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Journal articles on the topic 'Parseval's Theorem'

Author: Grafiati

Published: 4 June 2021

Last updated: 7 September 2023

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1

FINE, JONATHAN. "A NOTE ON BRAIDS AND PARSEVAL'S THEOREM." Journal of Knot Theory and Its Ramifications 21, no. 05 (April 2012): 1250024. http://dx.doi.org/10.1142/s0218216511009546.

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In 1986 Falk and Randell, based on Arnold's 1969 paper on braids, proved that the pure braid groups are residually nilpotent. They also proved that the quotients in the lower central series are free abelian groups. This brief note uses an example to provide evidence for a much stronger conjectural statement: That each braid b can be written as an infinite sum [Formula: see text], where each bi is a linear function of the ith Vassiliev–Kontsevich Zi(b) invariant of b. The example is pure braids on two strands. This leads to solving eτ = q for τ a Laurent series in q. We set [Formula: see text] and use Fourier series and Parseval's theorem to prove eτ = q. For more than two strands the stronger statement seems to rely on an as yet unstated Plancherel theorem for braid groups, which is likely both to be deep and to have deep consequences.

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2

Melin, Jan O. "Interpreting ISAR Images by Means of Parseval's Theorem." IEEE Transactions on Antennas and Propagation 55, no. 2 (February 2007): 498–500. http://dx.doi.org/10.1109/tap.2006.889993.

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3

Bonatti, Ivanil S., Pedro L. D. Peres, and Amauri Lopes. "Velocity of Propagation in Transmission Lines." International Journal of Electrical Engineering & Education 35, no. 1 (January 1998): 79–86. http://dx.doi.org/10.1177/002072099803500107.

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This paper discusses the skin effect on lossy transmission lines in the context of undergraduate electrical engineering courses. A new definition for propagation time derived from Parseval's theorem is proposed. In lossless transmission lines the proposed definition produces the conventional results and for lossy lines it matches quite exactly with the time simulation results, as shown by an illustrative example.

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4

Han, Feng, Yao Lin Liu, Zhen Liu, and Hai Dong Zeng. "Comments on Errors of DFT Spectrum." Applied Mechanics and Materials 568-570 (June 2014): 189–92. http://dx.doi.org/10.4028/www.scientific.net/amm.568-570.189.

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Discrete Fourier transform (DFT/FFT) spectrums contain a variety of inherent errors in asynchronous sampling. Spectrum analysis with the accuracy above 10-3 are generally challenging issues. This work divides the DFT procedure into four signal transforms and exams six spectrum errors originated from these distortions. Besides the review of traditional errors, a so-called energy loss-gain (ELG) error is briefly introduced, which is proved to be a considerable error on the basis of Parseval's theorem. With the help of full error analysis mentioned here and the further development of analytical error estimators, it is expectable to obtain a DFT spectrum with a specified accuracy.

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5

Urbancic, T. I., C. I. Trifu, R. A. Mercer, A. J. Feustel, and J. A. G. Alexander. "Automatic time-domain calculation of source parameters for the analysis of induced seismicity." Bulletin of the Seismological Society of America 86, no. 5 (October 1, 1996): 1627–33. http://dx.doi.org/10.1785/bssa0860051627.

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Abstract A time-domain methodology for the automatic (on-line) calculation of source parameters is introduced. The method is based on applying Parseval's theorem to the spectral equations proposed by Andrews (1986). The appropriateness of the technique for the analysis of induced seismicity was tested by comparing on-line values of seismic moment, seismic energy, and static stress drop with those obtained in the frequency domain (off-line) for an aftershock sequence of events recorded underground at Creighton mine, Ontario. Our analysis shows that the calculated on-line and off-line source parameters are in good agreement, with an average slope offset of 4% from the 1:1 line, suggesting that the proposed method is reliable. Additionally, by applying empirical on-line attenuation corrections (coda Q), similar results are obtained to those calculated using off-line corrections (spectral decay), improving the average slope offset to 1%.

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6

Liu, Yao Lin, Feng Han, Zhen Liu, and Min Chen Zhai. "Analysis of Energy Loss-Gain Error in Discrete Fourier Transform." Applied Mechanics and Materials 568-570 (June 2014): 172–75. http://dx.doi.org/10.4028/www.scientific.net/amm.568-570.172.

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In asynchronous sampling, discrete Fourier transform (DFT) spectrum involves errors. Scholars have done great investigations on the correction techniques of DFT spectrum, but the errors have not been completely eliminated all along. In this paper, spectrums were examined from the principle of conservation of energy. It is unnoticed before that the energy of the digital signal, which is the analysis object of DFT, isn't equal to that of the finite continuous signal truncated by rectangular window. Thus the energy of their spectrums are different according to the Parseval's theorem. The Energy Loss-Gain (ELG) error was introduced to express the energy difference between these two spectrums. The ELG error is zero if the observed continuous signal is truncated in integral multiple of half cycle and it is related to the cycle number and sampling number in one cycle. Analysis show that the ELG error decreases with the increment of these two parameters, which are helpful to the engineering.

