S-transform Research Articles

Beckner Logarithmic Uncertainty Principle for the Stockwell Transform Associated with the Singular Partial Differential Operators

Beckner Logarithmic Uncertainty Principle for the Stockwell Transform Associated with the Singular Partial Differential Operators

A Fast Adaptive S-Transform for Complex Quality Disturbance Feature Extraction

This paper proposes a fast adaptive S-transform (FAST) to improve the time-frequency resolution and computational efficiency of power quality disturbances (PQDs) feature extraction. By directly controlling the standard deviation instead of other parameters, FAST can reduce the difficulty of optimizing time-frequency resolution. Based on the frequency spectrum of PQD signals, FAST only needs to calculate characteristic frequency points determined by maximum envelope curve, which can eliminate redundant calculation without losing effective feature information. In fact, the computational complexity of parameter optimization step is often higher than that of S-transform (ST) calculation step. To address this problem, a window matching spectrum (WMS) method is proposed to optimize the time-frequency resolution. Matching the effective window width with the main spectrum energy interval of signals, WMS determines the standard deviation without iterative calculation. Based on the time-frequency representation of FAST, four features are extracted as the feature vectors and applied to the support vector machine (SVM), probabilistic neural network (PNN), extreme learning machine (ELM), convolutional neural network (CNN), decision tree (DT-C4.5) and random forest (RF) classifiers. Classification results of the six classifiers show that FAST has better time-frequency resolution and lower computational complexity than that of generalized S-transform (GST) and ST. In addition, the FAST-ELM method has stronger noise immunity and better performance than other combination methods with the simulation signals and experimental signals.

Quadratic phase S-Transform: Properties and uncertainty principles

In this paper, a novel quadratic phase S-transform (QPST) is proposed, by generalizing the S-transform (ST) with five parameters a, b, c,d and e. QPST displays the time and quadratic phase domain-frequency information jointly in the time-frequency plane. Firstly, we define the novel QPST and give its relation with quadratic phase Fourier transform (QPFT). Secondly, several important properties of newly defined QPST, such as conjugation, translation, modulation, orthogonality relation and reconstruction formula are derived. Finally, we formulate several classes of uncertainty inequalities, such as the Heisenberg uncertainty inequality and logarithmic uncertainty inequality.

Fault Detection and Classification to Design a Protection Scheme for Utility Grid with High Penetration of Wind and Solar Energy

This paper designed a protection scheme for utility grid with high share of renewable energy (RE) generated from wind energy and solar energy plants. This is based on extraction of features from the current using Stockwell transform (ST), Hilbert transform (HT), and alienation coefficient. A Stockwell index (SI) is designed by extracting current features using ST, a Hilbert index (HI) is designed by extracting current features using HT, and an alienation index (AI) is designed by extracting current features using an alienation coefficient. A fault index (FI) is formulated by multiplying the SI, HI, AI, and WF (weight factor). This FI is implemented for fault detection. Fault classification is achieved considering number of fault phases and ground fault index (GFI). This GFI is designed by processing zero sequence currents applying ST. GFI effectively identifies the ground involved during event of a fault. A designed protection scheme is effective to identify faults in the scenario of high RE share and during various cases of study which includes the variations of fault impedance, different fault occurrence angles (FOA), fault incident at different nodes, and noisy condition. This protection scheme effectively discriminates the fault events from the operational events such as feeder operation, load, and capacitor switching. Performance of hybrid protection method formulated in this paper is better relative to alienation coefficient-based protection scheme (ACPS) reported in literature. The ACPS has maximum error and mean error of fault detection equal to 9.54% and 5.99%, respectively, which is relatively high compared to the respective values for the proposed method which are 1.89% and 0.978%, respectively. ACPS is effective for detecting the fault events in noise level of 30 dB SNR (signal-to-noise ratio) whereas the proposed method effectively identifies the faults in the high noise scenario of 20 dB SNR.

