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Journal articles on the topic 'Fourier decomposition'

Author: Grafiati

Published: 4 June 2021

Last updated: 21 February 2023

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1

Batyanovskii, A. V., V. A. Namiot, I. V. Filatov, V. G. Tumanyan, N. G. Esipova, and I. D. Volotovsky. "Fourier transformation in spherical systems as a tool of structural biology." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 56, no. 4 (December 31, 2020): 496–503. http://dx.doi.org/10.29235/1561-2430-2020-56-4-496-503.

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Applications of the most common adaptation of Fourier analysis in spherical coordinate systems used to solve a number of problems in structural biology, namely, flat wave decomposition (flat waves are represented as spherical functions decomposition), are herein considered. Arguments in favor of this decomposition are compared with other decompositions in superposition of special functions. A more general justification for the correctness of this decomposition is obtained than that existing today. A method for representing groups of atoms in the form of a Fourier object is proposed. It is also considered what opportunities give such a representation. The prospects for the application of Fourier analysis in structural biophysics are discussed.

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2

Batyanovskii, A. V., V. A. Namiot, I. V. Filatov, V. G. Tumanyan, N. G. Esipova, and I. D. Volotovsky. "Fourier transformation in spherical systems as a tool of structural biology." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 56, no. 4 (December 31, 2020): 496–503. http://dx.doi.org/10.29235/1561-2430-2020-56-4-496-503.

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Applications of the most common adaptation of Fourier analysis in spherical coordinate systems used to solve a number of problems in structural biology, namely, flat wave decomposition (flat waves are represented as spherical functions decomposition), are herein considered. Arguments in favor of this decomposition are compared with other decompositions in superposition of special functions. A more general justification for the correctness of this decomposition is obtained than that existing today. A method for representing groups of atoms in the form of a Fourier object is proposed. It is also considered what opportunities give such a representation. The prospects for the application of Fourier analysis in structural biophysics are discussed.

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3

Wang, Yanbo, and Tao Qian. "Adaptive Fourier decomposition in." Mathematical Methods in the Applied Sciences 42, no. 6 (January 22, 2019): 2016–24. http://dx.doi.org/10.1002/mma.5494.

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4

ZHANG, LIMING, and HONG LI. "A NOVEL SIGNAL DECOMPOSITION APPROACH — ADAPTIVE FOURIER DECOMPOSITION." Advances in Adaptive Data Analysis 03, no. 03 (July 2011): 325–38. http://dx.doi.org/10.1142/s1793536911000702.

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This paper presents a novel signal decomposition approach — adaptive Fourier decomposition (AFD), which decomposes a given signal based on its physical characters. The algorithm is described in detail, that is based on recent theoretical studies on analytic instantaneous frequencies and stands as a realizable variation of the greedy algorithm. The principle of the algorithm gives rise to fast convergence in terms of energy. Effectiveness of the algorithm is evaluated by comparison experiments with the classical Fourier decomposition (FD) algorithm. The results are promising.

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5

Qian, Tao, Liming Zhang, and Zhixiong Li. "Algorithm of Adaptive Fourier Decomposition." IEEE Transactions on Signal Processing 59, no. 12 (December 2011): 5899–906. http://dx.doi.org/10.1109/tsp.2011.2168520.

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6

Kutay, M. Alper, Hakan Özaktaş, Haldun M. Ozaktas, and Orhan Arıkan. "The fractional Fourier domain decomposition." Signal Processing 77, no. 1 (August 1999): 105–9. http://dx.doi.org/10.1016/s0165-1684(99)00063-8.

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7

QIAN, TAO, and YANBO WANG. "REMARKS ON ADAPTIVE FOURIER DECOMPOSITION." International Journal of Wavelets, Multiresolution and Information Processing 11, no. 01 (January 2013): 1350007. http://dx.doi.org/10.1142/s0219691313500070.

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This is a continuation of the study of adaptive Fourier decomposition (AFD).15 Under a mild condition not in terms of smoothness, a convergence rate is provided. We prove that the selection of the parameters corresponding to Fourier series in the average sense is optimal. We also present the transformation matrices between the adaptive rational orthogonal system and the related sequence of the shifted Cauchy kernels and their derivatives.