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7

Karpov, Eduard G., Larry A. Danso, and John T. Klein. "Anomalous strain energy transformation pathways in mechanical metamaterials." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2226 (June 2019): 20190041. http://dx.doi.org/10.1098/rspa.2019.0041.

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This discussion starts with a mechanics version of Parseval's energy theorem applicable to any discrete lattice material with periodic internal structure: a microtruss, grid, frame, origami or tessellation. It provides a simple relationship between the strain energy volumetric/usual and spectral distributions in the reciprocal space. The spectral energy distribution leads directly to a spectral entropy of lattice deformation (Shannon's type), whose variance with a material coordinate represents the decrease of information about surface loads in the material interior. Spectral entropy is also a basic measure of complexity of mechanical responses of metamaterials to surface and body loads. Considering transformation of the energy volumetric and spectral distributions with a material coordinate pointed away from a surface load, several interesting anomalies are seen even for simple lattice materials, when compared to continuum materials. These anomalies include selective filtering of surface Raleigh waves (sinusoidal pressure patterns), Saint–Venant effect inversion illustrated by energy spectral distribution contours, occurrence of ‘hiding pockets’ of low deformation, and redirection of strain energy maximum away from axis of a concentrated surface load. The latter phenomenon can be significant for impact protection applications of mechanical metamaterials.

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8

PUTHANKATTIL, SUBHA D., and PAUL K. JOSEPH. "CLASSIFICATION OF EEG SIGNALS IN NORMAL AND DEPRESSION CONDITIONS BY ANN USING RWE AND SIGNAL ENTROPY." Journal of Mechanics in Medicine and Biology 12, no. 04 (September 2012): 1240019. http://dx.doi.org/10.1142/s0219519412400192.

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EEG is useful for the analysis of the functional activity of the brain and a detailed assessment of this non-stationary waveform can provide crucial parameters indicative of the mental state of patients. The complex nature of EEG signals calls for automated analysis using various signal processing methods. This paper attempts to classify the EEG signals of normal and depression patients using well-established signal processing techniques involving relative wavelet energy (RWE) and artificial feedForward neural network. High frequency noise present in the recorded signal is removed using total variation filtering (TVF). Classification of the frequency bands of EEG signals into appropriate detail levels and approximation level is carried out using an eight-level multiresolution decomposition method of discrete wavelet transform (DWT). Parseval's theorem is used for calculating the energy at different resolution levels. RWE analysis gives information about the signal energy distribution at different decomposition levels. Both RWE and feedforward Network are used to classify the signals from normal controls and depression patients. The performance of the artificial neural network was evaluated using the classification accuracy and its value of 98.11% indicates a great potential for classifying normal and depression signals.

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9

Bhatnagar, R. M. "Noise reduction in linear variable differential transformer data of recoil motion measurement by numerical methods." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 220, no. 2 (February 1, 2006): 159–66. http://dx.doi.org/10.1243/09544062jmes140.

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The measurement of recoil distance versus time by various methods such as the recoil potentiometer, photo electric transducer, slide wire, accelerometer, revolving drum system, and linear variable differential transformer (LVDT) has been used for gross muzzle brake efficiency measurements and recoil system performance evaluation by the calculation of recoil velocities. For a long recoil-length gun system, a combination of recoil potentiometer and LVDT is used extensively. In order to dispense with the use of recoil potentiometer in the above combination, the article proposes the use of the least-square-fit-based Richardson's extrapolation method and mean square velocity calculation for the accurate determination of free recoil velocity. The mean square velocity calculation is based on Parseval's theorem. The proposed method is based on the comparative evaluation of second- and third-order finite difference method, Richardson's fourth-order method, and the least-square-fit-based Richardson's extrapolation. The least-square-fit-based Richardson's extrapolation gives the lowest value of residual entropy. This is because the maximum likelihood estimators for Gauss probability distribution function and least-square estimators for the coefficients of polynomial representing recoil velocity time curve are coincident. The results of each of the four methods combined with the mean square velocity method were compared, and the least-square-fit-based Richardson's extrapolation was found to be accurate and consistent. The method can be used even when low pass filter is included in the LVDT circuit for stand-alone use.

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10

Hassanzadeh, Mohammad, and Behnam Shahrrava. "Linear Version of Parseval’s Theorem." IEEE Access 10 (2022): 27230–41. http://dx.doi.org/10.1109/access.2022.3157736.

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11

Ryabtsov, I. S. "REPRESENTATION OF PARSEVAL FRAMES IN HILBERT SPACES." Vestnik of Samara University. Natural Science Series 17, no. 5 (June 14, 2017): 60–70. http://dx.doi.org/10.18287/2541-7525-2011-17-5-60-70.

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In this paper we introduce two new classes of Parseval frames in arbitraryHilbert spaces of finite or infinite dimension: simple and composite Parsevalframes. Theorems of representation of composite Parseval frames by summationof simple ones are proved. Few classes of simple frames are described: orthonormal basis, equiangular Parseval frames and some other examples.

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12

Yu, Chii-Huei. "Application of Parseval’s Theorem on Evaluating Some Definite Integrals." Turkish Journal of Analysis and Number Theory 2, no. 1 (January 22, 2016): 1–5. http://dx.doi.org/10.12691/tjant-2-1-1.

