Showing papers on "Fractional Fourier transform published in 1982"
TL;DR: A method is presented for computing an orthonormal set of eigenvectors for the discrete Fourier transform (DFT) based on a detailed analysis of the eigenstructure of a special matrix which commutes with the DFT.
Abstract: A method is presented for computing an orthonormal set of eigenvectors for the discrete Fourier transform (DFT). The technique is based on a detailed analysis of the eigenstructure of a special matrix which commutes with the DFT. It is also shown how fractional powers of the DFT can be efficiently computed, and possible applications to multiplexing and transform coding are suggested.
243 citations
TL;DR: In this article, the Laplace transform of the solution for TEM soundings over an N-layer earth was derived and used to invert it numerically using the Gaver-Stehfest algorithm.
Abstract: Calculations for the transient electromagnetic (TEM) method are commonly performed by using a discrete Fourier transform method to invert the appropriate transform of the solution. We derive the Laplace transform of the solution for TEM soundings over an N-layer earth and show how to use the Gaver-Stehfest algorithm to invert it numerically. This is considerably more stable and computationally efficient than inversion using the discrete Fourier transform.
149 citations
TL;DR: In this article, the authors solved the direct scattering problem numerically in closed form in the time domain and spatial Fourier transform space, by casting the general basic global laws (i.e., the second-order partial differential wave equation or its integral representation) as a local algebraic equation in the space, and leaving the specific local constitutive equations (e.g., the algebraic boundary conditions, which specify a given structure, which are conventionally imposed on the differential or integral representation of the generalized global wave equation).
Abstract: The arbitrary direct scattering problem is solved numerically in closed form in the time domain and spatial Fourier transform space. This solution consists of casting the general basic global laws (i.e., the second‐order partial differential wave equation or its integral representation) as a local algebraic equation in the spatial Fourier transform space, and leaving the specific local constitutive equations (i.e., the algebraic boundary conditions, which specify a given structure, which are conventionally imposed on the differential or integral representation of the general basic global wave equation) as a local algebraic equation in real space, thereby reducing the scattering problem to a statement of two simultaneous local algebraic equations in two unknowns (the fields and the induced sources) in two spaces connected by the spatial Fourier transform. A temporally local representation in both spaces is obtained with the aid of an introduced auxilliary field and two propagators. By virtue of causality, ...
117 citations
90 citations
01 May 1982
TL;DR: This paper presents various conditions that are sufficient for reconstructing a discrete-time signal from samples of its short-time Fourier transform magnitude, for applications such as speech processing.
Abstract: This paper presents various conditions that are sufficient for reconstructing a discrete-time signal from samples of its short-time Fourier transform magnitude. For applications such as speech processing, these conditions place very mild restrictions on the signal as well as the analysis window of the transform. Examples of such reconstruction for speech signals are included in the paper.
79 citations
TL;DR: In this paper, the conditions générales d'utilisation (http://www.numdam.org/legal.php) of a fichier do not necessarily imply a mention of copyright.
Abstract: © Annales de l’institut Fourier, 1982, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
65 citations
TL;DR: The fast sine transform is accomplished by the existing implementation of the fast cosine transform by a method of composing the discrete sinetransform from the discrete cosinetransform.
Abstract: A method of composing the discrete sine transform from the discrete cosine transform is demonstrated. As a result of this method, the fast sine transform is accomplished by the existing implementation of the fast cosine transform.
55 citations
54 citations
TL;DR: In this article, an exact alternative approach to the diagrammatic technique for treating scattering from rough surfaces is developed, where the magnetic field integral equation for the current induced on the rough perfectly conducting surface is multiplied by a Fourier kernel involving all orders of surface height derivatives and their associated transform variables.
Abstract: An exact alternative approach to the diagrammatic technique for treating scattering from rough surfaces is developed. The magnetic field integral equation for the current induced on the rough perfectly conducting surface is multiplied by a Fourier kernel involving all orders of surface height derivatives and their associated transform variables. Averages of this weighted equation are converted to convolations in the transform domain. The result of this operation is a singular integral equation of the first kind of infinite dimensions (because of the infinite number of height derivatives) for the stochastic Fourier transform of the current. A procedure is developed for estimating the effects of ignoring one or more surface height derivatives in terms of the range of validity of the resulting approximate solution. Special limiting cases of very gently undulating surfaces and uniformly rough surfaces are examined. New and illuminating results are obtained for the latter case.
