Denoising of Magnetocardiography Based on Improved Variational Mode Decomposition and Interval Thresholding Method

This paper proposes a novel method to improve the variational mode decomposition (VMD) method and to automatically acquire the sensitive intrinsic mode function (IMF). First, since fault signals are impulsive and periodic, a weighted autocorrelative function maximum (AFM) indicator is constructed based on the Gini index and autocorrelation function to serve as the optimization objective function. The mode number K and the penalty parameter α of VMD are automatically obtained through an optimal parameter searching process underpinned by the improved particle swarm optimization (PSO) algorithm with a variety of inertia weights. This improvement solves one of the major drawbacks of the conventional VMD method, that is, the need to manually set parameters. Then, an optimal IMF automatic selecting process is performed for single-failure faults and compound faults, according to the principles of the maximum weighted AFM indicator and maximum spectrum peak ratio (SPR), respectively. The sensitive IMFs are then subjected to an envelope demodulation analysis to obtain the fault characteristic frequency. The results of simulations and experiments show that the proposed method can effectively identify fault characteristics early, especially compound faults, demonstrating great potential for real-world applications.

The variational mode decomposition (VMD) method for signal decomposition is severely affected by the number of components of the VMD method. In order to determine the decomposition modal number, K, in the VMD method, a new center frequency method of the multi-threshold is proposed in this paper. Then, an improved VMD (MTCFVMD) algorithm based on the center frequency method of the multi-threshold is obtained to decompose the vibration signal into a series of intrinsic modal functions (IMFs). The Hilbert transformation is used to calculate the envelope signal of each IMF component, and the maximum frequency value of the power spectral density is obtained in order to effectively and accurately extract the fault characteristic frequency and realize the fault diagnosis. The rolling element vibration data of the motor bearing is used to test the effectiveness of proposed methods. The experiment results show that the center frequency method of the multi-threshold can effectively determine the number, K, of decomposed modes. The proposed fault diagnosis method based on MTCFVMD and Hilbert transformation can effectively and accurately extract the fault characteristic frequency, rotation frequency, and frequency doubling, and can obtain higher diagnostic accuracy.

A motor bearing system is a nonlinear dynamics system with nonlinear support stiffness. It is an asymmetry system, which plays an extremely important role in rotating machinery. In this paper, a center frequency method of double thresholds is proposed to improve the variational mode decomposition (VMD) method, then an adaptive VMD (called DTCFVMD) method is obtained to extract the fault feature. In the DTCFVMD method, a center frequency method of double thresholds is a symmetry method, which is used to determine the decomposed mode number of VMD according to the power spectrum of the signal. The proposed DTCFVMD method is used to decompose the nonlinear and non-stationary vibration signals of motor bearing in order to obtain a series of intrinsic mode functions (IMFs) under different scales. Then, the Hilbert transform is used to analyze the envelope of each mode component and calculate the power spectrum of each mode component. Finally, the power spectrum is used to extract the fault feature frequency for determining the fault type of the motor bearing. To test and verify the effectiveness of the DTCFVMD method, the actual fault vibration signal of the motor bearing is selected in here. The experimental results show that the center frequency method of double thresholds can effectively determine the mode number of the VMD method, and the proposed DTCFVMD method can accurately extract the clear time frequency characteristics of each mode component, and obtain the fault characteristics of characteristics; frequency, rotating frequency, and frequency doubling and so on.

