Showing papers on "Fourier transform published in 1982"

NMR chemical shift imaging in three dimensions

Abstract: A method for obtaining the three-dimensional distribution of chemical shifts in a spatially inhomogeneous sample using Fourier transform NMR is presented. The method uses a sequence of pulsed field gradients to measure the Fourier transform of the desired distribution on a rectangular grid in (k,t) space. Simple Fourier inversion then recovers the original distribution. An estimated signal/noise ratio of 20 in 10 min is obtained for an "image" of the distribution of a 10 mM phosphorylated metabolite in the human head at a field of 20 kG with 2-cm resolution.

Analysis of the phase unwrapping algorithm.

Abstract: Several workers have recently proposed digital techniques for high-resolution imaging through the turbulent atmosphere. The basic concept of these algorithms is to calculate and average phase angles of a series of image Fourier transforms to suppress the unwanted atmospheric effects on image resolution. Since computed phase angles contain the ambiguities of integral multiples of 2 pirad, it is necessary to obtain continuous phase curves without the ambiguities before averaging. This process of eliminating the ambiguities is called phase tracking or unwrapping. A similar problem has been discussed by Oppenheim and Schafer and Tribolet in the context of realization of a certain homomorphic signal processing system.

New theory of partial coherence in the space–frequency domain. Part I: spectra and cross spectra of steady-state sources

Abstract: It is shown that, under very general conditions, the cross-spectral density of a steady-state source of any state of coherence may be expressed in terms of certain new modes of oscillations, each of which represents a completely spatially coherent elementary excitation. Making use of this result, a statistical ensemble of strictly monochromatic oscillations, all of the same temporal frequency, is then introduced that yields the cross-spectral density as a correlation function in the space–frequency domain. From these results two new expressions for the Wiener–Khintchine spectrum of the source and also a new mode representation of the cross-correlation function of the source follow at once.

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.

The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform

Abstract: This paper addresses two fundamental issues involved in the reconstruction of a multidimensional sequence from either the phase or magnitude of its Fourier transform The first issue relates to the uniqueness of a multidimensional sequence in terms of its phase or magnitude Although phase or magnitude information alone is not sufficient, in general, to uniquely specify a sequence, a large class of sequences are shown to be recoverable from their phase or magnitude The second issue which is addressed in this paper concerns the actual reconstruction of a multidimensional sequence from its phase or magnitude For those sequences which are uniquely specified by their phase, several practical algorithms are described which may be used to reconstruct a sequence from its phase Several examples of phase-only reconstruction are also presented Unfortunately, however, even for those sequences which are uniquely defined by their magnitude, it appears that a practical algorithm is yet to be developed for reconstructing a sequence from only its magnitude Nevertheless, an iterative procedure which has been proposed is briefly discussed and evaluated

Introduction to the theory of distributions

Abstract: 1. Test functions and distributions 2. Differentiation and multiplication 3. Distributions and compact support 4. Tensor products 5. Convolution 6. Distribution kernels 7. Co-ordinate transforms and pullbacks 8. Fourier transforms 9. Plancherel's theorem 10. The Fourier-Laplace transform Appendix. Topological vector spaces 11. The calculus of wavefront sets.

Electron density images from imperfect data by iterative entropy maximization

Abstract: Information theory provides a uniquely powerful apparatus for reconstructing an image from imperfect data in its dual space. An iterative procedure based on constrained entropy maximization is presented here for reconstruction of electron density from imperfect single-crystal X-ray diffraction data. In the ideal situation, continuous electron density, its periodicity given by the lattice, is the Fourier transform of an infinite set of structure factors whose squared moduli are observables. An ideal electron density is unattainable from experiment because of the common experimental inadequacies of incomplete and noisy data discussed by Gull and Daniell1. These inadequacies are partially overcome in the present procedure which yields true super-resolution and is economical even for protein crystal structures as illustrated.

