Showing papers on "Discrete-time Fourier transform published in 1982"

Forward modeling by a Fourier method

Abstract: A Fourier or pseudospectral forward-modeling algorithm for solving the two-dimensional acoustic wave equation is presented. The method utilizes a spatial numerical grid to calculate spatial derivatives by the fast Fourier transform. time derivatives which appear in the wave equation are calculated by second-order differcncing. The scheme requires fewer grid points than finite-diffcrcnce methods to achieve the same accuracy. It is therefore believed that the Fourier method will prove more efficient than finitedifference methods. especially when dealing with threedimensional models. The Fourier forward-modeling method was tested against two problems, a single-layer problem with a known analytic solution and a wedge problem which was also tested by physical modeling. The numerical results agreed with both the analytic and physical model results. Furthermore, the numerical model facilitates the explanation of certain events on the time section of the physical model which otherwise could not easily be taken into account.

Eigenvectors and functions of the discrete Fourier transform

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.

Random Fourier Series with Applications to Harmonic Analysis. (AM-101), Volume 101

Abstract: In this book the authors give the first necessary and sufficient conditions for the uniform convergence a.s. of random Fourier series on locally compact Abelian groups and on compact non-Abelian groups. They also obtain many related results. For example, whenever a random Fourier series converges uniformly a.s. it also satisfies the central limit theorem. The methods developed are used to study some questions in harmonic analysis that are not intrinsically random. For example, a new characterization of Sidon sets is derived. The major results depend heavily on the Dudley-Fernique necessary and sufficient condition for the continuity of stationary Gaussian processes and on recent work on sums of independent Banach space valued random variables. It is noteworthy that the proofs for the Abelian case immediately extend to the non-Abelian case once the proper definition of random Fourier series is made. In doing this the authors obtain new results on sums of independent random matrices with elements in a Banach space. The final chapter of the book suggests several directions for further research.

A Complete Set of Fourier Descriptors for Two-Dimensional Shapes

Abstract: A set of Fourier descriptors for two-dimensional shapes is defined which is complete in the sense that two objects have the same shape if and only if they have the same set of Fourier descriptors. It also is shown that the moduli of the Fourier coefficients of the parameterizing function of the boundary of an object do not contain enough information to characterize the shape of an object. Further a relationship is established between rotational symmetries of an object and the set of integers for which the corresponding Fourier coefficients of the parameterizing function are nonzero.

Signal reconstruction from the short-time Fourier transform magnitude

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.

Continuous and discrete Fourier transforms of steplike waveforms

Abstract: A steplike waveform which has attained its final value is converted into a duration-limited one which preserves the spectrum of the original waveform and is suitable for discrete Fourier transform (DFT) computations. The method, which is based upon the response of a time-invariant linear system excited by a rectangular pulse of suitable duration, is first applied to continuous waveforms and then to discrete (sampled) waveforms. The difference (errors) between the spectra of a continuous waveform and a discrete representation of it are reviewed.

Geometric Fourier analysis

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.

The Fast Fourier Transform

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.

Reconciliation of the discrete and integral Fourier transforms

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...

The Carleson-Hunt theorem on fourier series

Abstract: Fourier SeriesSystems, Approximation, Singular Integral Operators, and Related TopicsTopics in Analysis and Its ApplicationsAmerican journal of mathematicsModern Fourier AnalysisTransactions of the American Mathematical SocietyThe Abel PrizeThe Carleson-Hunt Theorem on Fourier SeriesClassical and Modern Fourier AnalysisAn Introduction to Non-Harmonic Fourier Series, Revised Edition, 93A Course in Functional AnalysisModern Fourier AnalysisHarmonic AnalysisCommutative Harmonic Analysis ICarleson Curves, Muckenhoupt Weights, and Toeplitz OperatorsPointwise Convergence of Fourier SeriesBrownian MotionDifferentiation of Integrals in RnDirichlet SeriesExplorations in Harmonic AnalysisChaos in Classical and Quantum MechanicsMartingales in Banach SpacesFourier AnalysisTrigonometric SeriesClassical Fourier AnalysisThe Geometry of Fractal SetsPerspectives in AnalysisCommutative Harmonic Analysis IVFourier Analysis on Local Fields. (MN-15)Fourier Restriction, Decoupling and ApplicationsWave Packet AnalysisMeasure and IntegralRevue Roumaine de Mathématiques Pures Et AppliquéesFractals in Probability and AnalysisA Panorama of Harmonic AnalysisInterpolation of OperatorsA Course in Abstract Harmonic AnalysisThe Carleson-Hunt theorem on Fourier seriesOrlicz Spaces and Generalized Orlicz SpacesDiophantine Approximation and Dirichlet Series

Iterative Procedures For Signal Reconstruction From Fourier Transform Phase

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.

