Showing papers on "Fourier series published in 2015"

Periodograms for Multiband Astronomical Time Series

Abstract: This paper introduces the multiband periodogram, a general extension of the well-known Lomb-Scargle approach for detecting periodic signals in time-domain data In addition to advantages of the Lomb-Scargle method such as treatment of non-uniform sampling and heteroscedastic errors, the multiband periodogram significantly improves period finding for randomly sampled multiband light curves (eg, Pan-STARRS, DES and LSST) The light curves in each band are modeled as arbitrary truncated Fourier series, with the period and phase shared across all bands The key aspect is the use of Tikhonov regularization which drives most of the variability into the so-called base model common to all bands, while fits for individual bands describe residuals relative to the base model and typically require lower-order Fourier series This decrease in the effective model complexity is the main reason for improved performance We use simulated light curves and randomly subsampled SDSS Stripe 82 data to demonstrate the superiority of this method compared to other methods from the literature, and find that this method will be able to efficiently determine the correct period in the majority of LSST's bright RR Lyrae stars with as little as six months of LSST data A Python implementation of this method, along with code to fully reproduce the results reported here, is available on GitHub

Power System Harmonics

Abstract: The French mathematician J. B. J. Fourier showed that arbitrary periodic functions could be represented by an infinite series of sinusoids of harmonically related frequencies. This chapter first defines periodic functions and orthogonal functions. A periodic function can be expanded in a Fourier series. The Fourier series of a periodic function is the sum of sinusoidal components of different frequencies. The chapter then illustrates the functions of odd or skew symmetry, even symmetry and half-wave symmetry. The odd and even symmetry has been obtained with the triangular function by shifting the origin. Fourier analysis of a continuous periodic signal in the time domain gives a series of discrete frequency components in the frequency domain. The chapter describes Dirichlet conditions and notion of power spectrum. Finally, it explains the function of convolution, which is generally carried out in the frequency domain.

Algebraic twists of modular forms and Hecke orbits

Abstract: We consider the question of the correlation of Fourier coefficients of modular forms with functions of algebraic origin. We establish the absence of correlation in considerable generality (with a power saving of Burgess type) and a corresponding equidistribution property for twisted Hecke orbits. This is done by exploiting the amplification method and the Riemann Hypothesis over finite fields, relying in particular on the l-adic Fourier transform introduced by Deligne and studied by Katz and Laumon.

Static and free vibration analyses of carbon nanotube-reinforced composite plate using differential quadrature method

Abstract: This work presents bending and free vibration behaviour of carbon nanotubes reinforced composite (CNTRC) plates using the three dimensional theory of elasticity. The single-walled carbon nanotubes reinforcement is either uniformly distributed or functionally graded (FG) along the thickness direction indicated with FG-V, FG-O and FG-X. In the present study the effective material properties of CNTRC plates, are estimated according to the rule of mixture along with considering the CNT efficiency parameters. For the plate with simply supported edges we used Fourier series expansion across the in plane coordinates as well as the state space technique across the thickness direction to obtain closed form solution. Since in the case of plate with non-simply supported boundary conditions it is not possible to use Fourier series along the longitudinal and width directions, therefore it should be employed numerical method along the above mentioned coordinates. In this investigation we used semi analytical technique, differential quadrature method along the in-plane coordinates and state-space technique across the thickness direction. Present approach is validated by comparing the numerical results with those published results. Furthermore, effect of types of CNT distributions in the polymer matrix, volume fraction of CNT, edges boundary conditions and width-to-thickness ratio on the bending and free vibration behaviour of FG-CNTRC plate are discussed.

The recoverability limit for superresolution via sparsity.

