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Showing papers on "Fractional Fourier transform published in 1987"


Journal ArticleDOI
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


Journal ArticleDOI
TL;DR: It is shown that the DRT can be used to compute various generalizations of the classical Radon transform (RT) and, in particular, the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves.
Abstract: This paper describes the discrete Radon transform (DRT) and the exact inversion algorithm for it. Similar to the discrete Fourier transform (DFT), the DRT is defined for periodic vector-sequences and studied as a transform in its own right. Casting the forward transform as a matrix-vector multiplication, the key observation is that the matrix-although very large-has a block-circulant structure. This observation allows construction of fast direct and inverse transforms. Moreover, we show that the DRT can be used to compute various generalizations of the classical Radon transform (RT) and, in particular, the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves. In fact, we describe not a single transform, but a class of transforms, representatives of which correspond in one way or another to discrete versions of the RT and its generalizations. An interesting observation is that the exact inversion algorithm cannot be obtained directly from Radon's inversion formula. Given the fact that the RT has no nontrivial one-dimensional analog, exact invertibility makes the DRT a useful tool geared specifically for multidimensional digital signal processing. Exact invertibility of the DRT, flexibility in its definition, and fast computational algorithm affect present applications and open possibilities for new ones. Some of these applications are discussed in the paper.

426 citations


PatentDOI
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


Journal ArticleDOI
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


Book
01 Sep 1987

215 citations


Book ChapterDOI
31 Dec 1987

189 citations


Journal ArticleDOI
TL;DR: In this paper, the eigenvalues and eigenvectors of the n×n unitary matrix of finite Fourier transform whose j, k element is (1/(n)1/2)exp[(2πi/n)jk], i=(−1) 1/2, is determined.
Abstract: The eigenvalues and eigenvectors of the n×n unitary matrix of finite Fourier transform whose j, k element is (1/(n)1/2)exp[(2πi/n)jk], i=(−1)1/2, is determined. In doing so, a multitude of identities, some of which may be new, are encountered. A conjecture is advanced.

121 citations


Journal ArticleDOI
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


Journal ArticleDOI
TL;DR: In this article, the Fourier transform approach to EBK quantization is extended to the case of strongly resonant classical motion, and the relation between spectral frequency assignments and the choice of good action-angle variables investigated.
Abstract: The Fourier transform approach to EBK quantization, previously applied to nonresonant systems with up to four degrees of freedom [J. Chem. Phys. 83, 2990 (1985)], is extended to the case of strongly resonant classical motion. The classical mechanics of systems with 3:4, 1:2, and 1:1 resonances is examined in detail from the Fourier transform point of view, and the results of nonlinear resonance analysis used to interpret numerical trajectory Fourier spectra. Calculation of classical actions and numerical construction of the angle parametrization of invariant tori is described, and the relation between spectral frequency assignments and the choice of good action‐angle variables investigated. It is shown that correct quantization conditions for arbitrary resonant motion can be determined by direct numerical evaluation of Maslov indices. Semiclassical eigenvalues are reported for the 3:4, 1:2, and 1:1 resonant systems.

79 citations


Journal ArticleDOI
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


Journal ArticleDOI
01 May 1987
TL;DR: The results show that each set of coded information is transparent to any other and that each shape-indicative distribution may be located using a convolution mask peculiar to that distribution.
Abstract: A general method is presented that uses the Radon transform as a means of defining a two-dimensional transform space in which information about different, analytically defined shape primitives in an edge image space may be encoded simultaneously. Examples are given illustrating how the shape-indicative distributions within the transform space may be deduced. The results show that each set of coded information is transparent to any other and that each shape-indicative distribution may be located using a convolution mask peculiar to that distribution.

Journal ArticleDOI
TL;DR: A new and simple algorithm for computing a discrete Hankel transform, which does not rely on the fast Fourier transform, can provide a major improvement in speed and accuracy over previously described methods.

