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Journal articles on the topic 'Spectral Fourier Method'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Kawata, S., T. Noda, and S. Minami. "Spectral Searching by Fourier-Phase Correlation." Applied Spectroscopy 41, no. 7 (September 1987): 1176–82. http://dx.doi.org/10.1366/0003702874447536.

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A new idea of spectral library searching is proposed, in which only the phase components of Fourier transforms of both the sample and the reference spectra are used for spectral identification. Compared with the conventional methods, this method has essentially a very high discrimination ability for distinguishing between similar spectra and is very resistant to the peak-height variation and the peak-position shift due to the experimental conditions. The principle and the algorithm of this library search method are described, along with the experimental results for infrared absorption spectra. The results demonstrate that the proposed method gives much lower scores for wrong reference spectra than does the ordinary correlation method, and gives high scores for the sample spectra measured by the various slit widths of the spectrophotometer.

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2

Wang, Li-lian, and Ben-yu Guo. "Mixed Fourier–Jacobi spectral method." Journal of Mathematical Analysis and Applications 315, no. 1 (March 2006): 8–28. http://dx.doi.org/10.1016/j.jmaa.2005.09.088.

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3

Eskola, Sisko Maria, and Folke Stenman. "Interpolation of Spectral Data Using the Shift Theorem of the Discrete Fourier Transform." Applied Spectroscopy 51, no. 8 (August 1997): 1179–84. http://dx.doi.org/10.1366/0003702971941719.

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We present a method for interpolating discrete spectral data using the shift theorem of the discrete Fourier transform. The advantages of the method as compared to ordinary curve fitting and similar direct interpolation methods are that, being based on the discrete Fourier transform, our method is also very fast for large sample volumes if used with the fast Fourier transform (FFT) algorithm. In contrast to direct interpolation methods in spectral space, our method does not modify the modulus of the inverse Fourier transform of the shifted profile, so that the spectral content of the signal is preserved. The basis of the method also makes it suitable for analyzing hot-band structures in vibrational spectra. The method is illustrated with two examples.

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4

Pelz, Richard B. "Fourier spectral method on ensemble architectures." Computer Methods in Applied Mechanics and Engineering 89, no. 1-3 (August 1991): 529–42. http://dx.doi.org/10.1016/0045-7825(91)90058-e.

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5

Petibois, Cyril, Georges Cazorla, André Cassaigne, and Gérard Déléris. "Plasma Protein Contents Determined by Fourier-Transform Infrared Spectrometry." Clinical Chemistry 47, no. 4 (April 1, 2001): 730–38. http://dx.doi.org/10.1093/clinchem/47.4.730.

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Abstract Background: Fourier-transform infrared (FT-IR) spectrometry has been used to measure small molecules in plasma. We wished to extend this use to measurement of plasma proteins. Methods: We analyzed plasma proteins, glucose, lactate, and urea in 49 blood samples from 35 healthy subjects and 14 patients. For determining the concentration of each biomolecule, the method used the following steps: (a) The biomolecule was sought for which the correlation between spectral range areas of plasma FT-IR spectra and concentrations determined by comparison method was greatest. (b) The IR absorption of the biomolecule at the most characteristic spectral range was calculated by analyzing pure samples of known concentrations. (c) The plasma concentration of the biomolecule was determined using the FT-IR absorption of the pure compound and the integration value obtained for the plasma FT-IR spectra. (d) The spectral contribution of the biomolecule was subtracted from the plasma FT-IR spectra, and the resulting spectra were saved for further analyses. (e) The same method was then applied to determining the concentrations of other biomolecules by sequentially comparing the resulting FT-IR spectra. Results: Results agreed with those obtained by clinical methods for the following biomolecules when analyzed in the following order: albumin, glucose, fibrinogen, IgG2, lactate, IgG1, α1-antitrypsin, α2-macroglobulin, transferrin, apolipoprotein (Apo)-A1, urea, Apo-B, IgM, Apo-C3, IgA, IgG4, IgG3, IgD, haptoglobin, and α1-acid glycoprotein. Conclusion: FT-IR spectrometry is a useful tool for determining concentrations of several plasma biomolecules.

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6

Meek, R. A., and A. A. Vassiliou. "Characterizing three‐dimensional wavefields with high‐resolution spectra." GEOPHYSICS 57, no. 4 (April 1992): 522–31. http://dx.doi.org/10.1190/1.1443266.

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Three‐dimensional spectra (frequency‐x‐wavenumber‐y‐wavenumber or [Formula: see text] spectra) can be used to determine the frequency content, velocity, and direction of waves entering an array of receivers. This information is important in detecting aliasing problems, understanding coherent noise, designing arrays, and determining parameters for coherent noise filters. Because of the limited spatial dimensions of most arrays the discrete Fourier transform produces an estimate of the three‐dimensional (3-D) spectrum with severe wavenumber distortion. We extend a 2-D hybrid spectral estimation method to three dimensions by combining a temporal Fourier transform with a spatial 2-D maximum entropy spectral estimation technique. The method produces [Formula: see text] spectra with higher wavenumber resolution and less spectral distortion than corresponding 3-D Fourier spectra. The 2-D maximum entropy spectral estimation algorithm uses a sequence of Fourier transforms to extrapolate the estimated autocorrelation function of the data. We assume the wavenumber spectrum of the data comprises a sum of a few poles. Field and synthetic data are used to demonstrate how 3-D wavefields can be characterized with this method of spectral analysis. From these results we conclude that the method gives excellent wavenumber resolution but performs poorly in detecting small signals in the presence of high amplitude signals. We feel this limitation is not serious for characterizing strong amplitude coherent energy recorded by an array of receivers.