Abstract: We consider the problem of robustly recovering a $k$-sparse coefficient vector from the Fourier series that it generates, restricted to the interval $[- \Omega, \Omega]$. The difficulty of this problem is linked to the superresolution factor SRF, equal to the ratio of the Rayleigh length (inverse of $\Omega$) by the spacing of the grid supporting the sparse vector. In the presence of additive deterministic noise of norm $\sigma$, we show upper and lower bounds on the minimax error rate that both scale like $(SRF)^{2k-1} \sigma$, providing a partial answer to a question posed by Donoho in 1992. The scaling arises from comparing the noise level to a restricted isometry constant at sparsity $2k$, or equivalently from comparing $2k$ to the so-called $\sigma$-spark of the Fourier system. The proof involves new bounds on the singular values of restricted Fourier matrices, obtained in part from old techniques in complex analysis.

The Fourier Decomposition Method for nonlinear and nonstationary time series analysis

Abstract: Since many decades, there is a general perception in literature that the Fourier methods are not suitable for the analysis of nonlinear and nonstationary data In this paper, we propose a Fourier Decomposition Method (FDM) and demonstrate its efficacy for the analysis of nonlinear (ie data generated by nonlinear systems) and nonstationary time series The proposed FDM decomposes any data into a small number of `Fourier intrinsic band functions' (FIBFs) The FDM presents a generalized Fourier expansion with variable amplitudes and frequencies of a time series by the Fourier method itself We propose an idea of zero-phase filter bank based multivariate FDM (MFDM) algorithm, for the analysis of multivariate nonlinear and nonstationary time series, from the FDM We also present an algorithm to obtain cutoff frequencies for MFDM The MFDM algorithm is generating finite number of band limited multivariate FIBFs (MFIBFs) The MFDM preserves some intrinsic physical properties of the multivariate data, such as scale alignment, trend and instantaneous frequency The proposed methods produce the results in a time-frequency-energy distribution that reveal the intrinsic structures of a data Simulations have been carried out and comparison is made with the Empirical Mode Decomposition (EMD) methods in the analysis of various simulated as well as real life time series, and results show that the proposed methods are powerful tools for analyzing and obtaining the time-frequency-energy representation of any data

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 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 which 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 super-resolution recovery of MRI phantoms and real MRI data from low-pass Fourier samples, which shows benefits over standard approaches for single-image super-resolution MRI.

A series solution for the in-plane vibration analysis of orthotropic rectangular plates with non-uniform elastic boundary constraints and internal line supports

Abstract: In this investigation, the free in-plane vibration analysis of orthotropic rectangular plates with non-uniform boundary conditions and internal line supports is performed with a modified Fourier solution. The exact solution for the problem is obtained using improved Fourier series method, in which both two in-plane displacements of the orthotropic rectangular plates are represented by a double Fourier cosine series and four supplementary functions, in the form of the product of a polynomial function and a single cosine series expansion, introduced to remove the potential discontinuities associated with the original displacement functions along the edges when they are viewed as periodic functions defined over the entire x–y plane. The unknown expansion coefficients are treated as the generalized coordinates and determined using the Rayleigh–Ritz procedure. The change of the boundary conditions can be easily achieved by only varying the stiffness of the two sets of the boundary springs at the all boundary of the orthotropic rectangular plates without the need of making any change to the solutions. The excellent accuracy of the current result is validated by comparison with those obtained from other analytical approach as well as the finite element method (FEM). Numerical results are presented to illustrate the current method that is applied not only to the homogeneous boundary conditions but also to other interesting and practically important boundary restraints on free in-plane vibrations of the orthotropic rectangular plates, including varying stiffness of boundary springs, point supported, partially supported boundary conditions and internal line supports. In addition to this, the effects of locations of line supports are also investigated and reported. New results for free vibration of moderately orthotropic rectangular plates with various edge conditions and internal line supports are presented, which may be used for benchmarking of researchers in the field.

