Showing papers on "Fourier series published in 1989"

Convergence of spectral methods for nonlinear conservation laws

Abstract: The convergence of the Fourier method for scalar nonlinear conservation laws which exhibit spontaneous shock discontinuities is discussed. Numerical tests indicate that the convergence may (and in fact in some cases must) fail, with or without post-processing of the numerical solution. Instead, a new kind of spectrally accurate vanishing viscosity is introduced to augment the Fourier approximation of such nonlinear conservation laws. Using compensated compactness arguments, it is shown that this spectral viscosity prevents oscillations, and convergence to the unique entropy solution follows.

A robust orthogonal algorithm for system identification and time-series analysis

Abstract: We describe and illustrate methods for obtaining a parsimonious sinusoidal series representation or model of biological time-series data. The methods are also used to identify nonlinear systems with unknown structure. A key aspect is a rapid search for significant terms to include in the model for the system or the time-series. For example, the methods use fast and robust orthogonal searches for significant frequencies in the time-series, and differ from conventional Fourier series analysis in several important respects. In particular, the frequencies in our resulting sinusoidal series need not be commensurate, nor integral multiples of the fundamental frequency corresponding to the record length. Freed of these restrictions, the methods produce a more economical sinusoidal series representation (than a Fourier series), finding the most significant frequencies first, and automatically determine model order. The methods are also capable of higher resolution than a conventional Fourier series analysis. In addition, the methods can cope with unequally-spaced or missing data, and are applicable to time-series corrupted by noise. Fially, we compare one of our methods with a wellknown technique for resolving sinusoidal signals in noise using published data for the test time-series.

Adaptive Fourier estimation of time-varying evoked potentials

Abstract: In this estimation procedure the evoked response is modeled as a dynamic Fourier series and the Fourier coefficients are estimated adaptively by the least-mean-square algorithm. Approximate expressions have been developed for the estimation error and time constant of adaptation. A procedure for optimizing the estimator performance is also presented. The effectiveness of the estimator with simulated as well as actual evoked responses is demonstrated. >

Approximation of infinite-dimensional systems

Abstract: A Fourier series-based method for approximation of stable infinite-dimensional linear time-invariant system models is discussed. The basic idea is to compute the Fourier series coefficients of the associated transfer function T/sub d/(Z) and then take a high-order partial sum. Two results on H/sup infinity / convergence and associated error bounds of the partial sum approximation are established. It is shown that the Fourier coefficients can be replaced by the discrete Fourier transform coefficients while maintaining H/sup infinity / convergence. Thus, a fast Fourier transform algorithm can be used to compute the high-order approximation. This high-order finite-dimensional approximation can then be reduced using balanced truncation or optimal Hankel approximation leading to the final finite-dimensional approximation to the original infinite-dimensional model. This model has been tested on several transfer functions of the time-delay type with promising results. >

Fourier series expansion of the transfer equation in the atmosphere-ocean system

Abstract: We consider radiative transfer in a plane-parallel atmosphere bounded by a rough ocean surface. The problem is solved by using a Fourier series decomposition of the radiation field. For the case of a Lambertian surface as a boundary condition, this decomposition is classically achieved by developing the scattering phase matrix in a series of Legendre functions. For the case of a rough ocean surface, we obtain the decomposition by developing both the Fresnel reflection matrix and the wave facet distribution function in Fourier series. This procedure allows us to derive the radiance field for the case of the ruffled ocean surface, with a computation time only a few percent larger than for the case of a Lambertian surface.

Certain computational aspects of vector diffraction problems

Abstract: Fourier decomposition of a given amplitude distribution into plane waves and the subsequent superposition of these waves after propagation is a powerful yet simple approach to diffraction problems. Many vector diffraction problems can be formulated in this way, and the classical results are usually the consequence of a stationary-phase approximation to the resulting integrals. For situations in which the approximation does not apply, a factorization technique is developed that substantially reduces the required computational resources. Numerical computations are based on the fast-Fourier-transform algorithm, and the practicality of this method is shown with several examples.

Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis

Abstract: A method for calculating the modes of arbitrarily shaped dielectric waveguides is presented. It consists of expanding the field in a two-dimensional Fourier series. The expansion is used to convert the scalar wave equation into a matrix eigenvalue equation. To facilitate calculation of the matrix elements, the waveguide geometry is approximated by a number of rectangles of constant refractive index. The accuracy of the method is demonstrated by calculating the dominant mode of a circular optical fiber and comparing it with the exact solution. The method well-fitted the experimental data on dielectric film waveguides on silicon, including data on waveguide-to-fiber butt-coupling loss, waveguide far-field angles, and the coupling length of directional couplers. >

The Rankin–Selberg Method: A Survey

Abstract: Publisher Summary This chapter presents a survey of the Rankin-Selberg method. The earliest applications of the Rankin-Selberg method were to the estimation of the Fourier coefficients of modular forms. The Rankin-Selberg method gives an improvement over the “trivial estimate”.. Hecke introduced operators on cusp forms for each p that play a role similar to the Laplacian. These operators are self-adjoint and mutually commutative, hence might be simultaneously diagonalized. Rankin and, independently at around the same time, Selberg introduced a new tool into the study of cusp forms, which is known as the Rankin-Selberg method. This gives the functional equation of a new kind of Euler product, and gives new estimates for the Fourier coefficients.

The synthesis of pulse sequences yielding arbitrary magnetization vectors.

Abstract: A new procedure and algorithm are presented to allow the synthesis of a pulse sequence which will generate an arbitrary frequency-dependent spin excitation. This procedure is a generalization of our previous paper, where this was done subject to the restriction that the spin excitation was symmetric about zero offset frequency, and pulses were restricted to being about a fixed axis. The required final z-magnetization vector (Mz) is expressed as a function of the off-resonance frequency as an Nth order complex Fourier series. We then form a consistent Fourier series for (Mxy). As many as 2 2N different pulse sequences may be directly generated all of which produce a different Mxy(f), but the same Mz(f). A pulse sequence is then generated which will yield the desired Mz(f) and Mxy(f). This is done by an analytic inversion of the Bloch equation, not by the classical Fourier approximation. This technique enables us to generate any Mz which is potentially realizable by a pulse sequence. © 1989 Academic Press, Inc.

Analysis of the spectral vanishing viscosity method for periodic conservation laws

Abstract: The convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid Burgers’ equation is analyzed. It is proved that this kind of vanishing viscosity is responsible for spectral decay of those Fourier coefficients located toward the end of the computed spectrum; consequently, the discretization error is shown to be spectrally small, independently of whether or not the underlying solution is smooth. This in turn implies that the numerical solution remains uniformly bounded and convergence follows by compensated compactness arguments.

Nonparametric time series analysis for periodically correlated processes

Abstract: Correlation functions of continuous-time periodically correlated processes can be represented by a Fourier series with coefficient functions. It is shown that the usual estimator for stationary covariances, formed from a single sample path of the process, can be simply modified to provide a consistent (in quadratic mean) estimator for any of the coefficient functions resulting from the aforementioned representation. It is shown that, if the process is Gaussian and B/sub k/( tau ) is a Fourier integral with respect to a density function g/sub k/( lambda ), a two-dimensional periodogram, formed from a single sample function, can be smoothed along a line of constant difference frequency to provide a consistent estimator for g/sub k/( lambda ). This natural extension of the well-known procedure for stationary processes provides a method for nonparametric spectral analysis of periodically correlated processes. >

A central limit theorem of fourier transforms of strongly dependent stationary processes

Abstract: . We consider a limiting distribution of the finite Fourier transforms of observations drawn from a strongly dependent stationary process. It is proved that the finite Fourier transforms at different frequencies are asymptotically independent and normally distributed. Our result can apply to a fractional autoregressive integrated moving-average process and a fractional Gaussian noise, two examples of strongly dependent stationary processes.

Method for analyzing complex two-dimensional forms: Elliptical Fourier functions.

Abstract: A generalized procedure, elliptical Fourier analysis, for accurately characterizing the shape of complex morphological forms of the type commonly encountered in the biological sciences, is described. Elliptical Fourier functions are derived as a parametric formulation from conventional Fourier analysis, i.e., as a pair of equations that are functions of a third variable. The use of elliptical Fourier functions circumvents three restrictions that have limited conventional Fourier analysis to certain classes of shapes. These restrictions are (1) equal divisions over the interval or period; (2) dependency on the coordinate system, i.e., conventional Fourier functions are not "coordinate free"; and (3) the presence of shapes with outlines that curve back on themselves, a common occurrence. These three limitations are effectively removed with the utilization of elliptical Fourier functions, facilitating the analysis of a much larger class of two-dimensional forms.

Essentially nonoscillatory spectral Fourier methods for shock wave calculations

Abstract: In this paper, we present an essentially nonoscillatory spectral Fourier method for the solution of hyperbolic partial differential equations. The method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the Fourier method. The high accuracy away from the shock is enhanced by using filters. Numerical results confirm that essentially no oscillations develop in the solution. Also, the accuracy of the spectral solution of the inviscid Burgers equation is shown to be higher than a fixed order.

