Showing papers on "Fourier series published in 1994"

Estimating frequency by interpolation using Fourier coefficients

Abstract: The periodogram of a time series that contains a sinusoidal component provides a crude estimate of its frequency parameter, the maximizer over the Fourier frequencies being within O(T/sup -1/) of the frequency as the sample size T increases. In the paper, a technique for obtaining an estimator that has root mean square error of order T/sup -3/2/ is presented, which involves only the Fourier components of the time series at three frequencies, The asymptotic variance of the estimator varies between, roughly, the asymptotic variance of the maximizer of the periodogram over all frequencies (the Cramer-Rao lower bound) and three times this variance. The advantage of the new estimator is its computational simplicity. >

Wavelets and other orthogonal systems with applications

Abstract: Chapter 1. Orthogonal Series General Theory Examples Problems Chapter 2. A Primer on Tempered Distributions Tempered Distributions Fourier Transforms Periodic Distributions Analytic Representations Sobolev Spaces Problems Chapter 3. An Introduction to Orthogonal Wavelet Theory Multiresolution Analysis Mother Wavelet Reproducing Kernels and a Moment Condition Regularity of Wavelets as a Moment Condition Mallat's Decomposition and Reconstruction Algorithm Filters Problems Chapter 4. Convergence and Summability of Fourier Series Pointwise Convergence Summability Gibbs' Phenomenon Periodic Distributions Problems Chapter 5. Wavelets and Tempered Distributions Multiresolution Analysis of Tempered Distributions Wavelets Based on Distributions Distributions with Point Support Problems Chapter 6. Orthogonal Polynomials General Theory Classical Orthogonal Polynomials Problems Chapter 7. Other Orthogonal Systems Self Adjoint Eigenvalue Problems on a Finit e Interval Hilbert-Schmidt Integral Operators An Anomaly-The Prolate Spheroidal Functions A Lucky Accident? Rademacher Functions Walsh Functions Periodic Wavelets Local Sine or Cosine Bases Biorthogonal Wavelets Problems Chapter 8. Pointwise Convergence of Wavelet Expansions Quasi-Positive Delta Sequences Local Convergence of Distribution Expansions Convergence almost Everywhere Rate of Convergence of the Delta Sequence Other Partial Sums of the Wavelet Expansion Gibbs' Phenomenon Problems Chapter 9. A Shannon Sampling Theorem in Vm A Riesz Basis of Vm The Sampling Sequence in Vm Examples of Sampling Theorems The Sampling Sequence in Tm Shifted Sampling Oversampling with Scaling Functions Cardinal Scaling Functions Problems Chapter 10. Translation and Dilation Invariance in Orthogonal Systems Trigonometric System Orthogonal Polynomials An Example Where Everything Works An Example Where Nothing Works Weak Translation Invariance Dilations and Other Operations Problems Chapter 11. Analytic Representations via Orthogonal Series Trigonometric Series Hermite Series Legendre Polynomial Series Analytical and Harmonic Wavelets Analytic Solutions to Dilation Equations Analytic Representation of Distributions by Wavelets Problems Chapter 12. Orthogonal Series in Statistics Fourier Series Density Estimators Hermite Series Density Estimators The Histogram as a Wavelet Estimator Smooth Wavelet Estimators of Density Local Convergence Positive Density Estimators Other Estimation with Wavelets Problems Chapter 13. Orthogonal Systems and Stochastic Processes K-L Expansions Stationary Processes and Wavelets A Series with Uncorrelated Coefficients Wavelets Based on Band Limited Processes Nonstationary processes Problems Bibliography Index

The cumulant theory of cyclostationary time-series. I. Foundation

Abstract: The problem of characterizing the sine-wave components in the output of a polynomial nonlinear system with a cyclostationary random time-series input is investigated. The concept of a pure nth-order sine wave is introduced, and it is shown that pure nth-order sine-wave strengths in the output time-series are given by scaled Fourier coefficients of the polyperiodic temporal cumulant of the input time-series. The higher order moments and cumulants of narrowband spectral components of time-series are defined and then idealized to the case of infinitesimal bandwidth. Such spectral moments and cumulants are shown to be characterized by the Fourier transforms of the temporal moments and cumulants of the time-series. It is established that the temporal and spectral cumulants have certain mathematical and practical advantages over their moment counterparts. To put the contributions of the paper in perspective, a uniquely comprehensive historical survey that traces the development of the ideas of temporal and spectral cumulants from their inception is provided. >

