Showing papers on "Fourier series published in 2001"

Introduction to Fourier analysis and wavelets

Abstract: This book provides a concrete introduction to a number of topics in harmonic analysis, accessible at the early graduate level or, in some cases, at an upper undergraduate level. Necessary prerequisites to using the text are rudiments of the Lebesgue measure and integration on the real line. It begins with a thorough treatment of Fourier series on the circle and their applications to approximation theory, probability, and plane geometry (the isoperimetric theorem). Frequently, more than one proof is offered for a given theorem to illustrate the multiplicity of approaches. The second chapter treats the Fourier transform on Euclidean spaces, especially the author's results in the three-dimensional piecewise smooth case, which is distinct from the classical Gibbs - Wilbraham phenomenon of one-dimensional Fourier analysis. The Poisson summation formula treated in Chapter 3 provides an elegant connection between Fourier series on the circle and Fourier transforms on the real line, culminating in Landau's asymptotic formulas for lattice points on a large sphere. Much of modern harmonic analysis is concerned with the behavior of various linear operators on the Lebesgue spaces Lp (Rn). Chapter 4 gives a gentle introduction to these results, using the Riesz - Thorin theorem and the Marcinkiewicz interpolation formula. One of the long-time users of Fourier analysis is probability theory. In Chapter 5 the central limit theorem, iterated log theorem, and Berry - Esseen theorems are developed using the suitable Fourier-analytic tools. The final chapter furnishes a gentle introduction to wavelet theory, depending only on the L2 theory of the Fourier transform (the Plancherel theorem). The basic notions of scale and location parameters demonstrate the flexibility of the wavelet approach to harmonic analysis. The text contains numerous examples and more than 200 exercises, each located in close proximity to the related theoretical material.

Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media.

Abstract: We establish the most general differential equations that are satisfied by the Fourier components of the electromagnetic field diffracted by an arbitrary periodic anisotropic medium. The equations are derived by use of the recently published fast-Fourier-factorization (FFF) method, which ensures fast convergence of the Fourier series of the field. The diffraction by classic isotropic gratings arises as a particular case of the derived equations; the case of anisotropic classic gratings was published elsewhere. The equations can be resolved either through classic differential theory or through the modal method for particular groove profiles. The new equations improve both methods in the same way. Crossed gratings, among which are grids and two-dimensional arbitrarily shaped periodic surfaces, appear as particular cases of the theory, as do three-dimensional photonic crystals. The method can be extended to nonperiodic media through the use of a Fourier transform.

An introduction to Laplace transforms and Fourier series

Abstract: 1. The Laplace Transform.- 1.1 Introduction.- 1.2 The Laplace Transform.- 1.3 Elementary Properties.- 1.4 Exercises.- 2. Further Properties of the Laplace Transform.- 2.1 Real Functions.- 2.2 Derivative Property of the Laplace Transform.- 2.3 Heaviside's Unit Step Function.- 2.4 Inverse Laplace Transform.- 2.5 Limiting Theorems.- 2.6 The Impulse Function.- 2.7 Periodic Functions.- 2.8 Exercises.- 3. Convolution and the Solution of Ordinary Differential Equations.- 3.1 Introduction.- 3.2 Convolution.- 3.3 Ordinary Differential Equations.- 3.3.1 Second Order Differential Equations.- 3.3.2 Simultaneous Differential Equations.- 3.4 Using Step and Impulse Functions.- 3.5 Integral Equations.- 3.6 Exercises.- 4. Fourier Series.- 4.1 Introduction.- 4.2 Definition of a Fourier Series.- 4.3 Odd and Even Functions.- 4.4 Complex Fourier Series.- 4.5 Half Range Series.- 4.6 Properties of Fourier Series.- 4.7 Exercises.- 5. Partial Differential Equations.- 5.1 Introduction.- 5.2 Classification of Partial Differential Equations.- 5.3 Separation of Variables.- 5.4 Using Laplace Transforms to Solve PDEs.- 5.5 Boundary Conditions and Asymptotics.- 5.6 Exercises.- 6. Fourier Transforms.- 6.1 Introduction.- 6.2 Deriving the Fourier Transform.- 6.3 Basic Properties of the Fourier Transform.- 6.4 Fourier Transforms and PDEs.- 6.5 Signal Processing.- 6.6 Exercises.- 7. Complex Variables and Laplace Transforms.- 7.1 Introduction.- 7.2 Rudiments of Complex Analysis.- 7.3 Complex Integration.- 7.4 Branch Points.- 7.5 The Inverse Laplace Transform.- 7.6 Using the Inversion Formula in Asymptotics.- 7.7 Exercises.- A. Solutions to Exercises.- B. Table of Laplace Transforms.- C. Linear Spaces.- C.1 Linear Algebra.- C.2 Gramm-Schmidt Orthonormalisation Process.