Determination of crystallite size and lattice distortions through X-ray diffraction line profile analysis

Abstract: Methods for the determination of crystallite size and lattice strain from X-ray diffraction line broadening are discussed. The subsequent steps of measurement, data correction and evaluation are elucidated; alternatives are indicated. Emphasis is laid on the rigorous analysis of line profiles in terms of Fourier coefficients. For the analysis in terms of integral breadth and full width at half maximum a powerful method exists which adopts a Voigt function for describing the shape of the profiles. Size broadening, strain broadening and single-line methods are commented. A practical example is given of the influence of a non-ideal standard line profile and of different background estimates when a Fourier deconvolution and a Warren-Averbach size-strain analysis are performed. It is expected that line profile analysis will become an automated routine-like analytical method soon, since the tools are available: non-expensive computers, error calculations and commercially available software.

Fast polarization modulated ellipsometer using a microprocessor system for digital Fourier analysis

Abstract: A new type of spectroscopic automatic ellipsometer using a piezobirefringent element for polarization modulation at 50 kHz is described. Instead of lock‐in amplifiers the data‐acquisition system consists of a 12.8‐MHz digital sampling of the detected signal with a high word rate 8‐bit ADC, followed by on line Fourier transformation of the accumulated data with a short instruction cycle (∼200 ns) microprocessor, driven by a commercial microcomputer. The absolute minimum time required for measuring one set of Fourier coefficients is thus reduced to the modulation period of 20 μs. For digital error reduction purposes and signal‐to‐noise ratio improvement a basic 5 ms sequence of 256 accumulated periods per point is chosen. At this data‐acquisition rate a precision of 5×10−4 is obtained. Further accumulation over 10 s leads to 10−5 precision capability. A detailed analysis of various sources of inaccuracy leads to an estimate of 0.5° maximum systematic error on the ellipsometric angles and ψ and Δ. Applicatio...

Collision-induced dissociation with Fourier transform mass spectrometry

Abstract: Collision-induced dissociations (CID) is demonstrated on a number of primary and secondary ions using a Nicolet prototype Fourier transform mass spectrometer (FT-MS). Like the triple quadrupole technique, CID using FT-MS is a relatively low energy and efficient process. The ability to study a wide range of ion-molecule reaction products is exemplified by results on proton-bound dimers and transition metal containing ionic species. Variation of collision energy by varying the RF irradiation level can provide information about product distributions as a function of energy as well as yield ion structural information. Like the triple quadrupole technique, no slits are employed and virtually all of the fragment ions formed by the CID process may be detected. Unlike all previous mass spectrometric techniques for studying CID, a tandem instrument is not required, and different experiments are performed by making software modifications rather than hardware modifications.

Optical information processing

Abstract: Linear System Theory and Fourier Transformation. Introduction to Diffraction. Fraunhofer and Fresnel Diffraction. Introduction to Partial Coherence Theory. Basic Properties of Recording Materials. Fourier Transform Properties of Lenses and Optical Information Processing. Techniques and Applications of Coherent Optical Processing. Optical Processing with Incoherent Source. Polychromatic Processing with Noncoherent Light. Introduction to Holography. Analysis of Nonlinear Holograms. Rainbow Holography. Applications of Holography. Applications of Rainbow Holography. References.

Reconstruction of the support of an object from the support of its autocorrelation

Abstract: The phase-retrieval problem consists of the reconstruction of an object from the modulus of its Fourier transform or, equivalently, from its autocorrelation. This paper describes a number of results relating to the reconstruction of the support of an object from the support of its autocorrelation. Methods for reconstructing the object’s support are given for objects whose support is convex and for certain objects consisting of collections of distinct points. The uniqueness of solutions is discussed. In addition, for the objects consisting of collections of points, a simple method is shown for completely reconstructing the object functions.

