Non-stationary Signal Processing Research Articles

Fault feature extraction method based on maximum envelope spectrum PFGI2 and empirical wavelet transform

Fast kurtogram (FK) is an efficient method for processing non-stationary signals, widely recognized by scholars as a rapid and effective approach for fault diagnosis. However, it has limitations in distinguishing between periodic pulse and random interference pulses due to the drawbacks in its frequency band segmentation methods and the inherent shortcomings of the kurtosis index itself. To address this, this paper proposes a fault feature extraction method based on the maximum envelope spectrum power function-based Gini indices (PFGI2) and empirical wavelet transform. This method, inspired by the concept of FK, constructs a series of band-pass filters following the principles of empirical wavelet decomposition. It applies envelope spectrum analysis to a series of sub-bands and calculates the PFGI2 value for each, to identify the optimal sub-band. The effectiveness of the proposed method is validated through simulations of vibration signals and experimental data.

Estimation of realized volatility of cryptocurrencies using CEEMDAN-RF-LSTM

Predicting cryptocurrency volatility is crucial for investors, traders, and decision-makers but is complicated by the market’s high non-linearity, volatility, and noise. This paper presents a novel approach, the CEEMDAN-RF-LSTM hybrid model, which is the first to combine the strengths of Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN), Random Forest (RF), and Long Short-Term Memory Network (LSTM) to predict the Realized Volatility (RV) of mainstream cryptocurrencies. The model exploits CEEMDAN’s proficiency in processing non-linear and non-stationary signals, RF’s exceptional feature selection capabilities, and LSTM’s distinctive advantages in dealing with time-series problems. Applied to actual transaction data for Bitcoin (BTC), Ethereum (ETH), and Binance Coin (BNB), empirical results show the superior performance of our model in predicting actual cryptocurrency volatility. These findings contribute to the academic understanding of cryptocurrency volatility and provide practical guidance for quantitative trading strategy development, offering fresh insights and methodologies for related research fields.

The Synchrosqueezed Method and Its Theory-Analysis-Based Novel Short-Time Fractional Fourier Transform for Chirp Signals

Time–frequency analysis is an important tool used for the processing and interpretation of non-stationary signals, such as seismic data and remote sensing data. In this paper, based on the novel short-time fractional Fourier transform (STFRFT), a new modified STFRFT is first proposed which can also generalize the properties of the modified short-time Fourier transform (STFT). Then, in the modified STFRFT domain, we derive the instantaneous frequency estimator for the chirp signal and present a new type of synchrosqueezing STFRFT (FRSST). The proposed FRSST presents many results similar to those of the synchrosqueezing STFT (FSST), and it extends the harmonic signal to a chirp signal that offers attractive new features. Furthermore, we provide a detailed analysis of the signal reconstruction, theories, and some properties of the proposed FRSST. Several experiments are conducted, and all of the results illustrate that the proposed FRSST is more effective than the FSST. Finally, based on the linear amplitude modulation and frequency modulation signal, we present a derivation for analyzing the limitations of the FRSST.

Permutation fuzzy entropy based ICEEMDAN de-noising for inertial sensors

As one of the critical factors affecting the accuracy of inertial navigation system (INS) positioning, random noise will lead to the accumulation of INS positioning errors, even the failure of the system. Previous studies mainly focus on achieving noise reduction by adopting either the wavelet decomposition method or some relevant methods of the empirical mode decomposition (EMD). However, the former is limited by Heisenberg’s uncertainty principle when processing non-stationary signals, and the latter exhibits numerous mode-mixing phenomena during the decomposition process. In this paper, a permutation fuzzy entropy (PFE) based improved complete ensemble EMD with adaptive noise (ICEEMDAN) de-noising method is proposed to utilize the ICEEMDAN method for signal decomposition, and the PFE as an index to distinguish the effective intrinsic mode functions (IMFs) and the noisy IMFs during the decomposition process. In addition, simulated signals with different noise levels and field data with different durations are utilized to design a group of experiments to test the de-noising performance of the PFE-based ICEEMDAN de-noising method. The experimental results demonstrate that the advantages of our proposed method consists of three main parts: (1) The ICEEMDAN method more effectively alleviates the mode-mixing problem compared to the complete ensemble EMD with adaptive noise (CEEMDAN), thereby enhancing the accuracy of inertial measurement unit (IMU) signal reconstruction; (2) the PFE is more preferable than the permutation entropy for classify IMFs decomposed by the ICEEMDAN method; (3) our proposed method can effectively reduce IMU signal noise and improve the positioning accuracy of inertial sensor.

