Numerical Fourier Transformation Research Articles

On the deficiencies of the anharmonic one-particle potential model

The temperature factors of Cd and Zn for h00 reflexions calculated by numerical Fourier transformation are compared with the temperature factors determined with the help of series expansion of the anharmonic term of the probability density function (p.d.f.). The anharmonic parameters used have been derived by least-squares fit of measured Bragg intensities in the framework of the anharmonic one-particle potential (OPP) model. For Cd a deviation of up to 7% is found for the results obtained for the symmetric part and up to 50% for the antisymmetric part of the temperature factor. It is shown that numerical Fourier transformation of the p.d.f. using the anharmonic parameters given in the literature for Zn, is not always possible, because the p.d.f. is divergent for some of these parameters.

On the role of inertial effects and dipole-dipole coupling in the theory of the Debye and far-infrared absorption of polar fluids. II. Comparison with experimental observations

A general two-friction itinerant-oscillator model is given for molecular motion in dipolar fluids. The polarizability and relevant time-correlation functions are calculated by numerical methods. It is shown how the results of an earlier paper are obtained when the two frictions per unit inertia are equal. The model is compared with recent experimental measurements on hexanone-2 and acetonitrile obtained by using a new quasi-optical technique for frequencies intermediate between the MW and FIR spectral regions. This technique is described briefly; in particular, for a reasonable fitting of the theory to experiment the moment of inertia of the outer cage of molecules isalwaysgreater than that of the inner dipole in agreement with the original concept, namely a cage of dipoles surrounding a typical dipole. This is at variance with the earlier concept of the model where the friction on the encaged dipole is zero. There, for a good fit to experiment, the moment of inertia of the cage is always less than that of the dipole. This suggests that the two friction version should be used for such comparison. An analytical formula (analogous to the Rocard equation) for the polarizability derived earlier is shown to agree well with the result of numerical Fourier transformation of the response function. A distinct harmonic peak structure is noted in the FIR region of the spectrum when friction does not act on the inner dipole. The structure is damped out as this friction is increased. Thus only the first harmonic peak is significant when the frictions per unit inertia are approximately equal. Measurements in the frequency region (the intermediate region) lying between the microwave and the FIR for hexanone-2 in dilute solutions of cyclohexane are reported. These show a significant excess absorption over that predicted by the model in this region. It is suggested that the excess absorption arises from jumping of dipoles from well to well of the cosine potential of dipole-dipole interaction indicating that the effects of finite potential well depth must be incorporated in the model to explain the intermediate frequency absorption.

High resolution Fourier transform Raman spectroscopy with ultrashort laser pulses

Coherent Raman scattering of delayed probing pulses after stimulated Raman excitation of molecular transitions is applied for a novel spectroscopy. Measuring the signal transient over long delay time intervals ≳ 10 -8 s numerical Fourier transformation yields precise spectroscopic information with a resolution ≲ 10 -3 cm -1. Experimental data are presented for the Q-band of the ν 1 vibration of CH 4 in the supersonic expansion. The potential of our Fourier transform Raman spectroscopy is demonstrated resolving the tensor splitting of the J = 2 transitions.

The Autocorrelation Function for Multi-h Coded Signals

The autocorrelation function of a binary multi- <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</tex> coded signal is determined analytically. This is done by considering the probability density of the "phase change variable" and then deriving the characteristic function. From the autocorrelation, the power spectrum can be found by an analytical or numerical Fourier transformation. Two examples are given.

A spectral method of solving Boltzmann's kinetic equation numerically

A spectral algorithm, conservative with respect to mass flow, is proposed for solving the homogeneous relaxation problem in the method of splitting into physical processes for Boltzmann's kinetic equation with Maxwell interaction potential. It is shown that, by realizing the algorithm by a numerical fast Fourier transformation procedure, the amount of calculation can be greatly reduced. The computational scope of the method is studies for the Cauchy problem for the non-linear kinetic equation with velocity space R 1 .

Dielectric relaxation measurements in the ultralow frequency region

Refinements in the dc transient current method are described for precision measurements of dielectric permittivity from 50 mHz down to 1 μHz. Relaxation increment, relaxation time, and distribution of the relaxation times for a test sample of poly(vinyl acetate) have been measured with satisfactory accuracy as judged by comparison with results from steady-state measurements. Convenient procedures for numerical Fourier transformation of the observed current are described to easily obtain the complex permittivity.