The variational mode decomposition (VMD) method has been widely applied in the field of mechanical fault diagnosis as an excellent non-recursive signal processing tool. The performance of VMD depends on its inherent prior parameters. Searching for the key parameters of VMD using intelligent optimization algorithms poses challenges for the internal essence and fitness function selection of intelligent optimization algorithm. Moreover, the computational complexity of optimization is high. Meanwhile, such methods are not competitive in evaluating orthogonality between intrinsic mode functions and the reconstruction error of the signal as a joint indictor for the termination of decomposition. Therefore, this paper proposes a new auto-regulative sparse variational mode decomposition method (ASparse–VMD) to achieve accurate feature extraction. The regularization term of the VMD handles sparsification by constructing an L2-norm with a damping coefficient ε, and mode number K is set adaptively in a recursive manner to ensure appropriateness. The penalty parameter α is dynamically selected according to the number of modes and sampling frequency. The update step τ of the VMD algorithm is set using the signal-to-noise ratio to ensure the singleness and orthogonality of the modal components and suppress mode aliasing. The experimental results of the simulation signal and measured signal demonstrate the effectiveness of the proposed strategies for improving the inherent defects of VMD. Extensive comparisons with state-of-the-art methods show that the proposed algorithm is more effective and practical for hybrid feature extraction in mechanical faults.

Machines learning models have recently been proposed for predicting rivers water temperature (Tw) using only air temperature (Ta). The proposed models relied on a nonlinear relationship between the Tw and Ta and they have proven to be robust modelling tools. The main motivation for this study was to evaluate how the variational mode decomposition (VMD) contributed to the improvement of machines learning performances for river Tw modelling. Measured data collected at five stations located in Poland from 1987 to 2014 were acquired and used for the analysis. Six machines learning models were used and compared namely, K-nearest neighbor's regression (KNNR), least square support vector machine (LSSVM), generalized regression neural network (GRNN), cascade correlation artificial neural networks (CCNN), relevance vector machine (RVM), and locally weighted polynomials regression (LWPR). The six models were developed according to three scenarios. First, the models were calibrated using only the Ta as input and obtained results show that the models were able to predict consistently water temperature, showing a high determination coefficient (R2) and Nash-Sutcliffe efficiency (NSE) with values near or above 0.910 and 0.915, respectively, and in overall the six models worked equally without clear superiority of one above another. Second, the air temperature was combined with the periodicity (i.e., day, month and year number) as input variable and a significant improvement was achieved. Both models show their ability to accurately predict river Tw with an overall accuracy of 0.956 for R2 and 0.955 for NSE values, but the LSSVM2 have some advantages such as a small errors metrics, and high fitting capabilities and it slightly surpasses the others models. Thirdly, air temperature was decomposed into several intrinsic mode functions (IMF) using the VMD method and the performances of the models were evaluated. The VMD parameters appeared to cause much influence on the prediction accuracy, exhibiting an improvement of about 40.50% and 39.12% in terms of RMSE and MAE between the first and the third scenarios, however, some models, i.e., GRNN and KNNR have not benefited from the VMD. This research has demonstrated the high capability of the VMD algorithm as a preprocessing approach in improving the accuracies of the machine learning models for river water temperature prediction.

The research about online monitoring and leakage automatic location of water distribution networks (WDN) has a wide range of applications that include water resource protection, monitoring, and allocation. Variational mode decomposition (VMD) and cross-correlation (CC) based leakage location is a popular and effective method in WDN. However, the value of K intrinsic mode functions (IMFs) based on VMD decomposition needs to be determined artificially, which affects the separation effect of signal frequency band characteristics directly. Hence, this work proposes an adaptive method to determine the parameter K of leakage vibration signal’s IMFs, which will be applied to automatic leakage location in WDN. Firstly, the number of saddle points in the frequency domain envelope of the sampled signal in different step sizes is calculated. The parameter K is determined according to the curvature change of the number of saddle points and the sampled signal. Finally, the selective IMFs are reconstituted into a new signal, which can determine a leak position using CC based time-delay estimation (TDE). To verify the effectiveness of the proposed algorithm, the different methods based on EMD and Fast ICA are compared. The experimental results demonstrate that the proposed parameter K value adaptive VMD (KVA-VMD) decomposition method is more suitable for leakage location in WDN.