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.

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.

ENSEMBLE OF ADAPTIVE PREDICTORS FOR MULTIVARIATE NONSTATIONARY SEQUENCES AND ITS ONLINE LEARNING

Context. In this research, we explore an ensemble of metamodels that utilizes multivariate signals to generate forecasts. The ensemble includes various traditional forecasting models such as multivariate regression, exponential smoothing, ARIMAX, as well as nonlinear structures based on artificial neural networks, ranging from simple feedforward networks to deep architectures like LSTM and transformers.
 Objective. A goal of this research is to develop an effective method for combining forecasts from multiple models forming metamodels to create a unified forecast that surpasses the accuracy of individual models. We are aimed to investigate the effectiveness of the proposed ensemble in the context of forecasting tasks with nonstationary signals.
 Method. The proposed ensemble of metamodels employs the method of Lagrange multipliers to estimate the parameters of the metamodel. The Kuhn-Tucker system of equations is solved to obtain unbiased estimates using the least squares method. Additionally, we introduce a recurrent form of the least squares algorithm for adaptive processing of nonstationary signals.
 Results. The evaluation of the proposed ensemble method is conducted on a dataset of time series. Metamodels formed by combining various individual models demonstrate improved forecast accuracy compared to individual models. The approach shows effectiveness in capturing nonstationary patterns and enhancing overall forecasting accuracy.
 Conclusions. The ensemble of metamodels, which utilizes multivariate signals for forecast generation, offers a promising approach to achieve better forecasting accuracy. By combining diverse models, the ensemble exhibits robustness to nonstationarity and improves the reliability of forecasts.

Motor Bearing Fault Diagnosis in an Industrial Robot Under Complex Variable Speed Conditions

Abstract Motor bearing is the key vulnerable part of the servomotor in an industrial robot, which is always arranged at the joint that is the main load area. In the movement process of the robot, motor bearing bears a great impact due to the frequent movement of joints, which is easily damaged. The fault characteristic information of a bearing in these complex conditions shows strong nonstationary characteristics. Early nonstationary fault signals are often weak and submerged in background noise. The nonstationary signal processing method using computed order analysis and the weak signal enhancement method using adaptive stochastic resonance both show good performances for the above problems. Inspired by these, a hybrid diagnosis strategy for motor bearing under these speed conditions is proposed. Firstly, the nonstationary fault signals of the motor bearing are transformed into stationary angular signals via computed order analysis. Then, the fault modes are identified via resonance demodulation and variational mode decomposition in the order spectrum. Finally, adaptive stochastic resonance is used to extract the fault features reflecting the bearing operation state. Two types of typical speed conditions are considered, which are representative of the joint. Numerical simulation analysis and experiments verify the effectiveness of the diagnosis method.

Generalized convolution and product theorems associated with the free metaplectic transformation and their applications

Convolution theorems for the free metaplectic transformation (FMT) have great significance in the fields of fractional domain non-stationary signal processing theory and method. The existing results have found many applications in high-dimensional non-stationary signal sampling and filtering. However, a widely accepted closed-form expression has not yet been established. In this paper, we introduce the definitions of generalized convolution and product operators in the FMT domain, and use them to formulate the generalized convolution and product theorems for the FMT. The derived results include particular cases the existing convolution and product theorems for the FMT and also the convolution and product theorems for the Fourier transform, the fractional Fourier transform, and the linear canonical transform. We employ the generalized convolution theorem for the FMT in designing an optimal filter in the FMT domain. We also validate the correctness and effectiveness of the theoretical analysis through numerical experiments, demonstrating the superiority of the proposed optimal filtering method over the conventional ones.

