Showing papers on "Discrete-time Fourier transform published in 1987"
TL;DR: In this article, it was shown that it is possible to reconstruct a complex-valued object from the modulus of its Fourier transform if one has a sufficiently strong support constraint.
Abstract: Previously it was shown that one can reconstruct an object from the modulus of its Fourier transform (solve the phase-retrieval problem) by using the iterative Fourier-transform algorithm if one has a nonnegativity constraint and a loose support constraint on the object. In this paper it is shown that it is possible to reconstruct a complex-valued object from the modulus of its Fourier transform if one has a sufficiently strong support constraint. Sufficiently strong support constraints include certain special shapes and separated supports. Reconstruction results are shown, including the effect of tapered edges on the object’s support.
529 citations
TL;DR: The present invention is a nonlinear joint transform image correlator which employs a spatial modulator operating in a binary mode at the Fourier plane which produces a correlation output formed by an inverse Fourier transform of this binarized Fouriers transform interference intensity.
Abstract: The present invention is a nonlinear joint transform image correlator which employs a spatial modulator operating in a binary mode at the Fourier plane. The reference and input images are illuminated by a coherent light at the object plane of a Fourier transform lens system. A image detection device, such as a charge coupled device, is disposed at the Fourier plane of this Fourier transform lens system. A thresholding network detects the median intensity level of the imaging cells of the charge coupled device at the Fourier plane and binarizes the Fourier transform interference intensity. The correlation output is formed by an inverse Fourier transform of this binarized Fourier transform interference intensity. In the preferred embodiment this is achieved via a second Fourier transform lens system. This binary data is then applied to spatial light modulator device operating in a binary mode located at the object plane of a second Fourier transform lens system. This binary mode spatial light modulator device is illuminated by coherent light producing the correlation output at the Fourier plane of the second Fourier transform lens system. The inverse Fourier transform may also be formed via a computer. In an alternative embodiment, the Fourier transform interference intensity is thresholded into one of three ranges. An inverse Fourier transform of this trinary Fourier transform interference intensity produces the correlation output.
333 citations
TL;DR: In this article, the wave function at various times during the propagation was split into two parts, one localized in the interaction region and the other in the force free region; the first is propagated by a fast Fourier transform method on a grid whose size barely exceeds the interaction area, and the latter by a single application of a free particle propagator.
Abstract: Various methods using fast Fourier transform algorithms or other ‘‘grid’’ methods for solving the time‐dependent Schrodinger equation are very efficient if the wave function remains spatially localized throughout its evolution. Here we present and test an extension of these methods which is efficient even if the wave function spreads out, provided that the potential remains localized. The idea is to split the wave function at various times during the propagation into two parts, one localized in the interaction region and the other in the force free region; the first is propagated by a fast Fourier transform method on a grid whose size barely exceeds the interaction region, and the latter by a single application of a free particle propagator. This splitting is performed whenever the interaction region wave function comes close to the end of the grid. The total asymptotic wave function at a given time t is reconstructed by adding coherently all the asymptotic wave function pieces which were split at earlier times, after they have been propagated to the common time t. The method is tested by studying the wave function of a diatomic molecule dissociated by a strong laser field. We compute the rate of energy absorption and dissociation and the momentum distribution of the fragments.
261 citations
TL;DR: An algorithm for the in-place computation of the discrete Fourier transform on real data: a decimation-in-time split-radix algorithm, more compact than the previously published one and a new fast Hartley transform algorithm with a reduced number of operations.
Abstract: This paper highlights the possible tradeoffs between arithmetic and structural complexity when computing cyclic convolution of real data in the transform domain. Both Fourier and Hartley-based schemes are first explained in their usual form and then improved, either from the structural point of view or in the number of operations involved. Namely, we first present an algorithm for the in-place computation of the discrete Fourier transform on real data: a decimation-in-time split-radix algorithm, more compact than the previously published one. Second, we present a new fast Hartley transform algorithm with a reduced number of operations. A more regular convolution scheme based on FFT's is also proposed. Finally, we show that Hartley transforms belong to a larger class of algorithms characterized by their "generalized" convolution property.
131 citations
TL;DR: In this paper, the role of the truncation/apodization function used to obtain the phase correction is discussed, as well as the special problem of emission spectra for one-sided interferograms.
