Showing papers on "Fourier transform published in 1984"

Signal estimation from modified short-time Fourier transform

Abstract: In this paper, we present an algorithm to estimate a signal from its modified short-time Fourier transform (STFT). This algorithm is computationally simple and is obtained by minimizing the mean squared error between the STFT of the estimated signal and the modified STFT. Using this algorithm, we also develop an iterative algorithm to estimate a signal from its modified STFT magnitude. The iterative algorithm is shown to decrease, in each iteration, the mean squared error between the STFT magnitude of the estimated signal and the modified STFT magnitude. The major computation involved in the iterative algorithm is the discrete Fourier transform (DFT) computation, and the algorithm appears to be real-time implementable with current hardware technology. The algorithm developed in this paper has been applied to the time-scale modification of speech. The resulting system generates very high-quality speech, and appears to be better in performance than any existing method.

Space charge effects in Fourier transform mass spectrometry. Mass calibration.

Abstract: Developpement d'une nouvelle relation entre la masse ionique et la frequence cyclotron efficace, pour des ions stockes dans une cellule cubique et detectes par spectrometrie de masse a transformee de Fourier

Gauss and the history of the fast fourier transform

Abstract: THE fast Fourier transform (Fm has become well known . as a very efficient algorithm for calculating the discrete Fourier Transform (Om of a sequence of N numbers. The OFT is used in many disciplines to obtain the spectrum or . frequency content of a Signal, and to facilitate the computation of discrete convolution and correlation. Indeed, published work on the FFT algorithm as a means of calculating the OFT, by J. W. Cooley and J. W. Tukey in 1965 [1], was a turning point in digital signal processing and in certain areas of numerical analysis. They showed that the OFT, which was previously thought to require N 2 arithmetic operations, could be calculated by the new FFT algorithm using only N log Noperations. This algorithm had a revolutionary effect on many digital processing methods, and remains the most Widely used method of computing Fourier transforms [2]. In their original paper, Cooley and Tukey referred only to I. J. Good's work published in 1958 [3] as having influenced their development. However, It was soon discovered there are major differences between the Cooley-Tukey FFT and the algorithm described by Good, which is now commonly referred to as the prime factor algorithm (PFA). Soon after the appearance of the CooleyTukey paper, Rudnick [4] demonstrated a similar algorithm, based on the work of Danielson and Lanczos [5] which had appeared in 1942. This discovery prompted an investigation into the history of the FFT algorithm by Cooley, Lewis, and Welch [6]. They discovered that the Oanielson-Lanczos paper referred to work by Runge published at the tu rn of the centu ry [7, 8]. The algorithm developed by Cooley and Tukey clearly had its roots in, though perhaps not a direct influence from, the early twentieth century. In a recently published history of numerical analysis [9], H. H. Goldstine attributes to Carl Friedrich Gauss, the eminent German mathematician, an algorithm similar to the FFT for the computation of the coefficients of a finite Fourier series. Gauss' treatise describing the algorithm was not published in his lifetime; it appeared only in his collected works [10] as an unpublished manuscript. The presumed year of the composition of this treatise is 1805, thereby suggesting that efficient algorithms for evaluating

The Fft, Fundamentals and Concepts

Abstract: Simple and concise discussion of both Fourier Theory and the FFT (fast fourier transform) are provided.

Phase and amplitude recovery from bispectra

Abstract: The bispectrum is the Fourier transform of the triple correlation, sometimes also referred to as triple product integral. We are concerned here with the bispectrum of the autotriple correlation. Bispectrum analysis can be used to solve phase problems in signal processing, since the knowledge of the bispectrum of a signal usually allows one to reconstruct both amplitude and phase of the Fourier transform signal. We present mathematical proof based on the theory of analytic functions and discuss the restrictions involved. A recursive algorithm is outlined for the reconstruction from sampled data. In addition, possibilities for noise reduction by averaging redundant information will be described. Examples are included for 1-D signals.

Maximum entropy signal processing in practical NMR spectroscopy

Abstract: NMR spectroscopy is intrinsically insensitive, a frequently serious limitation especially in biochemical applications where sample size is limited and compounds may be too insoluble or unstable for data to be accumulated over long periods. Fourier transform (FT) NMR was developed by Ernst1 to speed up the accumulation of useful data, dramatically improving the quality of spectra obtained in a given observing time by recording the free induction decay (FID) data directly in time, at the cost of requiring numerical processing. Ernst also proposed that more information could be obtained from the spectrum if the FID was multiplied by a suitable apodizing function before being Fourier transformed. For example (see ref. 2), an increase in sensitivity can result from the use of a matched filter1, whereas an increase in resolution can be achieved by the use of gaussian multiplication1,3, application of sine bells4–8 or convolution difference9. These methods are now used routinely in NMR data processing. The maximum entropy method (MEM)10 is theoretically capable of achieving simultaneous enhancement in both respects11, and this has been borne out in practice in other fields where it has been applied. However, this technique requires relatively heavy computation. We describe here the first practical application of MEM to NMR, and we analyse 13C and 1H NMR spectra of 2-vinyl pyridine. Compared with conventional spectra, MEM gives considerable suppression of noise, accompanied by significant resolution enhancement. Multiplets in the 1H spectra are resolved better leading to improved visual clarity.

