Showing papers on "Fourier series published in 1993"

Learning decision trees using the Fourier spectrum

Abstract: This work gives a polynomial time algorithm for learning decision trees with respect to the uniform distribution. (This algorithm uses membership queries.) The decision tree model that is considered is an extension of the traditional boolean decision tree model that allows linear operations in each node (i.e., summation of a subset of the input variables over $GF(2)$).This paper shows how to learn in polynomial time any function that can be approximated (in norm $L_2 $) by a polynomially sparse function (i.e., a function with only polynomially many nonzero Fourier coefficients). The authors demonstrate that any function f whose $L_1 $-norm (i.e., the sum of absolute value of the Fourier coefficients) is polynomial can be approximated by a polynomially sparse function, and prove that boolean decision trees with linear operations are a subset of this class of functions. Moreover, it is shown that the functions with polynomial $L_1 $-norm can be learned deterministically.The algorithm can also exactly identi...

Voigt-function modeling in Fourier analysis of size- and strain-broadened X-ray diffraction peaks

Abstract: With the assumption that both size- and strain-broadened profiles of the pure-specimen function are described with a Voigt function, it is shown that the analysis of Fourier coefficients leads to the Warren–Averbach method of separation of size and strain contributions. The analysis of size coefficients shows that the `hook' effect occurs when the Cauchy content of the size-broadened profile is underestimated. The ratio of volume-weighted and surface-weighted domain sizes can change from ~1.31, for the minimum allowed Cauchy content, to 2, when the size-broadened profile is given solely by a Cauchy function. If the distortion Subscripts coefficient is approximated by a harmonic term, mean-square strains decrease linearly with increasing the averaging distance. The local strain is finite only in the case of purely Gaussian strain broadening, because strains are then independent of averaging distance.

Dithered quantizers

Abstract: A theory of overall quantization noise for nonsubtractive dither was originally developed in unpublished work by J.N. Wright and by T.J. Stockham and subsequently expanded by L.K. Brinton, S.P. Lipshitz, J. Vanderkooy, and R.A. Wannamaker. It is suggested that since these latter results are not as well known as the original results, misunderstanding persists in the literature. New proofs of the properties of quantizer dither, both subtractive and nonsubtractive, are provided. The new proofs are based on elementary Fourier series and Rice's characteristic function method and do not require the use of generalized functions (impulse trains of Dirac delta functions) and sampling theorem arguments. The goal is to provide a unified derivation and presentation of the two forms of dithered quantizer noise based on elementary Fourier techniques. >

The FBD-method, a generally applicable tool for analyzing power relations

Abstract: Starting from the most general case of m-wire unbalanced multiphase power systems with unsymmetrical loads under nonsinusoidal conditions, it is explained how the relations between currents, voltages and power quantities are analyzed in the time domain using the FBD method. Treatment in the frequency domain on the basis of Fourier series expansion of the time functions is also possible. It is shown that components of the total nonactive currents may be compensated without any time delay and without changing the collective instantaneous power. Rules are given for deriving simple equivalent circuits with m-equally structured branches from given voltages and currents. >

