Showing papers on "Fourier transform published in 1981"

The importance of phase in signals

Abstract: In the Fourier representation of signals, spectral magnitude and phase tend to play different roles and in some situations many of the important features of a signal are preserved if only the phase is retained. Furthermore, under a variety of conditions, such as when a signal is of finite length, phase information alone is sufficient to completely reconstruct a signal to within a scale factor. In this paper, we review and discuss these observations and results in a number of different contexts and applications. Specifically, the intelligibility of phase-only reconstruction for images, speech, and crystallographic structures are illustrated. Several approaches to justifying the relative importance of phase through statistical arguments are presented, along with a number of informal arguments suggesting reasons for the importance of phase. Specific conditions under which a sequence can be exactly reconstructed from phase are reviewed, both for one-dimensional and multi-dimensional sequences, and algorithms for both approximate and exact reconstruction of signals from phase information are presented. A number of applications of the observations and results in this paper are suggested.

Computer Simulation Using Particles

Abstract: Computer experiments using particle models A one-dimensional plasma model The simulation program Time integration schemes The particle-mesh force calculation The solution of field equations Collisionless particle models Particle-particle/particle-mesh algorithms Plasma simulation Semiconductor device simulation Astrophysics Solids, liquids and phase changes Fourier transforms Fourier series and finite Fourier transforms Bibliography Index

Fabry–Perot cavity pulsed Fourier transform microwave spectrometer with a pulsed nozzle particle source

Abstract: We describe the design, construction, and operation of a new type of microwave spectrograph which allows the measurement of the resonant transitions of transient or otherwise short‐lived species. The spectrograph is composed of three parts: a Fabry–Perot cavity, a pulsed supersonic nozzle as a source for the sample, and the pulsed microwave Fourier transform method. Following a detailed discussion of the three above components in the spectrograph, the operation of the entire system is described and several examples are given.

Fourier Self-Deconvolution: A Method for Resolving Intrinsically Overlapped Bands

Abstract: The general theory of Fourier self-deconvolution, i.e., spectral deconvolution using Fourier transforms and the intrinsic line-shape, is developed. The method provides a way of computationally resolving overlapped lines that can not be instrumentally resolved due to their intrinsic linewidth. Examples of the application of the technique to synthetic and experimental infrared spectra are presented, and potential applications are discussed. It is shown that lines in spectra having moderate signal/noise ratios (∼1000) can readily be reduced in width by a factor of 3. The method is applicable to a variety of spectroscopic techniques.

An Orthogonally Multiplexed QAM System Using the Discrete Fourier Transform

Abstract: An orthogonally multiplexed QAM (O-QAM) system is a multichannel system with a baud rate spacing between adjacent carrier frequencies; this property is desirable to digitally implement the system using the discrete Fourier transformation (DFT). This paper provides a novel digital signal processing method based on an N /2-point DFT processing in the O-QAM system. A complexity comparison between a digital O-QAM system and a digital singlechannel QAM system shows that the digital O-QAM system using the new method is more economical than the digitally implemented conventional single-channel data transmission system.

Analysis of dispersive waves by wave field transformation

Abstract: The dispersive waves in a common‐shot wave field can be transformed into images of the dispersion curves of each mode in the data. The procedure consists of two linear transformations: a slant stack of the data produces a wave field in the phase slowness‐time intercept (p — τ) plane in which phase velocities are separated. The spectral peak of the one‐dimensional (1-D) Fourier transform of the p — τ wave field then gives the frequency associated with each phase velocity. Thus, the data wave field is linearly transformed from the time‐distance domain into the slowness‐frequency (p — ω) domain, where dispersion curves are imaged. All the data are present throughout the transformations. Dispersion curves for the mode overtones as well as the fundamental are directly observed in the transformed wave field. In the p — ω domain, each mode is separated from the others even when its presence is not visually detectable in the untransformed data. The resolution achieved in the result is indicated in the p — ω wave ...

Beam-propagation method: analysis and assessment

Abstract: A method for the calculation of the propagation of a light beam through an inhomogeneous medium is presented. A theoretical analysis of this beam-propagation method is given, and a set of conditions necessary for the accurate application of the method is derived. The method is illustrated by the study of a number of integrated-optic structures, such as thin-film waveguides and gratings.

