Showing papers on "Fourier series published in 1995"

Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems

Abstract: From the Publisher: This book presents the newly developed theory of non-harmonic Fourier series and its applications to the control of distributed parameter systems The book begins with the modern theory of exponentials, using an operator theory approach, then extends this to the method of moments--one of the most powerful tools in control theory Applications to various types of control problems are discussed Researchers in control theory, functional analysis, and partial differential equations will find much to interest them in this treatise

Elliptic Fourier shape analysis of fossil bivalves: some practical considerations

Abstract: Elliptic Fourier shape analysis is a powerful, though under-utilized, biometric tool that is particularly suited for the description of fossils lacking many homologous landmarks, such as several common bivalve groups. The method is conceptually more parsimonious than more traditional biometric methods based on discrete linear and angular measurements. Most importantly, however, shape analysis captures a much higher proportion of the morphological information resident in any fossil than analyses based on discrete measurements. The number of harmonics required in an elliptic Fourier analysis can be estimated from a series of inverse Fourier reconstructions, or from the power spectrum. In most studies it is appropriate to normalize Fourier coefficients for size, although this information can be reincorporated at a later stage. The coefficients should probably not be standardized, unless there is evidence to suggest that high-frequency information was genetically as important as low-frequency information. Depending upon the aims of a particular study and the morphological disparity of the fossils in question, it might be appropriate to eliminate the first harmonic (‘best-fitting’) ellipse from an analysis. Meaningful comparison of the left and right valves of bivalves requires the digitized coordinates of one or other to be mirrored prior to computation of the Fourier coefficients. □Biometric analysis, Bivalvia, elliptic Fourier analysis, morphometrics.

Fractional Fourier optics

Abstract: There exists a fractional Fourier-transform relation between the amplitude distributions of light on two spherical surfaces of given radii and separation. The propagation of light can be viewed as a process of continual fractional Fourier transformation. As light propagates, its amplitude distribution evolves through fractional transforms of increasing order. This result allows us to pose the fractional Fourier transform as a tool for analyzing and describing optical systems composed of an arbitrary sequence of thin lenses and sections of free space and to arrive at a general class of fractional Fourier-transforming systems with variable input and output scale factors.

Self-healing slip pulse on a frictional surface

Abstract: Guided by seismic observations of short-duration radiated pulses in earthquake ruptures, Heaton (1990) has postulated a mechanism for the frictional sliding of two identical elastic solids that consists in the subsonic propagation of a self-healing slip velocity pulse of finite duration along the interface. The same type of pulse may be conjectured for inhomogeneous slip along sufficiently large, and compliant, technological surfaces. We analyze such pulses, first as steady traveling waves which move at constant speed, and without alteration of shape, on the interface between joined elastic half-spaces, and later as transient disturbances along such an interface, arising as slip rupture propagates spontaneously from an over-stressed nucleation site. The study is conducted in the framework of antiplane elastodynamics; normal stress is uniform and alteration of it is not considered. We show that not all constitutive models allow for steady traveling wave pulses: the static friction threshold subsequent to the relocking of the fault must increase with time. That is, such solutions do not exist for pure velocity-dependent constitutive models, in which the stress-resisting slip on the ruptured surface is a continuously decreasing function of the instantaneous sliding rate (but not of its previous history or of other measures of the evolving state of the surface). Further, even for constitutive models that include both the rate- and state-dependence of friction, such as the laboratory-based constitutive models for friction as developed by Dieterich (1979, 1981) and Ruina (1983), steady pulse solutions do not exist for versions, like one discussed by Ruina (1983), which do not allow (rapid) restrengthening in truly stationary contact. For a particular class of rate- and state-dependent laws which includes such restrengthening, we establish parameter ranges for which steady pulse solutions exist, and use a numerical method stabilized by a Tikhonov-style regularization to construct the solutions. The numerical method used for the transient analysis adopts Fourier series representations for the spatial dependence of stress and slip along the interface, with the (time-dependent) coefficients in those Fourier series being related to one another in a way which obtains from exact solution to the equations of elastodynamics. This allows an efficient numerical method, based on use of the Fast Fourier Transform in each time step, with the frictional constitutive law enforced at the FFT sample points along the interface. Solutions based on a law that includes restrengthening in stationary contact show that spontaneous rupture propagation will occur either in the self-healing slip pulse mode (but not generally as a steady pulse) or in the classical enlarging-crack mode, depending on the values of parameters which enter the constitutive law. This analysis suggests that the strictly steady, traveling wave pulse solutions may either be unstable or have a limited basin of attraction.