Journal ArticleDOI
TL;DR: It is shown that the lower bound for the computation of the multidimensional transform is O(n2 log2 n) and an optimal architecture based on arrays of processors computing one-dimensional Fourier transforms and a rotation network or rotation array is proposed.
Abstract: It is often desirable in modern signal processing applications to perform two-dimensional or three-dimensional Fourier transforms. Until the advent of VLSI it was not possible to think about one chip implementation of such processes. In this paper several methods for implementing the multidimensional Fourier transform together with the VLSI computational model are reviewed and discussed. We show that the lower bound for the computation of the multidimensional transform is O(n2 log2 n). Existing nonoptimal architectures suitable for implementing the 2-D transform, the RAM array transposer, mesh connected systolic array, and the linear systolic matrix vector multiplier are discussed for area time tradeoff. For achieving a higher degree of concurrency we suggest the use of rotators for permutation of data. With ``hybrid designs'' comprised of a rotator and one-dimensional arrays which compute the one-dimensional Fourier transform we propose two methods for implementation of multidimensional Fourier transform. One design uses the perfect shuffle for rotations and achieves an AT2 p of O(n2 log2 n· log2 N). An optimal architecture for calculation of multidimensional Fourier transform is proposed in this paper. It is based on arrays of processors computing one-dimensional Fourier transforms and a rotation network or rotation array. This architecture realizes the AT2 p lower bound for the multidimensional FT processing.

Journal ArticleDOI
TL;DR: In this paper, the use and application of the Hilbert transform for identifying and quantifying nonlinearity associated with simulated and experimental frequency response functions is described, and the results show that both procedures give similar trends in the extracted modal parameters, with consistently lower damping estimates from the causalisation procedure.

Journal ArticleDOI
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°.

Journal ArticleDOI
TL;DR: In this paper, a method to solve the phase retrieval problem from two intensities observed at the Fourier transform of an object function in one dimension is proposed, which involves the solution of the linear equations consisting of the data of two data points, obtained with and without an exponential filter at the object plane, and unknown coefficients in the phase expansion of phase.
Abstract: A method to solve the phase-retrieval problem from two intensities observed at the Fourier transform of an object function in one dimension is proposed. This method involves the solution of the linear equations consisting of the data of two intensities, obtained with and without an exponential filter at the object plane, and unknown coefficients in the Fourier series expansion of phase. There is no need to treat the nonlinear equation for zero location in the complex plane. The usefulness of the method is shown in computer simulation studies of the reconstruction of the one-dimensional phase object from the observable moduli at the Fourier-transform plane of the object.

Journal ArticleDOI
R.C. Agarwal1, J.W. Cooley
01 Sep 1987
TL;DR: The algorithm formulation and implementation described here not only achieves full vector utilization but successfully copes with the problems of hierarchical storage.
Abstract: A number of previous attempts at the vectorization of the fast Fourier transform (FFT) algorithm have fallen somewhat short of achieving the full potential speed of vector processors. The algorithm formulation and implementation described here not only achieves full vector utilization but successfully copes with the problems of hierarchical storage. In the present paper, these techniques are described and extended to the general mixed radix algorithms, prime factor algorithm (PFA), the multidimensional discrete Fourier transform (DFT), the rectangular transform convolution algorithms, and the Winograd fast Fourier transform algorithm. Some of the methods were used in the Engineering Scientific Subroutine Library for the IBM 3090 Vector Facility. Using this approach, very good and consistent performance was obtained over a very wide range of transform lengths.

Journal ArticleDOI
TL;DR: In this paper, the numerical inversion of Laplace transforms by means of the finite Fourier cosine transform, as presented by Dubner and Abate, was analyzed, and it was found that the proper inversion formula should contain the Fourier sine series as well.

Journal ArticleDOI
M.G. Amin1
01 Nov 1987
TL;DR: In the recursive Fourier transform, the data window can be chosen such that the number of computations required to update the transform at each frequency upon reception of a new data sample is independent of the transform block length.
Abstract: In the recursive Fourier transform, the data window can be chosen such that the number of computations required to update the transform at each frequency upon reception of a new data sample is independent of the transform block length.

Journal ArticleDOI
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.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the case where S is the graph of a suitably convex function, homogeneous of degree d, and estimate the Fourier transform of S. They showed that if S is convex, with no tangent lines of infinite order, then /*(*(£) decays as |i|_n''2 provided a > ((n + 3)/2).
Abstract: We suppose that S is a smooth hypersurface in Rn+1 with Gaussian curvature re and surface measure dS, it) is a compactly supported cut-off function, and we let pa be the surface measure with dsa = u>Ka dS. In this paper we consider the case where S is the graph of a suitably convex function, homogeneous of degree d, and estimate the Fourier transform sa. We also show that if S is convex, with no tangent lines of infinite order, then /*(*(£) decays as |i|_n''2 provided a > ((n + 3)/2). The techniques involved are the estimation of oscillatory integrals; we give applications involving maximal functions. 1. Introduction. The purpose of this paper is to obtain estimates for the decay at infinity of certain oscillatory integrals related to the Fourier transform of surface carried measures. Let S denote a smooth hypersurface in Rn+1 with Gaussian curvature n and element of surface measure dS induced by the Lebesgue measure of Rn+1. We fix a smooth function w with compact support in Rn+1 and a nonnegative number a and consider the finite Borel measure p.a, with dp,a = \K.\awdS, which is carried by S. We seek conditions on S and a that guarantee that the Fourier transform {p,a)~ of p,a satisfies the estimate