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8

Qian, Tao. "Two-dimensional adaptive Fourier decomposition." Mathematical Methods in the Applied Sciences 39, no. 10 (February 29, 2016): 2431–48. http://dx.doi.org/10.1002/mma.3649.

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9

Kožar, Ivica, Željko Jeričević, and Tatjana Pecak. "Approximate modal analysis using Fourier decomposition." IOP Conference Series: Materials Science and Engineering 10 (June 1, 2010): 012119. http://dx.doi.org/10.1088/1757-899x/10/1/012119.

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10

Zhou, Yicong, Weijia Cao, Licheng Liu, Sos Agaian, and C. L. Philip Chen. "Fast Fourier transform using matrix decomposition." Information Sciences 291 (January 2015): 172–83. http://dx.doi.org/10.1016/j.ins.2014.08.022.

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11

Ma, LiQun, LiHao Feng, Chong Pan, Qi Gao, and JinJun Wang. "Fourier mode decomposition of PIV data." Science China Technological Sciences 58, no. 11 (August 19, 2015): 1935–48. http://dx.doi.org/10.1007/s11431-015-5908-y.

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12

Gao, You, Min Ku, Tao Qian, and Jianzhong Wang. "FFT formulations of adaptive Fourier decomposition." Journal of Computational and Applied Mathematics 324 (November 2017): 204–15. http://dx.doi.org/10.1016/j.cam.2017.04.029.

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13

Wang, Ze, Feng Wan, Chi Man Wong, and Liming Zhang. "Adaptive Fourier decomposition based ECG denoising." Computers in Biology and Medicine 77 (October 2016): 195–205. http://dx.doi.org/10.1016/j.compbiomed.2016.08.013.

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14

Hansen, L., and J. O. Petersen. "Fourier Decomposition Parameters for the Halo RR Lyrae Variables U and V Caeli." International Astronomical Union Colloquium 82 (1985): 272–75. http://dx.doi.org/10.1017/s0252921100109509.

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AbstractUBVRI light curves are obtained for the two halo RR Lyrae variables U Caeli with period 0.420 days (73 observations) and V Caeli with period 0.571 days (42 observations). It is shown that their light curve characteristics are very similar to those of field RR Lyrae stars.Fourier decompositions are studied for all five magnitudes and the resulting amplitude ratios and phase differences are discussed. The differences in the Fourier decomposition parameters between the five magnitudes are shown to be relatively small. Comparisons of the Fourier decomposition parameters for the two halo RR Lyrae stars with recently published data for field RR Lyrae stars show no systematic differences.

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15

Petersen, J. O. "On the Application of Fourier Decomposition Parameters." International Astronomical Union Colloquium 134 (1993): 157–58. http://dx.doi.org/10.1017/s0252921100014081.

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The application of Fourier decomposition parameters has revolutionized important areas of investigations of Cepheid type variables since the introduction of Fourier analysis in its modern form by Simon and Lee (1981).In the literature several different representations of the results of Fourier analysis have been utilized. In view of the growing interest for applications of Fourier decomposition it is important to use and publish Fourier data in an optimal way. Most studies until now have used amplitude ratios and phase differences derived from traditional light curves giving the light variation in magnitudes, following the original recipe of Simon and Lee (1981). However, Stellingwerf and Donohoe (1986) advocated the use of phases rather than phase differences. Recently, Buchler et al. (1990) argued that the standard Simon & Lee form contains all relevant physics, and suggested analysis of flux-values rather than of magnitudes, because this removes the distorting effects of constant, false light. Thus there are many choices to be made in practical applications of Fourier analysis, and there is at present no convincing argument for preferring one specific representation.

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16

Elbi, Mehmet Dogan, and Aydin Kizilkaya. "Multicomponent signal analysis: Interwoven Fourier decomposition method." Digital Signal Processing 104 (September 2020): 102771. http://dx.doi.org/10.1016/j.dsp.2020.102771.

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17

Petersen, J. O. "On the application of Fourier decomposition parameters." Astrophysics and Space Science 210, no. 1-2 (December 1993): 157–58. http://dx.doi.org/10.1007/bf00657888.