Advanced Online Broadband Impedance Spectrum Acquisition of Fuel Cells by S-Transform

Electrochemical impedance spectrum (EIS) applied in fuel cell characterization and diagnosis facilitates its system management in the vehicular application, but the first consideration of impedance-based management is how to obtain impedance online. The traditional laboratory method based on sine sweep is unsuitable for vehicular EIS measurement as limited by measured time and equipment price. In this article, we design an excitation device that can generate the square wave signal containing many harmonics as the excitation current. Besides, introduce S-transform (ST) for online broadband fuel cell impedance spectrum fast calculation, whose window width can be adaptively changed with frequency, and meanwhile, preserving high computational efficiency by fast Fourier transform operation. Based on this, the performance of ST and the previous method based on Morlet wavelet transform (MWT) is compared under different operating conditions. Extensive experimental results show that the ST can calculate impedance with the mean relative error of less than 3% based on appropriate excitation, while the computed time can be saved by nearly 95% as compared to MWT. Furthermore, the effectiveness and robustness of ST against different current densities, cell temperatures, fault scenarios with various fault degrees, and stable time are also verified through experimentation.

Early and Intelligent Bearing Fault Detection Using Adaptive Superlets

Faults in bearings interrupt the operation of three-phase induction motors (IMs) and disturb factory production. Vibration signal analysis, feature extraction, and incorporation of artificial intelligence can enhance the ability of fault detection in bearings. However, implementing these techniques is challenging in industries due to harsh environmental conditions and massive computational power. Prioritizing the issue, this article presents a fusion method considering the time-frequency contents of vibration signals via the super-resolution property of wavelets and a two-dimensional convolutional neural network (2-D-CNN). First, a laboratory-based real-time process is considered with Variable Frequency Drive (VFD) driven IM which is also exposed to environmental vibration. In the present work, a piezoelectric sensor is used to collect the vibration of IM during operation. Then the collected time-series signals are stored in a computer through a Data Acquisition System (DAS). Further, signals are segmented in small windows to extract respective feature images using Continuous Wavelet Transform (CWT), Stockwell Transform (ST), and Adaptive Superlet Transform (ASLT). Thereafter, the image resolution is optimally resized and fed to a 2-D-CNN model for multi-label classification of the bearings. It is observed that ASLT-2-D-CNN with 32×32×3 resolution outperforms the other two fusion methods as it reports high accuracy and reduces runtime (training and testing) along with memory usage. The robustness of ASLT-2-D-CNN is also validated with two open-source bearing datasets. Thus, the result indicates that the proposed method for bearing fault detection can be deployed in the field to make IMs safe which reduces the maintenance cost of the industry.

Utilization of Stockwell Transform, Support Vector Machine and D-STATCOM for the Identification, Classification and Mitigation of Power Quality Problems

Power Quality (PQ) has become a significant issue in power networks. Power quality disturbances must be precisely and appropriately identified. This activity involves identifying, classifying, and mitigating power quality problems. A case study of the Awada industrial zone in Ethiopia is taken into consideration to show the practical applicability of the proposed work. It is found that the current harmonic distortion levels exceed the restrictions with a maximum percentage Total Harmonic Distortion of Current (THDI) value of up to 23.09%. The signal processing technique, i.e., Stockwell Transform (ST) is utilized for the identification of power quality issues, and it covers the most important and common power quality issues. The Support Vector Machine (SVM) method is used to categorize power quality issues, which enhances the classification procedure. The ST scored better in terms of accuracy than the Wavelet Transform (WT), Fourier Transform (FT), and Hilbert Transform (HT), obtaining 97.1%, as compared to 91.08%, 88.91%, and 86.8%, respectively. The maximum classification accuracy of SVM was 98.3%. To lower the current level of harmonic distortion in the industrial sector, a Distribution Static Compensator (D-STATCOM) is developed in the current control mode. To evaluate the performance of the D-STATCOM, the performance of the distribution network with and without D-STATCOM is simulated. The simulation results show that THDI is reduced to 4.36% when the suggested D-STATCOM is applied in the system.

