Novel Fourier Quadrature Transforms and Analytic Signal Representations for Nonlinear and Non-stationary Time Series Analysis

The Hilbert transform (HT) and associated Gabor analytic signal (GAS) representation are well-known and widely used mathematical formulations for modeling and analysis of signals in various applications. In this study, like the HT, to obtain quadrature component of a signal, we propose the novel discrete Fourier cosine quadrature transforms (FCQTs) and discrete Fourier sine quadrature transforms (FSQTs), designated as Fourier quadrature transforms (FQTs). Using these FQTs, we propose sixteen Fourier-Singh analytic signal (FSAS) representations with following properties: (1) real part of eight FSAS representations is the original signal and imaginary part is the FCQT of the real part, (2) imaginary part of eight FSAS representations is the original signal and real part is the FSQT of the real part, (3) like the GAS, Fourier spectrum of the all FSAS representations has only positive frequencies, however unlike the GAS, the real and imaginary parts of the proposed FSAS representations are not orthogonal to each other. The Fourier decomposition method (FDM) is an adaptive data analysis approach to decompose a signal into a set of small number of Fourier intrinsic band functions which are AM-FM components. This study also proposes a new formulation of the FDM using the discrete cosine transform (DCT) with the GAS and FSAS representations, and demonstrate its efficacy for improved time-frequency-energy representation and analysis of nonlinear and non-stationary time series.Comment: 22 pages, 13 figure

Variational mode decomposition: mode determination method for rotating machinery diagnosis

Variational mode decomposition (VMD) is a modern decomposition method used for many engineering monitoring and diagnosis recently, which replaced traditional empirical mode decomposition (EMD) method. However, the performance of VMD method specifically depends on the parameter that need to pre-determine for VMD method especially the mode number. This paper proposed a mode determination method using signal difference average (SDA) to determine the mode number for the VMD method by taking the advantages of similarities concept between sum of variational mode functions (VMFs) and the input signals. Online high-speed gear and bearing fault data were used to validate the performance of the proposed method. The diagnosis result using frequency spectrum has been compared with traditional EMD method and the proposed method has been proved to be able to provide an accurate number of mode for the VMD method effectively for rotating machinery applications

A Comparative Analysis of Signal Decomposition Techniques for Structural Health Monitoring on an Experimental Benchmark

Signal Processing is, arguably, the fundamental enabling technology for vibration-based Structural Health Monitoring (SHM), which includes damage detection and more advanced tasks. However, the investigation of real-life vibration measurements is quite compelling. For a better understanding of its dynamic behaviour, a multi-degree-of-freedom system should be efficiently decomposed into its independent components. However, the target structure may be affected by (damage-related or not) nonlinearities, which appear as noise-like distortions in its vibrational response. This response can be nonstationary as well and thus requires a time-frequency analysis. Adaptive mode decomposition methods are the most apt strategy under these circumstances. Here, a shortlist of three well-established algorithms has been selected for an in-depth analysis. These signal decomposition approaches—namely, the Empirical Mode Decomposition (EMD), the Hilbert Vibration Decomposition (HVD), and the Variational Mode Decomposition (VMD)—are deemed to be the most representative ones because of their extensive use and favourable reception from the research community. The main aspects and properties of these data-adaptive methods, as well as their advantages, limitations, and drawbacks, are discussed and compared. Then, the potentialities of the three algorithms are assessed firstly on a numerical case study and then on a well-known experimental benchmark, including nonlinear cases and nonstationary signals

Blind source separation using statistical nonnegative matrix factorization

PhD ThesisBlind Source Separation (BSS) attempts to automatically extract and track a signal of interest in real world scenarios with other signals present. BSS addresses the problem of recovering the original signals from an observed mixture without relying on training knowledge. This research studied three novel approaches for solving the BSS problem based on the extensions of non-negative matrix factorization model and the sparsity regularization methods. 1) A framework of amalgamating pruning and Bayesian regularized cluster nonnegative tensor factorization with Itakura-Saito divergence for separating sources mixed in a stereo channel format: The sparse regularization term was adaptively tuned using a hierarchical Bayesian approach to yield the desired sparse decomposition. The modified Gaussian prior was formulated to express the correlation between different basis vectors. This algorithm automatically detected the optimal number of latent components of the individual source. 2) Factorization for single-channel BSS which decomposes an information-bearing matrix into complex of factor matrices that represent the spectral dictionary and temporal codes: A variational Bayesian approach was developed for computing the sparsity parameters for optimizing the matrix factorization. This approach combined the advantages of both complex matrix factorization (CMF) and variational -sparse analysis. BLIND SOURCE SEPARATION USING STATISTICAL NONNEGATIVE MATRIX FACTORIZATION ii 3) An imitated-stereo mixture model developed by weighting and time-shifting the original single-channel mixture where source signals can be modelled by the AR processes. The proposed mixing mixture is analogous to a stereo signal created by two microphones with one being real and another virtual. The imitated-stereo mixture employed the nonnegative tensor factorization for separating the observed mixture. The separability analysis of the imitated-stereo mixture was derived using Wiener masking. All algorithms were tested with real audio signals. Performance of source separation was assessed by measuring the distortion between original source and the estimated one according to the signal-to-distortion (SDR) ratio. The experimental results demonstrate that the proposed uninformed audio separation algorithms have surpassed among the conventional BSS methods; i.e. IS-cNTF, SNMF and CMF methods, with average SDR improvement in the ranges from 2.6dB to 6.4dB per source.Payap Universit

Identification of time-varying cable forces based on parameter optimization variational mode decomposition.

