Non-stationary Signals Research Articles

Instantaneous frequency identification for nonstationary signals of time-varying structures using enhanced synchroextracting wavelet transform and dynamic optimization

Civil engineering structures under ambient excitations are essentially time-varying and nonlinear structural systems and the resultant response signals exhibit non-stationarity. To reveal the time-varying characteristics of non-stationary signals, time frequency analysis methods with high resolutions are required. Although synchroextracting short time Fourier transform (SESTFT) is able to generate more energy concentrated time frequency representations and allow signal reconstruction, its major disadvantage is the window function is fixed. To address this issue, an enhanced synchroextracting wavelet transform (ESEWT) method is proposed to refine frequency bands by combing synchroextracting and continuous wavelet transform. After that, dynamic optimization (DO) is used to extract instantaneous frequency (IF) curves within the refined frequency bands. Two numerical examples and an experimental study case are investigated to illustrate the effectiveness and accuracy of the proposed method. The results demonstrate that the proposed ESEWT is capable of extracting frequency bands more accurately and its combination with DO identifies IFs of non-stationary signals better than current time frequency analysis methods such as SESTFT.

Optimal tuning of three deep learning methods with signal processing and anomaly detection for multi-class damage detection of a large-scale bridge

Long-span bridges play a crucial role in urbanization, connecting communities across vast obstacles. Structural health monitoring techniques have been deployed on these bridges, generate large amounts of data through sensor measurements, requiring data-driven approaches like deep learning (DL) for effective analysis. However, feature extraction from time-domain vibration response signals poses challenges for DL methods. To address this, the study proposes utilizing signal processing techniques such as the multivariate empirical mode decomposition (MEMD) and Wavelet transform (WT) to extract essential features for damage classification. The incorporation of MEMD and WT aims to overcome limitations and process nonstationary and nonlinear signals effectively. Three DL techniques, long-short-term memory (LSTM), one dimensional convolutional neural network (1D-CNN), multi-layer perceptron (MLP) are tuned and applied to Structural Health Monitoring of Tianjin Yonghe Bridge (located in China) as a real-world case study, in order to detect its condition by Deep signal anomaly detection and identify types of the damage. A powerful meta-heuristic algorithm called Observer-Teacher-Learner-Based Optimization, is used to optimize both hyperparameters and architecture of each DL models. The results demonstrate that the optimally tuned DLs are successful in identifying types of damage, as well as the condition of the structure, for the Tianjin Yonghe Bridge. The average accuracy values are obtained as 98.13, 97.96, and 97.79 for 1D-CNN, LSTM, and MLP, respectively. Such optimally tuned DLs are evaluated as effective solutions for detecting damage on large-scale bridges by extracting statistical time-domain and time–frequency domain features using the WT and MEMD.

A novel feature extraction method: iterative Ramanujan Fourier mixture spectrum

As a classical demodulation method, envelope spectrum (ES) has been used in rotating machinery fault diagnosis. However, for strong noise or multiple faults, the feature extraction ability of ES is not outstanding, even if it is unable to extract obvious fault features. Fourier transform (FT) in ES analysis is also not suitable for analyzing non-stationary signals. Considering the shortcomings of FT, this paper defines the Ramanujan Fourier mixture transform (RFMT). The RFMT has outstanding period feature extraction ability and can accurately extract the periodic features in the envelope signal. Based on this, this paper proposes the iterative Ramanujan Fourier mixture spectrum (IRFMS) method. The IRFMS enhances the fault features in the signal by iterative envelope and selects the envelope signals with obvious fault features by cyclic kurtosis. The results of signal analysis verify that IRFMS has excellent fault feature extraction ability, which is suitable for various fields of rotating machinery fault diagnosis.

