Showing papers on "Fourier series published in 1991"

E-map improvement in direct procedures

Abstract: The quality of an E map is usually affected by systematic and/or random phase errors, by amplitude truncation effects in the series representation of the electron density, by the experimental uncertainty in the estimation of |E| and by the intrinsic nature of the Fourier coefficients used (i.e. the E's). It is shown that simple supplemental calculations can improve the quality of an E map. Large molecular fragments can often be localized in the new map even when the original one is not easily interpretable.

Fast Fourier Transforms

Abstract: Basic aspects of fourier series definition of fourier series examples of fourier series fourier series of real functions pointwise convergence of fourier series further aspects of convergence of fourier series fourier sine series and cosine series convergence of fourier sine and cosine series the discrete fourier transform (DFT) the fast fourier transform (FFT) some applications of fourier series fourier transforms properties of fourier transforms inversion of fourier transforms convolution - an introduction the convolution theorem an application of convolution in quantum mechanics filtering, frequency detection, and removal of noise summation kernals arising from poisson summation fourier optics fresnel diffraction fraunhofer diffraction circular apertures the phase transformation induced by a thin lens imaging with a single lens user's manual for fourier analysis software some computer programmes the schwarz inequality.

Nodal surfaces of Fourier series : fundamental invariants of structured matter

Abstract: Periodic Nodal Surfaces (PNS) of Fourier series are derived and classified as fundamental invariants of structured matter. Relationships to periodic minimal surfaces PMS and to periodic zero potential surfaces (POPS) are given. A basic set of cubic PNS is represented in arithmetic form. The special importance of the invariance of the zeros to the type of the potential is stressed.

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Abstract: A theory of overall quantization noise for nonsubtractive dither was originally developed in unpublished work by J.N. Wright and by T.J. Stockham and subsequently expanded by L.K. Brinton, S.P. Lipshitz, J. Vanderkooy, and R.A. Wannamaker.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 evaluation of error probabilities for intersymbol and cochannel interference

Abstract: An efficient series that can be used to calculate the probability of error in a binary symmetric channel with intersymbol interference and additive noise is derived. The series is derived by representing the noise complementary probability distribution function by an exact or approximate Fourier series. Bounds on the accuracy of the estimate are derived for Gaussian noise. Examples show that only a small numerical effort may be required to compute error probabilities of interest using the series. Applications to both finite and infinite intersymbol interference systems are discussed. A similar technique is used to derive series representations for the probability of error of additive noise channels with cochannel interference or with combined intersymbol and cochannel interferences. The accuracy of the results is bounded for Gaussian noise. >