Vectorizing the FFTs

Abstract: Publisher Summary This chapter provides an overview on vectorizing the FFTs. The fast Fourier transform (FFT) is the most well known of all algorithms. It is superior to the slow transform and has applications in all areas of scientific computing. The term FFT was applied to a specific algorithm for the rapid computation of the discrete complex Fourier transform; however, it has become a generic term that is applied to any one of a large number of algorithms that compute the complex as well as other Fourier transforms. Many algorithms exist for a given Fourier transform, and when they are applied to a particular sequence, the result is the same. However, the algorithms differ in the ways that intermediate results are computed and stored. It is these important differences that provide the algorithms with unique properties that make one or the other more attractive for a particular application.

A survey of the physical optics inverse scattering identity

Abstract: The physical optics approximation is applied to the acoustic and electromagnetic direct scattering integral representation, yielding an inverse scattering identity which relates the characteristic function of a scatterer to the three-dimensional spatial Fourier transform of the augmented far-field scattering amplitude. This identity requires full scattering information for all frequencies and aspect angles. An integral equation for incomplete scattering information for this identity is developed. This integral equation is for the unknown characteristic function of the scatterer in terms of the known incomplete scattering information. The kernel of this integral equation is the three-dimensional spatial Fourier transform of the known characteristic function of the scattering information aperture. A regularized analytic closed-form solution to this integral equation is obtained. Synthesized numerico-experimental results verifying this solution are presented. The details of some special cases, consisting of a priori knowledge about the scatterer or the scattering information aperture, are presented.

Inversion formula for inverse scattering within the Born approximation

Abstract: The problem of determining a localized scattering potential V(r) from its associated scattering amplitude f(s, s0) is addressed within the first Born approximation. The conventional methods, based on Fourier synthesis, for obtaining approximate solutions to the problem are reviewed briefly. A new reconstruction method is proposed that is in the form of an integral transform over the scattering amplitude, assumed to be specified over all observation directions s and over all incident unit wave vectors s0. The proposed method is shown also to be applicable to the problem of determining the interatomic distance function ∫d3r′V(r + r′) V*(r′) from the magnitude square of the scattering amplitude.

Transmitting data on the phase of speech

Abstract: The present invention relates to a means for achieving simultaneous transmission of data and speech with only a minimal expansion of the bandwidth of the speech signal. A Fourier transform (14) is performed on the speech signal and a predetermined number of phase components are replaced with data (d(n)) in an appropriate form. The number of phase components replaced with data is determined by approximately classifying the speech (16) as either "silence", no data inserted; "unvoiced" speech, M phase components convey data; and "voiced" speech, J phase components convey data; where J is less than M, and M is not greater than the number of phase components in the message band of the speech signal. An inverse Fourier transform (22) is subsequently performed on the combined data and speech signal. The combined message signal (G(t)) will comprise approximately the same bandwidth as the original speech signal, by virtue of the frequency domain insertion of the data into the speech. At the receiver the signal is inspected and a classifier (38) determines if data is embedded in the received signal. If data is deemed embedded, a Fourier transformation is performed, the data carrying phase components are inspected, and the data signal regenerated in an appropriate form. The phase components used for the conveyance of data are replaced by random phase components, and the inverse Fourier transformation performed. Median filtering is employed to mitigate the effects of end-of-block distortion and yield the recovered speech signal.

The k-space formulation of the scattering problem in the time domain

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, ...

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.

On the analysis of diffusion length measurements by SEM

Abstract: A simpler analysis is given of the diffusion problem related to the scanning electron microscope measurements of bulk diffusion lengths in semiconductors using scanning normal to a p - n junction or a Schottky barrier. The current profile due to a point source is obtained in form of the Fourier transform of an expression containing elementary functions only. It is shown that this form can be readily adapted to include the presence of a back ohmic contact and allows an easier discussion of the case of an extended generation.

