Decomposition Of Non-stationary Signals Research Articles

Estimation of Time Varying Boundaries and Filtering of Empirical Wavelet Transform with Application to Adaptive Nonstationary Signal Decomposition

Estimation of Time Varying Boundaries and Filtering of Empirical Wavelet Transform with Application to Adaptive Nonstationary Signal Decomposition

Signal Separation Operator Based on Wavelet Transform for Non-Stationary Signal Decomposition.

This paper develops a new frame for non-stationary signal separation, which is a combination of wavelet transform, clustering strategy and local maximum approximation. We provide a rigorous mathematical theoretical analysis and prove that the proposed algorithm can estimate instantaneous frequencies and sub-signal modes from a blind source signal. The error bounds for instantaneous frequency estimation and sub-signal recovery are provided. Numerical experiments on synthetic and real data demonstrate the effectiveness and efficiency of the proposed algorithm. Our method based on wavelet transform can be extended to other time-frequency transforms, which provides a new perspective of time-frequency analysis tools in solving the non-stationary signal separation problem.

An Iterative Adaptive Vold–Kalman Filter for Nonstationary Signal Decomposition in Mechatronic Transmission Fault Diagnosis Under Variable Speed Conditions

An Iterative Adaptive Vold–Kalman Filter for Nonstationary Signal Decomposition in Mechatronic Transmission Fault Diagnosis Under Variable Speed Conditions

Estimating the first and second derivatives of discrete audio data

A new method for estimating the first and second derivatives of discrete audio signals intended to achieve higher computational precision in analyzing the performance and characteristics of digital audio systems is presented. The method could find numerous applications in modeling nonlinear audio circuit systems, e.g., for audio synthesis and creating audio effects, music recognition and classification, time-frequency analysis based on nonstationary audio signal decomposition, audio steganalysis and digital audio authentication or audio feature extraction methods. The proposed algorithm employs the ordinary 7 point-stencil central-difference formulas with improvements that minimize the round-off and truncation errors. This is achieved by treating the step size of numerical differentiation as a regularization parameter, which acts as a decision threshold in all calculations. This approach requires shifting discrete audio data by fractions of the initial sample rate, which was obtained by fractional delay FIR filters designed with modified 11-term cosine-sum windows for interpolation and shifting of audio signals. The maximum relative error in estimating first and second derivatives of discrete audio signals are respectively in order of 10-13\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-13}$$\\end{document} and 10-10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-10}$$\\end{document} over the entire audio band, which is close to double-precision floating-point accuracy for the first and better than single-precision floating-point accuracy for the second derivative estimation. Numerical testing showed that this performance of the proposed method is not influenced by the type of signal being differentiated (either stationary or nonstationary), and provides better results than other known differentiation methods, in the audio band up to 21 kHz.

Modulated Short-Time Fourier-Transform-Based Nonstationary Signal Decomposition for Dual-Comb Ranging Systems

Modulated Short-Time Fourier-Transform-Based Nonstationary Signal Decomposition for Dual-Comb Ranging Systems

Extension and convergence analysis of Iterative Filtering to spherical data

Many real-life signals are defined on spherical domains, in particular in geophysics and physics applications. In this work, we tackle the problem of extending the iterative filtering algorithm, developed for the decomposition of non-stationary signals defined in Euclidean spaces, to spherical domains. We review the properties of the classical Iterative Filtering method, present its extension, and study its convergence in the discrete setting. In particular, by leveraging the Generalized Locally Toeplitz sequence theory, we are able to characterize spectrally the operators associated with the spherical extension of Iterative Filtering, and we show a counterexample of its convergence. Finally, we propose a convergent version, called Spherical Iterative Filtering, and present numerical results of its application to spherical data.

Research on rotor fault diagnosis technology of three-phase asynchronous motor based on NA-MEMD mutual information and SVM

Aiming at the problem of mode aliasing in the adaptive decomposition of nonlinear and non-stationary current signals generated by three-phase asynchronous motor faults, and the fault features contained in signals collected by a single sensor can not be accurately and comprehensively extracted and characterized when early rotor bar breakage and air gap eccentricity faults occur, A fault diagnosis method for three-phase asynchronous motor based on noise assisted multivariate empirical mode decomposition (NA-MEMD) and mutual information is proposed. Firstly, the NA-MEMD algorithm is used to decompose the three-phase stator current signal of the asynchronous motor to obtain multi-scale intrinsic mode functions (IMFs). Then, the correlation algorithm is used to screen the IMFs containing useful information. Then, the filtered IMF components are reconstructed into new signals and their features are extracted, Finally, support vector machines (SVM) are used to identify the rotor broken bars and air gap eccentric faults of the three-phase asynchronous motor. The experimental results show that the NA-MEMD method has a higher recognition rate than the traditional empirical mode decomposition (EMD) and the ensemble empirical mode decomposition (EEMD) methods.