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7

Zhang, Lei, Weihua Ou Yang, Xuan Liu, and Haidong Qu. "Fourier Spectral Method for a Class of Nonlinear Schrödinger Models." Advances in Mathematical Physics 2021 (July 1, 2021): 1–11. http://dx.doi.org/10.1155/2021/9934858.

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In this paper, Fourier spectral method combined with modified fourth order exponential time-differencing Runge-Kutta is proposed to solve the nonlinear Schrödinger equation with a source term. The Fourier spectral method is applied to approximate the spatial direction, and fourth order exponential time-differencing Runge-Kutta method is used to discrete temporal direction. The proof of the conservation law of the mass and the energy for the semidiscrete and full-discrete Fourier spectral scheme is given. The error of the semidiscrete Fourier spectral scheme is analyzed in the proper Sobolev space. Finally, several numerical examples are presented to support our analysis.

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8

Yoon, Sungha, Darae Jeong, Chaeyoung Lee, Hyundong Kim, Sangkwon Kim, Hyun Geun Lee, and Junseok Kim. "Fourier-Spectral Method for the Phase-Field Equations." Mathematics 8, no. 8 (August 18, 2020): 1385. http://dx.doi.org/10.3390/math8081385.

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In this paper, we review the Fourier-spectral method for some phase-field models: Allen–Cahn (AC), Cahn–Hilliard (CH), Swift–Hohenberg (SH), phase-field crystal (PFC), and molecular beam epitaxy (MBE) growth. These equations are very important parabolic partial differential equations and are applicable to many interesting scientific problems. The AC equation is a reaction-diffusion equation modeling anti-phase domain coarsening dynamics. The CH equation models phase segregation of binary mixtures. The SH equation is a popular model for generating patterns in spatially extended dissipative systems. A classical PFC model is originally derived to investigate the dynamics of atomic-scale crystal growth. An isotropic symmetry MBE growth model is originally devised as a method for directly growing high purity epitaxial thin film of molecular beams evaporating on a heated substrate. The Fourier-spectral method is highly accurate and simple to implement. We present a detailed description of the method and explain its connection to MATLAB usage so that the interested readers can use the Fourier-spectral method for their research needs without difficulties. Several standard computational tests are done to demonstrate the performance of the method. Furthermore, we provide the MATLAB codes implementation in the Appendix A.

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9

Owolabi, Kolade M., and Ayodeji A. Adejola. "Fourier Spectral Method for Solving Fractional-order System." Universal Journal of Computational Mathematics 5, no. 3 (July 2017): 75–85. http://dx.doi.org/10.13189/ujcmj.2017.050304.

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10

Jiao, Yujian, Tianjun Wang, Xiandong Shi, and Wenjie Liu. "MIXED JACOBI-FOURIER SPECTRAL METHOD FOR FISHER EQUATION." Mathematical Modelling and Analysis 23, no. 2 (April 18, 2018): 240–61. http://dx.doi.org/10.3846/mma.2018.016.

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In this paper, we propose a mixed Jacobi-Fourier spectral method for solving the Fisher equation in a disc. Some mixed Jacobi-Fourier approximation results are established, which play important roles in numerical simulation of various problems deﬁned in a disc. We use the generalized Jacobi approximation to simulate the singularity of solution at the regional center. This also simpliﬁes the theoretical analysis and provides a sparse system. The stability and convergence of the proposed scheme are proved. Numerical results demonstrate the eﬃciency of this new algorithm and coincide well with the theoretical analysis.

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11

Pareschi, Lorenzo, and Benoit Perthame. "A Fourier spectral method for homogeneous boltzmann equations." Transport Theory and Statistical Physics 25, no. 3-5 (April 1996): 369–82. http://dx.doi.org/10.1080/00411459608220707.

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12

Buslov, Dmitrii K., and Nataliya A. Nikonenko. "Regularized Method of Spectral Curve Deconvolution." Applied Spectroscopy 51, no. 5 (May 1997): 666–72. http://dx.doi.org/10.1366/0003702971941070.

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A regularized method of spectral curve deconvolution is proposed. This method is based on three fundamental principles: the regularized method of solving the convolution equation; the use, instead of the apodization function, of the digital low-pass filter, which permits exact knowledge of its characteristics; and the use of the Fourier transform modulus of the spectrum being treated for obtaining a priori information about the frequency characteristics of the solution and noise, required for determination of the optimum parameters of the regularizing operator. The regularized method of de-convolution permits the acquisition of an approximately stable solution for the deconvolution problem of spectral curves, which moves toward an exact solution with the decrease of the experimental spectrum error. Examples are given of the application of the regularized method of deconvolution to simulated and experimental IR spectra. A conclusion about the expediency of using the given method for resolution enhancement in complex spectra is made.

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13

Jiang, Eric Y., William J. McCarthy, David L. Drapcho, and Richard A. Crocombe. "Generalized Two-Dimensional Fourier Transform Infrared Photoacoustic Spectral Depth-Profiling Analysis." Applied Spectroscopy 51, no. 11 (November 1997): 1736–40. http://dx.doi.org/10.1366/0003702971939442.

