising alternative, offering unobtrusive technique. In this study, an attempt has been made to classify fall detection using smoothed pseudo wigner-ville distribution (SPWVD) images and XGBoost learning. For this, online publicly available radar database (N=15) is considered. Radar signals is employed to SPWVD for time-frequency representation images. Ten features are extracted and applied to XGBoost learning. Experiments are performed and performance is evaluated using 10-fold cross validation. The proposed approach is able to discriminate elderly fall. Using XGBoost learning, the approach yields a maximum average classification accuracy, f1-score, precision, sensitivity, specificity, and kappa scores of 87.47%, 87.38%, 88.12%, 86.81%, 88.31% and 74.94% respectively. The combination of conventional features with concentration measures and median frequency obtained the second best performance. Thus, the proposed framework could be utilized for accurate and efficient detection of falls among the elderly population in their private spaces.

On the entanglement of co-ordinate and momentum degrees of freedom in noncommutative space

In this paper, we investigate the quantum entanglement induced by phase-space noncommutativity. Both the position–position and momentum–momentum noncommutativity are incorporated to study the entanglement properties of coordinate and momentum degrees of freedom under the shade of oscillators in noncommutative space. Exact solutions for the systems are obtained after the model is re-expressed in terms of canonical variables, by performing a particular Bopp’s shift to the noncommuting degrees of freedom. It is shown that the bipartite Gaussian state for an isotropic oscillator is always separable. To extend our study for the time-dependent system, we allow arbitrary time dependency on parameters. The time-dependent isotropic oscillator is solved with the Lewis–Riesenfeld invariant method. It turns out that even for arbitrary time-dependent scenarios, the separability property does not alter. We extend our study to the anisotropic oscillator, which provides an entangled state even for time-independent parameters. The Wigner quasi-probability distribution is constructed for a bipartite Gaussian state. The noise matrix (covariance matrix) is explicitly studied with the help of Wigner distribution. Simon’s separability criterion (generalized Peres–Horodecki criterion) has been employed to find the unique function of the (mass and frequency) parameters, for which the bipartite states are separable. In particular, we show that the mere inclusion of non-commutativity of phase-space is not sufficient to generate the entanglement, rather anisotropy is important at the same footing. We explore the experimental viability of our result through the computation of extractable work for the current situation.

A Multi-Cycle Echo Energy Concentration Method for High-Mobility Targets Enveloped by Time-Varying Plasma Sheath

When a target moves at hypersonic speed, the aerodynamic thermal effect will cause air molecules to form a plasma sheath that envelopes the outer surface of the target, which consists of a large number of charged particles. The plasma sheath imposes a complicated modulation effect on the radar echo signal in terms of amplitude, phase, and frequency. When the plasma sheath is time-varying, the inter-pulse coherence of the multi-cycle echo signals is severely disrupted, resulting in the failure of coherent accumulation. To address the problem of abnormal inter-pulse energy accumulation in targets covered with time-varying plasma sheaths, we analyzed the dynamic modulation effects of time-varying plasma sheaths on echo signals and constructed a radar echo model enveloped with time-varying plasma sheaths. Based on this, we propose a method for inter-pulse energy concentration of multi-cycle echo signals based on range-frequency inversion, second-order Wigner–Ville distribution (WVD), and slow-time symmetric auto-correlation. The proposed method is capable of realizing energy concentration for targets enveloped with time-varying plasma sheaths and can accurately estimate the motion parameters of the target. The effectiveness of our proposed method has been verified via simulation analysis of multi-cycle echo signals from targets enveloped with time-varying plasma sheaths, and the reliability of the method has been further validated through statistical experimental analysis.

Structural damage classification in composite materials using the Wigner-Ville distribution and convolutional neural networks

Detecting damage in composite materials is crucial and has received considerable attention in the field of machine learning. However, challenges remain in addressing backbone issues and classifying specific types of damage. This paper presents a novel deep-learning approach for automatically distinguishing delamination and microcracks in Carbon Fiber-Reinforced Polymer (CFRP). The methodology utilizes signals from piezoelectric transducers transformed into time–frequency representations based on the Wigner-Ville Distribution (WVD). Backbone issues were successfully addressed by transitioning the time series classification problem into a computer vision (CV) context through Convolutional Neural Networks (CNN). The analysis included a thorough examination of delamination and microcracks datasets produced experimentally by the National Aeronautics and Space Administration (NASA), focusing on these two types of damage. The proposed methodology achieved a precision range of 98.3 % to 100 % in damage classification, demonstrating its effectiveness for structural health monitoring in composite materials.