Molecular dynamics simulation of the frequency spectrum of amorphous silica

Abstract: The dynamic behavior of atoms in bulk amorphous silica SiO2 has been investigated by using the molecular dynamics computer simulation technique to generate the frequency spectrum. A modified Born–Mayer–Huggins equation was used as the interatomic potential function. Due to the covalency of the Si–O bond, the ability of using a central‐force model to reproduce the short time atomic motion in SiO2 was evaluated. The frequency spectrum was generated from the Fourier transform of the velocity autocorrelation function and was compared with the experimentally obtained spectrum presented in the literature. Results show that the frequency spectrum generated here has the three major peaks which are characteristic of silica— i.e., peaks at ∼400, ∼800, and ∼1100 cm−1. Changes in the Si–Si or O–O repulsive parameters in the potential function can be used to alter the frequency spectrum. The 800 cm−1 peak, due to oxygen bending and Si motion, and the 150 cm−1 correlated motion peak are the most affected by the alterat...

Fourier, Hadamard, and Hilbert Transforms in Chemistry

Abstract: Advantages of Transform Methods in Chemistry.- Hadamard and Other Discrete Transforms in Spectroscopy.- Processing Software for Fourier Transform Spectroscopies.- Dispersion versus Absorption (DISPA): Hilbert Transforms in Spectral Line Shape Analysis.- Fourier Transform Ion Cyclotron Resonance Spectroscopy.- Fourier Transform Nuclear Quadrupole Resonance Spectroscopy.- Fourier Transform Dielectric Spectroscopy.- Pulsed Fourier Transform Microwave Spectroscopy.- Two-Dimensional Fourier Transform NMR Spectroscopy.- Endor Spectroscopy by Fourier Transformation of the Electron Spin Echo Envelope.- Advances in FT-NMR Methodology for Paramagnetic Solutions: Detection of Quadrupolar Nuclei in Complex Free Radicals and Biological Samples.- Fourier Transform ?SR.- Fourier Transform Infrared Spectrometry.- Aspects of Fourier Transform Visible/UV Spectroscopy.- Fourier Transform Faradaic Admittance Measurements (FT-FAM): A Description and Some Applications.- Optical Diffraction by Electrodes: Use of Fourier Transforms in Spectroelectrochemistry.- List of Contributors.

Spectral approach to geophysical inversion by Lorentz, Fourier, and Radon transforms

Abstract: Geophysical inversion seeks to determine the structure of the interior of the earth from data obtained at the surface. In reflection seismology, the problem is to find inverse methods that give structure, composition, and source parameters by processing the received seismograms. The pioneering work of Jack Cohen and Norman Bleistein on general inverse methods has caused a revolution in the direction of research on long-standing unsolved geophysical problems. This paper does not deal with such general methods, but instead gives a survey of some production-type data processing methods in everyday use in geophysical exploration. The unifying theme is the spectral approach which provides methods for the approximate solution of some simplified inverse problems of practical importance. This paper is divided into two parts, one dealing with one-dimensional (1-D) inversion, the other with two-dimensional (2-D) inversion. The 1-D case treated is that of a horizontally layered earth (Goupillaud model) with seismic raypaths only in the vertical direction. This model exhibits a lattice structure which corresponds to the lattice methods of spectral estimation. It is shown that the lattice structure is mathematically equivalent to the sturcture of the Lorentz transformation of the special theory of relativity. The solution of this 1-D inverse problem is the discrete counterpart of the Gelfand-Levitan inversion method in physics. A practical computational scheme to carry out the inversion process is the method of dynamic deconvolution. It is based on a generalization of the Levinson recursion, and involves the interacting recursions of two polynomials P and Q. This paper treats only much simplified 2-D models. One 2-D method gives the forward and inverse solution for a horizontally layered earth (Goupillaud model) with slanting seismic raypaths. This method involves the Radon transform which is often called "slant stacking" by geophysicists. The other 2-D methods given in this paper are concerned with the process of wavefield reconstruction and imaging known as "migration" in the geophysical industry. A major breakthrough occurred in 1978 when Stolt introduced spectral migration which makes the use of the fast Fourier transform. Another method, the slant-stack migration of Hubral, is based on the Radon transform.

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.