Early remaining-useful-life prediction applying discrete wavelet transform combined with improved semi-empirical model for high-fidelity in battery energy storage system

The recycling of lithium-ion batteries (LIBs) from electric vehicles (EVs) for augmenting the capacity of battery energy storage systems (BESS) presents a sustainable approach to leverage investment in LIBs, mitigating economic losses. This study highlights the critical role of accurate capacity estimation and remaining-useful-life (RUL) prediction in determining optimal replacement and augmentation schedules. We introduce a semi-empirical (SE) model that integrates experimental aging data with an adaptive method and an electrochemical model to capture the degradation trends of LIBs accurately. The adoption of the Moore-Penrose pseudoinverse (Pinv) method, suited for the SE model's multi-parameter, single-output nature, significantly improves capacity estimation accuracy, achieving a maximum error below 0.2 Ah. This improved SE (ISE) model accounts for capacity trends and addresses non-stationary phenomena, including noise and signal distortion during parameter identification and capacity recovery phases.To address these challenges, we propose using discrete wavelet transform (DWT) for the decomposition of non-stationary signals, adjusting the window size to signal characteristics for enhanced noise reduction. The application of DWT to denoise α and β parameters, characterized by their non-constant periodicity and noise interference, facilitates the development of DWT combined with the ISE (DWT-ISE) aging model for early RUL prediction. Experimental validation reveals that our DWT-based approach minimizes root-mean square error (RMSE) and mean absolute error (MAE) for early RUL prediction, demonstrating the model's superiority at 300 cycles for identification and achieving the lowest prediction error at 400 cycles. Furthermore, comparison results of other models (Particle filter, and long short-term memory) reveal that our DWT-ISE aging model exhibits the best RMSE and MAE on limited data. Specifically, the RMSE and MAE values for RUL prediction, underscored as 0.032 % and 0.027 % respectively at 300 cycles, are confirmed as the highest evaluation metrics.

Generalized singular spectrum analysis for the decomposition and analysis of non-stationary signals

Singular spectrum analysis (SSA) has been verified to be an effective method for decomposing non-stationary signals. The decomposition and reconstruction stages can be interpreted as a zero-phase filtering process where reconstructed components are obtained by inputting a signal through moving average filters. However, mathematical analysis indicates that the use of a default rectangular window in the embedding stage would corrupt the frequency characteristics of the trajectory matrix, resulting in spectral leakage. To attenuate the effect of spectral leakage and to obtain more concentrated SSA components, this study introduces a windowing technique in SSA, called generalized singular spectrum analysis (GSSA). In GSSA, the default rectangular window is replaced with adjustable taper windows, which are widely used for attenuating spectral leakage. Through windowing, GSSA can achieve less spectral leakage, and produce more energy-concentrated reconstructed components compared with conventional SSA. Grouped spectral entropy (GSE) is used as the metric for evaluating the performance of the proposed GSSA algorithm. Results from experiments, which were conducted on a synthetic signal and two real electroencephalogram signals, show that GSSA outperforms the conventional SSA and the baseline in the reduction of spectral leakage. Compared with the baseline, the proposed GSSA achieves a lower averaged GSE, resulting in reduction of 0.4 for eigenfilters and 0.11 for reconstructed components, respectively. Our results reveal the effectiveness of GSSA in the decomposition and analysis of non-stationary signals.

New theoretical insights in the decomposition and time-frequency representation of nonstationary signals: The IMFogram algorithm

The analysis of the time–frequency content of a signal is a classical problem in signal processing, with a broad number of applications in real life. Many different approaches have been developed over the decades, which provide alternative time–frequency representations of a signal each with its advantages and limitations. In this work, following the success of nonlinear methods for the decomposition of signals into intrinsic mode functions (IMFs), we first provide more theoretical insights into the so–called Iterative Filtering decomposition algorithm, proving an energy conservation result for the derived decompositions. Furthermore, we present a new time–frequency representation method based on the IMF decomposition of a signal, which is called IMFogram. We prove theoretical results regarding this method, including its convergence to the spectrogram representation for a certain class of signals, and we present a few examples of applications, comparing results with some of the most well-known approaches available in the literature.

