Fourier-Bessel Series Expansion Research Articles

Improved modeling of anisotropic effects on seismic waves in layered transversely isotropic half-spaces: Implications for velocity profile inversion challenges

In this paper, we first derive the dynamic solution in a transversely isotropic (TI) elastic and layered half-space, induced by a time-harmonic vertical load on its surface. The solution is obtained via the efficient and unconditionally stable dual variable and position (DVP) matrix method along with the fast Fourier-Bessel series (FBS) expansion approach. In order to study the effect of material anisotropy, the degree of anisotropy, namely the ratio of the horizontal Young's modulus over vertical Young's modulus, is introduced as the key parameter, with a very reasonable assumption on other involved elastic constants in the layered TI half-space. This enables us to investigate the effect of material anisotropy on the wave Green's functions, dispersion curves, ellipticity, and polarity. By comparing with the effect of Poisson's ratio in the corresponding layered isotropic (ISO) half-space, we observe that nearly all the wave features with varying degrees of anisotropy in the TI layered half-space are similar to those with varying Poisson's ratio in the corresponding ISO half-space. This indicates the challenge or non-uniqueness in the inversion of TI velocity profiles using vertical surface loading only.

AFM signal model for dysarthric speech classification using speech biomarkers.

Neurological disorders include various conditions affecting the brain, spinal cord, and nervous system which results in reduced performance in different organs and muscles throughout the human body. Dysarthia is a neurological disorder that significantly impairs an individual's ability to effectively communicate through speech. Individuals with dysarthria are characterized by muscle weakness that results in slow, slurred, and less intelligible speech production. An efficient identification of speech disorders at the beginning stages helps doctors suggest proper medications. The classification of dysarthric speech assumes a pivotal role as a diagnostic tool, enabling accurate differentiation between healthy speech patterns and those affected by dysarthria. Achieving a clear distinction between dysarthric speech and the speech of healthy individuals is made possible through the application of advanced machine learning techniques. In this work, we conducted feature extraction by utilizing the Amplitude and frequency modulated (AFM) signal model, resulting in the generation of a comprehensive array of unique features. A method involving Fourier-Bessel series expansion is employed to separate various components within a complex speech signal into distinct elements. Subsequently, the Discrete Energy Separation Algorithm is utilized to extract essential parameters, namely the Amplitude envelope and Instantaneous frequency, from each component within the speech signal. To ensure the robustness and applicability of our findings, we harnessed data from various sources, including TORGO, UA Speech, and Parkinson datasets. Furthermore, the classifier's performance was evaluated based on multiple measures such as the area under the curve, F1-Score, sensitivity, and accuracy, encompassing KNN, SVM, LDA, NB, and Boosted Tree. Our analyses resulted in classification accuracies ranging from 85 to 97.8% and the F1-score ranging between 0.90 and 0.97.

Unsteady double-diffusive Brinkman–Bénard convection in cylindrical enclosure saturated with hybrid bi-viscous Bingham nanoliquid

This paper aims to present an analytical as well as a comparative study to investigate the effect of the Brinkman porous medium on the onset of regular and chaotic motion in cylindrical enclosures of different heights. The hybrid bi-viscous Bingham nanoliquid is considered as the working fluid. Modified Brinkman–Buongiorno and bi-viscous Bingham fluid models are incorporated to obtain the flow governing dynamics. The thermophysical properties of the hybrid nanoliquid are calculated using phenomenological laws and the mixture theory. The study is carried out for the axisymmetric mode, and the Bessel functions are taken as the eigenfunctions of the problem. Double Fourier–Bessel series expansions are used for weakly non-linear stability analysis. The limiting cases of the study are obtained, and the results on the onset of convection, heat, and mass transport are discussed graphically. The behavior of the dynamical system is analyzed using the maximum Lyapunov exponent plot, the bifurcation diagram, and phase plots. Outcomes suggest that convection sets in earlier in the water-based hybrid nanoliquid than in the bi-viscous Bingham hybrid nanoliquid. The use of Single-walled carbon nanotubes enhances the heat transfer rate by approximately 17%. Further, it is concluded that a tall cylindrical enclosure is the most favorable geometry for achieving a higher heat transfer rate among the others.