Frequency-dependent damping in structural vibration analysis by use of complex series expansion of transfer functions and numerical Fourier transformation

The response of damped linear finite systems (discrete and continuous) to harmonic, stationary random, and transient excitations is studied. The assumed damping may be light or heavy, viscous and/or hysteretic (the latter being frequency-independent or frequency-dependent), and proportionally or non-proportionally distributed over the structure. Closed-form analytic transfer functions are derived for beam systems. In order to rationalize subsequent numerical calculations, the transfer functions are approximated by using truncated series. Complex eigenfrequencies of the structure and complex residues of the actual transfer function are used. Special interest is paid to the mathematical modelling of experimentally measured damping. Causality requirements are considered. Numerical examples are given. An efficient method (by which aliasing is eliminated) for numerical Fourier transformation has been developed and applied.

Application of numerical Fourier transformation on measurements made on board rotating spacecrafts

Signals obtained on board rocket payloads or satellites are often “spin modulated”. The relevant information is then found within the envelope and the phase of the signal. The most efficient method of extracting this information is the numerical Fourier analysis.

Mathematical Modelling of the Modulation Transfer Function of X-ray Film/Screen Systems

An exponential equation can be used to characterize the line spread function (LSF) of the X-ray film/screen combination. The analytic Fourier transform of the LSF yields the modulation transfer function (MTF). Predictions using the theoretical model show good agreement with results btained by numerical Fourier transformation. A single figure of merit parameter may be assigned to the film/screen system. This parameter determines the system spatial frequency response.

Fourier analysis of spherically averaged momentum densities for some gaseous molecules

The spherically averaged autocorrelation function, B( r), of the position-space wavefunction, ψ( r ), is calculated by numerical Fourier transformation from spherically averaged momentum densities, ϱ( p), obtained from either theoretical wavefunctions or (e,2e) electron-impact ionization experiments. Inspection of B( r) for the π molecular orbitals of C 4 H 6 establishes that autocorrelation function differences, Δ B( r), can be qualitatively related to bond lengths and numbers of bonding interactions. Differences between B( r) functions obtained from different approximate wavefunctions for a given orbital can be qualitatively understood in terms of wavefunction difference,Δψ( r ), maps for these orbitals. Comparison of the B( r) function for the 1 a u orbital of C 4H 6 obtained from (e,2e) momentum densities with that obtained from an ab initio SCF MO wavefunction shows differences consistent with expected correlation effects. Thus, B( r) appears to be a useful quantity for relating spherically averaged momentum distributions to position-space wavefunction differences.

A Variable-Depth Green's Function for Shelf Edge Tides

Abstract The Green's function for a semi-infinite ocean with depth a function of distance from the boundary is developed numerically for the M2 frequency and with Coriolis frequency and depth profile appropriate to the continental slope off the Gulf of Maine. This involves numerical integration of the linearized shallow water equations for all longshore wavenumbers, followed by numerical Fourier transformation. This variable-depth Green's function is approximately equal to Buchwald's (1971) constant-depth Green's function for distances along the boundary greater than the width of the slope, and at very short range tends to limiting values which can be approximated analytically. The Green's function, when combined with currents from Greenberg's (1979) numerical model of the Bay of Fundy and Gulf of Maine, is used to explain substantial observed variations in M2 amplitude and phase along the edge of the shelf off the Gulf of Maine; the variable-depth Green's function produces significantly better results th...

Emission noise spectra from premixed sheathed C 2H 2-air flames and the detection limit

Noise spectra in the frequency range 0.02 Hz < f < 20 kHz are reported for the thermal emission of the Na- D doublet and the background from premixed, sheathed, quasi-laminar C 2H 2-air flames at 1 atm under various experimental conditions. Below 1.5 Hz noise spectra were obtained by numerical Fourier transformation of a digitized noise signal; above 1.5 Hz a frequency-analyzer and a lock-in analyzer were used. The measuring procedure is theoretically explained and its experimental testing is described. The role of statistical cross-correlation is emphasized. The flame-emission noise proper was obtained by subtracting the shot-noise power, measured with the aid of an incandescent lamp, from the total noise power. The shot noise level exceeded the theoretical value as a consequence of the reduction in the number of amplifying stages of the photomultiplier used. Below 1 Hz the noise spectra showed roughly a 1/ f γ-dependence with γ values above and below unity for the Na emission and the background, respectively. Between 5 and 50 Hz pronounced peaked structures were observed for both emissions. At higher frequencies a steep roll-off with γ > 2 was seen. A dip in the noise spectrum was not found; but the flame-emission noise was already drowned in the shot noise above 500 Hz as a result of the low photocathode currents used. Some tentative explanations of the noise observations are given. A standardized figure-of-merit for the effect of flame-emission noise in FES, FAAS and FAFS is introduced and conclusions are drawn for the optimum choice of modulation frequency. The general theory of the analytical detection limit for paired readings, given in a previous paper, is worked out for the present case. Allowance is made for the presence of a high-pass filter (blocking any drifting d.c. component) and for the deviation of the background noise spectrum from a strict 1/ f-dependence. The application of the theory is illustrated by calculating the optimized detection limit for Na in FES with the simple flame equipment described in this paper.

