Showing papers on "Fractional Fourier transform published in 1985"
263 citations
TL;DR: In this paper, a technique reduisant le temps de calcul d'une transformation de Fourier discrete d'un facteur 4 a 6, sans perte significative de precision, is presented.
Abstract: On presente une technique reduisant le temps de calcul d'une transformation de Fourier discrete d'un facteur 4 a 6, sans perte significative de precision
173 citations
TL;DR: In this article, it was shown that the composition of the Fourier transform in Rn followed by restriction to r defines a bounded operator from LP(Rn) to Lq(F) for certain p, q. The curvature hypotheses are the weakest under which this could hold, and p is optimal for a range of q.
Abstract: For smooth curves F in Rn with certain curvature properties it is shown that the composition of the Fourier transform in Rn followed by restriction to r defines a bounded operator from LP(Rn) to Lq(F) for certain p, q. The curvature hypotheses are the weakest under which this could hold, and p is optimal for a range of q. In the proofs the problem is reduced to the estimation of certain multilinear operators generalizing fractional integrals, and they are treated by means of rearrangement inequalities and interpolation between simple endpoint estimates.
136 citations
TL;DR: The relationship among different versions of DWT and their relation with the discrete Fourier transform (DFT) are given and Convolution theorems represented by different version of the DWT are derived.
120 citations
26 Apr 1985
TL;DR: A fast radix-2 two dimensional discrete cosine transform (DCT) is presented and a reduction of more than 50% in the number of multiplications and a comparable amount of additions is obtained in comparison to other algorithm.
Abstract: A fast radix-2 two dimensional discrete cosine transform (DCT) is presented. First, the mapping into a 2-D discrete Fourier transform (DFT) of a real signal is improved. Then an usual polynomial transform approach is used in order to map the 2-D DFT into a reduced size 2-D DFT and one dimensional odd DFT's. Finally, optimized odd DFT algorithms for real signals are developped. All together, a reduction of more than 50% in the number of multiplications and a comparable amount of additions is obtained in comparison to other algorithm.
117 citations
TL;DR: RDFT has better performance than DFT in the computation of real convolution because of the reduced number of operations, and the fact that forward and inverse transforms can be implemented with the same signal flowgraph, thereby facilitating hardware and software design.
Abstract: The real discrete Fourier transform (RDFT) corresponds to the Fourier series for sampled periodic signals with sampled periodic frequency responses just as discrete Fourier transform (DFT) corresponds to the complex Fourier series for the same type of signals RDFT has better performance than DFT in data compression and filtering for all signals in the sense that Pearl's measure for RDFT is less than Pearl's measure for DFT by an amount ΔW RDFT also has better performance than DFT in the computation of real convolution because of the reduced number of operations, and the fact that forward and inverse transforms can be implemented with the same signal flowgraph, thereby facilitating hardware and software design
77 citations
TL;DR: In this paper, it was shown that the Fourier transform belongs to Lq(r, da) for a certain natural measure on the su.rface of a circular cone in R3.
Abstract: Let r be the su.rface of a circular cone in R3. We show that if 1 < p < 4/3, 1/q = 3(1-1/p) and f E LP(R3), then the Fourier transform of f belongs to Lq(r, da) for a certain natural measure a on r. Following P. Tomas we also establish bounds for restrictions of Fourier transforms to conic annuli at the endpoint p = 4/3, with logarithmic growth of the bound as the thickness of the annulus tends to zero.
77 citations
TL;DR: In this paper, the authors present a Hankel transform algorithm using a fast (linear time) Abel transform, followed by an FFT, which maps an axisymmetric two-dimensional function into a line integral projection, and a one-dimensional Fourier transform.
Abstract: The Hankel, or Fourier-Bessel, transform is an important computational tool for optics, acoustics, and geophysics. It may be computed by a combination of an Abel transform, Which maps an axisymmetric two-dimensional function into a line integral projection, and a one-dimensional Fourier transform. This paper presents a Hankel transform algorithm using a fast (linear time) Abel transform, followed by an FFT.
67 citations
TL;DR: This paper deals with the problem of retrieving a finite-extent function from the magnitude of its Fourier transform and presents a brief review of the main results known for both discrete and continuous phase retrieval models giving special emphasis to the algebraic problem of the uniqueness of the solution.
