Showing papers on "Fourier transform published in 1976"

Position, rotation, and scale invariant optical correlation

Abstract: A new optical transformation that combines geometrical coordinate transformations with the conventional optical Fourier transform is described. The resultant transformations are invariant to both scale and rotational changes in the input object or function. Extensions of these operations to optical pattern recognition and initial experimental demonstrations are also presented.

An eigenfunction expansion method for the analysis of exponential decay curves

Abstract: A method is developed for the analysis of data composed of random noise, plus an unknown constant ’’baseline,’’ plus a sum (or an integral over a continuous distribution) of exponential decay functions. It is based on the expansion of the solution of a Fredholm integral equation of the first kind in the eigenfunctions of the kernel. In contrast to the Fourier transform solution [Gardner et al., J. Chem. Phys. 31, 978 (1959)], the finite time range of the data is exactly accounted for, and no extrapolation or iteration is necessary. A computer program is available for the analysis of sums of exponentials. It is completely automatic in that the only input are the data (not necessarily in equal intervals of time); no potentially biased initial estimates of either the number or values of the amplitudes and decay constants are needed. These parameters and their standard deviations are decided with a linear hypothesis test corrected approximately for nonlinearity. Tests with simulated two‐, three‐, and four‐com...

Filter response of nonuniform almost-periodic structures

Abstract: The filter response of nonuniform, almost-periodic structures, such as corrugated optical waveguides, is investigated theoretically. The filter process, leading to reflection of a band of frequencies near the Bragg frequency, is treated as a contradirectional coupled-wave interaction and shown to obey a Riccati differential equation. The nonuniformity of the structure is represented by a tapering in the coupling strength (e.g., the depth of the corrugation) and by a chirp in the period of the structure. For small reflectivities, the filter response is a Fourier transform of the taper function. For large reflectivities, the Riccati equation was evaluated numerically and plots are given for the response of filters with linear and quadratic tapers and with linear and quadratic chirps.

New apodizing functions for Fourier spectrometry

Abstract: A new class of apodizing functions suitable for Fourier spectrometry (and similar applications) is introduced. From this class, three specific functions are discussed in detail, and the resulting instrumental line shapes are compared to numerous others proposed for the same purpose.

Stokes flow for a stokeslet between two parallel flat plates

Abstract: Velocity and pressure fields for Stokes flow due to a force singularity of arbitrary orientation and arbitrary distance between two parallel plates are found, using the image technique and a Fourier transform. Two alternative expressions for the solution, one in terms of infinite integrals and the other in terms of infinite series, are given. The infinite series solution is especially suitable for computation purposes being an exponentially decreasing series. From the series the “far field” behaviour is extracted. It is found that a force singularity parallel to the two planes has a far field behaviour of source and image for the parallel components (a two dimensional source doublet of height-dependent strength) whereas the normal component, and all fields due to a force singularity normal to the planes, die out exponentially. Velocity fields are compared with those of the one plane case. An estimate of the influence of the second wall and when its effect can be disregarded is obtained.

A laser diagnostic technique for the measurement of droplet and particle size distribution

Abstract: : In order to verify the various theoretical models of practical combustion systems, experimental measurements are urgently required. A number of laser instrumentation systems which do not disturb the flow are being devised to satisfy this need. This study describes a technique which has been developed for the measurement of droplet or solid particle size distribution. These measurements are required, for example, as the input to mathematical models of combustion which include evaporation. The technique is based on the Fraunhofer diffraction of a parallel beam of mono-chromatic light by the moving droplets. A Fourier transform lens is used to focus a stationary light pattern onto a multielement photo-detector to measure the diffracted light energy distribution. A mini-computer program translates the light energy distribution into the corresponding, unique, droplet size distribution. The droplets or particles are classified into 31 size groups spanning two decades of diameter, (e.g. 5 micrometers to 500 micrometers using a 300 mm focal length Fourier transform lens).

Phase retrieval from modulus data

Abstract: Phase retrieval implies extraction of the phase of a complex signal f from its modulus |f|. We give examples where an additional constraint is imposed: knowledge of the modulus of F, the Fourier transform of f. The retrieval is accomplished by computer processing of samples of |f| and |F|. The problems of noisy data and nonuniqueness are addressed.

A new principle for fast Fourier transformation

Abstract: An alternative form of the fast Fourier transform (FFT) is developed. The new algorithm has the peculiarity that none of the multiplying constants required are complex-most are pure imaginary. The advantages of the new form would, therefore, seem to be most pronounced in systems for which multiplication are most costly.

