Showing papers on "Fourier series published in 2016"

Accurate estimators of correlation functions in Fourier space

Abstract: Efficient estimators of Fourier-space statistics for large number of objects rely on Fast Fourier Transforms (FFTs), which are affected by aliasing from unresolved small scale modes due to the finite FFT grid. Aliasing takes the form of a sum over images, each of them corresponding to the Fourier content displaced by increasing multiples of the sampling frequency of the grid. These spurious contributions limit the accuracy in the estimation of Fourier-space statistics, and are typically ameliorated by simultaneously increasing grid size and discarding high-frequency modes. This results in inefficient estimates for e.g. the power spectrum when desired systematic biases are well under per-cent level. We show that using interlaced grids removes odd images, which include the dominant contribution to aliasing. In addition, we discuss the choice of interpolation kernel used to define density perturbations on the FFT grid and demonstrate that using higher-order interpolation kernels than the standard Cloud in Cell algorithm results in significant reduction of the remaining images. We show that combining fourth-order interpolation with interlacing gives very accurate Fourier amplitudes and phases of density perturbations. This results in power spectrum and bispectrum estimates that have systematic biases below 0.01% all the way to the Nyquist frequency of the grid, thus maximizing the use of unbiased Fourier coefficients for a given grid size and greatly reducing systematics for applications to large cosmological datasets.

Arbitrary-Order Trigonometric Fourier Collocation Methods for Multi-Frequency Oscillatory Systems

Abstract: We rigorously study a novel type of trigonometric Fourier collocation methods for solving multi-frequency oscillatory second-order ordinary differential equations (ODEs) $$q^{\prime \prime }(t)+Mq(t)=f(q(t))$$q?(t)+Mq(t)=f(q(t)) with a principal frequency matrix $$M\in \mathbb {R}^{d\times d}$$M?Rd×d. If $$M$$M is symmetric and positive semi-definite and $$f(q) = - abla U(q)$$f(q)=-?U(q) for a smooth function $$U(q)$$U(q), then this is a multi-frequency oscillatory Hamiltonian system with the Hamiltonian $$H(q,p)=p^{T}p/2+q^{T}Mq/2+U(q),$$H(q,p)=pTp/2+qTMq/2+U(q), where $$p = q'$$p=q?. The solution of this system is a nonlinear multi-frequency oscillator. The new trigonometric Fourier collocation method takes advantage of the special structure brought by the linear term $$Mq$$Mq, and its construction incorporates the idea of collocation methods, the variation-of-constants formula and the local Fourier expansion of the system. The properties of the new methods are analysed. The analysis in the paper demonstrates an important feature, namely that the trigonometric Fourier collocation methods can be of an arbitrary order and when $$M\rightarrow 0$$M?0, each trigonometric Fourier collocation method creates a particular Runge---Kutta---Nystrom-type Fourier collocation method, which is symplectic under some conditions. This allows us to obtain arbitrary high-order symplectic methods to deal with a special and important class of systems of second-order ODEs in an efficient way. The results of numerical experiments are quite promising and show that the trigonometric Fourier collocation methods are significantly more efficient in comparison with alternative approaches that have previously appeared in the literature.

Off-the-Grid Recovery of Piecewise Constant Images from Few Fourier Samples

Abstract: We introduce a method to recover a continuous domain representation of a piecewise constant two-dimensional image from few low-pass Fourier samples. Assuming the edge set of the image is localized to the zero set of a trigonometric polynomial, we show that the Fourier coefficients of the partial derivatives of the image satisfy a linear annihilation relation. We present necessary and sufficient conditions for unique recovery of the image from finite low-pass Fourier samples using the annihilation relation. We also propose a practical two-stage recovery algorithm that is robust to model-mismatch and noise. In the first stage we estimate a continuous domain representation of the edge set of the image. In the second stage we perform an extrapolation in Fourier domain by a least squares two-dimensional linear prediction, which recovers the exact Fourier coefficients of the underlying image. We demonstrate our algorithm on the superresolution recovery of MRI phantoms and real MRI data from low-pass Fourier sam...