An optimized VMD method and its applications in bearing fault diagnosis

A new denoising algorithm and feature extraction algorithm that combine a new kind of permutation entropy (NPE) and variational mode decomposition (VMD) are put forward in this paper. VMD is a new self-adaptive signal processing algorithm, which is more robust to sampling and noise, and also can overcome the problem of mode mixing in empirical mode decomposition (EMD) and ensemble EMD (EEMD). Permutation entropy (PE), as a nonlinear dynamics parameter, is a powerful tool that can describe the complexity of a time series. NPE, a new version of PE, is interpreted as distance to white noise, which shows a reverse trend to PE and has better stability than PE. In this paper, three kinds of ship-radiated noise (SN) signal are decomposed by VMD algorithm, and a series of intrinsic mode functions (IMF) are obtained. The NPEs of all the IMFs are calculated, the noise IMFs are screened out according to the value of NPE, and the process of denoising can be realized by reconstructing the rest of IMFs. Then the reconstructed SN signal is decomposed by VMD algorithm again, and one IMF containing the most dominant information is chosen to represent the original SN signal. Finally, NPE of the chosen IMF is calculated as a new complexity feature, which constitutes the input of the support vector machine (SVM) for pattern recognition of SN. Compared with the existing denoising algorithms and feature extraction algorithms, the effectiveness of proposed algorithms is validated using the numerical simulation signal and the different kinds of SN signal.

This paper aims to present a non-destructive, optimized variational mode decomposition (VMD)-based ground-penetrating radar (GPR) method developed for identifying void defects in reinforced concrete (RC) structures. This study also presents an enhanced framework for defect detection in RC by integrating advanced spectrum analysis with deep learning techniques. A GPR investigation was conducted on an RC bridge deck with known structural defects to generate a representative dataset reflecting both intact and void-defective conditions. In addition to conventional spectral techniques such as fast Fourier transform (FFT), spectrogram, and scalogram, an optimized variational mode decomposition (VMD) method was implemented. The VMD approach decomposes GPR signals into intrinsic mode functions, enabling refined feature extraction beyond traditional spectral methods and allowing clear differentiation between intact and defective signals. The limited availability and quality of GPR small datasets have restricted the application of a functional 1D-CNN which generally requires at least several hundred datasets. To address this challenge, a data augmentation strategy is adopted. FFT-based features were successfully utilized to train a one-dimensional convolutional neural network (1D-CNN) for automated defect identification. The results demonstrate that both the advanced spectrum-based approach and the hybrid framework combining spectral analysis with deep learning significantly improve defect detection performance. Overall, the proposed methodology provides an effective and intelligent solution to support timely, data-driven decision-making for maintenance and safety assurance of bridge infrastructure.

As the gear is one of the most critical components of the high-speed rotating machinery, the fault identification of the gear has attracted increasing attention recently. However, the nonlinear and non-stationary vibration signals with mode mixing phenomenon and heavy background noise make it difficult to excavate the hidden weak fault feature. Variational Mode Decomposition (VMD), which can decompose the non-linear and non-stationary signals into a couple of Intrinsic Mode Functions (IMF) adaptively and non-recursively, bring a feasible tool. While the heavy background noise seriously affects the setting of the model number, which may lead to information loss or over decomposition problem. In this paper, a new improved VMD method, namely, Generalized Variational Mode Decomposition (GVMD) method is proposed to extract the fault characteristic information buried in the vibration signals adaptively. For the gear fault quantitative diagnosis, the corresponding relationship between the characteristic parameters and the working conditions is established under a specific condition, and it is no longer applicable to other conditions. Based on the dynamic model of the gear transmission system, the characteristic parameters that are sensitive and correlative to the gear tooth crack level, but not sensitive to the working conditions, can be extracted from the dynamic responses. Then the Normalized Feature Vector (NFV) can be formed with the extracted parameters. Based on the methods mentioned above, a novel intelligent method that combines the GVMD with NFV extraction to realize the quantitative diagnosis of the gear fault under variable working conditions is proposed. The NFV can be extracted from the experimental vibration signals preprocessed by the GVMD, and establish the relationship between the NFV and the gear tooth crack level. Then the quantitative diagnosis of the gear fault can be realized by the Support Vector Machine (SVM). The results show that the intelligent quantitative diagnosis method proposed in this paper is a potential quantitative diagnosis tool for the gear fault.