Objective assessment of image quality. II. Fisher information, Fourier crosstalk, and figures of merit for task performance

Abstract: Figures of merit for image quality are derived on the basis of the performance of mathematical observers on specific detection and estimation tasks. The tasks include detection of a known signal superimposed on a known background, detection of a known signal on a random background, estimation of Fourier coefficients of the object, and estimation of the integral of the object over a specified region of interest. The chosen observer for the detection tasks is the ideal linear discriminant, which we call the Hotelling observer. The figures of merit are based on the Fisher information matrix relevant to estimation of the Fourier coefficients and the closely related Fourier crosstalk matrix introduced earlier by Barrett and Gifford [Phys. Med. Biol. 39, 451 (1994)]. A finite submatrix of the infinite Fisher information matrix is used to set Cramer-Rao lower bounds on the variances of the estimates of the first N Fourier coefficients. The figures of merit for detection tasks are shown to be closely related to the concepts of noise-equivalent quanta (NEQ) and generalized NEQ, originally derived for linear, shift-invariant imaging systems and stationary noise. Application of these results to the design of imaging systems is discussed.

A spectral method for three-dimensional elastodynamic fracture problems

Abstract: We present a numerical formulation for three-dimensional elastodynamic problems of fracture on planar cracks and faults. Stress and displacement components are given a spectral representation as finite Fourier series in space coordinates parallel to the fracture plane. The formulation is based on an exact representation, involving a convolution integral for each Fourier mode, of the elastodynamic relation existing between the time-dependent Fourier coefficients for the tractions acting on the fracture plane and for the resulting displacement discontinuities. A wide range of constitutive models can be used to relate the local value of the strength on the fracture plane with the displacement and velocity history. Efficiency of the code is achieved by using an explicit time integration scheme and by computing the conversion between the spatial and spectral distributions through a FFT algorithm. The method is particularly suited to implementation on massively parallel computers; a CM-5 was used in this work. The stability and precision of the formulation are discussed for tensile (mode 1) situations in a detailed modal analysis, and numerical results are compared with existing three-dimensional elastodynamic solutions. The adequacy of the method to investigate various three-dimensional dynamic fracture problems involving non-propagating and propagating tensile cracks is illustrated, including crack growth along a plane of heterogeneous fracture toughness.

Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials

Abstract: A new technique is proposed for the recovery of optical phase from intensity information. The method is based on the decomposition of the transport-of-intensity equation into a series of Zernike polynomials. An explicit matrix formula is derived, expressing the Zernike coefficients of the phase as functions of the Zernike coefficients of the wave-front curvature inside the aperture and the Fourier coefficients of the wave-front boundary slopes. Analytical expressions are given, as well as a numerical example of the corresponding phase retrieval matrix. This work lays the basis for an effective algorithm for fast and accurate phase retrieval.

Classification and Approximation of Periodic Functions

Abstract: Preface. Introduction. 1. Classes of periodic functions. 2. Integral representations of deviations of linear means of Fourier series. 3. Approximations by Fourier sums in the spaces c and L1. 4. Simultaneous approximation of functions and their derivatives by Fourier sums. 5. Convergence rate of Fourier series and best approximations in the spaces Lp. 6. Best approximations in the spaces C and L. Bibliographical notes. References. Index.

Optimal filtering in fractional Fourier domains

Abstract: The ordinary Fourier transform is suited best for analysis and processing of time-invariant signals and systems. When we are dealing with time-varying signals and systems, filtering in fractional Fourier domains might allow us to estimate signals with smaller minimum mean square error (MSE). We derive the optimal fractional Fourier domain filter that minimizes the MSE for given non-stationary signal and noise statistics, and time-varying distortion kernel. We present an example for which the MSE is reduced by a factor of 50 as a result of filtering in the fractional Fourier domain, as compared to filtering in the conventional Fourier or time domains. We also discuss how the fractional Fourier transformation can be computed in O(N log N) time, so that the improvement in performance is achieved with little or no increase in computational complexity.

Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions

Abstract: Kowledge of a truncated Fourier series expansion for a 2π-periodic function of finite regularity, which is assumed to be piecewise smooth in each period, is used to accurately reconstruct the corresponding function. An algebraic equation of degree M is constructed for the M singularity locations in each period for the function in question. The M coefficients in this algebraic equation are obtained by solving an algebraic system of M equations determined by the coefficients in the known truncated expansion. If discontinuities in the derivatives of the function are considered, in addition to discontinuities in the function itself, that algebraic system will be nonlinear with respect to the M unknown coefficients. The degree of the algebraic system will depend on the desired order of accuracy for the reconstruction, i.e., a higher degree will normally lead to a more accurate determination of the singularity locations. By solving an additional linear algebraic system for the jumps of the function and its derivatives up to the arbitrarily specified order at the calculated singularity locations, we are able to reconstruct the 2π-periodic function of finite regularity as the sum of a piecewise polynomial function and a function which is continuously differentiab1e up to the specified order

The way of analysis

Abstract: The Way of Analysis gives a thorough account of real analysis in one or several variables, from the construction of the real number system to an introduction of the Lebesgue integral The text provides proofs of all main results, as well as motivations, examples, applications, exercises, and formal chapter summaries Additionally, there are three chapters on application of analysis, ordinary differential equations, Fourier series, and curves and surfaces to show how the techniques of analysis are used in concrete settings

Quantitative Evaluation of Soybean (Glycine max L. Merr.) Leaflet Shape by Principal Component Scores Based on Elliptic Fourier Descriptor

Abstract: Leaflet shape of thirty-nine soybean cultivars/strains selected to cover the possible diversity of leaf shape, was quantitatively evaluated by principal components scores based on the elliptic Fourier descriptor of contours. After central leaflets of fully expanded compound-leaves of the cultivars/strains were videotaped, binary images of the leaflets were obtained from those video images by image processing. Then, the closed contour of each leaflet was extracted from the binary images and chain-coded by image processing. Because the first twenty harmonics could sufficiently represent soybean leaf contours, 77 elliptic Fourier coefficients were calculated for each chain-coded contour. Then, the Fourier coefficients were standardized so that the coefficients were invariant of the size, rotation, shift and chain-code starting-point of any contour. The principal component analysis about the standardized Fourier coefficients, showed that the cumulative contribution at the fifth principal component was about 96 o/o' Moreover, the effect of each principal component on the leaf shape was clarified by drawing the contours of leaflets using the Fourier coefficients inversely estimated under some typical values of the principal component scores. Consequently, it was indicated that the principal components scores about the standardized elliptic Fourier coefficients gave us powerful quantitative measures to evaluate soybean leaf shape. The analysis of variance and multiple comparison indicated that the genotypic differences on the first, the second and the fifth principal components were significantly large. Because the variations of those principal components were con-tinuous, the effects of the polygenes on the (size-invariant) shape were also suggested.

Fourier series and wavelets

Abstract: OF THESIS Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science Mathematics The University of New Mexico Albuquerque, New Mexico

Fourier-based Separation Technique for Shape Grading of Potatoes Using Machine Vision

Abstract: A Fourier-based shape separation method was developed for shape grading of potatoes using machine vision for automated inspection. The relationship between object shape and its boundary spectrum values in Fourier domain was explored for shape extraction. A new and fast method of using Greens theorem and boundary Fourier coefficients was given for estimating elongation of an object. A shape separator based on harmonics of the transform was defined for potato shape separation. Tests showed the shape separator was effective and efficient for difficult shape separation. The machine vision system developed has a great potential to assist humans for automated potato grading.

Monolithic optical waveguide filters based on Fourier expansion

Abstract: In accordance with the invention, a new type of monolithic optical waveguide filter comprises a chain of optical couplers of different effective lengths linked by differential delays of different lengths. The transfer of the chain of couplers and delays is the sum of contributions from all possible optical paths, each contribution forming a term in a Fourier series whose sum forms the optical output. A desired frequency response is obtained by optimizing the lengths of the couplers and the delay paths so that the Fourier series best approximates the desired response. The filter is advantageously optimized so that it is insensitive to uncontrolled fabrication errors and is short in length. The wavelength dependence of practical waveguide properties is advantageously incorporated in the optimization. Consequently, the filter is highly manufacturable by mass production. Such filters have been shown to meet the requirements for separating the 1.3 and 1.551 μm telecommunications channels and for flattening the gain of Er amplifiers.

