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Showing papers on "Fourier transform published in 1979"


Journal ArticleDOI
TL;DR: In this paper, it was shown that rotational spin echoes provide a convenient means of studying very slow random molecular rotations (τc≲1 sec), which must be described by a proper average Hamiltonian theory.
Abstract: The NMR free induction decay from a spinning sample having inhomogeneous anisotropic interactions (chemical shifts, first order quadrupole couplings) takes the form of a train of rotational spin echoes. The Fourier transform of the echo envelope is a sharp spectrum from which the effects of anisotropy have been removed. The Fourier transform of the echo shape contains information concerning the anisotropies: This information can be extracted by a moment analysis. The effects of localized homonuclear spin–spin interactions are to convert the ’’isotropic’’ spectrum into a characteristic powder pattern. Second order quadrupole coupling produces a similar effect. It is shown in this case that the usual second‐order level shifts cannot be used to calculated this pattern, which must be described by a proper average Hamiltonian theory. Finally it is shown that rotational spin echoes provide a convenient means of studying very slow random molecular rotations (τc≲1 sec).

1,224 citations


Journal ArticleDOI
D. H. Kelly1
TL;DR: The spatio-temporal threshold surface for stabilized vision is constructed, and its properties are displayed in terms of the usual frequency parameters; e.g., at low spatial frequencies, the temporal response becomes nearly independent of spatial frequency, while at low temporal frequency, the spatial response becomes independent of temporal frequency.
Abstract: The stabilized contrast-sensitivity function measured at a constant retinal velocity is tuned to a particular spatial frequency, which is inversely related to the velocity chosen The Fourier transforms of these constant-velocity passbands have the same form as retinal receptive fields of various sizes At low velocities, in the range of the natural drift motions of the eye, the stabilized contrast-sensitivity function matches the normal, unstablized result At higher velocities (corresponding to motions of objects in the environment), this curve maintains the same shape but shifts toward lower spatial frequencies The constant-velocity passband is displaced across the spatio-temporal frequency domain in a manner that is almost symmetric about the constant-velocity plane at v = 2 deg/s Interpolating these diagonal profiles by a suitable analytic expression, we construct the spatio-temporal threshold surface for stabilized vision, and display its properties in terms of the usual frequency parameters; eg, at low spatial frequencies, the temporal response becomes nearly independent of spatial frequency, while at low temporal frequencies, the spatial response becomes independent of temporal frequency

761 citations


Journal ArticleDOI
TL;DR: The Wigner distribution function of optical signals and systems can be interpreted directly in terms of geometrical optics as mentioned in this paper, which can be applied to partially coherent light as well.
Abstract: The Wigner distribution function of optical signals and systems has been introduced. The concept of such functions is not restricted to deterministic signals, but can be applied to partially coherent light as well. Although derived from Fourier optics, the description of signals and systems by means of Wigner distribution functions can be interpreted directly in terms of geometrical optics: (i) for quadratic-phase signals (and, if complex rays are allowed to appear, for Gaussian signals, too), it leads immediately to the curvature matrix of the signal; (ii) for Luneburg’s first-order system, it directly yields the ray transformation matrix of the system; (iii) for the propagation of quadratic-phase signals through first-order systems, it results in the well-known bilinear transformation of the signal’s curvature matrix. The zeroth-, first-, and second-order moments of the Wigner distribution function have been interpreted in terms of the energy, the center of gravity, and the effective width of the signal, respectively. The propagation of these moments through first-order systems has been derived. Since a Gaussian signal is completely described by its three lowest-order moments, the propagation of such a signal through first-order systems is known as well.

411 citations


Journal ArticleDOI
TL;DR: In this paper, a linear system, Fourier transform and Optica Acta: International Journal of Optics: Vol. 26, No. 7, pp. 836-836.
Abstract: (1979). Linear Systems, Fourier Transforms and Optics. Optica Acta: International Journal of Optics: Vol. 26, No. 7, pp. 836-836.

332 citations


Journal ArticleDOI
TL;DR: A new family of unitary transforms is introduced and it is shown that the well-known discrete Fourier, cosine, sine, and the Karhunen-Loeve (KL) (for first-order stationary Markov processes) transforms are members of this family.
Abstract: A new family of unitary transforms is introduced. It is shown that the well-known discrete Fourier, cosine, sine, and the Karhunen-Loeve (KL) (for first-order stationary Markov processes) transforms are members of this family. All the member transforms of this family are sinusoidal sequences that are asymptotically equivalent. For finite-length data, these transforms provide different approximations to the KL transform of the said data. From the theory of these transforms some well-known facts about orthogonal transforms are easily explained and some widely misunderstood concepts are brought to light. For example, the near-optimal behavior of the even discrete cosine transform to the KL transform of first-order Markov processes is explained and, at the same time, it is shown that this transform is not always such a good (or near-optimal) approximation to the above-mentioned KL transform. It is also shown that each member of the sinusoidal family is the KL transform of a unique, first-order, non-stationary (in general), Markov process. Asymptotic equivalence and other interesting properties of these transforms can be studied by analyzing the underlying Markov processes.