A novel time-frequency analysis method for fault diagnosis based on generalized S-transform and synchroextracting transform

Rotating machinery usually operates under variable-speed conditions, and how to effectively handle nonstationary signal in fault diagnosis is a critical task. The time-frequency analysis (TFA) method is widely used in the processing of nonstationary signal. To improve the time-frequency resolution and clearly identify instantaneous frequency (IF) characteristics, the adaptive generalized S-synchroextracting transform (AGSSET), which is a novel TFA method proposed in this paper. Firstly, a new transform named adaptive generalized S-transform (AGST) is put forward by optimizing the window function of generalized S transform. In this paper, an adaptive window function optimization method based on the frequency spectrum of the vibration signal is introduced, and the energy concentration measure is used to determine the window function’s parameters in AGST. Simultaneously, the synchrony extraction idea is incorporated into the AGST, then the AGSSET is derived. To address more complex IF characteristics, the synchronous extraction operator (SEO) is reconstructed. In the simulation experiment, the GMLC signal model is selected to represent nonstationary signal and to verify the effectiveness of the proposed method. In addition, bearing fault data is also used for fault diagnosis experiments. The results of both numerical simulation and experimental analysis indicate that AGSSET performs well in identifying the time-varying IF characteristic in nonstationary signals. It can also efficiently detect faults with high accuracy and strong stability.

Multipath time delay estimation based on the improved Wigner-ville distribution

The accuracy of underwater multipath channel time delay is crucial in underwater acoustic signal processing. The temporal resolution of the traditional matched filter is restricted by the transmitted signal bandwidth. Wigner-Ville distribution (WVD) which has excellent energy concentration characteristics and effectively avoids the interaction between temporal and frequency resolutions, and is widely used in non-stationary signal processing. However, considering the multipath signal processing, the cross-term in echo WVD would cause inaccurate time delay estimation. Therefore, in this paper, a novel multipath signal time delay estimation method based on the improved WVD is proposed. By utilizing the difference in the continuity of energy changes between auto-term and cross-term in the time-frequency domain, the Seeded Region Growing algorithm is applied and improved to extract auto-term in the WVD. The simulation results show that the method removes cross-term and maintains high concentration characteristics of auto-term, compared with pseudo-WVD and smooth pseudo-WVD, etc. In addition, it improves the accuracy of multipath signal delay estimation and has some anti-noise ability.

Vibration amplitude normalization enhanced fault diagnosis under conditions of variable speed and extremely limited samples

Intelligent fault diagnosis of rotating equipment is increasingly reliant on algorithms that are driven by big data. By contrast, signal processing was once widely utilized for fault diagnosis in machinery as a classical tool for signal analysis due to its capability to investigate the fault-related mechanism and almost no demand on the number of data samples. This investigation was motivated by the notion that signal processing and data-driven algorithms are combined to exploit their respective characteristics and strengths. Furthermore, in engineering practice, numerous complex factors such as time-variable operating conditions of equipment, non-stationary properties of signals, and extremely limited samples available for model training, can make it difficult to learn discriminative features from input data, thereby diminishing the diagnostic accuracy. In this paper, a novel framework of vibration amplitude normalization (VAN) enhanced fault diagnosis is proposed. Firstly, after dissects deeply the effects of the time-varying speed conditions on vibration signal and its characteristics, VAN technique is proposed for non-stationary signal processing to obtain the approximate stationary signal, so as to facilitate the subsequent state characteristics mining from the vibration signal. Then, two VAN enhanced fault diagnosis methods—i.e. signal amplitude normalization integrated with shallow learning by cascade and VAN integrated with deep learning by embedding—are developed to capture discriminative features from approximate stationary signal for fault diagnosis under conditions of variable speed and extremely limited samples. Finally, the feasibility and effectiveness of the proposed methods are verified using actual vibration datasets measured on test rig and in-site wind turbines. The number of samples required to achieve the same diagnostic accuracy is reduced by an average of 60%, demonstrating the superiority.

Climate oscillation impacts on water supply augmentation planning

Climate oscillations ranging from years to decades drive precipitation variability in many river basins globally. As a result, many regions will require new water infrastructure investments to maintain reliable water supply. However, current adaptation approaches focus on long-term trends, preparing for average climate conditions at mid- or end-of-century. The impact of climate oscillations, which bring prolonged and variable but temporary dry periods, on water supply augmentation needs is unknown. Current approaches for theory development in nature-society systems are limited in their ability to realistically capture the impacts of climate oscillations on water supply. Here, we develop an approach to build middle-range theory on how common climate oscillations affect low-cost, reliable water supply augmentation strategies. We extract contrasting climate oscillation patterns across sub-Saharan Africa and study their impacts on a generic water supply system. Our approach integrates climate model projections, nonstationary signal processing, stochastic weather generation, and reinforcement learning-based advances in stochastic dynamic control. We find that longer climate oscillations often require greater water supply augmentation capacity but benefit more from dynamic approaches. Therefore, in settings with the adaptive capacity to revisit planning decisions frequently, longer climate oscillations do not require greater capacity. By building theory on the relationship between climate oscillations and least-cost reliable water supply augmentation, our findings can help planners target scarce resources and guide water technology and policy innovation. This approach can be used to support climate adaptation planning across large spatial scales in sectors impacted by climate variability.