Abstract: In this paper, we deal with the problem of retrieving a finite-extent function from the magnitude of its Fourier transform. This so-called phase retrieval problem will first be posed under its different underlying models. We will present a brief review of the main results known in this area for both discrete and continuous phase retrieval models giving special emphasis to the algebraic problem of the uniqueness of the solution. Several important issues which are yet unresolved will be pointed out and discussed. We will then consider the discrete phase retrieval problem as a special case of a more general problem which consists of recovering a real-valued sequence x from the magnitude of the output of a linear distortion: $| Hx | ( j ),\, j = 1, \cdots ,n$. A number of important results will be obtained for this general setting by means of algebraic-geometric techniques. In particular, the problems of the existence of a solution for phase retrieval, number of feasible solutions, stability of the (essential...
59 citations
TL;DR: In this article, it has been shown that the range and phase normalized far field is proportional to the three-dimensional spatial Fourier transform of the source distribution, taken on the surface of the Ewald sphere at a fixed frequency.
Abstract: It has been known for some time that the range and phase normalized far field is proportional to the three‐dimensional spatial Fourier transform of the source distribution, taken on the surface of the Ewald sphere at a fixed frequency. Unfortunately, however, this very useful relationship is neither widely known, nor does there even exist for it a good, short, simple, and direct derivation in the open literature (thus making references to the relationship difficult). The purpose of this paper is to alleviate these difficulties by presenting a brief derivation of the relationship for both the scalar acoustic as well as the vector electromagnetic cases. A brief discussion of the implications of this relationship on the inverse source and inverse scattering problem is also presented.
55 citations
TL;DR: The discrete Hartley transform as mentioned in this paper is a new tool for the analysis, design and implementation of digital signal processing algorithms and systems, which is strictly symmetric concerning the transformation and its inverse.
Abstract: The discrete Hartley transform is a new tool for the analysis, design and implementation of digital signal processing algorithms and systems. It is strictly symmetrical concerning the transformation and its inverse. A new fast Hartley transform algorithm has been developed. Applied to real signals, it is faster than a real fast Fourier transform, especially in the case of the inverse transformation. The speed of operation for a fast convolution can thus be increased.
01 Jan 1985
TL;DR: In this paper, it was shown that bit rates as low as 0.3 bit per pixel can be achieved by encoding a combination of the Fourier phase and amplitude data, which is achieved by low-pass filtering together with a clustering procedure in Fourier plane which seeks out the more important Fourier amplitude coefficients and their associated phases.
Abstract: The scientific advantages are pointed out from the Fourier transform encoding optical and electron microscope images and source data for computer-plotted Fourier-plane holograms, especially if bit compression ratios may be achieved, with comparable reconstructions, at the level found for the adaptive cosine transform. The relative importance is considered of image reconstruction based on the Fourier phase data alone and on combined phase and amplitude data. It is shown that bit rates as low as 0.3 bit per pixel can be achieved by encoding a combination of the Fourier phase and amplitude data. This is achieved by low-pass filtering together with a clustering procedure in the Fourier plane which seeks out the more important Fourier amplitude coefficients and their associated phases.
TL;DR: In this article, the nilpotent Fourier transform on spaces of distributions is used to prove the equivalence between *-products on g* for nil-potent g.
Abstract: We study the nilpotent Fourier transform on spaces of distributions. We use it to prove the equivalence between *-products on g* for nilpotent g.
TL;DR: In this article, a fully discretized projection method with Fourier series is proposed, which is based on a modification of the fast Fourier transform and is applied to systems of integro-differential equations with the Cauchy kernel, boundary integral equations from the boundary element method and, more generally, to certain elliptic pseudodifferential equations on closed smooth curves.
Abstract: Here we present a fully discretized projection method with Fourier series which is based on a modification of the fast Fourier transform. The method is applied to systems of integro-differential equations with the Cauchy kernel, boundary integral equations from the boundary element method and, more generally, to certain elliptic pseudodifferential equations on closed smooth curves. We use Gaussian quadratures on families of equidistant partitions combined with the fast Fourier transform. This yields an extremely accurate and fast numerical scheme. We present complete asymptotic error estimates including the quadrature errors. These are quasioptimal and of exponential order for analytic data. Numerical experiments for a scattering problem, the clamped plate and plane estatostatics confirm the theoretical convergence rates and show high accuracy.
TL;DR: In this article, the analytical and numerical properties of the Fourier transform of a two-center product of exponentially declining functions (exponential-type functions, ETFs) are derived with the help of Fourier convolution theorem and Feynman's identity.
TL;DR: In this paper, a relationship between the Fourier transform of a potential field at the Earth's surface and the transform of the inducing source distribution is derived, which can be used to determine all possible source distributions compatible with the data.