The Fractional Laplacian

Abstract: The fractional Laplacian in one dimension Random walkers with constant steps Ordinary diffusion Random jumpers Central limit theorem and stable distributions Power-law probability jump lengths A principal-value integral Wires and springs The fractional Laplacian Fourier transform Effect of fractional order Numerical computation of the fractional Laplacian Green's function of the fractional Laplace equation Fractional Poisson equation in a restricted domain Green's function of unsteady fractional diffusion Numerical discretization in one dimension Computation of a principal-value integral Fractional Laplacian differentiation matrix Fractional Poisson equation Evolution under fractional diffusion Differentiation by spectral expansion Further concepts in one dimension Fractional first derivative Properties of the fractional first derivative Laplacian potential Fractional derivatives from finite-difference stencils Fractional third derivative Fractional fourth derivative Periodic functions Sine, cosines, and the complete Fourier series Cosine Fourier series Sine Fourier series Green's functions Integral representation of the periodic Laplacian Numerical discretization Periodic differentiation matrix Differentiation by spectral expansion Embedding of the fractional Poisson equation The fractional Laplacian in three dimensions Stipulation on the Fourier transform Integral representation Fractional gradient Laplacian potential Green's function of the fractional Laplace equation The Riesz potential Triply periodic Green's function Fractional Poisson equation Evolution under fractional diffusion Periodic functions and arbitrary domains Fractional Stokes flow The fractional Laplacian in two dimensions Stipulation on the Fourier transform Integral representation Fractional gradient Laplacian potential Green's function of the fractional Laplace equation The Riesz potential Doubly periodic Green's function Fractional Poisson equation Evolution due to fractional diffusion Periodic functions and arbitrary domains Appendix A: Selected definite integrals Appendix B: The Gamma function Appendix C: The Gaussian distribution Appendix D: The fractional Laplacian in arbitrary dimensions Appendix E: Fractional derivatives Appendix F: Aitken extrapolation of an infinite sum References Index

An Algorithm for the Machine Calculation of

Abstract: An efficient method for the calculation of the interactions of a 2' factorial experiment was introduced by Yates and is widely known by his name. The generalization to 3' was given by Box et al. [1]. Good [2] generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series. In their full generality, Good's methods are applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices, where m is proportional to log N. This results inma procedure requiring a number of operations proportional to N log N rather than N2. These methods are applied here to the calculation of complex Fourier series. They are useful in situations where the number of data points is, or can be chosen to be, a highly composite number. The algorithm is here derived and presented in a rather different form. Attention is given to the choice of N. It is also shown how special advantage can be obtained in the use of a binary computer with N = 2' and how the entire calculation can be performed within the array of N data storage locations used for the given Fourier coefficients. Consider the problem of calculating the complex Fourier series N-1

A unified solution for vibration analysis of functionally graded circular, annular and sector plates with general boundary conditions

Abstract: The vibrations of functionally graded circular plates, annular plates, and annular, circular sectorial plates have been traditionally treated as different boundary value problems, which results in numerous specific solution algorithms and procedures. It is the problem itself that has been an overwhelming task for a new researcher or application engineer to comprehend. Furthermore each type of plate usually needs treating separately when different boundary conditions are involved. In this paper, a unified method is presented for the vibration analysis of the plates mentioned above with general boundary conditions based on the first-order shear deformation theory and Ritz procedure. The material properties are assumed to vary continuously through the thickness according to the general four-parameter power-law distribution. Regardless of the shapes of the plates and the types of boundary conditions, the displacements of the plates are described as an improved Fourier series expansion which is composed of a double Fourier cosine series and several auxiliary functions. As an innovative point of this work, the auxiliary functions are introduced to eliminate all the relevant discontinuities with the displacement and its derivatives at the boundaries and to accelerate the convergence of series representations. The accuracy, reliability and versatility of the current solution are fully demonstrated and verified through numerical examples involving plates with various shapes and boundary conditions. Some new results of functionally graded circular, annular and sector plates with various boundary conditions are presented, which may serve as datum solutions for future computational methods. In addition, the influence of boundary conditions, the material and geometric parameters on the vibration characteristics of the plates are also reported.

Fourier Series In Control Theory

Abstract: Thank you very much for downloading fourier series in control theory. Maybe you have knowledge that, people have search hundreds times for their favorite readings like this fourier series in control theory, but end up in infectious downloads. Rather than enjoying a good book with a cup of coffee in the afternoon, instead they juggled with some malicious bugs inside their computer. fourier series in control theory is available in our digital library an online access to it is set as public so you can download it instantly. Our book servers spans in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the fourier series in control theory is universally compatible with any devices to read.