Time-synchroextracting of generalized S-transform and its application in fault identification

Seismic signals are typical non-stationary signals, and time-frequency analysis method is a powerful tool for processing non-stationary signals, which is also widely used in seismic signal processing. S-transform is a frequently used time-frequency analysis method. However, the time and frequency resolution of S-transform is limited due to Heisenberg uncertainty principle. The post-processing method of time-frequency analysis can solve this problem by further calculation on the basis of the original time-frequency spectrum of the signal. In order to improve the time and frequency resolution, we propose a new time-frequency analysis post-processing method - time synchroextracting of generalized S-transform algorithm. We first deduce the group delay operator expression of the generalized S-transform, and then remove a large amount of fuzzy energy by extracting only the time-frequency coefficients at the group delay operator in the time-frequency spectrum to improve the time-frequency resolution. The comparative analysis of synthetic signals shows that the proposed algorithm has better time-frequency aggregation and can more accurately describe the time-frequency characteristics of seismic signals. We apply the proposed method to the fault identification of the field seismic data, and the results show that the coherent attribute slice extracted by the time-synchroextracting of the generalized S-transform can well describe the fault.

Vibration-based looseness identification of bolted structures via quasi-analytic wavelet packet and optimized large margin distribution machine

Bolted joints are the most widely utilized connection types in industries, and therein looseness identification of bolted structures is of great significance to guarantee structural reliability. In this article, a comprehensive study of bolt looseness identification under random excitation is presented. To fulfill this task, this research focuses on three prominent difficulties, including nonstationary signal processing, subtle feature extraction, and robust state classification. First, a novel filter bank structure of quasi-analytic dual-tree complex wavelet packet transform is constructed to analyze the measured vibration response signals, for purpose of capturing subtle feature information. Then, multiple features are extracted from subband signals to capture the variations of dynamic characteristics, and sensitive features are selected by Laplacian score to construct the low-dimensional feature set. Subsequently, a novel classifier with better generalization performance, named large margin distribution machine, is optimized with the wavelet kernel function and the whale optimization algorithm, in order to handle the intrinsic uncertainty related to the looseness states of bolted structures. After feeding the low-dimensional feature set, the proposed classifier is trained to identify looseness states of bolted structures. Finally, experiments of a two-bolt lapped beam under random excitation are conducted to verify the effectiveness of the proposed method, and two typical loading conditions (paired-bolt looseness and single-bolt looseness) are considered. Besides, the superiority of the proposed method is demonstrated by comparing with other analogical methods. This research can provide a promising implement in practical applications of bolt looseness identification under random excitation.

Advanced Video-Based Processing for Low-Cost Damage Assessment of Buildings under Seismic Loading in Shaking Table Tests.

This paper explores the potential of a low-cost, advanced video-based technique for the assessment of structural damage to buildings caused by seismic loading. A low-cost, high-speed video camera was utilized for the motion magnification processing of footage of a two-story reinforced-concrete frame building subjected to shaking table tests. The damage after seismic loading was estimated by analyzing the dynamic behavior (i.e., modal parameters) and the structural deformations of the building in magnified videos. The results using the motion magnification procedure were compared for validation of the method of the damage assessment obtained through analyses of conventional accelerometric sensors and high-precision optical markers tracked using a passive 3D motion capture system. In addition, 3D laser scanning to obtain an accurate survey of the building geometry before and after the seismic tests was carried out. In particular, accelerometric recordings were also processed and analyzed using several stationary and nonstationary signal processing techniques with the aim of analyzing the linear behavior of the undamaged structure and the nonlinear structural behavior during damaging shaking table tests. The proposed procedure based on the analysis of magnified videos provided an accurate estimate of the main modal frequency and the damage location through the analysis of the modal shapes, which were confirmed using advanced analyses of the accelerometric data. Consequently, the main novelty of the study was the highlighting of a simple procedure with high potential for the extraction and analysis of modal parameters, with a special focus on the analysis of the modal shape's curvature, which provides accurate information on the location of the damage in a structure, while using a noncontact and low-cost method.