Walsh Series and Transforms: Theory and Applications

Abstract: 1 Walsh Functions and Their Generalizations.- 1.1 The Walsh functions on the interval [0, 1).- 1.2 The Walsh system on the group.- 1.3 Other definitions of the Walsh system. Its connection with the Haar system.- 1.4 Walsh series. The Dirichlet kernel.- 1.5 Multiplicative systems and their continual analogues.- 2 Walsh-Fourier Series Basic Properties.- 2.1 Elementary properties of Walsh-Fourier series. Formulae for partial sums.- 2.2 The Lebesgue constants.- 2.3 Moduli of continuity of functions and uniform convergence of Walsh-Fourier series.- 2.4 Other tests for uniform convergence.- 2.5 The localization principle. Tests for convergence of a Walsh-Fourier series at a point.- 2.6 The Walsh system as a complete, closed system.- 2.7 Estimates of Walsh-Fourier coefficients. Absolute convergence of Walsh-Fourier series.- 2.8 Fourier series in multiplicative systems.- 3 General Walsh Series and Fourier-Stieltjes Series Questions on Uniqueness of Representation of Functions by Walsh Series.- 3.1 General Walsh series as a generalized Stieltjcs series.- 3.2 Uniqueness theorems for representation of functions by pointwise convergent Walsh series.- 3.3 A localization theorem for general Walsh series.- 3.4 Examples of null series in the Walsh system. The concept of U-sets and M-sets.- 4 Summation of Walsh Series by the Method of Arithmetic Mean.- 4.1 Linear methods of summation. Regularity of the arithmetic means.- 4.2 The kernel for the method of arithmetic means for Walsh- Fourier series.- 4.3 Uniform (C, 1) summability of Walsh-Fourier series of continuous functions.- 4.4 (C, 1) summability of Fourier-Stieltjes series.- 5 Operators in the Theory of Walsh-Fourier Series.- 5.1 Some information from the theory of operators on spaces of measurable functions.- 5.2 The Hardy-Littlewood maximal operator corresponding to sequences of dyadic nets.- 5.3 Partial sums of Walsh-Fourier series as operators.- 5.4 Convergence of Walsh-Fourier series in Lp[0, 1).- 6 Generalized Multiplicative Transforms.- 6.1 Existence and properties of generalized multiplicative transforms.- 6.2 Representation of functions in L1(0, ?) by their multiplicative transforms.- 6.3 Representation of functions in Lp(0, ?), 1 < p ? 2, by their multiplicative transforms.- 7 Walsh Series with Monotone Decreasing Coefficient.- 7.1 Convergence and integrability.- 7.2 Series with quasiconvex coefficients.- 7.3 Fourier series of functions in Lp.- 8 Lacunary Subsystems of the Walsh System.- 8.1 The Rademacher system.- 8.2 Other lacunary subsystems.- 8.3 The Central Limit Theorem for lacunary Walsh series.- 9 Divergent Walsh-Fourier Series Almost Everywhere Convergence of Walsh-Fourier Series of L2 Functions.- 9.1 Everywhere divergent Walsh-Fourier series.- 9.2 Almost everywhere convergence of Walsh-Fourier series of L2[0, 1) functions.- 10 Approximations by Walsh and Haar Polynomials.- 10.1 Approximation in uniform norm.- 10.2 Approximation in the Lp norm.- 10.3 Connections between best approximations and integrability conditions.- 10.4 Connections between best approximations and integrability conditions (continued).- 10.5 Best approximations by means of multiplicative and step functions.- 11 Applications of Multiplicative Series and Transforms to Digital Information Processing.- 11.1 Discrete multiplicative transforms.- 11.2 Computation of the discrete multiplicative transform.- 11.3 Applications of discrete multiplicative transforms to information compression.- 11.4 Peculiarities of processing two-dimensional numerical problems with discrete multiplicative transforms.- 11.5 A description of classes of discrete transforms which allow fast algorithms.- 12 Other Applications of Multiplicative Functions and Transforms.- 12.1 Construction of digital filters based on multiplicative transforms.- 12.2 Multiplicative holographic transformations for image processing.- 12.3 Solutions to certain optimization problems.- Appendices.- Appendix 1 Abelian groups.- Appendix 2 Metric spaces. Metric groups.- Appendix 3 Measure spaces.- Appendix 4 Measurable functions. The Lebesgue integral.- Appendix 5 Normed linear spaces. Hilbert spaces.- Commentary.- References.

Modern signals and systems

Abstract: Overview of signals and systems an introduction to signals an introduction to systems difference and differential systems state description of systems expansion theory and Fourier series Fourier transforms the z- and laplace transforms applications to signal processing and digital filtering applications to communication feedback and applications to automatic control supplement reviews and SIGSYS tutorial.

A uniqueness theorem of Beurling for Fourier transform pairs

Abstract: There are many theorems known which state that a function and its Fourier transform cannot simultaneously be very small at infinity, such as various forms of the uncertainty principle and the basic results on quasianalytic functions. One such theorem is stated on page 372 in volume II of the collected works of Arne Beurling [1]. Although it is not in every respect the most precise result of its kind, it has a simplicity and generality which make it very attractive. The editors state that no proof has been preserved. However, in my files I have notes which I made when Arne Beurling explained this result to me during a private conversation some time during the years 1964---1968 when we were colleagues at the Institute for Advanced Study, I shall reproduce these notes here in English translation with onIy minor details added where my notes are too sketchy. Theorem. Let fELl (R) and assume that

Partial differential equations and boundary-value problems with applications

Abstract: Preliminaries Fourier series boundary-value problems in rectangular co-ordinates boundary-value problems in cylindrical co-ordinates boundary-value problems in spherical co-ordinates Fourier transforms and applications asymptotic analysis numerical analysis Green's functions appendices.