Status Recognition of Marine Centrifugal Pumps Based on a Stacked Sparse Auto-Encoder

Marine centrifugal pumps (MCPs) are widely used in ships, so it is important to identify their status accurately for their maintenance. Due to the influence of load, friction, and other non-linear factors, the vibration signal of an MCP shows non-linear and non-stationary characteristics, and it is difficult to extract the state characteristics contained in the vibration signal. To solve the difficulty of feature extraction of non-linear non-stationary vibration signals generated by MCPs, a novel MCP frequency domain signal feature extraction method based on a stacked sparse auto-encoder (SSAE) is proposed. The characteristic parameters of MCP frequency domain signals are extracted via the SSAE model for classification training, and different statuses of MCPs are identified. The vibration signals in different MCP statuses were collected for feature extraction and classification training, and the MCP status recognition accuracy based on the time domain feature and fuzzy entropy feature was compared. According to the test data, the accuracy of MCP status recognition based on the time domain feature is 71.2%, the accuracy of MCP status recognition based on the fuzzy entropy feature is 87.7%, and the accuracy of MCP status recognition based on the proposed method is 100%. These results show that the proposed method can accurately identify each status of an MCP under test conditions.

Relaxed evolutionary power spectral density functions: A probabilistic approach to model uncertainties of non-stationary stochastic signals

The identification of patterns and underlying characteristics of natural or engineering time-varying phenomena poses a challenging task, especially in the scope of simulation models and accompanying stochastic models. Because of their complex nature, time-varying processes such as wind speed, seismic ground motion, or vibrations of machinery in the presence of degradation oftentimes lack a closed-form description of their underlying Evolutionary Power Spectral Density (EPSD) function. To overcome this issue, a wide range of measurements exist for these types of processes. This opens up the path to a data-driven stochastic representation of EPSD functions. Rather than solely relying on time–frequency transform methods like the familiar short-time Fourier transform or wavelet transform for EPSD estimation, a probabilistic representation of the EPSD can provide valuable insights into the epistemic uncertainty associated with these processes. To address this problem, the evolutionary EPSD function is relaxed based on multiple similar data to account for these uncertainties and to provide a realistic representation of the time data in the time–frequency domain. This results is the so-called Relaxed Evolutionary Power Spectral Density (REPSD) function, which serves as a modular probabilistic representation of the time–frequency content of stochastic signals. For this purpose, truncated normal distributions and kernel density estimates are used to determine a probability density function for each time–frequency component. The REPSD function enables the sampling of individual EPSD functions, facilitating their direct application to the simulation model through stochastic simulation techniques like Monte Carlo simulation or other advanced methods. Even though the accuracy is highly dependant on the data available and the time–frequency transformation method used, the REPSD representation offers a stochastic representation of characteristics used to describe stochastic signals and can reduce epistemic uncertainty during the modelling of such time-varying processes. The method is illustrated by numerical examples involving the analysis of dynamic behaviour under random loads. The results show that the method can be successfully employed to account for uncertainties in the estimation of the EPSD function and represent the accuracy of the time–frequency transformation used.

Application of matrix in signal processing

Signal processing, a foundational discipline in modern technology, encompasses a diverse array of applications, ranging from audio and image processing to communication systems and medical imaging. This review investigates how matrix-based techniques are widely used to advance signal processing methodologies. In order to discretize continuous-time signals for digital processing, which occurs in the first section of the paper, matrices play a crucial role in signal sampling. A key principle, the Nyquist-Shannon Sampling Theorem, directs appropriate sampling rates to prevent aliasing, with matrices permitting effective signal representation. The effectiveness of matrix-based filtering methods for frequency modulation and noise reduction, such as convolution and correlation, is then investigated. By utilising matrix operations, these methods enable real-time signal processing. The Fourier Transform and Wavelet Transform are also featured in matrix-driven signal transformation, providing insights into frequency analysis and non-stationary signal characterization. By reducing noise components, matrix-based approaches, particularly Singular Value Decomposition (SVD) denoising, are essential for improving signal quality. Additionally, image compression employs SVD. Matrix-based compressive sensing revolutionises signal recovery from sparse data and results in data-efficient reconstruction. Signal processing has been transformed by matrix-based approaches, which have enabled previously unheard-of levels of efficiency, accuracy, and adaptability. The review highlights their significant influence on several signal processing fields.