A spectral-iteration technique for analyzing a corrugated-surface twist polarizer for scanning reflector antennas

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.

Some approaches for location of centroids of quartz grain outlines to increase homology between Fourier amplitude spectra

Abstract: The ability to test for similarities and differences among families of shapes by closed-form Fourier expansion is greatly enhanced by the concept of homology. Underlying this concept is the assumption that each term of a Fourier series, when compared to the same term in another series, represents the “same thing”. A method that ensures homology is one which minimizes the “centering error,” as reflected in the first harmonic term of the Fourier expansion. The problem is to chose a set of edge points derived from a much larger, but variable, number of edge points such that a valid homologous Fourier series can be calculated. Methods are reviewed and criteria given to define a “proper” solution. An algorithm is presented which takes advantage of the fact that minimization of the “error term” can be accomplished by minimizing the distance between the origin of the polar coordinate system in the calculation of the Fourier series and the shape centroid. The use of this algorithm has produced higher quality solutions for quartz grain provenance studies.

Susceptibilities for the brownian motion in a cosine potential with application to the rotation of a dipole in a constant external field

Abstract: The susceptibility for the brownian motion in a cosine potential is proportional to the one sided Fourier transform of the velocity autocorrelation function whereas the polarizability for the rotation of a dipole in a constant external field is proportional to the one sided Fourier transform of the time derivative of the sin- and cos-autocorrelation function. The one sided Fourier transform of these autocorrelation functions can be expressed by matrix continued fractions. They are evaluated for large, medium and even for very small damping constants, thus obtaining various susceptibilities practically in the whole region of friction constants. Furthermore the connection to the zero friction limit case is discussed.

Optical Fourier transform: what is the optimal setup?

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.

The discrete fourier transform applied to time domain signal processing

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.

Calculating Detection Probabilities for Systems Employing Noncoherent Integration

Abstract: Cumulative probability distributions that occur in radar and sonar detection problems are calculated directly from the characteristic function by using a Fourier series. The error in the result is controlled by two parameters which can be adjusted to suit the application. The technique is applied to the problem of determining the detection performance of consecutive discrete Fourier transforms (DFTs) for a narrowband Gaussian signal with a rectangular spectrum. Since the characteristic function is used directly in its product form this technique does not suffer from the numerical problems associated with the partial fraction approach. The technique can handle many different problems in a single computational structure making it a valuable tool in system performance studies.

Spectral information from fourier analysis of digitized quartz grain profiles

Abstract: Fourier analysis of closed curves defining two-dimensional images has emerged as a promising new tool for the quantification of projected shape in detrital quartz grains. The Fourier method has received added impetus with the introduction of microprocessor-controlled video imaging systems that permit automated acquisition and processing of digital information for hundreds of grains in a few hours. Although new methods and technology have removed much of the tedium and subjectivity associated with older representations of particle shape, other problems are introduced during the acquisition and interpretation of Fourier shape spectra that find their analog in conventional time series analysis. These include the aliasing problem and the treatment of quantization errors which attend the digitization process, along with decisional problems regarding the selection of harmonics and the smoothing of spectral power estimates from individual grain shapes. Our work with quartz grain shapes suggests that bandwidth averaging of power estimates over four adjacent harmonics is an effective smoothing procedure and that harmonics 18 through 21 may constitute a suitable standard band for multiple study comparisons.

New algorithm for the calculation of the Fourier transform of discrete signals

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.

M/M/1 Transient State Occupancy Probabilities Via the Discrete Fourier Transform

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.

Generalized running discrete transforms

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.