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This paper reports the first application of a generalized two-dimensional (G2D) correlation method in photoacoustic spectral depth-profiling analysis of laminate/heterogeneous samples. In this method, photoacoustic magnitude spectra at different modulation frequencies are used to generate two-dimensional (2D) correlation spectra. The relative spatial origins of correlated signals are then determined from the signs of the corresponding contours. The unique features and advantages of this technique over previously reported depth-profiling methods are demonstrated and discussed. These include simplification of step-scan photoacoustic phase-modulation experiments as well as 2D correlation-analysis rules, avoidance of phase-related negative-band ambiguities, and enhancement of both spectral and depth resolutions, etc. The recently developed software-based digital signal-processing (DSP) technique for step-scan photoacoustic measurements offers an efficient means (sampling-depth multiplexing advantage) to collect within a single scan all necessary data for this type of 2D correlation analysis. Index Headings: Photoacoustic; PAS; Two-dimensional; Step-scan; FT-IR; Depth profiling; Polymer films; DSP.

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14

Donahue, Steven M., Chris W. Brown, and Robert J. Obremski. "Multicomponent Analysis Using Fourier Transform Infrared and UV Spectra." Applied Spectroscopy 42, no. 2 (February 1988): 353–59. http://dx.doi.org/10.1366/0003702884428176.

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Two- and three-component mixtures of methylated benzenes were analyzed with the use of both infrared and UV spectra. The spectra of known mixtures were Fourier transformed and coefficients from the transforms selected to form coordinates of vectors. The resulting vectors were subjected to factor analysis to obtain representations for multicomponent analysis. A total of eight data sets were analyzed by factor analysis after preprocessing by taking the Fourier transforms of the spectra. The eight data sets were also analyzed by the P-matrix method (inverse Beer's law) in the spectral domain after preprocessing of the data to allow selection of the optimum analytical wavenumbers. This spectral method was compared to the Fourier transform method using cross-validation, in which one sample at a time was left out of the standards and treated as an unknown. The Standard Error of Prediction (SEP) was calculated for the two methods for all possible numbers of vectors and numbers of wavenumbers, starting with the number equal to the number of components and increasing up to a total number of standards or some reasonable cut-off value. Processing in the Fourier domain clearly produced the best results for seven of the data sets and equal results for the other set.

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15

Puryear, Charles I., Oleg N. Portniaguine, Carlos M. Cobos, and John P. Castagna. "Constrained least-squares spectral analysis: Application to seismic data." GEOPHYSICS 77, no. 5 (September 1, 2012): V143—V167. http://dx.doi.org/10.1190/geo2011-0210.1.

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An inversion-based algorithm for computing the time-frequency analysis of reflection seismograms using constrained least-squares spectral analysis is formulated and applied to modeled seismic waveforms and real seismic data. The Fourier series coefficients are computed as a function of time directly by inverting a basis of truncated sinusoidal kernels for a moving time window. The method resulted in spectra that have reduced window smearing for a given window length relative to the discrete Fourier transform irrespective of window shape, and a time-frequency analysis with a combination of time and frequency resolution that is superior to the short time Fourier transform and the continuous wavelet transform. The reduction in spectral smoothing enables better determination of the spectral characteristics of interfering reflections within a short window. The degree of resolution improvement relative to the short time Fourier transform increases as window length decreases. As compared with the continuous wavelet transform, the method has greatly improved temporal resolution, particularly at low frequencies.

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16

Zhang, Jiaqi, Ruigang Zhang, and Liangui Yang. "Topographic solitary waves by the shooting method and Fourier spectral method." Results in Physics 16 (March 2020): 102944. http://dx.doi.org/10.1016/j.rinp.2020.102944.

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17

Zhan, Lei, Feng Liu, and Dimitri Papamoschou. "Fourier time spectral method for subsonic and transonic flows." Acta Mechanica Sinica 32, no. 3 (January 12, 2016): 380–96. http://dx.doi.org/10.1007/s10409-015-0547-x.

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18

Wang, Jian-Ping. "Finite spectral method based on non-periodic Fourier transform." Computers & Fluids 27, no. 5-6 (June 1998): 639–44. http://dx.doi.org/10.1016/s0045-7930(97)00056-x.

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19

Waxler, Roger, and Doru Velea. "Modeling infrasound propagation using the Fourier pseudo‐spectral method." Journal of the Acoustical Society of America 129, no. 4 (April 2011): 2444. http://dx.doi.org/10.1121/1.3588004.

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20

Cai, Wei, David Gottlieb, and Chi-Wang Shu. "Essentially Nonoscillatory Spectral Fourier Method for Shocks Wave Calculations." Mathematics of Computation 52, no. 186 (April 1989): 389. http://dx.doi.org/10.2307/2008473.

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21

Qiu, Zhouhua, Zhong Zeng, Huan Mei, Liang Li, Liping Yao, and Liangqi Zhang. "A Fourier–Legendre spectral element method in polar coordinates." Journal of Computational Physics 231, no. 2 (January 2012): 666–75. http://dx.doi.org/10.1016/j.jcp.2011.10.003.

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22

Ye, Xingde, and Xiaoliang Cheng. "The Fourier spectral method for the Cahn–Hilliard equation." Applied Mathematics and Computation 171, no. 1 (December 2005): 345–57. http://dx.doi.org/10.1016/j.amc.2005.01.050.