Interpreting azimuthal Fourier coefficients for anisotropic and fracture parameters

Abstract: Amplitude variation with offset and azimuth (AVOAz) analysis can be separated into two separate parts: amplitude variation with offset (AVO) analysis and amplitude variation with azimuth (AVAz) analysis. Useful information about fractures and anisotropy can be obtained just by examining the AVAz. The AVAz can be described as a sum of sinusoids of different periodicities, each characterized by its magnitude and phase. This sum is mathematically equivalent to a Fourier series, and hence the coefficients describing the AVAz response are azimuthal Fourier coefficients (FCs). This FC parameterization is purely descriptive. The aim of this paper is to help the interpreter understand what these coefficients mean in terms of anisotropic and fracture parameters for the case of P-wave reflectivity using a linearized approximation. The FC representation is valid for general anisotropy. However, to gain insight into the significance of FCs, more restrictive assumptions about the anisotropy or facture system m...

Fourier method for solving the multi- frequency inverse source problem for the Helmholtz equation

Abstract: We consider an inverse source problem for the Helmholtz equation. This is concerned with the reconstruction of an unknown source from multi-frequency data obtained from the radiated fields. Based on a Fourier expansion of the source, a numerical method is proposed to solve the inverse problem. Stability is analyzed and numerical experiments are presented to show the effectiveness of our method.

Single-Carrier OFDMA

Abstract: An OFDM transmitter comprises a Discrete Fourier Transform (DFT) spreader configured to spread reference-signal symbols with Fourier coefficients to generate DFT-spread reference symbols, which are mapped to OFDM subcarriers. An OFDM modulator, such as an inverse-DFT, modulates the DFT-spread reference symbols onto the OFDM subcarriers to produce an OFDM transmission signal with low peak-to-average power.

A Generalized Spatial Correlation Model for 3D MIMO Channels Based on the Fourier Coefficients of Power Spectrums

Abstract: Previous studies have confirmed the adverse impact of fading correlation on the mutual information (MI) of two-dimensional (2D) multiple-input multiple-output (MIMO) systems. More recently, the trend is to enhance the system performance by exploiting the channels degrees of freedom in the elevation, which necessitates the derivation and characterization of three-dimensional (3D) channels in the presence of spatial correlation. In this paper, an exact closed-form expression for the Spatial Correlation Function (SCF) is derived for 3D MIMO channels. The proposed method resorts to the spherical harmonic expansion (SHE) of plane waves and the trigonometric expansion of Legendre and associated Legendre polynomials. The resulting expression depends on the underlying arbitrary angular distributions and antenna patterns through the Fourier Series (FS) coefficients of power azimuth and elevation spectrums. The novelty of the proposed method lies in the SCF being valid for any 3D propagation environment. The developed SCF determines the covariance matrices at the transmitter and the receiver that form the Kronecker channel model. In order to quantify the effects of correlation on system performance, the information-theoretic deterministic equivalents of the MI for the Kronecker model are utilized in both mono-user and multi-user cases. Numerical results validate the proposed analytical expressions and elucidate the dependence of the system performance on azimuth and elevation angular spreads and antenna patterns. Some useful insights into the behavior of MI as a function of downtilt angles are provided. The derived model will help evaluate the performance of correlated 3D MIMO channels in the future.

Free vibration of four-parameter functionally graded spherical and parabolic shells of revolution with arbitrary boundary conditions

Abstract: The objective of this work is to present a Haar Wavelet Discretization (HWD) method-based solution approach for the free vibration analysis of functionally graded (FG) spherical and parabolic shells of revolution with arbitrary boundary conditions. The first-order shear deformation theory is adopted to account for the transverse shear effect and rotary inertia of the shell structures. Haar wavelet and their integral and Fourier series are selected as the basis functions for the variables and their derivatives in the meridional and circumferential directions, respectively. The constants appearing in the integrating process are determined by boundary conditions, and thus the equations of motion as well as the boundary condition equations are transformed into a set of algebraic equations. The proposed approach directly deals with nodal values and does not require special formula for evaluating system matrices. Also, the convenience of the approach is shown in handling general boundary conditions. Numerical examples are given for the free vibrations of FG shells with different combinations of classical and elastic boundary conditions. Effects of spring stiffness values and the material power-law distributions on the natural frequencies of shells are also discussed. Some new results for the considered shell structures are presented, which may serve as benchmark solutions.