Signals, Systems, and Transforms

Abstract: For sophomore/junior-level signals and systems courses in Electrical and Computer Engineering departments. This text provides a clear, comprehensive presentation of both the theory and applications in signals, systems, and transforms. It presents the mathematical background of signals and systems, including the Fourier transform, the Fourier series, the Laplace transform, the discrete-time and the discrete Fourier transforms, and the z-transform. The text integrates MATLAB examples into the presentation of signal and system theory and applications.

Wavelets and other orthogonal systems

Abstract: ORTHOGONAL SERIES General Theory Examples A PRIMER ON TEMPERED DISTRIBUTIONS Intuitive Introduction Test Functions Tempered Distribution Fourier Transforms Periodic Distributions Analytic Representations Sobolev Spaces AN INTRODUCTION TO ORTHOGONAL WAVELET THEORY Multiresolution Analysis Mother Wavelet Reproducing Kernels and a Moment Condition Regularity of Wavelets as a Moment Condition Mallat's Decomposition and Reconstruction Algorithm Filters CONVERGENCE AND SUMMABILITY OF FOURIER SERIES Pointwise Convergence Summability Gibbs' Phenomenon Periodic Distributions WAVELETS AND TEMPERED DISTRIBUTIONS Multiresolution Analysis of Tempered Distributions Wavelets Based on Distributions Distributions with Point Support Approximation with Impulse Trains ORTHOGONAL POLYNOMIALS General Theory Classical Orthogonal Polynomials Problems OTHER ORTHOGONAL SYSTEMS Self-Adjoint Eigenvalue Problems on Finite Intervals Hilber-Schmnidt Integral Operators An Anomaly: The Prolate Spheroidal Functions A Lucky Accident? Rademacher Functions Walsh Function Periodic Wavelets Local Sine or Cosine Base Biorthogonal Wavelets POINTWISE CONVERGENCE OF WAVELET EXPANSIONS Reproducing Kernel Delta Sequences Positive and Quasi-Positive Delta Sequences Local Convergence of Distribution Expansions Convergence Almost Everywhere Rate of Convergence of the Delta Sequence Other Partial Sums of the Wavelet Expansion Gibbs' Phenomenon Positive Scaling Functions A SHANNON SAMPLING THEOREM IN WAVELET SUBSPACES A Riesz Basis of Vm The Ampling Sequence in Vm Examples of Sampling theorems The Sampling Sequence in Tm Shifted Sampling Gibbs' Phenomenon for Sampling Series Irregular Sampling in Wavelet Subspaces EXTENSIONS OF WAVELET SAMPLING THEOREMS Oversampling with Scaling Functions Hybrid Sampling Series Positive Hybrid Sampling The Convergence of the Positive Hybrid Series Cardinal Scaling Functions Interpolating Multiwavelets Orthogonal Finite Element Multiwavelets TRANSLATION AND DILATION INVARIANCE IN ORTHOGONAL SYSTEMS Trigonometric System Orthogonal Polynomials An Example Where Everything Works An Example Where Nothing Works Weak Translation Invariance Dilations and Other Operations ANALYTIC REPRESENTATIONS VIA ORTHOGONAL SERIES Trigonometric Series Hermite Series Legendre Polynomial Series Analytic and Harmonic Wavelets Analytic Solutions to Dilation Equations Analytic Representation of Distributions by Wavelets Wavelets Analytic in the Entire Complex Plane ORTHOGONAL SERIES IN STATISTICS Fourier Series Density Estimators Hermite Series Density Estimators The Histogram as a Wavelet Estimator Smooth Wavelet Estimators of Density Local Convergence Positive Density Estimators Based on Characteristic Functions Positive Estimators Based on Positive Wavelets Density Estimation with Noisy Data Other Estimation with Wavelets Threshold Methods ORTHOGONAL SYSTEMS AND STOCHASTIC PROCESSES K-L Expansions Stationary Processes and Wavelets A Series with Uncorrelated Coefficients Wavelets Based on Band Limited Processes Nonstationary Processes Each chapter also contains a Problems section