52 citations
01 Jan 1982
TL;DR: In this article, the main properties of the discrete Fourier transform (DFT) are summarized and various fast DFT computation techniques known collectively as the Fast Fourier Transform (FFT) algorithm are presented.
Abstract: The object of this chapter is to briefly summarize the main properties of the discrete Fourier transform (DFT) and to present various fast DFT computation techniques known collectively as the fast Fourier transform (FFT) algorithm. The DFT plays a key role in physics because it can be used as a mathematical tool to describe the relationship between the time domain and frequency domain representation of discrete signals. The use of DFT analysis methods has increased dramatically since the introduction of the FFT in 1965 because the FFT algorithm decreases by several orders of magnitude the number of arithmetic operations required for DFT computations. It has thereby provided a practical solution to many problems that otherwise would have been intractable.
50 citations
TL;DR: In this article, the authors evaluated the relationship derived in terms of the continuous Fourier integral transform with the discrete Fourier transform for potential field geophysical studies and showed that the discrete transform can be made essentially equivalent to the integral transform if, before sampling, the continuous aperiodic input function is made periodic by shifting the function by integer multiples of the data interval and summi...
Abstract: In the application of harmonic analysis to potential‐field geophysical studies, relationships derived in terms of the continuous Fourier integral transform are evaluated in terms of the discrete Fourier transform. The discrete transform, obtained by transforming a finite number of equispaced samples of the actual aperiodic continuous function, is too low at the dc level and increasingly too high in the high frequencies, compared with the theoretical integral transform. As a consequence, overly restrictive limitations must be placed on high‐frequency‐amplifying operators such as differentiation and downward continuation. Also, a spurious and troublesome azimuthal distortion occurs in the discrete Fourier analysis of three‐dimensional (3-D) (map) data represented as grids. The discrete transform can be made essentially equivalent to the integral transform if, before sampling, the continuous aperiodic input function is made periodic by shifting the function by integer multiples of the data interval and summi...
TL;DR: In this article, the Fourier transform was used to analyze the self-potential anomaly due to a two-dimensional inclined sheets of finite depth extent using the frequency domain using Fourier Transform.
Abstract: The self-potential anomaly due to a two-dimensional inclined sheets of finite depth extent has been analysed in the frequency domain using the Fourier transform. Expression for the Fourier amplitude and phase spectra are derived. The Fourier amplitude and phase spectra are analysed so as to evaluate the parameters of the sheet. Application of this method on two anomalies (synthetic and field data) has given good results.
TL;DR: An algorithm that reduces the computational effort to two-thirds of the effort required by most radix-2 algorithms and its structure is particularly appealing when transforming pure real or imaginary sequences and/or symmetric or antisymmetric sequences.
Abstract: This paper develops and presents a radix-2 fast Fourier transform (FFT) algorithm that reduces the computational effort (as measured by the number of multiplications) to two-thirds of the effort required by most radix-2 algorithms. The resulting algorithm is similar to one obtained by applying a principle suggested by Rader and Brenner; however, its structure is particularly appealing when transforming pure real or imaginary sequences and/or symmetric or antisymmetric sequences; furthermore, memory requirements (other than those for storing the input data) do not grow with the size of the transform.
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01 Feb 1982TL;DR: Five schemes for the adaptive transform coding of video signals are presented and photographic results are presented to see the effects of the distortions introduced by different transforms on different source pictures due to data compression.
Abstract: In the paper, five schemes for the adaptive transform coding of video signals are presented. Adaptation is based on adaptive quantisation and adaptive bit selection. A Max quantiser having a Laplacian density distribution is used to achieve adaptive quantisation, whereas classification according to the activity within the transform block and based on the human visual characteristics are used for adaptive bit selection. The discrete cosine transform and the Hadamard transform are used to transform three source pictures of different statistics, and results are compared to find the best scheme for each transform and picture. Photographic results are presented to see the effects of the distortions introduced by different transforms on different source pictures due to data compression. Owing to the limitations of the photographic process, the subjective quality of the pictures as perceived from the video monitor cannot be accurately represented. However they are useful for comparison purposes.