Harmonic analysis and boundary value problems in the complex domain

Abstract: 1 Preliminary results. Integral transforms in the complex domain -- 1.1 Introduction -- 1.2 Some identities -- 1.3 Integral representations and asymptotic formulas -- 1.4 Distribution of zeros -- 1.5 Identities between some Mellin transforms -- 1.6 Fourier type transforms with Mittag-Leffler kernels -- 1.7 Some consequences -- 1.8 Notes -- 2 Further results. Wiener-Paley type theorems -- 2.1 Introduction -- 2.2 Some simple generalizations of the first fundamental Wiener-Paley theorem -- 2.3 A general Wiener-Paley type theorem and some particular results -- 2.4 Two important cases of the general Wiener-Paley type theorem -- 2.5 Generalizations of the second fundamental Wiener-Paley theorem -- 2.6 Notes -- 3 Some estimates in Banach spaces of analytic functions -- 3.1 Introduction -- 3.2 Some estimates in Hardy classes over a half-plane -- 3.3 Some estimates in weighted Hardy classes over a half-plane -- 3.4 Some estimates in Banach spaces of entire functions of exponential type -- 3.5 Notes -- 4 Interpolation series expansions in spacesW1/2,?p,?of entire functions -- 4.1 Introduction -- 4.2 Lemmas on special Mittag-Leffler type functions -- 4.3 Two special interpolation series -- 4.4 Interpolation series expansions -- 4.5 Notes -- 5 Fourier type basic systems inL2(0, ?) -- 5.1 Introduction -- 5.2 Biorthogonal systems of Mittag-Leffler type functions and their completeness inL2(0, ?) -- 5.3 Fourier series type biorthogonal expansions inL2(0, ?) -- 5.4 Notes -- 6 Interpolation series expansions in spacesWs+1/2,?p,?of entire functions -- 6.1 Introduction -- 6.2 The formulation of the main theorems -- 6.3 Auxiliary relations and lemmas -- 6.4 Further auxiliary results -- 6.5 Proofs of the main theorems -- 6.6 Notes -- 7 Basic Fourier type systems inL2spaces of odd-dimensional vector functions -- 7.1 Introduction -- 7.2 Some identities -- 7.3 Biorthogonal systems of odd-dimensional vector functions -- 7.4 Theorems on completeness and basis property -- 7.5 Notes -- 8 Interpolation series expansions in spacesWs,?p,?of entire functions -- 8.1 Introduction -- 8.2 The formulation of the main interpolation theorem -- 8.3 Auxiliary relations and lemmas -- 8.4 Further auxiliary results -- 8.5 The proof of the main interpolation theorem -- 8.6 Notes -- 9 Basic Fourier type systems inL2spaces of even-dimensional vector functions -- 9.1 Introduction -- 9.2 Some identities -- 9.3 The construction of biorthogonal systems of even-dimensional vector functions -- 9.4 Theorems on completeness and basis property -- 9.5 Notes -- 10 The simplest Cauchy type problems and the boundary value problems connected with them -- 10.1 Introduction -- 10.2 Riemann-Liouville fractional integrals and derivatives -- 10.3 A Cauchy type problem -- 10.4 The associated Cauchy type problem and the analog of Lagrange formula -- 10.5 Boundary value problems and eigenfunction expansions -- 10.6 Notes -- 11 Cauchy type problems and boundary value problems in the complex domain (the case of odd segments) -- 11.1 Introduction -- 11.2 Preliminaries -- 11.3 Cauchy type problems and boundary value problems containing the operators $$ {\mathbb{L}_{s + 1/2}}$$ and $$ \mathbb{L}_{s + 1/2}^*$$ -- 11.4 Expansions inL2{?2s+1(?)} in terms of Riesz bases -- 11.5 Notes -- 12 Cauchy type problems and boundary value problems in the complex domain (the case of even segments) -- 12.1 Introduction -- 12.2 Preliminaries -- 12.3 Cauchy type problems and boundary value problems containing the operators $${{\mathbb{L}}_{s}} $$ and $$ \mathbb{L}_{s}^*$$ -- 12.4 Expansions inL2{?2s(?)} in terms of Riesz bases -- 12.5 Notes.

Affine theorem for two-dimensional Fourier transform

Abstract: The well known shift and similarity theorems for the Fourier transform generalise to two dimensions but new theorems come into existence in two dimensions. Simple theorems for rotation and shear distortion are examples. A theorem is presented which determines what the Fourier transform becomes when the function domain is subjected to an affine co-ordinate transformation. The full theorem contains a variety of simpler theorems as special cases. It may prove useful in its general form in image processing where sequences of affine transformations are applied.

Absolute testing of flats by using even and odd functions.

Abstract: We describe a modified three-flat method. In a Cartesian coordinate system, a flat can be expressed as the sum of even-odd, odd-even, even-even, and odd-odd functions. The even-odd and the odd-even functions of each flat are obtained first, and then the even-even function is calculated. All three functions are exact. The odd-odd function is difficult to obtain. In theory, this function can be solved by rotating the flat 90°, 45°, 22.5°, etc. The components of the Fourier series of this odd-odd function are derived and extracted from each rotation of the flat. A flat is approximated by the sum of the first three functions and the known components of the odd-odd function. In the experiments, the flats are oriented in six configurations by rotating the flats 180°, 90°, and 45° with respect to one another, and six measurements are performed. The exact profiles along every 45° diameter are obtained, and the profile in the area between two adjacent diameters of these diameters is also obtained with some approximation. The theoretical derivation, experiment results, and error analysis are presented.

A new Walsh domain technique of harmonic elimination and voltage control in pulse-width modulated inverters

Abstract: A method for selective harmonic elimination in pulse-width-modulated (PWM) inverter waveforms by the use of Walsh functions is presented. The Walsh operational matrix of PWM is introduced as a means of obtaining the Walsh spectral equations of PWM waveforms. The slope and intercept Fourier operational matrices of PWM are also introduced as a means of obtaining Fourier spectral equations of PWM waveforms. A noniterative algorithm that produces piecewise-linear, global solutions between angles and for the angles is proposed. The algorithm also produces the full range of variation of fundamental voltage for given harmonic elimination constraints. The set of systems of linear equations obtained replaces the system of nonlinear transcendental equations used in the Fourier series harmonic elimination approach. In general, the algorithm makes possible the synthesis of two-state PWM inverter waveforms with specified old harmonic content. >

How to Make Wavelets

Abstract: (1993) How To Make Wavelets The American Mathematical Monthly: Vol 100, No 6, pp 539-556

A Spectral Limited-Area Model Formulation with Time-dependent Boundary Conditions Applied to the Shallow-Water Equations