Exact image theory for the Sommerfeld half-space problem, part II: Vertical electrical dipole

Abstract: Applying the Laplace transform, the exact distributed image current function is obtained for the classical Sommerfeld half-space problem with vertical magnetic current source. The resulting field integral is well behaved when the image current is situated in complex space. Unlike previous approximate images, the present theory is valid for any distance, height of the source, frequency, and half-space parameters. It is demonstrated that the present image theory reduces to the well-known dipole image at complex depth for large dielectric parameters of the half-space. Also, the reflection-coefficient method is obtained as a farfield approximation. Calculation of fields through exact image integration is seen to be simple and accurate and require modest computer capacity and time. In an appendix, some properties of the multivalued Green's function arising from a dipole source in complex space are also studied.

Seismic waves in stratified anisotropic media

Abstract: Summary. The response of a structure composed of anisotropic strata can be built up from the reflection and transmission properties of individual interfaces using a slightly modified version of the recursion scheme of Kennett. This scheme is conveniently described in terms of scatterer operators and scatterer products. The effects of a free surface and the introduction of a simple point source at any depth can be accommodated in a manner directly analogous to the treatment for isotropic structures. As in the isotropic case the results so obtained are stable to arbitrary wavenumbers. For isotropic media, synthetic seismograms can be constructed by computing the structure response as a function of frequency and radial wavenumber, then performing the appropriate Fourier and Hankel transforms to obtain the wavefield in time-distance space. Such a scheme is convenient for any system with cylindrical symmetry (including transverse isotropy). Azimuthally anisotropic structures, however, do not display cylindrical symmetry; for these the transverse component of the wavenumber vector will, in general, be non-zero, with the result that phase, group, and energy velocities may all diverge. The problem is then much more conveniently addressed in Cartesian coordinates, with the frequency-wavenumber to time-distance transformation accomplished by 3-D Fourier transform.

Elastic wave calculations by the Fourier method

Abstract: We introduce a two-dimensional forward modeling algorithim based on a Fourier method. In order to be able to handle the free surface boundary condition with the Fourier method, a new set of wave equations are derived which contain the stresses as unknowns instead of the displacements. The solution algorithm includes a discretization in both space and time. Spatial derivatives are approximated with the use of the Fast Fourier Transform, whereas temporal derivatives are calculated with second order differencing. The numerical method is tested against the analytic solution for Lamb's problem in two dimensions.

Significance of group delay functions in signal reconstruction from spectral magnitude or phase

Abstract: In this paper we discuss the problem of signal reconstruction from spectral magnitude or phase using group delay functions. We define two separate group delay functions for a signal, one is derived from the magnitude and the other from the phase of the Fourier transform of the signal. The group delay functions offer insight into the problem of signal reconstruction and suggest methods for reconstructing signals from partial information such as spectral magnitude or phase. We examine the problem of signal reconstruction from spectral magnitude or phase on the basis of these two group delay functions and derive the conditions for signal reconstruction. Based on existing iterative and noniterative algorithms for signal reconstruction, we propose new algorithms for some special classes of signals. The algorithms are illustrated with several examples. Our study shows that the relative importance of spectral magnitude and phase depends on the nature of signals. Speech signals are used to illustrate the importance of spectral magnitude and picture signals are used to illustrate the importance of phase in signal reconstruction problems. Using the group delay functions, we explain the convergence behavior of the existing iterative algorithms for signal reconstruction.

Improvements in Real-Space Deconvolution of Small-Angle Scattering Data

Abstract: In the cases of spherical, cylindrical or lamellar symmetry it is possible to calculate the radial density distribution from small-angle scattering intensities via indirect Fourier transformation and a convolution square-root technique avoiding the phase problem. The density function is expressed in terms of step functions. All this is accomplished by a method described previously. Firstly, the necessary modifications of the method allowing for arbitrary step widths are discussed. Additional background to the scattering intensities is an important problem in practical application, so it is shown that remaining background scattering is mainly eliminated automatically and does not essentially influence the results in the convolution square-root technique. Finally, the introduction of a least-squares variation algorithm allows for the optimization of simple step models.

