Parseval's Theorem Research Articles

Wavelet Based Energy Extraction Method for Determining Power Quality Disturbances

In this paper, energy of wavelet coefficients are used for determining power quality disturbances (PQD) which are important for power systems. These power quality disturbances are voltage with harmonics, transient and flicker. After analyzing energy coeffients based on wavelet transform, these PQD are compared with healty condition. 50 Hz pure sine is chosen as reference. These signals are generated by using MATLAB. Sampling frequency is 25.6 kHz. Energy distribution of detail coefficients are obtained by using 12 level Daubechies 4 discrete wavelet filter. Parseval theorem is applied to wavelet coefficients for obtaining energy distribution of detail coefficients of these relevant disturbances in different resolution levels. Satisfactory results are taken visually when examining energy distrubutions of these PQD. Also it is observed that these PQD can be distinguished visually when their amplitudes are increased. Energy coefficients of PQD and amplitude of PQD are in direct propotion.

New sparse regularization approach for extracting transient impulses from fault vibration signal of rotating machinery

Fault impulses induced by localized failure is critical for fault diagnosis and degradation prognostics of rotating machinery, however, weak transient fault impulses are always submerged in heavy noise and unrelated components. To overcome this bottleneck issue, a novel approach named smoothing sparse low-rank matrix (SSLRM) associated with asymmetric and singular value decomposition (SVD) penalty regularizers is proposed, for the first time. To be specific, the asymmetric and SVD penalty regularizers are utilized to alleviate the issues of underestimation deficiency of traditional sparse regularization, where the Parseval theorem in wavelet framework and smoothing operation are conducted, so as to enhance the sparsity of the estimated transient impulses while restraining the unrelated interference. Meanwhile, the strict convexity of the cost function is proved theatrically, and the effective optimization algorithm with fast convergence speed based upon the alternating direction method of multipliers (ADMM) is proposed through splitting the optimization function into two simple sub-parts. The simulated case and two experimental cases regarding bearings failure in corn thresher and gear-root crack failure in reducer are investigated for theoretical verification of the proposed approach, the results indicate that diagnosis accuracy and fault impulses amplitude of the algorithm are superior to the state-of-the-art benchmarks.

Fractional Fourier transform associated with polar coordinates

The fractional Fourier transform (FRFT) is a generalized form of the Fourier transform (FT), it is another important class of time–frequency analysis tool in signal processing. In this paper, we study the two-dimensional (2D) FRFT in the polar coordinates setting. First, Parseval theorem of the 2D FRFT in the polar coordinates is obtained. Then, according to the relationship between 2D FRFT and fractional Hankel transform (FRHT), the convolution theorem for the 2D FRFT in polar coordinates is obtained. It shows that the FRFT of the convolution of two functions is the product of their respective FRFTs. Moreover, the fast algorithm for the convolution theorem of the 2D FRFT is discussed. Finally, the sampling theorem for signal is explored.

Nonlinear oscillations of a cylindrical shell with a circumferentially discontinuous elastic base

This work investigates the linear and nonlinear vibrations of a functionally graded (FG) transversally excited cylindrical shell and in contact along its length with a circumferentially discontinuous elastic base, a configuration frequently found in practical applications. The elastic medium is modeled as a Winkler foundation. The adopted distribution of the FG material along the thickness is similar to that of a sandwich shell, with one material predominating in the core and the other on both faces. First a finite element analysis (FEA) is performed to obtain the natural frequencies and vibration modes of the FG cylindrical shell and a parametric analysis is conducted to study the influence of the contact region and foundation stiffness on natural frequencies and mode shapes. Even though the problem is a symmetric, the cylindrical shell displays a sequence of symmetric and antisymmetric vibrations modes and these modes may interact due to nonlinear coupling. For much geometry the lowest vibration modes have very close natural frequencies, leading to internal resonances, which influence strongly the nonlinear shell response. To obtain a reliable reduced order model (ROM) for the nonlinear dynamic analysis, Donnell’s nonlinear shallow shell theory is considered. To discretize the nonlinear equations of motion by the Galerkin method, a consistent modal solution is obtained considering nonlinear couplings and interactions. Here, first the displacements are consistently derived from a perturbation procedure proposed in previous papers. To start the perturbation procedure, a seed solution is necessary for the transversal displacement field. For this, first the linear vibration modes obtained from the finite element analysis are expanded in Fourier series in terms of the circumferential coordinate. Then, the essential modes to be used in the ROM are identified using a mathematical procedure that combines Fourier transform and Parseval theorem. The proposed modal solution for the transversal displacements is also used to obtain the modal expansions for the axial and circumferential displacements and their modal amplitudes are determined as a function of the modal amplitudes of the transversal displacements, thus reducing even further the number of unknows. These expansions are finally used to discretize the nonlinear equation of motion in the transversal direction and the resulting equations of motion are solved numerically using continuation software AUTO. A geometry displaying multiple internal resonances is selected and resonance curves, time-responses and phase-portraits are obtained to analyze the nonlinear dynamics of the shell. Depending on the excitation, different bifurcation scenarios are obtained, with important changes in stability and resonant peaks due to modal interaction. Competition between coexisting stable attractors is investigated using basins of attraction, which exhibit a complex topology, often with fractal boundaries.