Fourier and Taylor series on fitness landscapes

Abstract: Holland's "hyperplane transform" of a "fitness landscape", a random, real valued function of the verticies of a regular finite graph, is shown to be a special case of the Fourier transform of a function of a finite group. It follows that essentially all of the powerful Fourier theory, which assumes a simple form for commutative groups, can be used to characterize such landscapes. In particular, an analogue of the KarhunenLoeve expansion can be used to prove that the Fourier coefficients of landscapes on commutative groups are uncorrelated and to infer their variance from the autocorrelation function of a random walk on the landscape. There is also a close relationship between the Fourier coefficients and Taylor coefficients, which provide information about the landscape's local properties. Special attention is paid to a particularly simple, but ubiquitous class of landscapes, so-called "AR(1) landscapes".

Comprehensive theory of the Fourier transform ion cyclotron resonance signal for all ion trap geometries

Abstract: An analytic solution for the charge induced on each of the detection electrodes of a Fourier transform ion cyclotron resonance (FT/ICR) ion trap has been derived from basic electrostatics for both tetragonal and cylindrical traps of arbitrary aspect ratio, by use of a Green’s function formalism. Dunbar has shown that the result of that calculation is in general equivalent to that obtained from prior ‘‘reciprocity’’‐based methods (see text). A primary advantage of the present treatment is its variety of functional forms arising from the various forms of the Green’s function, some of which may converge much more rapidly in numerical evaluation. (Moreover, because the Green’s function is the potential field of a unit point charge, the Green’s function must be employed in any treatment of ion–ion repulsions.)The present results (a) exactly confirm prior analyses of the cubic and tetragonal traps; (b) provide the first complete analysis of the cylindrical trap; and (c) may be extended to any trap geometry for which the Green’s function is known. In the absence of an available Green’s function, the reciprocity‐based treatment, either analytic or numerical, is the method of choice to solve for the induced charge for any ion trap geometry (e.g, unbroken or segmented hyperbolic). For circular orbits centered on the longitudinal axis of the trap, the presence of spectral components at odd multiples of the fundamental ICR orbital frequency is explained and a closed form solution for the relative magnitudes of these components is presented for tetragonal and cylindrical traps. The ratio of the spectral peak height at the third harmonic to that at the first (i.e., the ratio of the third to first Fourier coefficients) is a strong monotonic function of orbital radius; thus, measurement of that ratio provides a simple and direct means for determining the cyclotron orbital radius and hence its orbital translational energy.The presence and location of magnetron and trapping sidebands of the fundamental peak are also predicted. In addition, we show that a cylindrical trap whose ring electrode is divided into equal quadrants is only slightly more sensitive than a tetragonal trap of the same aspect ratio. Finally, we develop a general circuit model which relates the charge induced on one or more detection electrodes to the detected voltage (i.e., the unamplified signal). Since the effect of trapped‐ion motion on each detection electrode is modeled as a charge (or current) source relative to ground, the net signal from any given electrode arrangement and interconnection scheme can be accommodated simply by adding, subtracting, or grounding the signal from each detection electrode, e.g., single‐electrode detection, Comisarow’s differential detection between two electrodes, or any of various multiple‐electrode configurations. From the measured ICR signal and ICR orbital radius, the number of coherently orbiting ions may be determined.

Biomedical signal processing (in four parts). Part 2. The frequency transforms and their inter-relationships.

Abstract: This is the second in a series of four tutorial papers on biomedical signal processing, and it concerns the relationships between commonly used frequency transforms. It begins with the Fourier series and Fourier transform for continuous time signals and extends these concepts for aperiodic discrete time data and then periodic discrete time data. The Laplace transform is discussed as an extension of the Fourier transform. The z-transform is introduced and the ideas behind the chirp-z transform are described. The equivalence between the time and frequency domains is described in terms of Parseval's theorem and the theory of convolution. The use of the FFT for fast convolution and fast correlation is described for both short recordings and long recordings that must be processed in sections.

Local and Global Descriptions of Periodic Pseudodifferential Operators

Abstract: Periodic pseudodifferential operators can be defined either globally, via FOURIER series, or else locally, via partitions of unity and FOURIER integrals. Here, it is proved that the two definitions are equivalent.