On unifying the concepts of scale, instrumentation, and stochastics in the development of multiphase transport theory

Abstract: A generalized theory of multiphase transport is presented which combines the concepts of scale, instrumentation, stochastics, and time series with the development of transport equations. By defining the filtering process as a convolution of a measure P with a field property ψ, we are able to exploit the Fourier transform to place plausible restrictions on an instrument in frequency space so as to make its measurement relevant to its physical environment. An ideal instrument is defined which filters out high-frequency noise (corresponding to short distances) and yet does not alter the structure of low frequencies. Using ideal instruments we successively filter out lower frequency noise in a multiscale, multiphase environment. Formulas are developed to relate the autocorrelation of a field property on one scale of motion to that on any other scale while taking into account the types of instruments used in the measuring process. An equation relating the integral scale on one scale of motion to the integral scale on any other scale of motion is developed. Power spectra are developed which relate spectra on different scales to the measuring instrument used. By successively applying filtering theorems, a hierarchy of multiscale transport equations is developed. Filtered properties in the transport equations are mass averages. Different properties are allowed to be measured by different instruments and different instruments are allowed on different scales of motion and for different phases. The concept of a wide sense stationary, ergodic process is introduced to develop mass average autocorrelations and spectra over scales of motion as a function of measuring devices.

On Fourier multiplier transformations of Banach-valued functions

Abstract: We obtain analogues of the Mihlin multiplier theorem and Littlewood-Paley inequalities for functions with values in a suitable Banach space B. The requirement on B is that it have the unconditionality property for martingale difference sequences.

Fourier transform spectrometer with a self-scanning photodiode array.

Abstract: A Fourier transform spectrometer with no mechanical moving parts is described. The interferogram is generated spatially by a triangle common-path interferometer and is detected by a self-scanning photodiode array. The spectrum is reconstructed by fast Fourier transform in a microcomputer system. Since no moving part is used and a common-path interferometer is employed for simple, stable, and easy alignment, this spectrometer may be built in a relatively small size and with moderate cost. The self-scanning photodiode array as a multichannel detector may lead this spectrometer to the application to time-resolved spectroscopy. The optical throughput is much larger than that of a multichannel dispersion-type spectrometer, because in the system neither a slit nor an aperture is necessary. The emission spectra of a low pressure mercury lamp and a LED are shown to demonstrate the system performance.

Estimating Fried’s parameter from a time series of an arbitrary resolved object imaged through atmospheric turbulence

Abstract: A method to obtain an estimate of Fried’s seeing parameter r0 from time series of an arbitrarily shaped, resolved structure that exhibits degradation resulting from atmospheric turbulence is presented. The basic idea is to evaluate the ratio of the observed squared modulus of the average Fourier transform and the observed average power spectrum. The theory of the method is developed, and the influence of noise on the ratio is discussed. The method has been applied to five consecutive time series of observations of solar granulation under different seeing conditions. The power spectra, which are reconstructed with appropriate theoretical modulation transfer functions, converge.

Dynamical atom/surface effects: Quantum mechanical scattering and desorption

Abstract: Desorption and scattering of a helium atom from dynamic surfaces are studied using a generalized Langevin equation formalism for the bulk solid and time dependent quantum mechanical propagation of the helium. The motion of the bulk solid enters the Schrodinger equation as a time dependent potential coupling the solid surface to the atom. The propagation of the atomic wave packet is then followed in coordinate space using a Fourier transform technique to evaluate the kinetic energy operator. An attenuating grid is used to remove the wave packet in the asymptotic region thus permitting stable calculations for long duration times. Rates for one dimensional desorption from tungsten as a function of temperature are computed which display a change in mechanism. Three dimensional scattering from platinum at two temperatures shows significant nonspecular amplitude. These results indicate the utility of this time dependent technique.

Finite periodic structure approach to large scanning array problems

Abstract: There are two conventional techniques dealing with mutual coupling problems for antenna arrays. The "element-by-element" method is useful for small to moderate size arrays. The "infinite periodic structure" method deals with one cell of infinite periodic structures, including all the mutual coupling effects. It cannot, however, include edge effects, current tapers, and nonuniform spacings. A new technique called the "finite periodic structure" method, is presented and applied to represent the active impedance of an array, it involves two operations. The first is to convert the discrete array problem into a series of continuous aperture problems by the use of Poisson's sum formula. The second is to use spatial Fourier transforms to represent the impedance in a form similar to the infinite periodic structure approach. The active impedance is then given by a convolution integral involving the infinite periodic structure solution and the Fourier transform of the equivalent aperture distribution of the current over the entire area of the array. The formulation is particularly useful for large finite arrays, and edge effects, current tapers, and nonuniform spacings can also be included in the general formulation. Although the general formulation is valid for both the free and forced modes of excitation, the forced excitation problem is discussed to illustrate the method.