An adaptive estimation of periodic signals using a Fourier linear combiner

Abstract: Presents an adaptive algorithm for estimating from noisy observations, periodic signals of known period subject to transient disturbances. The estimator is based on the LMS algorithm and works by tracking the Fourier coefficients of the data. The estimator is analyzed for convergence, noise misadjustment and lag misadjustment for signals with both time invariant and time variant parameters. The analysis is greatly facilitated by a change of variable that results in a time invariant difference equation. At sufficiently small values of the LMS step size, the system is shown to exhibit decoupling with each Fourier component converging independently and uniformly. Detection of rapid transients in data with low signal to noise ratio can be improved by using larger step sizes for more prominent components of the estimated signal. An application of the Fourier estimator to estimation of brain evoked responses is included. >

Multidimensional Transform Inversion with Applications to the Transient M/G/1 Queue

Abstract: We develop an algorithm for numerically inverting multidimensional transforms. Our algorithm applies to any number of continuous variables (Laplace transforms) and discrete variables (generating functions). We use the Fourier-series method; that is, the inversion formula is the Fourier series of a periodic function constructed by aliasing. This amounts to an application of the Poisson summation formula. By appropriately exponentially damping the given function, we control the aliasing error. We choose the periods of the multidimensional periodic function so that each infinite series is a finite sum of nearly alternating infinite series. Then we apply the Euler transformation to compute the infinite series from finitely many terms. The multidimensional inversion algorithm enables us, evidently for the first time, to calculate probability distributions quickly and accurately from several classical transforms in queueing theory. For example, we apply our algorithm to invert the two-dimensional transforms of the joint distribution of the duration of a busy period and the number served in that busy period, and the time-dependent transient queue-length and workload distributions in the M/G/1 queue. In other related work, we have applied the inversion algorithms here to calculate time-dependent distributions in the transient BMAP/G/1 queue (with a batch Markovian arrival process) and the piecewise-stationary $M_t/G_t/1$ queue.

Fourier series for analysis of temporal sequences of satellite sensor imagery

Abstract: Fourier Series and the derivative were used in this study for analysing time series of remotely-sensed data. The technique allows fundamental characteristics of time series data to be quantified. In Fourier analysis a function in space or time is broken down into sinusoidal components, or harmonics. The first and second harmonics are a function of the mono or bi-modality of the curve, demonstrated in the study on Global Vegetation Index data classified into typical mono and bi-modal vegetation index zones. The last harmonic explains close to 100 per cent of the variance in the curve. Other important parameters of the time series, such as extreme points and rate of change, can be extracted from the derivative of the Fourier Series. Fourier Series may form a basis for a quantitative approach to the problem of handling temporal sequences of remotely-sensed data.

Analysis of neutron and x-ray reflectivity data. II. Constrained least-squares methods

Abstract: Two methods for the determination of scattering length density profiles from specular reflectivity data are described. Both kinematical and dynamical theory can be used for calculating the reflectivity. In the first method, the scattering density is parameterized using cubic splines. The coefficients in the series are determined by constrained nonlinear least-squares methods, in which the smoothest solution that agrees with the data is chosen. The method is a further development of the two-step approach of Pedersen [J. Appl. Cryst. (1992), 25, 129–145]. The second approach is based on a method introduced by Singh, Tirrell & Bates [J. Appl. Cryst. (1993), 26, 650–659] for analyzing reflectivity data from periodic profiles. In this approach, the profile is expressed as a series of sine and cosine terms. Several new features have been introduced in the method, of which the most important is the inclusion of a smoothness constraint, which reduces the coefficients of the higher harmonics in the Fourier series. This makes it possible to apply the method also to aperiodic profiles. For the analysis of neutron reflectivity data, the instrumental smearing of the model reflectivity is important and a method for fast calculation of smeared reflectivity curves is described. The two methods of analyzing reflectivity data have been applied to sets of simulated data based on examples from the literature, including an amphiphilic monolayer and block copolymer thin films. The two methods work equally well in most situations and are able to recover the original profiles. In general, the method using splines as the basis functions is better suited to aperiodic than to periodic structures, whereas the sine/cosine basis is well suited to periodic and nearly periodic structures.