Preliminary study of radio frequency waves in hypervelocity impact plasma

Orbital debris and meteoroids can cause mechanical and electrical anomalies by impacting spacecraft. Although mechanical damage tends to occur only from particles with masses >10−3g, electrical damage can be a result of hypervelocity (> 10 km/s) impacts (HVI) even with smaller particles. In HVI, the projectile speed exceeds the speed of sound in the material and has enough energy to create plasma by ionizing the material near the surface, which can then emit radio frequency (RF) waves.In this paper, we conduct a preliminary study on the characteristics of impact-generated RF emission with the ultimate goal of building an RF predictive model for risk assessment of HVI electrical effects that can damage spacecraft. First, we introduce wavelet analysis as a method for denoising and quantifying the total RF signal energy. An appropriate mother wavelet and the number of decomposition levels are selected using a synthetic signal. Second, the total energy of the RF emission is calculated using Parseval's theorem. Results suggest that the total RF signal energy is proportional to the energy flux of the impactor. Finally, spectrograms of the RF emission are analyzed to characterize features that can be used in the future to predict impacts that can cause electrical damage to satellites.

A physics‐informed learning technique for fault location of DC microgrids using traveling waves

Fast and accurate fault location in DC power systems is of particular importance to ensure their reliable operation. One of the approaches for implementing a fast-tripping protection scheme is to use Traveling waves (TW) initiated by a fault scenario. This paper proposes a physics-informed machine learning approach that utilizes TWs for fault location in DC microgrids. TWs are extracted by the so-called multiresolution analysis which identifies the TW's wavelet coefficients for multiple frequency ranges. This paper deploys Parseval's theorem to find the energy of wavelet coefficients as a quantitative metric for describing TWs. The hypothesis of this paper is that once the Parseval energy curves for a specific cable are extracted, they can be utilized to locate faults along with that cable regardless of the DC system in which the cable is deployed. The fault location algorithm uses Parseval energy curves to train a Gaussian Process (GP) estimator. With the Parseval energy values of measured current at the protection device location, the GP estimator is able to estimate fault locations with high accuracy. The effectiveness of the proposed algorithm is verified by simulating a DC microgrid system in PSCAD/EMTDC.

Novel windowed linear canonical transform: Definition, properties and application

Linear canonical transform (LCT) has emerged as a powerful tool in nonstationary signal processing. However, it fails to reveal the information of local frequency varying with time due to the global kernel function. In this paper, a novel windowed linear canonical transform (NWLCT) is presented to give a clear time-frequency representation for time-varying signals. Firstly, the definition of NWLCT is proposed by a generalized convolution operator not a sliding window. Some basic properties of NWLCT are derived, such as inverse transform, Parseval theorem and reproducing kernel. Moreover, time-frequency resolution of NWLCT is analyzed theoretically, and the optimal window is derived. Finally, the applications of NWLCT in local spectral analysis for linear frequency modulated (LFM) signals are offered, including mono- and multi-component signals with similar local features. Theoretical and simulation results demonstrate that the proposed NWLCT preserves a crucial physical interpretation similar to classical short-time Fourier transform, which extends low-pass filter banks in the LCT domain. Compared with other time-frequency methods, NWLCT outperforms in improving the energy concentration and being robust to noise in terms of LFM signal processing.