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18

Niu, Qiang, and Lin-Zhang Lu. "Fourier analysis of frequency filtering decomposition preconditioners." Applied Mathematics and Computation 216, no. 6 (May 2010): 1805–18. http://dx.doi.org/10.1016/j.amc.2009.12.024.

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19

Cho, Z. H., K. J. Jung, and Y. M. Ro. "MR fourier transform arteriography using spectral decomposition." Magnetic Resonance in Medicine 16, no. 2 (November 1990): 226–37. http://dx.doi.org/10.1002/mrm.1910160204.

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20

Shi, Ruonan, Jae-Hun Jung, and Ferdinand Schweser. "Two-dimensional local Fourier image reconstruction via domain decomposition Fourier continuation method." PLOS ONE 14, no. 1 (January 9, 2019): e0197963. http://dx.doi.org/10.1371/journal.pone.0197963.

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21

GYONGYOSI, LASZLO, and SANDOR IMRE. "QUANTUM SINGULAR VALUE DECOMPOSITION BASED APPROXIMATION ALGORITHM." Journal of Circuits, Systems and Computers 19, no. 06 (October 2010): 1141–62. http://dx.doi.org/10.1142/s0218126610006797.

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Singular Value Decomposition (SVD) is one of the most useful techniques for analyzing data in linear algebra. SVD decomposes a rectangular real or complex matrix into two orthogonal matrices and one diagonal matrix. The proposed Quantum-SVD algorithm interpolates the non-uniform angles in the Fourier domain. The error of the Quantum-SVD approach is some orders lower than the error given by ordinary Quantum Fourier Transformation. Our Quantum-SVD algorithm is a fundamentally novel approach for the computation of the Quantum Fourier Transformation (QFT) of non-uniform states. The presented Quantum-SVD algorithm is based on the singular value decomposition mechanism, and the computation of Quantum Fourier Transformation of non-uniform angles of a quantum system. The Quantum-SVD approach provides advantages in terms of computational structure, being based on QFT and multiplications.

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22

DÜZ, Murat. "Solution of Lane-Emden Equation with Fourier Decomposition Method." Süleyman Demirel Üniversitesi Fen Edebiyat Fakültesi Fen Dergisi 17, no. 2 (November 25, 2022): 247–60. http://dx.doi.org/10.29233/sdufeffd.978260.

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In this article, we tried to get the solution of a class of Lane Emden type equations by using the Fourier Decomposition Method. This method is obtained by using the Fourier transform and the Adomian Decomposition method (FADM) together.

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23

Lorsolo, Sylvie, and Altuğ Aksoy. "Wavenumber Analysis of Azimuthally Distributed Data: Assessing Maximum Allowable Gap Size." Monthly Weather Review 140, no. 6 (June 1, 2012): 1945–56. http://dx.doi.org/10.1175/mwr-d-11-00219.1.

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Abstract Performing wavenumber decomposition on azimuthally distributed data such as those in tropical cyclones can be challenging when data gaps exist in the signal. In the literature, ad hoc approaches are found to determine maximum gap size beyond which not to perform Fourier decomposition. The goal of the present study is to provide a more objective and systematic method to choose the maximum gap size allowed to perform a Fourier analysis on observational data. A Monte Carlo–type experiment is conducted where signals of various wavenumber configurations are generated with gaps of varying size, then a simple interpolation scheme is applied and Fourier decomposition is performed. The wavenumber decomposition is evaluated in a way that requires retrieval of at least 80% of the original amplitude with less than 20° phase shift. Maximum allowable gap size is then retrieved for wavenumbers 0–2. When prior assessment of signal configuration is available, the authors believe that the present study can provide valuable guidance for gap size beyond which Fourier decomposition is not advisable.

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24

Wang, Wenlong, George A. McMechan, Chen Tang, and Fei Xie. "Up/down and P/S decompositions of elastic wavefields using complex seismic traces with applications to calculating Poynting vectors and angle-domain common-image gathers from reverse time migrations." GEOPHYSICS 81, no. 4 (July 2016): S181—S194. http://dx.doi.org/10.1190/geo2015-0456.1.