An accurate theoretical model for the thin-wire circular half-loop antenna

Abstract: The conventional Fourier series analysis for the thin-wire circular transmitting loop, or its image equivalent to the half-loop, uses a delta-function generator for excitation. This method of excitation introduces two problems: it does not correspond to any realizable method of feeding the antenna, so an accurate comparison with measurement is not possible, and it produces a divergent series for the input susceptance. To overcome these problems, a new theoretical model is used for the antenna: a half-loop driven through an image plane by a coaxial transmission line, with a transverse electromagnetic mode assumed in the aperture of the coaxial line. This model is solved in a manner that preserves the simplicity of the original Fourier series analysis. All coefficients are obtained as closed-form expressions. Input admittances calculated from this new theoretical model are in excellent agreement with accurate measurements. >

Uncertainty principles invariant under the fractional Fourier transform

Abstract: Uncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.

Analysis of mode propagation in optical waveguide devices by Fourier expansion

Abstract: A general method for calculating the solution of the scalar wave equation for the field propagating through integrated optical devices is presented. The method is capable of a three-dimensional description and of treating problems with reflected waves. It consists of dividing the device into a series of sections of axially uniform waveguides. The modes in each section are found by expansion of the field in a two-dimensional Fourier series and solving the associated matrix eigenvalue problem. Propagation is then described by relating the mode amplitudes of each section to the previous one. The amplitudes are related by a matrix that is the product of the eigenvector matrices of the two sections. The method is illustrated by the analysis of an adiabatic mode transformer, the coupling of light from a semiconductor laser through free space to a waveguide, and the propagation through an adiabatic 3 dB coupler and Y branch. >

Phase-mixing and surface waves: a new interpretation

Abstract: The classical incompressible MHD or cold plasma phase mixing problem, which involves Alfven or plasma waves in inhornogeneous media, is re-examined using a spatial Fourier series rather than the usual temporal Fourier or Laplace transform approach. A number of exact and near-exact analytic and numerical results are derived which reveal an attractive picture of energy cascading to smaller length-scales in a manner reminiscent of turbulence. Furthermore, we present a simple and unambiguous description of how a surface wave arises in the limit in which the inhomogeneity becomes a discontinuity.

Theory and Application of Morphological Analysis: Fine Particles and Surfaces

Abstract: INTRODUCTION: WHAT IS A SCIENTIFIC THEORY? The Theory of Morphology. The Development of a Theory. Elements of a Theory. Morphology as a Theory. A CRITIQUE OF CLASSICAL SIZE AND SHAPE ANALYSIS. Classical Size. Classical Shape. CLASSICAL FOURIER ANALYSIS OF A SINGLE PARTICLE. Fourier Series. The R(q) Method. The f*(L) Method. The (R,S) Method. Classical Feature Extraction. THE THEORY OF MORPHOLOGY. The Need. Particle Characterization. Morphological Analysis. Characterization of a Physical Process. Characterization of a Physical Process-An Example in Fluid Flow. THE BOUNDARY FUNCTIONS OF PARTICLE REPRESENTATION. A Unified Approach to Particle Representation. Boundary Functions and the Variational Principle. The Particle Profile, R(q). The Extended Surface, G(x,y). The Extended Surface, G(r,q). The Three-Dimensional "Bulky" Particle. The Finite Fiber. The Repeating Fiber. Infinite Fibers and Threads. Flakes. The Future. DEVELOPMENT OF THE MORPHOLOGICAL VARIATIONAL PRINCIPLE AND DERIVATION OF THE BOUNDARY FUNCTIONS. The Concept of Shape. The Development of the Morphological Variational Principle. Derivation of the Boundary Function, R(q). The Boundary Function, G(r,q), for Extended Surfaces. Extending the Morphological Variational Principle to Higher Dimensions. The Theory of Continuous Functions. The Morphological Variational Principle. The Surface Area for an Irregular Surface. Image Analysis. The Derivation of the Gray Level Function for Extended Surfaces. Applications of the Gray Level Function in Microscopy. The Bessel-Fourier Coefficients for Extended Surfaces. Limitations of the Gray Level Function for Extended Surfaces. The Three Dimensional Particle, R(q,Ae). The Finite Fiber, R(q,Z). The Repeating Fiber, R(q,Z). The Infinite Fiber, R*(q,Z). Flakes, R(q), Independent of Z. Commentary on the Derivations of the Boundary Functions as Particle Representations. Closure-The Boundary Functions of the Theory of Morphology. FEATURE EXTRACTION FROM PARTICLE REPRESENTATIONS. Guiding Principles of Feature Extraction in the Theory of Morphology. The Concept of Particle Shape. The Concept of Particle Size. Extracting the Size Feature. The Morphological Variational Principle as a Fundamental Law or Hypothesis in the Theory of Morphology. The Equivalent Radius as a Derived Law in the Theory of Morphology. Statistical Features of the Particle Profile. Rotational Invariance of the Moments of the Radial Distribution. Invariant Fourier Shape Features. Symmetry Analysis of a Particle Profile. Partial Symmetry Operations. The Rotational Partial Symmetry Element, Cm. The Reflectional Partial Symmetry Element, m. Group Properties of the Classical Symmetry Elements. A Symmetry Classification Scheme When a Plane of Symmetry Exists in the Particle Profile. A Symmetry Classification Scheme Utilizing Group Properties When a Plane of Symmetry Does Not Exist in the Profile. Physical Features Associated With Boundary Functions. Feature Extraction From Sets of Particles. The Future of the Theory of Morphology. THE GENERALIZED R(q) METHOD-THE REENTRANT PARTICLE. The Reentrant Particle Profile. Reparameterization of the Reentrant Particle. Derivation of the Boundary Function, R(t), for the Reentrant Particle. Closure. AN INTRODUCTION TO THE MORPHOLOGICAL ANALYSIS OF THE REGULAR FIGURES. The Circle. The Cardioid. The Lemniscate. The Triangle. The Square. The Pentagon. The Hexagon. The Ellipse. The Rectangle. APPLICATIONS OF THE THEORY OF MORPHOLOGY. The Effect of Particle Morphology of the Flow of a Dextrose Powder. Quality Assessment of Industrial Sieve Mesh. Effects of Powder Production and Material Processing on the Morphology of Adipic Acid. Differentiation Between Three Races of Giraffe Based on the Morphic Features of Trunk Spots. The Morphological Features of the Lower 48. Closure. GENERAL REMARKS ON THE THEORY OF MORPHOLOGY. The Theory of Morphology as a Scientific Theory. Future Technical Developments for the Theory of Morphology. The Future for the Theory of Morphology. Opportunity. APPENDIX I: A COMPUTER PROGRAM TO CALCULATE FOURIER COEFFICIENTS FOR NON-REENTRANT PARTICLE PROFILES. APPENDIX II: THE GENERAL PROPERTIES OF THE FOURIER MOMENT FUNCTION. The Fourier Moment Function. The Derivation of the General Moment Function of the Fourier Series. The Second Moment Function.