Abstract: A relationship is derived between the Fourier transform of a potential field at the Earth’s surface and the transform of the inducing source distribution. The Fourier transform of the field is the Laplace transform of the source distribution spectrum when the Laplace transform variable p is equal to the wavenumber. This relationship can be used to determine all possible source distributions compatible with the data. The solution is the superposition of a particular solution to an inhomogeneous problem and of the general solution to the homogeneous problem (i.e., for which the field vanishes at the surface). Source distribution can be expanded into a set of known functions; coefficients of the expansion are determined by solving a system of linear equations. Physical constraints can be introduced to restrict the variation range of the coefficients of expansion. Two examples are presented to illustrate the method: a synthetic gravity profile and a heat flow profile are inverted to determine density or heat ...
TL;DR: In this article, the inverse spectral transform for the periodic Korteweg-de Vries equation is investigated in the limit for small-amplitude waves and the inverse Fourier transform is recovered.
Abstract: The inverse spectral transform for the periodic Korteweg-de Vries equation is investigated in the limit for small-amplitude waves and the inverse Fourier transform is recovered. In the limiting process we find that the widths of the forbidden bands approach the amplitudes of the Fourier spectrum. The number of spectral bands is estimated from Fourier theory and depends explicitly on the assumed spatial discretization in the wave amplitude function (potential). This allows one to estimate the number of degrees of freedom in a discrete (and, therefore, finite-banded) potential. An essential feature of the calculations is that all results for the periodic problem are cast in terms of the infinite-line reflection and transmission coefficientsb(k), a(k). Thus the connection between the whole-line and periodic problems is clear at every stage of the computations.
TL;DR: In this article, it was shown that the infinite dimensional integral equation for the stochastic Fourier transform of the surface current can be reduced to the three dimensions associated with the random surface height and slopes.
Abstract: Further developments in the application of the stochastic Fourier transform approach (SFTA) to random surface scattering are presented. It is first shown that the infinite dimensional integral equation for the stochastic Fourier transform of the surface current can be reduced to the three dimensions associated with the random surface height and slopes. A three-dimensional integral equation of the second kind is developed for the average scattered field in stochastic Fourier transform space using conditional probability density functions. Various techniques for determining the transformed current (and, subsequently, the incoherent scattered power) from the average scattered field in stochastic Fourier transform space are developed and studied from the point of view of computational suitability. The case of vanishingly small surface correlation length is reexamined and the SFTA is found to provide erroneous results for the average scattered field due to the basic failure of the magnetic field integral equation (MFIE) in this limit.
TL;DR: This chapter has described the mathematical basis of the fast Fourier transform as applied to the calculation of crystallographic Fourier syntheses, and program organizations have been presented for performing general crystallography Fourier transforms on computer systems ranging from the very largest systems down to minicomputers.
Abstract: This chapter has described the mathematical basis of the fast Fourier transform as applied to the calculation of crystallographic Fourier syntheses. The relationship between real space and reciprocal space symmetry operators has been described. Finally, program organizations have been presented for performing general crystallographic Fourier transforms on computer systems ranging from the very largest systems down to minicomputers. Programs are available from the author, written in FORTRAN IV and in Ratfor, which are suitable for building blocks in these program designs.
01 Jan 1985
TL;DR: This chapter is for establishing the basis of this combined approach in dealing with computer tomography, computer holography and hologram matrix radar.
Abstract: The Fast Fourier Transform (FFT) is one of the most frequently used mathematical tools for digital signal processing. Techniques that use a combination of digital and analogue approaches have been increasing in numbers. This chapter is for establishing the basis of this combined approach in dealing with computer tomography, computer holography and hologram matrix radar.
TL;DR: In this article, an alternative discrete Fourier transform algorithm with suppressed aliasing is presented, inspired by work done by Sorella and Ghosh [Rev. Sci. Instrum.
Abstract: An alternative discrete (fast) Fourier transform algorithm with suppressed aliasing is presented. It is inspired by work done by Sorella and Ghosh [Rev. Sci. Instrum. 55, 1348 (1984)]. While using their idea of expanding the time function as a series (as Schutte [Rev. Sci. Instrum. 52, 400 (1981)] and Makinen [Rev. Sci. Instrum. 53, 627 (1982)] have done), it corrects a flaw in their method. The remarkable quality of the calculation is illustrated for an exponential decay by comparing the results to analytical values.
TL;DR: A coherent optical processor composed of two nonsymmetrical Fourier transformers is introduced which allows rotation-variant linear filtering operations and matched filtering and two configurations for such a processor are proposed.
Abstract: In this paper, we consider the properties of the nonsymmetrical Fourier transformation which is space-variant in both rectangular and polar coordinates A coherent optical processor composed of two nonsymmetrical Fourier transformers is introduced This processor allows rotation-variant linear filtering operations and matched filtering Two configurations for such a processor are proposed For certain parameters of both nonsymmetrical Fourier transformers it is possible to obtain a space-invariant processor with both lateral magnifications equal to unity However, introducing any filter operation results in a rotation-variant performance
Patent•
19 Mar 1985
TL;DR: In this article, a method and apparatus for representing a multi-dimensional, finite extent information containing signal in a locally sensitive, frequency domain representation employs transforming the digital signal using a Short Space Fourier transform having overlapping basis functions.
