Showing papers on "Fourier transform published in 1973"
TL;DR: In this paper, it is shown how a series of Fourier transforms can be used to calculate the magnetic or gravitational anomaly caused by an uneven, non-uniform layer of material.
Abstract: Summary It is shown how a series of Fourier transforms can be used to calculate the magnetic or gravitational anomaly caused by an uneven, non-uniform layer of material. Modern methods for finding Fourier transforms numerically are very fast and make this approach attractive in situations where large quantities of observations are available.
1,365 citations
TL;DR: In this article, the sensitivity of the solutions of large sets of coupled nonlinear rate equations to uncertainties in the rate coefficients is investigated, and it is shown via an application of Weyl's ergodic theorem that a subset of the Fourier coefficients is related to ∂ci/∂kl ǫ, the rate of change of the concentration of species i with respect to the rate constant for reaction l averaged over the uncertainties of all the other rate coefficients.
Abstract: A method has been developed to investigate the sensitivity of the solutions of large sets of coupled nonlinear rate equations to uncertainties in the rate coefficients. This method is based on varying all the rate coefficients simultaneously through the introduction of a parameter in such a way that the output concentrations become periodic functions of this parameter at any given time t. The concentrations of the chemical species are then Fourier analyzed at time t. We show via an application of Weyl's ergodic theorem that a subset of the Fourier coefficients is related to 〈∂ci/∂kl〉, the rate of change of the concentration of species i with respect to the rate constant for reaction l averaged over the uncertainties of all the other rate coefficients. Thus a large Fourier coefficient corresponds to a large sensitivity, and a small Fourier coefficient corresponds to a small sensitivity. The amount of numerical integration required to calculate these Fourier coefficients is considerably less than that requi...
954 citations
TL;DR: In this paper, a method for estimating the magnitude-squared coherence function for two zero-mean wide-sense-stationary random processes is presented, which utilizes the weighted overlapped segmentation fast Fourier transform approach.
Abstract: A method for estimating the magnitude-squared coherence function for two zero-mean wide-sense-stationary random processes is presented. The estimation technique utilizes the weighted overlapped segmentation fast Fourier transform approach. Analytical and empirical results for statistics of the estimator are presented. The analytical expressions are limited to the nonoverlapped case; empirical results show a decrease in bias and variance of the estimator with increasing overlap and suggest a 50-percent overlap as being highly desirable when cosine (Hanning) weighting is used.
521 citations
Journal Article•
464 citations
TL;DR: In this paper, mathematical expressions for the Wiener spectrum of the image of a point source were obtained for angular frequencies much less than, and much greater than, the conventional seeing limit.
Abstract: A new technique (speckle interferometry) has been developed by Gezari, Labeyrie, and Stachnik, which allows the measurement of stellar diameters from a series of photographs obtained from large-aperture ground-based telescopes. The series of photographs is processed to obtain the Weiner spectrum of the photographic image, i.e., the ensemble-averaged modulus-squared Fourier transform obtained from the series of images. Gezari, Labeyrie, and Stachnik have measured stellar diameters as small as 0″.05, about 20 times better than is usually possible. In this paper, mathematical expressions are obtained for the Wiener spectrum of the image of a point source. As is well known, the Wiener spectrum of the image of an extended, incoherently radiating object, is expressible as a product of this point-source spectrum and the object spectrum. Calculations are performed using the Rytov approximation and assuming that the underlying atmospheric turbulence is describable by a Kolmogorov spectrum. Asymptotic closed-form expressions are obtained for angular frequencies much less than, and much greater than, the conventional seeing limit. In the latter case, the Wiener spectrum is found to be proportional to the optical transfer function.
234 citations
TL;DR: The Fourier amplitude method developed in this article as a diagnostic tool for determining the sensitivity of the results of complex calculations to the parameters which enter these calculations has been applied to two chemical reaction systems involving sets of coupled, nonlinear rate equations.