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13

Yürekli, O., and I. Sadek. "A Parseval-Goldstein type theorem on the widder potential transform and its applications." International Journal of Mathematics and Mathematical Sciences 14, no. 3 (1991): 517–24. http://dx.doi.org/10.1155/s0161171291000704.

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In this paper a Parseval-Goldstein type theorem involving the Widder potential transform and a Laplace type integral transform is given. The theorem is then shown to yield a relationship between the𝒦-transform and the Laplace type integral transform. The theorem yields some simple algorithms for evaluating infinite integrals. Using the theorem and its results, a number of new infinite integrals of elementary and special functions are presented. Some illustrative examples are also given.

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14

Bhat, Mohammad Younus, Shahbaz Rafiq, and Mohra Zayed. "Wigner–Ville Distribution Associated with Clifford Geometric Algebra Cln,0, n=3(mod 4) Based on Clifford–Fourier Transform." Symmetry 15, no. 7 (July 15, 2023): 1421. http://dx.doi.org/10.3390/sym15071421.

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In this study, the Wigner–Ville distribution is associated with the one sided Clifford–Fourier transform over Rn, n = 3(mod 4). Accordingly, several fundamental properties of the WVD-CFT have been established, including non-linearity, the shift property, dilation, the vector differential, the vector derivative, and the powers of τ∈Rn. Moreover, powerful results on the WVD-CFT have been derived such as Parseval’s theorem, convolution theorem, Moyal’s formula, and reconstruction formula. Eventually, we deduce a directional uncertainty principle associated with WVD-CFT. These types of results, as well as methodologies for solving them, have applications in a wide range of fields where symmetry is crucial.

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15

Ibrahimov, Elman J. "On Gegenbauer transformation on the half-line." gmj 18, no. 3 (May 20, 2011): 497–515. http://dx.doi.org/10.1515/gmj.2011.0024.

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Abstract In this paper, the Gegenbauer transformation is constructed and some of its properties similar to the Fourier transformation are proved. An equation of Parseval–Plancherel type is obtained. The inversion theorem for the Gegenbauer transformation is proved.

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16

Aylikci, Fatih, Nese Dernek, and Gulesin Balaban. "New identities on some generalized integral transforms and their applications." Filomat 36, no. 9 (2022): 2947–60. http://dx.doi.org/10.2298/fil2209947a.

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In this paper the authors gave an iteration identity for the generalized Laplace transform L2n and the generalized Glasser transform G2n. Using this identity a Parseval-Goldstein type theorem for the L2n-transform and the G2n-transform is given. By making use of these results a number of new Parseval-Goldstein type identities are obtained for these and many other well-known integral transforms. The identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions. Some examples are also given.

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17

Karjanto, Natanael. "Bright Soliton Solution of the Nonlinear Schrödinger Equation: Fourier Spectrum and Fundamental Characteristics." Mathematics 10, no. 23 (December 1, 2022): 4559. http://dx.doi.org/10.3390/math10234559.

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We derive exact analytical expressions for the spatial Fourier spectrum of the fundamental bright soliton solution for the 1+1-dimensional nonlinear Schrödinger equation. Similar to a Gaussian profile, the Fourier transform for the hyperbolic secant shape is also shape-preserving. Interestingly, this associated hyperbolic secant Fourier spectrum can be represented by a convergent infinite series, which can be achieved using Mittag–Leffler’s expansion theorem. Conversely, given the expression of the series of the spectrum, we recover its closed form by employing Cauchy’s residue theorem for summation. We further confirm that the fundamental soliton indeed satisfies essential characteristics such as Parseval’s relation and the stretch-bandwidth reciprocity relationship. The fundamental bright soliton finds rich applications in nonlinear fiber optics and optical telecommunication systems.

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18

Bohner, Martin, and Ayça Çetinkaya. "Uniqueness for an Inverse Quantum-Dirac Problem with Given Weyl Function." Tatra Mountains Mathematical Publications 84, no. 2 (June 1, 2023): 1–18. http://dx.doi.org/10.2478/tmmp-2023-0011.

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Abstract In this work, we consider a boundary value problem for a q-Dirac equation. We prove orthogonality of the eigenfunctions, realness of the eigenvalues, and we study asymptotic formulas of the eigenfunctions. We show that the eigenfunctions form a complete system, we obtain the expansion formula with respect to the eigenfunctions, and we derive Parseval’s equality. We construct the Weyl solution and the Weyl function. We prove a uniqueness theorem for the solution of the inverse problem with respect to the Weyl function.

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19

Lu, Shiue-Der, Hong-Wei Sian, Meng-Hui Wang, and Rui-Min Liao. "Application of Extension Neural Network with Discrete Wavelet Transform and Parseval’s Theorem for Power Quality Analysis." Applied Sciences 9, no. 11 (May 30, 2019): 2228. http://dx.doi.org/10.3390/app9112228.

