Showing papers on "Fourier series published in 1990"

Convex models of uncertainty in applied mechanics

Abstract: 1. Probabilistic Modelling: Pros and Cons. Preliminary considerations. Probabilistic modelling in mechanics. Reliability of structures. Sensitivity of failure probability. Some quotations on the limitations of probabilistic methods. 2. Mathematics of Convexity. Convexity and Uncertainty. What is convexity? Geometric convexity in the Euclidean plane. Algebraic convexity in Euclidean space. Convexity in function spaces. Set-convexity and function-convexity. The structure of convex sets. Extreme points and convex hulls. Extrema of linear functions on convex sets. Hyperplane separation of convex sets. Convex models. 3. Uncertain Excitations. Introductory examples. The massless damped spring. Excitation sets. Maximum responses. Measurement optimization. Vehicle vibration. Introduction. The vehicle model. Uniformly bounded substrate profiles. Extremal responses on uniformly bounded substrates. Duration of acceleration excursions on uniformly bounded substrates. Substrate profiles with bounded slopes. Isochronous obstacles. Solution of the Euler-Lagrange equations. Seismic excitation. Vibration measurements. Introduction. Damped vibrations: full measurement. Example: 2-dimensional measurement. Damped vibrations: partial measurement. Transient vibrational acceleration. 4. Geometric Imperfections. Dynamics of thin bars. Introduction. Analytical formulation. Maximum deflection. Duration above a threshold. Maximum integral displacements. Impact loading of thin shells. Introduction. Basic equations. Extremal displacement. Numerical example. Buckling of thin shells. Introduction. Bounded Fourier coefficients: first-order analysis. Bounded Fourier coefficients: second-order analysis. Uniform bounds on imperfections. Envelope bounds on imperfections. Estimates of the knockdown factor. First and second-order analyses. 5. Concluding Remarks. Bibliography. Index.

Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers

Abstract: A description is given of a relatively simple derivation of the bit-error probability for a lightwave communications system using an amplitude-shift-keying (ASK) pulse modulation format and employing optical amplifiers such that amplified spontaneous emission noise dominates all other noise sources Mathematically, this noise is represented as a Fourier series expansion with Fourier coefficients that are assumed to be independent Gaussian random variables The bit-error probability is given in a closed analytical form that is derived by the approximate evaluation of several integrals appearing in the analysis The author uses the theory to derive the Gaussian approximation and finds that it overestimates the bit-error rate by one to two orders of magnitude >

The discrete picard condition of discrete ill-posed problems

Abstract: We investigate the approximation properties of regularized solutions to discrete ill-posed least squares problems. A necessary condition for obtaining good regularized solutions is that the Fourier coefficients of the right-hand side, when expressed in terms of the generalized SVD associated with the regularization problem, on the average decay to zero faster than the generalized singular values. This is the discrete Picard condition. We illustrate the importance of this condition theoretically as well as experimentally.

Neural networks for system identification

Abstract: Two approaches are presented for utilization of neural networks in identification of dynamical systems. In the first approach, a Hopfield network is used to implement a least-squares estimation for time-varying and time-invariant systems. The second approach, which is in the frequency domain, utilizes a set of orthogonal basis functions and Fourier analysis to construct a dynamic system in terms of its Fourier coefficients. Mathematical formulations are presented, along with simulation results. >

Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables

Abstract: Using the theta correspondence we construct liftings from the cohomology with compact supports of locally symmetric spaces associated to O(p, q) (resp. U(p, q)) of degreenq (resp. Hodge typenq, nq) to the space of classical holomorphic Siegel modular forms of weight (p +q)/2 and genusn (resp. holomorphic hermitian modular forms of weightp +q and genusn). It is important to note that the cohomology with compact supports contains the cuspidal harmonic forms by Borel [3]. We can express the Fourier coefficients of the lift of η in terms of periods of η over certain totally geodesic cycles—generalizing Shintani’s solution [21] of a conjecture of Shimura. We then choose η to be the Poincare dual of a (finite) cycle and obtain a collection of formulas analogous to those of Hirzebruch-Zagier [8]. In our previous work we constructed the above lifting but we were unable to prove that it took values in theholomorphic forms. Moreover, we were unable to compute the indefinite Fourier coefficients of a lifted class. By Koecher’s Theorem we may now conclude that all such coefficients are zero.

Dispersion Investigation in the Split Hopkinson Pressure Bar

Abstract: Dispersion of an elastic wave propagating in a 76.2-mm-diameter (3 in.) Split Hopkinson Pressure Bar system was investigated with two consecutive pulses recorded in the transmitter bar. Assuming that the dispersive high frequency oscillatory components riding on the top of the main pulse originate from the first mode vibration, the dispersion was corrected by using the Fast Fourier Transform and Fourier series expansion numerical schemes

Group Theoretical Methods in Image Processing

Abstract: Preliminaries.- Representations of groups.- Representations of somes matrix groups.- Fourier series on compact groups.- Applications.