Abstract: The Fourier amplitude method developed in Paper I as a diagnostic tool for determining the sensitivity of the results of complex calculations to the parameters which enter these calculations has been applied to two chemical reaction systems involving sets of coupled, nonlinear rate equations. These were: (a) a five reaction set describing the high temperature (6000 °K) dissociation of air and (b) a nine reaction set describing the constant temperature (2000 °K) combustion of H2 and O2. We have evaluated the Fourier amplitudes for all the species at a number of different times for both reaction systems. The analysis of these results verifies the claims made in Paper I. The relative magnitudes of the Fourier amplitudes showed a several order of magnitude distribution which permitted a clear distinction of the relative sensitivity of the species concentration to uncertainties in the rate coefficients. The conclusions based on the Fourier amplitude method for these two reaction systems are in excellent agreement with sensitivity predictions which could be made on the basis of previous kinetic studies of these systems.
214 citations
01 Oct 1973
TL;DR: The smoothed coherence transform is defined and examples of its uses and shortcomings are given and Computation of this function shows promise for measuring time delays between weak broad-band correlated noises received at two sensors.
Abstract: The smoothed coherence transform (SCOT) is defined and examples of its uses and shortcomings are given. Computation of this function shows promise for measuring time delays between weak broad-band correlated noises received at two sensors.
203 citations
TL;DR: Methods for incorporating crystallographic symmetry properties into complex Fourier transforms in a form particularly well suited for use with the Cooley-Tukey fast Fourier transform algorithm are presented.
Abstract: This paper presents methods for incorporating crystallographic symmetry properties into complex Fourier transforms in a form particularly well suited for use with the Cooley-Tukey fast Fourier transform algorithm. The crystallographic transforms are expressed in terms of a small number of one-dimensional special cases. The algebra presented here has been used to write computer programs for both Fourier syntheses and Fourier inversions. Even for some quite large problems (7000 structure factors and 149000 grid points in the asymmetric unit) the rate-limiting step is output of the answers.
199 citations
TL;DR: In this paper, a method for exciting a nuclear magnetic resonance spectrum with a radio frequency source having any desired frequency spectrum is described, which is applied to the problem of suppressing a strong solvent line while simultaneously observing all other resonance lines in a spectrum and to homonuclear decoupling while observing a complete spectrum.
Abstract: A method is described for exciting a nuclear magnetic resonance spectrum with a radio frequency source having any desired frequency spectrum. The frequency spectrum of the source is first specified and then Fourier synthesized to define a function which is used to modulate a radio frequency carrier. The NMR spectrum is obtained by Fourier transforming the response of the spin system to this excitation. The technique is applied to the problem of suppressing a strong solvent line while simultaneously observing all other resonance lines in a spectrum and to homonuclear decoupling while observing a complete spectrum.
186 citations
TL;DR: In this article, the second and third virial coefficients of a classical monatomic adsorbate are put into a form suitable for numerical evaluation, which is accomplished by expanding the Boltzmann factors in the general expressions as Fourier series in the translational symmetry coordinates of the surface lattice.
TL;DR: In this article, the numerical results from a computer solution of the time-dependent thin-wire electric-field integral equation described in Part I of this paper are validated by their Fourier transform of the frequency domain, where they are compared with independently computed data.
TL;DR: In this article, an ellipsometry technique based upon on-line Fourier analysis of the transmitted light intensity of a rotating-analyzer ellipsometer by means of a minicomputer is presented and discussed.
TL;DR: In this article, the authors define the time response function of an optical system, that is the result given by any pupil receiving a very short impulse of light, which can be used to generalize the explanation of the channelled spectra and bring an argument to the parallelism between space and time domains.