A Novel Energy Flow Analysis and Its Connection With Modal Analysis for Investigating Electromechanical Oscillations in Multi-Machine Power Systems

In this paper, a novel energy flow analysis (EFA) is proposed based on the signal reconstruction and decomposition to investigate the electromechanical oscillations. In contrast to the conventional EFA, the connection between the proposed EFA and modal analysis (MA) can be quantitatively revealed for arbitrary models of synchronous generators in multi-machine power systems. Firstly, the time-domain implementation (TDI) of the proposed EFA is designed. Specifically, the measurements at the terminal of a local generator are reconstructed through an exponential operator and then decomposed with respect to an angular frequency. Then, the mode-screened damping torque coefficient is defined to extract the damping feature with respect to an electromechanical oscillation mode. After that, the frequency-domain implementation (FDI) is derived. Specifically, the Parseval's Theorem is applied to transform the proposed EFA from the time domain to frequency domain. On this basis, the consistency between the proposed EFA and MA is strictly proved, which is applicable for arbitrary models of synchronous generators in multi-machine power systems. Additionally, the application procedure of the proposed EFA in investigating electromechanical oscillations is given. Finally, the proposed EFA are substantially demonstrated in multiple case studies.

Application of energy flow analysis in investigating machine‐side oscillations of full converter‐based wind generation systems

The machine-side oscillation (MSO) of the full converter-based wind generation (FCWG) systems is a critical threat to the reliable wind power supply. The introduction of the auxiliary resonance controller (ARC) to the machine-side converter (MSC) controls of FCWG can effectively improve the converter-driven stability of the power grid, but it would also complicate MSOs of FCWG. Thus, it is of great significance to study the damping feature of MSOs of FCWG. In this paper, the energy flow analysis (EFA) is applied to quantitatively investigate MSOs of FCWG. Firstly, the configuration and machine-side control loops of FCWG are briefly introduced. Then, a mode screening-based EFA is proposed for evaluating the damping feature of MSOs of FCWG in the time domain by applying the principle of the Laplace transform. After that, the eigenvalue analysis is conducted for MSOs of FCWG in the frequency domain to lay a foundation for revealing the essence of EFA. On this basis, the consistency of EFA with the eigenvalue analysis is strictly proved based on the Parseval's Theorem, which is applicable for arbitrary control schemes of FCWG. Finally, the proposed EFA is applied in numerically investigating multiple types of MSOs of FCWG in case studies.

Frequency-based NMPC for Flexible Systems with Parametric Resonance

Many mechatronic systems, especially nonlinear oscillatory systems with variable structures, are prone to parametric resonance caused by time-varying characteristics. This paper addresses the vibration damping problem of a flexible nonlinear system with parametric resonance studied on a stacker crane with a variable load. It tackles the related challenges of established approaches for both feedforward and feedback. Accordingly, this paper proposes a novel scheme of nonlinear model predictive control (NMPC). In particular, it addresses parametric resonance explicitly and exploits the Parseval theorem to enforce specific requirements on relevant frequencies. Recursive feasibility and stability are guaranteed by using an appropriate invariant set. In addition, experimental validation is performed to demonstrate the significant advantages of the proposed NMPC scheme over two established vibration damping control approaches.

Data-driven simulation of multivariate nonstationary wind velocity with explicit introduction of the time-varying coherence functions

An S-transform (ST) based data-driven simulation methodology is proposed of multivariate nonstationary wind velocity with explicit introduction of the time-varying coherence functions. This framework differs from that based on the evolutionary spectral theory that uniform or non-uniform modulation functions needs to be assigned. In defining the time-frequency power spectral density (TFPSD) in accordance with the ST coefficients, the Parseval's theorem is considered to ensure energy preservation. The proposed method is firstly applied to simulated nonstationary wind velocity. For comparisons, the results of the discrete orthonormal S transform (DOST) are also included. The research demonstrates that the ST based results agree well with the original, but a bias exists between the results simulated by DOST and the original, indicating that the feasibility of the proposed method. Employing four measured downburst records to simulate multivariate nonstationary wind velocity by the proposed method further validates its feasibility. In order to enhance the simulation fidelity, the inverse S-transform (IST) based regulation and control method (IST-RCM) of high precision is developed, where the balancing parameter is defined and formulated to express quantitatively the relationship between accuracy and efficiency. It is found that the results with IST-RCM get more close to the actual wind velocity records.