Networks with Learned Unit Response Functions

Abstract: Feedforward networks composed of units which compute a sigmoidal function of a weighted sum of their inputs have been much investigated We tested the approximation and estimation capabilities of networks using functions more complex than sigmoids Three classes of functions were tested: polynomials, rational functions, and flexible Fourier series Unlike sigmoids, these classes can fit non-monotonic functions They were compared on three problems: prediction of Boston housing prices, the sunspot count, and robot arm inverse dynamics The complex units attained clearly superior performance on the robot arm problem, which is a highly non-monotonic, pure approximation problem On the noisy and only mildly nonlinear Boston housing and sunspot problems, differences among the complex units were revealed; polynomials did poorly, whereas rationals and flexible Fourier series were comparable to sigmoids

Measurement of signal parameters by using nonlinear observers

Abstract: Recursive measurements of signal parameters that are nonlinear functions of observations require nonlinear observers. A method to design absolutely stable observers for multiple-output time-varying nonlinear free dynamic systems is presented. Because of the high computational burden of absolutely stable observers the authors have developed a computationally efficient globally, but not absolutely, stable observer measuring the frequency and the Fourier coefficients of bandlimited periodic signals. Simulation and realization results are also provided. >

Generalized in-plane circuit averaging

Abstract: The authors present an unified approach to in-place circuit averaging that is applicable to resonant-type circuits as well as pulse-width-modulated (PWM) circuits. The approach allows the refinement of an averaged circuit model to obtain an arbitrary degree of accuracy. In the context of a particular circuit, an approximate averaged representation for each branch variable consists of a subset of the Fourier coefficients (index-k averages). The selection of this subset is determined by the dominant harmonic content of the circuit waveforms. For instance, in a series resonant DC-DC converter, the index-1 averages would be selected for the resonant tank variables, whereas the index-0 averages would be selected for the load side elements. In a PWM converter one would normally use the index-0 averages to obtain a low-frequency approximate model. These models can then be refined by including additional coefficients. This procedure is illustrated for a PWM up-down converter and for a DC-DC series resonant converter. >