The generalized Peierls–Nabarro model

Abstract: In the original Peierls-Nabarro model the core structure of a dislocation is determined as the solution of an integrodifferential equation. This equation describes the balance between the forces resulting from the deformation of two elastic half-spaces and from a one-dimensional periodic lattice potential acting across the glide plane. A method is described here which allows the core structure and core energy to be obtained for a straight dislocation with arbitrary Burgers vector in an arbitrary glide plane in a crystal of arbitrary anisotropy, for which the displacement potential is represented by a two-dimensional Fourier series. This is accomplished by describing the internal displacements by appropriate trial functions with a set of free parameters whose value is then determined by minimizing the total energy. The method is applied to obtain the core configuration of a screw dislocation dissociated in a {111} plane of a f.c.c. lattice.

Convolution Analysis of Milling Force Pulsation

Abstract: George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405 This paper presents the establishment of a closed form expression for the dynamic forces as explicit functions of cutting parameters and tool/workpiece geometry in milling processes. Based on the existing local cutting force model, the generation of total cutting forces is formulated as the angular domain convolution of three cutting process component functions, namely the elementary cutting function, the chip width density function, and the tooth sequence function. The elemental cutting force function is related to the chip formation process in an elemental cutting area and it is characterized by the chip thickness variation, and radial cutting configuration. The chip width density function defines the chip width per unit cutter rotation along a cutter flute within the range of axial depth of cut_ The tooth sequence function represents the spacing between flutes as well as their cutting sequence as the cutter rotates. The analysis of cutting forces is extended into the Fourier domain by taking the frequency multiplication of the transforms of the three component functions. Fourier series coefficients of the cutting forces are shown to be explicit algebraic functions of various tool parameters and cutting conditions. Numerical simulation results are presented in the frequency domain to illustrate the effects of various process parameters. A series of end milling experiments are performed and their results discussed to validate the analytical model.

Generalized thermoelastic problem of a thick plate under axisymmetric temperature distribution

Abstract: The two-dimensional problem of a thick plate whose lower and upper surfaces are traction free and subjected to a given axisymmetric temperature distribution is considered within the context of the theory of generalized thermoelasticity with one relaxation time. Potential functions together with Laplace and Hankel transform techniques are used to derive the solution in the transformed domain. The Hankel transforms are inverted analytically. The inversion of the Laplace transforms are carried out using the inversion formula of the transform together with Fourier expansion techniques. Numerical methods are used to accelerate the convergence of the resulting series and to evaluate the improper integrals involved to obtain the temperature and stress distributions in the physical domain. Analysis of wave propagation in the medium is presented. Numerical results are represented graphically and discussed. A comparison is made with the solution of the corresponding coupled problem.

An Application of Fourier Series to the Most Significant Digit Problem

Abstract: (1994). An Application of Fourier Series to the Most Significant Digit Problem. The American Mathematical Monthly: Vol. 101, No. 9, pp. 879-886.

A wavelet-based KL-like expansion for wide-sense stationary random processes

Abstract: We describe a wavelet-based series expansion for wide-sense stationary processes. The expansion coefficients are uncorrelated random variables, a property similar to that of a Karhunen-Loeve (KL) expansion. Unlike the KL expansion, however, the wavelet-based expansion does not require the solution of the eigen equation and does not require that the process be time-limited. This expansion also has advantages over Fourier series, which is often used as an approximation to the KL expansion, in that it completely eliminates correlation and that the computation for its coefficients are more stable over large time intervals. The basis functions of this expansion can be obtained easily from wavelets of the Lemaire-Meyer (1990) type and the power spectral density of the process. Finally, the expansion can be extended to some nonstationary processes, such as those with wide-sense stationary increments. >

Determination of equivalent current sources from spherical cap harmonic models of geomagnetic field variations.