The determination of crystal size and disorder from X-ray diffraction photographs of polymer fibres. 1. The accuracy of determination of Fourier coefficients of the intensity profile of a reflection

Abstract: Methods which determine the number and disorder of lattice planes in a crystal from the Fourier cosine coefficients of the intensity profile of an X-ray reflection use only the low harmonics and require that the coefficients be normalized so that the zero harmonic is unity. Experimentally, the profiles can only be recorded over a smaller range of scattering angle than required by the theory, and it is necessary to subtract background, which is likely to be estimated with considerable error, before determining the coefficients. It is shown that with polymer fibres this causes serious errors in the normalization, and in the values of those low harmonics used in the size and disorder determination, and prevents reliable values being obtained. Methods which avoid normalization and use only high harmonics are needed. It is shown that disorder may be obtained in such a way, but not size, for which low-order normalized coefficients are essential. A method of extrapolation is described and tested which enables the accurate high harmonics to be used to improve the estimates of the low ones. Whilst this will yield more reliable values of crystal size than are obtainable from existing methods, the accuracy depends entirely on the validity of the extrapolation, which cannot be tested in many cases of interest.

On optimal global rate of convergence of some nonparametric identification procedures

Abstract: A study of orthogonal series procedures for function fitting, motivated by an interest in nonparametric identification of linear dynamic systems, is reported It is shown that the procedures attain the optimal rate of convergence for the Fourier and the Walsh orthonormal systems These rates cannot be exceeded by any method of estimation This is the first analytical result giving a clear answer to the question of which orthonormal system is the best one for the purpose of system identification >

Applications of fast orthogonal search: time-series analysis and resolution of signals in noise.

Abstract: In this paper a technique is examined for obtaining accurate and parsimonious sinusoidal series representations of biological time-series data, and for resolving sinusoidal signals in noise. The technique operates via a fast orthogonal search method discussed in the paper, and achieves economy of representation by finding the most significant sinusoidal frequencies first, in a least squares fit sense. Another reason for the parsimony in representation is that the identified sinusoidal series model is not restricted to frequencies which are commensurate or integral multiples of the fundamental frequency corresponding to the record length. Biological applications relate to spectral analysis of noisy time-series data such as EEG, ECG, EMG, EOG, and to speech analysis. Simulations are provided to demonstrate precise detection of component frequencies and weights in short data records, coping with missing or unequally spaced data, and recovery of signals heavily contaminated with noise. The technique is also shown to be capable of higher frequency resolution than is achievable by conventional Fourier series analysis.

An introduction to the numerical analysis of spectral methods

Abstract: This is a very lucid introduction to spectral methods emphasizing the mathematical aspects of the theory rather than the many applications in numerical analysis and the engineering sciences. The first part is a fairly complete introduction to Fourier series while the second emphasizes polynomial expansion methods like Chebyshev's. The author gives rigorous proofs of fundamental results related to one-dimensional advection and diffusions equations. The book addresses students as well as practitioners of numerical analysis.

Identification of discrete Hammerstein systems by the Fourier series regression estimate

Abstract: The identification of a single-input, single-output (SISO) discrete Hammerstein system is studied. Such a system consists of a non-linear memoryless subsystem followed by a dynamic, linear subsystem. The parameters of the dynamic, linear subsystem are identified by a correlation method and the Newton-Gauss method. The main results concern the identification of the non-linear, memoryless subsystem. No conditions are imposed on the functional form of the non-linear subsystem, recovering the non-linear using the Fourier series regression estimate. The density-free pointwise convergence Of the estimate is proved, that is.algorithm converges for all input densities The rate of pointwise convergence is obtained for smooth input densities and for non-linearities of Lipschitz type.Globle convergence and its rate are also studied for a large class of non-linearities and input densities

Application of multiple Fourier series to identification of multivariable non-stationary systems

Abstract: An algorithm for the on-line identification of multivariable non-stalionary systems is presented. The procedure is based on the first arithmetic means (C, 1) of multiple Fourier series. Convergence in the probability and with probability one is proved. Systems with multiplicative non-stalionarity are discussed in details.