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23

Jia-dong, Zheng, Zhang Ru-fen, and Guo Ben-yu. "The Fourier pseudo-spectral method for the SRLW equation." Applied Mathematics and Mechanics 10, no. 9 (September 1989): 843–52. http://dx.doi.org/10.1007/bf02013752.

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24

Ridder, T., T. Warneke, and J. Notholt. "Source brightness fluctuation correction of solar absorption Fourier Transform mid infrared spectra." Atmospheric Measurement Techniques Discussions 4, no. 1 (January 20, 2011): 443–59. http://dx.doi.org/10.5194/amtd-4-443-2011.

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Abstract. Solar absorption Fourier Transform infrared spectrometry is considered a precise and accurate method for the observation of trace gases in the atmosphere. The precision and accuracy of such measurements are dependent on the stability of the light source. Fluctuations in the source brightness reduce the precision and accuracy of the trace gas concentrations, but cannot always be avoided. Thus, a strong effort is made within the community to reduce the impact of source brightness fluctuations by applying a correction on the spectra following the measurements. So far, it could be shown that the precision and accuracy of CO2 total column concentrations could be improved by applying a source brightness fluctuation correction to spectra in the near infrared spectral region. The analysis of trace gas concentrations obtained from spectra in the mid infrared spectral region is fundamental. However, spectra below 2000 cm−1 are generally considered uncorrectable, if they are measured with a MCT detector. Such measurements introduce an unknown offset to MCT interferograms, which prevents a source brightness fluctuation correction. Here, we show a method of source brightness fluctuation correction, which can be applied on spectra in the whole infrared spectral region including spectra measured with a MCT detector. We present a solution to remove the unknown offset in MCT interferograms allowing MCT spectra for an application of source brightness fluctuation correction. This gives an improvement in the quality of MCT spectra and we demonstrate an improvement in the retrieval of O3 profiles and total column concentrations. For a comparison with previous studies, we apply our source brightness fluctuation correction method on spectra in the near infrared spectral region and show an improvement in the retrieval of CO2 total column concentrations.

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25

Cohen, Joel E., Charles M. Newman, Adam E. Cohen, Owen L. Petchey, and Andrew Gonzalez. "Spectral mimicry: A method of synthesizing matching time series with different Fourier spectra." Circuits Systems and Signal Processing 18, no. 4 (July 1999): 431–42. http://dx.doi.org/10.1007/bf01200792.

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26

Wood, Bayden R., Brian Tait, and Donald McNaughton. "Fourier Transform Infrared Spectroscopy as a Method for Monitoring the Molecular Dynamics of Lymphocyte Activation." Applied Spectroscopy 54, no. 3 (March 2000): 353–59. http://dx.doi.org/10.1366/0003702001949627.

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In this paper we report the application of Fourier transform infrared (FT-IR) microspectroscopy to monitor the molecular dynamics of lymphocyte activation. Infrared spectra of lymphocytes stimulated with the mitogen phytohaemagglutinin-L show spectral features 15 min after initial stimulation that are not apparent in resting lymphocytes. By analyzing the second-order derivatives of the raw spectra and applying principal components analysis (PCA), we conclude that the major spectral changes observed in the first hour result from an increase in overall RNA synthesis. Bands characteristic of RNA at 1244, 1080, 1050, 970, 1160, and 1120 cm−1 appear progressively more intense over time in the spectra of activated lymphocytes. The magnitude of these changes increases over time as the cell differentiates into a blast cell. The sensitivity of infrared spectroscopy to RNA moieties and the rapidity of the technique suggest a possible future role for FT-IR spectroscopy in histocompatibility testing.

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27

Kataieva, Mariia, and Vladimir Kvasnikov. "METHOD OF OBTAINING THE SPECTRAL CHARACTERISTICS OF THE SCANNING PROBE MICROSCOPE." Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska 11, no. 2 (June 30, 2021): 52–55. http://dx.doi.org/10.35784/iapgos.2646.

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The article discusses methods and algorithms for digital processing and filtering when carrying out nano-measurements using a scanning probe microscope. The paper discusses frequency methods for improving images, in particular, the use of the Fourier transforms with various filtering methods to improve the quality of the resulting image. Stable computational algorithms have been developed for transforming discrete signals based on the Fourier transform. Methods for the interpretation of the numerical results of the discrete Fourier transform in such packages as Matlab, MathCad, Matematica are presented. It is proposed to use a window transform, developed based on the Fourier transform, which makes it possible to single out the informative features of the signal and to reduce the influence of the destabilizing factors that arise when processing signals from a scanning gold microscope in real conditions.

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28

Sherman, Jeffrey W., James A. de Haseth, and David G. Cameron. "A Window Fourier-Domain Infrared Search System." Applied Spectroscopy 43, no. 8 (November 1989): 1311–16. http://dx.doi.org/10.1366/0003702894204290.

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A search system is presented that utilizes Fourier-domain infrared data. This system removes or nulls features from infrared absorbance spectra that are found to be dominant and similar in the spectra of interest. This method enhances spectral differences. An example of one such region that is commonly nulled is the C-H stretch region of the spectra. These bands occur in organic compounds and, while generally being quite strong, contribute little to differentiation between spectra. More than one region may be nulled with this system, and the regions are chosen by visual inspection of the spectra of interest. The nulled spectra are then transformed into the Fourier domain, where data in the region that is 55–155 points displaced from the centerburst are extracted and normalized. Finally, a complete library search is performed on the normalized Fourier-domain data. The actual matching algorithm uses a simple dot product calculation, and excellent search results are obtained. This method can be used to distinguish similar compounds that are not easily distinguished by other infrared search methods. In the most difficult cases, a minimum of two-orders-of-magnitude improvement in the difference in dot products is observed when the results of full bandpass spectra are compared with the results of spectra that had one or more regions nulled.

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29

Akter, Sayeda Irin, Md Shahriar Mahmud, Md Kamrujjaman, and Hazrat Ali. "Global Spectral Collocation Method with Fourier Transform to Solve Differential Equations." GANIT: Journal of Bangladesh Mathematical Society 40, no. 1 (July 14, 2020): 28–42. http://dx.doi.org/10.3329/ganit.v40i1.48193.

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Numerical analysis is the area of mathematics that creates, analyzes, and implements algorithms for solving numerically the problems from real-world applications of algebra, geometry, and calculus, and they involve variables which vary continuously. Till now, numerous numerical methods have been introduced. Spectral method is one of those techniques used in applied mathematics and scientific computing to numerically solve certain differential equations, potentially involving the use of the Fast Fourier Transform (FFT). This study presents some of the fundamental ideas of spectral method. Orthogonal basis are used to establish computational algorithm. The accuracy and efficiency of proposed model are discussed. This manuscript estimates for the error between the exact and approximated discrete solutions. This paper shows that, grid points for polynomial spectral methods should lie approximately in a minimal energy configuration associated with inverse linear repulsion between points. The wave equation, linear and non-linear boundary value problems are solved using spectral method on the platform of MATLAB language. GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 28-42

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30

Duan, Yanting, Chaodong Wu, Xiaodong Zheng, Yucheng Huang, and Jian Ma. "Coherence based on spectral variance analysis." GEOPHYSICS 83, no. 3 (May 1, 2018): O55—O66. http://dx.doi.org/10.1190/geo2017-0158.1.

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The eigenstructure-based coherence attribute is a type of efficient and mature tool for mapping geologic edges such as faults and/or channels in the 3D seismic interpretation. However, the eigenstructure-based coherence algorithm is sensitive to low signal-to-noise ratio seismic data, and the coherence results are affected by the dipping structures. Due to the large energy gap between the low- and high-frequency components, the low-frequency components play the principal role in coherence estimation. In contrast, the spectral variance balances the difference between the low- and high-frequency components at a fixed depth. The coherence estimation based on amplitude spectra avoids the effect of the time delays resulting from the dipping structures. Combining the spectral variance with the amplitude spectra avoids the effect of dipping structures and enhances the antinoise performance of the high-frequency components. First, we apply the short-time Fourier transform to obtain the time-frequency spectra of seismic data. Next, we compute the variance values of amplitude spectra. Then, we apply the fast Fourier transform to obtain the amplitude spectra of spectral variance. Finally, we calculate the eigenstructure coherence by using the amplitude spectra of spectral variance as the input. We apply the method to the theoretical models and practical seismic data. In the Marmousi velocity model, the coherence estimation using the amplitude spectra of the spectral variance as input shows more subtle discontinuities, especially in deeper layers. The results from field-data examples demonstrate that the proposed method is helpful for mapping faults and for improving the narrow channel edges’ resolution of interest. Therefore, the coherence algorithm based on the spectral variance analysis may be conducive to the seismic interpretation.

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31

Brown, Chris W., Robert J. Obremski, and Philip Anderson. "Infrared Quantitative Analysis in the Fourier Domain: Processing Vector Representations." Applied Spectroscopy 40, no. 6 (August 1986): 734–42. http://dx.doi.org/10.1366/0003702864508368.

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The methodologies for forming orthogonal vector representations of transformed infrared spectra of pure components in multicomponent systems are developed. Either the Gram-Schmidt orthogonalization method or factor analysis can be used to generate the vector representations; both give equivalent results when one vector representation per component is generated. The results of processing data in the Fourier domain using both orthogonalization methods are compared with those obtained by processing in the spectral domain for the following three systems: (1) simulated spectra; (2) real spectra with dissimilar features; and (3) real spectra with very similar features. In all cases, processing in the Fourier domain produces the optimum results while requiring a minimum amount of operator contributions.

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32

Choi, Jae-Young, Dong Kyun Im, Jangho Park, and Seongim Choi. "Prediction of Dynamic Stability Using Mapped Chebyshev Pseudospectral Method." International Journal of Aerospace Engineering 2018 (August 1, 2018): 1–14. http://dx.doi.org/10.1155/2018/2508153.

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A mapped Chebyshev pseudospectral method is extended to solve three-dimensional unsteady flow problems. As the classical Chebyshev spectral approach can lead to numerical instabilities due to ill conditioning of the spectral matrix, the Chebyshev points are evenly redistributed over the domain by an inverse sine mapping function. The mapped Chebyshev pseudospectral method can be used as an alternative time-spectral approach that uses a Chebyshev collocation operator to approximate the time derivative terms in the unsteady flow governing equations, and the method can make general applications to both nonperiodic and periodic problems. In this study, the mapped Chebyshev pseudospectral method is employed to solve three-dimensional periodic problem to verify the spectral accuracy and computational efficiency with those of the Fourier pseudospectral method and the time-accurate method. The results show a good agreement with both of the Fourier pseudospectral method and the time-accurate method. The flow solutions also demonstrate a good agreement with the experimental data. Similar to the Fourier pseudospectral method, the mapped Chebyshev pseudospectral method approximates the unsteady flow solutions with a precise accuracy at a considerably effective computational cost compared to the conventional time-accurate method.