The Classical-Quantum Limit

The standard notion of a classical limit, represented schematically by ℏ→0, provides a method for approximating a quantum system by a classical one. In this work, we explain why the standard classical limit fails when applied to subsystems, and show how one may resolve this by explicitly modeling the decoherence of a subsystem by its environment. Denoting the decoherence time by τ, we demonstrate that a double scaling limit in which ℏ→0 and τ→0 such that the ratio Ef=ℏ/τ remains fixed leads to an irreversible open-system evolution with well-defined classical and quantum subsystems. The main technical result is showing that, for arbitrary Hamiltonians, the generators of partial versions of the Wigner, Husimi, and Glauber-Sudarshan quasiprobability distributions may all be mapped in the above-mentioned double scaling limit to the same completely positive classical-quantum generator. This provides a regime in which one can study effective and consistent classical-quantum dynamics. Published by the American Physical Society 2024

Uncertainty principles for coupled fractional Wigner-Ville distribution.

The coupled fractional Wigner-Ville distribution is a more general version of the fractional Wigner-Ville distribution. Main properties including boundedness, Moyal's formula and inversion formula are studied in detail for the transformation. Additionally, the relation of the coupled fractional Wigner-Ville distribution with the two-dimensional Fourier transform is studied. We also present the relationship between the coupled fractional Wigner-Ville distribution with the two-dimensional Wigner-Ville distribution. We show how the properties and relations allow us to derive several versions of the uncertainty inequalities related to the coupled fractional Wigner-Ville distribution.

Diagnosis of Al-CFRTP TA-FSLW defect using acoustic emission signal based on SPWVD and ResNet

As one of the important development directions in modern railway and automobile, lightweight requires reliable connection technology of dissimilar materials. Thermal-assisted friction stir lap welding (TA-FSLW) is an available welding technology for the joining of dissimilar materials, which can solve the problems of physical and chemical properties in the welding process of dissimilar materials. Unfortunately, defects may occur during the welding process. This paper proposes a defect diagnosis method based on smooth pseudo Wigner-Ville distribution (SPWVD) and ResNet18 for Al-CFRTP TA-FSLW. Firstly, the acoustic emission (AE) signal and the temperature of the weld nugget are acquired by the AE sensor and thermography camera in the FSLW process. Then, the time–frequency domain features of AE signal are extracted by SPWVD, and the maximum temperature of the weld nugget is extracted by exponential moving average (EMA) from thermography. Moreover, the dataset is established by SPWVD features, maximum temperature of welding nugget, and several time–frequency features. Finally, the ResNet18-attention network is employed to classify welding defects and stages. The Al-CFRTP TA-FSLW experiments are carried out based on the proposed method. Experimental result indicates that the SPWVD spectrum can effectively extract the feature frequency of deep and shallow hole defects in the welding process. In the section of classification, the accuracy of the dataset with SPWVD features is improved by 2% to 4% compared to other methods, which demonstrates that the SPWVD feature is superior in defect feature extraction.

Vehicle Response-Based Bridge Modal Identification Using Different Time–Frequency Analysis Methods

Modal identification of bridges based on the responses of moving vehicles, termed as the indirect method, has drawn much attention in recent years. This indirect method has been proven to be capable of identifying bridge frequencies and mode shapes in the feasibility research. However, in the presence of additional factors such as vehicle speed and road surface roughness, it is difficult to identify high-order frequencies and high-resolution mode shapes of the bridge. The authors previously proposed a tractor-double-trailers model and the time–domain subtraction method to obtain the relative displacement of two trailers for bridge modal identification. In this study, the authors develop a methodology to improve the identification performance by combining the EMD technique and the time–frequency analysis method. The EMD technique is applied to separate the multi-source response signals of the trailers to obtain the signal component that contains only the bridge vibration. Then, different time–frequency analysis methods such as the short-time Fourier transform (STFT), Wigner–Ville distribution (WVD), and continuous wavelet transform (CWT) are adopted to construct the high-resolution bridge mode shapes. Numerical studies with one-dimensional and three-dimensional bridge models are conducted to verify the proposed method and investigate the effects of different factors. Finally, a laboratory test is implemented on a scaled bridge to verify the performance of the proposed method. The results show that the proposed method is able to separate the vehicle frequency and bridge frequencies, which facilitates the identification of bridge frequencies from the dynamic response of moving vehicles. Meanwhile, the CWT method performs the best in the identification of modal shapes among the three time–frequency analysis methods.

