Showing papers on "Fourier transform published in 1968"

Application of fourier analysis to the visibility of gratings

Abstract: 1. The contrast thresholds of a variety of grating patterns have been measured over a wide range of spatial frequencies.2. Contrast thresholds for the detection of gratings whose luminance profiles are sine, square, rectangular or saw-tooth waves can be simply related using Fourier theory.3. Over a wide range of spatial frequencies the contrast threshold of a grating is determined only by the amplitude of the fundamental Fourier component of its wave form.4. Gratings of complex wave form cannot be distinguished from sine-wave gratings until their contrast has been raised to a level at which the higher harmonic components reach their independent threshold.5. These findings can be explained by the existence within the nervous system of linearly operating independent mechanisms selectively sensitive to limited ranges of spatial frequencies.

Computer "Experiments" on Classical Fluids. II. Equilibrium Correlation Functions

Abstract: : Equilibrium correlation functions for a dense classical fluid are obtained by integrating the equation of motion of a system of 864 particles interacting through a Lennard-Jones potential. The behaviour of the correlation function at large distance, and that of its Fourier transform at large wave number are discussed in detail and shown to be related to the existence of a strong repulsion in the potential. A simple hard sphere model is shown to reproduce very well the Fourier transform of those correlations functions at high density, the only parameter of the model being the diameter a of the hard spheres. (Author)

The Spectral Function of an Elliptic Operator

Abstract: In this paper we shall obtain the best possible estimates for the remainder term in the asymptotic formula for the spectral function of an arbitrary elliptic (pseudo-)differential operator. This is achieved by means of a complete description of the singularities of the Fourier transform of the spectral function for low frequencies.

Signal energy distribution in time and frequency

Abstract: The Fourier representation of signals and its relation to the signal structure in time and frequency, and more generally the inherent properties of phase-modulated signals, have received considerable attention in the past. These topics have led to such seemingly unrelated studies as the representation of a signal in a combined time-frequency plane, "instantaneous power spectra," and the ambiguity function and its transform relations. It is shown in this paper that the studies can be unified by the introduction of the concept of the complex energy density function of a signal. The function is an extension and combination of the one-dimensional energy density functions in time and frequency, the energy density spectrum |\Psi(f)|^{2} , and energy density waveform |\psi (t)|^{2} . On the basis of the complex energy density function, the significance of complicated-appearing transform relations is readily understood. The new concept also conveys a good insight into the internal structure of phase-modulated signals.

The Estimation of Probability Densities and Cumulatives by Fourier Series Methods

Abstract: A class of estimators (referred to as the Fourier estimators m and m) of the probability density function f and the associated cumulative distribution function F are considered. Here m = Σmk=0 âkψk and m = Σmk=0 Âk ψk where the functions {ψk} comprise an orthogonal set with respect to weight function w(x), and the statistics âk and Âk are formed from the n unordered observations.Simple expressions are found for the mean integrated square errors, M.I.S.E., of the estimators m and m, i.e., E∫{ƒ(x) – m(x)}2ω(x)dx and E∫{F(x) – m(x)}2w(x)dx in terms of the variances of âk and Âk and the Fourier coefficients of f and F.For Fourier estimators based upon the trigonometric orthogonal functions the âk are the sample trigonometric moments. The variances and covariances of the statistics âk and Âk for these special cases are shown to be linear functions of the density f's Fourier coefficients. Therefore, simple expressions are obtained which relate the M.I.S.E. of the Fourier estimators m and m to the Fourier coeffi...

Variational Method for the Analysis of Microstrip Lines

Abstract: This paper reports a method for computing the line capacitance of a microstrip line based on the application of Fourier transform and variational techniques. The characteristic impedance, guide wavelength, and the surface potential distribution in the microstrip line are obtained for a range of structure parameters and the dielectric constant. The results calculated from the expressions developed in the paper are compared with the theoretical results presently available in the literature and good agreement is found. Comparison with available experimental results is also made where feasible. Possible applications and limitations of the method are discussed.