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33

Ito, Izumi. "A New Pseudo-Spectral Method Using the Discrete Cosine Transform." Journal of Imaging 6, no. 4 (March 28, 2020): 15. http://dx.doi.org/10.3390/jimaging6040015.

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The pseudo-spectral (PS) method on the basis of the Fourier transform is a numerical method for estimating derivatives. Generally, the discrete Fourier transform (DFT) is used when implementing the PS method. However, when the values on both sides of the sequences differ significantly, oscillatory approximations around both sides appear due to the periodicity resulting from the DFT. To address this problem, we propose a new PS method based on symmetric extension. We mathematically derive the proposed method using the discrete cosine transform (DCT) in the forward transform from the relation between DFT and DCT. DCT allows a sequence to function as a symmetrically extended sequence and estimates derivatives in the transformed domain. The superior performance of the proposed method is demonstrated through image interpolation. Potential applications of the proposed method are numerical simulations using the Fourier based PS method in many fields such as fluid dynamics, meteorology, and geophysics.

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34

Popov, S. P. "Application of the quasi-spectral fourier method to soliton equations." Computational Mathematics and Mathematical Physics 50, no. 12 (December 2010): 2064–70. http://dx.doi.org/10.1134/s0965542510120080.

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35

Wei, Liang, and Zhiping Li. "Fourier-Chebyshev spectral method for cavitation computation in nonlinear elasticity." Frontiers of Mathematics in China 13, no. 1 (December 13, 2017): 203–26. http://dx.doi.org/10.1007/s11464-017-0664-x.

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36

Guo, Benyu, Jian Li, and Heping Ma. "Fourier-Chebyshev spectral method for solving three-dimensional vorticity equation." Acta Mathematicae Applicatae Sinica 11, no. 1 (January 1995): 94–109. http://dx.doi.org/10.1007/bf02012626.

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Ju, Wei, Changhua Lu, Chun Liu, Weiwei Jiang, Yujun Zhang, and Feng Hong. "Rapid Identification of Atmospheric Gaseous Pollutants Using Fourier-Transform Infrared Spectroscopy Combined with Independent Component Analysis." Journal of Spectroscopy 2020 (January 22, 2020): 1–14. http://dx.doi.org/10.1155/2020/8920732.

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Fourier-transform infrared (FTIR) spectroscopy is a rapid and nondestructive technology for monitoring atmospheric quality. The identification of each component from the FTIR spectra is a prerequisite for the accurate quantitative analysis of gaseous pollutants. Due to the overlap of different gas absorption peaks and the interference of water vapor in the actual measurement, the existing identification methods of gas spectra have drawbacks of low identification rate and the inability to carry out real-time online analysis in atmospheric quality monitoring. In this work, independent component analysis (ICA) is applied to the spectral separation of heavily overlapped spectra of gaseous pollutants. The proposed method is validated by the analysis of mixture spectra obtained in laboratory and actual atmospheric spectra collected from stationary source. The average time consumption of separation process is less than 0.2 seconds, and the identification rate of experimental gases is up to 100%, as shown by the results of peak searching and the analysis of the correction coefficient between the separated spectra and the standard spectra database. The identification results of actual atmospheric spectra demonstrated that the proposed method can effectively identify the gaseous pollutants whose concentration changes in the measured spectra, and it is a promising qualitative spectral analysis tool that can shorten the identification time, as well as increase the identification rate. Therefore, this method can be a useful alternative to traditional qualitative identification methods for real-time online atmospheric pollutant detection.

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Rashid, Abdur, and Ahmad Izani Bin Md Ismail. "A Fourier Pseudospectral Method for Solving Coupled Viscous Burgers Equations." Computational Methods in Applied Mathematics 9, no. 4 (2009): 412–20. http://dx.doi.org/10.2478/cmam-2009-0026.

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AbstractThe Fourier pseudo-spectral method has been studied for a one- dimensional coupled system of viscous Burgers equations. Two test problems with known exact solutions have been selected for this study. In this paper, the rate of con- vergence in time and error analysis of the solution of the first problem has been studied, while the numerical results of the second problem obtained by the present method are compared to those obtained by using the Chebyshev spectral collocation method. The numerical results show that the proposed method outperforms the conventional one in terms of accuracy and convergence rate.

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Leible, Benedikt, Daniel Plabst, and Norbert Hanik. "Back-to-Back Performance of the Full Spectrum Nonlinear Fourier Transform and Its Inverse." Entropy 22, no. 10 (October 6, 2020): 1131. http://dx.doi.org/10.3390/e22101131.

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In this paper, data-transmission using the nonlinear Fourier transform for jointly modulated discrete and continuous spectra is investigated. A recent method for purely discrete eigenvalue removal at the detector is extended to signals with additional continuous spectral support. At first, the eigenvalues are sequentially detected and removed from the jointly modulated received signal. After each successful removal, the time-support of the resulting signal for the next iteration can be narrowed, until all eigenvalues are removed. The resulting truncated signal, ideally containing only continuous spectral components, is then recovered by a standard NFT algorithm. Numerical simulations without a fiber channel show that, for jointly modulated discrete and continuous spectra, the mean-squared error between transmitted and received eigenvalues can be reduced using the eigenvalue removal approach, when compared to state-of-the-art detection methods. Additionally, the computational complexity for detection of both spectral components can be decreased when, by the choice of the modulated eigenvalues, the time-support after each removal step can be reduced. Numerical simulations are also carried out for transmission over a Raman-amplified, lossy SSMF channel. The mutual information is approximated and the eigenvalue removal method is shown to result in achievable rate improvements.