Two-dimensional rotational spin-echo nuclear magnetic resonance in solids: correlation of chemical shift and dipolar interactions

Abstract: Two-dimensional NMR techniques which sep. the chem. shift and heteronuclear dipolar interactions were applied to samples spinning at the magic angle. Because of the inhomogeneous nature of the 2 interactions, rotational echoes were obsd. in the time domain of each dimension. The corresponding Fourier transforms yield rotational sideband spectra which provide information on the principal values and relative orientations of the shift and dipolar tensors, and from the latter, internuclear distances may be calcd. The techniques therefore provide a means for obtaining structural data, for example, 13C-1H and 15N-1H distances, in powder samples. [on SciFinder (R)]

Numerical Methods Based on Whittaker Cardinal, or Sinc Functions

Abstract: This paper summarizes the results known to date for using sine functions composed with other functions as bases for approximations in numerical analysis. Described in this paper are methods of interpolation and approximation of functions and their derivatives, quadrature, the approximate evaluation of transforms (Hilbert, Fourier, Laplace, Hankel and Mellin) and the approximate solution of differential and integral equations. The methods have many advantages over classical methods which use polynomials as bases. In addition, all of the methods converge at an optimal rate, if singularities on the boundary of approximation are ignored.

Reconstruction of a sparse spike train from a portion of its spectrum and application to high-resolution deconvolution

Abstract: An algorithm is proposed for the reconstruction of a sparse spike train from an incomplete set of its Fourier components. It is shown that as little as 20–25 percent of the Fourier spectrum is sufficient in practice for a high‐quality reconstruction. The method employs linear programming to minimize the L1‐norm of the output, because minimization of this norm favors solutions with isolated spikes. Given a wavelet, this technique can be used to perform deconvolution of noisy seismograms when the desired output is a sparse spike series. Relative reliability of the data is assessed in the frequency domain, and only the reliable spectral data are included in the calculation of the spike series. Equations for the unknown spike amplitudes are solved to an accuracy compatible with the uncertainties in the reliable data. In examples with 10 percent random noise, the output is superior to that obtained using conventional least‐squares techniques.

Noise in Fourier self-deconvolution.

Abstract: A general formula for computing changes in the signal-to-noise ratio of a spectrum resulting from the Fourier self-deconvolution procedure is derived. Self-deconvolution reduces the intrinsic halfwidths of lines by a factor K, which is in practice limited by the noise in the spectrum. With the help of the derived formula, the rate of decrease in the SNR as a function of K for eight different smoothing (apodization) functions is studied. With high K values there are significant differences in the SNR as a result of the use of different smoothing functions. With K = 4 difference of more than 1 order of magnitude between two extreme cases is demonstrated, and with K = 5 a difference of almost 2 orders of magnitude in the SNR is predicted.

Criteria for automatic x‐ray absorption fine structure background removal

Abstract: Criteria for terminating smoothing to remove a cubic spline background from the x‐ray absorption coefficient are stated in terms of three parameters obtained from the k3 weighted Fourier transform of the resulting x‐ray absorption fine structure data. The three parameters are HR, the average value of the transform magnitude between 0 and 0.25 A, HM, the maximum value in the transform magnitude between 1 and 5 A, and HN, the average value of the transform magnitude between 9 and 10 A. The termination criteria are HR−HN ⩾0.05HM; or if HN ≳0.1HM, then HR ⩾0.1HM. The incorporation of the criteria into a computer program to facilitate automatic background removal is discussed. Examples of application of the technique to copper, β‐PtO2, and ferritin samples are presented.

Linear systems, Fourier transforms, and optics

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Direct Fourier reconstruction in computer tomography

Abstract: Direct Fourier transform (FT) reconstruction of images in computerized tomography (CT) is not widely used because of the difficulty of precisely interpolating from polar to Cartesian samples. In this paper, an exact interpolation scheme is proposed which, in practice, can be approached with arbitrary accuracy using well-conditioned algorithms. Several features of the direct FT method are discussed. A method that allows angular band limiting of the data before processing -to avoid angular aliasing artifacts in the reconstructed image-is discussed and experimentally verified. The experimental results demonstrate the feasibility of direct FT reconstruction of CT data.

Pressure fluctuations in a fluidized bed

Abstract: An on-line statistical study of the pressure fluctuations in fluidized beds was conducted by using pressure transducers, a correlation and probability analyzer and a Fourier transform analyzer. The causes of the fluctuations were explored, and the effects of the gas velocity, bed height, particle size and distributor design on the major frequency and amplitude of the fluctuations were investigated. The results indicate that the motion of bubbles appears to be the major cause of the pressure fluctuations in the upper portion of a fluidized bed. In the lower portion, the combined effects of the formation of large bubbles in the middle portion of the bed, the formation of small bubbles near the distributor, and the jet flow immediately above the distributor appear to be the major causes of pressure fluctuations.

