Showing papers on "Fourier transform published in 1971"

Data Transmission by Frequency-Division Multiplexing Using the Discrete Fourier Transform

Abstract: The Fourier transform data communication system is a realization of frequency-division multiplexing (FDM) in which discrete Fourier transforms are computed as part of the modulation and demodulation processes. In addition to eliminating the bunks of subcarrier oscillators and coherent demodulators usually required in FDM systems, a completely digital implementation can be built around a special-purpose computer performing the fast Fourier transform. In this paper, the system is described and the effects of linear channel distortion are investigated. Signal design criteria and equalization algorithms are derived and explained. A differential phase modulation scheme is presented that obviates any equalization.

Three-dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions instead of Fourier Transforms

Abstract: A new technique is proposed for the mathematical process of reconstruction of a three-dimensional object from its transmission shadowgraphs; it uses convolutions with functions defined in the real space of the object, without using Fourier transforms. The object is rotated about an axis at right angles to the direction of a parallel beam of radiation, and sections of it normal to the axis are reconstructed from data obtained by scanning the corresponding linear strips in the shadowgraphs at different angular settings. Since the formulae in the convolution method involve only summations over one variable at a time, while a two-dimensional reconstruction with the Fourier transform technique requires double summations, the convolution method is much faster (typically by a factor of 30); the relative increase in speed is larger where greater resolution is required. Tests of the convolution method with computer-simulated shadowgraphs show that it is also more accurate than the Fourier transform method. It has good potentialities for application in electron microscopy and x-radiography. A new method of reconstructing helical structures by this technique is also suggested.