Automated Bundle Branch Block Detection using Multivariate Fourier-Bessel Series Expansion based Empirical Wavelet Transform

Automated Bundle Branch Block Detection using Multivariate Fourier-Bessel Series Expansion based Empirical Wavelet Transform

A New Automated Classification Framework for Gear Fault Diagnosis Using Fourier–Bessel Domain-Based Empirical Wavelet Transform

Gears are the most important parts of a rotary system, and they are used for mechanical power transmission. The health monitoring of such a system is needed to observe its effective and reliable working. An approach that is based on vibration is typically utilized while carrying out fault diagnostics on a gearbox. Using the Fourier–Bessel series expansion (FBSE) as the basis for an empirical wavelet transform (EWT), a novel automated technique has been proposed in this paper, with a combination of these two approaches, i.e., FBSE-EWT. To improve the frequency resolution, the current empirical wavelet transform will be reformed utilizing the FBSE technique. The proposed novel method includes the decomposition of different levels of gear crack vibration signals into narrow-band components (NBCs) or sub-bands. The Kruskal–Wallis test is utilized to choose the features that are statistically significant in order to separate them from the sub-bands. Three classifiers are used for fault classification, i.e., random forest, J48 decision tree classifiers, and multilayer perceptron function classifier. A comparative study has been performed between the existing EWT and the proposed novel methodology. It has been observed that the FBSE-EWT with a random forest classifier shows a better gear fault detection performance compared to the existing EWT.

A semi-analytical model of an array of moored floating flexible offshore net cages under current loads

An analytical model of an array of moored floating flexible net cage systems is developed to study the dynamics and wave loads on the net cages under the linear water wave theory and small-amplitude wave response assumption. The Fourier–Bessel series expansion of the velocity potential is derived for the regions enclosed under the open-water and cage systems and their proximity. The theoretical solution is obtained by employing the least squares approximation method along with matching the velocity and pressure at the fluid-structure interfaces. The present result of horizontal displacement and horizontal wave load are compared with a published finite element analysis and with experimental data sets, respectively. Further, the results of cage displacements, cage accelerations, and wave loads on the different arrangements for mooring stiffness, cage radius, cage heights, and current speeds are compared and analysed. Moreover, the present analysis will be helpful to understand designing further a multiple floating fish cage system with flexible floaters application to offshore aquaculture.

Performance of a two-body heaving cylindrical wave energy converter on stepped sea bottom

The present study deals with the surface gravity wave interaction with a heaving dual concentric cylindrical wave energy converter (WEC) buoy system on the stepped sea bottom. The energy captured by linear electric generator (LEG), is directly proportional to relative heave motion of the present WEC buoy system. In the present study, the analytical diffraction and radiation problems are solved using the Fourier–Bessel series expansion method and matched eigenfunction expansion method (MEFEM) under the assumption of small amplitude water wave theory and structural responses. Wave exciting force acting on both vertical cylinders is obtained. The role of the step type bottom on the relative heave response amplitude operator (RAO) of the buoy system, is analyzed for various structural configurations and wave characteristics. A series of experiments are carried out in a two-dimensional wave flume both with and without the presence of step-type bottom. The computed results are in good agreement with the experimental results. It is found that the inclusion of artificial step at the sea bottom leads to higher wave energy harnessing. The findings of the present study are likely to be of immense help in the design and development of various point absorber-type WEC buoy systems.

Diffraction of plane P waves by a circular-arc canyon in an unsaturated poroelastic half-space

Based on the theory of unsaturated porous elastic medium, the diffraction around the circular-arc canyon in unsaturated half-space under the incidence of plane P wave is studied by using the Fourier-Bessel series expansion method of wave function. Through numerical examples, the influence of parameters such as incident wave frequency, incident angle, arc depth-width ratio and saturation on the surface displacement amplitude in unsaturated half-space circular-arc canyon is analyzed. The results show that the saturation has a significant effect on the variation of surface displacement amplitude, and the influence law of saturated and unsaturated site soils on the variation of surface displacement amplitude is very different under different depth-to-width ratio and incident frequency. In the case of different depth-width ratios of unsaturated half-space circular-arc canyon, the displacement amplitude change of semi-circular canyon (h/a = 1) is more sensitive to the change of saturation than that of shallow circular canyon (h/a = 0.5). When the incident frequency is low, the maximum displacement amplitude of the half-space circular-arc-canyon in both saturated and unsaturated cases increases with the increase of the incident angle. It is worth noting that when the saturation Sr = 0.99, i.e., when the soil is close to saturation, the maximum surface displacement amplitude caused by unsaturated depressed terrain (Sr = 0.8) is greater than that caused by the saturated soil case as the incidence angle increases.