Solution of the equations of dynamic elasticity by a Chebychev spectral method

Abstract: We present a spectral method for solving the two‐dimensional equations of dynamic elasticity, based on a Chebychev expansion in the vertical direction and a Fourier expansion for the horizontal direction. The technique can handle the free‐surface boundary condition more rigorously than the ordinary Fourier method. The algorithm is tested against problems with known analytic solutions, including Lamb’s problem of wave propagation in a uniform elastic half‐space, reflection from a solid‐solid interface, and surface wave propagation in a haft‐space containing a low‐velocity layer. Agreement between the solutions is very good. A fourth example of wave propagation in a laterally heterogeneous structure is also presented. Results indicate that the method is very accurate and only about a factor of two slower than the Fourier method.

Analysis of Simply-Supported Orthotropic Cylindrical Shells Subject to Lateral Impact Loads

Abstract: The analysis is based on an expansion of the loads, displacements, and rotations in a double Fourier series which satisfies the end boundary conditions of simple support. By neglecting in-plane and rotary inertia the problem becomes a second-order ordinary differential equation in time for the Fourier coefficients of the radial deflection

Advanced Engineering Mathematics

Abstract: Part I. Review materials Part II. Vectors and matrices Part III. Ordinary differential equations Part IV. Fourier Series, integrals, and the Fourier transform Part V. Vector calculus Part VI. Complex analysis Part VIII. Numerical mathematics Answers References.

The theoretical, discrete, and actual response of the Barnes objective analysis scheme for one- and two-dimensional fields

Abstract: This paper examines the response of the Barnes objective analysis scheme as a function of wavenumber or wavelength and extends previous work in two primary areas. First, the first- and second-pass theoretical response functions for continuous two-dimensional (2-D) fields are derived using Fourier transforms and compared with Barnes' (1973) responses for one-dimensional (1-D) waves. All responses are nondimensionalized with respect to a smoothing scale length, such that the first-pass responses are a function only of nondimensional wavelength. The 2-D response is of the same functional form as the 1-D response, with the 2-D wavenumber substituted for the 1-D wavenumber. The 2-D response departs significantly from the 1-D value (for the same x-component of the wavelength) when the y-component of the wavelength is less than approximately ten scale lengths, a condition applying to most fields with closed centers as well as open waves with a significant latitudinal variation. Second, the continuous th...

Fourier-Based Optimal Control of Nonlinear Dynamic Systems

Abstract: A method for generating near optimal trajectories of linear and nonlinear dynamic systems, represented by deterministic, lumped-parameter models, is proposed. The method is based on a Fourier series approximation of each generalized coordinate that converts the optimal control problem into an algebraic nonlinear programming problem. The results of computer simulation studies compare favorably to optimal solutions obtained by closed-form analyses and/or by other numerical schemes

The wavelet transform - Some applications to fluid dynamics and turbulence

Abstract: The basic definitions and the most attractive properties of the wavelet transform are reviewed and explained using the classical language of turbulence The wavelet transform appears to be a natural alternative to the decompositions commonly used in fluid dynamics and turbulence (mainly the Fourier decomposition)

Electrical impedance tomography of translationally uniform cylindrical objects with general cross-sectional boundaries

Abstract: An algorithm is developed for electrical impedance tomography (EIT) of finite cylinders with general cross-sectional boundaries and translationally uniform conductivity distributions. The electrodes for data collection are assumed to be placed around a cross-sectional plane; therefore, the axial variation of the boundary conditions and the potential field are expanded in Fourier series. For each Fourier component a two-dimensional (2-D) partial differential equation is derived. Thus the 3-D forward problem is solved as a succession of 2-D problems, and it is shown that the Fourier series can be truncated to provide substantial savings in computation time. The finite element method is adopted and the accuracy of the boundary potential differences (gradients) thus calculated is assessed by comparison to results obtained using cylindrical harmonic expansions for circular cylinders. A 1016-element and 541-node mesh is found to be optimal. The algorithm is applied to data collected from phantoms, and the errors incurred from the several assumptions of the method are investigated. >

Cascade model for fluvial geomorphology

Abstract: Erosional landscapes are generally scale invariant and fractal. Spectral studies provide quantitative confirmation of this statement. Linear theories of erosion will not generate scale-invariant topography. In order to explain the fractal behavior of landscapes a modified Fourier series has been introduced that is the basis for a renormalization approach. A nonlinear dynamical model has been introduced for the decay of the modified Fourier series coefficients that yield a fractal spectra. It is argued that a physical basis for this approach is that a fractal (or nearly fractal) distribution of storms (floods) continually renews erosional features on all scales.