Energy distribution and chaotic pressure pulsation analysis of vortex ropes in Francis-99

Francis turbines, essential for stability in diverse operating conditions and variable-speed scenarios, encounter efficiency-compromising vortex rope formations in the draft tube, leading to substantial pressure fluctuations. This research delves into the analysis of energy loss and pressure fluctuations associated with these vortex ropes. Employing the local entropy generation rate (LEGR) method and chaos theory, we scrutinize the behaviour of vortex ropes and their resultant pressure fluctuations. Notably, vortex ropes exhibit maximum LEGR near the runner cone, with secondary vortices escalating instability downstream. In the elbow section, the collision of vortex ropes with the outer elbow amplifies LEGR, primarily driven by fluctuating velocities (approximately 90%). Leveraging the GWO-VMD algorithm, non-stationary signals are decomposed, unveiling a significant 1.6 Hz vortex rope frequency under partial load (PL) conditions and isolating external noise frequencies, such as the prominent 300 Hz. Following decomposition, chaos theory tools, including phase space reconstruction and phase trajectory graphs, unveil the chaotic nature of PL conditions attributed to spiral vortex ropes, resulting in profound pressure fluctuations. This study enhances our understanding of such systems and provides methodologies for improved noise reduction and optimization of turbine performance.

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.

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.

Time-varying modal identification of structures under seismic excitations using a novel time-frequency method

Civil engineering structures are susceptible to damage when subjected to seismic excitations, resulting in presenting nonlinear features and exhibiting time-varying dynamic behavior of structural modal properties. The instantaneous frequency (IF) of structures is thus an essential indicator to evaluate structural damage state. Time-varying filtering empirical mode decomposition (TVF-EMD) holds promise for non-stationary signals decomposition and IF identification due to its robustness against noise interference. However, the application of TVF-EMD is limited by the requirement for manually presetting the decomposition parameters (i.e., B-spline order and bandwidth threshold). In this study, a novel IF identification method based on an improved TVF-EMD algorithm is proposed. In this method, TVF-EMD is firstly improved to accurately decompose the measured responses into mono-component modal responses, which adopts orthogonal coefficient and energy kurtosis as the objective function of salp swarm algorithm (SSA) to adaptively determine the optimal decomposition parameters. Then, the resulting mono-component modal responses are analyzed by local maximum synchrosequencing transform (LMSST) to obtain structural IF under seismic excitations. Examples of non-stationary signals of sinusoidal function, Duffing system, and simulated two-story shear building model under El Centro seismic excitations are used to illustrate the advantages of the proposed method, which proves to be successful in signal decomposition and IF identification. Furthermore, the proposed method is applied to a shaking table test and a real building-Van Nuys hotel under seismic excitations to identify their IF and evaluate structural damage states. The results reveal that the proposed method is efficient for structural IF identification, and can be utilized to qualitatively evaluate the damage severity of structures subjected to strong seismic excitations.

Improved lightweight federated learning network for fault feature extraction of reciprocating machinery

Abstract The working environment of reciprocating machinery is complex, characterized by nonlinear and non-stationary signals. Deep learning can solve the above problems, but it has its own problems such as complex model and large amount of parameters. Additionally, privacy considerations among enterprises prevent data sharing, leading to the emergence of ‘data islands’ and inadequate training of deep learning models. Based on the above analysis, this paper proposes a reciprocating mechanical feature extraction method based on an improved federated lightweight network. A lightweight network SqueezeNet model is used to solve the problems such as long training time of deep learning. By establishing a federated learning framework, the reciprocating mechanical data can be collectively diagnosed across various enterprises, thereby addressing the problem of limited model training caused by insufficient data. Furthermore, to enhance the accuracy of network training and diagnosis, modifications are made to the SqueezeNet network to reduce the number of model parameters while increasing the number and variety of feature extractions. Experimental results demonstrate that when the number of 1 × 1 and 3 × 3 channels is 1–7, the fault diagnosis accuracy is the highest, about 97.96%, which enriches the categories of feature extraction. The number of parameters in In-SqueezeNet is 56% of that in SqueezeNet network model, and the training time is reduced by nearly 15%. The fault diagnosis accuracy is increased from 95.1% to 97.3%, and the diversity of extracted features is increased. Compared with other network models such as ResNet, the improved lightweight federated learning network has a fault diagnosis accuracy of 96.6%, an improvement of 10.6%. At the same time, the training time was reduced to 1982s, a reduction of about 41.5%. The validity of the proposed model is further verified.

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.

Gated recurrent unit predictor model-based adaptive differential pulse code modulation speech decoder

Speech coding is a method to reduce the amount of data needs to represent speech signals by exploiting the statistical properties of the speech signal. Recently, in the speech coding process, a neural network prediction model has gained attention as the reconstruction process of a nonlinear and nonstationary speech signal. This study proposes a novel approach to improve speech coding performance by using a gated recurrent unit (GRU)-based adaptive differential pulse code modulation (ADPCM) system. This GRU predictor model is trained using a data set of speech samples from the DARPA TIMIT Acoustic-Phonetic Continuous Speech Corpus actual sample and the ADPCM fixed-predictor output speech sample. Our contribution lies in the development of an algorithm for training the GRU predictive model that can improve its performance in speech coding prediction and a new offline trained predictive model for speech decoder. The results indicate that the proposed system significantly improves the accuracy of speech prediction, demonstrating its potential for speech prediction applications. Overall, this work presents a unique application of the GRU predictive model with ADPCM decoding in speech signal compression, providing a promising approach for future research in this field.

Synchronous odd symmetric transform for rolling bearing fault diagnosis

Time-frequency analysis (TFA) is an important tool to process non-stationary signals and detect bearing faults. For considering energy concentration, computational efficiency, and computational accuracy, a novel TFA technique, termed synchronous odd symmetric transform (SOST) is proposed in this paper. In the SOST, the odd symmetry extraction operator (OSEO) is defined according to the odd symmetric window based short-time Fourier transform (OSTFT). The main idea of OSEO is to obtain accurate instantaneous frequency (IF) ridges in a satisfactory computation time. Meanwhile, the STFT result is retained at the IF ridges, and the time–frequency representation (TFR) with high energy concentration is calculated. The performance of the developed technique is verified by a multi-component signal with strong frequency-modulated and amplitude-modulated (FM-AM) laws. Experimental analysis results show that the proposed SOST can display bearing fault-related time–frequency trajectories with high time–frequency resolution.

Fault diagnosis using signal processing and deep learning-based image pattern recognition

Abstract The vibration signal is a typical non-stationary signal, making it challenging to use traditional time-frequency analysis techniques for fault diagnosis. Therefore, this work investigates the processing of vibration signals and proposes a deep learning method based on processed signals for the fault diagnosis of ball bearings. In this work, the fault diagnosis is formulated as an image classification problem and solved with deep learning networks. The intrinsic mode functions (IMFs), converted from the vibration signals in the time domain, are then transformed into symmetrized dot pattern (SDP) images. In order to increase classification accuracy, the SDP parameters in this study are chosen by optimizing image similarity. The feasibility and accuracy of the proposed approach are examined experimentally.

Quaternion empirical Ramanujan Fourier decomposition and its application in gear fault diagnosis

In the field of gear health monitoring, the measured signals are traditionally derived from a single sensor or a single direction. However, with the increasing complexity and size of equipment structures nowadays, a single-channel signal often falls short in providing comprehensive status information. Furthermore, fault-induced vibrations are randomly indeterminate in both location and direction, which indicates the necessity of using multi-channel signals for effective gear fault diagnosis. As a single-channel method, Ramanujan Fourier transform (RFT) has been recognized for its effectiveness in extracting periodic components from signals. However, the RFT is limited to handling single-channel signals and lacks the ability to analyze nonstationary signals. To overcome these challenges, this paper proposes a novel multi-channel signal processing method called quaternion empirical Ramanujan Fourier decomposition (QERFD), which extends the RFT to the multi-channel field using the quaternion method. By fusing multi-channel signals while preserving the advantages of RFT, the proposed QERFD method enhances fault information extraction. In addition, QERFD shows high computational efficiency. Therefore, QERFD is of great practical value for the health monitoring of complex equipment in the present industrial context of big data. To substantiate the superiority of QERFD, simulations and experiments are carried out and compared against both single-channel and multi-channel methods.

Wavelet-based feature extraction with hidden Markov model classification of Antarctic blue whale sounds

An approach for analyzing blue whale vocalizations in continuous acoustic recordings using wavelet transform (WT) algorithm as a feature extraction method is proposed in this study. The WT-based feature extraction technique offers a unique perspective on the frequency-time properties of whale vocalizations, enabling a comprehensive understanding of the complexity of non-stationary signals. The WT-based features are seamlessly adopted with a Hidden Markov Model (HMM) for efficient and accurate classification. Through extensive theoretical evaluation and comparison with conventional methods such as Principal Component Analysis (PCA) and Dynamic Mode Decomposition (DMD), the proposed approach demonstrates superior performance in accurately detecting and classifying blue whale calls from background noise. Furthermore, when benchmarked against state-of-the-art machine learning techniques such as Artificial Neural Network (ANN), Support Vector Machine (SVM), and Spectrogram Correlation Detector (SCD), the proposed WT-HMM detector exhibits the highest average accuracy, affirming its applicability for real-time scenarios. This study presents a valuable contribution to the field of marine bioacoustics, offering an effective solution for robust blue whale vocalization analysis.