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The development of renewable energy and the increase of intermittent fluctuating loads have affected the power quality of power systems, and in the long run, damage the power equipment. In order to effectively analyze the quality of power signals, this paper proposes a method of signal feature capture and fault identification, as based on the extension neural network (ENN) algorithm combined with discrete wavelet transform (DWT) and Parseval’s theorem. First, the original power quality disturbance (PQD) transient signal was subjected to DWT, and its spectrum energy was calculated for each order of wavelet coefficients through Parseval’s theorem, in order to effectively intercept the eigenvalues of the original signal. Based on the features, the extension neural algorithm was used to establish a matter-element model of power quality disturbance identification. In addition, the correlation degree between the identification data and disturbance types was calculated to accurately identify the types of power failure. To verify the accuracy of the proposed method, five common power quality disturbances were analyzed, including voltage sag, voltage swell, power interruption, voltage flicker, and power harmonics. The results were then compared with those obtained from the back-propagation network (BPN), probabilistic neural network (PNN), extension method and a learning vector quantization network (LVQ). The results showed that the proposed method has shorter computation time (0.06 s), as well as higher identification accuracy at 99.62%, which is higher than the accuracy rates of the other four types.

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20

YüREKLI, OSMAN. "A Parseval-type Theorem Applied to Certain Integral Transforms." IMA Journal of Applied Mathematics 42, no. 3 (1989): 241–49. http://dx.doi.org/10.1093/imamat/42.3.241.

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21

Gennat, Marc, and Bernd Tibken. "An Interval Arithmetic Approach for Multilevel Converter Harmonic Minimization Using Parseval’s Theorem." IFAC-PapersOnLine 53, no. 2 (2020): 13430–35. http://dx.doi.org/10.1016/j.ifacol.2020.12.250.

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22

Iwasaki, Atsushi. "Deriving the Variance of the Discrete Fourier Transform Test Using Parseval’s Theorem." IEEE Transactions on Information Theory 66, no. 2 (February 2020): 1164–70. http://dx.doi.org/10.1109/tit.2019.2947045.

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23

O. Akcay and Kh. R. Mamedov. "On the spectral expansion formula for a class of Dirac operators." Malaya Journal of Matematik 4, no. 02 (April 1, 2016): 297–304. http://dx.doi.org/10.26637/mjm402/014.

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This paper deals with a problem for the canonical Dirac differential equations system with piecewise continuous coefficient and spectral parameter dependent in boundary conditions. The resolvent operator is constructed. The completeness theorem for eigenvector functions is proved. The spectral expansion formula with respect to eigenvector functions is obtained and Parseval equality is given.

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24

Hoyo-Montaño, José Antonio, Jesús Naim Leon-Ortega, Guillermo Valencia-Palomo, Rafael Armando Galaz-Bustamante, Daniel Fernando Espejel-Blanco, and Martín Gustavo Vázquez Palma. "Non-Intrusive Electric Load identification using Wavelet Transform." Ingeniería e Investigación 38, no. 2 (May 1, 2018): 42–51. http://dx.doi.org/10.15446/ing.investig.v38n2.70550.

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This paper shows the development of a decision tree for the classification of loads in a non-intrusive load monitoring (NILM) system implemented in a simple board computer (Raspberry Pi 3). The decision tree uses the total energy value of the power signal of an equipment, which is generated using a discrete wavelet transform and Parseval’s theorem. The power consumption data of different types of equipment were obtained from a public access database for NILM applications. The best split point for the design of the decision tree was determined using the weighted average Gini index. The tree was validated using loads available in the same public access database.

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25

Ahmadi-Tabatabaei, A., A. Fatehi, and A. Khaki-Sedigh. "An Improved Input-Output Pairing Method Based on Concept of Energy." Advanced Materials Research 403-408 (November 2011): 2065–72. http://dx.doi.org/10.4028/www.scientific.net/amr.403-408.2065.

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In this paper input-output pairing is done based on concept of energy. Parseval theorem and cross-covariance samples of input-output are used for estimation of energy. After approximating interaction energy between input and output of the plant, input-output pairing is fulfilled. Through examples, it is illustrated that proposed method is appropriate for input-output pairing. The result is compared with Effective Relative Energy Array (EREA) as another energy based approach for input-output pairing.

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26

Wang, Xiaotong, Guanlei Xu, Yue Ma, Lijia Zhou, and Longtao Wang. "Generalized Parseval’s Theorem on Fractional Fourier Transform for Discrete Signals and Filtering of LFM Signals." Journal of Signal and Information Processing 04, no. 03 (2013): 274–81. http://dx.doi.org/10.4236/jsip.2013.43035.

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27

Siruvallur Vasudevan, Viswajith, Salim Olia, Harvey S. Borovetz, and James Antaki. "BIO30: A Method to Monitor Ventricular Recovery Using VAD Signals Based on Parseval’s Power Theorem." ASAIO Journal 69, Supplement 2 (June 2023): 34. http://dx.doi.org/10.1097/01.mat.0000943432.17503.66.

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28

Ding, Qiao Lin, Yi Min Yuan, and Zhong Li. "A Wavelet Analysis of Three-Dimensional Surface Vibration Signal of Running Power Transformer." Advanced Materials Research 1049-1050 (October 2014): 634–37. http://dx.doi.org/10.4028/www.scientific.net/amr.1049-1050.634.