Current- and Voltage-Actuated Transmission Line Protection Scheme Using a Hybrid Combination of Signal Processing Techniques

This paper presents a current- and voltage-driven protection scheme for transmission lines based on a hybrid mix of Stockwell transform (ST) and Hilbert transform (HT). Use of both current and voltage waveforms to detect and categorize faults, improves the reliability of this protection scheme and avoids false tripping. Current and voltage waveforms captured during a period of fault are analyzed using ST to compute a median intermediate fault index (MIFI), a maximum value intermediate fault index (MVFI), and a summation intermediate fault index (SIFI). Current and voltage signals are analyzed via applying HT to compute a Hilbert fault index (HFI). The proposed hybrid current and voltage fault index (HCVFI) is obtained from the MIFI, MVFI, SIFI, and HFI. A threshold magnitude for this hybrid current and voltage fault index (HCVFITH) is set to 500 to identify the faulty phase. The HCVFIT is selected after testing the method for various conditions of different fault locations, different fault impedances, different fault occurrence angles, and reverse flows of power. Fault classification is performed using the number of faulty phases and an index for ground detection (IGD). The ground involved in a fault is detected by comparison of peak IGD magnitude with a threshold for ground detection (THGD). THGD is considered equal to 1000 in this study. The study is carried out using a two-terminal transmission line modeled in MATLAB software. The performance of the proposed technique is better compared to a discrete wavelet transform (DWT)-based technique, a time–frequency approach, and an alienation method. Our algorithm effectively detected an AG fault, observed on a practical transmission line.

Application of signal processing techniques and intelligent classifiers for high-impedance fault detection in ensuring the reliable operation of power distribution systems

High-impedance fault (HIF) is always a threat and the biggest challenge in the power transmission and distribution system (PTDS). For a PTDS to operate effectively, HIF diagnosis is essential. However, given the HIF’s nature and the involved complexity, detection, identification, and fault location are difficult. This will be even more complicated in conventional PTDSs as they are inefficient and highly vulnerable. Given the importance and urgent need for HIF diagnosis in PTDS, this study reviews state-of-the-art HIF phenomenon and detection techniques and proposes the use of “various signal processing techniques for fault feature extraction” and “ different classifiers for identifying HIF.” First, HIF current/voltage signals are analyzed using signal processing techniques, which include the discrete wavelet transform (DWT), pattern recognition, Kalman filtering, TT transform, mathematical morphology (MM), S transform (ST), fast Fourier transform (FFT), principal component analysis (PCA), linear discriminant analysis (LDA), and wavelet transforms, such as dual-tree, maximum overlap discrete wavelet transform (MODWT), and lifting wavelet transform (LWT). Second, the various HIF and non-HIF faults are classified using intelligent classifiers. The intelligent classifiers include artificial neural networks (ANNs), probabilistic neural networks (PNNs), genetic algorithms (GAs), fuzzy logic, adaptive neuro-fuzzy interface system, support vector machine (SVM), extreme learning machine (ELM), adaptive resonance theory, random forests (RFs), decision trees (DTs), and convolution neural networks (CNNs). In addition to the comparative discussion of various classifier techniques, their evaluation criterion and performance are prioritized. Third, this review also studied different test systems, such as radial distribution network, mesh distribution network, IEEE 4 node, IEEE 13 node feeder, IEEE 34 node feeder, IEEE 39 node feeder, IEEE 123 node feeder, Palash feeder, and test microgrid systems, to assess the pertinence of various HIF detection schemes and the behavior along with methods to locate the HIF. Overall, we believe this review would serve as a comprehensive compendium of advanced techniques for HIF diagnosis in different test systems.

Characterization of infrasonic wind noise using the Discrete Orthonormal Stockwell Transform

The Discrete Orthonormal Stockwell Transform (DOST) (Stockwell, 2007) projects a finite-length, periodically sampled signal onto a set of orthonormal basis vectors that are localized in time and scale such that all basis functions at each scale are represented by a finite number of Discrete Fourier Transform (DFT) basis functions. It is based on the Stockwell Transform (Stockwell et al., 1996), sometimes referred to as the S-transform, which is applicable to continuously defined waveforms. Furthermore, the DFT basis functions corresponding to each DOST scale are non-overlapping. The shorter DOST scales correspond to higher-frequency DFT basis functions and greater bandwidths than the longer DOST scales. Thus, the DOST is a type of time-frequency transform and is suitable for non-stationary time-series analysis. Additionally, the DOST of an N-point record can be calculated efficiently in order N log N operations (Wang and Orchard, 2009). Several applications of the S-transform and the DOST have been documented including beamforming with array data (Collar and Frazier, 2022; Frazier, 2022). In this presentation, the DOST used is to examine the statistics of data measured with infrasound sensors as a function of the DOST scales, adaptively characterize the amplitude distributions of the DOST coefficients at each scale and perform near-optimal detection of transient signals. Results corresponding to infrasound data recorded under various conditions are presented.