Improving EEG major depression disorder classification using FBSE coupled with domain adaptation method based machine learning algorithms

Major depression disorder (MDD) has become the leading mental disorder worldwide. Medical reports have shown that people with depression exhibit abnormal wave patterns in EEG signals compared with the healthy subjects when they are exposed to positive and negative stimuli. In this paper, we proposed an intelligent MDD detection model based on Fourier-Bessel series expansion (FBSE) coupled with domain adaptation (DA). First, EEG signals are segmented into intervals and each segment is passed through FBSE. Two types of features, including statistical and nonlinear features are investigated and extracted from each FBSE coefficient to detect MDD. Student t-test and Wilcoxon test are employed to remove noisy and bad features that can compromise the performance of data-driven learners. Then, DA method named Independence Domain Adaptation was applied to reduce the difference of feature distributions among subjects. The selected features are sent to a least square support vector machine (LS-SVM), and other classifiers named SVM, k-nearest (KNN), ransom forest, Bagged ensemble, boosted ensemble, decision tree, gradient boosting and stacked ensemble for the comparison purpose. Our experiments are simulated by using publicly available dataset. The performance of the proposed model is evaluated in both subject dependence experiment by 10-fold cross validation, subject independence experiment by leave-one-subject-out cross-validation, and Confidence interval respectively. Results showed that the features reduction method can significantly improve the mean accuracy by 4.20. The proposed model is compared with previous studies and the results show that the proposed model outperforms the other methods.

An intelligent protection scheme for series-compensated transmission lines connecting large-scale wind farms

Series-compensated transmission lines (SCTLs) are increasingly preferred for transmitting bulk amounts of electricity generated from the present-day large-scale wind farm to the utility grid due to several technical and economic benefits. However, when a fault occurs in such a wind farm-integrated SCTL, the impedance across the metal oxide varistor (MOV)-protected series capacitor varies non-linearly. Also, the fault current contributed from the wind farm side is quite different compared to the grid side. Consequently, the widely used fixed impedance-based distance relaying schemes showed limitations when used for protecting such crucial TLs. In this paper, the impacts of series compensation and wind farm integration on distance relay are investigated, and this paper proposes an intelligent relaying scheme using only the current measurements. In the proposed scheme, the fault detection task is performed using the signs of the half-cycle magnitude differences of the line end positive-sequence currents, and the fault classification task is performed using only the local current measurements processed through the Fourier–Bessel series expansion (FBSE) bagging ensemble (BE) classifier. The non-stationary components present in the current signal at the initiation of a fault are captured by calculating FBSE coefficients, and the singular value decomposition is applied for dimensionality reduction of the feature set. Finally, the extracted features are used by the BE classifier for fault classification. The method is evaluated in MATLAB/Simulink® on numerous fault and non-fault data simulated in two-bus systems and also validated through the OPAL-RT (OP4510) manufactured real-time digital simulation platform. The obtained results (response time for fault detection and classification <10 ms), including the comparative assessment results (fault detection accuracy =100% and fault classification accuracy =99.37%), justify the effectiveness of the proposed relaying scheme in protecting the wind farm-integrated SCTLs.