On the Pressure Gradient Force Error in σ-Coordinate Spectral Models

Abstract: The pressure gradient force error of the spectral technique used in combination with the σ vertical coordinate was examined in an idealized case of an atmosphere at rest and in hydrostatic equilibrium. Small-scale (one-point and three-point) mountains were used in the tests. With such definitions of topography, difficulties could be expected with the spectral method due to slow convergence of the Fourier series. For reference, the finite-difference pressure gradient force errors were also computed. In the examples considered, it. was found that the errors of the spectral method can be large. In the rms sense, the spectral pressure gradient force errors were larger than those of the finite-difference method.

A simple series for personal computer computation of the error function Q(.)

Abstract: A simple series for computation of the error function Q(.) is derived. It is well suited for implementation on a personal computer, having six or more significant figure accuracy over a wide range of argument and requiring few lines of code to program. Its advantage over other series is its rapid convergence over a wide range of argument. >

Estimating high speed circuit interconnect performance

Abstract: A technique requiring a minimum of computational effort is presented for providing engineering estimates of the effect of package mismatch, line loss, and crosstalk. The technique entails modifying the Fourier series of a symmetrical trapezoid (which serves as a reasonable approximation to an actual digital signal) by transfer functions representing mismatch, loss, or crosstalk. Transfer functions are introduced which utilize frequently-domain data (measured or calculated) and provide the necessary transition to the time-domain performance. Sample calculations are given along with an interpretation of the results. >

Elementary Differential Equations with Boundary Value Problems

Abstract: Preface 1 First-Order Differential Equations 1.1 Differential Equations and Mathematical Models 1.2 Integrals as General and Particular Solutions 1.3 Slope Fields and Solution Curves 1.4 Separable Equations and Applications 1.5 Linear First-Order Equations 1.6 Substitution Methods and Exact Equations 1.7 Population Models 1.8 Acceleration-Velocity Models 2 Linear Equations of Higher Order 2.1 Introduction: Second-Order Linear Equations 2.2 General Solutions of Linear Equations 2.3 Homogeneous Equations with Constant Coefficients 2.4 Mechanical Vibrations 2.5 Nonhomogeneous Equations and Undetermined Coefficients 2.6 Forced Oscillations and Resonance 2.7 Electrical Circuits 2.8 Endpoint Problems and Eigenvalues 3 Power Series Methods 3.1 Introduction and Review of Power Series 3.2 Series Solutions Near Ordinary Points 3.3 Regular Singular Points 3.4 Method of Frobenius: The Exceptional Cases 3.5 Bessel's Equation 3.6 Applications of Bessel Functions 4 LaplaceTransform Methods 4.1 Laplace Transforms and Inverse Transforms 4.2 Transformation of Initial Value Problems 4.3 Translation and Partial Fractions 4.4 Derivatives, Integrals, and Products of Transforms 4.5 Periodic and Piecewise Continuous Input Functions 4.6 Impulses and Delta Functions 5 Linear Systems of Differential Equations 5.1 First-Order Systems and Applications 5.2 The Method of Elimination 5.3 Matrices and Linear Systems 5.4 The Eigenvalue Method for Homogeneous Systems 5.5 Second-Order Systems and Mechanical Applications 5.6 Multiple Eigenvalue Solutions 5.7 Matrix Exponentials and Linear Systems 5.8 Nonhomogeneous Linear Systems 6 Numerical Methods 6.1 Numerical Approximation: Euler's Method 6.2 A Closer Look at the Euler Method 6.3 The Runge-Kutta Method 6.4 Numerical Methods for Systems 7 Nonlinear Systems and Phenomena 7.1 Equilibrium Solutions and Stability 7.2 Stability and the Phase Plane 7.3 Linear and Almost Linear Systems 7.4 Ecological Models: Predators and Competitors 7.5 Nonlinear Mechanical Systems 7.6 Chaos in Dynamical Systems 8 Fourier Series Methods 8.1 Periodic Functions and Trigonometric Series 8.2 General Fourier Series and Convergence 8.3 Fourier Sine and Cosine Series 8.4 Applications of Fourier Series 8.5 Heat Conduction and Separation of Variables 8.6 Vibrating Strings and the One-Dimensional Wave Equation 8.7 Steady-State Temperature and Laplace's Equation 9 Eigenvalues and Boundary Value Problems 9.1 Sturm-Liouville Problems and Eigenfunction Expansions 9.2 Applications of Eigenfunction Series 9.3 Steady Periodic Solutions and Natural Frequencies 9.4 Cylindrical Coordinate Problems 9.5 Higher-Dimensional Phenomena References for Further Study Appendix: Existence and Uniqueness of Solutions Answers to Selected Problems Index I-1