Linear Circuit Analysis

Abstract: Lapice transfrom analysis- basics Laplace transfrom analysis - circuit applications Laplace transform analysis - transfer function applications time domain circuit response computations the convolution method resonant and bandpass circuits magnetically coupled circuits and transformers two-ports analysis of interconnected two-ports principles of basic filtering Fourier series with applications to electronic circuits. Appendices: introduction to magnetic circuit analysis SPICE for basic circuits.

Self-diffusion in a periodic porous medium: A comparison of different approaches.

Abstract: We compare two different approaches for computing the propagator for a particle diffusing in a fluid filled porous medium, where the pore space has a periodic structure and some absorption of the particle can occur at the pore-matrix interface. One of these approaches is based on computer simulations of a random walker in this structure, while the other is based on an explicit calculation of the diffusion eigenstates using a Fourier series expansion of the diffusion equation. Both methods are applied to the same nondilute model systems in order to calculate the wave-vector and time-dependent nuclear magnetization measured in pulsed-field-gradient-spin-echo experiments. When the physical parameters are confined to the range of values found in most systems of interest, good quantitative agreement is found between the two methods. However, as the interfacial relaxation strength, the time, or the wave vector becomes large, calculations based on eigenstate expansion are more stable and less subject to the sampling problems inherent in random walk simulations. In the absence of surface relaxation, our calculations are also used to test the results predicted by a recently proposed ansatz for the behavior of the diffusion propagator. Finally, a problem is identified and discussed regarding the relation between random walk and continuum diffusion treatments of interface absorption. © 1995 The American Physical Society.

The inverse scattering transform: Tools for the nonlinear fourier analysis and filtering of ocean surface waves

Abstract: Surface wave data from the Adriatic Sea are analysed in the light of new data analysis techniques which may be viewed as a nonlinear generalization of the ordinary Fourier transform. Nonlinear Fourier analysis as applied herein arises from the exact spectral solution to large classes of nonlinear wave equations which are integrable by the inverse scattering transform (IST). Numerical methods are discussed which allow for implementation of the approach as a tool for the time series analysis of oceanic wave data. The case for unidirectional propagation in shallow water, where integrable nonlinear wave motion is governed by the Korteweg-deVries (KdV) equation with periodic/quasi-periodic boundary conditions, is considered. Numerical procedures given herein can be used to compute a nonlinear Fourier representation for a given measured time series. The nonlinear oscillation modes (the IST ‘basis functions’) of KdV obey a linear superposition law, just as do the sine waves of a linear Fourier series. However, the KdV basis functions themselves are highly nonlinear, undergo nonlinear interactions with each other and are distinctly non-sinusoidal. Numerical IST is used to analyse Adriatic Sea data and the concept of nonlinear filtering is applied to improve understanding of the dominant nonlinear interactions in the measured wavetrains.

Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff

Abstract: By transforming the infinite x-y plane onto a unit square and using two-dimensional Fourier series expansions, the modal fields and propagation constants of dielectric waveguides are accurately determined within the scalar (weak-guidance) regime. The new method is reliable down to modal cutoff and gives cutoff V-values directly. Numerical cutoff values for the LP/sub 11/ modes of square- and rectangular-core waveguides are determined as a function of core aspect ratio, and are found to agree with those obtained by the finite element method to within 0.1%. >

Uniform strict convexity of a cost functional for three-dimensional inverse scattering problem

Abstract: An inverse problem of determination of the coefficient $a(x)$ in the equation $u_{tt} = \Delta u + a(x)u,x \in \mathbb{R}^3 ,t \in (0,T)$ is considered with initial conditions $u(x,0) = 0,u_t (x,0) = \delta (x)$, and some additional data that can be treated as backscattering information. The goal is to develop a finite-dimensional technique that would be a basis for future computations. We reduce our inverse scattering problem to an equivalent Cauchy problem for a nonlinear hyperbolic integrodifferential equation with the data on the lateral side of a time cylinder. It is assumed that the solution $v(x,t)$ of this equation has the form $v(x,t) = p(x,t) + w(x,t)$, where function $p(x,t)$ is given and function $w(x,t)$ is unknown and has a finite number of nonzero Fourier coefficients. In particular, function $p(x,t)$ can be considered as a first guess. A special cost-functional $J_\lambda (w)$ dependent on a large parameter $\lambda $ is introduced. The main result of this paper is Theorem 1.1. By this the...