314 citations


Journal ArticleDOI
Wai-Hon Lee1
TL;DR: An accurate numerical method which circumvents two difficulties in using binary Fourier transform holograms and three different techniques for storing amplitude information in the binary computer-generated holograms are discussed.
Abstract: Binary computer-generated holograms are similar to interferograms with fringe patterns hardclipped by a photographic process. Therefore the fringe locations in the binary hologram can be determined by solving a grating equation. However, there are two difficulties in using this approach to make binary Fourier transform holograms. First the discrete Fourier transform provides only data at discrete sampling locations. Second, the phase angles thus calculated are given in terms of the residues of the original phase angles after multiples of 2pi rad are removed. In this paper an accurate numerical method which circumvents these two difficulties is described. Also discussed are three different techniques for storing amplitude information in the binary computer-generated holograms. The different solution methods discussed in this paper are further illustrated by a number of computer-generated holograms and their reconstructed images.

281 citations


Journal ArticleDOI
TL;DR: In this paper, a relation from vector scattering theory has been used to predict the angular distribution of scattered light from optical surfaces as a function of the wavelength, optical constants of the material, and spectral density function.
Abstract: A relation from vector scattering theory has been used to predict the angular distribution of scattered light from optical surfaces as a function of the wavelength, optical constants of the material, and spectral density function. For calculations of one-dimensional (two-dimensional) scattering, the spectral density function of the surface roughness is obtained from the Fourier transform (Hankel transform) of the autocovariance function, which in turn is determined from surface-profile data. Measured statistics presented for various types of optical surfaces indicate that there are three basic components of surface structure: long-range waviness, short-range random roughness, and periodicity; one or more of which may be present on a given surface. Averaged and unaveraged surface-profile data for the same surface are shown to be consistent. Experimental data are presented that yield an exponential autocovariance function, and give a reasonably good fit to a Poisson distribution of zero crossings. Finally, angular scattering values calculated using measured surface statistics with vector scattering theory are compared to scattering values measured on the same surface. The shapes of the measured and calculated curves are similar, but the magnitudes are not. However, the rms surface roughnesses calculated from total integrated scattering measurements are in excellent agreement with values measured directly on these same surfaces.

273 citations


Journal ArticleDOI
TL;DR: In this article, a new method of obtaining NMR images is described which retains the inherent sensitivity of the two-dimensional Fourier transform while obviating the need for any changes of field gradient.

250 citations


Journal ArticleDOI
TL;DR: The orientation tuning, spatial‐frequency tuning and responsiveness of cells to a plaid pattern were found to be predictable from the pattern's two‐dimensional Fourier spectrum, and both simple and complex striate cortex cells can be characterized as two-dimensional spatial‐ frequencies filters.
Abstract: 1. Cells in visual cortex have been alternately considered as bar and edge detectors, or as spatial-frequency filters responding to the two-dimensional Fourier component of patterns. 2. The responses to gratings and to checkerboards allow one to test these alternate models: the Fourier components of a checkerboard pattern do not occur at the same orientation as the edges, nor do the checkerboard spatial frequencies correspond to the check widths. 3. Knowing the orientation tuning of a cell for gratings, one can precisely predict its orientation tuning to checkerboards from the orientation of the fundamental Fourier components of the patterns, not from the orientation of their edges. This was found for both square and rectangular checkerboards, and held for both simple and complex cortical cells. 4. Knowing the spatial tuning of a cell for sine-wave gratings, one can precisely predict its spatial tuning to square and rectangular checkerboards from the spatial frequencies of the fundamental Fourier components of the patterns, not from the widths of their checks. 5. When presented with checkerboards in which not the fundamental but the upper harmonics were within its spatial bandpass, a cell's orientation tuning was found to be predictable from the (quite different) orientation of the higher Fourier harmonic components, but not from the orientation of the edges. 6. Knowing a cell's contrast sensitivity for gratings, one can predict the cell's contrast sensitivity for checkerboards much more accurately from the amplitudes of the two-dimensional Fourier components of the patterns than from the contrasts of the patterns. 7. The orientation tuning, spatial-frequency tuning and responsiveness of cells to a plaid pattern were also found to be predictable from the pattern's two-dimensional Fourier spectrum. 8. Both simple and complex striate cortex cells can thus be characterized as two-dimensional spatial-frequency filters. Since different cells responsive to the same region in the visual field are tuned to different spatial frequencies and orientations, the ensemble of such cells would fairly precisely encode the two-dimensional Fourier spectrum of a patch of visual space.