Three Dimensional Fractional Fourier-Mellin Transform, and its Applications

The main objective of the paper is to study the three-dimensional fractional Fourier Mellin transforms (3DFRFMT), their basic properties and applicability due to mainly use in the radar system, reconstruction of grayscale images, in the detection of the human face, etc. Only the fractional Fourier transform is based on time-frequency distribution, whereas only the fractional Mellin transform is on scale covariant transformation. Both transforms can discover action in the definite assortment. The fractional Fourier transform is applicable for controlling the range of shift, whereas the fractional Mellin transform is accustomed to managing the range of rotation and scaling of the function. So, combining both transformations, we get an elegant expression for 3DFRFMT, which can be used in several fields. The paper introduces the concept of three-dimensional fractional Fourier Mellin transforms and their applications. Modulation property is the most useful concept in the signal system, radar technology, pattern reorganization, and many more in the integral transform. Parseval's identity applies to the conservation of energy in the universe. Thus we establish the modulation theorem, Parseval's theorem, scaling theorem, analytic theorem for three-dimensional fractional Fourier Mellin transform. We also give some examples of three-dimensional fractional Fourier-Mellin transform on some functions. Finally, we provide three-dimensional fractional Fourier-Mellin transform applications for solving homogeneous and non-homogeneous Mboctara partial differential equations that we can apply with advantages to solve the different types of problems in signal processing systems. The transform is beneficial in a maritime strategy as a co-realtor to control moments in any specific three-dimensional space. The concept is the most powerful tool to deal with any information system problems. After obtaining the generalization, we can explore many more ideas in applying three-dimensional fractional Fourier-Mellin transformations in many real word problems.

Electromagnetic Compatible Energy Measurements Using the Orthogonality of Nonfundamental Power Components

Measurement bandwidth of energy measurements are increasing to incorporate all the harmonics created by nonlinear loads and distributed generators making the measurement electronics complex and sensitive to electromagnetic interference. This article proposes to change the accuracy paradigm by focusing on fundamental active power and lower harmonics for energy metering, simplifying the electronics and making them robust against electromagnetic interference. Using the orthogonality of power flow via Parseval's theorem, a theoretical analysis, simulations, and measurements on power calculations using the fundamental active power are presented. It is shown that a perfect power measurement is achieved with a pure 50 Hz supply voltage, regardless of the nonlinear current. Even with the highest allowed harmonic distortion of the voltage and the current as listed in the international standards EN 50160 and IEC 61000-3-2, more than 97.5% of the active power is contained in the fundamental active power. Negligible active power is contained in the higher frequency components. This error margin falls within limits for electricity meters. Filtering the current with a basic low-pass filter can prevent the erroneous measurements that have appeared with static energy meters. In other words, it has been proven that negligible energy flows in the higher frequency components.

Finite frequency fault estimation observer design for T-S Fuzzy discrete-time descriptor systems

This paper deals with the problem of fault estimation observer design for T-S fuzzy nonlinear discrete-time descriptor systems in the finite frequency domain. First, a novel fault estimation observer for the augmented descriptor system is given. Based on Parseval's theorem and Lyapunov theory, the performance in the finite frequency domain is derived via time-domain method. Then, new sufficient conditions of admissibility and robust performance index for the dynamic error system are obtained as bilinear matrix inequalities. Meanwhile, an iterative linear matrix inequality algorithm is given to obtain the optimal solution. Finally, an example is given to demonstrate the effectiveness of our method.

Wavelet Energy Fuzzy Neural Network-Based Fault Protection System for Microgrid

To perform the fault protection for the microgrid in grid-connected mode, the wavelet energy fuzzy neural network-based technique (WEFNNBT) is proposed in this paper. Through the accurate activation of protective relay, the microgrid can be effectively isolated from the utility power system to prevent serious voltage fluctuation when the power quality of power system is disturbed. The proposed WEFNNBT can be divided into three stages—feature extraction (FE), feature condensation (FC), and disturbance identification (DI). In the FE stage, the feature of power signal at the point of common coupling (PCC) between microgrid and utility power system would be extracted with discrete wavelet transform (DWT). Then, the wavelet energy and variation of singular power signal can be obtained according to Parseval Theorem. To determine the dominant wavelet energy and enhance the robustness to the noise, the feature information is integrated in the FC stage. The feature information then would be processed in the DI stage to perform the fault identification and activate the protective relay if necessary. From the experimental results, it is realized that the proposed WEFNNBT can effectively perform the fault protection of microgrid.