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Separations of up- and down-going as well as of P- and S-waves are often a part of processing of multicomponent recorded data and propagating wavefields. Most previous methods for separating up/down propagating wavefields are expensive because of the requirement to save time steps to perform Fourier transforms over time. An alternate approach for separation of up-and down-going waves, based on extrapolation of complex data traces is extended from acoustic to elastic, and combined with P- and S-wave decomposition by decoupled propagation. This preserves all the information in the original data and eliminates the need for a Fourier transform over time, thereby significantly reducing the storage cost and improving computational efficiency. Wavefield decomposition is applied to synthetic elastic VSP data and propagating wavefield snapshots. Poynting vectors obtained from the particle velocity and stress fields after P/S and up/down decompositions are much more accurate than those without because interference between the corresponding wavefronts is significantly reduced. Elastic reverse time migration with the P/S and up/down decompositions indicated significant improvement compared with those without decompositions, when applied to elastic data from a portion of the Marmousi2 model.

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25

Hole, M. J., and L. C. Appel. "Fourier decomposition of magnetic perturbations in toroidal plasmas using singular value decomposition." Plasma Physics and Controlled Fusion 49, no. 12 (October 25, 2007): 1971–88. http://dx.doi.org/10.1088/0741-3335/49/12/002.

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26

Le Clainche, Soledad, and José M. Vega. "Analyzing Nonlinear Dynamics via Data-Driven Dynamic Mode Decomposition-Like Methods." Complexity 2018 (December 12, 2018): 1–21. http://dx.doi.org/10.1155/2018/6920783.

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This article presents a review on two methods based on dynamic mode decomposition and its multiple applications, focusing on higher order dynamic mode decomposition (which provides a purely temporal Fourier-like decomposition) and spatiotemporal Koopman decomposition (which gives a spatiotemporal Fourier-like decomposition). These methods are purely data-driven, using either numerical or experimental data, and permit reconstructing the given data and identifying the temporal growth rates and frequencies involved in the dynamics and the spatial growth rates and wavenumbers in the case of the spatiotemporal Koopman decomposition. Thus, they may be used to either identify and extrapolate the dynamics from transient behavior to permanent dynamics or construct efficient, purely data-driven reduced order models.

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27

Kou, Kit Ian, and Hong Li. "Greedy adaptive decomposition of signals based on nonlinear Fourier atoms." International Journal of Wavelets, Multiresolution and Information Processing 14, no. 03 (May 2016): 1650014. http://dx.doi.org/10.1142/s0219691316500144.

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This paper aims at adaptive decomposition of signals in terms of nonlinear Fourier atoms. Each nonlinear Fourier atom is analytic and mono-component. The algorithm is considered as an adaptive greedy procedure based on nonlinear Fourier atoms. The convergence results for the proposed algorithms show that it is suitable to approximate a signal by a linear combinations of nonlinear Fourier atoms. Experiments are presented to illustrate the proposed algorithm and theory.

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28

Hrivnák, Jiří, Mariia Myronova, and Jiří Patera. "Central Splitting of A2 Discrete Fourier–Weyl Transforms." Symmetry 12, no. 11 (November 4, 2020): 1828. http://dx.doi.org/10.3390/sym12111828.

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Two types of bivariate discrete weight lattice Fourier–Weyl transforms are related by the central splitting decomposition. The two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group A2 constitute the kernels of the considered transforms. The central splitting of any function carrying the data into a sum of components governed by the number of elements of the center of A2 is employed to reduce the original weight lattice Fourier–Weyl transform into the corresponding weight lattice splitting transforms. The weight lattice elements intersecting with one-third of the fundamental region of the affine Weyl group determine the point set of the splitting transforms. The unitary matrix decompositions of the normalized weight lattice Fourier–Weyl transforms are presented. The interpolating behavior and the unitary transform matrices of the weight lattice splitting Fourier–Weyl transforms are exemplified.

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29

Mendez, M. A., M. Balabane, and J. M. Buchlin. "Multi-scale proper orthogonal decomposition of complex fluid flows." Journal of Fluid Mechanics 870 (May 15, 2019): 988–1036. http://dx.doi.org/10.1017/jfm.2019.212.