Modeling of the acoustic wave equation with transform methods

Abstract: Numerical methods are described for the simulation of wave phenomena with application to the modeling of seismic data. Two separate topics are studied. The first deals with the solution of the acoustic wave equation. The second topic treats wave phenomena whose direction of propagation is restricted within ±90 degrees from a given axis. In the numerical methods developed here, the wave field is advanced in time by using standard time differencing schemes. On the other hand, expressions including space derivative terms are computed by Fourier transform methods. This approach to computing derivatives minimizes truncation errors. Another benefit of transform methods becomes evident when attempting to restrict propagation to upward moving waves, e.g., to avoid multiple reflections. Constraints imposed on the direction of the wave propagation are accomplished most precisely in the wavenumber domain. The error analysis of the algorithms shows that truncation errors are due mainly to time discretization. Such er...

Computer‐generated pulse signal applied for sound measurement

Abstract: A computer‐generated pulse signal for sound measurement is discussed. A pulse signal whose power spectrum is flat is generated by inverse Fourier transformation. The generation of a time‐stretched pulse and its compression method are also discussed. Computer‐controlled measurements enable time averaging and the elimination of reflected sound is made in the computer memory by the operator’s instruction monitoring acquired waveform on CRT.

Iterative techniques for minimum phase signal reconstruction from phase or magnitude

Abstract: In this paper, we develop iterative algorithms for reconstructing a minimum phase sequence from the phase or magnitude of its Fourier transform. These iterative solutions involve repeatedly imposing a causality constraint in the time domain and incorporating the known phase or magnitude function in the frequency domain. This approach is the basis of a new means of computing the Hilbert transform of the log-magnitude or phase of the Fourier transform of a minimum phase sequence which does not require phase unwrapping. Finally, we discuss the potential use of this iterative computation in determining samples of the unwrapped phase of a mixed phase sequence.

Forward and backward projection of acoustic fields using FFT methods

Abstract: The forward and backward propagation of harmonic acoustic fields using Fourier transform methods is presented. In particular, the forward propagation of a velocity distribution to obtain a pressure field and the backward propagation of a pressure field to obtain a velocity distribution are addressed. Numerical examples are presented to illustrate the nearfield behavior of the pressure field from complex planar vibrators, e.g,—an ultrasonic transducer or plate, with nonuniform velocity distributions. The numerical results, which were obtained via the use of FFT algorithms, are presented for vibrators which are operating above and below coincidence. These results illustrate the acoustic nearfield as a function of distance from the vibrator. Numerical results are also presented to illustrate the backward projection method. The pressure field of a 3×3 focused array is back projected to obtain the velocity distribution for several cases of interest. These results illustrate the utility of the transform method and the effect of spatial windows or filters in its implementation using FFT algorithms.

The Phase Retrieval Problem

Abstract: A method for solving the phase retrieval problem is proposed. The method consists of measuring the modulus of the Fourier transform of the unknown function and the modulus of the Fourier transform of the product of the unknown function and an exponential. From these two measurements, the location of the complex zeros of the analytic continuation of the Fourier transform of the unknown function may easily and quickly be deduced and the unknown function constructed.

An Investigation of Computerized Tomography by Direct Fourier Inversion and Optimum Interpolation

Abstract: In this paper an exact interpolation formula forms the basis for reconstructing computerized tomographic (CT) imagery by direct Fourier methods. Practical variations of exact interpolation are compared with other interpolation methods (i.e., nearest neighbor, etc.) and are shown to yield superior imagery. Images produced by the direct Fourier approach using near-exact interpolation are shown to be equal in quality with those produced by filtered convolution backprojection (FCBP). Moreover, the direct Fourier approach computes an image in O(N2 log N) time versus O(N3) for the FCBP method.

Tomographic image reconstruction from limited projections using iterative revisions in image and transform spaces

Abstract: An iterative technique is proposed for improving the quality of reconstructions from projections when the number of projections is small or the angular range of projections is limited. The technique consists of transforming repeatedly between image and transform spaces and applying a priori object information at each iteration. The approach is a generalization of the Gerchberg-Papoulis algorithm, a technique for extrapolating in the Fourier domain by imposing a space-limiting constraint on the object in the spatial domain. A priori object data that may be applied, in addition to truncating the image beyond the known boundaries of the object, include limiting the maximum range of variation of the physical parameter being imaged. The results of computer simulations show clearly how the process of forcing the image to conform to a priori object data reduces artifacts arising from limited data available in the Fourier domain.