235 citations


Journal ArticleDOI
TL;DR: In this article, the authors discuss the discrete Fourier transform and point out some computational problems in complex analysis where it can be fruitfully applied, such as trigonometric interpolation, conjugate periodic functions, and numerical inversion of Laplace transforms.
Abstract: In this paper we discuss the discrete Fourier transform and point out some computational problems in (mainly) complex analysis where it can be fruitfully applied. We begin by describing the elementary properties of the transform and its efficient implementation, both in the one-dimensional and in the multi-dimensional case, by the reduction formulas of Cooley, Lewis, and Welch (IBM Res, paper, 1967).The following applications are then discussed: Calculation of Fourier coefficients using attenuation factors; solution of Symm’s integral equation in numerical conformal mapping; trigonometric interpolation; determination of conjugate periodic functions and their application to Theodorsen’s integral equation for the conformal mapping of simply and of doubly connected regions; determination of Laurent coefficients with applications to numerical differentiation, generating functions, and the numerical inversion of Laplace transforms; determination of the “density” of the zeros of high degree polynomials. We then...

228 citations


Journal ArticleDOI
TL;DR: In this paper, the authors developed a general theory for numerical evaluation of integrals of the Hankel type and showed that the absolute error on the output function is less than (K(ω 0)/r) · exp (−ρω 0/Δ), Δ being the logarthmic sampling distance.
Abstract: Inspired by the linear filter method introduced by D. P. Ghosh in 1970 we have developed a general theory for numerical evaluation of integrals of the Hankel type: Replacing the usual sine interpolating function by sinsh (x) =a· sin (ρx)/sinh (aρx), where the smoothness parameter a is chosen to be “small”, we obtain explicit series expansions for the sinsh-response or filter function H*. If the input function f(λ exp (iω)) is known to be analytic in the region o < λ < ∞, |ω|≤ω0 of the complex plane, we can show that the absolute error on the output function is less than (K(ω0)/r) · exp (−ρω0/Δ), Δ being the logarthmic sampling distance. Due to the explicit expansions of H* the tails of the infinite summation ((m−n)Δ) can be handled analytically. Since the only restriction on the order is ν > − 1, the Fourier transform is a special case of the theory, ν=± 1/2 giving the sine- and cosine transform, respectively. In theoretical model calculations the present method is considerably more efficient than the Fast Fourier Transform (FFT).



Journal ArticleDOI
TL;DR: In this article, analytical expressions for Fourier transform ion cyclotron resonance (FT-ICR) line shape [absorption mode, dispersion mode, and magnitude (absolute value) mode] are derived for coherently excited ions that undergo both reactive and nonreactive ion-molecule collisions.
Abstract: Analytical expressions for Fourier transform ion cyclotron resonance (FT‐ICR) line shape [absorption mode, dispersion mode, and magnitude (absolute value) mode] are derived for coherently excited ions that undergo both reactive and nonreactive ion–molecule collisions. The expressions are valid at arbitrary sample pressure, and reduce to particularly simple form in the ’’zero‐pressure’’ limit (essentially no ion–molecule collisions during the data acquisition period) or ’’high‐pressure’’ limit (many ion–molecule collisions during the data acquisition period). The zero‐pressure line shape has been analyzed in earlier papers; in this paper, various useful properties of the high‐pressure line shape (e.g., linewidth, mass resolution, and upper mass limit) are tabulated for various choices of the fraction of maximal absorption (or magnitude) peak height at which linewidth is to be measured. Absorption, dispersion, and magnitude spectra are plotted for zero‐pressure and high‐pressure limits, and also for an inte...