Very high frequency radiowave scattering by a disturbed sea surface Part I: Scattering from a slightly disturbed boundary

Abstract: This paper considers the scattering of very high frequency (VHF) electromagnetic waves from a random weakly corrugated surface by the perturbation method. The calculations show that the scattering has a resonant nature, i.e., only certain Fourier components of the surface shape are responsible for scattering in every given direction. Experiments carried out in a water basin confirmed the results of the calculations. The backscattered intensity is proportional to the spectral density of those Fourier components of the surface oscillation that have a resonant space period. In these experiments, resonant maxima of the reflected signal corresponding to the second-order approximation of the perturbation method were also observed. The frequency spectrum of the scattered electromagnetic field is also investigated. It is shown that the spectrum of the scattered radiation is shifted from the incident frequency by a certain value related to the phase velocity of the resonantly scattering Fourier component of the surface shape. The experimentally observed dependence of the scattered intensity on frequency and the theoretically predicted one are very much alike.

Design of monopulse antenna difference patterns with low sidelobes

Abstract: The flexibility of modern monopulse radar antenna systems makes possible the independent optimization of sum and difference patterns. The two parameter difference pattern, developed here for the circular aperature antenna, is designed to have nearly equal sidelobes similar to those of the Taylor sum pattern. The difference pattern is asymptotic to a model difference pattern which has the greatest slope (angle sensitivity) for a given sidelobe level. The model function is unrealizable because it has uniform sidelobes which are infinite in extent. The two parameter difference pattern is realizable and is expressed in a Fourier — Bessel series of N terms in a manner similar to Taylor's treatment of the sum pattern. The other parameter, A, controls sidelobe level. Comprehensive tables of the Fourier — Bessel coefficients are given for both the circular aperture series and the difference pattern series. Directivity and angle sensitivity are investigated and found to have maximum values that decrease as sidelobe level decreases. The monopulse system performance using the asymptotic difference pattern and the Taylor sum pattern compares favorably with a maximum likelihood angle estimation system. Development of a line source difference pattern is presented in the appendix.

Very High Frequency Radiowave Scattering by a Disturbed Sea Surface

Abstract: Tbis paper considers the scattering of very high frequency 0 electromagnetic waves from a random weakly corrugated surface by the perturbation method. The calculations show that the scattering has a resonant nature, i.e., only certain Fourier components of the surface shape are responsible for scattering in every given direction. Experiments carried out in a water basin confkmed the results of the calculations. The backscattered intensity is proportional to the spectral density of those Fourier components of the surface oscillation that have a resonant space period. In these experiments, resonant maxima of the reflected si,d cor- responding to the second-ordw approximation of the perturbation method were also observed. The frequency spectrum of the scattered electromagnetic field is also investigated. It is shown that the spectrum of the scattered radiation is shifted from the incident frequency by a certain value related to the phase velocity of the resonantly scattering Fourier component of the surface shape. The experimentally observed dependence of the scattered intensity on frequency and the theoretically predicted one are very much alike.

Sampling Theorem for the Fourier Transform of a Distribution with Bounded Support

Abstract: This result is due originally to Whittaker [1]. Either the expansion (2) or the fact that a bandlimited function is detered by its sample values f (nwr/Q) is frequently called the sampling theorem. Kotelnikov, Shannon and later workers have exploited the sampling theorem and its extensions in developing communication theory. It is also possible to reverse the implication in proceeding from (1) to (2) and to use (2) as an interpolation series. That is, given a suitably restricted sequence of numbers {f(nar/2)}, one can construct a function f(t) by the series (2) which interpolates values between these numbers. The function so constructed is bandlimited; that is, it satisfies (1). It has been suggested by Hamming 12, p. 276] that this kind of bandlimited interpolation function is more useful than a polynomial for some purposes. It is natural to inquire whether the sampling theorem is still true if f (t) is the Fourier transform of a distribution with support in the interval (-Q, Q2). If we put g (co) = a (co a), where a is the Dirac distribution and where < a < Q, the Fourier transform of g (w) is f (t) = e'it. Substitution in (2) produces the equation

Sequential encoding with multislit spectrometers

Abstract: A new method for optically encoding the spectral output of multislit spectrometers is proposed. It is based on sequential measurements of the total intensity of light in combinations of selected spectral bands. The resulting encoded optical information is obtained as a set of simultaneous linear algebraic equations, and spectrum reconstruction is accomplished through the use of matrix inversion techniques. This encoding method avoids the problems associated with frequency transform encoding, because it is not based on the usual Fourier or Fresnel transforms. Rather, it makes use of the theory of simultaneous linear algebraic equations.