Abstract: In high precision Fourier transform spectrometry, proper handling of the phase correction is essential if the full potential wave number accuracy of the data is to be preserved. One-sided interferograms are shown to be especially sensitive to phase error, and this sensitivity is quantitatively related to the signal-to-noise ratio. The role of the truncation/apodization function used to obtain the phase correction is also discussed, as is the special problem of emission spectra.
94 citations
TL;DR: A single-mode star network, made from polarization-preserving components, can perform the spatial discrete Fourier transform of coherent light patterns presented at the inputs.
Abstract: A single-mode star network, made from polarization-preserving components, can perform the spatial discrete Fourier transform of coherent light patterns presented at the inputs. This can be accomplished with passive components, such as 2 x 2 couplers, and propagation delays. The Hadamard transform can be performed similarly.
83 citations
TL;DR: The theoretical basis of the selective Fourier transform technique is developed and experimental results are presented, including comparisons of spectral localization using either the selective fourier transform method or conventional multidimensional Fouriertransform chemical‐shift imaging.
Abstract: We have introduced the selective Fourier transform technique for spectral localization. This technique allows the acquisition of a high-resolution spectrum from a selectable location with control over the shape and size of the spatial response function. The shape and size of the spatial response are defined during data acquisition and the location is selectable through processing after the data acquisition is complete. The technique uses pulsed-field-gradient phase encoding to define the spatial coordinates. In this paper the theoretical basis of the selective Fourier transform technique is developed and experimental results are presented, including comparisons of spectral localization using either the selective Fourier transform method or conventional multidimensional Fourier transform chemical-shift imaging. © Academic Press, Inc.
79 citations
60 citations
TL;DR: This paper presents a new direct solution to the problem of reconstructing a two-dimensional discrete signal of finite support from knowledge of only its Fourier transform magnitude and support using the autocorrelation function of the unknown signal.
Abstract: In this paper we present a new direct solution to the problem of reconstructing a two-dimensional discrete signal of finite support from knowledge of only its Fourier transform magnitude and support. Using the autocorrelation function of the unknown signal, zeros are calculated of a polynomial whose coefficients correspond to the unknown image or the image reversed. From these zeros, a set of linear equations is developed whose solution yields either the original image or the image rotated by 180°.
44 citations
TL;DR: A new reconstruction algorithm has been developed which performs the image assembly operation in Fourier space, rather than in image space by backprojection, which represents a significant decrease in the number of operations required to assemble the image.
Abstract: Filtered backprojection reconstruction algorithms are based upon the relationship between the Fourier transform of the imaged object and the Fourier transforms of its projections. A new reconstruction algorithm has been developed which performs the image assembly operation in Fourier space, rather than in image space by backprojection. This represents a significant decrease in the number of operations required to assemble the image. The new Fourier domain algorithm has resolution comparable to the filtered backprojection algorithm, and, after correction by a pointwise multiplication, demonstrates proper recovery throughout image space. Although originally intended for three-dimensional imaging applications, the Fourier domain algorithm can also be developed for two-dimensional imaging applications such as planar positron imaging systems.
TL;DR: Inverse Fourier transform has been used to derive the gradient-index profiles of inhomogeneous films having spectral requirements and results show a good agreement with the theory and evidences the reliability of the technology used to produce inhomogeneity media.
Abstract: Inverse Fourier transform has been used to derive the gradient-index profiles of inhomogeneous films having spectral requirements. Two examples are given, and the corresponding experimental designs are presented. Results show a good agreement with the theory and evidences the reliability of the technology used to produce inhomogeneous media.
TL;DR: A range and error analysis is developed for a discrete Fourier transform computed using the ring of cyclotomic integers, and derivations of both deterministic and statistical upper bounds for the range are presented.
Abstract: A range and error analysis is developed for a discrete Fourier transform (fast Fourier transform) computed using the ring of cyclotomic integers. Included are derivations of both deterministic and statistical upper bounds for the range of the resulting processor and formulas for the ratio of the mean square error to mean square signal, in terms of the pertinent parameters. Comparisons of theoretical predictions with empirical results are also presented.
01 Apr 1987
TL;DR: A new algorithm useful for extrapolation and Fourier analysis of discrete signals that are given by a relative small number of samples that can be applied to higher-dimensional problems.
Abstract: This paper describes a new algorithm useful for extrapolation and Fourier analysis of discrete signals that are given by a relative small number of samples. The extrapolation is based on the assumption that the discrete Fourier spectrum shows dominant spectral lines. Involving only FFT, the iterative algorithm is not restricted to one-dimensional signals but can also be applied to higher-dimensional problems. Additional knowledge on the signal like band-limitedness or positivity can easily be taken into account.