Improving the Gaussian Process Sparse Spectrum Approximation by Representing Uncertainty in Frequency Inputs

Abstract: Standard sparse pseudo-input approximations to the Gaussian process (GP) cannot handle complex functions well. Sparse spectrum alternatives attempt to answer this but are known to over-fit. We suggest the use of variational inference for the sparse spectrum approximation to avoid both issues. We model the covariance function with a finite Fourier series approximation and treat it as a random variable. The random covariance function has a posterior, on which a variational distribution is placed. The variational distribution transforms the random covariance function to fit the data. We study the properties of our approximate inference, compare it to alternative ones, and extend it to the distributed and stochastic domains. Our approximation captures complex functions better than standard approaches and avoids over-fitting.

Static and free vibration analyses of functionally graded sandwich plates using state space differential quadrature method

Abstract: This study presents static and free vibration behaviors of two type of sandwich plates based on the three dimensional theory of elasticity. The core layer of one type is functionally graded material (FGM) with the isotropic face sheets whereas in second type, the core layer is isotropic with the face sheets FGM. The effective material properties of FGM layers are estimated to vary continuously through the thickness direction according to a power-law distribution of the volume fractions of the constituents. By using differential equilibrium equations and/or equations of motion as well as constitutive relations, state-space differential equation can be derived. In the case of simply supported condition, applying Fourier series to the quantities along the in-plane coordinates, governing equation can be solved analytically and for the other edges condition, a semi analytical solution can be obtained by using differential quadrature method (DQM) along the in-plane coordinate as well as state spaces technique in the thickness direction. Accuracy and exactness of the present approach is validated by comparing the numerical results with the results of published literature. Moreover, the influences of volume fraction, width-to-thickness ratios and aspect ratio on the vibration and static behaviors of plate are investigated.

Exact Analytical Solution for Large-Amplitude Oscillatory Shear Flow

Abstract: When polymeric liquids undergo large-amplitude shearing oscillations, the shear stress responds as a Fourier series, the higher harmonics of which are caused by fluid nonlinearity. Previous work on large-amplitude oscillatory shear flow has produced analytical solutions for the first few harmonics of a Fourier series for the shear stress response (none beyond the fifth) or for the normal stress difference responses (none beyond the fourth) [JNNFM, 166, 1081 (2011)], but this growing subdiscipline of macromolecular physics has yet to produce an exact solution. Here, we derive what we believe to be the first exact analytical solution for the response of the extra stress tensor in large-amplitude oscillatory shear flow. Our solution, unique and in closed form, includes both the normal stress differences and the shear stress for both startup and alternance. We solve the corotational Maxwell model as a pair of nonlinear-coupled ordinary differential equations, simultaneously. We choose the corotational Maxwell model because this two-parameter model (η0 and λ) is the simplest constitutive model relevant to large-amplitude oscillatory shear flow, and because it has previously been found to be accurate for molten plastics (when multiple relaxation times are used). By relevant we mean that the model predicts higher harmonics. We find good agreement between the first few harmonics of our exact solution, and of our previous approximate expressions (obtained using the Goddard integral transform). Our exact solution agrees closely with the measured behavior for molten plastics, not only at alternance, but also in startup.

Extension of Chebfun to periodic functions

Abstract: Algorithms and underlying mathematics are presented for numerical computation with periodic functions via approximations to machine precision by trigonometric polynomials, including the solution of linear and nonlinear periodic ordinary differential equations. Differences from the nonperiodic Chebyshev case are highlighted.

Real Analysis and Applications: Theory in Practice

Abstract: Analysis.- Review.- The Real Numbers.- Series.- Topology of.- Functions.- Differentiation and Integration.- Norms and Inner Products.- Limits of Functions.- Metric Spaces.- Applications.- Approximation by Polynomials.- Discrete Dynamical Systems.- Differential Equations.- Fourier Series and Physics.- Fourier Series and Approximation.- Wavelets.- Convexity and Optimization.