The modelling of the opening and closure of a crack

Abstract: The paper presents a method which utilizes substructure normal modes to predict the vibration properties of a cantilever beam with a breathing transverse crack. The two segments of the cantilever beam, separated by the crack are related to one another by time varying connection matrices representing the interaction forces. The connection matrices are expanded in a Fourier series leading to a classical eigenvalue problem. Subsequently, the initial formulation is extended to avoid interference of the crack interfaces with a time domain formulation. The Lagrange multipliers, used to enforce the exact continuity constraints when the crack is closed, produce the interfaces forces needed for the modelling of interface dry friction.

The use of Fourier descriptors in the classification of particle shape

Abstract: A new quantitative method for characterizing quartz grain shape is presented. The method employs a harmonic analysis based upon Fourier descriptors which is a distinct variation of the traditional and widely used Fourier series. Quartz grain images from a scanning electron microscope were ‘frame grabbed’and converted to a digitized grey-level image. The image processing techniques of enhancement, segmentation and boundary tracking were applied to remove all features except the image boundary. This boundary was sampled at uniform intervals of are length and represented mathematically on the complex plane. In this way problems associated with the location of particle centroid and re-entrant values were avoided. The resulting data was standardized relative to scale, rotation and starting position. Hence the discrete Fourier transform was applied using modern fast Fourier transform techniques and the modulus of the resulting harmonic amplitude used to characterize the grain shape. The technique was applied to a sample of 0–5-m quartz grains from three distinct populations: desert quartz, beach grains (Fire Island, New York) and Brazilian crushed quartz. Whilst plots of average amplitude vs. harmonic number for each population appeared similar, discriminant analysis applied to each grain sample distinguished characteristic grain shape with an excellent degree of success. The problems of location of the centroid and re-entrant values were eliminated. This allowed the technique to be applied to a much wider group of irregularly shaped sedimentary particles such as loess.

Fourier analysis of human soft tissue facial shape : sex differences in normal adults

Abstract: Sexual dimorphism in human facial form involves both size and shape variations of the soft tissue structures. These variations are conventionally appreciated using linear and angular measurements, as well as ratios, taken from photographs or radiographs. Unfortunately this metric approach provides adequate quantitative information about size only, eluding the problems of shape definition. Mathematical methods such as the Fourier series allow a correct quantitative analysis of shape and of its changes. A method for the reconstruction of outlines starting from selected landmarks and for their Fourier analysis has been developed, and applied to analyse sex differences in shape of the soft tissue facial contour in a group of healthy young adults. When standardised for size, no sex differences were found between both cosine and sine coefficients of the Fourier series expansion. This shape similarity was largely overwhelmed by the very evident size differences and it could be measured only using the proper mathematical methods.

Analytical Expressions for the Relaxation Moduli of Linear Viscoelastic Composites With Periodic Microstructure

Abstract: Many micromechanical models have been used to estimate the overall stiffness of heterogeneous- materials and a large number of results and experimental data have been obtained. However, few theoretical and experimental results are available in the field of viscoelastic behavior of heterogeneous media. In this paper the viscoelastostatic problem of composite materials with periodic microstructure is studied. The matrix is assumed linear viscoelastic and the fibers elastic. The correspondence principle in viscoelasticity is applied and the problem in the Laplace domain is solved by using the Fourier series technique and assuming the Laplace transform of the homogenization eigenstrain piecewise constant in the space. Formulas for the Laplace transform of the relaxation functions of the composite are obtained in terms of the properties of the matrix and the fibers and in function of nine triple series which take in account the geometry of the inclusions. The inversion to the time domain of the relaxation and the creep functions of composites reinforced by long fibers is carried out analytically when the four parameters model is used to represent the viscoelastic behavior of the matrix. Finally, comparisons with experimental results are presented.

An integral equation method for elastostatics of periodic composites

Abstract: An interface integral equation is presented for the elastostatic problem in a two-dimensional isotropic composite. The displacement is represented by a single layer force density on the component interfaces. In a simple numerical example involving hexagonal arrays of disks the force density is expanded in a Fourier series. This leads to an algorithm with superalgebraic convergence. The integral equation is solved to double precision accuracy with a modest computational effort. Effective moduli are extracted both for dilute arrays where previously three digit accurate results were available, and for dense arrays where previously no results were available.