Abstract: The concepts and experiments discussed in this article are scarcely used in physical optics although they can introduce new ways of processing the information content of any optical pupil. They are related to diffraction phenomena in polychromatic light. One aspect deals with the well-known channelled spectra. The other consists in the definition of the time response function of an optical system - that is the result given by any pupil receiving a very short impulse of light. The time response function allows to generalize the explanation of the channelled spectra. It is also deduced from Fourier techniques and brings an argument to the parallelism between space and time domains. If the entrance pupil of a spectrometer is set in the interference pattern of a two-beam device illuminated in white light, the coloured spectrum is crossed by dark bands that are closer and closer as the difference between the optical paths increases. Along a time frequency (or wave-number) scale, the dark bands are sinusoidal. In this manner, one can say that the time spectrum is the Fourier transform of the couple of the wave trains. The time spectrum of white light being broad, a beam of white light should be considered as juxtaposition of very short wave groups. So, one can consider two individual wave groups passing along each arm of the interferometer. The same reasoning holds as one deals with several well-arranged wave trains, issued for instance from a grating or a multiple beam interferometer, or even further, with a sequence of wave trains in complete disorder. The situation is summarized in the diagram (fig. i): an incident plane wave is diffracted when transmitted through various devices. The spectrometer slit is placed in the region where the parallel beams combine with each other, at infinity or not. One should note the Fourier relationships of the spectral modulation curves with each sequnce of wave trains drawn along a time axis. These considerations immediately open out into new possibilities of conveying and collecting information. While a familiar way of transmission is to let the signal modulate the carrier which is in optics a parallel beam of monochromatic light, here the carrier consists of a parallel beam of polychromatic light, the modulation law being of temporal type. In turn this law is determined by the configuration of the optical system. Therefore any given function can be transmitted to any receiver station having a spectrometer at disposal - the role of this spectrometer being the spectral analysis of the message (or of its Fourier Transform). As an example suppose the tansmission of a (sinX/X)2 function is desired. It is the intensity distribution in the far-field diffraction pattern of a rectangular aperture. Therefore one has to perform the diffraction of a beam of white light by a slit of suitable width, a. In a first approximation (fig. ii), in a direction θ, the sequence of wave trains is limited by a rectangle function of width a cosθ and the power spectrum is expected to be given by the square modulus of a sinc-function. Likewise another test signal for studying the transmission mechanism would be a pair of thin slits that yields a cos2 function, or else a pupil of gaussian transmittance would be nessary to build up a message described by the reciprocal gaussian law. The previous process suffers of limitations. It applies to real and positive function whose Fourier transforms are real and positive. For a wider class of functions, namely complex ones, one has to resort to an equivalent of holography. The transposition comes to send a reference wave-group before or after the information proper. The phase terms are thus kept, same as in conventional holography. Such a simple experiment was carried out by means of a narrow slit set in the same plane as that of the diffracting pupil. A temporal hologram was then recorded. It appears as an image of the time spectrum striped with thin dark fringes of sinusoidal profile, carrying the amplitude and phase temrs. The usual reconstruction process in monochromatic light applies to that temporal Fourier hologram. Information concerning the previous rectangular pupil has been transmitted in the form of a hologram recorded at the output of a spectrometer 10 meters away from the source. Incidentally a much more interesting and promising technique enables real-time operating. All one has to do is get the temporal autocorrelation function of the whole of the message with its reference waves group. The pair of antisymmetrical side band distribution represents the image and its conjugate. Practically such phenomena are observable as the amplitude and phase modulation of the visibility function of the fringes displayed at the output of a Michelson interferometer. An important requirement is the use of a spatially coherent source of high luminance, with a broad spectrum (its transform - the reference - would be narrow then). The input function of the Michelson interferometer is the sequence of wave trains. Therefore this represents the basic arrangement in Fourier transform spectroscopy, with a slight difference: instead of gathering a signal and making an interferogram whose spectrum would describe the source, one gets here the autocorrelation function of the information under test, that is of the time holographic signal coming from the pupil. As a matter of fact