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Theoretical analysis and practice experience have shown that, the vibration signal on power transformer surface can be used to analyze and judge the work state of the winding and iron core. For vibration analysis, to extract a precise feature of the vibration signal is a basic work. Based on wavelet multiresolution analysis, three-dimensional surface vibration signal of the running power transformer is analyzed. Based on the Parseval theorem, the feature of frequency bands-energy in X, Y, Z directions is computed and compared. It provides some reference for the power transformer vibration analysis.

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29

Adawi, A. "A Parseval-type theorem applied to certain integral transforms on generalized functions." IMA Journal of Applied Mathematics 68, no. 6 (December 1, 2003): 587–93. http://dx.doi.org/10.1093/imamat/68.6.587.

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30

de Goursac, Axel, and Jean-Philippe Michel. "Superunitary Representations of Heisenberg Supergroups." International Mathematics Research Notices 2020, no. 19 (August 14, 2018): 5926–6006. http://dx.doi.org/10.1093/imrn/rny184.

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Abstract Numerous Lie supergroups do not admit superunitary representations (SURs) except the trivial one, for example, Heisenberg and orthosymplectic supergroups in mixed signature. To avoid this situation, we introduce in this paper a broader definition of SUR, relying on a new definition of Hilbert superspace. The latter is inspired by the notion of Krein space and was developed initially for noncommutative supergeometry. For Heisenberg supergroups, this new approach yields a smooth generalization, whatever the signature, of the unitary representation theory of the classical Heisenberg group. First, we obtain Schrödinger-like representations by quantizing generic coadjoint orbits. They satisfy the new definition of irreducible SURs and serve as ground to the main result of this paper: a generalized Stone–von Neumann theorem. Then, we obtain the superunitary dual and build a group Fourier transformation, satisfying Parseval theorem. We eventually show that metaplectic representations, which extend Schrödinger-like representations to metaplectic supergroups, also fit into this definition of SURs.

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31

Lekesi̇z, Esra Güldoğan, Rabi̇a Aktaş, and Iván Area. "Fourier Transform of the Orthogonal Polynomials on the Unit Ball and Continuous Hahn Polynomials." Axioms 11, no. 10 (October 14, 2022): 558. http://dx.doi.org/10.3390/axioms11100558.

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Some systems of univariate orthogonal polynomials can be mapped into other families by the Fourier transform. The most-studied example is related to the Hermite functions, which are eigenfunctions of the Fourier transform. For the multivariate case, by using the Fourier transform and Parseval’s identity, very recently, some examples of orthogonal systems of this type have been introduced and orthogonality relations have been discussed. In the present paper, this method is applied for multivariate orthogonal polynomials on the unit ball. The Fourier transform of these orthogonal polynomials on the unit ball is obtained. By Parseval’s identity, a new family of multivariate orthogonal functions is introduced. The results are expressed in terms of the continuous Hahn polynomials.

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32

Abitha Memala, W., and V. Rajini. "Wavelet Based Induction Motor Fault Diagnosis Using Zero Sequence Current." Journal of Computational and Theoretical Nanoscience 14, no. 1 (January 1, 2017): 411–20. http://dx.doi.org/10.1166/jctn.2017.6336.

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Induction motor stator fault is diagnosed by applying Discrete Wavelet transform on zero sequence components. The single phasing stator fault is created and diagnosed in the induction motor model developed in stationary reference frame, under varying load conditions. The stator inter-turn incipient fault is created and diagnosed in the induction motor experimental setup as well under no load condition. The qdo components are calculated from Park’s equations. The faults can be diagnosed from wavelet transform of the zero sequence current components. PSD is used for diagnosing the fault and the statistical value is used for verifying the result. The energy is calculated using Parseval’s theorem. The energy and the statistical data calculated from the wavelet coefficients of zero sequence current components are used as fault indicators. The energy value is able to reveal the fault severity in the induction motor stator winding. Power spectral Density along with Discrete Wavelet Transform plays very important role in diagnosing the fault.

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33

Li, Haiyang, Qianghua Pan, Xiaotong Zhang, and Zhiwu An. "An Approach to Size Sub-Wavelength Surface Crack Measurements Using Rayleigh Waves Based on Laser Ultrasounds." Sensors 20, no. 18 (September 7, 2020): 5077. http://dx.doi.org/10.3390/s20185077.

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In this paper, the interaction of a broadband Rayleigh wave generated by a laser and an artificial rectangular notch is analyzed theoretically and experimentally. For the theoretical analysis, a Gaussian function is adopted to analyze the modulation of notch depth on the frequency spectrum via reflection and transmission coefficients. By the finite element method, the Rayleigh wave generated by pulsed laser beam irradiation and its scattering waves at cracks are calculated. A curve with a slope close to 4 fitted by crack depth and critical wavelength of the threshold phenomenon is obtained by the wavelet transform and Parseval’s theorem according to simulated and experimental results. Based on this relationship, the critical frequency at which the threshold phenomenon happens due to energy transformation of transmission/reflection Rayleigh waves is adopted to determine the size of sub-wavelength surface crack. The experimental results of artificial notch depth estimation on aluminum alloy specimens consistent with theoretical analysis validates the usefulness of the critical frequency method based on a broadband Rayleigh wave generated by laser ultrasonic.