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Data-driven decompositions are becoming essential tools in fluid dynamics, allowing for tracking the evolution of coherent patterns in large datasets, and for constructing low-order models of complex phenomena. In this work, we analyse the main limits of two popular decompositions, namely the proper orthogonal decomposition (POD) and the dynamic mode decomposition (DMD), and we propose a novel decomposition which allows for enhanced feature detection capabilities. This novel decomposition is referred to as multi-scale proper orthogonal decomposition (mPOD) and combines multi-resolution analysis (MRA) with a standard POD. Using MRA, the mPOD splits the correlation matrix into the contribution of different scales, retaining non-overlapping portions of the correlation spectra; using the standard POD, the mPOD extracts the optimal basis from each scale. After introducing a matrix factorization framework for data-driven decompositions, the MRA is formulated via one- and two-dimensional filter banks for the dataset and the correlation matrix respectively. The validation of the mPOD, and a comparison with the discrete Fourier transform (DFT), DMD and POD are provided in three test cases. These include a synthetic test case, a numerical simulation of a nonlinear advection–diffusion problem and an experimental dataset obtained by the time-resolved particle image velocimetry (TR-PIV) of an impinging gas jet. For each of these examples, the decompositions are compared in terms of convergence, feature detection capabilities and time–frequency localization.

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30

Chou, Jung-Hua, and Raylin Wu. "Fourier Series With Spectral and Wave Number Decomposition." American Journal of Applied Sciences 1, no. 4 (April 1, 2004): 258–60. http://dx.doi.org/10.3844/ajassp.2004.258.260.

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31

Wang, Ze, Chi Man Wong, Agostinho Rosa, Tao Qian, and Feng Wan. "Adaptive Fourier Decomposition for Multi-Channel Signal Analysis." IEEE Transactions on Signal Processing 70 (2022): 903–18. http://dx.doi.org/10.1109/tsp.2022.3143723.

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32

Cheng Liu, 刘诚, 金东永 Dug Young Kim, and 朱健强 Jianqiang Zhu. "Spatial Fourier-decomposition optical fluorescence tomography-theoretical investigation." Chinese Optics Letters 6, no. 9 (2008): 665–68. http://dx.doi.org/10.3788/col20080609.0665.

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33

Young, Leslie A. "RAPID COMPUTATION OF OCCULTATION LIGHTCURVES USING FOURIER DECOMPOSITION." Astronomical Journal 137, no. 2 (January 29, 2009): 3398–403. http://dx.doi.org/10.1088/0004-6256/137/2/3398.

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34

Lu, Yufeng, Alireza Kasaeifard, Erdal Oruklu, and Jafar Saniie. "Fractional Fourier Transform for Ultrasonic Chirplet Signal Decomposition." Advances in Acoustics and Vibration 2012 (July 25, 2012): 1–13. http://dx.doi.org/10.1155/2012/480473.

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A fractional fourier transform (FrFT) based chirplet signal decomposition (FrFT-CSD) algorithm is proposed to analyze ultrasonic signals for NDE applications. Particularly, this method is utilized to isolate dominant chirplet echoes for successive steps in signal decomposition and parameter estimation. FrFT rotates the signal with an optimal transform order. The search of optimal transform order is conducted by determining the highest kurtosis value of the signal in the transformed domain. A simulation study reveals the relationship among the kurtosis, the transform order of FrFT, and the chirp rate parameter in the simulated ultrasonic echoes. Benchmark and ultrasonic experimental data are used to evaluate the FrFT-CSD algorithm. Signal processing results show that FrFT-CSD not only reconstructs signal successfully, but also characterizes echoes and estimates echo parameters accurately. This study has a broad range of applications of importance in signal detection, estimation, and pattern recognition.

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35

Wan, Qun, Li Hong Guo, Ding Wang, Lin Zou, and Ji Hao Yin. "Relationship between Discrete Fourier Transformation and Eigenvalue Decomposition." International Journal of Information and Education Technology 9, no. 1 (2019): 74–77. http://dx.doi.org/10.18178/ijiet.2019.9.1.1177.

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36

Moskalik, P., T. Krzyt, N. A. Gorynya, and N. N. Samus. "Accurate Fourier Decomposition of Cepheid Radial Velocity Curves." International Astronomical Union Colloquium 176 (2000): 233–34. http://dx.doi.org/10.1017/s0252921100057614.

