Showing papers on "Fourier series published in 1998"

An accurate algorithm for nonuniform fast Fourier transforms (NUFFT's)

Abstract: Based on the (m, N, q)-regular Fourier matrix, a new algorithm is proposed for fast Fourier transform (FFT) of nonuniform (unequally spaced) data. Numerical results show that the accuracy of this algorithm is much better than previously reported results with the same computation complexity of O(N log/sub 2/ N). Numerical examples are shown for the applications in computational electromagnetics.

Experimental study of interfacial solitary waves

Abstract: A small-scale experiment was conducted (in a 3 m long flume) to study interfacial long-waves in a two-immiscible-fluid system (water and petrol were used). Experiments and nonlinear theories are compared in terms of wave profiles, phase velocity and mainly frequency--amplitude relationships. As expected, the KdV solitary waves match the experiments for small-amplitude waves for all layer thickness ratios. The characteristics of 'large'-amplitude waves (that is when the crest is close to the critical level - approximately located at mid-depth) asymptotically tend to be predicted by a 'KdV-mKdV' equation containing both quadratic and cubic nonlinear terms. In addition a numerical solution of the complete Euler equations, based on Fourier series expansions, is devised to describe solitary waves of intermediate amplitude. In all cases, solitary interfacial waves in this numerical theory tally with the experimental data. When the layer thicknesses are almost equal (ratio of lower layer to total depth equal to 0.4 or 0.63) both the KdV-mKdV and the numerical solutions match the experimental points.

Wavelets Made Easy

Abstract: Part 1 Algorithms for wavelet transforms: Haar's simple wavelets multidimensional wavelets and applications algorithms for Daubechie's wavelets. Part 2 Basic Fourier analysis: inner products and orthogonal projections discrete and fast Fourier transforms Fourier series for periodic functions. Part 3 Computation and design of wavelets: Fourier transforms on the line and space Duabechies' wavelets design signal representations with wavelets.

Fast Approximate Fourier Transforms for Irregularly Spaced Data

Abstract: Several algorithms for efficiently evaluating trigonometric polynomials at irregularly spaced points are presented and analyzed. The algorithms can be viewed as approximate generalizations of the fast Fourier transform (FFT), and they are compared with regard to their accuracy and their computational efficiency.

A new approach to model switched reluctance motor drive application to dynamic performance prediction, control and design

Abstract: In this paper a new model for the switched reluctance motor (SRM) based on phase currents as state variables is presented. Position dependency of the phase inductance is represented by a limited number of Fourier series terms and the nonlinear variation of the inductance with phase current is expressed by means of polynomial functions. The coefficients of the terms in the Fourier series are determined by the aligned position inductance, the unaligned position inductance and the inductance at the midway point from the aligned position. The main advantage of the proposed model is that it requires minimum amount of measurements and predicts the complete dynamic performance of the drive system, viz., constant torque, constant power and natural mode regions. Any type of control strategy can be incorporated in the simulation. A low-voltage, high current SRM drive has been simulated and the results are validated by comparing with finite element and experimental results.

Greedy Algorithm and m -Term Trigonometric Approximation

Abstract: We study the following nonlinear method of approximation by trigonometric polynomials in this paper. For a periodic function f we take as an approximant a trigonometric polynomial of the form \(G_m(f) := \sum_{k \in \Lambda} \hat f(k) e^{i(k,x)}\) , where \(\Lambda \subset {\bf Z}^d\) is a set of cardinality m containing the indices of the m biggest (in absolute value) Fourier coefficients \(\hat f(k)\) of function f . We compare the efficiency of this method with the best m -term trigonometric approximation both for individual functions and for some function classes. It turns out that the operator G m provides the optimal (in the sense of order) error of m -term trigonometric approximation in the L p -norm for many classes.

On a high order numerical method for functions with singularities

Abstract: By splitting a given singular function into a relatively smooth part and a specially structured singular part, it is shown how the traditional Fourier method can be modified to give numerical methods of high order for calculating derivatives and integrals. Singular functions with various types of singularities of importance in applications are considered. Relations between the discrete and the continuous Fourier series for the singular functions are established. Of particular interest are piecewise smooth functions, for which various important applications are indicated, and for which numerous numerical results are presented.

Total variation based interpolation

Abstract: We propose an reversible interpolation method for signals or images, in the sense that the original image can be deduced from its interpolation by a sub-sampling. This imposes some constraints on the Fourier coefficients of the interpolated image. Now, there still exist many possible interpolations that satisfy these constraints. The zero-padding method is one of them, but gives very oscillatory images. We propose to choose among all possibilities, the one which is the most "regular". We justify the total variation of the image as a good candidate as measure of regularity. This yields to minimise a regularisation functional defined in space domain, with a constraint defined in the frequency domain.