the time-hologram itself is never formed; no intermediate recording is needed as one directly reaches three terms: the central one being the superposition of the autocorrelation of the signal and the reference respectively, the other two the symetrical side-bands displaying, as said before, the cross correlations of the signal by the reference which was assumed as a narrow impulse along a time axis. A more rigourous treatment has proved necessary to give a full account of the mechanisms involved in the transmission of a time message. In particular a time impulse response and a time transfer function can be defined for any pupil. Looking at figure iii, one notes that the scale of the Fraunhofer spatial diffraction pattern of a rectangular aperture varies as the reciprocal of the time frequency ν. As ν of the incident plane wave is tuned continuously from 0 to ∞, an observer set in a fixed position at infinity will see the « breathing » of the pattern according to the law sin Kν/Kν To complete the determination of the amplitude H(ν) at the observation point, one must add an important fact: usually, in the calculation of the amplitude diffracted by any pupil by means of the Fresnel-Kirchhoff integral, one omits the multiplicative factor, - j/λ. Therefore: H(ν) = (j2πν sin Kν/Kν. The reciprocal of ν in the time domain is t. The time response of the system to a unit impulse h(u, t) in the u-direction is then the Fourier Transform of H(ν). After a well-known derivative theorem, the transform of H(ν) is the derivative of a rectangle function. The same procedure would show that if one considers any pupil, whatever its contour and amplitude distribution, the time impulse response is the derivative of the pupil function. A similar result is obtained with phase pupils. Generally, the time impulse response of a complex pupil in a given direction is described by the derivative of the complex pupil function projected on this direction. This is illustrated by various drawings of functions corresponding to one or several slit apertures. Applications are envisaged, namely in metrology - for instance in the assessment of the quality of surfaces and the measurement of thicknesses.
IBM1
TL;DR: In this article, a numerical procedure for the solution of partial differential equations for the convection of scalar variables is developed for the forward marching problem, where the space derivatives are computed with very high accuracy by means of Fourier transform methods.
TL;DR: In this article, a study of faradaic admittance versus frequency measurements based on Fourier analysis of the response to multiple-frequency test signal waveforms is presented, where data are acquired using complex periodic signals, almost periodic signals; transient inputs; and bandwidth-limited white noise.
IBM1
TL;DR: In this paper, a numerical model is presented allowing calculation of the dc, ac, and large-signal parameters of field effect transistors (FETs) based on finite-difference approximations to the full time-dependent set of equations.
Abstract: A numerical model is presented allowing calculation of the dc, ac, and large-signal parameters of field-effect transistors (FET's). The numerical procedure is based on finite-difference approximations to the full time-dependent set of equations. The scheme presented uses centered difference quotients and an implicit treatment of the continuity equation. It is shown to be absolutely stable and accurate for time steps below 1 ps. A set of numerical data calculated for one typical example is compared systematically with experimental values. Excellent agreement between measured and computed values is found for the dc characteristics. Small-signal solutions, obtained by Fourier transform methods are also close to the empirical values. The good fit between experiment and numerical simulation is a thorough validation of both the physical model and the numerical procedure.
TL;DR: The theoretical basis for representation of a speech signal by its short-time Fourier transform is discussed and the design tradeoffs necessary to achieve moderate information rate reductions are elucidated.
Abstract: This paper discusses the theoretical basis for representation of a speech signal by its short-time Fourier transform. The results of the theoretical studies were used to design a speech analysis-synthesis system which was simulated on a general-purpose laboratory digital computer system. The simulation uses the fast Fourier transform in the analysis stage and specially designed finite duration impulse response filters in the synthesis stage. The results of both the theoretical and computational studies lead to an understanding of the effect of several design parameters and elucidate the design tradeoffs necessary to achieve moderate information rate reductions.
TL;DR: In this article, the water vapour spectrum recorded with a resolution of 5 × 10-3 cm-1 between 2930 cm 1 and 4255 cm 1 on a Fourier transform spectrometer is presented.
Abstract: The water vapour spectrum recorded with a resolution of 5 × 10-3 cm-1 between 2930 cm-1 and 4255 cm-1 on a Fourier transform spectrometer is presented. Wavenumbers, attributions and equivalent widths of about 1500 lines are given. Lines of the 2v 2, v 1 and v 3 bands of H2 16O, lines of the hot band v 2 + v 3 - v 2 of H2 16O and lines of the v 1 and v 3 bands of H2 18O and H2 17O are observed.