Journal ArticleDOI
TL;DR: Both the mode eigenvalues and mode weights, which are necessary for determining the impulse response, can be obtained with high accuracy from a numerical Fourier transform of the complex field-correlation function by the use of digital-filtering techniques.
Abstract: Methods are developed for extracting from a numerical propagating-beam solution of a scalar wave equation the information necessary to compute the impulse-response function and the pulse dispersion for a multimode graded-index fiber. It is shown that the scalar Helmholtz equation and the parabolic wave equation have the same set of eigenfunctions in common and that the eigenvalues for the two equations are simply related. Thus one can work exclusively with the simpler parabolic equation. Both the mode eigenvalues (propagation constants) and mode weights, which are necessary for determining the impulse response, can be obtained with high accuracy from a numerical Fourier transform of the complex field-correlation function by the use of digital-filtering techniques. It is shown how a solution obtained in the absence of profile dispersion can be simply corrected for the presence of profile dispersion. In an illustrative example a graded-index fiber with a central dip in its profile is considered.

Journal ArticleDOI
TL;DR: In this paper, a general treatment of scaler diffraction theory is presented and some interesting concepts are discussed which yield new insight into the phenomena of diffraction throughout the whole space in which it occurs.
Abstract: A general treatment of scaler diffraction theory is presented and some interesting concepts are discussed which yield new insight into the phenomena of diffraction throughout the whole space in which it occurs. The direct application of Fourier transform theory to the diffraction process results in two equivalent descriptions of the diffracted wave field: one describes the wave field as a superposition of plane‐wave components and corresponds to the transfer function approach in linear systems theory, and the other describes the wave field as a superposition of hemispherical‐wave components (or Huygen wavelets) and corresponds to the impulse response approach in linear systems theory. The convolution of the initial disturbance with the impulse response results in the well‐known Rayleigh‐Sommerfeld formula for near‐field diffraction. This formula is then rewritten in the form of the Fourier transform integral of a generalized pupil function which includes phase variations in the diffracting aperture. Any d...

Journal ArticleDOI
TL;DR: A new method of deriving very fast Fourier transform algorithms that do not employ multiplication and have a form suitable for high performance hardware implementations is described.
Abstract: A new method of deriving very fast Fourier transform (FFT) algorithms is described. The resulting algorithms do not employ multiplication and have a form suitable for high performance hardware implementations. The complexity of the algorithms compares favorably to the recent results of Winograd [1].


Journal ArticleDOI
15 Aug 1979
TL;DR: In this paper, the Fourier transform of the electron momentum density (B(r) is examined and a number of theoretical results relating to this new observable are given, and a wave-mechanical representation with (natural) orbitals is employed for the subsequent analysis of B(r).
Abstract: Recent work has shown that the one-dimensional projection of the electron momentum density, the Compton profile, can be usefully interpreted as a position space quantity. This has led to an examination of B(r), the Fourier transform of the momentum density. A number of theoretical results relating to this new observable are given. The wave-mechanical representation with (natural) orbitals is employed, and this forms the basis for the subsequent analysis of B(r). The relationship of B(r) to overlap integrals and more generally to other electron density functions is considered. Atomic wavefunctions for krypton are used to illustrate the potential of this new approach to the analysis of momentum density data. General expressions are derived for atoms and molecules, and the radial and angular dependence of B(r) for various orbitals is displayed. The possibility of extracting accurate bond lengths from B(r) is assessed, and an example is given using some recent theoretical data for the fluorine molecule.

Journal ArticleDOI
01 Dec 1979
TL;DR: In this article, Fan-Beam reconstruction is applied to the problem of reconstructing density distributions from arbitrary fan-beam data, where the kernel of the general linear operator is factored and rewritten as a function of the difference of coordinates only and the superposition integral consequently simplifies into a convolution integral.
Abstract: In a previous paper a technique was developed for finding reconstruction algorithms for arbitrary ray-sampling schemes. The resulting algorithms use a general linear operator, the kernel of which depends on the details of the scanning geometry. Here this method is applied to the problem of reconstructing density distributions from arbitrary fan-beam data. The general fan-beam method is then specialized to a number of scanning geometries of practical importance. Included are two cases where the kernel of the general linear operator can be factored and rewritten as a function of the difference of coordinates only and the superposition integral consequently simplifies into a convolution integral. Algorithms for these special cases of the fan-beam problem have been developed previously by others. In the general case, however, Fourier transforms and convolutions do not apply, and linear space-variant operators must be used. As a demonstration, details of a fan-beam method for data obtained with uniform ray-sampling density are developed.