Fourier analysis and approximation

Abstract: 0 Preliminaries.- 0 Preliminaries.- 0.1 Fundamentals on Lebesgue Integration.- 0.2 Convolutions on the Line Group.- 0.3 Further Sets of Functions and Sequences.- 0.4 Periodic Functions and Their Convolution.- 0.5 Functions of Bounded Variation on the Line Group.- 0.6 The Class BV2?.- 0.7 Normed Linear Spaces, Bounded Linear Operators.- 0.8 Bounded Linear Functional, Riesz Representation Theorems.- 0.9 References.- I Approximation by Singular Integrals.- 1 Singular Integrals of Periodic Functions.- 1.0 Introduction.- 1.1 Norm-Convergence and-Derivatives.- 1.1.1 Norm-Convergence.- 1.1.2 Derivatives.- 1.2 Summation of Fourier Series.- 1.2.1 Definitions.- 1.2.2 Dirichlet and Fejer Kernel.- 1.2.3 Weierstrass Approximation Theorem.- 1.2.4 Summability of Fourier Series.- 1.2.5 Row-Finite ?-Factors.- 1.2.6 Summability of Conjugate Series.- 1.2.7 Fourier-Stieltjes Series.- 1.3 Test Sets for Norm-Convergence.- 1.3.1 Norms of Some Convolution Operators.- 1.3.2 Some Applications of the Theorem of Banaeh-Steinhaus.- 1.3.3 Positive Kernels.- 1.4 Pointwise Convergence.- 1.5 Order of Approximation for Positive Singular Integrals.- 1.5.1 Modulus of Continuity and Lipschitz Classes.- 1.5.2 Direct Approximation Theorems.- 1.5.3 Method of Test Functions.- 1.5.4 Asymptotic Properties.- 1.6 Further Direct Approximation Theorems, Nikolski? Constants.- 1.6.1 Singular Integral of Fejer-Korovkin.- 1.6.2 Further Direct Approximation Theorems.- 1.6.3 Nikolski? Constants.- 1.7 Simple Inverse Approximation Theorems.- 1.8 Notes and Remarks.- 2 Theorems of Jackson and Bernstein for Polynomials of Best Approximation and for Singular Integrals.- 2.0 Introduction.- 2.1 Polynomials of Best Approximation.- 2.2 Theorems of Jackson.- 2.3 Theorems of Bernstein.- 2.4 Various Applications.- 2.5 1.- 4.2.1 The Case p = 2.- 4.2.2 The Case p ? 2.- 4.3 Finite Fourier-Stieltjes Transforms.- 4.3.1 Fundamental Properties.- 4.3.2 Inversion Theory.- 4.3.3 Fourier-Stieltjes Transforms of Derivatives.- 4.4 Notes and Remarks.- 5 Fourier Transforms Associated with the Line Group.- 5.0 Introduction.- 5.1 L1-Theory.- 5.1.1 Fundamental Properties.- 5.1.2 Inversion Theory.- 5.1.3 Fourier Transforms of Derivatives.- 5.1.4 Derivatives of Fourier Transforms, Moments of Positive Functions Peano and Riemann Derivatives.- 5.1.5 Poisson Summation Formula.- 5.2 Lp-Theory, 1 < p ? 2.- 5.2.1 The Case p = 2.- 5.2.2 The Case 1 2.- 5.2.3 Fundamental Properties.- 5.2.4 Summation of the Fourier Inversion Integral.- 5.2.5 Fourier Transforms of Derivatives.- 5.2.6 Theorem of Plancherel.- 5.3 Fourier-Stieltjes Transforms.- 5.3.1 Fundamental Properties.- 5.3.2 Inversion Theory.- 5.3.3 Fourier-Stieltjes Transforms of Derivatives.- 5.4 Notes and Remarks.- 6 Representation Theorems.- 6.0 Introduction.- 6.1 Necessary and Sufficient Conditions.- 6.1.1 Representation of Sequences as Finite Fourier or Fourier-Stieltjes Transforms.- 6.1.2 Representation of Functions as Fourier or Fourier-Stieltjes Transforms.- 6.2 Theorems of Bochner.- 6.3 Sufficient Conditions.- 6.3.1 Quasi-Convexity.- 6.3.2 Representation as L1/2? Transform.- 6.3.3 Representation as L1-Transform.- 6.3.4 A Reduction Theorem.- 6.4 Applications to Singular Integrals.- 6.4.1 General Singular Integral of Weierstrass.- 6.4.2 Typical Means.- 6.5 Multipliers.- 6.5.1 Multipliers of Classes of Periodic Functions.- 6.5.2 Multipliers on LP.- 6.6 Notes and Remarks.- 7 Fourier Transform Methods and Second-Order Partial Differential Equations.- 7.0 Introduction.- 7.1 Finite Fourier Transform Method.- 7.1.1 Solution of Heat Conduction Problems.- 7.1.2 Dirichlet's and Neumann's Problem for the Unit Disc.- 7.1.3 Vibrating String Problems.- 7.2 Fourier Transform Method in L1.- 7.2.1 Diffusion on an Infinite Rod.- 7.2.2 Dirichlet's Problem for the Half-Plane.- 7.2.3 Motion of an Infinite String.- 7.3 Notes and Remarks.- III Hilbert Transforms.