Oscillations and Waves

Abstract: 1 Introduction.- 2 Free Oscillations.- 3 Forced Oscillations.- 4 Kinematics of Systems.- 5 Transfer Systems.- 6 Instability and Chaos.- 7 Linear Waves.- 8 Nonlinear Waves.- 9 Standing Waves.- A Appendix.- A.1 Fourier Series.- A.1.1 General Rules.- A.1.2 Real Periodic Functions.- A.2 Fourier Transformation.- A.2.1 General Rules.- A.2.2 Real Functions.- A.3 Laplace Transformation.- A.3.1 General Rules.- A.3.2 Heaviside and Dirac Functions.- A.3.3 Real Functions.- A.4 Convolution (Faltung).- A.4.1 General Rules.- A.4.2 Heaviside Unit Step.- A.4.3 Special Real Functions.- A.4.4 Hilbert Transformation.- References.- B Books.- J Publications in Journals.

Fourier analysis and partial differential equations

Abstract: Part I. Fourier Series and Periodic Distributions: 1. Preliminaries 2. Fourier series: basic theory 3. Periodic distributions and Sobolev spaces Part II. Applications to Partial Differential Equations: 4. Linear equations 5. Nonlinear evolution equations 6. The Korteweg-de Vries Part III. Some Nonperiodic Problems: 7. Distributions, Fourier transforms and linear equations 8. KdV, BO and friends Appendix A. Tools from the theory of ODEs Appendix B. Commutator estimates Bibliography Index.

A Padé-based algorithm for overcoming the Gibbs phenomenon

Abstract: Truncated Fourier series and trigonometric interpolants converge slowly for functions with jumps in value or derivatives The standard Fourier–Pade approximation, which is known to improve on the convergence of partial summation in the case of periodic, globally analytic functions, is here extended to functions with jumps The resulting methods (given either expansion coefficients or function values) exhibit exponential convergence globally for piecewise analytic functions when the jump location(s) are known Implementation requires just the solution of a linear system, as in standard Pade approximation The new methods compare favorably in experiments with existing techniques

Electrothermal CAD of power devices and circuits with fully physical time-dependent compact thermal modeling of complex nonlinear 3-d systems