Impulse wave diffraction by rigid wedges and plates

The impulse wave solution of point source radiation which is diffracted by an infinite rigid corner or wedge, [M. A. Biot and I. Tolstoy, J. Acoust. Soc. Am. 29, 381–391 (1957)], describes a closed form time‐domain solution to an old problem that is generally studied in the frequency domain. The theory is tested by an experiment using models of wedges and plates in an anechoic chamber, and the source regions of diffraction are identified and compared with theoretical predictions. In addition, numerical Fourier transformation of the dominant part of the solution provides the transmission loss as a function of frequency without further assumptions or approximations. [Work supported by the Office of Naval Research.]

Numerical fourier transformation: Singularity expansions and product integration

Numerical fourier transformation: Singularity expansions and product integration

Phase refinement by a Fourier transformation of difference electron density peaks

A method to refine phases by a numerical Fourier transformation of difference electron density peaks is described. It was tested with an artificial two-dimensional structure of NaNO2. Starting from the phases of a spherical-free-atom model, those of the hypothetical true structure of aspherical atoms were reproduced fairly well after five cycles. The scale factor of the structure factors was also improved by the refinement.

Transmission of a Transient electromagnetic plane wave through a grating of circular cylinders

An analysis of the transmission of a transient electromagnetic plane wave through a grating formed by an infinite array of circular cylinders is presented. Emphasis is placed on the transient shielding characteristics of the grating for various cylinder spacings and angles of incidence. The transient results are obtained by first obtaining analytical experssions for the fields in the frequency domain and then performing a numerical inverse Fourier transformation. Some physical interpretations of the shielding effectiveness of the grating are given.

A Numerical Fourier Transform Technique and Its Application to Acoustic-Surface-Wave Bandpass Filter Synthesis and Design

A Numerical Fourier Transform Technique and Its Application to Acoustic-Surface-Wave Bandpass Filter Synthesis and Design

Radial Electron–Electron Distributions and the Coulomb Hole for Be

Individual radial electron–electron distribution functions, Pi(r12)'s, are defined in terms of the eigen-functions (the natural geminals, NG's) of the spin-free 2-matrix, P(2)(r1, r2 | r1′, r2′). The total radial electron–electron distribution function, P0(r12), is defined in terms of P(2). It is demonstrated that the structure of individual two-particle eigenfunctions can easily be determined from the corresponding Pi(r12). Distributions are obtained for the S1 ground state of the Be atom by numerical Fourier transformation of x-ray intensity values. The natural spin geminal (NSG) analysis by Barnett and Shull of Weiss' configuration interaction (CI) wavefunction is used for the determination of correlated Pi(r12)'s, while the 2-matrix analysis of Clementi's Restricted Hartree–Fock (RHF) function is employed in the evaluation of independent particle model (IPM) Pi(r12)'s. The correlated and RHF P0(r12)'s are compared with that obtained from a two-configuration wavefunction of Watson, which takes into account the near degeneracy of 2s and 2p orbitals. By subtracting the RHF from the Weiss P0(r12), a description of the Coulomb hole function, Δ(r12), is obtained for the S1 Be ground state. A measure of the Fermi correlation may be obtained with Pi(r12)'s constructed from those S1 and S3 RHF geminals which involve the spatially symmetric or the spatially antisymmetric combination of 1s and 2s orbitals.

Intermediate scattering functions obtained by fast fourier transformation of cold neutron time-of-flight spectra

The influence of the finite resolution of the spectrometer in cold neutron time-of-flight spectra is studied. It will be shown how by direct numerical fast Fourier transformation of these spectra, intermediate scattering functions can be obtained from which the resolution is removed. Under certain conditions depending on the scattering law of the sample and the resolution function it is possible to calculate reliable double-differential scattering cross sections. A short description of the Fortran program used in the calculations is given, together with some applications of the method.

Closure to “Numerical Fourier Transform in Random Vibration”

Closure to “Numerical Fourier Transform in Random Vibration”