- 8 Hilbert Transforms on the Real Line.- 8.0 Introduction.- 8.1 Existence of the Transform.- 8.1.1 Existence Almost Everywhere.- 8.1.2 Existence in L2-Norm.- 8.1.3 Existence in Lp-Norm, 1 ?.- 8.2 Hilbert Formulae, Conjugates of Singular Integrals, Iterated Hilbert Transforms.- 8.2.1 Hilbert Formulae.- 8.2.2 Conjugates of Singular Integrals: 1 ?.- 8.2.3 Conjugates of Singular Integrals: p = 1.- 8.2.4 Iterated Hilbert Transforms.- 8.3 Fourier Transforms of Hilbert Transforms.- 8.3.1 Signum Rule.- 8.3.2 Summation of Allied Integrals.- 8.3.3 Fourier.- 8.3.4 Norm-Convergence of the Fourier Inversion Integral.- 8.4 Notes and Remarks.- 9 Hilbert Transforms of Periodic Functions.- 9.0 Introduction.- 9.1 Existence and Basic Properties.- 9.1.1 Existence.- 9.1.2 Hilbert Formulae.- 9.2 Conjugates of Singular Integrals.- 9.2.1 The Case 1 ?.- 9.2.2 Convergence in C2? and L1/2?.- 9.3 Fourier Transforms of Hilbert Transforms.- 9.3.1 Conjugate Fourier Series.- 9.3.2 Fourier Transforms of Derivatives of Conjugate Functions, the Classes (W~)xr2?'(V~)rx2?.- 9.3.3 Norm-Convergence of Fourier Series.- 9.4 Notes and Remarks.- IV Characterization of Certain Function Classes 355.- 10 Characterization in the Integral Case.- 10.0 Introduction.- 10.1 Generalized Derivatives, Characterization of the Classes Wrx2?.- 10.1.1 Riemann Derivatives in X2?-Norm.- 10.1.2 Strong Peano Derivatives.- 10.1.3 Strong and Weak Derivatives, Weak Generalized Derivatives.- 10.2 Characterization of the Classes Vr2?.- 10.3 Characterization of the Classes (V~)rx2?.- 10.4 Relative Completion.- 10.5 Generalized Derivatives in Lp-Norm and Characterizations for 1 ? p ?2.- 10.6 Generalized Derivatives in X(R)-Norm and Characterizations of the Classes Wrx(R) and Vrx(R).- 10.7 Notes and Remarks.- 11 Characterization in the Fractional Case.- 11.0 Introduction.- 11.1 Integrals of Fractional Order.- 11.1.1 Integral of Riemann-Liouville.- 11.1.2 Integral of M. Riesz.- 11.2 Characterizations of the Classes W[LP |?|?], V[LP |?|?], 1 ? p ? 2.- 11.2.1 Derivatives of Fractional Order.- 11.2.2 Strong Riesz Derivatives of Higher Order, the Classes V[LP |?|? ].- 11.3 The Operators R?{?} on Lp 1 ? p ? 2.- 11.3.1 Characterizations.- 11.3.2 Theorems of Bernstein-Titchmarsh and H. Weyl.- 11.4 The Operators R?(?} on 2?.- 11.5 Integral Representations, Fractional Derivatives of Periodic Functions.- 11.6 Notes and Remarks.- V Saturation Theory.- 12 Saturation for Singular Integrals on X2? and Lp, 1 ? p ? 2 433.- 12.0 Introduction.- 12.1 Saturation for Periodic Singular Integrals, Inverse Theorems.- 12.2 Favard Classes.- 12.2.1 Positive Kernels.- 12.2.2 Uniformly Bounded Multipliers.- 12.2.3 Functional Equations.- 12.3 Saturation in Lp, 1 ? p ? 2.- 12.3.1 Saturation Property.- 12.3.2 Characterizations of Favard Classes: p = 1.- 12.3.3 Characterizations of Favard Classes: 1 < p? 2.- 12.4 Applications to Various Singular Integrals.- 12.4.1 Singular Integral of Fejer.- 12.4.2 Generalized Singular Integral of Picard.- 12.4.3 General Singular Integral of Weierstrass.- 12.4.4 Singular Integral of Bochner-Riesz.- 12.4.5 Riesz Means.- 12.5 Saturation of Higher Order.- 12.5.1 Singular Integrals on the Real Line.- 12.5.2 Periodic Singular Integrals.- 12.6 Notes and Remarks.- 13 Saturation on X(R).- 13.0 Introduction.- 13.1 Saturation of D?(f x t) in X(R), Dual Methods.- 13.2 Applications to Approximation in Lp, 2 ?.- 13.2.1 Differences.- 13.2.2 Singular Integrals Satisfying (12.3.5).- 13.2.3 Strong Riesz Derivatives.- 13.2.4 The Operators R?{?}.- 13.2.5 Riesz and Fejer Means.- 13.3 Comparison Theorems.- 13.3.1 Global Divisibility.- 13.3.2 Local Divisibility.- 13.3.3 Special Comparison Theorems with no Divisibility Hypothesis.- 13.3.4 Applications to Periodic Continuous Functions.- 13.4 Saturation on Banach Spaces.- 13.4.1 Strong Approximation Processes.- 13.4.2 Semi-Groups of Operators.- 13.5 Notes and Remarks.- List of Symbols.- Tables of Fourier and Hilbert Transforms.