Introductory Signal Processing

Abstract: A valuable introduction to the fundamentals of continuous and discrete time signal processing, this book is intended for the reader with little or no background in this subject. The emphasis is on development from basic principles. With this book the reader can become knowledgeable about both the theoretical and practical aspects of digital signal processing.Some special features of this book are: (1) gradual and step-by-step development of the mathematics for signal processing, (2) numerous examples and homework problems, (3) evolutionary development of Fourier series, Discrete Fourier Transform, Fourier Transform, Laplace Transform, and Z-Transform, (4) emphasis on the relationship between continuous and discrete time signal processing, (5) many examples of using the computer for applying the theory, (6) computer based assignments to gain practical insight, (7) a set of computer programs to aid the reader in applying the theory.

Stability and harmonics in thyristor controlled reactors

Abstract: Harmonics that arise from the interaction of thyristor controlled reactors (TCRs) and power systems can sometimes cause stability problems. The classical method for calculating harmonics is to calculate the harmonic current assuming an infinite bus at the high side of the TCR transformer. This current is then used as a harmonic current source on the AC system. The basic problem with this method is that many of the interactions between the AC system and the TCR are neglected. Two methods for studying the neglected interactions are described. The first uses state variables to analyze the circuit containing the TCR. The resulting equations are linear differential equations with periodic coefficients. This formulation allows the study of stability, periodic operation, and resonance, which cannot be achieved by other methods. The second method uses a Fourier matrix description of the TCR. In this model the coupling between the different harmonics due to the switching is clearly shown. >

The Helgason Fourier transform for compact Riemannian symmetric spaces of rank one

Abstract: For Riemannian symmetric spaces (RSS) of noncompact type Helgason [2], [3], [5], found the analog of classical Fourier analysis. This paper concerns the counterpart of Helgason's theory for RSS of compact type. Together with classical Fourier theory these results constitute a unified Fourier analysis on RSS related to, but distinct from the established alternatives of representation theory and the spherical Fourier transform. An advantage of this style of Fourier theory is that the transform kernel has the same kind of simplicity as the functions e/xy of classical Fourier theory. In particular, the transform kernel employs scalar-valued eigenfunctions of first order differential operators. On the other hand, the theory given here for the compact RSS involves a severe singularity in part of the transform kernel. One may avoid this singularity by confining consideration to the half of the RSS closest to the origin; call this the local theory. The local theory is given in Section 1 for any compact RSS. The rest of this paper is devoted to the global theory for compact RSS of rank one. (It is not clear that a global theory exists for the higher rank compact RSS.) In broad outline, Helgason's transform comes from the wedding of the spherical Fourier transform with an integral formula for the Poisson kernel. In Helgason's notation ([5], p. 418) this formula is

Optimal Design for Plate Buckling

Abstract: This paper extends Keller’s classic solution for the optimal design of columns to the case of plates. After the introduction of nondimensional variables, a calculus of variations technique is used to derive an optimality condition, which states that “the strain-energy density is proportional to the thickness in an optimal plate design.” The case of a simply supported plate is then discussed, using a truncated Fourier series solution. For a square, isotropic plate, the buckling load of the optimal design is larger than that of the uniform plate by a factor of 1.71 + 1.37 υ\N (υ\N is Poisson’s ratio). An appendix is included which discusses Keller’s original solution and shows how it can be applied when a fourth-order differential equation is used rather than the second-order one used by Keller. This appendix also discusses the use of a truncated Fourier series and Rayleigh’s method as an approximation of Keller’s result, thus laying the groundwork for the use of Fourier series in the plate problem.

Optimal control of linear time-varying systems via Fourier series

Abstract: The optimal control of linear time-varying systems with quadratic cost functional is obtained by Fourier series approximation. The properties of Fourier series are first briefly presented and the operational matrix of backward integration together with the product operational matrix are utilized to reduce the optimal control problem to a set of simultaneous linear algebraic equations. An illustrative example is included to demonstrate the validity and applicability of the technique.

About the Coulomb potential in crystals

Abstract: The Coulomb potential in a crystal is discussed. It is shown that its Fourier series expansion has a singularity for the V(0, 0, 0) component, which is important when comparing different compounds, or when using the Coulomb potential as a probe for reactivity. Methods to calculate this term are discussed. Sum rules for multipolar moments of crystals in terms of structure factors are derived, which are of interest for the comparison of microscopic and macroscopic dielectric properties.

Equivalence of Green’s Function and the Fourier Series Representation of Composites with Periodic Microstructure

Abstract: This work is concerned with modeling the nonlinear mechanical deformation of composites comprised of a periodic microstructure under small displacement conditions at elevated temperatures. The practical motivation for such work stems from the need to design and optimize new multiphase materials and to predict their micromechanical and bulk material behavior under in-service thermomechanical loading conditions.