Synthesis for OAM Vortex Electromagnetic Waves Based on Fourier–Bessel Expansion

In this article, we proposed a method for low sidelobe vortex wave synthesis. Using Fourier–Bessel series expansion, arbitrary radiation pattern function can be synthesized. Fortunately, the radiation pattern of concentric annulus array for vortex wave is a finite series involving Bessel functions of the first kind, which exactly suits for the Fourier–Bessel series expansion. Considering that a radiation pattern of vortex beam has a null depth in the center, it is similar to the difference pattern of monopulse radar. Therefore, we use the Bayliss difference pattern function as the objective pattern. By calculating Fourier–Bessel expansion coefficients, concentric annulus arrays can be used to effectively generate low sidelobe vortex beams. Our synthesis method may provide a theoretical reference for secure vortex communication and vortex radars in the future.

Detection and classification of drones using Fourier-Bessel series representation of acoustic emissions

Detection, classification, and line-of-sight range estimation of drones are vital for security, safety, and privacy reasons. Representation of the audio emissions of drones in a Fourier-Bessel (FB) series expansion is proposed for the identification of a drone and/or the prediction of its range from an observation point. A deep learning network employing the FB series coefficients as the preprocessed input has been shown to classify accurately each of seven drones flying in a controlled environment in about 84 % of cases. For the case of any one of three drones flying outdoors, presence of the drone—as opposed to background noise—was detected correctly with few false positive and false negative results. Additionally, the range of the drone—from 2.5 m to 935.6 m—was estimated to be within ±50 cm of actual line-of-sight distance in over 85 % of the available test cases.

Automated Emotion Identification Using Fourier-Bessel Domain-Based Entropies.

Human dependence on computers is increasing day by day; thus, human interaction with computers must be more dynamic and contextual rather than static or generalized. The development of such devices requires knowledge of the emotional state of the user interacting with it; for this purpose, an emotion recognition system is required. Physiological signals, specifically, electrocardiogram (ECG) and electroencephalogram (EEG), were studied here for the purpose of emotion recognition. This paper proposes novel entropy-based features in the Fourier–Bessel domain instead of the Fourier domain, where frequency resolution is twice that of the latter. Further, to represent such non-stationary signals, the Fourier–Bessel series expansion (FBSE) is used, which has non-stationary basis functions, making it more suitable than the Fourier representation. EEG and ECG signals are decomposed into narrow-band modes using FBSE-based empirical wavelet transform (FBSE-EWT). The proposed entropies of each mode are computed to form the feature vector, which are further used to develop machine learning models. The proposed emotion detection algorithm is evaluated using publicly available DREAMER dataset. K-nearest neighbors (KNN) classifier provides accuracies of 97.84%, 97.91%, and 97.86% for arousal, valence, and dominance classes, respectively. Finally, this paper concludes that the obtained entropy features are suitable for emotion recognition from given physiological signals.

EEG-based cross-subject emotion recognition using Fourier-Bessel series expansion based empirical wavelet transform and NCA feature selection method

Automated emotion recognition using brain electroencephalogram (EEG) signals is predominantly used for the accurate assessment of human actions as compared to facial expression or speech signals. Various signal processing methods have been used for extracting representative features from EEG signals for emotion recognition. However, the EEG signals are non-stationary and vary across the subjects as well as in different sessions of the same subject; hence it exhibits poor generalizability and low classification accuracy for an emotion classification of cross subjects. In this paper, EEG signals-based automated cross-subject emotion recognition framework is proposed using the Fourier-Bessel series expansion-based empirical wavelet transform (FBSE-EWT) method. This method is used to decompose the EEG signals from each channel into four sub-band signals. Manually ten channels are selected from the frontal lobe, from which entropy and energy features are extracted from each sub-band signal. The subject variability is reduced using an average moving filter method on each channel to obtain the smoothened feature vector of size 80. The three feature selection techniques, such as neighborhood component analysis (NCA), relief-F, and mRMR, are used to obtain an optimal feature vector. The machine learning models, such as artificial neural network (ANN), k-nearest neighborhood (k-NN) with two (fine and weighted) functions, and ensemble bagged tree classifiers are trained by the obtained feature vectors. The experiments are performed on two publicly accessible databases, named SJTU emotion EEG dataset (SEED) and dataset for emotion analysis using physiological signals (DEAP). The training and testing of the models have been performed using 10-fold cross-validation and leave-one-subject-out-cross-validation (LOSOCV). The proposed framework based on FBSE-EWT and NCA feature selection approach shows superior results for classifying human emotions compared to other state-of-art emotion classification models.