Abstract: An original, fully analytical, spectral domain decomposition approach is presented for the time-dependent thermal modeling of complex nonlinear (3-D) electronic systems, from metallized power FETs and MMICs, through MCMs, up to circuit board level. This solution method offers a powerful alternative to conventional numerical thermal simulation techniques, and is constructed to be compatible with explicitly coupled electrothermal device and circuit simulation on CAD timescales. In contrast to semianalytical, frequency space, Fourier solutions involving DFT-FFT, the method presented here is based on explicit, fully analytical, double Fourier series expressions for thermal subsystem solutions in Laplace transform s-space (complex frequency space). It is presented in the form of analytically exact thermal impedance matrix expressions for thermal subsystems. These include double Fourier series solutions for rectangular multilayers, which are an order of magnitude faster to evaluate than existing semi-analytical Fourier solutions based on DFT-FFT. They also include double Fourier series solutions for the case of arbitrarily distributed volume heat sources and sinks, constructed without the use of Green's function techniques, and for rectangular volumes with prescribed fluxes on all faces, removing the adiabatic sidewall boundary condition. This combination allows treatment of arbitrarily inhomogeneous complex geometries, and provides a description of thermal material nonlinearities as well as inclusion of position varying and non linear surface fluxes. It provides a fully physical, and near exact, generalized multiport network parameter description of nonlinear, distributed thermal subsystems, in both the time and frequency domains. In contrast to existing circuit level approaches, it requires no explicit lumped element, RC-network approximation or nodal reduction, for fully coupled, electrothermal CAD. This thermal impedance matrix approach immediately gives rise to minimal boundary condition independent compact models for thermal systems. Implementation of the time-dependent thermal model as N-port netlist elements within a microwave circuit simulation engine, Transim (NCSU), is described. Electrothermal transient, single-tone, two-tone, and multitone harmonic balance simulations are presented for a MESFET amplifier. This thermal model is validated experimentally by thermal imaging of a passive grid array representative of one form of spatial power combining architecture.

High‐frequency bistatic cross sections of the ocean surface

Abstract: An analysis leading to the first and second-order bistatic cross sections of the ocean surface in the context of high-frequency ground wave radar operation is presented. Initially, a pulsed dipole source is introduced into the previously derived electric field expressions for the bistatic reception of vertically polarized radiation scattered from rough surfaces that do not vary with time. To make application to the ocean, a time-varying surface is introduced via a three-dimensional Fourier series with two spatial variables and one temporal variable. The surface randomness is accounted for by allowing the Fourier coefficients to be zero-mean Gaussian random variables. Fourier transformation of the autocorrelations of the resulting fields gives the appropriate power spectral densities. The latter are used in the bistatic radar range equation to produce the cross sections. The features of the bistatic case are seen to reduce to the well-known monostatic results when the appropriate geometry is introduced. Illustrative comparisons of monostatic and bistatic reception are presented.

Bounds for automorphic L-functions. II

Abstract: We continue our study of GL2 L–functions with the aim of providing upper bounds for their order of magnitude. As is familiar it suffices to provide such bounds on the critical line and, both for the sake of applications and for the ideas involved, we are most interested in breaking the convexity bound and this with respect to the conductor. In this paper we are interested primarily in L–functions attached to characters of the class group of the imaginary quadratic field K = Q(√−D). We are motivated by our paper [DFI4]. That work was not included in the current series because the class group L–functions are treated there directly. They may however be viewed as L–functions associated to cusp forms of weight 1, level D and character (the nebentypus) χD(n) = (−D n )

Principles of Fourier Analysis

Abstract: PRELIMINARIES The Starting Point Basic Terminology, Notation, and Conventions Basic Analysis I: Continuity and Smoothness Basic Analysis II: Integration and Infinite Series Symmetry and Periodicity Elementary Complex Analysis Functions of Several Variables FOURIER SERIES Heuristic Derivation of the Fourier Series Formulas The Trigonometric Fourier Series Fourier Series over Finite Intervals (Sine and Cosine Series) Inner Products, Norms, and Orthogonality The Complex Exponential Fourier Series Convergence and Fourier's Conjecture Convergence and Fourier's Conjecture: The Proofs Derivatives and Integrals of Fourier Series Applications CLASSICAL FOURIER TRANSFORMS Heuristic Derivation of the Classical Fourier Transform Integrals on Infinite Intervals The Fourier Integral Transforms Classical Fourier Transforms and Classically Transformable Functions Some Elementary Identities: Translation, Scaling, and Conjugation Differentiation and Fourier Transforms Gaussians and Other Very Rapidly Decreasing Functions Convolution and Transforms of Products Correlation, Square-Integrable Functions, and the Fundamental Identity of Fourier Analysis Identity Sequences Generalizing the Classical Theory: A Naive Approach Fourier Analysis in the Analysis of Systems Gaussians as Test Functions, and Proofs of Some Important Theorems GENERALIZED FUNCTIONS AND FOURIER TRANSFORMS A Starting Point for the Generalized Theory Gaussian Test Functions Generalized Functions Sequences and Series of Generalized Functions Basic Transforms of Generalized Fourier Analysis Generalized Products, Convolutions, and Definite Integrals Periodic Functions and Regular Arrays General Solutions to Simple Equations and the Pole Functions THE DISCRETE THEORY Periodic, Regular Arrays Sampling and the Discrete Fourier Transform APPENDICES