TL;DR: In this paper, a simple technique for constructing signals with Wigner distributions that are linear transformations of the wigner distribution of a known signal is developed, which gives insight into the tradeoffs in time-frequency leakage between various windows and allows quick and accurate estimates of the leakage in the short-time Fourier transform.
Abstract: Conventional frequency-domain window-leakage analysis accurately describes the leakage in the short-time Fourier transform only for stationary signals. Leakage in the time-frequency plane from concentrated transient or nonstationary signals can be effectively analyzed by use of a time-frequency window-leakage envelope with rectangular contours. This envelope is obtained from the Wigner distribution of the analysis window, with appropriate corrections for the sidelobe leakage. The time-frequency window-leakage envelope gives insight into the tradeoffs in time-frequency leakage between various windows and allows quick and accurate estimates of the leakage in the short-time Fourier transform. A simple technique for constructing signals with Wigner distributions that are linear transformations of the Wigner distribution of a known signal is developed. With this technique, windows with a variety of time-frequency orientations and leakage behavior can be developed.
TL;DR: This tutorial is an outgrowth of a course in signal processing given by Julius O. Smith at Stanford University in the fall of 1984 and provides an elementary mathematical introduction to spectrum analysis.
Abstract: This tutorial is an outgrowth of a course in signal processing given by Julius O. Smith at Stanford University in the fall of 1984 (see Smith 1981, as well). It provides an elementary mathematical introduction to spectrum analysis. This is the first of two parts. In part one, the discrete Fourier transform is introduced and analyzed in depth. In part two, some fundamental spectrum analysis theorems and applications are discussed. The only mathematical background assumed is high school trigonometry, algebra, and geometry. No calculus is required. Familiarity with summation formulae, complex numbers, and vectors is helpful, although not essential.
01 Feb 1987
TL;DR: In this paper, it was shown that the result of Wiener's theorem holds for functions on the circle with positive Fourier coefficients that are pth power integrable near 0, 1 < P < 2, and 1 < p < oo.
Abstract: Extending a result of N. Wiener, it is shown that functions on the circle with positive Fourier coefficients that are pth power integrable near 0, 1 < P < 2, have Fourier coefficients in lp . The following result was proved (but never published) by Norbert Wiener in the early 1950's. (See [1, pp. 242, 250] and [3].) WIENER'S THEOREM. // YLcneint is the Fourier series of a function f € L1(—7T, 7t) with cn > 0 for all n, and f restricted to a neighborhood (—6,6) of the origin belong to L2(—6,6), then f belongs to L2(—tt,tt). A question which immediately arises in connection with this result is the following: does the theorem remain true if one replaces L2(—6,6) and L2(—tt,tt) in its statement respectively by Lp(—6,6) and Lp(—-k,tt), with 1 < p < oo? In 1969 Stephen Wainger showed, by ingenious counterexamples, that the answer is negative for 1 < p < 2 [4]. If p is an even integer or oo it is very easy to see that the answer is "yes." For every other exponent between 2 and oo it is "no," as was shown in 1975 by Harold S. Shapiro [3]. These negative results have been extended to compact abelian groups [2]. However, the conclusion of Wiener's theorem can be stated equivalently as "then J2 cn < °°-" This suggested the following theorem. THEOREM. If¿~2cneint is the Fourier series of a function f S Lx(—tt, tt) with cn > 0 for all n, and f restricted to a neighborhood (—6,6) of the origin belongs to Lp(—6,6) with 1 < p < 2, then ^cn < °°> where p' = p/(p — 1). PROOF. (See [3, p. 12].) Let h(t) be the 27r-periodic function which for |t| < -k is defined by M-KIA M Si, I 0, 6 < \t\ < it. Received by the editors March 2, 1987. 1980 Mathematics Subject Classification. Primary 42A32, 42A16; Secondary 43A15. 1 The research presented here was supported in part by a grant from the University Research Council of DePaul University. ©1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page
06 Apr 1987
TL;DR: It is pointed out that a correspondence between RN+1space and discrete signals with duration of N+1 points can be set up, and the signals that can't be reconstructed from its end point and FT magnitude correspond to a subset of RN-1 with measure zero.