Synchronous fault feature extraction for rolling bearings in a generalized demodulation framework

Generalized demodulation (GD) has the potential to process non-stationary vibration signals since it can demodulate a signal with a curved time–frequency (TF) ridge into a signal, with a TF ridge parallel to a time axis with an improved time-frequency representation (TFR) energy concentration level. However, current GD methods require iteration operations and cannot simultaneously deal with vibrations from multiple components of rolling bearings. This paper proposes a method based on the GD framework, which can simultaneously demodulate multiple components of interest using the Hadamard product between matrices. A synchronous extractor is also constructed to post-process TFRs of generalized demodulated signals to further improve the TF aggregation. Unlike the conventional synchronous extraction transform, the synchronous extractor in this paper can be directly applied to TF ridges parallel to the time axis without the estimation of instantaneous frequencies (IFs). Then, the post-processed TF ridges are backward demodulated to restore the actual IF. The proposed synchronous fault feature extraction method in the GD framework also allows for the signal reconstruction. Both simulated and experimental signals are applied to validate the effectiveness of the proposed method for rolling bearing fault diagnosis.

Local discrete fractional fourier transform: An algorithm for calculating partial points of DFrFT

Fractional Fourier transform (FrFT) is widely utilized in nonstationary signal processing owing to its linearity and flexibility. Various discrete algorithms focus on the global discrete FrFT (DFrFT) points. However, only a part of the DFrFT points are considered because of a priori knowledge of applications and limited memory space of digital machines. Therefore, a local DFrFT (LDFrFT) was proposed in this study. LDFrFT for the first Q continuous DFrFT points is proposed, and its complexity was reduced from O(Mlog2(M)) to O(Mlog2(Q)) for an M-point sequence with Q being a factor of M. It is further extended to the LDFrFT for any Q continuous DFrFT points having complexity less than that of the DFrFT. These two algorithms were then generalized to local discrete linear canonical transform algorithms. The LDFrFT uses all sampling points, thus maintaining the resolution of the DFrFT. Moreover, LDFrFT is not limited by the starting point and is more flexible than the recently proposed local DFrFT algorithm. Finally, simulated frequency-modulated continuous-wave radar signals, gravitational waves, and real-world bat echolocation chirp data were utilized to demonstrate the effectiveness of the LDFrFT, which works well when the SNR is over -30 dB.

Filtering GRACE temporal gravity field solutions using ensemble empirical mode decomposition approach

Due to the strong noise that exists in GRACE (Gravity Recovery and Climate Experiment) temporal gravity field solutions, geophysical signals are normally drowned which need many effective filtering approaches. Considering the advantage of the ensemble empirical mode decomposition (EEMD) approach, we used the EEMD to filter the noise in this study together with the empirical mode decomposition (EMD) for comparisons. EMD method is a spectrum analysis method, which is very effective for non-stationary signals. EMD process is essentially a means to process non-stationary signals. It has been applied in many fields in recent years. Considering the characteristics of the spherical harmonic coefficient model that the noise level higher with the increasing degree, we divided the gravity field solutions into two parts (degrees 2–28 and degrees 29–60) based on the ratios of the latitude-weighted root mean square (RMS) over the land and ocean signals when adopting different truncated degrees. For the real GRACE solution experiments, the results show that the fitting errors of EEMD approach are always smaller than those of EMD approach, and the mean RMS ratio of EEMD is 3.45, larger than 3.40 of EMD. The simulation results show that the latitude weighted root mean square errors for EEMD approach are smaller than those of EMD, indicating that EEMD can extract the geophysical signals more accurately. Therefore, it is reasonable to conclude that EEMD performs better than EMD for filtering GRACE solutions.

Demodulated synchrosqueezing S-transform and its application to machine-fault diagnosis

The time–frequency analysis (TFA) technique has been viewed as a useful tool for processing non-stationary signals in the field of industrial machinery. Rub-impacts of a rotor system will cause vibration of the rotor and stator, thus any vibration signal with rub-impacts will be accompanied by high-frequency oscillation characteristics. In this paper, a novel TFA algorithm, termed a demodulated synchrosqueezing S-transform (DSSST), is proposed to extract the strong time-varying features in rub-impact vibration signals. The DSSST method is based on a modified S-transform, and introduces a pre-processing technique, a demodulation algorithm, to partially demodulate the oscillated modes for rub-impact identification. Meanwhile, a synchrosqueezing transform is utilized to further sharpen the time–frequency representation. Assisted by the proposed method, the rub-impact phenomenon and its impact frequency are clearly recognized through experimental and real validations.