Fourier Coding of Image Boundaries

Abstract: A transform coding scheme for closed image boundaries on a plane is described The given boundary is approximated by a series of straight line segments Depending on the shape, the boundary is represented by the (x-y) coordinates of the endpoints of the line segments or by the magnitude of the successive radii vectors that are equispaced in angle around the given boundary Due to the circularity present in the data, the discrete Fourier transform is used to exactly decorrelate the finite boundary data By fitting a Gaussian circular autoregressive model to represent the boundary data, estimates of the variances of the Fourier coefficients are obtained Using the variances of the Fourier coefficients and the MAX quantizer, the coding scheme is implemented The scheme is illustrated by an example

Fourier Transformed Finding Propagation Matrix Method of Characteristics of Complex Anisotropic Layered Media .

Abstract: A planar structure having arbitrarily located conductor lines immersed in complex anisotropic media presents one with a very general guided wave problem. This problem is solved here by a rigorous formulation technique characterizing each layer by a 6 x 6 tensor and finding the appropriate Fourier transformed Green's function matrix G of 2n x 2n size. From G, a method-of-moments solution for the propagation characteristics follows, including propagation constant eigenvalues and field eigenvectors at all spatial Iocations. The method is very versatile and can handle a huge class of microwave or millimeter-wave integrated circuit or monolithic circuit problems, no matter how simple or complex as long as they possess planar symmetry.

Image reconstruction from frequency-offset Fourier data

Abstract: Motivated by the ability of synthetic-aperture radar and related imaging systems to produce images of surprisingly high quality, we consider the problem of reconstructing the magnitude of a complex signal f from samples of the Fourier transform of f located in a small region offset from the origin. It is shown that high-quality speckle reconstructions are possible so long as the phase of f is highly random. In this case, the quality of the reconstruction is insensitive to the location of the known Fourier data, and edges at all orientations are reproduced equally well. A large number of computer examples are presented demonstrating these attributes. Methods for improving image quality are also briefly discussed.

Asymptotic evaluation of high-frequency fields near a caustic: An introduction to Maslov's method

Abstract: It is well known that the geometrical optics approximation, generally valid for high-frequency fields, fails in the vicinity of a caustic. A systematic procedure of V. P. Maslov that remedies this situation will be reviewed in this paper. Maslov's method makes use of a representation of the geometrical optics field in the phase space M = X × K, where a point m = (x, k) is a pair of a position vector x e X and a wave vector k e K. A Lagrangian submanifold of M, Λ, that lies in the dispersion surface and is a union of the phase space trajectories selected by the initial conditions is constructed. It can be considered as a global representation of the phase. The phase space amplitudes (half densities) satisfy transport equations defined along those trajectories in Λ. Since trajectories in M never form a caustic, a globally defined amplitude can be established on Λ. The field on X is related to the resultant field on Λ by the “canonical operator,” an operator introduced by Maslov. It generates an integral form of the solution near a caustic that can be evaluated analytically, numerically, or with uniform asymptotic techniques. Away from the caustic it recovers the geometrical optics field. Alternatively, the phase space field can be projected on a hybrid space Y where some of the space coordinates have been replaced by the corresponding wave vector components. For any caustic point in X, one such hybrid space Y where this projection does not encounter a caustic exists. A geometrical optics field results in Y that is related to the original in X by an asymptotic Fourier transform. The solution in X near a caustic can be represented as the Fourier transform to X of that hybrid space geometrical optics solution. These techniques are illustrated with two simple but revealing problems: continuation of the field through a fold caustic in a linear layer medium and through a caustic with a cusp point in a homogeneous medium.

Prediction of stable processes: Spectral and moving average representations

Abstract: For stable processes which are Fourier transforms of processes with independent increments, we obtain a Wold decomposition, we characterize their regularity and singularity, and, in the discrete-parameter case, we derive their linear predictors. In sharp contrast with the Gaussian case, regular stable processes which are Fourier transforms of processes with independent increments are not moving averages of stable motion.

Image restoration from non-uniform magnetic field influence for direct Fourier NMR imaging.

Abstract: A new technique is proposed for NMR image restoration from the influence of main magnetic field non-uniformities. This technique is applicable to direct Fourier NMR imaging. The mathematical basis and details of this technique are fully described. Modification to include image restoration from non-linear field gradient influence is also presented. Computer simulation demonstrates the effectiveness of this technique for both Fourier zeugmatography and spin-warp imaging.