A unified solution for the vibration analysis of FGM doubly-curved shells of revolution with arbitrary boundary conditions

Abstract: This paper describes a unified solution for the vibration analysis of functionally graded material (FGM) doubly-curved shells of revolution with arbitrary boundary conditions. The solution is derived by means of the modified Fourier series method on the basis of the first order shear deformation shell theory considering the effects of the deepness terms. The material properties of the shells are assumed to vary continuously and smoothly along the normal direction according to general three-parameter power-law volume fraction functions. In summary, the energy functional of the shells is expressed as a function of five displacement components firstly. Then, each of the displacement components is expanded as a modified Fourier series. Finally, the solutions are obtained by using the variational operation. The convergence and accuracy of the solution are validated by comparing its results with those available in the literature. A variety of new vibration results for the circular toroidal, paraboloidal, hyperbolical, catenary, cycloidal and elliptical shells with classical and elastic boundary conditions as well as different geometric and material parameters are presented, which may serve as benchmark solution for future researches. Furthermore, the effects of the boundary conditions, shell geometric and material parameters on the frequencies are carried out.

Vibration analysis of coupled conical-cylindrical-spherical shells using a Fourier spectral element method.

Abstract: This paper presents a Fourier spectral element method (FSEM) to analyze the free vibration of conical-cylindrical-spherical shells with arbitrary boundary conditions. Cylindrical-conical and cylindrical-spherical shells as special cases are also considered. In this method, each fundamental shell component (i.e., cylindrical, conical, and spherical shells) is divided into appropriate elements. The variational principle in conjunction with first-order shear deformation shell theory is employed to model the shell elements. Since the displacement and rotation components of each element are expressed as a linear superposition of nodeless Fourier sine functions and nodal Lagrangian polynomials, the global equations of the coupled shell structure can be obtained by adopting the assembly procedure. The Fourier sine series in the displacement field is introduced to enhance the accuracy and convergence of the solution. Numerical results show that the FSEM can be effectively applied to vibration analysis of the coupled shell structures. Numerous results for coupled shell structures with general boundary conditions are presented. Furthermore, the effects of geometric parameters and boundary conditions on the frequencies are investigated.

Analytical determination of orbital elements using Fourier analysis - I. The radial velocity case

Abstract: We describe an analytical method for computing the orbital parameters of a planet from the periodogram of a radial velocity signal. The method is very efficient and provides a good approximation of the orbital parameters. The accuracy is mainly limited by the accuracy of the computation of the Fourier decomposition of the signal which is sensitive to sampling and noise. Our method is complementary with more accurate (and more expensive in computer time) numerical algorithms (e.g. Levenberg-Marquardt, Markov chain Monte Carlo, genetic algorithms). Indeed, the analytical approximation can be used as an initial condition to accelerate the convergence of these numerical methods. Our method can be applied iteratively to search for multiple planets in the same system.

On Fast Bilateral Filtering Using Fourier Kernels

Abstract: It was demonstrated in earlier work that, by approximating its range kernel using shiftable functions, the nonlinear bilateral filter can be computed using a series of fast convolutions. Previous approaches based on shiftable approximation have, however, been restricted to Gaussian range kernels. In this work, we propose a novel approximation that can be applied to any range kernel, provided it has a pointwise-convergent Fourier series. More specifically, we propose to approximate the Gaussian range kernel of the bilateral filter using a Fourier basis, where the coefficients of the basis are obtained by solving a series of least-squares problems. The coefficients can be efficiently computed using a recursive form of the QR decomposition. By controlling the cardinality of the Fourier basis, we can obtain a good tradeoff between the run-time and the filtering accuracy. In particular, we are able to guarantee subpixel accuracy for the overall filtering, which is not provided by the most existing methods for fast bilateral filtering. We present simulation results to demonstrate the speed and accuracy of the proposed algorithm.

A general Fourier formulation for vibration analysis of functionally graded sandwich beams with arbitrary boundary condition and resting on elastic foundations

Abstract: Free vibration analysis of functionally graded sandwich beams with general boundary conditions and resting on a Pasternak elastic foundation is presented by using strong form formulation based on modified Fourier series. Two types of common sandwich beams, namely beams with functionally graded face sheets and isotropic core and beams with isotropic face sheets and functionally graded core, are considered. The bilayered and single-layered functionally graded beams are obtained as special cases of sandwich beams. The effective material properties of functionally graded materials are assumed to vary continuously in the thickness direction according to power-law distributions in terms of volume fraction of constituents and are estimated by Voigt model and Mori–Tanaka scheme. Based on the first-order shear deformation theory, the governing equations and boundary conditions can be obtained by Hamilton’s principle and can be solved using the modified Fourier series method which consists of the standard Fourier cosine series and several supplemented functions. A variety of numerical examples are presented to demonstrate the convergence, reliability and accuracy of the present method. Numerous new vibration results for functionally graded sandwich beams with general boundary conditions and resting on elastic foundations are given. The influence of the power-law indices and foundation parameters on the frequencies of the sandwich beams is also investigated.