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34

Wu, Chenhao, Jiguang Yue, Li Wang, and Feng Lyu. "Detection and Classification of Recessive Weakness in Superbuck Converter Based on WPD-PCA and Probabilistic Neural Network." Electronics 8, no. 3 (March 5, 2019): 290. http://dx.doi.org/10.3390/electronics8030290.

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This paper proposes a detection and classification method of recessive weakness in Superbuck converter through wavelet packet decomposition (WPD) and principal component analysis (PCA) combined with probabilistic neural network (PNN). The Superbuck converter presents excellent performance in many applications and is also faced with today’s demands, such as higher reliability and steadier operation. In this paper, the detection and classification issue to recessive weakness is settled. Firstly, the performance of recessive weakness both in the time and frequency domain are demonstrated to clearly show the actual deterioration of the circuit system. The WPD and Parseval’s theorem are utilized in this paper to feature the extraction of recessive weakness. The energy discrepancy of the fault signals at different wavelet decomposition levels are then chosen as the feature vectors. PCA is also employed to the dimensionality reduction of feature vectors. Then, a probabilistic neural network is applied to automatically detect and classify the recessive weakness from different components on the basis of the extracted features. Finally, the classification accuracy of the proposed classification algorithm is verified and tested with experiments, which present satisfying classification accuracy.

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35

Bardaro, Carlo, Paul L. Butzer, and Ilaria Mantellini. "The Mellin–Parseval formula and its interconnections with the exponential sampling theorem of optical physics." Integral Transforms and Special Functions 27, no. 1 (September 21, 2015): 17–29. http://dx.doi.org/10.1080/10652469.2015.1087401.

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36

Li-Yun, Hu, and Fan Hong-Yi. "Inversion formula and Parseval theorem for complex continuous wavelet transforms studied by entangled state representation." Chinese Physics B 19, no. 7 (July 2010): 074205. http://dx.doi.org/10.1088/1674-1056/19/7/074205.

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37

Butzer, P. L., P. J. S. G. Ferreira, J. R. Higgins, G. Schmeisser, and R. L. Stens. "The sampling theorem, Poisson's summation formula, general Parseval formula, reproducing kernel formula and the Paley–Wiener theorem for bandlimited signals – their interconnections." Applicable Analysis 90, no. 3-4 (March 2011): 431–61. http://dx.doi.org/10.1080/00036811003627567.

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Bekbolat, B., and N. Tokmagambetov. "Well-posedness results for the wave equation generated by the Bessel operator." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 101, no. 1 (March 30, 2021): 11–16. http://dx.doi.org/10.31489/2021m1/11-16.

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In this paper, we consider the non-homogeneous wave equation generated by the Bessel operator. We prove the existence and uniqueness of the solution of the non-homogeneous wave equation generated by the Bessel operator. The representation of the solution is given. We obtained a priori estimates in Sobolev type space. This problem was firstly considered in the work of M. Assal [1] in the setting of Bessel-Kingman hypergroups. The technique used in [1] is based on the convolution theorem and related estimates. Here, we use a different approach. We study the problem from the point of the Sobolev spaces. Namely, the conventional Hankel transform and Parseval formula are widely applied by taking into account that between the Hankel transformation and the Bessel differential operator there is a commutation formula [2].

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Hu, Guang-Da. "An algorithm to design a stabilizing feedback controller for linear delay systems via Parseval ’s theorem." Journal of Computational and Applied Mathematics 415 (December 2022): 114502. http://dx.doi.org/10.1016/j.cam.2022.114502.

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Celik, Mehmet, and Necati Olgun. "An Introduction to Neutrosophic Real Banach and Hillbert Spaces." Galoitica: Journal of Mathematical Structures and Applications 2, no. 1 (2022): 08–13. http://dx.doi.org/10.54216/gjmsa.020101.

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Banach and Hillbert spaces are the main important concepts in the study of classical functional analysis. This paper generalizes these two kinds of functional spaces into neutrosophic systems, where the concept of neutrosophic Banach space and neutrosophic Hillbert space will be defined and discussed for the first time over partial ordered neutrosophic spaces. Also, many related concepts such as neutrosophic Cauchy sequence, neutrosophic Bessel's inequality, and neutrosophic Parseval's identity will be established and proved.

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Jiménez-Aparicio, Miguel, Matthew J. Reno, and Felipe Wilches-Bernal. "Traveling Wave Energy Analysis of Faults on Power Distribution Systems." Energies 15, no. 8 (April 8, 2022): 2741. http://dx.doi.org/10.3390/en15082741.