Vibration of Strongly Nonlinear Discontinuous Systems

Abstract: 1 Operators of Linear Systems.- 1. Dynamic Compliance.- 1.1. Operator of Mechanical System.- 1.2 Fundamental Features of the Generalised Dirac ?-function.- 1.3. Green Functions for Systems with Lumped Parameters.- 1.4. Operator of Dynamic Compliance.- 1.5. The Eigenmode Decomposition of the Dynamic Compliance Operator.- 1.6. Linear System as a Low-pass Filter.- 1.7. Linear Single-Degree-of-Freedom System.- 1.8. Operators of Rod Systems.- 1.9. Expression of Forces Through Operator Functions.- 1.10. Some Generalisations.- 2. Periodic Green Functions.- 2.1. Periodic Generalised Functions.- 2.2. Periodic Green Functions.- 2.3. Features of Periodic Green Functions.- 2.4. Periodic Green Function on the Interval of Periodicity.- 2.5. Single-Degree-of-Freedom System.- 2.6. Eigenfunction Expansion of PGF.- 2.7. Steady-state Motion.- 2.8. Representation of PGF in the Form of Fast Convergent Fourier Series.- 3 Parametric Periodic Green Functions.- 3.1. Integral Equations of Periodic Vibration.- 3.2. Integral Fredholm Equations.- 3.3. Description of Parametric Periodic Green Functions.- 3.4. Excitation of Parametric Vibration by Impacts.- 2 Strongly Nonlinear Single-Degree-of Freedom Systems.- 4 Conservative Systems.- 4.1. Classification of Nonlinear Systems.- 4.2. Equations of Conservative Systems.- 4.3. Vibro-impact Systems.- 4.4. Singular Force of Impact.- 4.5. Motions of Vibro-impact Systems.- 4.6. Strongly Nonlinear Systems.- 4.7. Strongly Nonlinear Systems of Threshold Type.- 4.8. Singularisation.- 4.9. Improved Singularisation.- 4.10. Piecewise Linear Force of Threshold Type.- 4.11. Threshold-type Force Defined by the Power Function.- 4.12. Symmetric Threshold-type Forces.- 5 Forced Vibration.- 5.1. Problem Statement.- 5.2. Change of Variables.- 5.3. Resonant Processes.- 5.4. Averaging in Systems with Impact Interactions.- 5.5. Steady-state Vibration and Stability.- 5.6. Vibro-impact Systems Under Harmonic Excitation.- 5.7. Exact Laws of Motion of Vibro-impact Systems.- 5.8. Resonance Vibration of a System with Piecewise Restoring Force.- 5.9. Piecewise Power Restoring Force.- 5.10. Principle of Energy Balance.- 5.11. Conditions of Existence of Resonant Regimes under Harmonic Excitation.- 5.12. Bifurcation of Fundamental Resonant Regimes under Polyharmonic Excitation.- 5.13. Bifurcation of Solutions in Vibro-impact System.- 5.14. Analysis of Superperiodic and Combination Resonances.- 6 Vibration in Autonomous Systems.- 6.1. Preliminary Considerations.- 6.2. Analysis of Autonomous Systems using the Averaging Method.- 6.3. Chatter.- 6.4. Analysis of the Autoresonant System.