Accumulated fatigue damage is one of the main causes of damage and destruction of actual bridge structures. for cable bridges, the cable is the key force component. the cumulative fatigue damage of the cable seriously threatens the safety of the bridge structure. the traditional cable force test based on the vibration method can only identify the average cable force of a bridge cable over a period of time. however, due to vehicle load and environmental factors, the cable force of the bridge cable is time-varying. time-varying cable force is the main cause of fatigue damage, and it is also the basis for the safety assessment of cable limit state and the evaluation of cumulative fatigue damage. to this end, this paper studies the identification method of bridge cable time-varying cable force based on variational mode decomposition. the main research contents of this article include: based on the time-frequency analysis method of variational mode decomposition, a new method for identifying time-varying cable forces is proposed. the time-frequency analysis method of variational modal decomposition is a new development method in the field of current signal processing. its principle is to obtain a limited number of imf and extract the instantaneous frequency of the time-varying system by performing hilbert transform on the obtained imf. firstly, according to the time-frequency analysis method of variational modal decomposition, the time-varying modal frequency is identified from the measured cable acceleration. then, the bridge cable is simplified into an ideal tension string, and the cable force is identified based on the relationship between the cable force and frequency established by the classic string vibration theory. the frequency-doubling relationship of the vibration of the cable is used to reduce the optimization variables of the method, improve the calculation efficiency, reduce the influence of noise on the different instantaneous frequencies of the cable, and improve the accuracy of frequency identification of time-varying modal. finally, the time-varying modal frequency of the identified cable is substituted into the cable force formula to obtain the time-varying cable force of the cable. in the practical application of variational modal decomposition, the choice of penalty factors and the number of components has a great influence on the final signal decomposition results. in order to automatically determine the best parameter combination, the particle swarm optimization algorithm is used to search for these two influencing parameters in parallel . simulation signal and engineering signal processing results show that the proposed method can achieve identification of time-varying frequency. for the end effect of the instantaneous frequency curve, the signal extension method is used. reduce the error of the instantaneous frequency at the end point. design and build a tilting cable vibration test platform, and compare the test results with the calculation results to further verify the correctness of the method in this paper. the results show that the cable force error is about ± 5.0%

Variational mode decomposition: mode determination method for rotating machinery diagnosis

Variational mode decomposition (VMD) is a modern decomposition method used for many engineering monitoring and diagnosis recently, which replaced traditional empirical mode decomposition (EMD) method. However, the performance of VMD method specifically depends on the parameter that need to pre-determine for VMD method especially the mode number. This paper proposed a mode determination method using signal difference average (SDA) to determine the mode number for the VMD method by taking the advantages of similarities concept between sum of variational mode functions (VMFs) and the input signals. Online high-speed gear and bearing fault data were used to validate the performance of the proposed method. The diagnosis result using frequency spectrum has been compared with traditional EMD method and the proposed method has been proved to be able to provide an accurate number of mode for the VMD method effectively for rotating machinery applications

Application of variational mode decomposition in vibration analysis of machine components

Monitoring and diagnosis of machinery in maintenance are often undertaken using vibration analysis. The machine vibration signal is invariably complex and diverse, and thus useful information and features are difficult to extract. Variational mode decomposition (VMD) is a recent signal processing method that able to extract some of important features from machine vibration signal. The performance of the VMD method depends on the selection of its input parameters, especially the mode number and balancing parameter (also known as quadratic penalty term). However, the current VMD method is still using a manual effort to extract the input parameters where it subjects to interpretation of experienced experts. Hence, machine diagnosis becomes time consuming and prone to error. The aim of this research was to propose an automated parameter selection method for selecting the VMD input parameters. The proposed method consisted of two-stage selections where the first stage selection was used to select the initial mode number and the second stage selection was used to select the optimized mode number and balancing parameter. A new machine diagnosis approach was developed, named as VMD Differential Evolution Algorithm (VMDEA)-Extreme Learning Machine (ELM). Vibration signal datasets were then reconstructed using VMDEA and the multi-domain features consisted of time-domain, frequency-domain and multi-scale fuzzy entropy were extracted. It was demonstrated that the VMDEA method was able to reduce the computational time about 14% to 53% as compared to VMD-Genetic Algorithm (GA), VMD-Particle Swarm Optimization (PSO) and VMD-Differential Evolution (DE) approaches for bearing, shaft and gear. It also exhibited a better convergence with about two to nine less iterations as compared to VMD-GA, VMD-PSO and VMD-DE for bearing, shaft and gear. The VMDEA-ELM was able to illustrate higher classification accuracy about 11% to 20% than Empirical Mode Decomposition (EMD)-ELM, Ensemble EMD (EEMD)-ELM and Complimentary EEMD (CEEMD)-ELM for bearing shaft and gear. The bearing datasets from Case Western Reserve University were tested with VMDEA-ELM model and compared with Support Vector Machine (SVM)-Dempster-Shafer (DS), EEMD Optimal Mode Multi-scale Fuzzy Entropy Fault Diagnosis (EOMSMFD), Wavelet Packet Transform (WPT)-Local Characteristic-scale Decomposition (LCD)- ELM, and Arctangent S-shaped PSO least square support vector machine (ATSWPLM) models in term of its classification accuracy. The VMDEA-ELM model demonstrates better diagnosis accuracy with small differences between 2% to 4% as compared to EOMSMFD and WPT-LCD-ELM but less diagnosis accuracy in the range of 4% to 5% as compared to SVM-DS and ATSWPLM. The diagnosis approach VMDEA-ELM was also able to provide faster classification performance about 6 40 times faster than Back Propagation Neural Network (BPNN) and Support Vector Machine (SVM). This study provides an improved solution in determining an optimized VMD parameters by using VMDEA. It also demonstrates a more accurate and effective diagnostic approach for machine maintenance using VMDEA-ELM