TL;DR: A lower bound for the number of additions necessary to compute a family of linear functions by a linear algorithm is given when an upper bound c can be assigned to the modulus of the complex numbers involved in the computation.
Abstract: A lower bound for the number of additions necessary to compute a family of linear functions by a linear algorithm is given when an upper bound c can be assigned to the modulus of the complex numbers involved in the computation. In the case of the fast Fourier transform, the lower bound is (n/2) log2n when c = 1.
TL;DR: In this article, the Fourier transform of the spin echo using the pulsed gradient, spin echo technique was used to determine the self-diffusion coefficient of each component in a multicomponent system.
TL;DR: In this article, the Fourier transform of a blurred image is used to identify the optical transfer function, provided that it is one of the suitable common simple forms, or to determine certain parameters of the transfer function if its form is known.
Abstract: In cases in which there are zeros in the optical transfer function, display of the absolute value of the Fourier transform of a blurred image may allow these zeros to be seen, if the noise level is low enough. This can be used to identify the optical transfer function, provided that it is one of the suitable common simple forms, or to determine certain parameters of the optical transfer function if its form is known. Examples of the use of this technique in the generation of restoring filters for image enhancement are presented.
TL;DR: The fast Fourier transform is employed in the most efficient measurement of the Wiener kernels of a non-linear system so far discovered.
Abstract: A new method is presented for the measurement of the Wiener kernels of a non-linear system. The method uses the complex exponential functions as a set of orthogonal functions with which to expand the kernels. The fast Fourier transform is then employed in the most efficient measurement of the kernels so far discovered.
01 May 1973
TL;DR: In this paper, a computer program for earthquake analysis of gravity dams including effects of hydrodynamic interaction is presented, where the dam-water system is idealized as two-dimensional in geometry, the material behavior is assumed to be linearly elastic, and the transverse horizontal as well as the vertical component of ground motion are considered.
Abstract: : A computer program for earthquake analysis of gravity dams including effects of hydrodynamic interaction is presented. The dam-water system is idealized as two-dimensional in geometry, the material behavior is assumed to be linearly elastic, and the transverse horizontal as well as the vertical component of ground motion are considered. The dam is represented as a finite element system, whereas water in the reservoir is treated as a continum of infinite length in the upstream direction governed by the wave equation. Compressibility of water is considered resulting in governing equations for the dam depending on the excitation frequency. The analysis is performed in the frequency domain, first obtaining the frequency responses and then Fourier Synthesizing them by Fast Fourier Transform procedures to obtain responses to arbitrary ground motion. A listing of the computer program is included and the usage and capabilities are illustrated by examples. (Author)
TL;DR: In this article, it was shown that the truncation error can also be removed at discrete frequencies, simply by first subtracting a ramp from the step response of a network, with consequent noise enhancement.
Abstract: If the discrete Fourier transform of the step response of a network is taken, a large truncation error results, since only a finite number of samples is used. This error is usually removed by first differentiating the waveform, with consequent noise enhancement. The letter shows that the error may also be removed at discrete frequencies, simply by first subtracting a ramp from the step response.
TL;DR: A technique for the digital simulation of multicorrelated Gaussian random processes is described, based upon generating discrete frequency functions which correspond to the Fourier transform of the random processes and then using the fast Fourier Transform algorithm to obtain the actual random processes.
Abstract: A technique for the digital simulation of multicorrelated Gaussian random processes is described. This technique is based upon generating discrete frequency functions which correspond to the Fourier transform of the random processes and then using the fast Fourier transform (FFT) algorithm to obtain the actual random processes. The main advantage of this method over other methods is computation time; it appears to be more than an order of magnitude faster than present methods of simulation. One of the main uses of multicorrelated simulated random processes is in solving nonlinear random vibration problems by numerical integration of the governing differential equations. [This research is supported in part by NASA.]