- 6.5. Quasi-isochronous Approximation.- 6.6. Symmetric Systems.- 7 Parametric Vibration.- 7.1. Preliminary Considerations.- 7.2. Resonant Regimes Outside the Zones of Instability of a Linear System.- 7.3. Integral Equation of Parametric Vibration.- 7.4. Resonant Regimes within the Zone of Instability of Linear Systems.- 7.5. Parametric Systems with Force Excitation.- 7.6. Energy Condition of Instability.- 7.7. Mathieu Equation with Strong Nonlinearity.- 7.8. System with Symmetric Nonlinearity.- 7.9. Calculations for Systems under Combined Excitation.- 7.10. Bifurcation of Regimes in Parametric Systems.- 7.11. Explicit Solutions to a Specific Class of Model Problems.- 8 Random Vibration.- 8.1. Preliminary Considerations.- 8.2. Some Exact Solutions.- 8.3. Random Vibration in Self-sustained System with Small Clearance.- 8.4. Contact Damping.- 8.5. Deviations from Solutions of Averaged Systems.- 8.6. Quasi-resonant Regimes.- 8.7. Parametric Systems in a Quasi-isochronous Approximation.- 8.8. Perturbed Periodic Green Functions.- 8.9. Application of Perturbed Periodic Green Functions to the Analysis of a Vibro-impact system.- 8.10. Narrowband Excitation.- 3 Multiple-Degree-of-Freedom Systems.- 9 Forced Vibration in Multiple-Degree-of-Freedom Systems.- 9.1. Preliminary Considerations.- 9.2. Integro-differential Equation of Periodic Regimes in System of Two Strongly Interacting Linear Subsystems.- 9.3. Newtonian Interaction.- 9.4. Principle of Energy Balance.- 9.5. Singularisation.- 9.6. Interaction of Two Systems with Lumped Parameters.- 9.7. Interaction of Rod Systems.- 9.8. Resonant Regimes in Systems with Arbitrary Dynamic Compliance Operators.- 9.9. Quasi-resonant Regimes.- 9.10. Analysis of Multi-dimensional Systems using Markov Processes.- 9.11 Systems with Relaxation.- 9.12. Single-frequency Vibration in Systems Given by the Operator Equation.- 9.13. Analysis of Symmetric Systems.- 10 Parametric Vibration in the Multiple-Degree-of-Freedom Systems.- 10.1. Method of Analysis.- 10.2. Equation of Energy Balance.- 10.3. Auxiliary Analysis.- 10.4. The Second Approximation for Impact Impulse.- 10.5. Parametric Vibration of an Oscillator Suspended Inertially Inside a Container.- 10.6. Dynamics of Vibro-impact Mechanisms Mounted on a Vibrating Base.- Additional Bibliography.- Appendix I The Averaging Method in Systems with Impacts.- Appendix II On the Analysis of Resonant Vibration of Vibro-impact Systems Using the Averaging Technique.- Appendix III Structure-borne Vibroimpact Resonances and Periodic Green Functions.- Appendix IV Nonlinear Correction of a Vibration Protection System Containing Tuned Dynamic Absorber.