Journal ArticleDOI
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.

Journal ArticleDOI
31 Dec 1987-Nature
TL;DR: In this paper, it was shown that the two-dimensional Hartley transform is mathematically equivalent to the Fourier transform, but is real valued; amplitude alone fully represents everything.
Abstract: When the two-dimensional Fourier transformation is performed with a lens the optical amplitude and phase in the output plane represent the complex transform. It can be shown that the two-dimensional Hartley transform is mathematically equivalent to the Fourier transform, but is real valued; amplitude alone fully represents everything. This is significant because ordinary optical detectors do not respond to phase. Here we describe the construction of an optical system in the form of a modified Michelson interferometer which physically demonstrates that it is possible to produce the Hartley transform of a plane luminous object. It is thus possible to encode in the form of amplitude the half of the information in a diffraction pattern that normally is carried in the form of phase.

Journal ArticleDOI
M.G. Perkins1
01 Aug 1987
TL;DR: The cas-cas transform as mentioned in this paper is a real-to-real transform for convolutional arrays and power spectra, which can be used to compute 2D power spectrum.
Abstract: This letter introduces a discrete, separable, real-to-real transform, called the cas-cas transform. Theorems for the two-dimensional (2-D) case are presented, and the cas-cas transform is compared to the Hartley transform as an alternative way to convolve 2-D arrays and compute 2-D power spectra.

Journal ArticleDOI
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.


Proceedings ArticleDOI
01 Jan 1987
TL;DR: A new Fourier transform algorithm for almost-periodic functions (APFT) is developed that is both efficient and accurate and uses the theoretically minimum number of time points.
Abstract: Harmonic balance is a powerful technique for the simulation of mildly nonlinear microwave circuits. This technique has had limited application for tie analysis of almost-periodic circuits, such as mixers, due to the difficulties of transforming waveforms from the time domain to the frequency domain and vice versa. In this paper, a new Fourier transform algorithm for almost-periodic functions (APFT) is developed that is both efficient and accurate. Unlike previous attempts to solve this problem, the new afgorithm does not constmin the input frequencies and uses the theoretically minimum number of time points.

Journal ArticleDOI
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.

Journal ArticleDOI
TL;DR: In this paper, a fast algorithm for computing the discrete Hartley transform (DHT) via the Walsh-Hadamard transform (WHT) is proposed, which is carried out on an interframe basis in (N × N) data blocks, where N is an integer power of two.
Abstract: A new fast algorithm is proposed to compute the discrete Hartley transform (DHT) via the Walsh-Hadamard transform (WHT). The processing is carried out on an interframe basis in (N × N) data blocks, where N is an integer power of two. The WHT coefficients are obtained directly, and then used to obtain the DHT coefficients. This is achieved by a transform matrix, the H-transform matrix, which is ortho-normal and has a block-diagonal structure. A complete derivation of the block-diagonal structure for the H-transform matrix is given.

Journal ArticleDOI
TL;DR: In this paper, the authors introduced the discrete Fourier transform (DFT) and showed that the DFT can produce a sequence of spectral components equally spaced in frequency, with a length equal to that of the original waveform.
Abstract: In part one of this tutorial (Jaffe 1987), we introduced the discrete Fourier transform (DFT). To review, the DFT takes a waveform as input and produces as output the spectrum of that waveform. One way to understand this process is to consider the samples of the waveform as a vector and to see the DFT as the projection of this vector onto a set of complex sinusoidal basis vectors. In this manner, the DFT produces a sequence of spectral components equally spaced in frequency, with a length equal to that of the original waveform. Each element of the spectrum is a coefficient of the projection given by the inner product of the waveform with one of the basis sinusoids. This coefficient can be represented in polar coordinates to give the amplitude and phase of the corresponding sinusoid. The equation for the DFT is: