Showing papers on "Fourier transform published in 1991"

Selection of a convolution function for Fourier inversion using gridding (computerised tomography application)

Abstract: In the technique known as gridding, the data samples are weighted for sampling density and convolved with a finite kernel, then resampled on a grid preparatory to a fast Fourier transform. The authors compare the artifact introduced into the image for various convolving functions of different sizes, including the Kaiser-Bessel window and the zero-order prolate spheroidal wave function (PSWF). They also show a convolving function that improves upon the PSWF in some circumstances. >

Calculation of a constant Q spectral transform

Abstract: The frequencies that have been chosen to make up the scale of Western music are geometrically spaced. Thus the discrete Fourier transform (DFT), although extremely efficient in the fast Fourier transform implementation, yields components which do not map efficiently to musical frequencies. This is because the frequency components calculated with the DFT are separated by a constant frequency difference and with a constant resolution. A calculation similar to a discrete Fourier transform but with a constant ratio of center frequency to resolution has been made; this is a constant Q transform and is equivalent to a 1/24‐oct filter bank. Thus there are two frequency components for each musical note so that two adjacent notes in the musical scale played simultaneously can be resolved anywhere in the musical frequency range. This transform against log (frequency) to obtain a constant pattern in the frequency domain for sounds with harmonic frequency components has been plotted. This is compared to the conventio...

A two-dimensional Fourier transform method for the measurement of propagating multimode signals

Abstract: A technique for the analysis of propagating multimode signals is presented. The method involves a two-dimensional Fourier transformation of the time history of the waves received at a series of equally spaced positions along the propagation path. The technique has been used to measure the amplitudes and velocities of the Lamb waves propagating in a plate, the output of the transform being presented using an isometric projection which gives a three-dimensional view of the wave-number dispersion curves. The results of numerical and experimental studies to measure the dispersion curves of Lamb waves propagating in 0.5-, 2.0-, and 3.0-mm-thick steel plates are presented. The results are in good agreement with analytical predictions and show the effectiveness of using the two-dimensional Fourier transform (2-D FFT) method to identify and measure the amplitudes of individual Lamb modes.

Oscillatory integrals and regularity of dispersive equations

Abstract: This paper is concerned with oscillatory integrals and their relationship with the local and global smoothing properties of dispersive equations. Also we shall study some applications of these smoothing properties to nonlinear problems and their link with restriction theorems for the Fourier transform and pointwise convergence results

Multidimensional orientation estimation with applications to texture analysis and optical flow

Abstract: The problem of detection of orientation in finite dimensional Euclidean spaces is solved in the least squares sense. The theory is developed for the case when such orientation computations are necessary at all local neighborhoods of the n-dimensional Euclidean space. Detection of orientation is shown to correspond to fitting an axis or a plane to the Fourier transform of an n-dimensional structure. The solution of this problem is related to the solution of a well-known matrix eigenvalue problem. The computations can be performed in the spatial domain without actually doing a Fourier transformation. Along with the orientation estimate, a certainty measure, based on the error of the fit, is proposed. Two applications in image analysis are considered: texture segmentation and optical flow. The theory is verified by experiments which confirm accurate orientation estimates and reliable certainty measures in the presence of noise. The comparative results indicate that the theory produces algorithms computing robust texture features as well as optical flow. >

The ‘infinity cell’: A new trapped‐ion cell with radiofrequency covered trapping electrodes for fourier transform ion cyclotron resonance mass spectrometry

Abstract: The excitation event in Fourier transform ion cyclotron resonance mass spectrometry is optimized in order to obtain more reliable relative signal intensities and enhance the ion-selection performance in multi-tandem mass spectrometric experiments. Standard trapped-ion cells suffer from the undesirable ejection of ions along the symmetry axis (z-axis) of the cell, which is oriented parallel to the magnetic field lines. This z-eiection effect is difficult to predict, and thus difficult to avoid, especially when complicated broad-band excitation schemes are applied to the ions. An improved trapped-ion cell design, referred to as the ‘Infinity Cell’, is discussed which eliminates z-ejection. The Infinity Cell concept is based on the finding that it is possible to model the electric excitation field of an infinitely long cell with a cell of finite dimensions. The virtual elimination of z-ejection effects is demonstrated in several suitable experiments by comparing the operation of the standard cell and the Infinity Cell.

A digital recursive measurement scheme for online tracking of power system harmonics

Abstract: An optimal measurement scheme for tracking the harmonics in power system voltage and current waveforms is presented. The scheme does not require an integer number of samples in an integer number of cycles. It is not limited to stationary signals, but it can track harmonics with time-varying amplitudes. A review is first presented of the common frequency domain techniques for harmonics measurement. The frequency domain techniques are based on the discrete Fourier transform and the fast Fourier transform. Examples of pitfalls in the common techniques are given. The authors then introduce the concepts of the new scheme. This scheme is based on Kalman filtering theory for the optimal estimation of the parameters of time-varying harmonics. The scheme was tested on simulated and actual recorded data sets. It is concluded that the Kalman filtering algorithm is more accurate than the other techniques. >

The fractional Fourier transform and applications

Abstract: This paper describes the “fractional Fourier transform,” which admits computation by an algorithm that has complexity proportional to the fast Fourier transform algorithm. Whereas the discrete Fourier transform (DFT) is based on integral roots of unity $e^{{{ - 2\pi i} / n}} $, the fractional Fourier transform is based on fractional roots of unity $e^{ - 2\pi i\alpha } $ where $\alpha $ is arbitrary. The fractional Fourier transform and the corresponding fast algorithm are useful for such applications as computing DFTs of sequences with prime lengths, computing DFTs of sparse sequences, analyzing sequences with noninteger periodicities, performing high-resolution trigonometric interpolation, detecting lines in noisy images, and detecting signals with linearly drifting frequencies. In many cases, the resulting algorithms are faster by arbitrarily large factors than conventional techniques.

Application of fast-Fourier-transform techniques to the discrete-dipole approximation.

Abstract: We show how fast-Fourier-transform methods can be used to accelerate computations of scattering and absorption by particles of arbitrary shape using the discrete-dipole approximation.

A new finite element formulation for computational fluid dynamics: IX. Fourier analysis of space-time Galerkin/least-squares algorithms

Abstract: A Fourier stability and accuracy analysis of the space-time Galerkin/least-squares method as applied to a time-dependent advective-diffusive model problem is presented. Two time discretizations are studied: a constant-in-time approximation and a linear-in-time approximation. Corresponding space-time predictor multi-corrector algorithms are also derived and studied. The behavior of the space-time algorithms is compared to algorithms based on semidiscrete formulations.

Wavenumber calibration tables from heterodyne frequency measurements

Abstract: This new calibration atlas is based on frequency rather than wavelength calibration techniques for absolute references. Since a limited number of absolute frequency measurements is possible, additional data from alternate methodology are used for difference frequency measurements within each band investigated by the frequency measurements techniques. Data from these complementary techniques include the best Fourier transform measurements available. Included in the text relating to the atlas are a description of the heterodyne frequency measurement techniques and details of the analysis, including the Hamiltonians and least-squares-fitting and calculation. Also included are other relevant considerations such as intensities and lincshape parameters. A 390-entry bibliography which contains all data sources used and a subsequent section on errors conclude the text portion. The primary calibration molecules are the linear triatomics, carbonyl sulfide and nitrous oxide, which cover portions of the infrared spectrum ranging from 488 to 3120 cm-1. Some gaps in the coverage afforded by OCS and N2O are partially covered by NO, CO, and CS2. An additional region from 4000 to 4400 cm-1 is also included. The tabular portion of the atlas is too lengthy to include in an archival journal. Furthermore, different users have different requirements for such an atlas. In an effort to satisfy most users, we have made two different options available. The first is NIST Special Publication 821, which has a spectral map/facing table format. The spectral maps (as well as the facing tables) are calculated from molecular constants derived for the work. A complete list of all of the molecular transitions that went into making the maps is too long (perhaps by a factor of 4 or 5) to include in the facing tables. The second option for those not interested in maps (or perhaps to supplement Special Publication 821) is the complete list (tables-only) which is available in computerized format as NIST Standard Reference Database #39, Wavelength Calibration Tables.

Motivic decomposition of abelian schemes and the Fourier transform.

Abstract: such that the /-adic realization of h (A) is H (A, Qt). Recall that Chow motives are obtained from the category of smooth projective varieties over a field by a construction of Grothendieck using äs intersection theory the Chow ring tensored with Q. See [Ma] and [Mur], 1.1 and l .6, remark 2 for details. By an intricate argument Shermenev also shows how to express h *(A) in terms of h (A). For Chow motives a decomposition äs in (0.1) with /-adic realization äs above is by no means unique. Shermenev's decomposition in particular depends on choices. In this paper we establish a canonicalfunctorial decomposition äs above not only for abelian varieties but also for abelian schemes over a smooth quasiprojective base 5 over a field. In order to formulate such a Statement we extend parts of Manin's paper [Ma] on motives over a field to the case where the base is S. This is straightforward and we obtain a category of relative Chow motives over S.

The van Cittert–Zernike theorem in pulse echo measurements

Abstract: A classical theorem of statistical optics, the van Cittert–Zernike theorem, is generalized to pulse echo ultrasound. This theorem fully describes the second‐order statistics of the spatial fluctuations (the spatial covariance) of the field produced by an incoherent source. As a random scattering medium is insonified, it behaves as an incoherent source. The van Cittert–Zernike theorem can thus predict the spatial covariance of the pressure field backscattered by a random medium. It is shown that this spatial covariance and the incident energy diagram are Fourier pairs. In the case of a focused illumination, the spatial covariance of the backscattered pressure field is proportional to the autocorrelation of the transmitting aperture function. This is independent of frequency and of F/ number. Experimental results obtained with a linear array are in good agreement with theoretical expectations. The implications of this theorem in speckle reduction and in focusing in nonhomogenous media are discussed.

Inverse Q filtering by Fourier transform

Abstract: Although some attention has been paid to the idea that seismic migration is equivalent to a type of deconvolution (of the spatial wavelet), less thought has been given to the opposite perspective: that deconvolution (of the earth Q filter) might itself be equivalent to a form of migration. The key point raised in this paper is that a dispersive 1-D backward propagation can form the basis of a number of different algorithms for inverse Q filtering, each of which is akin to a particular migration algorithm. An especially efficient algorithm can be derived by means of a coordinate transformation equivalent to that in the Stolt frequency-wavenumber migration.This fast algorithm, valid for Q constant with depth, can be extended to accommodate depth-variable Q by cascading a series of constant Q compensations, as in cascaded migration. By combining a cascaded phase compensation with a windowed approach to amplitude compensation, we obtain an algorithm that is sufficiently efficient to be used routinely for prestack data processing. Data examples compare the results of conventional processing with the more stable phase treatment that can be obtained by including prestack inverse Q filtering in the processing.

Nodal surfaces of Fourier series : fundamental invariants of structured matter

Abstract: Periodic Nodal Surfaces (PNS) of Fourier series are derived and classified as fundamental invariants of structured matter. Relationships to periodic minimal surfaces PMS and to periodic zero potential surfaces (POPS) are given. A basic set of cubic PNS is represented in arithmetic form. The special importance of the invariance of the zeros to the type of the potential is stressed.

Dithered Quantizers

Abstract: A theory of overall quantization noise for nonsubtractive dither was originally developed in unpublished work by J.N. Wright and by T.J. Stockham and subsequently expanded by L.K. Brinton, S.P. Lipshitz, J. Vanderkooy, and R.A. Wannamaker. It is suggested that since these latter results are not as well known as the original results, misunderstanding persists in the literature. New proofs of the properties of quantizer dither, both subtractive and nonsubtractive, are provided. The new proofs are based on elementary Fourier series and Rice's characteristic function method and do not require the use of generalized functions (impulse trains of Dirac delta functions) and sampling theorem arguments. The goal is to provide a unified derivation and presentation of the two forms of dithered quantizer noise based on elementary Fourier techniques. >

Biomedical signal processing (in four parts). Part 3. The power spectrum and coherence function.