Natural Abundance Carbon‐13 Partially Relaxed Fourier Transform Nuclear Magnetic Resonance Spectra of Complex Molecules

Abstract: Proton‐decoupled partially relaxed Fourier transform (PRFT) NMR of carbon‐13 in natural abundance was used to determine spin–lattice relaxation times (T1) of individual carbons in solutions of cholesteryl chloride, sucrose, and adenosine 5′‐monophosphate (AMP) at 15.08 MHz and 42°C. With the exception of a few side‐chain groups, all protonated carbons have T1 values of less than 1 sec. Some side‐chain carbons on cholesteryl chloride show evidence of internal reorientation and have relaxation times of up to 2 sec. Nonprotonated carbons have T1 values in the range 2–8 sec. These relaxation times are sufficiently short to make ordinary Fourier transform NMR a very sensitive technique in the study of complex molecules without the need for spin‐echo refocusing schemes. Integrated intensities and nuclear Overhauser enhancements prove that, except for two of the three nonprotonated carbons in AMP, all 13C nuclei in these compounds relax mainly through 13C–1H dipolar interactions. Measured T1 values of protonated...

Measurement and compensation of defocusing and aberrations by Fourier processing of electron micrographs

Abstract: The effects of defocusing and spherical aberration in the electron microscope image are most simply and directly displayed in the Fourier transform of the image. We have investigated the process of image formation by determining the changes in the transform of the image of a thin crystal of catalase, which has discrete diffraction maxima in the resolution range of 10 to 2.5 nm, as a function of defocusing. The changes in amplitude and phase of these diffraction maxima have been measured and compared with the predictions of a first-order theory of image formation. The theory is generally confirmed, and the transfer function of the microscope is completely determined by finding the relative contributions from phase and amplitude contrast. A 'true' maximum contrast image of the catalase crystal, compensated for the effects of defocusing, is reconstructed from the set of micrographs in the focal series. The relation of this compensated image to individual underfocused micrographs, and the use of underfocus contrast enhancement in conventional electron microscopy, are discussed. This approach and the experimental methods can be extended to high resolution in order to compensate for spherical aberration as well as defocusing. In as much as spherical aberration is the factor presently limiting the resolution of electron lenses, this could provide a considerable extension of the resolution of the electron microscope.

A Frequency Dependent Solution for Microstrip Transmission Lines

Abstract: Theoretical and experimental results of "open" microstrip propagation on both a pure dielectric and a demagnetized ferrite substrate are presented. The theory enables one to obtain the frequency dependence of phase velocity and characteristic impedance, and also to obtain the electromagnetic field quantities around the microstrip line. It utilizes a Fourier transform method in which the hybrid-mode solutions for a "fictitious" surface current at the substrate-air interface are summed in such a way as to represent the fields caused by a current distribution that is finite only over the region occupied by the conducting strip and is assumed equal to that for the quasi-static case.

Image Coding by Linear Transformation and Block Quantization

Abstract: The feasibility of coding two-dimensional data arrays by first performing a two-dimensional linear transformation on the data and then block quantizing the transformed data is investigated. The Fourier, Hadamard, and Karhunen-Loeve transformations are considered. Theoretical results for Markov data and experimental results for four pictures comparing these transform methods to the standard method of raster scanning, sampling, and pulse-count modulation code are presented.

VIII Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions

Abstract: Publisher Summary This chapter discusses applications, specifically dealing with the performance characteristics of optical systems. The linear prolate functions are a set of band limited functions, which like the trigonometric functions, are orthogonal and complete over a finite interval. However, unlike the trig functions, they are also complete and orthogonal over the infinite interval. Fourier transform of a linear prolate function is proportional to the same prolate function. The mathematical properties of the circular prolates, which make them readily applicable to the analysis of optical imagery, are summarized in the chapter. The optical applications arise from the modern use of the prolates as a convenient set of one-dimensional, orthogonal functions. These applications have been on the general subject of optical systems analysis. Performance characteristics of the laser have been established, along with the ultimate ability of lens and lensless systems to form high quality images. There is one physical phenomenon that unifies these applications—diffraction at a finite aperture—and it can be said that this phenomenon is optimally analyzed by use of the prolate functions.

Shearing interferometer using the grating as the beam splitter.