Noise source separation and identification of excavator cab based on EEMD-PCA-ICA-ST

In order to effectively separate and identify the noise source of excavator cab, this paper adopts the methods of ensemble empirical mode decomposition (EEMD), principal component analysis (PCA), independent component analysis (ICA) and S-transform (ST) to study. Firstly, EEMD is used to decompose the single channel noise signal collected at the driver’s position into multiple intrinsic mode functions (IMF); Secondly, PCA is used to reduce the dimension of IMFs to remove the redundant IMF; Next, ICA processing is performed on the remaining IMFs to obtain independent IC components; Finally, S-transform is used to analyze the time-frequency of IC component, and the source of IC component is identified by prior experience. The results show that the main noise sources are the vibration radiation noise of related parts in the cab, the combustion noise of diesel engine and the fan noise. After identifying the main noise sources, the next step is to study the contribution of these noise sources, which provides a new idea for noise control.

Power Quality Detection and Categorization Algorithm Actuated by Multiple Signal Processing Techniques and Rule-Based Decision Tree

This paper introduces a power quality (PQ) detection and categorization algorithm actuated by multiple signal processing techniques and rule-based decision tree (RBDT). This is aimed to recognize PQ events of simple nature and higher order multiplicity with less computational time using hybridization of the signal processing techniques. A voltage waveform with a PQ event (PQE) is processed using the Stockwell transform (ST) to compute the Stockwell PQ detection index (SPDI). The voltage waveform is also processed using the Hilbert transform (HT) to compute the Hilbert PQ detection index (HPDI). A voltage waveform is also decomposed using the Discrete Wavelet transform (DWT) to compute the classification feature index (CFI) [CFI1 to CFI4]. A combined PQ detection index (CPDI) is computed by multiplication of the SPDI, the HPDI and CFI1 to CFI4. Incidence of a PQE on a voltage signal is located with the help of a location PQ disturbance index (LPDI) which is computed by differentiating the CPDI with respect to time. CFI5, CFI6 and CFI7 are computed from the SPDI, the HPDI and the CPDI, respectively. Categorization of PQ events is performed using CFI1 to CFI7 by the rule-based decision tree (RBDT) with the help of simple decision rules. We conclude that the proposed algorithm is effective to identify the PQE with an accuracy of 98.58% in a noise-free environment and 97.62% in the presence of 20 dB SNR (signal-to-noise ratio) noise. Ten simple nature PQEs and eight combined PQ events (CPQEs) with multiplicity of two, three and four are effectively detected and categorized using the algorithm. The algorithm is also tested to detect a sag PQ event due to a line-to-ground (LG) fault incident on a practical distribution utility network. The performance of the investigated method is compared with a DWT-based technique in terms of accuracy of classification with and without noise, maximum computational time of PQ detection and multiplicity of PQE which can be effectively detected. A simulation is performed using the MATLAB software. MATLAB codes are used for modelling the PQE disturbances and the proposed algorithm using mathematical formulations.

Traveling Wave-Based Fault Localization in FACTS-Compensated Transmission Line via Signal Decomposition Techniques

Modern power systems are structurally complex and are vulnerable to undesirable events like faults. In the event of faults in transmission line, accurate fault location improves restoration process, thereby enhancing the reliability of the overall system. Fault location methods (FLMs) are tools which assist in identifying fault locations quickly. However, the accuracy of these FLMs gets affected in the presence of flexible alternating current transmission system (FACTS) devices. Therefore, in this work, the performance of four different signal decomposition techniques aided traveling wave aided FLMs are qualitatively compared in the context of fault localization in FACTS-compensated systems. FLMs based on intrinsic time decomposition (ITD), empirical mode decomposition (EMD), S-transform (ST), and estimation of signal parameters via rotational invariance technique (ESPRIT) are investigated. The accuracy of FLMs is tested for different cases of series, shunt, and series-shunt FACTS-compensated systems. A 500 kV system employed with 100 MVAr FACTS device is used for simulation. The instant of arrival wave at end of transmission line is from all aforementioned FLMs. The obtained ATWs are used in fault localization. Further, the associated percentage errors are calculated. The results suggest that EMD and ESPRIT-based FLMs are more accurate than others.