Vibration analysis and transient response of a functionally graded piezoelectric curved beam with general boundary conditions

Abstract: The paper presents a unified solution for free and transient vibration analyses of a functionally graded piezoelectric curved beam with general boundary conditions within the framework of Timoshenko beam theory. The formulation is derived by means of the variational principle in conjunction with a modified Fourier series which consists of standard Fourier cosine series and supplemented functions. The mechanical and electrical properties of functionally graded piezoelectric materials (FGPMs) are assumed to vary continuously in the thickness direction and are estimated by Voigt's rule of mixture. The convergence, accuracy and reliability of the present formulation are demonstrated by comparing the present solutions with those from the literature and finite element analysis. Numerous results for FGPM beams with different boundary conditions, geometrical parameters as well as material distributions are given. Moreover, forced vibration of the FGPM beams subjected to dynamic loads and general boundary conditions are also investigated.

The Henry problem: New semianalytical solution for velocity-dependent dispersion

Abstract: A new semi-analytical solution is developed for the velocity-dependent dispersion Henry problem using the Fourier-Galerkin method (FG). The integral arising from the velocity-dependent dispersion term is evaluated numerically using an accurate technique based on an adaptive scheme. Numerical integration and nonlinear dependence of the dispersion on the velocity render the semi-analytical solution impractical. To alleviate this issue, and to obtain the solution at affordable computational cost, a robust implementation for solving the nonlinear system arising from the FG method is developed. It allows for reducing the number of attempts of the iterative procedure and the computational cost by iteration. The accuracy of the semi-analytical solution is assessed in terms of the truncation orders of the Fourier series. An appropriate algorithm based on the sensitivity of the solution to the number of Fourier modes is used to obtain the required truncation levels. The resulting Fourier series are used to analytically evaluate the position of the principal isochlors and metrics characterizing the saltwater wedge. They are also used to calculate longitudinal and transverse dispersive fluxes and to provide physical insight into the dispersion mechanisms within the mixing zone. The developed semi-analytical solutions are compared against numerical solutions obtained using an in house code based on variant techniques for both space and time discretization. The comparison provides better confidence on the accuracy of both numerical and semi-analytical results. It shows that the new solutions are highly sensitive to the approximation techniques used in the numerical code which highlights their benefits for code benchmarking. This article is protected by copyright. All rights reserved.

The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory

Abstract: We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined with a classical criterion due to Jain and Monrad [Ann. Probab. 11 (1983) 46–57]. This condition is verified in many examples, even in absence of explicit expressions for the covariance or Volterra kernels. Of special interest are random Fourier series, with covariance given as Fourier series itself, and we formulate conditions directly in terms of the Fourier coefficients. We also establish convergence and rates of convergence in rough path metrics of approximations to such random Fourier series. An application to SPDE is given. Our criterion also leads to an embedding result for Cameron–Martin paths and complementary Young regularity (CYR) of the Cameron–Martin space and Gaussian sample paths. CYR is known to imply Malliavin regularity and also Ito-like probabilistic estimates for stochastic integrals (resp., stochastic differential equations) despite their (rough) pathwise construction. At last, we give an application in the context of non-Markovian Hormander theory.

An improved Fourier series solution for the dynamic analysis of laminated composite annular, circular, and sector plate with general boundary conditions

Abstract: In this article, the authors presented a unified solution for the dynamic analysis of laminated composite annular, circular, and sector plate with general boundary conditions. The first-order shear deformation theory is employed to formulate the theoretical model. Regardless of the shapes of the plates and the types of boundary conditions, each displacement and rotation component of the elements is expanded as an improved Fourier series expansion which is composed of a double Fourier cosine series and several auxiliary functions introduced to eliminate all the relevant discontinuities with the displacement and its derivatives at the boundaries and to accelerate the convergence of series representations. Since the displacement fields are constructed adequately smooth throughout the entire solution domain, an exact solution is obtained based on the Rayleigh–Ritz procedure by the energy functions of the plates. The accuracy, reliability, and versatility of the current solution is fully demonstrated and verif...