Programming and analysis for digital time series data

Abstract: : Contents: Preprocessing of data; Digital filtering; Fourier series and Fourier transform computations; Correlation function computations; Spectral density function computations; Frequency response function and coherence function computations; Probability density function computations; Nonstationary processes; and Test case and examples.

Introduction to Fourier Transform Spectroscopy

Abstract: This paper provides an introduction to Fourier transform spectroscopy. The basic concepts of the Fourier transform technique are reviewed. These include the characteristics and generation of interferograms and the generation of spectra by the Fourier transformation of interferograms. The multiplex and throughput advantages and the data handling disadvantages are also discussed. The optical, electronic, and data-handling instrumentation and techniques necessary for a Fourier transform spectrometer system are, in general, quite different from those of a conventional prism or grating spectrometer. The requirements of these areas are outlined and recent developments are reviewed. The Fourier transform technique has been successfully applied to a number of spectral measurements from the visible through to the far infrared. Some of the more recent applications are discussed.

Formation and inversion of pseudoscopic images.

Abstract: The reconstruction from an integral photograph is pseudoscopic. It is usually inverted into an orthoscopic image by a second recording and reconstruction process. Some schemes are described which perform this inversion without the need for a second recording. All schemes use some form of autocollimating screen, i.e., a screen which reflects a ray of light back onto itself. The inversion can either be formed on the scene (inversion from orthoscopic to pseudoscopic) before the integral photograph is recorded or it can be performed on the pseudoscopic reconstruction. Some theoretical considerations concerning the optimum screen are given. The relation between these schemes and integral photography,image dissection, and a previously proposed scheme are given.

Computed Response of an Acoustic Logging Tool

Abstract: Transient waveforms have been computed for a family of source and receiver types and for geometry corresponding to plausible acoustic logging tools in oil wells. The acoustic tool now in commercial use is one member of the family. The computations also treat flexural waves and torsional waves. All quantities are expressed in terms of appropriate vector and scalar potentials, and output waveforms are obtained by numerical evaluation of triple Fourier transforms. When loss parameters are included in the stress‐strain relations, the Fourier inversion formulas are free of singularities and numerical integration is straightforward. For the purely elastic case, it is necessary to cope with the singularities of the integrands in performing the numerical integrations. Dispersion curves and other features agree with earlier publications on wave propagation along concentric cylinders.

Analysis of gravity anomalies of two-dimensional faults using fourier transforms*

Abstract: The Fourier transform formula for a two-dimensional fault truncating a horizontal bed at an arbitrary angle of inclination is derived. The amplitude spectrum of the Fourier transform is found to give information about the depth to the top of the upper part of the faulted bed and the inclination of the fault-plane. Under suitable conditions the thickness and the displacement of the bed involved can be obtained. With actual field data, these transforms can be obtained at discrete points by a Fourier analysis of the gravity anomaly. A field example from the Logan fault area near Montreal, Que., Canada, is given.

Resolution enhancement of ESR spectra from irradiated single crystals of glycine.

Abstract: The electron spin resonance spectrum of +H3NĊHCOO− in single crystals of glycine γ irradiated at room temperature has been re‐examined at X band and evaluated by resolution‐enhancement technique. The resolution enhancement is obtained by reducing the linewidth by a factor of maximum 0.55 at the expense of the signal‐to‐noise ratio. The calculations, which require a high‐speed digital computer, consist of three steps: (1) transformation of the experimental spectrum to a complex Fourier plane, (2) multiplication of this Fourier spectrum with a suitable linewidth‐reduction function, and (3) inverse transformation back to the real plane. From the obtained constants, the hyperfine coupling tensors have been calculated, including contributions from second‐order effects (ΔMI = ± 1). One of the principal values of one of the tensors was appreciably affected by these second‐order effects. The results fully confirm the structure of the radical.

Spectral Recovery in Fourier Spectroscopy

Abstract: The general problem of obtaining an acceptable estimate of a spectrum in Fourier spectroscopy is discussed for four specific cases. The two known techniques currently used for the spectral recovery from an asymmetric interferogram are shown to be mathematically equivalent. A discussion of their practical merit is also included.