Train\u2013track coupled dynamics analysis: system spatial variation on geometry, physics and mechanics

This paper aims to clarify the influence of system spatial variability on train–track interaction from perspectives of stochastic analysis and statistics. Considering the spatial randomness of system properties in geometry, physics and mechanics, the primary work is therefore simulating the uncertainties realistically, representatively and efficiently. With regard to the track irregularity simulation, a model is newly developed to obtain random sample sets of track irregularities by transforming its power spectral density function into the equivalent track quality index for representation based on the discrete Parseval theorem, where the correlation between various types of track irregularities is accounted for. To statistically clarify the uncertainty of track properties in physics and mechanics in space, a model combining discrete element method and finite element method is developed to obtain the spatially varied track parametric characteristics, e.g. track stiffness and density, through which the highly expensive experiments in situ can be avoided. Finally a train–track stochastic analysis model is formulated by integrating the system uncertainties into the dynamics model. Numerical examples have validated the accuracy and efficiency of this model and illustrated the effects of system spatial variability on train–track vibrations comprehensively.

Detection of the natural frequency of transmission power lines applying S-transform on Transient Recovery Voltage

This paper presents a new approach for detecting the natural frequency of transmission power lines applying S-transform on Transient Recovery Voltage (TRV). TRV usually appears between terminals of a circuit breaker during an isolation of a fault. The TRV phenomenon contains wideband frequency information such as the natural frequency of the system that depends not only on the electrical transmission line characteristics but also on the location of the fault. S-transform was applied to the TRV signal and the energy content in the time-frequency contours was evaluated using Parseval's theorem. This analysis ensures the accurate identification of the natural frequency of the resulting system by locating the maximum relative of the energy content. The case studies studied involved taking actual systems and using frequency-dependent phase models. The experiments were performed to show that the proposed approach can robustly determine the frequency associated with the parameters of the line. Different cases were considered: distances, inception angles, fault resistances, circuit breakers and models of electric arc. The simulation results demonstrate that using S-transform to analyse TRV provides a simple and reliable method for determining the natural frequency and opens a new way for studying the behaviour of protections and faults in power systems.

Deriving the Variance of the Discrete Fourier Transform Test Using Parseval’s Theorem

The discrete Fourier transform test is a randomness test included in NIST SP800-22. However, the variance of the test statistic is smaller than expected and the theoretical value of the variance is not known. Hitherto, the mechanism explaining why the former variance is smaller than expected has been qualitatively explained based on Parseval's theorem. In this paper, we explore this quantitatively and derive the variance using Parseval's theorem under particular assumptions. Numerical experiments are then used to show that this derived variance is robust.

Sensor fault estimation in finite-frequency domain for nonlinear time-delayed systems by T–S fuzzy model approach with local nonlinear models

This paper investigates the sensor fault estimation problem in finite-frequency domain for nonlinear time-delayed systems via T–S fuzzy approach with local nonlinear models. First, to estimate the fault, an augmented system is constructed, where the auxiliary state vector consists of the states of the system and the auxiliary filter. Second, an unknown input observer with finite-frequency specifications is designed to achieve fault estimation. Then, by utilising the Parseval's theorem, the sufficient conditions of the presented observer are established. Different from some existing methods, the state estimation error dynamics are decoupled from the local nonlinear dynamics via the designed observer such that the observer synthesis is simplified and the conservatism can be reduced. Meanwhile, compared with some conventional methods in the entire-frequency domain, a less restrictive analysis condition could be derived by the developed observer scheme. Finally, simulation results are given to illustrate the merit of the proposed approach.

An efficient numerical method for the generalised Kolmogorov equation

ABSTRACTAn efficient algorithm for computing the terms appearing in the Generalised Kolmogorov Equation (GKE) written for the indefinite plane channel flow is presented. The algorithm, which features three distinct strategies for parallel computing, is designed such that CPU and memory requirements are kept to a minimum, so that high-Re wall-bounded flows can be afforded. Computational efficiency is mainly achieved by leveraging the Parseval's theorem for the two homogeneous directions available in the plane channel geometry. A speedup of 3-4 orders of magnitude, depending on the problem size, is reported in comparison to a key implementation used in the literature. Validation of the code is demonstrated by computing the residual of the GKE, and example results are presented for channel flows at and , where for the first time they are observed in the whole four-dimensional domain. It is shown that the space and scale properties of the scale-energy fluxes change for increasing values of the Reynolds number. Among all scale-energy fluxes, the wall-normal flux is found to show the richest behaviour for increasing streamwise scales.