Abstract: SUMMARY The intrinsic ability of the method of spherical cap harmonic analysis to separate external and internal sources allows the calculation of equivalent ionospheric and induced currents that are able to explain variations of the geomagnetic field over a portion of the earth's surface. Formulations for current densities and current functions are derived and found to be analogous to those derived for the global case from conventional spherical harmonic analysis. Although spherical cap formulations for current density have been given by another worker, they were incorrect because of an error in defining the equivalent current. An example of the use of current functions is given by modelling variations from hourly mean values recorded at 40 geomagnetic observatories over Europe during a very quiet day in 1978. The modelling can be done spatially for each of the 24 hours separately, or spatially and temporally either by expressing each spatial coefficient as a Fourier series or by smoothing the spatial coefficients obtained from the separate hourly models.

Wave Intensity Analysis of High Frequency Vibrations

Abstract: In the statistical energy analysis (SEA) approach to high frequency dynamics it is assumed that the vibrational wavefield in each component of an engineering structure is diffuse. In some instances the directional filtering effects of structural joints can lead to highly non-diffuse wavefields, and in such cases SEA will yield a very poor estimate of the vibrational response. An alternative approach is presented here in which the directional dependency of the vibrational wavefield in each component is modelled by using a Fourier series. It is shown that, if required, the resulting energy balance equations may be cast in the form of conventional SEA with the addition of `non-direct' coupling loss factors. The method is applied to the bending and in-plane vibrations of various plate structures and a comparison is made with exact results yielded by the dynamic stiffness method. A significant improvement over conventional SEA is demonstrated.

Probability for Analysts

Abstract: Fourier transforms on Rd Weak convergence in M1 (Rd) Independende Infinite series of random vectors Normal distributions and central limits Martingales Projective limits and infinite products of probability measures Brownian motions Random Fourier series of continuous functions Fourier coefficients of continuous functions

Bifurcations of limit cycles in periodically forced nonlinear systems: the harmonic balance approach

Abstract: The harmonic balance approach is used to analyze tangent (saddle-node) and flip (period-doubling) bifurcations of limit cycles in periodically forced nonlinear dynamical systems. An algebraic system of equations, whose unknowns are the coefficients of a truncated Fourier series, is defined and the relationships between the bifurcations of the solutions of this algebraic system and the tangent and flip bifurcations of the limit cycles are pointed out. Some examples are presented to illustrate the method and its accuracy. >

Reconstruction of crack shape by optimization using eddy current field measurement

Abstract: The eddy current field perturbation due to a thin crack may be described as the field generated by a current dipole layer located on the surface of the crack. In this paper the Fourier-transform of the eddy current field by a known dipole layer is evaluated analytically if the dipole layer density function is given as, for example, a Taylor's or Fourier series. This result is used for the calculation of the impedance change of the exciting coil due to a crack by solving an integral equation. In the case of an unknown crack the measured impedance is used for reconstruction. By zeroth order optimization the shape of the crack is varied to fit the calculated impedance data to the measured ones. Several local minima of the objective function are found and statistically processed to give reliable approximation of the crack shape even in the case of sparse and noisy data. >

Simple method for calculating Fourier coefficients of experimentally obtained waveforms

Abstract: An algorithm is presented for calculating the Fourier series coefficients of experimentally obtained waveforms. The implementation of this algorithm requires only simple basic mathematical operations. Application of the algorithm is illustrated with an example.

Analysis of neutron and x-ray reflectivity data by constrained least-squares methods

Abstract: Two methods for the determination of scattering length density profiles from specular reflectivity data are discussed. For either method kinematical or dynamical theory can be used to calculate the reflectivity. In the first method the scattering density is parametrized using cubic splines. The coefficients in the series are determined by constrained nonlinear least-squares methods, in which the smoothest solution that agrees with the data is chosen. In the second approach the profile is expressed as a series of sine and cosine terms. A smoothness constraint is used which reduces the coefficients of the higher harmonics in the Fourier series. The two methods work equally well in most situations, and they are able to recover the original profiles. In general, the method using splines as the basis functions is better suited for aperiodic than for periodic structures, whereas the sine/cosine basis is well suited for periodic and nearly periodic structures.