Zero-Phase Filter-Based Adaptive Fourier Decomposition and Its Application to Fault Diagnosis of Rolling Bearing

Empirical wavelet transform (EWT) is a recently developed non-stationary signal decomposition approach. However, the number of spectrum divisions needs to be preset first, and the performance of spectrum division is inferior for a strong noisy or non-stationary signal caused by the mode mixing problem. To address this issue, a novel non-stationary signal decomposition method termed zero-phase filter-based adaptive Fourier decomposition (ZPF-AFD) is proposed in this paper. In the ZPF-AFD method, the number of spectrum divisions is adaptively determined first using the envelope entropy (EE) metric. Next, the spectral envelope processing (SEP) is applied to achieve an adaptive spectrum division. Last, the zero-phase filter (ZPF) is utilized to filter the frequency domain signal to obtain the components. The ZPF can effectively eliminate the mode mixing problem of EWT because of its no transition phase. The proposed ZPF-AFD approach is contrasted with the existing EMD, VMD, and EWT approaches through analyzing the simulated and measured signals in rolling bearing with local failures. The experimental results demonstrate that the proposed ZPF-AFD method has the best anti-noise and diagnosis accuracy, and its diagnosis performance is superior to the compared methods.

Research on non-stationary characteristic test and decomposition for dynamic response of floating structures

Considering the complex influences of harsh marine environments, auxiliary structures, and various operational processes, motion response of floating structures will inevitably exhibit non-stationary behaviors. Therefore, grasping the non-stationary characteristics of motion responses is imperative for a comprehensive understanding of the operational mechanisms. This paper presents a strategy for testing non-stationary characteristics and decomposing dynamic responses, with the objective of investigating the influence of non-stationary characteristics on motion responses of floating structures. The primary contributions of this study are: (1) Proposed a cointegration analysis method for characteristics testing applicable to floating structures to investigate potential non-stationary variations. It explains the statistics of non-stationary characteristics through dimensional analysis, establishing a connection between the test statistics and the engineering intuition of non-stationary motion response. (2) Proposed a non-stationary signal decomposition method based on adaptive segmentation and decomposition control techniques to overcome mode mixing issues in the decomposition of structural motion responses. The validity and applicability of the developed strategy for testing and decomposing non-stationary responses are verified through a synthesized signal and a numerical semi-submersible platform. Numerical results demonstrate that this strategy can effectively test the non-stationarity, and the accuracy of decomposition results is greater than 99%, and the maximum normalization error is less than 2%. Finally, physical experiments are conducted on a semi-submersible platform under regular and irregular waves action, respectively, to verify the feasibility of the developed strategy when applied to measured data. Experimental results indicate that this strategy can effectively detect the non-stationary characteristics of the measured response, and realize an accurate decomposition of structural motion responses. These results will provide valuable insights into understanding non-stationary characteristics relevant to various engineering applications.

System and approach to detecting of gastric slow wave and environmental noise suppression based on optically pumped magnetometer

Gastric slow waves (SWs) are commonly used for the quantitative assessment of gastric functional disorders. Compared with surface electrogastrography, using of magnetic signals to record SWs can achieve higher-quality signal recording. In this study, we discovered that optically pumped magnetometers (OPM) based on the spin exchange relaxation-free method have comparable weak magnetic detection capabilities to superconducting quantum interference devices but without liquid helium cooling. However, owing to the inevitable interference of low-frequency environmental drift, the characteristic features of SW are obscured, greatly increasing the difficulty in detecting gastric magnetic signals. Therefore, in this study, we constructed an OPM Magnetogastrography (OPM-MGG). We proposed an adaptive filtering architecture combined with environmental drift suppression and a non-stationary signal decomposition method for extracting SW signals. Through controlled human experiments, the results demonstrated that our testing system successfully extracted SW signals in the frequency range of 2–4 cycles per minute. The extracted SW signals exhibited consistent power and time–frequency characteristics with the reported results. This study validates the feasibility of (1) using the OPM-MGG system for capturing SW signals and (2) the proposed processing strategies for identifying ultralow-frequency SW signals. In conclusion, the OPM-MGG system and the signal extraction strategies developed in this study have the potential to provide a wearable technology for bioweak magnetic field measurements, offering new opportunities for both research and clinical applications.