Algebraic methods to compute Mathieu functions

Abstract: The standard form of the Mathieu differential equation is y''(z) + (a-2 qcos 2z) y(z) = 0, where a is the characteristic number and q is a real parameter. The most useful solution forms are given in terms of expansions for either small or large values of q. In this paper we obtain closed formulae for the generic term of expansions of Mathieu functions in the following cases: (1) standard series expansion for small q; (2) Fourier series expansion for small q; (3) asymptotic expansion in terms of trigonometric functions for large q; and (4) asymptotic expansion in terms of parabolic cylinder functions for large q. We also obtain closed formulae for the generic term of expansions of characteristic numbers and normalization formulae for small and large q. Using these formulae one can efficiently generate high-order expansions that can be used for implementation of the algebraic aspects of Mathieu functions in computer algebra systems. These formulae also provide alternative methods for numerical evaluation of Mathieu functions.

Reconstructing the thermal Green functions at real times from those at imaginary times

Abstract: By exploiting the analyticity and boundary value properties of the thermal Green functions that result from the KMS condition in both time and energy complex variables, we treat the general (non-perturbative) problem of recovering the thermal functions at real times from the corresponding functions at imaginary times, introduced as primary objects in the Matsubara formalism The key property on which we rely is the fact that the Fourier transforms of the retarded and advanced functions in the energy variable have to be the `unique Carlsonian analytic interpolations' of the Fourier coefficients of the imaginary-time correlator, the latter being taken at the discrete Matsubara imaginary energies, respectively in the upper and lower half-planes Starting from the Fourier coefficients regarded as `data set', we then develop a method based on the Pollaczek polynomials for constructing explicitly their analytic interpolations

Combined FEM and Green's function analysis of periodic SAW structure, application to the calculation of reflection and scattering parameters

Abstract: Because of more and more stringent requirements on SAW filter performances, it is important to compute, with very good accuracy, the SAW propagation characteristics, which include the calculation of reflection and scattering parameters For that reason, the analysis of periodic structures on a semi-infinite piezoelectric substrate is one of the most important problems being investigated by SAW researchers For infinite periodic grating modeling, we developed numerical mixed FEM/BEM (finite element method-boundary element method) models using an efficient interpolation basis function that takes into account the singularity at both edges of each electrode In this paper, a review of the numerical program that has been developed during the past few years will be presented For an infinite periodic grating, it is convenient to solve the propagation problem in the Fourier domain (wave number space and harmonic excitation), and important efforts have been spent to properly integrate the so-called periodic harmonic Green function Using this numerical model together with the general P-matrix formalism, it is possible to compute all of the basic parameters with a very good accuracy These consist of the single strip reflectivity, acoustic wave-phase velocity, and position offset between reflection and transduction centers Simulations and comparisons with experiments are shown for each model

A Galerkin boundary integral method for multiple circular elastic inclusions

Abstract: The problem of an infinite, isotropic elastic plane containing an arbitrary number of circular elastic inclusions is considered. The analysis procedure is based on the use of a complex singular integral equation. The unknown tractions at each circular boundary are approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using the classical Galerkin method and the Gauss–Seidel algorithm is used to solve the system. Several numerical examples are considered to demonstrate the effectiveness of the approach. Copyright © 2001 John Wiley & Sons, Ltd.

Roundness modeling of machined parts for tolerance analysis

Abstract: Out-of-roundness of circular and cylindrical parts can greatly affect assembly accuracy. At present, the ASME Y14.5M-1994 specifies circularity tolerance based on two extreme circular boundaries to confine the highest peak and the lowest valley of a roundness profile; however, the profile variations within the two extreme boundaries are not accounted for. In this paper, we propose a harmonic roundness model using Fourier series expansion. In addition, a cutting profile model has been developed to illustrate the relationship between the radial error motion of a spindle and the resultant part profile. This relationship is then used to provide physical meanings to the harmonics generated by the proposed roundness model for a part profile, in particular those caused by fractional frequency spindle error motions. The proposed harmonic model has been verified statistically by a large number of real profiles produced by both turning and cylindrical grinding processes. The proposed roundness model is expected to provide insights into the advanced tolerance analysis of circular and cylindrical parts.

Comments on "Exact solutions of electromagnetic fields in both near and far zones radiated by thin circular-loop antennas: a general representation" [with reply]

Abstract: This paper presents an alternative vector analysis of the electromagnetic (EM) fields radiated from thin circular-loop antennas of arbitrary radius a. This method, which employs the dyadic Green's function in the derivation of the EM radiated fields, makes the analysis more general, compact, and straightforward than those two methods published recently by Werner (1996) and Overfelt (1996). Both near and far zones are considered so that the EM radiated fields are expressed in terms of the vector-wave eigenfunctions. Not only the exact solution of the EM fields in the near and far zones outside the region (where r>a) is derived by the use of the spherical Hankel function of the first kind, but also the closed-series form of the EM fields radiated in the near zone inside the region 0/spl les/r