Optical contrast enhancement system

Abstract: An optical system for enhancing the contrast of optical images. A sheet of photochromic material is disposed in the Fraunhofer diffraction or first Fourier transform plane of the optical system. Among the wavelengths of light transilluminating the object or transparency is a band which darkens the photochromic material especially at the center of the Fourier series display which contains contrast information (the zeroth order of the Fourier series display). This zeroth order term is attenuated to a much greater degree than the higher order terms which contain the information describing the object or transparency, thus producing an enhanced image.

Three-Dimensional Fourier-Transform Method for Synthesizing Binary Holograms*

Abstract: A new method for the synthesis of holograms, based on a Fourier transformation, has been theoretically derived and experimentally verified. The method extends the familiar two-dimensional optical Fourier-transform analysis to include objects located in a volume centered about the focal point of the transform lens, and is specifically developed to provide an easily synthesized binary hologram for the reconstruction of three-dimensional objects. The synthetic holograms produced display, in addition to three-dimensionality of the reconstructed image, all other properties associated with photographic holograms. This technique for synthesizing holograms has the potential advantage of computational simplicity and increased image resolution over the Fresnel-transform technique, and permits the synthesis of a variety of objects composed of discrete points, continuous lines, and continuous surfaces.

Pattern defect sensing using error free blocking spacial filter

Abstract: Defects in microcircuit patterns are sensed by illuminating the pattern with monochromatic collimated light. The illuminated pattern is imaged through a lens to produce substantially a twodimensional optical Fourier transform of the pattern at a plane on the output side of the lens. An optical filter (transparency) which includes substantially the negative of the Fourier transform of a defect-free specimen of the microcircuit is placed at the aforesaid plane to block the optical frequency components corresponding to the defect-free specimen. Light passing through the filter is processed to provide various indications of the pattern defects.

Numerical calculation of fourier integrals with cubic splines

Abstract: The most used formula for calculation of Fourier integrals is Filon's formula which is based on approximation of the function by a quadratic in each double interval. In order to obtain a better approximation we use the cubic spline fit. The method is not restricted to equidistant points, but the final formulas are only derived in this case. Test computations show that the spline formula may be superior to Filon's formula.

Diffusion Analysis for the Propagation of Mutual Coherence

Abstract: A new method of evaluating the mutual-coherence function for propagation in a randomly inhomogeneous medium like the atmosphere is presented. The new method, which is highly physical, as distinct from a mathematical approach, does not involve the treatment of any differential equations. Instead, the treatment is based on decomposition of a randomly distorted wavefront into a set of plane waves with random amplitudes. These plane waves constitute orthogonal modes. Propagation in a random medium is treated as the physical process of diffusion of amplitude (or energy) between the modes, and a short-path-propagator function for this diffusion is developed. From the short-path-propagator function, a long-path-propagator function is easily obtained, and from this the mutual-coherence function is computed. Starting from the known short-path mutual-coherence function, which is known to be accurate, the mutual-coherence function for long paths is obtained. The results are in agreement with previous results, all of whose derivations have recently been subject to criticism. Because this derivation is not a mathematical exercise, it should not be subject to any of these or similar criticisms, which were primarily questions of mathematical rigor.

Theory of Remote Sensing of Clear-Air Turbulence Profiles*

Abstract: A method is presented for determining the profile of atmospheric turbulence along the line of sight to a star or other plane-wave light source. The spectral density of the refractive-index fluctuations is assumed to be a product of a known function of the turbulent wavenumber, Φn(0)(κ), and an unknown function of position along the line of sight, Cn2(z). We demonstrate that Tatarski’s formula for calculating the correlation function of the fluctuations of the logarithmic amplitude of a plane electromagnetic wave, Bχ(ρ), from Cn2(z) is actually an integral equation for Cn2(z) with a unique solution. Consequently, the turbulence profile, Cn2(z), may be calculated by taking an integro-differential transform of the correlation function, Bχ(ρ). For Kolmogorov turbulence the kernel of the transform contains a confluent hypergeometric function. This result suggests that it may be possible to determine the magnitude and position of distant turbulence from a measurement of the correlation function of light which has passed through the turbulence.

Fast Fourier transforms and Butler matrices

Abstract: The direct equivalence between fast Fourier transforms and Butler matrices is pointed out. The parallel and extensive development of these techniques is discussed.