Abstract: The spectral technique is frequently used for the horizontal discretization in global atmospheric models. This paper presents a method where double Fourier series are used in a limited-area model (LAM). The method uses fast Fourier transforms (FFT) in both horizontal directions and takes into account time-dependent boundary conditions. The basic idea is to extend the time-dependent boundary fields into a zone outside the integration area in such a way that periodic fields are obtained. These fields in the extension zone and the forecasted fields inside the integration area are connected by use of a narrow relaxation zone along the boundaries of the limited area. The extension technique is applied to the shallow-water equations. A simple explicit (leapfrog) integration is shown to give results that are almost identical to the hemispherical forecast used as boundary fields. A nonlinear normal-mode initialization scheme developed in the framework of the spectral formulation is shown to work satisfac...

Feature‐recognizing MRI

Abstract: We describe a new theory of MR imaging that utilizes prior information in the form of a set of "training" images thought to be similar to the "unknown" objects to be scanned. First, the training images are processed to find an orthonormal series representation of these images that is more convergent than the usual Fourier series. The coefficients in this new series can be calculated from a subset of the phase-encoded signals needed to construct the Fourier image representation. The characteristics of the training images also determine exactly which phase-encoded signals should be measured in order to minimize error in the image reconstruction. The optimal phase-encodings are usually scattered nonuniformly in kappa-space. Good results were obtained when this theory was applied to imaging data from simulated objects and to experimental data from phantom scans. This theory provides the basis for developing efficient scanning and image reconstruction techniques that are "tailored" to each body part or to particular disease states.

Fiber Cross-Sectional Shape Analysis Using Image Processing Techniques:

Abstract: . Quantitative characterization of fiber cross sections has attracted considerable in terlst, since cross-sectional size and shape have an important impact on the physical and mechanical properties of fibers, as well as the performance of end-use products. We present one application of automated measurement using image processing tech niques that extract basic shape features from fiber cross sections. Cross-sectional shapes are characterized with the aid of geometric and Fourier descriptors. Geometric de scriptors measure attributes such as area, roundness, and ellipticity. Fourier descriptors are derived from the Fourier series for the cumulative angular function of the cross- sectional boundary and are used to characterize shape complexity and other geometric attributes. Shape reconstruction based on Fourier coefficients is also discussed. We present the results of shape analysis for a wide variety of fiber types.

A simple algorithm for fitting measured data to Fourier‐series models

Abstract: An algorithm is described for fitting measured data to Fourier‐series models of any order without recourse to discrete Fourier transform or curve‐fitting routines. The implementation of this algorithm requires only simple basic mathematical operations and can be easily implemented in microcomputer‐ or microprocessor‐based real‐time systems.

Multiple Fourier series procedures for extraction of nonlinear regressions from noisy data

Abstract: Three nonparametric procedures for the extraction of nonlinear regressions from noisy data are proposed. The procedures are based on the Dirichlet, Fejer, and de la Vallee Poussin multiple kernels. Convergence properties are investigated. In particular, it is shown that the algorithms are convergent in the mean-integrated-square-error sense. The appropriate theorem establishes a relation between the order of kernels and the number of observations. Special attention is focused on the two-dimensional case. It is proved that the procedures attain the optimal rate of convergence, which cannot be exceeded by any other nonparametric algorithm. >

Fundamental Solutions for a Poroelastic Half-Space With Compressible Constituents

Abstract: General solutions for equations of equilibrium expressed in terms of displacements and variation of fluid volume are derived by applying Fourier expansion, Hankel transforms, and Laplace transforms with respect to the circumferential, radial, and time coordinates, respectively. The general solutions are used to derive a set of fundamental solutions corresponding to circular ring loads and to a fluid source applied at a finite depth below the free surface of a poroelastic half space. The circumferential variation of the ring loads and the fluid source is descrived by appropriate trigonometric terms

Simulation of Spatially Incoherent Random Ground Motions

Abstract: Two techniques for the artificial generation of spatially incoherent Gaussian seismic ground motions are proposed and validated. The simulated motions are homogeneous and stationary, and may be one-, two-, or three-dimensional in space. They satisfy a prescribed, or target, local power spectrum and a target coherency function. Nonstationarity characteristics are introduced by superimposing a time-dependent envelope function to produce a uniformly modulated nonstationary process. The first technique is asymptotic and approximates the coherency function by a Fourier series: it is general and suits any form of spectrum and coherency. The second technique is approximate in the sense that it satisfies the autospectrum everywhere but satisfies the cross spectrum, or the coherency, between successive stations only. The latter technique is computationally very efficient, and may be accurate enough for discretely supported systems such as single-span structures and multispan, simply supported bridges. The techniques proposed may be useful in response analysis of structures, or structural components, for spatially incoherent random processes in the fields of earthquake, ocean, and wind engineering.