Image Registration, Optical Flow and Local Rigidity

Abstract: We address the theoretical problems of optical flow estimation and image registration in a multi-scale framework in any dimension. Much work has been done based on the minimization of a distance between a first image and a second image after applying deformation or motion field. Usually no justification is given about convergence of the algorithm used. We start by showing, in the translation case, that convergence to the global minimum is made easier by applying a low pass filter to the images hence making the energy “convex enough”. In order to keep convergence to the global minimum in the general case, we introduce a local rigidity hypothesis on the unknown deformation. We then deduce a new natural motion constraint equation (MCE) at each scale using the Dirichlet low pass operator. This transforms the problem to solving the energy minimization in a finite dimensional subspace of approximation obtained through Fourier Decomposition. This allows us to derive sufficient conditions for convergence of a new multi-scale and iterative motion estimation/registration scheme towards a global minimum of the usual nonlinear energy instead of a local minimum as did all previous methods. Although some of the sufficient conditions cannot always be fulfilled because of the absence of the necessary a priori knowledge on the motion, we use an implicit approach. We illustrate our method by showing results on synthetic and real examples in dimension 1 (signal matching, Stereo) and 2 (Motion, Registration, Morphing), including large deformation experiments.

Reconstructing the thermal Green functions at real times from those at imaginary times

Abstract: By exploiting the analyticity and boundary value properties of the thermal Green functions that result from the KMS condition in both time and energy complex variables, we treat the general (non-perturbative) problem of recovering the thermal functions at real times from the corresponding functions at imaginary times, introduced as primary objects in the Matsubara formalism. The key property on which we rely is the fact that the Fourier transforms of the retarded and advanced functions in the energy variable have to be the `unique Carlsonian analytic interpolations' of the Fourier coefficients of the imaginary-time correlator, the latter being taken at the discrete Matsubara imaginary energies, respectively in the upper and lower half-planes. Starting from the Fourier coefficients regarded as `data set', we then develop a method based on the Pollaczek polynomials for constructing explicitly their analytic interpolations.

High-precision forecast using grey models

Abstract: In recent years the grey theorem has been successfully used in many prediction applications. The proposed Markov-Fourier grey model prediction approach uses a grey model to predict roughly the next datum from a set of most recent data. Then, a Fourier series is used to fit the residual error produced by the grey model. With the Fourier series obtained, the error produced by the grey model in the next step can be estimated. Such a Fourier residual correction approach can have a good performance. However, this approach only uses the most recent data without considering those previous data. In this paper, we further propose to adopt the Markov forecasting method to act as a longterm residual correction scheme. By combining the short-term predicted value by a Fourier series and a long-term estimated error by the Markov forecasting method, our approach can predict the future more accurately. Three time series are used in our demonstration. They are a smooth functional curve, a curve for the stock market and th...

An algebraic approach for identifying operating point dependent parameters of synchronous machines using orthogonal series expansions

Abstract: This paper presents a method for identifying armature and field parameters of synchronous machines from digital fault recorder (DFR) data. The method uses operational properties of orthogonal series expansions such as the Hartley, Walsh and Fourier series to transform a set of differential equations into linear algebraic equations. The algebraic formulation and use of operational calculus reduce the problem of identifying parameters to the manipulation of matrices that may be easily performed in such computational packages as Matlab. The variation of machine parameters with operating point is considered.

Frequency response of the thyristor controlled series capacitor

Abstract: This paper introduces the application of a new method based on Fourier series expansion, to derive the frequency response of a thyristor controlled series capacitor (TCSC). The theory and the numerical approach are presented and the results are reported. Although this method is solely applied to the TCSC device in this context, it is completely general and may well be applied to any type of switched power device. The analysis shows the behavior of a TCSC at frequencies other than fundamental frequency. Such information is very important for sub-synchronous resonance (SSR) as well as other harmonic related studies.