Seismic data filtering using deconvolutive short-time Fourier transform

Cohen class distributions are well-known time-frequency distributions that have been widely used to analyze different signals. However, they are impractical for signal filtering because they only provide amplitude spectra and suffer from cross terms. In this paper, deconvolutive short-time Fourier transform (DSTFT) is developed by estimating phase spectra and updating moduli to address residual cross terms. The DSTFT moduli is used as weights and apply a weighted least-squares technique to estimate a high-resolution and almost cross-term-free Wigner-Ville distribution. Through numerical tests, it has been found that choosing the optimal window length can minimize cross terms in short-time Fourier transform and DSTFT spectrograms. Furthermore, using thresholding or reweighting techniques can effectively eliminate weights associated with noise. The performance of our method is demonstrated using synthetic and two real seismic wavefield separation problems, including ground-roll removal in seismic shot records and polarization analysis in seismology. The results indicate the high performance of our method in estimating phase spectra and filtering seismic data.

Automatic Wake and Deep-Sleep Stage Classification Based on Wigner-Ville Distribution Using a Single Electroencephalogram Signal.

This research paper outlines a method for automatically classifying wakefulness and deep sleep stage (N3) based on the American Academy of Sleep Medicine (AASM) standards. The study employed a single-channel EEG signal, leveraging the Wigner-Ville Distribution (WVD) for time-frequency analysis to determine EEG energy per second in specific frequency bands (δ, θ, α, and entire band). Particle Swarm Optimization (PSO) was used to optimize thresholds for distinguishing between wakefulness and stage N3. This process aims to mimic a sleep technician's visual scoring but in an automated fashion, with features and thresholds extracted to classify epochs into correct sleep stages. The study's methodology was validated using overnight PSG recordings from 20 subjects, which were evaluated by a technician. The PSG setup followed the 10-20 standard system with varying sampling rates from different hospitals. Two baselines, T1 for the wake stage and T2 for the N3 stage, were calculated using PSO to ascertain the best thresholds, which were then used to classify EEG epochs. The results showed high sensitivity, accuracy, and kappa coefficient, indicating the effectiveness of the classification algorithm. They suggest that the proposed method can reliably determine sleep stages, being aligned closely with the AASM standards and offering an intuitive approach. The paper highlights the strengths of the proposed method over traditional classifiers and expresses the intentions to extend the algorithm to classify all sleep stages in the future.

Online detection and assessment of cable insulation fault based on time–frequency domain reflectometry

The insulation performance of cables plays an important role in the safe and stable operation of aircraft. Time–frequency domain reflectometry (TFDR) is an effective method to detect the local insulation faults of cables. Usually, Wigner–Ville distribution (WVD) is used to analyze the energy distribution of reflected signals in TFDR. However, WVD can easily introduce cross-term interference to the detection results. In addition, most of the current studies have used TFDR to perform offline detection of cable insulation fault, and the performance of TFDR for online operation is unclear. In order to solve the above problems, first, this paper eliminates the interference of cross-term based on smoothed pseudo-Wigner–Ville distribution. Then, an online cable insulation fault detection platform based on non-contact signal injection is built. According to the requirements of the standard GJB181A, the interference degree of the incident signal of TFDR to the original signal of cable is analyzed, and the feasibility of using TFDR method to perform online detection is proved. Finally, the correlation between different features of reflected signals under different fault degrees of cable is analyzed. It is proved that time-domain correlation has a better correlation with the fault degree of cable compared to other four features.

Relating the Glauber-Sudarshan, Wigner, and Husimi quasiprobability distributions operationally through the quantum-limited amplifier and attenuator channels

Relating the Glauber-Sudarshan, Wigner, and Husimi quasiprobability distributions operationally through the quantum-limited amplifier and attenuator channels

A novel fluid identification method based on a high-precision spectral decomposition method