TL;DR: In this paper, a set of conditions has been developed under which a sequence is uniquely specified by the phase or samples of the phase of its Fourier transform, which are applicable to both one-dimensional and multi-dimensional sequences.
Abstract: Recently, a set of conditions has been developed under which a sequence is uniquely specified by the phase or samples of the phase of its Fourier transform. These conditions are distinctly different from the minimum or maximum phase requirement and are applicable to both one-dimensional and multi-dimensional sequences. Under the specified conditions, several numerical algorithms have been developed to reconstruct a sequence from its phase. In this paper, we review the recent theoretical results pertaining to the phase-only reconstruction problem, and we discuss in detail two iterative numerical algorithms for performing the reconstrucction.
TL;DR: A modified version of Burrus' prime factor fast Fourier transform program is described, which implements the in-place, in-order algorithm for variable transform sizes.
Abstract: This paper describes a modified version of Burrus' prime factor fast Fourier transform program. The modifications produce a general-purpose program which implements the in-place, in-order algorithm for variable transform sizes. Speed tests show the resulting program to be faster than a program using a separate reordering pass.
TL;DR: In this paper, an analysis of the corrugated-surface twist polarizer is presented, which finds application in the design of scanning reflector antennas using the spectraI-interation technique, a novel procedure which combines the use of the Fourier transform method with an iterative procedure.
Abstract: An analysis of the corrugated-surface twist polarizer which finds application in the design of scanning reflector antennas is presented. The spectraI-interation technique, a novel procedure which combines the use of the Fourier transform method with an iterative procedure is employed. The first step in the spectral-iteration method is the conversion of the original integral equation for the interface field into a form which is suitable for iteration using a method developed previously. An important feature of the technique is that it takes advantage of the discrete Fourier transform (DFT) type of kernel of the integral equation and evaluates the integral operators efficiently using the fast Fourier transform (FFT) algorithm. Thus, in contrast to the conventional techniques, e.g., the moment method, the spectral-iteration approach requires no matrix inversion and is capable of handling a large number of unknowns. Furthermore the method has a built-in check on the satisfaction of the boundary conditions at each iteration.
TL;DR: It is shown that the converging-beam illumination setup (CB-FT) is much simpler and works better than the classical parallel beam illumination setup within a restricted range of object size and lens aperture.
Abstract: The two basic optical Fourier transform configurations are examined with respect to component complexity, aberrations, and optical noise. It is shown that the converging-beam illumination setup (CB-FT) is much simpler and works better than the classical parallel beam illumination setup within a restricted range of object size and lens aperture. This range corresponds to many practical cases. Therefore, the CB-FT should be preferred in ordinary cases whereas the classical setup with a special purpose Fourier lens should be used only for a large space–bandwidth product. It is probably never a good solution to use the parallel beam configuration with a general purpose lens as the Fourier lens.
TL;DR: Two new algorithms that are more convenient for computation than existing ones for the slant transform are developed and reveal the close relationship between theSlant transform and the Walsh-Hadamard transform.
Abstract: Two new algorithms that are more convenient for computation than existing ones for the slant transform are developed. These algorithms reveal the close relationship between the slant transform and the Walsh-Hadamard transform and demonstrate that the slant transform may be approached by a series of steps which gradually change the transform from a Hadamard or Walsh transform to a slant transform.
TL;DR: A review of the discrete Fourier transform, emphasizing the use of DFT in direct and indirect methods of time domain signal processing and a number of applications to communications.
Abstract: A review of the discrete Fourier transform, emphasizing the use of DFT in direct and indirect methods of time domain signal processing. T HE discrete Fourier transform (DFT), implemented as a computationally efficient algorithm called the fast Fourier transform (FFT), has found application to all aspects of signal processing. These applications include time domain processing as well as frequency domain processing. The proper noun \"Fourier\" may elicit images of frequency domain data and, by these images, restrict the vista of applications of this important tool.. W e will review the DFT with emphasis on perspectives which facilitate time domain processing. In particular, we will review a number of applications to communications.
TL;DR: A new algorithm for the calculation of the Fourier transform of sampled time functions is described, based on second‐degree polynomial interpolations between the sample points, which was found to be significantly more accurate than the conventionally used discrete Fouriertransform (DFT).