A generalized steady-state analysis of resonant converters using two-port model and Fourier-series approach

Abstract: A generalized steady-state analysis of resonant converters using a two-port model and Fourier-series approach is presented. Analysis is presented for both voltage-source (VS) and current-source (CS)-type loads. Analysis can also be used either for variable-frequency (half- or full-bridge) or fixed-frequency (phase-shift control for full-bridge) operation. Steady-state solutions have been obtained. Particular cases are considered to show the method of application in analyzing different schemes. A simple design procedure is given for two particular cases to illustrate the use of analysis in obtaining design curves and in designing the converters. Experimental results obtained from a MOSFET-based 500-W fixed-frequency LCL-type resonant converter are presented to verify the analysis.

Variational methods for the homogenization of periodic heterogeneous media

Abstract: In this paper, several variational principles for the evaluation of the overall properties of composite materials with periodic microstructure are introduced. The two classical homogenization problems corresponding to assigned average strain or assigned average stress are considered. The periodicity of the variables governing the problem is enforced either by considering special representations (i.e. Fourier series) of the periodic part of the displacement and stress fields or by adopting appropriate boundary conditions on the unit cell. In particular, the boundary conditions ensuring the periodicity of the governing variable are introduced in the functionals by using Lagrangian multipliers. Once the variational principles are introduced, the Fourier series technique and the finite element method are adopted to obtain rational approximation procedures. Finally, numerical applications are carried out in order to assess the performances of the proposed methods in the computation of estimates or bounds on the overall elastic properties of a composite material, and in the determination of the displacement and stress distribution in the unit cell.

A Fast Poisson Solver of Arbitrary Order Accuracy in Rectangular Regions

Abstract: In this paper we propose a direct method for the solution of the Poisson equation in rectangular regions. It has an arbitrary order accuracy and low CPU requirements which makes it practical for treating large-scale problems. The method is based on a pseudospectral Fourier approximation and a polynomial subtraction technique. Fast convergence of the Fourier series is achieved by removing the discontinuities at the corner points using polynomial subtraction functions. These functions have the same discontinuities at the corner points as the sought solution. In addition to this, they satisfy the Laplace equation so that the subtraction procedure does not generate nonperiodic, nonhomogeneous terms. The solution of a boundary value problem is obtained in a series form in O(N log N) floating point operations, where N2 is the number of grid nodes. Evaluating the solution at all N2 interior points requires O(N2 log N) operations.

The higher order theory of generalized almost-cyclostationary time series

Abstract: In this paper, the class of generalized almost-cyclostationary (GACS) time series is introduced. Time series belonging to this class are characterized by multivariate statistical functions that are almost-periodic functions of time whose Fourier series expansions can exhibit coefficients and frequencies depending on the lag shifts of the time series. Moreover, the union over all the lag shifts of the lag-dependent frequency sets is not necessarily countable. Almost-cyclostationary (ACS) time series turn out to be the subclass of GACS time series for which the frequencies do not depend on the lag shifts and the union of the above-mentioned sets is countable. The higher order characterization of GACS time series in the strict and wide sense is provided. It is shown that the characterization in terms of cyclic moment and cumulant functions is inadequate for those GACS time series that are not ACS. Then, generalized cyclic moment and cumulant functions (in both the time and frequency domains) are introduced. Finally, the problem of estimating the introduced generalized cyclic statistics is addressed, and two examples of GACS time series are considered.

Basic analog of Fourier series on a $q$-quadratic grid

Abstract: We prove orthogonality relations for some analogs of trigonometric functions on a g-quadratic grid and introduce the corresponding g-Fourier series. We also discuss several other properties of this basic trigonometric system and the g-Fourier series.

Error-rate evaluation of linear equalization and decision feedback equalization with error propagation

Abstract: We propose applying an approximate Fourier series to evaluate efficiently the bit-error-rate (BER) performance of finite-length linear equalization (LE) and decision feedback equalization (DFE). By extending the Fourier series, we enable BER calculations for quadrature phase-shift keying (QPSK) transmission on complex channels with in-phase and crosstalk intersymbol interference (ISI). The BER calculation is based on determining the residual ISI samples and background Gaussian noise variance at the equalizer output for static channels or for realizations of quasi-static fading channels. A simple bound on the series error magnitude in terms of the Fourier series parameters ensures the required accuracy and precision. Improved state transition probability estimates are derived and verified by simulation for an approximate Markov model of the DFE error propagation for the case in which residual ISI exists even when the previous decisions stored in the feedback filter (FBF) are correct. We demonstrate the ease and widespread applicability of our approach by producing results which elucidate a variety of equalization tradeoffs. Our analysis includes symbol-spaced and fractionally spaced minimum mean-square error (MMSE)-LE, zero-forcing (ZF)-LE, and MMSE-DFE (with and without error propagation) on static ISI channels and multipath channels with quasi-static Rayleigh fading; a comparison between suboptimum and optimum receiver filtering in conjunction with equalization; and an assessment of the accuracy of some widely used equalization BER approximations and bounds.