High impedance fault detection in distribution network based on S-transform and average singular entropy

When a high impedance fault (HIF) occurs in a distribution network, the detection efficiency of traditional protection devices is strongly limited by the weak fault information. In this study, a method based on S-transform (ST) and average singular entropy (ASE) is proposed to identify HIFs. First, a wavelet packet transform (WPT) was applied to extract the feature frequency band. Thereafter, the ST was investigated in each half cycle. Afterwards, the obtained time-frequency matrix was denoised by singular value decomposition (SVD), followed by the calculation of the ASE index. Finally, an appropriate threshold was selected to detect the HIFs. The advantages of this method are the ability of fine band division, adaptive time-frequency transformation, and quantitative expression of signal complexity. The performance of the proposed method was verified by simulated and field data, and further analysis revealed that it could still achieve good results under different conditions.

Fall Feature Enhancement and Fusion Using the Stockwell Transform With Dual mmWave Radars

Fall detection based on radar technology has recently attracted increasing attention. Human motions are often multidirectional and have complex trajectories. Numerous time–frequency representation methods have been used to extract micro-Doppler fall features. Peak-to-peak Doppler frequency measurements are limited due to the relatively weak energy in the high-Doppler-frequency area. Fall features are not obvious when using a single millimeter-wave (mmWave) radar if human motions are nearly perpendicular to the radar. A fall feature enhancement and fusion method using the Stockwell transform with dual mmWave radars is proposed in this article. Two mmWave radars synchronously collect human motions in different directions, and the micro-Doppler features of these human motions are obtained by using the Stockwell transform. To enhance the features obtained for recognition, the ridgelines extracted from the two radars are fused in the time–frequency domain. In addition, a model trained by a convolutional neural network (CNN) is used for fall recognition; 960 fall motions and 720 nonfall motions are collected from 60 volunteers. The experimental results show that the proposed method can produce a recognition accuracy of up to 94.14%, which is beneficial for the realization of an indoor fall detection system.

Damage Detection in Reinforced Concrete Member Using Local Time-Frequency Transform Applied to Vibration Measurements

Signal processing and analysis of structural vibration measurements are key components of structural damage detection (SDD) in structural health monitoring (SHM). The goal of signal processing is to extract subtle changes in the measured signals, which can be used to infer changes in structural parameters and damage. Time-frequency analysis is one of the most popular characterization methods for studying non-stationary vibration signals. In this article, the local time-frequency transform (LTFT) is applied and evaluated to calculate the time-domain signals because of its excellent time-frequency energy distribution properties. The LTFT matches the input data by the Fourier basis in an inverse problem framework and uses the least squares method to solve the time-varying Fourier coefficients. Subsequently, it defines the time-frequency spectrum as the calculated time-varying Fourier coefficients. While the LTFT has been used in the field of geophysics for seismic data processing, its application to structural vibration signals is novel. Both synthetic signals as well as signals collected from a large-scale laboratory test of a reinforced concrete girder were processed with the LTFT and compared with Rényi entropy for quantifying the time-frequency spectrum, the time-frequency resolution abilities of short time Fourier transform (STFT), and S transform (ST). The results show that the LTFT is superior to the traditional time-frequency analysis schemes, in that it is more effective in identifying the energy changes in the time-frequency spectrum before and after structural damage in the form of cracking has occurred. At the same time, it provides high-precision time-frequency resolution and excellent noise suppression abilities. The effectiveness and feasibility of the LTFT applied to the synthetic and experimental signals are verified.