Abstract: A new algorithm for the calculation of the Fourier transform of sampled time functions is described. The algorithm is especially applicable to the Fourier analysis of nonperiodic signals which are not band limited. The method is based on second‐degree polynomial interpolations between the sample points. The obtained continuous approximation of the signal allows the determination of the Fourier transform analytically. In the case of exponentially decaying functions the algorithm was found to be significantly more accurate than the conventionally used discrete Fourier transform (DFT). The computing time is only about twice the time required by the fast Fourier transform (FFT) algorithm.
TL;DR: A procedure is given for the computation of the transient state occupancy probabilities of the M/M/1 queue, which makes use of the inverse discrete Fourier transform computed by means of the fast Fouriertransform.
Abstract: A procedure is given for the computation of the transient state occupancy probabilities of the M/M/1 queue. The method makes use of the inverse discrete Fourier transform, computed by means of the fast Fourier transform. It avoids the direct evaluation of modified Bessel functions and sidesteps difficulties due to the computation of very large and very small intermediate quantities.
TL;DR: A generalized running discrete transform with respect to arbitrary transform bases is introduced, and the generalized transform to the running discrete Fourier z and short-time discrete Fouriers transforms is related.
Abstract: This paper introduces a generalized running discrete transform with respect to arbitrary transform bases, and relates the generalized transform to the running discrete Fourier z and short-time discrete Fourier transforms. Concepts associated with the running and short-time discrete Fourier transforms such as 1) filter bank implementation, 2) synthesis of the original sequence by summation of the filter bank outputs, 3) frequency sampling, and 4) recursive implementations are all extended to the generalized transform case. A formula is obtained for computing the transform coefficients of a segment of data at time n recursively from the transform coefficients of the segment of data at time n - 1. The computational efficiency of this formula is studied, and the class of transforms requiring the minimum possible number of arithmetic operations per coefficient is described.
01 May 1982
TL;DR: In this paper a 2400 bit/s implementation of the DFT-vocoder is discussed and the harmonic-sieve technique for pitch extraction combines very well with this scheme because it is based on hopping-DFT as well.
Abstract: The DFT-vocoder is based on speech analysis and synthesis using the discrete Fourier transform (DFT). Analysis is done using hopping-DFT and spectral parameters are obtained by a piece-wise constant approximation of the amplitude spectrum. The harmonic-sieve technique for pitch extraction combines very well with this scheme because it is based on hopping-DFT as well. Synthesis is achieved by convolution of the generated excitation signal with the inverse-DFT of the reconstructed piece-wise constant amplitude spectrum. In this paper a 2400 bit/s implementation of the DFT-vocoder is discussed.
TL;DR: In this paper, two original methods are presented for deconvolving such transforms for signals containing significant noise, and the results of numerical experiments with noisy data are presented in order to demonstrate the capabilities and limitations of the methods.
Abstract: Time series or spatial series of measurements taken with nonuniform spacings have failed to yield fully to analysis using the Discrete Fourier Transform (DFT). This is due to the fact that the formal DFT is the convolution of the transform of the signal with the transform of the nonuniform spacings. Two original methods are presented for deconvolving such transforms for signals containing significant noise. The first method solves a set of linear equations relating the observed data to values defined at uniform grid points, and then obtains the desired transform as the DFT of the uniform interpolates. The second method solves a set of linear equations relating the real and imaginary components of the formal DFT directly to those of the desired transform. The results of numerical experiments with noisy data are presented in order to demonstrate the capabilities and limitations of the methods.
TL;DR: In this paper, the analysis of light with very low intensity was carried out by the measurement of Laplace or Fourier transform of the probability of the time-intervals between two successive photons.
TL;DR: In this paper, the Wigner transform of an integral kernel on the full line generalizes the Fourier transform of a translation kernel, and the eigenvalue spectra of Hermitian kernels are related to the topographic features of their WIGNer transforms.
Abstract: The Wigner transform of an integral kernel on the full line generalizes the Fourier transform of a translation kernel. The eigenvalue spectra of Hermitian kernels are related to the topographic features of their Wigner transforms. Two kernels whose Wigner transforms are equivalent under the unimodular afline group have the same spectrum of eigenvalues and have eigenfunctions related by an explicit linear transformation. Any kernel whose Wigner transform has concentric ellipses as contour lines, yields an eigenvalue problem which may be solved exactly.