Comparative Analysis of Artificial Neural Network (ANN) and Wavelet Integrated Artificial Neural Network (W-ANN) Approaches for Rainfall Modeling of Southern Rajasthan, India

This paper addresses the challenge of predicting erratic rainfall in Rajasthan state of India, particularly in southern regions. Reliable rainfall predictions are crucial for water resource management and agriculture planning. The research involved selecting 58 stations across seven districts of southern Rajasthan and identifying the best fit computational neural (ANN) and wavelet integrated computational neural (W-ANN) architectures based on performance metrics. Different combinations of input characters, hidden layer neurons, learning algorithms, and training cycles were tested to determine optimal models. Hybrid models, combining wavelet analysis with ANN, were explored to tackle non-stationary hydrologic signals effectively. Results showed that ANN Model C with ten input layer neurons performed best for 74% of stations, followed by Model B (21% of stations) and Model A (5% of stations). Models with increased input and hidden layer neurons performed better. Among the selected stations, 81% of stations demonstrated improved performance using W-ANN models due to effective signal decomposition and information extraction. The hybrid W-ANN models outperformed simple ANN models for rainfall prediction. Both ANN and W-ANN models accurately forecasted weekly rainfall, as observed in the comparison of actual and forecasted values.

Order-frequency Holo-Hilbert spectral analysis for machinery fault diagnosis under time-varying operating conditions

Holo-Hilbert spectral analysis (HHSA) has been demonstrated to be an effective instantaneous feature demodulation tool for revealing the coupling relationship between the frequency-modulated (FM) carriers and amplitude-modulated (AM) characteristics within nonlinear and non-stationary mechanical vibration signals. However, it is unable to acquire the time varying AM characteristics from the vibration signals of the equipment operates under variable speed conditions. To decode such signals, inspired by HHSA, a novel angle-time double-layer decomposition structure termed order-frequency HHSA (OFHHSA) is established to demodulate the fault information from the time varying vibration signals in this paper. The corresponding spectrogram, namely, order-frequency Holo-Hilbert spectrum (OFHHS) is acquired for describing the interaction relationship between time and angle domains. Besides, the order AM-marginal spectrum is derived from the OFHHS via integrating the carrier variable to exhibit the fault characteristic-related orders. Moreover, the differences between OFHHSA and angle-time cyclo-stationary framework-based order-frequency spectral correlation (OFSC) are analyzed for time varying machinery fault diagnosis. Finally, from the analyses of simulated and tested data of mechanical equipment, the OFHHSA method has avoided the limitations of the two OFSC estimators on periodic assumption and the maximum cut-off order, and the proposed method obtained a more accurate rate in fault identification and more robust ability of anti-noise.

A novel time-frequency slice extraction method for target recognition and local enhancement of non-stationary signal features

Blind deconvolution can remove the effects of complex paths and extraneous disturbances, thus recovering simple features of the original fault source, and is used extensively in the field of fault diagnosis. However, it can only identify and extract the statistical mean of the fault impact features in a single domain and is unable to simultaneously highlight the local features of the signal in the time-frequency domain. Therefore, the extraction effect of weak fault signals is generally not ideal. In this paper, a new time-frequency slice extraction method is proposed. The method first computes a high temporal resolution spectrum of the signal by short-time Fourier transform to obtain multiple frequency slices with distinct temporal waveforms. Subsequently, the constructed harmonic spectral feature index is used to quantify and target the intensity of feature information in each frequency slice and enhance their fault characteristics using maximum correlation kurtosis deconvolution. Enhancing the local features of selected frequency slice clusters can reduce noise interference and obtain signal components with more obvious fault signatures. Finally, the validity of the method was confirmed by a simulated signal and fault diagnosis of the rolling bearing outer and inner rings was accomplished sequentially. Compared with other common deconvolution methods, the proposed method obtains more accurate and effective results in identifying fault messages.