Dispersive effects for nonlinear geometrical optics with rectification

Abstract: Oscillating approximate solutions to nonlinear hyperbolic dispersive systems are studied. Ansatz of three scales are used in order to deal with diffractive effects. The scaling of the approximate solutions is chosen so that diffractive, dispersive effects and rectification are present in the leading term. The propagation along the rays of geometrical optics of the oscillating Fourier coefficients of the leading terms is corrected by a Schrodinger dispersion which appears for long times only. The propagation of the nonoscillating Fourier coefficient depends on the properties of a symmetric hyperbolic system, whose characteristic variety is the tangent cone at 0 to the characteristic variety of the initial operator. Equations determining the leading term require a sublinear growth condition for the corrector and the inroduction of the analytical "average operators" which convey this sublinear growth condition in a simple way and sort the nonlinearities out. In the last part, detailed physical examples are given.

The generalization of the Wiener-Khinchin theorem

Abstract: We generalize the Wiener-Khinchin theorem. A full generalization is presented where both the autocorrelation function and power spectral density are defined in terms of a general basis set. In addition, we present a partial generalization where the density is the Fourier transform of the autocorrelation function but the autocorrelation function is defined in terms of an arbitrary basis set. Both the deterministic and random cases are considered.

Modeling Hourly Energy Use in Commercial Buildings With Fourier Series Functional Forms

Abstract: Hourly energy use in commercial buildings shows periodic variation in daily and annual cycles. Moreover, the pattern of variation is such that Fourier series functional forms provide one of the best approaches for modeling this behavior. Results from numerous case studies have demonstrated the power of Fourier series as a modeling tool to represent hourly energy use in commercial buildings. This paper describes the Fourier series approaches that have been developed for modeling energy use in commercial buildings and presents the results of application to data collected from many sites.

Analysis of eddy-current brakes for high speed railway

Abstract: The analysis of eddy-current brakes for a high-speed railway is described that shows the operational characteristics of the brake as function of the train speed up to 300 km/hr and of the DC magnetizing current of magnetic poles. Braking forces are generated by employing a DC excitation field for the poles moving 7 mm above the rails. Results from the Fourier series method and the finite difference method are compared for showing the attraction and braking forces between the rails and the poles. The nonlinear B-H curve described by the Frohlich form is assumed for the rail. The calculated operating curves are compared to the published experimental data and also those measured in-house.

Boundary Controllability of a Hybrid System Consisting in Two Flexible Beams Connected by a Point Mass

Abstract: We consider a hybrid system consisting of two flexible beams connected by a point mass. The constant of rotational inertia is assumed to be nonzero. In a previous paper we have proved that, in the presence of the point mass, the system is well posed in asymmetric spaces in which solutions have one more degree of regularity to one side of the mass. We are interested in the problem of controllability when the control acts on the free extreme of one of the beams. We prove that when the control time is large enough the system is exactly controllable in an asymmetric space. This result is sharp. The proofs combine classical techniques from asymptotic analysis and the theory of nonharmonic Fourier series.

Approximate Solution of a Class of Nonlinear Oscillators in Resonance with a Periodic Excitation

Abstract: This paper deals with the most important characteristics of a generalized Van der Pol–Duffing oscillator in resonance with a periodic excitation. We use an asymptotic perturbation method based on Fourier expansion and time rescaling and demonstrate through a second order perturbation analysis the existence of one or two limit cycles. Moreover, we identify a sufficient condition to obtain a doubly periodic motion, when a second low frequency appears, in addition to the forcing frequency. Comparison with the solution obtained by the numerical integration confirms the validity of our analysis.

Obtaining order in a world of chaos [signal processing]

Abstract: Measurements of a physical or biological system result in a time series, s(t)=s(t/sub 0/+n/spl tau//sub s/)=s(n) sampled at intervals of /spl tau//sub s/ and initiated at t/sub 0/. When a signal can be represented as a superposition of sine waves with different amplitudes, its characteristics can be adequately described by Fourier coefficients of amplitude and phase. In these circumstances, linear and Fourier based methods for extracting information from the signal are appropriate and powerful. However, the signal may be generated by a nonlinear system. The waveform can be irregular and continuous and broadband in the frequency domain. The signal is noise-like, but is deterministic and may be chaotic. More information than the Fourier coefficients is required to describe the signal. This article describes methods for distinguishing chaotic signals from noise, and how to utilize the properties of a chaotic signal for classification, prediction, and control.

Generalization of the Wiener-Khinchin theorem

Abstract: We generalize the concept of the autocorrelation function and give the generalization of the Wiener-Khinchin theorem. A full generalization is presented where both the autocorrelation function and power spectral density are defined in terms of a general basis set. In addition, we present a partial generalization where the density is the Fourier transform of the characteristic function but the characteristic function is defined in terms of an arbitrary basis set. Both the deterministic and random cases are considered.