Adaptive spectrum segmentation Ramanujan decomposition and its application to gear fault diagnosis

Ramanujan Fourier mode decomposition (RFMD) is a novel non-stationary signal decomposition method, which can decompose a complex signal into several components and extract the periodic characteristics of the signal. However, the mode generation method adopted by RFMD does not consider the physical meaning of the component signal, which makes over-decomposition when dealing with real-life gear signals with complex modulation characteristics, thus destroying the integrity of the signal sideband, increasing the difficulty of subsequent analysis, and even losing key fault information. The iterative envelope-segmentation algorithm combines the modulation characteristics of the local fault gear signal and divides the original signal into a limited number of dominant frequency bands containing the modulation region in the Fourier spectrum, thereby ensuring that the obtained frequency bands contain rich fault information. Based on the above algorithm, a new adaptive decomposition method is proposed in this paper, which is adaptive spectrum segmentation Ramanujan decomposition (ASSRD). ASSRD uses fault envelope harmonic noise ratio as the index to evaluate the fault information content of component signals and uses it to assist the iterative envelope-segmentation algorithm to complete the adaptive segmentation of the Fourier spectrum. Finally, based on the segmentation result, the inverse RFT reconstruction of each frequency band is performed. Thus, the signal is decomposed into a finite number of component signals containing rich fault information. In addition, through the experiment on the gear simulation signal and the measured crack fault gear signal, the ASSRD method is compared with the original RFMD method and the existing ensemble empirical mode decomposition (EMD), variational mode decomposition, empirical wavelet transform, and singular spectrum decomposition method, verifying the feasibility and superiority of ASSRD in gear fault diagnosis. Besides, a comparative experiment based on compound faults diagnosis is carried out, in which ensemble EMD, Fourier decomposition method, empirical wavelet transform, and sparse decomposition are involved. The results show that the proposed method can extract the local fault information in the gear signal more effectively, and the performance is better than the comparison method.

One or two frequencies? The Iterative Filtering answers

The Iterative Filtering method is a technique aimed at the decomposition of non-stationary and non-linear signals into simple oscillatory components. This method, proposed a decade ago as an alternative technique to the Empirical Mode Decomposition, has been used extensively in many applied fields of research and studied, from a mathematical point of view, in several papers published in the last few years. However, even if its convergence and stability are now established both in the continuous and discrete setting, it is still an open problem to understand up to what extent this approach can separate two close-by frequencies contained in a signal.In this paper, first we recall previously discovered theoretical results about Iterative Filtering. Afterward, we prove a few new theorems regarding the ability of this method in separating two nearby frequencies both in the case of continuously and discrete sampled signals. Among them, we prove a theorem which allows to construct filters which captures, up to machine precision, a specific frequency. We run numerical tests to confirm our findings and to compare the performance of Iterative Filtering with the one of Empirical Mode Decomposition and Synchrosqueezing methods. All the results presented confirm the ability of the technique under investigation in addressing the fundamental “one or two frequencies” question.

State space model-based harmonic decomposition of pseudo-periodic non-stationary multivariate signals

This work is concerned with the harmonic decomposition of pseudo-periodic non-stationary multivariate signals. In this framework, a signal component corresponds to a single sinusoid with time-dependent amplitude and frequency. Thus, in the proposed harmonic decomposition, all the signal components are bound to share the same fundamental time-dependent frequency. Here, we establish a procedure to yield such decomposition by casting the multivariate signal into a block-diagonal state-space representation with multiple outputs. The harmonic components and instantaneous frequency are estimated using Kalman filtering. In addition, we provide an optimisation framework for the hyperparameters of the state space representation based on the Expectation-Maximisation algorithm. The proposed method provides an accurate, compact and physically explainable procedure for analysing pseudo-periodic non-stationary multiple-output systems, as demonstrated in the case of a simulated unbalanced rotor system and on real electrocardiographic records.

Adaptive nonlinear group delay mode estimation

The decomposition of non-stationary signals remains a challenge in a wide variety of fields. Especially, the impulse or cross-mode signals are difficult to be reconstructed by recent methods due to their transient characteristic. Moreover, most methods rely heavily on the user-defined settings of the regularized parameter for the convex optimization algorithm. In this work, an adaptive nonlinear group delay mode estimation (ANGDME) algorithm is proposed by exploiting the sparsity of signals formulated as the nonlinear group delay model. The ANGDME introduces a complex Bayesian compressive sensing (CBCS) framework to process the sparse reconstruction. Then, a hierarchical Laplace scale mixture (LSM) prior is utilized to model dependencies among coefficients and provides superior probabilistic predictions. Furthermore, the estimator of amplitudes and group delays (GDs) of signals are obtained from the posterior distribution by Bayesian inference instead of point estimation. Finally, the proposed method updates the dictionary matrix in a traditional data-driven manner, resulting in a high-resolution time-frequency representation. Both simulated and experimental results illustrate the adaptability and effectiveness of the proposed method.