Abstract: The theoretical interpretation of the shearing interferometer based on the moire method using the fourier image of the grating is described. To obtain a pattern with good contrast, the observing plane must coincide with the normal fourier image plane of the grating or with the reversed fourier image plane. The information obtained by this method is the first partial derivative and under certain conditions the second partial derivative of the distortion from the reference wavefront, which is planar or spherical. Applications to measurement of the phase gradient and lens aberration are shown.

Harmonic analysis of switching functions

Abstract: This chapter is a self-contained development of abstract harmonic analysis applied to a single-output combinational logic functions; linear algebra and elementary group theory are the only mathematical prerequisites. New analysis and synthesis techniques are developed, and the groundwork is laid for future extensions to multiple output combinational logic, sequential machines, and real- or complex-valued functions of binary arguments. Harmonic analysis adds novel conceptual insights and unifying principles, improved computational techniques, and new measures of complexity to the traditional approach to switching theory. The first section is a summary of the chapter. Section II surveys classical Fourier transform properties and introduces the canonical expansion of a switching function as an n-dimensional abstract Fourier transform over the finite two-element field. The two most important transform properties are the convolution theorem, which leads to tests for prime implicants and disjunctive decompositions, and spectrum invariance which is basic to further theoretical developments and to a new synthesis technique called encoded input logic. Section III develops a new algorithm which concurrently extracts prime implicants and detects disjunctive decompositions of a switching function. Implicants of both the function and its complement are detected simultaneously, and “core” implicants can be identified. The algorithm which is not sensitive to functional complexity has been programmed for a commercial time-sharing system. Section IV introduces the restricted affine group (RAG) whose elements, called prototype transformations, encode the arguments and outputs of combinational logic functions. This group partitions the space F of all two-valued functions on Zn into 3, 8, and 48 equivalence classes respectively for n=3, 4, and 5. Unique representatives are identified for each class when n=3 and 4 and for 46 of the 48 classes of 5-argument functions. Section V applies the tools of abstract harmonic analysis to the synthesis problem for large truth tables (many-input combinational logic). A general multilevel synthesis approach, called encoded input logic, is introduced which is compatible with large-scale-integrated circuit technology. Both the conventional macrocellular and the newer microcellular array approach are included as special cases. Prototype encoding transformations are used to reduce the complexity of an imbedded normal form realization. Practical synthesis algorithms are based on Fourier analysis. A realistic 6-argument example is treated in detail. Section V concludes with a list of fundamental problems whose solutions would extend the research presented herein.

On the Plancherel Formula and the Paley-Wiener Theorem for Spherical Functions on Semisimple Lie Groups

Abstract: One of the difficult points in the proof of Harish-Chandra's Plancherel formula for spherical functions on a semisimple Lie group is to show that an appropriate inversion formula exists for the Fourier transform f f, and that this inversion formula holds for sufficiently many functions in the space of spherical square-integrable functions on G. Briefly, let f be a square integrable spherical function on G for which the Fourier transform f(X)= 5f(x)cp_(x)dx is well-defined. Here, as in [5], qA is the elementary (zonal) G spherical function corresponding to the parameter X. The problem is to show that for f in a L2-dense subspace of square-integrable spherical functions, the following inversion formula holds:

Non-linear wave-number interaction in near-critical two-dimensional flows

Abstract: This paper deals with a system of equations which includes as special cases the equations governing such hydrodynamic stability problems as the Taylor problem, the Benard problem, and the stability of plane parallel flow. A non-linear analysis is made of disturbances to a basic flow. The basic flow depends on a single co-ordinate η. The disturbances that are considered are represented as a superposition of many functions each of which is periodic in a co-ordinate ξ normal to η and is independent of the third co-ordinate direction. The paper considers problems in which the disturbance energy is initially concentrated in a denumerable set of ‘most dangerous’ modes whose wave-numbers are close to the critical wave-number selected by linear stability theory. It is a major result of the analysis that this concentration persists as time passes. Because of this the problem can be reduced to the study of a single non-linear partial differential equation for a special Fourier transform of the modal amplitudes. It is a striking feature of the present work that the study of a wide class of problems reduces to the study of this single fundamental equation which does not essentially depend on the specific forms ofthe operators in the original system of governing equations. Certain general conclusions are drawn from this equation, for example for some problems there exist multi-modal steady solutions which are a combination of a number of modes with different spatial periods. (Whether any such solutions are stable remains an open question.) It is also shown in other circumstances that there are solutions (at least for some interval of time) which are non-linear travelling waves whose kinematic behaviour can be clarified by the concept of group speed.