Red deer optimized recurrent neural network for the classification of power quality disturbance

Power Quality Disturbance (PQD) in a power grid distribution destroys the quality of power to the user. Thus, early detection of disturbances in the power grid distribution is essential to diagnose the network before failure. Several disturbances in the power grid may cause voltage sag, voltage swell, or occurrence of both. In the proposed method deep recurrent neural network (DRNN) is used for classifying the PQD as well as Red Deer Optimization (RDO) algorithm is used for optimizing the weight from DRNN. Based on the behaviour of deer roaring rate will optimize the weight of DRNN from RDO. Signal processing is done by S-transform (ST) because of the better performance in signals detection in terms of a high order of noise. The proposed method is implemented in Simulink tool and the results are compared with the existing methods. The result shows that the power disturbances are classified with high accuracy of 99.95% and precision of 99.98% that are higher than the existing methods.

An Improved Unscale S-Transform in Frequency Domain

The time–frequency analysis methods are powerful tools in seismic interpretation and bright spot identification. The S-transform (ST), as a hybrid of the short-time Fourier transform (STFT) and continuous wavelet transform (CWT), can achieve progressive time–frequency resolution. However, limited by its linear-frequency-dependent term, the ST obtains a deviated frequency distribution. By removing this term, the frequency form of unscaled ST (FUST), as a variation of the ST, is proposed to preserve the reliable frequency distribution. The fly in the ointment is that the FUST decreases the temporal resolution of the low-frequency components, which may not be suitable for seismic reflection interpretation and reservoir location. In addition, the ST cannot tailor the time–frequency resolution for particular applications. To solve these problems, a simple and effective method is proposed by substituting the basis of the FUST. The proposed method can obtain the desired time resolution by adjusting two adjustable parameters. The corresponding inverse transform is also derived to guarantee its energy conservation and inevitable property. Numerical experiments and real data examples show better performance of the proposed method in improving temporal resolution and reservoir location over the ST and FUST.

SSN: Stockwell Scattering Network for SAR Image Change Detection

Recently, synthetic aperture radar (SAR) image change detection has become an interesting yet challenging direction due to the presence of speckle noise. Although both traditional and modern learning-driven methods attempted to overcome this challenge, deep convolutional neural networks (DCNNs)-based methods are still hindered by the lack of interpretability and the requirement of large computation power. To overcome this drawback, wavelet scattering network (WSN) and Fourier scattering network (FSN) are proposed. Combining respective merits of WSN and FSN, we propose Stockwell scattering network (SSN) based on Stockwell transform (ST), which is widely applied against noisy signals and shows advantageous characteristics in speckle reduction. The proposed SSN provides noise-resilient feature representation and obtains state-of-the-art performance in SAR image change detection as well as high computational efficiency. Experimental results on three real SAR image datasets demonstrate the effectiveness of the proposed method.

Detection and Classification of Power Quality Disturbances Using Stock Well Transform and Improved Grey Wolf Optimization-Based Kernel Extreme Learning Machine

Power quality disturbances (PQDs) in modern electrical power systems, caused by the integration of nonlinear power electronic devices and erratic distributed generation (DG), lead to interruptions and significant energy losses for end users. However, conventional methods for PQDs classification face challenges in dealing with noise interference and feature selection. To address these challenges, this research paper proposes a novel approach that combines the stockwell transform (ST) with an improved grey wolf optimization-based kernel extreme learning machine to enhance classification accuracy. The stockwell transform is utilized to extract meaningful features from the power quality (PQ) signals, which are subsequently input into the kernel extreme learning machine (KELM). Furthermore, the parameters of the KELM model are optimized using the improved grey wolf optimization (IGWO) approach to improve the accuracy of classification. To evaluate the performance of the proposed method, real-time implementation is considered by incorporating PQDs data with signal-to-noise ratios (SNR) of 20 dB, 30 dB, and 40 dB into the original synthetic signals. Multiple noise conditions are simulated to assess the proposed model ability to identify and classify disturbance signals. The results demonstrate a detection accuracy of 99.76% under noiseless conditions, indicating the model’s high accuracy. Moreover, the proposed method exhibits robustness to noise, achieving accuracies of 98.86%, 98.32%, and 97.3% at SNR levels of 40 dB, 30 dB, and 20 dB, respectively. In the end, this work has performed a comparative study with other previously published work. Compared with other classification methods, the algorithm proposed in this paper has higher accuracy, and it is an efficient and feasible classification method.