A distributed optical fiber acoustic sensing system (DAS) based on phase-sensitive optical time domain reflectometry ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Phi $ </tex-math></inline-formula> -OTDR) can detect the vibration information along an optical fiber and has been widely used for various engineering detections; however, the high-coherence detection light source used in a DAS will cause coherent fading in the fiber, leading to system fading noise and affecting the monitoring of the target vibration events. In order to solve the problem of the fading noise in the DAS, this study considered a method of suppressing the fading noise based on the genetic algorithm (GA) and variational mode decomposition (VMD). Furthermore, the parameter selection problem in the VMD method is improved by introducing the GA. By introducing sample entropy as a fitness function and combining it with the GA, this method adaptively searches for the optimal decomposition parameters by iterating the minimum value of the fitness function according to the type of input signal. Then, it decomposes the noisy DAS data into multiple intrinsic mode functions (IMFs), and the permutation entropy and time-frequency domain information of each component were used to extract the effective IMF components for signal reconstruction to obtain the denoised signal. Experimental results show that compared with empirical mode decomposition (EMD), ensemble EMD (EEMD), and complete EEMD (CEEMD) methods, the proposed method can eliminate the mode aliasing problem, effectively suppress the fading noise of the DAS, improve the signal-to-noise ratio, and reduce the root-mean-square error. It is, thus, possible to more accurately monitor vibration events, improving the practical application value of the DAS.

Underwater acoustic signal denoising model based on secondary variational mode decomposition

An operating condition information-guided iterative variational mode decomposition method based on Mahalanobis distance criterion for surge characteristic frequency extraction of the centrifugal compressor

Background/Objectives: This paper introduced an image fusion algorithm based on Variational Mode Decomposition (VMD). Methods/Statistical Analysis: Image fusion is one of the image enhancement methods which results the image with better quality derived from a set of degraded images. Fused image contains more information than input images and it is efficient for visual perception and computer vision applications. This paper proposed an image fusion technique based on VMD for multi focus images. VMD has been a recently introduced non-recursive decomposition method, which decomposes the image into separate spectral bands called Intrinsic Mode Function (IMF) or modes. The modes are generated with respect to the associated central frequencies and they are band limited. Findings: A fusion rule based on weighing scheme is performed at the decomposition level for increasing the features by decreasing the mutual information. The reconstruction of the IMFs results the final fused image. The performance analysis of the proposed fusion method is experimented using standard objective quality metrics. The efficiency of the proposed method is determined by comparing the method with some state of the art methods. Application/Improvements: The image fusion using VMD is applicable to multi-resolution, multi model multi-sensor images.

High-G MEMS accelerometer (HGMA) is a new type of sensor; it has been widely used in high precision measurement and control fields. Inevitably, the accelerometer output signal contains random noise caused by the accelerometer itself, the hardware circuit and other aspects. In order to denoise the HGMA’s output signal to improve the measurement accuracy, the improved VMD and TFPF hybrid denoising algorithm is proposed, which combines variational modal decomposition (VMD) and time-frequency peak filtering (TFPF). Firstly, VMD was optimized by the multi-objective particle swarm optimization (MOPSO), then the best decomposition parameters [kbest, abest] could be obtained, in which the permutation entropy (PE) and fuzzy entropy (FE) were selected for MOPSO as fitness functions. Secondly, the accelerometer voltage output signals were decomposed by the improved VMD, then some intrinsic mode functions (IMFs) were achieved. Thirdly, sample entropy (SE) was introduced to classify those IMFs into information-dominated IMFs or noise-dominated IMFs. Then, the short-window TFPF was selected for denoising information-dominated IMFs, while the long-window TFPF was selected for denoising noise-dominated IMFs, which can make denoising more targeted. After reconstruction, we obtained the accelerometer denoising signal. The denoising results of different denoising algorithms in the time and frequency domains were compared, and SNR and RMSE were taken as denoising indicators. The improved VMD and TFPF denoising method has a smaller signal distortion and stronger denoising ability, so it can be adopted to denoise the output signal of the High-G MEMS accelerometer to improve its accuracy.