Abstract Time-frequency decomposition technology is an effective tool for analyzing non-stationary signals. Improving resolution of spectral decomposition techniques is important to extract more useful information from the received signal. The Wigner-Ville distribution (WVD) has been widely applied in seismic signal analysis, it can better analyze seismic signals due to many excellent mathematical properties, but this method has a drawback that cross terms interference exists in the analyzing of multicomponent signals, which severely limits its application. The combination of the complex-domain matching pursuit (CDMP) with this approach effectively solves this problem. However, the conventional CDMP-WVD does not take the influence of the scale parameter on the Morlet wavelet waveform into account, which reduces the time-frequency resolution of CDMP-WVD. Therefore, to correct the defect that the atomic waveforms change only with the frequency parameter, we propose an improved spectral decomposition method ICDMP-WVD that considers the scale parameter. In this study, we first analyze influences of the scale parameter on Morlet wavelet waveform and make the scale parameter a search parameter that improves the computational efficiency and time-frequency resolution of the traditional CDMP-WVD method. Accordingly, the seismic dispersion-dependent attributes are calculated via combing the improved CDMP-WVD algorithm and the frequency-dependent AVO inversion. We adopt a two-step frequency-dependent AVO inversion method to improve the stability of the conventional frequency-dependent AVO inversion. Theoretical data and real data application show that the approach in this study can identify gas reservoirs efficiently and accurately.

The short-time Wigner–Ville distribution

The Wigner–Ville distribution (WD) and ambiguity function (AF) are serviceable tools for the global analysis of non-stationary signals. However, when dealing with time-varying signals that exhibit changing characteristics and require real-time signal processing, they have high limitations due to the existence of cross-items. In this paper, we propose the short-time Wigner–Ville distribution (STWD) as a novel approach that effectively analyzes time-varying signals with changing characteristics. First, the definition of STWD and short-time ambiguity function (STAF) and their properties are advanced. Next, three uncertainty principles that define the lower bound for the product of signal spread and its bandwidth are derived. Then, the discrete STWD (DSTWD) and the process of computerization are also proposed which can be used more widely in practice. Finally, the optimization of window functions based on Rényi entropy is discussed. Simulation experiments show that the STWD can be effectively applied for the analysis of local change characteristics and real-time processing of signals with valid and accurate results.

Wigner functions of the finite square-well bound states

The bound states of a particle confined in a one-dimensional finite square well cannot be solved analytically, since the eigen-energies are determined by transcendental equations. Here, we numerically calculate the bound states and show their non-classical properties, using Wigner's quasi-probability distribution (also called the Wigner functions) in the phase space (x, p). In contrast to the infinite-well case, we find that the Wigner functions spread over the space dimension x, get squeezed along the momentum dimension p, and show negativity outside the well. Negativity in a Wigner function indicates non-classical properties of the bound states.

Low frequency inversion method based on multi-layer holizontal variation shallow seafloor model

Sound propagation in shallow water is significantly influenced by geoacoustic properties. Estimating these geoacoustic parameters is essential for sound field analysis and sonar performance assessment. As a common practice, the seafloor is often treated as a single-layer or two-layer range-independent geoacoustic model to reduce the number of involved parameters. However, acoustic parameters inverted through these two geoacoustic models are typically limited in their applicability to a specific frequency range, thus posing challenges when applied across a broader frequency range. A range-dependent multi-layer geoacoustic model based on experimental measurements obtained with a sub-bottom profiler is proposed in this study. The inversion scheme combines three inversion methods to estimate geoacoustic parameters, considering the different sensitivities of geoacoustic parameters to different physical parameters within the acoustic field. Firstly, modal dispersion is used to invert the geoacoustic parameters of each layer, with the dispersion curve obtained through warping transform and the Wigner-Ville distribution. After that, both the localization using matched field processing and the dispersion curve fitting demonstrate the effectiveness of the inversion results for each layer, although the peak of the probability distribution of sound speed in the first layer is broader than in others. Secondly, matched field processing is employed to invert the geoacoustic parameters of the first layer. This method is based on the theory that as frequency increases, the depth of sound rays penetrating the seabed decreases, revealing changes in the first layer's sound speed with the seabed depth. Lastly, bottom attenuation coefficients at different frequencies are inverted by the transmission loss (TL), and a fitting relationship between the attenuation coefficient and the frequency is derived. The inversion results obtained by using the range-dependent multi-layer geoacoustic model are compared with results estimated by the single-layer geoacoustic model. The findings indicate that the transmission loss (TL) error from the range-dependent multi-layer geoacoustic model in this study is smaller than that from the single-layer geoacoustic model, especially in the lower frequency band. The range-dependent multi-layer geoacoustic model proves to be suitable for a broader frequency range, providing better precision in explaining various acoustic phenomena.