Abstract: In this paper, the reconstruction of discrete signal with finite time duration from its end point and Fourier Transform (FT) magnitude is considered. Based on one result of [1] that a class of discrete signal can be reconstructed from its FT magnitude and one end sample point, with the help of Measure Theory, furtherly we point out that a correspondence between RN+1space and discrete signals with duration of N+1 points can be set up, and the signals that can't be reconstructed from its end point and FT magnitude correspond to a subset of RN+1with measure zero. In other words, discrete signal with finite time duration can almost be uniquely reconstructed from its end point and FT magnitude.
TL;DR: In this article, the Fourier operational matrix of integration is introduced and computational algorithms analogous to those already developed for other orthogonal functions are proposed to solve the linear equations involved in the control problems.
Abstract: [n this paper, solution of the state-space model for linear control systems using the Fourier series expansion is presented. The Fourier operational matrix of integration is introduced and computational algorithms analogous to those already developed for other orthogonal functions are proposed. These algorithms are then applied to solve the linear equations involved in the control problems. Finally, some important results and comparisons are provided.
01 Jan 1987
TL;DR: The operator of Fourier operational calculus is constructed and its spectral properties are related to the construction of Korobov spaces to derive error estimates for Fourier product approximation, Fourier blending approximation, and the related Krylov-Lanczos approximation in these spaces.
Abstract: It is the objective of this paper to apply Boolean methods of approximation in combination with the theory of right invertible operators to bivariate Fourier expansions. We construct the operator of Fourier operational calculus and relate its spectral properties to the construction of Korobov spaces. We will derive error estimates for Fourier product approximation, Fourier blending approximation, Fourier hyperbolic approximation, and the related Krylov-Lanczos approximation in these spaces.
TL;DR: In this paper, the Fourier operational matrix is extended to reduce the system equations to a set of linear algebraic equations and to obtain the approximate solution, and an illustrative example is given and some useful results are provided.
Abstract: An optimal control problem for linear systems is solved by the use of the Fourier series approach. The Fourier operational matrix is extended to reduce the system equations to a set of linear algebraic equations and to obtain the approximate solution. An illustrative example is given and some useful results are provided.
TL;DR: In this article, the Fourier transform of light-beam-induced current profiles is expressed in a closed form and the influence of the experimental errors on the estimate of the diffusion length and of the recombination velocity is evaluated.
Abstract: The observation that the Fourier transform of light‐beam‐induced current profiles can be expressed in a closed form leads to a new method for their analysis. This method takes full advantage of the knowledge of the profile line shape and does not depend critically on the details of the current asymptotic behavior. The Fourier transform can be quickly calculated, making possible the use of standard minimization procedures on a personal computer. The influence of the experimental errors on the estimate of the diffusion length and of the recombination velocity is evaluated.
01 May 1987
TL;DR: In this article, an expression for the Fourier series spectrum of the transmission error arising from tooth-spacing errors on a single spur gear meshing with a perfect involute mating gear for the case of a contact ratio of unity and no elastic deformations present was derived.
Abstract: An expression is derived for the Fourier series spectrum of the transmission error arising from tooth-spacing errors on a single spur gear meshing with a perfect involute mating gear for the case of a contact ratio of unity and no elastic deformations present. This expression is found to be in exact agreement with previously derived results. The expression illustrates the role of the discrete Fourier transform in transmission error analysis and interpretation.
TL;DR: In this article, the Fourier transform of the one-particle distribution function for the rarefied, isotropic, three-dimensional Lorentz gas was studied.
Abstract: The Fourier transform of the one-particle distribution function for the rarefied, isotropic, three-dimensional Lorentz gas is studied. Assuming that the Fourier transform of the initial distribution function depends on the wave vector k , the modulus of the velocity v and the cosinus of the angle between them, the Fourier-Laplace transform of the distribution function is expanded into a series in Legendre polynomials. The explicit time dependence of these coefficients is studied.
TL;DR: In this article, a new method for estimating model parameters from pulse chromatography experiments was developed, where discrete and fast Fourier transforms were employed to analyze the discrete experimental data acquired with a computer-automated system.
Abstract: A new method for estimating model parameters from pulse chromatography experiments was developed. Discrete and fast Fourier transforms were employed to analyze the discrete experimental data acquired with a computer-automated system. The Kubin-Kucera model was used to describe dispersion and intraparticle diffusion in a single pellet string reactor and the model parameters were obtained by solving a minimization problem in the Fourier domain. Results from effective diffusivity measurements are presented and the advantages of the new method are discussed.