Fast Initial Trajectory Design for Low-Thrust Restricted-Three-Body Problems

Abstract: This paper presents an approach to generate initial trajectories in a three-body dynamical framework, assuming the use of a low-thrust propulsion system. Finite Fourier series were previously implemented successfully in approximating two-dimensional continuous-thrust trajectories in two-body dynamic models. In this paper, Finite Fourier series are implemented in a dual-level-solver strategy for low-thrust trajectory approximation, in the presence of thrust acceleration constraints, in the three-body dynamic model. This approximation enables the feasibility assessment of low-thrust trajectories, especially in the presence of thrust level constraint. The developed method demonstrates capability in generating trajectories using a fully automated procedure for various levels of thrust acceleration with a moderate to high number of revolutions. The suitability of using the approximated trajectories as initial guesses for high-fidelity solvers is demonstrated.

Structural vibration : a uniform accurate solution for laminated beams, plates and shells with general boundary conditions

Abstract: Fundamental equations of laminated beams, plates and shells.- The modified Fourier series and Ritz method.- Straight and curved beams.- Plate structures.- Closed and deep open cylindrical shells.- Closed and deep open conical shells.- Closed and deep open spherical shells.- Doubly-curved shallow shells.

Fourier Transformation for Pedestrians

Abstract: Introduction.- Fourier Series.- Continuous Fourier Transformation.- Window Functions.- Discrete Fourier Transformation.- Filter Effect in Digital Data Processing.- Data Streams and Fractional Delay.- Tomography: Back projection of Filtered Projections.

Traveling waves and their tails in locally resonant granular systems

Abstract: In the present study, we revisit the theme of wave propagation in locally resonant granular crystal systems, also referred to as mass-in-mass systems. We use three distinct approaches to identify relevant traveling waves. The first consists of a direct solution of the traveling wave problem. The second one consists of the solution of the Fourier tranformed variant of the problem, or, more precisely, of its convolution reformulation (upon an inverse Fourier transform) in real space. Finally, our third approach will restrict considerations to a finite domain, utilizing the notion of Fourier series for important technical reasons, namely the avoidance of resonances, which will be discussed in detail. All three approaches can be utilized in either the displacement or the strain formulation. Typical resulting computations in finite domains result in the solitary waves bearing symmetric non-vanishing tails at both ends of the computational domain. Importantly, however, a countably infinite set of anti-resonance conditions is identified for which solutions with genuinely rapidly decaying tails arise.

On the variation of Fourier parameters for Galactic and LMC Cepheids at optical, near-infrared and mid-infrared wavelengths

Abstract: We present a light curve analysis of fundamental-mode Galactic and Large Magellanic Cloud (LMC) Cepheids based on the Fourier decomposition technique. We have compiled light curve data for Galactic and LMC Cepheids in optical ({\it VI}), near-infrared ({\it JHK}$_s$) and mid-infrared (3.6 $\&$ 4.5-$\mu$m) bands from the literature and determined the variation of their Fourier parameters as a function of period and wavelength. We observed a decrease in Fourier amplitude parameters and an increase in Fourier phase parameters with increasing wavelengths at a given period. We also found a decrease in the skewness and acuteness parameters as a function of wavelength at a fixed period. We applied a binning method to analyze the progression of the mean Fourier parameters with period and wavelength. We found that for periods longer than about 20 days, the values of the Fourier amplitude parameters increase sharply for shorter wavelengths as compared to wavelengths longer than the $J$-band. We observed the variation of the Hertzsprung progression with wavelength. The central period of the Hertzsprung progression was found to increase with wavelength in the case of the Fourier amplitude parameters and decrease with increasing wavelength in the case of phase parameters. We also observed a small variation of the central period of the progression between the Galaxy and LMC, presumably related to metallicity effects. These results will provide useful constraints for stellar pulsation codes that incorporate stellar atmosphere models to produce Cepheid light curves in various bands.