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RAHMANI, Ramin Khosravi, Theo Gordon KEITH, and Russell COTTAM. "An Approximate Spectral Method for Accurate Solution of Fourier and Non-Fourier Heat Conduction Problems." JSME International Journal Series B 46, no. 4 (2003): 519–27. http://dx.doi.org/10.1299/jsmeb.46.519.

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Masutani, Koji, Hisashi Sugisawa, Akira Yokota, Yukio Furukawa, and Mitsuo Tasumi. "Asynchronous Time-Resolved Fourier Transform Infrared Spectroscopy." Applied Spectroscopy 46, no. 4 (April 1992): 560–67. http://dx.doi.org/10.1366/0003702924124871.

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A novel asynchronous time-resolved FT-IR spectrophotometer based on a conventional continuous-scan interferometer has been developed. In contrast to the existing methods, this method does not require the synchronization between the signal for time resolving and that for the sampling of the A/D converter. The signal-processing assembly for time-resolved measurements consists of a pulse generator, a pulse delay circuit, a gate circuit, and a low-pass filter. This assembly can be attached to any conventional FT-IR spectrophotometer. By this method, time-resolved spectra without any spectral distortion or artifacts can be obtained for repetitive fast phenomena. As an application of this method, time-resolved FT-IR measurements have been carried out for the reorientation process in a ferroelectric liquid crystal induced by the reversal of external electric field, and a time resolution of 0.5 μs or less has been achieved. The present system is best suited for observing repetitive transient phenomena with lifetimes in the range from approximately 1 μs to 1 ms.

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Yu, Xu-Hong, and Zhong-Qing Wang. "Mixed Fourier-Jacobi Spectral Method for Two-Dimensional Neumann Boundary Value Problems." East Asian Journal on Applied Mathematics 1, no. 3 (August 2011): 284–96. http://dx.doi.org/10.4208/eajam.281010.200411a.

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AbstractIn this paper, we propose a mixed Fourier-Jacobi spectral method for two dimensional Neumann boundary value problem. This method differs from the classical spectral method. The homogeneous Neumann boundary condition is satisfied exactly. Moreover, a tridiagonal matrix is employed, instead of the full stiffness matrix encountered in the classical variational formulation. For analyzing the numerical error, we establish the mixed Fourier-Jacobi orthogonal approximation. The convergence of proposed scheme is proved. Numerical results demonstrate the efficiency of this approach.

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Pardo-Ig�zquiza, E., and M. Chica-Olmo. "The Fourier Integral Method: An efficient spectral method for simulation of random fields." Mathematical Geology 25, no. 2 (February 1993): 177–217. http://dx.doi.org/10.1007/bf00893272.

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Saarinen, Pekka E. "Spectral Line Narrowing by Use of the Theoretical Impulse Response." Applied Spectroscopy 51, no. 2 (February 1997): 188–200. http://dx.doi.org/10.1366/0003702971940125.

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Fourier transform spectroscopy is nowadays able to produce spectra with extremely high signal-to-noise ratios, and thus extremely high information content. Unfortunately, this information is partially lost because of a lack of sufficiently effective line-narrowing methods to resolve overlapping spectral lines. A novel, and very promising, approach to the problem is the LOMEP line-narrowing method, based on consecutive linear prediction; the line narrowing is carried out in the signal domain by extrapolating the Fourier transform of the spectrum. However, LOMEP is not yet optimal. For example, it does not make use of the information contained in the output line shapes to correct the errors made in linear prediction. In fact, that procedure would not even be possible by using the prediction strategy adopted by LOMEP. Therefore it is possible to considerably improve the method by including the information contained in the distortions of the output spectral lines. In this paper a new method of line narrowing is presented, based on progressive improvement of the prediction until the output is free from distortions. The method is very easy to use and does not require a profound understanding of the underlying mathematics.

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Aboul-Enein, Hassan Y., Oana Mihaela Antochi, Gheorghe Nechifor, and Andrei A. Bunaciu. "Analysis of Raspberry Ketone in Nutraceutical Formulation Using Fourier Transform Infrared Spectrophotometric Method." Open Bioactive Compounds Journal 7, no. 1 (November 5, 2019): 8–13. http://dx.doi.org/10.2174/1874847301907010008.

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Aims: A Fourier Transform Infrared (FT-IR) spectrometric method was developed for the rapid, direct measurement of Raspberry Ketone (RK) and Caffeine (CAF) in a nutraceutical formulation. Methods: Conventional KBr-spectra and KBr+0.5 mg Microcrystalline Cellulose (MCC)-spectra were used as the basis for a better determination of active substances in the nutraceutical formulation. A calibration model was developed using caffeine and raspberry ketone standards of varying concentrations in the mid-infrared region (4000-400 cm-1). The Beer-Lambert law was used in data processing. Results: The results indicate that FT-IR spectrometry is applicable to the analytical quantification of RK and CAF in the nutraceutical formulation. Conclusion: The method proposed is simple, precise and not time-consuming compared to the chromatographic methods that are cited in the literature. Quantification is performed in about 10-15 minutes, including sample preparation and spectral acquisition.