Hydrodynamic Correlation Functions for Binary Mixtures

Abstract: Correlation functions of various local thermodynamic variables of binary mixtures are obtained from the linearized hydrodynamic equations following the method of Mountain and Deutch. “Non-Lorentzian” parts of the time Fourier transformed correlation functions which have not been included by these authors are obtained here and their effect on the light scattering spectrum considered. An extension of the treatment by Fox and Uhlenbeck of Landau and Lifshitz hydrodynamical fluctuation theory to binary mixtures has been accomplished and correlation functions of the "fluctuating forces" are obtained for this case.

Application of Speckling for In-plane Vibration Analysis

Abstract: Time averaged as well as doubly exposed speckle patterns of in-plane moving objects are studied. These speckle patterns are recorded photo-graphically; their Fourier transform shows, interference fringes corresponding to the lateral displacement of the object. It leads to a very simple engineering tool for the analysis of mechanical vibrations as well as lateral displacements. Theoretical and experimental results will be shown.

The LEED Instrument Response Function

Abstract: The effect of the measurement system is usually ignored in the interpretation of LEED patterns. This effect can, however, be treated analytically through the use of the instrument response function, which represents the intensity function the instrument would record for a perfect surface. We have measured the response function for a particular LEED system using a double aperture Faraday collector and have identified the various experimental factors which determine its detailed functional form. The Fourier transform of the response function defines the range over which the instrument behaves as an interference detector. The effect of long range correlations on the pattern is suppressed by this finite range, with the result that fairly untidy surfaces may not be easily distinguished from those which are nearly perfect.

A technique for recovering Doppler line profiles from fabry-perot interferometer fringes of very low intensity.

Abstract: A technique for recovering doppler line profiles from Fabry-Perot interferometer fringes of very low intensity is described. The technique is based on a fourier decomposition of the data and a subsequent nonlinear least squares fit of the low order fourier coefficients to the fourier decomposition of an ideal instrument function. The ideal instrument function is expressed by the convolution of various instrument broadening functions and includes a parametric representation of the actual instrument. The method for recovering doppler temperature, emission line intensity, and mass motion of the emitting molecules is described. A theoretical analysis of errors for doppler temperature and emission line intensity is made for a statistical noise distribution superimposed upon a fringe profile of very low intensity. These errors are related to the emission line intensity, number of data points per fringe, background continuum level, and instrument parameters. As a specific example, the errors in retrieving the doppler temperature from the 6300-A atomic oxygen emission line OI((1)D - (3)P) in the nightglow are determined for the 15-cm Fabry-Perot interferometer at the University of Michigan Airglow Observatory.

Measurement and Interpretation of Dynamic Spectrograms of Picosecond Light Pulses

Abstract: A dynamic spectrogram depicts intensity as a function of frequency and time simultaneously, subject to the classical uncertainty relation δωδt≈2π. For an optical pulse having the Fourier transform |g(ω)|eiφ(ω), high‐resolution spectroscopy gives |g(ω)|2, while the dynamic spectrogram gives knowledge of ∂φ(ω)/∂ω permitting reconstruction of many features of the amplitude and phase modulations of the original pulse. Linear, parabolic, and two sinusoidal spectrogram shapes are interpreted theoretically. An instrument for measuring dynamic spectrograms is described. It involves the use of a spectrometer with ultrafast time response and has been used in an antisymmetric mode to measure frequency sweep rate as a function of wavelength for picosecond laser pulses. The change in the dynamic spectrogram of a pulse brought about by linear pulse compression and the resultant change in pulse envelope shape have been computed for a typical picosecond pulse.

Method for Finding the Equation of State of Liquid Metals

Abstract: A variational method due to Mansoori and Canfield for finding the equation of state of a liquid has been extended to the case where atoms have a pair potential whose Fourier transform is known analytically. The method is used to calculate thermodynamic properties of sodium. The results are in good agreement with experiment.