Abstract: This is the third in a series of four tutorial papers on biomedical signal processing and concerns the estimation of the power spectrum (PS) and coherence function (CF) od biomedical data. The PS is introduced and its estimation by means of the discrete Fourier transform is considered in terms of the problem of resolution in the frequency domain. The periodogram is introduced and its variance, bias and the effects of windowing and smoothing are considered. The use of the autocovariance function as a stage in power spectral estimation is described and the effects of windows in the autocorrelation domain are compared with the related effects of windows in the original time domain. The concept of coherence is introduced and the many ways in which coherence functions might be estimated are considered.

Projection exposure method and apparatus

Abstract: In projection exposure of isolated pattern such as a contact hole, in order to increase the depth of focus a coherence reducing member is disposed on a Fourier transform plane in an image-forming optical path between a mask and a sensitized base, so that coherence is reduced between image-forming beams respectively passing through a plurality of different, concentric regions around the optical axis of the projection optical system on the Fourier transform plane. The coherence reducing member may be a polarization state control member for making a difference in polarization state, a member for making a difference in optical path length, or space filters with different shapes.

Polarization modulation Fourier transform infrared reflectance measurements of thin films and monolayers at metal surfaces utilizing real-time sampling electronics

Abstract: The technique of polarization modulation Fourier transform infrared (PM-FTIR) spectroscopy is applied to the reflectance spectra of thin polymer films and spontaneously organized monolayers adsorbed onto gold, sliver, and chromium surfaces. The differential PM-FTIR reflectance spectra are obtalned by the photoelastic modulation of the FTIR beam polarization and a novel real-time sampling methodology that generates the average and differential FTIR interferograms from measurements of the Infrared signal during each modulation cycle. In comparison with conventional electronics that utilize a lock-in amplifier, the real-time electronics permit the operation of the FTIR spectrometer at normal mirror velocities. The use of polarization-independent optics after the metal surface ensures that the true surface Infrared differential reflectance spectrum is obtained. The theoretical wavelength dependence of the PM-FTIR spectrum is rederlved for the case of the real-time sampling measurement and compared to the experimental data. Spectra of a 15-nm film of polyimide on chromium, a spontaneously Organized monolayer of octadecanethioi on gold, and a spontaneously organized monolayer of arachidic acid on silver are shown to demonstrate the applicability of the method to different metals, samples, and spectral regions.

Modern signals and systems

Abstract: Overview of signals and systems an introduction to signals an introduction to systems difference and differential systems state description of systems expansion theory and Fourier series Fourier transforms the z- and laplace transforms applications to signal processing and digital filtering applications to communication feedback and applications to automatic control supplement reviews and SIGSYS tutorial.

Comparison of finite-difference and Fourier-transform solutions of the parabolic wave equation with emphasis on integrated-optics applications

Abstract: The solution of the scalar wave equation in the parabolic approximation is considered through the finite-difference and the Fourier-transform (i.e., beam propagation method) techniques. Examples are taken from the field of integrated optics and include propagation in straight, tapered, Y-branched, and coupled waveguides. A comparison of numerical results obtained by the two methods is presented, and a comparison with other analytical or numerical methods is also given. In the numerous cases studied it is shown that the finite-difference method yields a large, order-of-magnitude range improvement in accuracy or computational speed when compared with the Fourier-transform method.

A uniqueness theorem of Beurling for Fourier transform pairs

Abstract: There are many theorems known which state that a function and its Fourier transform cannot simultaneously be very small at infinity, such as various forms of the uncertainty principle and the basic results on quasianalytic functions. One such theorem is stated on page 372 in volume II of the collected works of Arne Beurling [1]. Although it is not in every respect the most precise result of its kind, it has a simplicity and generality which make it very attractive. The editors state that no proof has been preserved. However, in my files I have notes which I made when Arne Beurling explained this result to me during a private conversation some time during the years 1964---1968 when we were colleagues at the Institute for Advanced Study, I shall reproduce these notes here in English translation with onIy minor details added where my notes are too sketchy. Theorem. Let fELl (R) and assume that

A restriction theorem for the Fourier transform

Abstract: In this note we will prove a (L , LP) -restriction theorem for certain submanifolds & of codimension / > 1 in an n— dimensional Euclidean space which arise as orbits under the action of a compact group K. As is well known such a result can in general only hold for 1 < p < j^y. We will show that for the submanifolds under consideration the inequality ^\\f(x)\\dKx)