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This paper explores the most important factors that define the Traveling Wave (TW) propagation on distribution systems. The factors considered in this work are: the distance to the fault location, the fault type, and the crossing of system elements (such as regulators, capacitor banks, laterals, and extra loads within the protection zones). This work uses a realistic, yet simplified, distribution system composed of two protection zones, in which, several combinations of the previously mentioned factors are considered. The simulated fault measurements undergo a signal processing stage in which, first, they are decomposed into independent modes using the Karrenbauer transform. Second, a time–frequency representation is obtained using the Stationary Wavelet Transform (SWT), dividing the signal into several frequency bands. Finally, the Parseval’s Energy (PE) theorem is applied to calculate the signal energy in each frequency band. A qualitative analysis is performed based on the previously calculated energies to outline which are the factors that most affect the TW energy during propagation. The results show that distance, the presence of regulators, either in the propagation path or upstream, and the type of fault are the main factors that affect TW propagation across the system, and therefore they should be considered for TW-based protection schemes for distribution systems.

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Dutkay, Dorin Ervin, and Rajitha Ranasinghe. "Parseval frames of piecewise constant functions." Operators and Matrices, no. 2 (2020): 317–31. http://dx.doi.org/10.7153/oam-2020-14-25.

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Jia, Huiqin, and Ruirong Dang. "Development of Contactless Sand Production Test Instrument." Open Petroleum Engineering Journal 10, no. 1 (January 12, 2017): 1–11. http://dx.doi.org/10.2174/1874834101701010001.

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Background:Sand production is an important factor affecting reservoir exploitation. Excessive sand production affects the oil well life, causes the damage to the oil recovery device and brings serious hidden danger in the process of safety production.Objective:This paper presents a sand content computation model for the crude oil, which uses the power of sand production and Parseval theorem, develops a system to monitor the sand content.Method:The main problem of designing the mechanical structure and matching layer for sand sensor is introduced. Furthermore a contactless Doppler ultrasonic method is used to develop sand amount detection system‚ finally the system measurement accuracy is calibrated by the developed calibration system.Conclusion:The sand production computation model can describe the relationship between sand monitor output and sand amount; compute the oil sand carrying ratio. The sand sensor is installed on the outside of pipe, not directly contacted with the sand particles. Measurement accuracy of the sand production is higher than the same type instrument.

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Urynbassarova, Didar, and Aajaz A. Teali. "Convolution, Correlation, and Uncertainty Principles for the Quaternion Offset Linear Canonical Transform." Mathematics 11, no. 9 (May 7, 2023): 2201. http://dx.doi.org/10.3390/math11092201.

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Quaternion Fourier transform (QFT) has gained significant attention in recent years due to its effectiveness in analyzing multi-dimensional signals and images. This article introduces two-dimensional (2D) right-sided quaternion offset linear canonical transform (QOLCT), which is the most general form of QFT with additional free parameters. We explore the properties of 2D right-sided QOLCT, including inversion and Parseval formulas, besides its relationship with other transforms. We also examine the convolution and correlation theorems of 2D right-sided QOLCT, followed by several uncertainty principles. Additionally, we present an illustrative example of the proposed transform, demonstrating its graphical representation of a given signal and its transformed signal. Finally, we demonstrate an application of QOLCT, where it can be utilized to generalize the treatment of swept-frequency filters.

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Mendez, Robert P., Bernhard G. Bodmann, Zachery J. Baker, Micah G. Bullock, and Jacob E. McLaney. "Binary Parseval frames from group orbits." Linear Algebra and its Applications 556 (November 2018): 265–300. http://dx.doi.org/10.1016/j.laa.2018.07.016.

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Zhang, Peixin, Jianxin Wang, Peng Ren, Shushu Yang, and Haiwei Song. "Improved ASM-TER Training Sequence Detection and Fine Doppler Frequency Estimation Methods from a Satellite." Journal of Sensors 2020 (June 8, 2020): 1–11. http://dx.doi.org/10.1155/2020/3625184.

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To detect terrestrial application-specific messages (ASM-TER) signals from a satellite, a novel detection method based on the fast computation of the cross ambiguity function is proposed in this paper. The classic cross ambiguity function’s computational burden is heavy, and we transform the classic cross ambiguity function to a frequency domain version to reduce the computational complexity according to Parseval’s theorem. The computationally efficient sliding discrete Fourier transform (SDFT) is utilized to calculate the frequency spectrum of the windowed received signal, from which the Doppler frequency could be estimated coarsely. Those subbands around the Doppler frequency are selected to calculate the ambiguity function for reducing the computational complexity. Furthermore, two local sequences with half length of the training sequence are utilized to acquire a better Doppler frequency tolerance; thus, the frequency search step is increased and the computational complexity could be further reduced. Once an ASM-TER signal is detected by the proposed algorithm, a fine Doppler frequency estimation could be obtained easily from the correlation peaks of the two local sequences. Simulation results show that the proposed algorithm shares almost the same performance with the classic cross ambiguity function-based method, and the computational complexity is greatly reduced. Simulation results also show that the proposed algorithm is more resistant to cochannel interference (CCI) than the differential correlation (DC) algorithm, and the performance of fine Doppler frequency estimation is close to that of the Cramér–Rao lower bound (CRLB).

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Ponomareva, Olga, and Aleksey Ponomarev. "Theoretical foundations of digital vector Fourier analysis of two-dimensional signals padded with zero samples." Information and Control Systems, no. 1 (March 3, 2021): 55–65. http://dx.doi.org/10.31799/1684-8853-2021-1-55-65.