An efficient full-wavefield computational approach for seismic testing in a layered half-space

Inversion of subsurface shear-wave velocity profile based on effective dispersion curve or full wavefield is still rare due to the lack of efficient forward computation kernel for the full wavefield in a layered medium. To facilitate exploiting the complete signal content in seismic testing, an efficient new computational framework for modeling synthetic full wavefields in horizontally layered systems is proposed. Dynamic response of an isotropic elastic layered system subjected to vertical loading is implemented and thoroughly validated. The new procedure is based the Fourier-Bessel series expansion (not the traditional Fourier-Bessel transform), and it is at least two-orders faster than the current full wavefield computation method based on integral transformation. It considers survey parameters and models all wave phenomena, including near-field effect and leaky waves, which cannot be captured by the modal summation method. Examples of full wavefield simulation are presented to demonstrate different levels of information provided by the experimentally available fundamental mode, broadband effective dispersion curve, and the full velocity spectrum. Different earth models may have similar fundamental modes or even effective dispersion curves; however, their difference manifests in different higher-mode contributions and can only be exploited by utilizing the entire velocity spectra including the vertical and radial components. The coupling between the vertical and radial components in terms of their spectral ratio and phase difference can provide additional constraints (data) in seismic testing.

Detection of cardiac arrhythmias from ECG signals using FBSE and Jaya optimized ensemble random subspace K-nearest neighbor algorithm

Cardiac Arrhythmias is defined as the sudden and unexpected occurrence of a rhythm in the heart. Arrhythmia detection is prevalent in developing countries that require systems-based screening solutions for diagnosis. A new rhythm-based approach is proposed to screen the patients with cardiac arrhythmia at the primary level. This method eliminates multiple tests, enabling faster and more accurate diagnosis. This method first segments the electrocardiogram signals to create a complete picture of the single heartbeat. Then the Fourier–Bessel series expansion (FBSE) is used to transform the sequences of each heartbeat into more meaningful ones that can characterize the structural integrity of arrhythmia. The FBSE sequence series are trained using the Jaya optimized ensemble random subspace KNN (JO-ERSKNN) model with 10-fold cross-validation for classifying five types of cardiac arrhythmia beats. We have achieved an accuracy of 99.49%, a sensitivity of 95.43%, and specificity of 99.48%. The results demonstrate that the proposed algorithm can detect differences in the five types of cardiac arrhythmia signals. It can also be utilized as a screening tool for detecting arrhythmia and can be made compatible with various wearable devices.

Inverse source problem for an equation of mixed parabolic-hyperbolic type with the time fractional derivative in a cylindrical domain

This article is devoted to the study of an inverse source problem for a mixed type equation with a fractional diffusion equation in the parabolic part and a wave equation in the hyperbolic part of a cylindrical domain. The solution is obtained in the form of FourierBessel series expansion using an orthogonal set of Bessel functions. The theorems of uniqueness and existence of a solution are proved.

Blind deconvolution criterion based on Fourier–Bessel series expansion for rolling element bearing diagnostics

In the last years, Blind Deconvolution methods demonstrated their effectiveness for the diagnostics of rotating machines through the extraction of impulsive signatures directly from noisy observations. Recently, in this scenario the explicit combination between Blind Deconvolution and cyclostationary theory strongly improved the fault detection ability of this diagnostic tool. This work presents a novel criterion based on the Fourier–Bessel series expansion instead of the common Fourier transform. This idea comes from the comparison between the mathematical nature of the Fourier–Bessel and the Fourier series, based on modulated and constant amplitude sinusoidal functions, respectively. The two criteria are compared through the analysis of both simulated and real vibration signals of faulty bearings. The results highlight the ability of the proposed criterion to detect the fault-related source with a lower number of characteristic cyclic frequency harmonics, strongly reducing the computational time required by the algorithm.