Nonlinear vibration feature extraction based on power spectrum envelope adaptive empirical Fourier decomposition

Analyzing the vibration features of a shipboard stabilized platform (SSP) is significant to the design or vibration compensation of a marine gravimetric survey vibration isolation system. Empirical Fourier decomposition (EFD) is a recently developed method for nonlinear and non-stationary signal decomposition. However, the spectral segmentation boundary needs to be set in advance according to sophisticated experience, and it is easy to be disturbed by noise, and the decomposition result is inaccurate. In order to accurately extract the nonlinear vibration characteristics of SSP, this paper proposes a new method called power spectrum envelope adaptive empirical Fourier decomposition (PSEEFD). Firstly, the number of selected modal decomposition is determined based on the mutual information to realize adaptive segmentation. Then, the improved power spectrum envelope segmentation method is adopted to effectively diminish the interference of noise since the segmentation boundary is formed by the minimum of the adjacent extreme points enveloped by the maximum value of the power spectrum. The spectrum segments obtained from segmentation contain less interference. Finally, the component signal in each frequency band is reconstructed by inverse fast Fourier transform, and the instantaneous frequency signal component with physical significance is obtained. Through the analysis of vibration simulation signals and measured data of SSP, the proposed method is compared with EMD, AFVMD, EWT and EFD. The results show that PSEEFD has a well suppression of noise interference and can effectively extract the characteristics of nonlinear vibration signals.

Data-driven nonstationary signal decomposition approaches: a comparative analysis

Signal decomposition (SD) approaches aim to decompose non-stationary signals into their constituent amplitude- and frequency-modulated components. This represents an important preprocessing step in many practical signal processing pipelines, providing useful knowledge and insight into the data and relevant underlying system(s) while also facilitating tasks such as noise or artefact removal and feature extraction. The popular SD methods are mostly data-driven, striving to obtain inherent well-behaved signal components without making many prior assumptions on input data. Among those methods include empirical mode decomposition and variants, variational mode decomposition and variants, synchrosqueezed transform and variants and sliding singular spectrum analysis. With the increasing popularity and utility of these methods in wide-ranging applications, it is imperative to gain a better understanding and insight into the operation of these algorithms, evaluate their accuracy with and without noise in input data and gauge their sensitivity against algorithmic parameter changes. In this work, we achieve those tasks through extensive experiments involving carefully designed synthetic and real-life signals. Based on our experimental observations, we comment on the pros and cons of the considered SD algorithms as well as highlighting the best practices, in terms of parameter selection, for the their successful operation. The SD algorithms for both single- and multi-channel (multivariate) data fall within the scope of our work. For multivariate signals, we evaluate the performance of the popular algorithms in terms of fulfilling the mode-alignment property, especially in the presence of noise.

Experimental detection of train wheel defects using wayside vibration signal processing

While many studies have been used onboard monitoring methods to detect train wheel defects, this paper aims to extract damage features from vibrational signals obtained by the wayside monitoring system. To provide an appropriate tool for the decomposition of experimental nonstationary vibration signals containing impacts, an analytical amplitude-based signal decomposition method is provided. A two-axle motor car with a flat wheel defect is studied at different operating conditions. Vibrational signals arising from wheel-rail interactions are gathered by the Wayside Data Collection Setup (WDCS) with five piezoelectric accelerometers. The tests are performed at the constant speeds of 10, 20, 30, and 40 km/h. The acquired signals are decomposed by the Multi-channel Singular Spectrum Analysis (MSSA) into the contributing components. Based on the results, increases in the amplitudes of the second reconstructed components (RCs) of the vibrational signals sensed by all the accelerometers can be observed at a specific time instance as a repeating pattern. Similar patterns are obtained at different motor car positions relative to the measurement area, as well as different motor car speeds. To verify the consistency of the second RCs, the crest factors are obtained for all the angular positions. The results show sudden increases in the crest factor at a fixed angular position that indicate the existence of a defect at that specific angular position of the wheel.