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Introduction: The practice of using Fourier-processing of finite two-dimensional signals (including images), having confirmed its effectiveness, revealed a number of negative effects inherent in it. A well-known method of dealing with negative effects of Fourier-processing is padding signals with zeros. However, the use of this operation leads to the need to provide information control systems with additional memory and perform unproductive calculations. Purpose: To develop new discrete Fourier transforms for efficient and effective processing of two-dimensional signals padded with zero samples. Method: We have proposed a new method for splitting a rectangular discrete Fourier transform matrix into square matrices. The method is based on the application of the modulus comparability relation to order the rows (columns) of the Fourier matrix. Results: New discrete Fourier transforms with variable parameters were developed, being a generalization of the classical discrete Fourier transform. The article investigates the properties of Fourier transform bases with variable parameters. In respect to these transforms, the validity has been proved for the theorems of linearity, shift, correlation and Parseval's equality. In the digital spectral Fourier analysis, the concepts of a parametric shift of a two-dimensional signal, and a parametric periodicity of a two-dimensional signal have been introduced. We have estimated the reduction of the required memory size and the number of calculations when applying the proposed transforms, and compared them with the discrete Fourier transform. Practical relevance: The developed discrete Fourier transforms with variable parameters can significantly reduce the cost of Fourier processing of two-dimensional signals (including images) padded with zeros.

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Chen, Cheng-I., Chien-Kai Lan, Yeong-Chin Chen, Chung-Hsien Chen, and Yung-Ruei Chang. "Wavelet Energy Fuzzy Neural Network-Based Fault Protection System for Microgrid." Energies 13, no. 4 (February 24, 2020): 1007. http://dx.doi.org/10.3390/en13041007.

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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.

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Perez-Portero, Adrian, Jorge Querol, Adriano Camps, Manuel Martin-Neira, Martin Suess, Juan Ignacio Ramirez, Alberto Zurita, Josep Closa, Roger Oliva, and Raul Onrubia. "RFI Detection and Mitigation for Advanced Correlators in Interferometric Radiometers." Remote Sensing 14, no. 18 (September 19, 2022): 4672. http://dx.doi.org/10.3390/rs14184672.

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This work presents the first RFI detection and mitigation algorithm for the interferometric radiometers that will be implemented in its correlator unit. The algorithm operates in the time and frequency domains, applying polarimetric and statistical tests in both domains, and exhibiting a tunable and arbitrary low probability of false alarm. It is scalable to a configurable number of receivers, and it is optimized in terms of quantization bits and the implementation of the cross-correlations in the time or frequency domains for hardware resource saving. New features of this algorithm are the computation of the Stokes parameters per frequency bin in the Short-Time Fourier Transform and a new parameter called Polarimetric Kurtosis. If RFI is detected in one domain or in both, it is removed using the calculated blanking masks. The optimum algorithm parameters are computed, such as length of the FFTs, the threshold selection for a given probability of false alarm, and the selection of the blanking masks. Last, an important result refers to the application of Parseval’s theorem for the computation of the cross-correlations in the frequency domain, instead of in the time domain, which is more efficient and leads to smaller errors even when using moderate quantization levels. The algorithm has been developed in the framework of the ESA’s technology preparation for a potential L-band radiometer mission beyond SMOS. However, it is also applicable to (polarimetric) real aperture radiometers, and its performance would improve if more than one bit is used in the signal quantization.

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Wang, Xinhai, Gong Zhang, Xiangmin Wang, Qingqing Song, and Fangqing Wen. "ECCM Schemes against Deception Jamming Using OFDM Radar with Low Global PAPR." Sensors 20, no. 7 (April 7, 2020): 2071. http://dx.doi.org/10.3390/s20072071.

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In this paper, a type of effective electronic counter-countermeasures (ECCM) technique for suppressing the high-power deception jamming using an orthogonal frequency division multiplexing (OFDM) radar is proposed. Concerning the velocity deception jamming, the initial phases of the pulses transmitted in a coherent processing interval (CPI) are designed to minimize the jamming power within a specific range, forming a notch around the jamming in the Doppler spectrum. For the purpose of suppressing the range deception jamming and the joint range-velocity deception jamming, the phase codes of the subcarriers belonging to the OFDM pulses are optimized to minimize the jamming power, distributing some specific bands in the range and the range-velocity domain, respectively. According to Parseval’s theorem, the phase encoding, acting as the coding manner of the OFDM subcarriers can ensure that the energy of each OFDM symbol stays the same. It is worth noticing that the phase codes of the OFDM subcarriers can influence the peak-to-average power ratio (PAPR). Thus, an optimization problem is formulated to optimize the phase codes of the subcarriers under the constraint of global PAPR, which can regulate the PAPRs of multiple OFDM symbols at the same time. The proposed problem is non-convex; therefore, it is a huge challenge to tackle. Then we present a method named by the phase-only alternating direction method multipliers (POADMM) to solve the aforementioned optimization problem. Some necessary simulation results are provided to demonstrate the effectiveness of the proposed radar signaling strategy

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