Epileptic-seizure classification using phase-space representation of FBSE-EWT based EEG sub-band signals and ensemble learners

Electroencephalogram (EEG) signals are non-linear and non-stationary in nature. The phase-space representation (PSR) method is useful for analysing the non-linear characteristics of EEG signals. Hence non-linear features based on a phase-space representation of EEG signals are effective in epileptic-seizure classification. In the past, various machine learning methods are used for classifying seizure EEG signals. However, they might fail to classify accurately due to complex data, so with effective non-linear features, ensemble learning classifiers can be investigated to improve the accuracy of an automated epileptic-seizure detection system. In this paper, the EEG signals are first decomposed in sub-bands using empirical wavelet transform (EWT) based on the Fourier Bessel series expansion (FBSE) which is termed as FBSE-EWT. Then, these sub-bands are reconstructed into three-dimensional (3D) PSR. Next, entropy-based features like line-length (LL), log-energy-entropy (LEEnt), and norm-entropy (NEnt) are computed from Euclidean distances of 3D PSR of sub-band signals. The extracted features are ranked based on p-values obtained from the Kruskal-Wallis statistical test for reducing the feature space. Experiments have been conducted using obtained ranked features on different ensemble learning classifiers, and the five best-performing classifiers are reported here, which are random forest (RF), extra tree (ET), extremegradientboosting tree (xgBT), bagged-SVM (B-SVM), and bagged-k-nearest neighbours (B-k-NN), for classifying epileptic-seizure EEG signals. The performance of the proposed framework is evaluated using a publically accessible Bonn university EEG database for classifying epileptic-seizure EEG signals on well-known classification problems such as C1 (seizure, normal), C2, and C3 (seizure, normal, and seizure-free). This dataset consists recording of 100 single-EEG signals from five seizure, seizure-free, and normal (healthy) subjects each. Model is trained and tested using 10-fold cross-validation to stave off from overfitting. The performance is also compared with other state-of-art methods. Obtained results confirm the superior performance of the proposed framework by achieving maximum classification accuracy of 100 %, 98.3 %, 97.8 % with ET, B-SVM, and ET classifiers from each studied classification problem, respectively. Hence, the proposed framework can assist medical professionals in analysing epileptic-seizure EEG signals more accurately.

Automated FBSE-EWT based learning framework for detection of epileptic seizures using time-segmented EEG signals.

Epilepsy is a neurological disorder that has severely affected many people's lives across the world. Electroencephalogram (EEG) signals are used to characterize the brain's state and detect various disorders. The EEG signals are non-stationary and non-linear in nature. Therefore, it is challenging to accurately process and learn from the recorded EEG signals in order to detect disorders like epilepsy. This paper proposed an automated learning framework using the Fourier-Bessel series expansion-based empirical wavelet transform (FBSE-EWT) method for detecting epileptic seizures from EEG signals. The scale-space boundary detection method was adopted to segment the Fourier-Bessel series expansion (FBSE) spectrum of multiple frame-size time-segmented EEG signals. Multiple frame-size time-segmented EEG signal's analysis was done using four different frame sizes: full, half, quarter, and half-quarter length of recorded EEG signals. Two different time-segmentation approaches were investigated on EEG signals: 1) segmenting signals based on multiple frame-size and 2) segmenting signals based on multiple frame-size with zero-padding the remaining signal. The FBSE-EWT method was applied to decompose the EEG signals into narrow sub-band signals. Features such as line-length (LL), log-energy-entropy (LEnt), and norm-entropy (NEnt) were computed from various frequency range sub-band signals. The relief-F feature ranking method was employed to select the most significant features; this reduces the computational burden of the models. The top-ranked accumulated features were used for classification using least square-support machine learning (LS-SVM), support vector machine (SVM), k-nearest neighbor (k-NN), and ensemble bagged tree classifiers. The proposed framework for epileptic seizure detection was evaluated on two publicly available benchmark EEG datasets: the Bonn EEG dataset and Children's Hospital Boston (CHB) and the Massachusetts Institute of Technology (MIT), well known as the CHB-MIT scalp EEG dataset. Training and testing of the models were performed using the 10-fold cross-validation technique. The FBSE-EWT based learning framework was compared with other state-of-the-art methods using both datasets. Experimental results showed that the proposed framework achieved 100% classification accuracy on the Bonn EEG dataset, whereas 99.84% classification accuracy on the CHB-MIT scalp EEG dataset.