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AbstractThe shapes of light curves and of radial velocity curves are two main predictions of the hydrodynamical models of Cepheids. Of the two, the velocity curves are more robust numerically and therefore, more suitable for comparison with the observations. In this report, we present accurate Fourier parameters for an extensive set of classical Cepheid velocity curves. Published radiative models reproduce the observations very well, with only small discrepancies present. We estimate the center of the ω2 = 2ω0 resonance to occur at Pr = 9.947 ± 0.051 day

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37

Qian, Tao, Jianzhong Wang, and Weixiong Mai. "An enhancement algorithm for cyclic adaptive Fourier decomposition." Applied and Computational Harmonic Analysis 47, no. 2 (September 2019): 516–25. http://dx.doi.org/10.1016/j.acha.2019.01.003.

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38

Li, Jingyu, Xuenan Yang, Tao Qian, and Qiwei Xie. "The adaptive Fourier decomposition for financial time series." Engineering Analysis with Boundary Elements 150 (May 2023): 139–53. http://dx.doi.org/10.1016/j.enganabound.2023.01.037.

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39

Nakajima, M. Y., Y. Kawano, and H. Sekigawa. "A new algorithm for producing quantum circuits using KAK decompositions." Quantum Information and Computation 6, no. 1 (January 2006): 67–80. http://dx.doi.org/10.26421/qic6.1-5.

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We provide a new algorithm that translates a unitary matrix into a quantum circuit according to the G=KAK theorem in Lie group theory. With our algorithm, any matrix decomposition corresponding to type-AIII KAK decompositions can be derived according to the given Cartan involution. Our algorithm contains, as its special cases, Cosine-Sine decomposition (CSD) and Khaneja-Glaser decomposition (KGD) in the sense that it derives the same quantum circuits as the ones obtained by them if we select suitable Cartan involutions and square root matrices. The selections of Cartan involutions for computing CSD and KGD will be shown explicitly. As an example, we show explicitly that our method can automatically reproduce the well-known efficient quantum circuit for the $n$-qubit quantum Fourier transform.

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40

Heinig, Hans P. "Fourier operators on weighted Hardy spaces." Mathematical Proceedings of the Cambridge Philosophical Society 101, no. 1 (January 1987): 113–21. http://dx.doi.org/10.1017/s0305004100066457.

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AbstractIn this note we utilize the atomic decomposition of weighted Hardy spaces to prove weighted versions of Hardy's inequality for the Fourier transform with Muckenhoupt weight. The result extends to certain integral operators with homogeneous kernels of degree −1.

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41

Avetisyan, Zhirayr G. "A unified mode decomposition method for physical fields in homogeneous cosmology." Reviews in Mathematical Physics 26, no. 03 (April 2014): 1430001. http://dx.doi.org/10.1142/s0129055x14300015.

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The methods of mode decomposition and Fourier analysis of classical and quantum fields on curved spacetimes previously available mainly for the scalar field on Friedman–Robertson–Walker (FRW) spacetimes are extended to arbitrary vector bundle fields on general spatially homogeneous spacetimes. This is done by developing a rigorous unified framework which incorporates mode decomposition, harmonic analysis and Fourier analysis. The limits of applicability and uniqueness of mode decomposition by separation of the time variable in the field equation are found. It is shown how mode decomposition can be naturally extended to weak solutions of the field equation under some analytical assumptions. It is further shown that these assumptions can always be fulfilled if the vector bundle under consideration is analytic. The propagator of the field equation is explicitly mode decomposed. A short survey on the geometry of the models considered in mathematical cosmology is given and it is concluded that practically all of them can be represented by a semidirect homogeneous vector bundle. Abstract harmonic analytical Fourier transform is introduced in semidirect homogeneous spaces and it is explained how it can be related to the spectral Fourier transform. The general form of invariant bi-distributions on semidirect homogeneous spaces is found in the Fourier space which generalizes earlier results for the homogeneous states of the scalar field on FRW spacetimes.

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42

Kulkarni, SS, AK Bewoor, and RB Ingle. "Vibration signature analysis of distributed defects in ball bearing using wavelet decomposition technique." Noise & Vibration Worldwide 48, no. 1-2 (January 2017): 7–18. http://dx.doi.org/10.1177/0957456517698318.