Book ChapterDOI
TL;DR: In this article, the authors discussed small-angle scattering experiments with particles in solution and showed that the correlation between the distance distribution function and the structure of the particle is also discussed.
Abstract: Publisher Summary This chapter discusses small-angle scattering experiments with particles in solution—i.e., the particles are nonoriented. A large number of particles contribute to the scattering and the resulting spatial average leads to a loss in information. The information contained in the three-dimensional electron density distribution is thereby reduced to the one-dimensional distance distribution function. This function is proportional to the number of lines with length, which connect any volume element i with any volume element k of the same particle. The spatial orientation of these connection lines is of no account to the function. The connection lines are weighted by the product of the number of electrons situated in the volume elements i and k , respectively. The correlation between the function and the structure of the particle is also discussed in the chapter. The connection between the distance distribution function and the measured experimental scattering curve is also shown. It is observed that the each distance between two electrons of the particle, which is part of the function, leads to an angular-dependent scattering intensity. This physical process of scattering can be mathematically expressed by a Fourier transformation, which defines the way in which the information in “real space” (distance distribution function) is transformed into “reciprocal space” (scattering function). The chapter also discusses monochromatization and the camera type developed in Graz.

Journal ArticleDOI
TL;DR: In this article, a general method for direct measurement of differential absorption intensities using a Fourier transform infrared spectrometer is described, and specific expressions are presented for the measurement of circular and linear dichroism.
Abstract: A general method for the direct measurement of differential absorption intensities using a Fourier transform infrared spectrometer is described. The differential intensities must be higher in frequency than the interferogram frequencies and may arise from a periodic variation of the absorption strength of the sample, or by dichroic response of the sample to alternate states of polarization of the infrared beam. Specific expressions are presented for the measurement of circular and linear dichroism. These expressions represent an extension of the Grosjean-Legrand polarization modulation technique to Fourier transform interferometry.

Journal ArticleDOI
TL;DR: In this article, an image processing technique that uses interferometer data (the modulus of the Fourier transform) to reconstruct diffraction limited images is discussed. But it is not shown how to find a real, non-negative object that agrees with the modulus data.
Abstract: For telescopes operating at optical wavelengths, the turbulence of the atmosphere limits the resolution of space objects to about one second of arc, although the diffraction limit of the largest telescopes is many times as fine. We discuss an image processing technique that uses interferometer data (the modulus of the Fourier transform) to reconstruct diffraction limited images. Data from a stellar speckle interferometer or from an amplitude interferometer can be used. The processing technique is an iterative method that finds a real, non-negative object that agrees with the Fourier modulus data. For complicated two-dimensional objects, the solutions found by this technique are surprisingly unique. New results are shown for simulated speckle interferometer data having realistic noise present.

Journal ArticleDOI
TL;DR: The theoretical formalism for the treatment of magnetic resonance in the presence of nonequilibrium chemical reactions is presented in this paper, which is applied to the study of fast transient chemical reactions by means of NMR Fourier experiments.

Journal ArticleDOI
TL;DR: The feasibility of applying the methods of factor analysis to Fourier transform infrared spectra is examined in this article, where the method is designed to determine the number of components contributing to a spectral region.
Abstract: The feasibility of applying the methods of factor analysis to Fourier transform infrared spectra is examined. The method is designed to determine the number of components contributing to a spectral region. The technique has been applied to a model system consisting of solutions of p-, o-, and m-xylenes. The method has also been used for calculating the number of independently absorbing species in several polymer systems. Spectral requirements and the general applicability of factor analysis are also discussed.

Journal ArticleDOI
TL;DR: It is shown that the impulse response for the composite of the projection and back-projection operators for fan-beam geometry is not spatially invariant if the data are taken over 180° as is true for parallel-beam projection data.

Journal ArticleDOI
TL;DR: Fourier transformed photoacoustic IR spectroscopy of a solid is demonstrated for the first time in this paper, where an uncorrected spectrum of polystyrene film is shown to be fairly readily obtained, although multiple interferometer-scan averaging is required.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the problem of computing the acoustic field generated by a moving point source in terms of the normal modes of a horizontally stratified ocean, where the source motion is assumed to be uniform (unaccelerated), but is not restricted to a path radial to the receiver.
Abstract: This paper considers the problem of computing the acoustic field generated by a moving point source. In particular, the acoustic field is obtained in terms of the normal modes of a horizontally stratified ocean. The source motion is assumed to be uniform (unaccelerated), but is not restricted to a path radial to the receiver. The structure of the Fourier inversion integral is carefully analyzed and an evaluation is carried out by the method of stationary phase. The stationary phase point is explicitly computed as an expansion in powers of the ratio of the source speed to the mode group velocity. The resulting expression for the velocity potential is examined for Doppler effects for both instantaneous (modal) Doppler as well as Doppler determined by a finite bandwidth Fourier transform.