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Carcione, José M. "A spectral numerical method for electromagnetic diffusion." GEOPHYSICS 71, no. 1 (January 2006): I1—I9. http://dx.doi.org/10.1190/1.2159050.

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I present a pseudospectral explicit scheme that can simulate low-frequency electromagnetic (EM) propagation in the earth. This scheme solves linear periodic parabolic equations, having accuracy within machine precision, both temporally and spatially. The method is based on a Chebyshev expansion of the evolution operator, with the spatial derivatives computed via a staggered Fourier pseudospectral technique. The results match analytical solutions of the initial-value problem and the Green's function. An example of the EM field produced by a set of magnetic sources in a heterogeneous model illustrates the algorithm's performance.

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Shang, Jing, Xinran Wu, Keqing Hu, Zongyao Huyan, Qi Li, and Xiuzhu Yu. "A simple and practical method to determine peroxide values in edible oilsviainfrared quartz cuvette-based Fourier transform infrared spectroscopy." Analytical Methods 10, no. 29 (2018): 3675–79. http://dx.doi.org/10.1039/c8ay00594j.

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A rapid and simple method was developed to determine the peroxide values (PVs) of edible oils by Fourier transform infrared spectroscopy. The spectra of edible oils were obtained using an infrared quartz cuvette (IQC) as a spectral accessory. The IQC-based FTIR method is more feasible and accurate than the traditional method.

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Cięszczyk, Sławomir. "A Multi-Band Integrated Virtual Calibration-Inversion Method for Open Path FTIR Spectrometry." Metrology and Measurement Systems 20, no. 2 (June 1, 2013): 287–98. http://dx.doi.org/10.2478/mms-2013-0025.

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Abstract This paper addresses problems arising from in situ measurement of gas content and temperature. Such measurements can be considered indirect. Transmittance or natural radiation of a gas is measured directly. The latter method (spectral radiation measurement) is often called spectral remote sensing. Its primary uses are in astronomy and in the measurement of atmospheric composition. In industrial processes, in situ spectroscopic measurements in the plant are often made with an open path Fourier Transform Infrared (FTIR) spectrometer. The main difficulty in this approach is related to the calibration process, which often cannot be carried out in the manner used in the laboratory. Spectral information can be obtained from open path spectroscopic measurements using mathematical modeling, and by solving the inverse problem. Determination of gas content based on spectral measurements requires comparison of the measured and modeled spectra. This paper proposes a method for the simultaneous use of multiple lines to determine the gas content. The integrated absorptions of many spectral lines permits calculation of the average band absorption. An inverse model based on neural networks is used to determine gas content based on mid-infrared spectra at variable temperatures.

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Jufriansah, Adi, Azmi Khusnani, Arief Hermanto, Mohammad Toifur, and Erwin Prasetyo. "The Existence of Fourier Coefficients and Periodic Multiplicity Based on Initial Values and One-Dimensional Wave Limits Requirements." Jurnal Penelitian Fisika dan Aplikasinya (JPFA) 10, no. 2 (December 31, 2020): 146. http://dx.doi.org/10.26740/jpfa.v10n2.p146-157.

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Physical systems in partial differential equations can be interpreted in a visual form using a wave simulation. In particular, the interpretation of the differential equations used is in the nonlinear hyperbolic model, but in its completion, there are some limitations to the stability requirements found. The aim of this study is to investigate the analytical and numerical analysis of a wave equation with a similar unit and fractal intervals using the Fourier coefficient. The method in this research is to use the analytical solution approach, the spectral method, and the finite difference method. The hyperbolic wave equation's analytical solution approach, illustrated in the Fourier analysis, uses a pulse triangle. The spectral method minimizes errors when there is the addition of the same sample grid points or the periodic domain's expansion with a trigonometric basis. Meanwhile, different ways offer a more efficient solution. Based on the research results, the information obtained is that the Fourier analysis illustrates the pulse triangle use to solve the solution. These results are also suitable for adding sample points to the same spectra. Fourier analysis requires a relatively long time to solve one pulse triangle graph to need another solution, namely the finite difference method. However, its use is still limited in terms of stability when faced with more complex problems.

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Alsberg, Bjørn K., William G. Wade, and Royston Goodacre. "Chemometric Analysis of Diffuse Reflectance-Absorbance Fourier Transform Infrared Spectra Using Rule Induction Methods: Application to the Classification of Eubacterium Species." Applied Spectroscopy 52, no. 6 (June 1998): 823–32. http://dx.doi.org/10.1366/0003702981944562.

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Strains representative of four Eubacterium species were analyzed by using diffuse reflectance-absorbance Fourier transform infrared spectroscopy. To identify important wavenumber regions for the classification of these 22 bacterial isolates, we investigated three rule induction methods and various spectral preprocessing regimes. In this study both univariate and multivariate classification and regression trees (CART) methods and the fuzzy multivariate rule-building expert system (FuRES) method were exploited. It was found that the FuRES method was superior in terms of prediction, whereas the rules proposed by the univariate CART method were easier to interpret in terms of which wavenumbers in the IR spectra were important for bacterial class separation. Scaled and detrended FT-IR spectra and first-order numerical differentiation preprocessing steps were necessary to obtain optimal classification models. Finally, a reduction in the classification error for the CART-based methods was observed by analyzing the compressed B-spline coefficients rather than the original spectra representation. The spectral interpretation of these rules is in agreement with analyses using the uncompressed representation.

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