Abstract: A method and apparatus for representing a multi-dimensional, finite extent, information containing signal in a locally sensitive, frequency domain representation employs transforming the digital signal using a Short-Space Fourier transform having overlapping basis functions. The theory and application of the Short-Space Fourier transform provide, in one particular application of picture image transmission, an improved image quality over previously employed block transform coding methods and apparatus. A particularly preferred window function for use in connection with image signal processing is the multi-dimensional sinc function which has the unique advantage of a rectangular bandpass signal in the frequency domain.
IBM1
TL;DR: This paper presents a brief review of the algebraic problem of the uniqueness of the solution for both discrete and continuous phase retrieval models and considers the discrete phase retrieval problem as a special case of a more general problem of recovering a real-valued signal x from the magnitude of the output of a linear distortion.
Abstract: In this paper we deal with the problem of retrieving a finite-extent signal from the magnitude of its Fourier transform. We will present a brief review of the algebraic problem of the uniqueness of the solution for both discrete and continuous phase retrieval models. Several important issues which are yet unresolved will be pointed out and discussed. We will then consider the discrete phase retrieval problem as a special case of a more general problem which consists of recovering a real-valued signal x from the magnitude of the output of a linear distortion: |Hx|(j), j = 1, ..., n . An important result concerning the conditioning of this problem will be obtained for this general setting by means of algebraic-geometric techniques. In particular, the problems of the existence of a solution for phase retrieval, conditioning of the problem and stability of the (essentially) unique solution will be addressed.
Patent•
19 Mar 1985TL;DR: The Discrete Bracewell Transform (DBT) as discussed by the authors is a special purpose computer and method of computation for performing an N-length real-number discrete transform, which is performed without need for employing real and imaginary parts, and in efficient embodiments, is executed efficiently and in less time than the Discrete Fourier Transform (DFT).
Abstract: A special purpose computer (35) and method of computation for performing an N-length real-number discrete transform. For a real-valued function f(tau) where tau has the values 0,1,....,(N-1), the Discrete Bracewell Transform (DBT) H (v) is as in (I), where, v = 0,1,....,N-1; cas = cos + sin. The DBT is performed without need for employing real and imaginary parts, and in efficient embodiments, is executed efficiently and in less time than the Discrete Fourier Transform (DFT). The process steps for the original transform and the inverse retransformation are the same.
TL;DR: A hybrid system has been constructed to perform the complex Fourier transform of real 2-D data based on the Radon transform, which is performed with SAW filters via the chirp transform algorithm.
Abstract: A hybrid system has been constructed to perform the complex Fourier transform of real 2-D data The system is based on the Radon transform; ie, operations are performed on 1-D projections of the data The projections are derived optically from transmissive or reflective objects, and the complex Fourier transform is performed with SAW filters via the chirp transform algorithm The real and imaginary parts of the 2-D transform are produced in two bipolar output channels
TL;DR: In this article, an extension of the Discrete Fourier Transform (DFT) is defined as a linear combination of the forward and inverse DF's of a sequence, and the coefficients of the linear combinations can be chosen to define a real transform for a real sequence.
Abstract: An extension of the Discrete Fourier Transform (DFT) is defined as a linear combination of the forward and inverse DF's of a sequence. The coefficients of the linear combinations can be chosen to define a real transform for a real sequence. A fast algorithm can be used to compute the transform for a sequence whose length is a power of two.
01 Jan 1985
TL;DR: The Quantum Inverse Problem Method (or Quantum Spectral Transform) was invented at the end of 1978/beginning of 1979 as discussed by the authors, and it has been a remarkable development of this method with many papers on QST published.
Abstract: The Quantum Inverse Problem Method (or Quantum Spectral Transform) was invented at the end of 1978/beginning of 1979. During these five years there has been a remarkable development of this method with many papers on QST published (cf. reviews [3–5, 10, 17, 28]) and many results obtained.
26 Apr 1985
TL;DR: A new multidimensional Hartley transform is defined and a vector-radix algorithm for fast computation of the transform is developed that is shown to be faster (in terms of multiplication and addition count) compared to other related algorithms.
Abstract: A new multidimensional Hartley Transform is defined and a vector-radix algorithm for fast computation of the transform is developed. The algorithm is shown to be faster (in terms of multiplication and addition count) compared to other related algorithms.