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The analysis of vibration signals acquired from a ball bearing with an extended type of distributed defects is carried out using wavelet decomposition technique. The influence of artificially generated defect and its location on outer and inner race of the ball bearing is observed using vibration data acquired from bearing housing. The comparison of diagnostic information from fast Fourier transform and time frequency decomposition method is made for inner and outer race of ball bearing with single as well as multiple extended defects. To decompose vibration signal acquired from bearing, db04 wavelet technique was implemented. It is observed that impulses appear with a time period corresponding to characteristic defect frequencies. The results observed from wavelet decomposition technique and fast Fourier transform reveal that the characteristic defect frequency is quite consistent even with change in location of defect. The extended type of distributed defects in the ball bearings can also be effectively diagnosed with the help of wavelet decomposition technique and fast Fourier transform.

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43

Zhang, Zhihua. "Fourier Expansions with Polynomial Terms for Random Processes." Journal of Function Spaces 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/763075.

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Based on calculus of random processes, we present a kind of Fourier expansions with simple polynomial terms via our decomposition method of random processes. Using our method, the expectations and variances of the corresponding coefficients decay fast and partial sum approximations attain the best approximation order. Moreover, since we remove boundary effect in our decomposition of random process, these coefficients can discover the instinct frequency information of this random process. Therefore, our method has an obvious advantage over traditional Fourier expansion. These results are also new for deterministic functions.

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44

Gaudard, Éric, Philippe Druault, Régis Marchiano, and François Van Herpe. "POD and Fourier analyses of a fluid-structure-acoustic interaction problem related to interior car noise." Mechanics & Industry 18, no. 2 (2017): 201. http://dx.doi.org/10.1051/meca/2016027.

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In order to approach a flow configuration revealing the aerodynamic noise contribution in the interior of road vehicles due to the A-pillar vortex, a numerical simulation of a Forward Facing Step (FFS) coupled with a vibrating structure is performed. This numerical study is based on a weak coupling of three solvers to compute (i) the flow field in interaction with the FFS, (ii) the vibration of the structure and (iii) the acoustic radiation in the open cavity. The purpose of this work is then to evaluate the ability of two different post-processing methods: Proper Orthogonal Decomposition and Fourier Decomposition to identify the origin of the noise radiated into a cavity surrounded by an unsteady flow. Fourier and POD decompositions are then successively performed to extract the part of the aeroacoustic wall pressure field impacting the upper part of an upward step mainly related to the radiated acoustic pressure in the cavity. It is observed that the acoustic part, extracted from the wavenumber frequency decomposition (Fourier analysis) of the wall pressure field generates a non-negligible part of the interior cavity noise. However, this contribution is of several orders smaller than the one related to the aerodynamic part of the pressure field. Moreover, it is shown that the most energetic part of the pressure field (POD analysis) is due to the shear flapping motion and mainly contributes to the low-frequency noise in the cavity. Such post-processing results are of particular interest for future analyzes related to the noise radiated inside a car.

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Wang, Wenlong, Biaolong Hua, George A. McMechan, and Bertrand Duquet. "P- and S-decomposition in anisotropic media with localized low-rank approximations." GEOPHYSICS 83, no. 1 (January 1, 2018): C13—C26. http://dx.doi.org/10.1190/geo2017-0138.1.

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We have developed a P- and S-wave decomposition algorithm based on windowed Fourier transforms and a localized low-rank approximation with improved scalability and efficiency for anisotropic wavefields. The model and wavefield are divided into rectangular blocks that do not have to be geologically constrained; low-rank approximations and P- and S-decomposition are performed separately in each block. An overlap-add method reduces artifacts at block boundaries caused by Fourier transforms at wavefield truncations; limited communication is required between blocks. Localization allows a lower rank to be used than global low-rank approximations while maintaining the same quality of decomposition. The algorithm is scalable, making P- and S-decomposition possible in complicated 3D models. Tests with 2D and 3D synthetic data indicate good P- and S-decomposition results.

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Abdel Kareem, Waleed, Mahmoud Abdel Aty, and Zafer M. Asker. "Fourier Decomposition and Anisotropic Diffusion Filtering of Forced Turbulence." International Journal of Applied Mechanics 09, no. 08 (December 2017): 1750121. http://dx.doi.org/10.1142/s1758825117501216.

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The Fourier decomposition and the anisotropic diffusion filtering model are used to extract various flow field scales and their coherent and incoherent parts. The different flow field scales are identified using the Fourier decomposition. Three cutoff wavenumbers are chosen to extract large, medium and fine scale velocity fields, respectively. Then, the anisotropic diffusion model is applied against the obtained velocity fields for each scale to define the coherent and incoherent parts. The forced turbulent velocities are simulated using the lattice Boltzmann method with resolutions [Formula: see text] and [Formula: see text], respectively. The Fourier decomposition of the velocity fields make the filtering process very difficult, so the anisotropic diffusion parameters should be chosen carefully to overcome the problems arising from the sharp cutoffs process. Although of such difficulties, results show that the anisotropic diffusion model successfully isolate the incoherent parts for each scale. It is shown that the incoherent parts are existed everywhere in the flow fields and they are not limited to the fine scales. The coherent fields that are identified by the anisotropic diffusion filtering method are found similar to the extracted scales by the Fourier decomposition. The incoherent regions are fewer in the large scale fields compared with that found in the intermediate and fine fields. The statistical characteristics of the three flow field scales as well as their coherent and incoherent parts are studied and compared with the universal characteristics of turbulence.

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Chupin, Maxime. "Animating Fourier series decomposition of a character with LuaTeX and MPLIB." TUGboat 42, no. 1 (2021): 67–71. http://dx.doi.org/10.47397/tb/42-1/tb130chupin-fourier.

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Kinsella, Karen, James R. Markham, Chad M. Nelson, and Thomas R. Burkholder. "Thermal Decomposition Products of Fiberglass Composites: A Fourier Transform Infrared Analysis." Journal of Fire Sciences 15, no. 2 (March 1997): 108–25. http://dx.doi.org/10.1177/073490419701500203.

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Decomposition products of fiberglass composites used in construc tion were identified using Fourier transform infrared (FT-IR) spectroscopy. This bench-scale study concentrated on identification and quantification of toxic species. Identifying compounds evolved during thermal decomposition provides data to develop early fire detection systems as well as evaluate product fire safety performance. Material fire behavior depends on many factors. Ventila tion, radiant heat flux, and chemical composition are three factors that can be modeled. Physical observations of composites during thermal decomposition with simultaneous identification and quantification of evolved gases offer re searchers in both material development and fire safety an advancement in the state-of-the-art for material testing. Gas analysis by FT-IR spectroscopy iden tified toxic effluent species over a wide range of composite exposure tempera tures (100 to 1000 ° C), during pyrolysis and combustion. Fiberglass composites with melamine, epoxy, and silicone resins were profiled. Formaldehyde, meth anol, carbon monoxide, nitric oxide, methane, and benzene were identified by the spectral analysis prior to physical evidence of decomposition. Toxic concen trations of formaldehyde, carbon monoxide, nitric oxide, ammonia, and hydro gen cyanide were observed as thermal decomposition progressed.

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Yarman, Evren, and Adriana C. Ramírez. "Directional wavefield decomposition." GEOPHYSICS 78, no. 2 (March 1, 2013): WA71—WA76. http://dx.doi.org/10.1190/geo2012-0324.1.

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We bring an alternative view to the computation of the upgoing wavefield and generalize it to directional wavefield decomposition. We start by comparing the integral representation of the upgoing wavefield to the dispersion relation of the wave equation. Developing the relationship between the two, we write a Fourier transform representation that allows to generalize wavefield decomposition to an arbitrary direction. Decomposing the wavefield for arbitrary incoming and outgoing directions at the source and the receiver allows a complete decomposition of the data that can be used for many purposes, such as selecting or enhancing illumination directions for imaging and inversion.

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Braun, D. C., A. C. Birch, and Y. Fan. "Probing the Solar Meridional Circulation Using Fourier Legendre Decomposition." Astrophysical Journal 911, no. 1 (April 1, 2021): 54. http://dx.doi.org/10.3847/1538-4357/abe7e4.

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