Preliminary empirical model for scaling Fourier amplitude spectra of strong ground accleration in terms of earthquake magnitude, source-to-station distance, and recording site conditions

Abstract: An empirical model for scaling Fourier Amplitude Spectra of strong earthquake ground acceleration in terms of magnitude, M, epicentral distance, R, and recording site conditions has been presented. The analysis based on this model implies that: It has been shown that the uncertainties which are associated with the forecasting of Fourier amplitude spectra in terms of magnitude, epicentral distance, site conditions, and component direction are considerable and lead to the range of spectral amplitudes which for an 80 per cent confidence interval exceed one order of magnitude. A model has been presented which empirically approximates the distribution of Fourier spectrum amplitudes and enables one to estimate the spectral shapes which are not exceeded by the presently available data more than 100 (1 - p) per cent of time where p represents the desired confidence level (0 < p <1).

Preliminary empirical model for scaling Fourier Amplitude Spectra of strong ground acceleration in terms of earthquake magnitude, source-to-station distance, and recording site conditions

Abstract: An empirical model for scaling Fourier Amplitude Spectra of strong earthquake ground acceleration in terms of magnitude, M, epicentral distance, R, and recording site conditions has been presented. The analysis based on this model implies that: 1.(a) the Fourier amplitude spectra of strong-motion accelerations are characterized by greater energy content and relatively larger amplitudes for long-period waves corresponding to larger magnitudes M, 2.(b) the shape of Fourier amplitude spectra does not vary appreciably for the distance range between about 10 and 100 km, and 3.(c) long-period spectral amplitudes (T > 1 sec) recorded on alluvium are on the average 2.5 times greater than amplitudes recorded on basement rocks, whereas short-period (T < 0.2 sec) spectral amplitudes tend to be larger on basement rocks. It has been shown that the uncertainties which are associated with the forecasting of Fourier amplitude spectra in terms of magnitude, epicentral distance, site conditions, and component direction are considerable and lead to the range of spectral amplitudes which for an 80 per cent confidence interval exceed one order of magnitude. A model has been presented which empirically approximates the distribution of Fourier spectrum amplitudes and enables one to estimate the spectral shapes which are not exceeded by the presently available data more than 100 (1 - p) per cent of time where p represents the desired confidence level (0 < p <1).

Depth estimation of magnetic sources by means of fourier amplitude spectra

Abstract: The Fourier transform of a square‐shaped section of a magnetic survey, digitized in a square grid, forms a rectangular matrix of coefficients which can be condensed to a series of average amplitudes dependent only on their frequency and no longer on the direction of the respective partial waves. These average amplitudes together represent a spectrum which–when plotted in a semilogarithmic coordinate system (log amplitude versus frequency)–often shows straight segments which decrease with increasing frequency. By continuing the given field downwards these straight segments become horizontal at a certain depth, the so‐called “white depth”. This white depth may be used as a first estimate for the depth of magnetic sources producing the respective part of the field. It is shown that the sources which correspond to such use of the white depth can be expected to be “randomly distributed with some positive autocorrelation”. As an example for such a depth estimation the interpretation of the aeromagnetic survey of NW‐Germany by a relief in 8–16 km depth is given. The relief divides the subsurface in an upper nonmagnetic layer and a lower layer with magnetization M= 2 Am−1.

Pulsed microwave Fourier transform spectrometer

Abstract: A pulsed microwave Fourier transform spectrograph is described in detail. The increase in sensitivity and resolution obtained in the time domain relative to standard spectroscopy in the frequency domain is described theoretically and experimentally. Fast switching microwave diodes with a large dynamic range for the generation of high‐power microwave pulses are essential for the operation of the spectrograph. A 512‐point analog–digital converter and averager with a repetition rate of up to 30 kHz is used for signal‐to‐noise improvement. The observed emission signals in the time domain are Fourier transformed to the frequency domain on‐line by a small laboratory computer. The spectra obtained for 13CH2O and CD2O show a marked improvement in signal‐to‐noise ratio and resolution. A 50‐MHz bandwidth can be covered by a single pulse train.

Fourier transforms with only real zeros

Abstract: The class of even, nonnegative, finite measures p on the real line such that for any b > 0 the Fourier transform of exp(bt2) dp(t) has only real zeros is completely determined. This result is then applied to the Riemann hypothesis. 1. Main results. The problem of determining whether a Fourier transform has only real zeros arises in two rather disparate areas of mathematics: number theory and mathematical physics. In number theory, the problem is intimately associated with the Riemann hypothesis [T, Chapter 10], while in mathematical physics it is closely connected with the Lee-Yang theorem of statistical mechanics and quantum field theory [SGj, [NI], [N3]; see Kac's remarks in [P, pp. 424-426] for a discussion of the historical connection between these two topics. The results of this paper developed out of the study of certain quantum field theoretic problems, but for pedagogical reasons, we present them in the context of the Riemann hypothesis. Following standard practice, we define the Riemann xi function as (1.1) _(z) = s(s 1)7T`s/2r(S/2) (s)./2; s = iz + 2 where t (s) is the Riemann zeta function. _ is the Fourier transform of the strictly positive, even function,

An integral equation solution to the scattering of electromagnetic radiation by dielectric spheroids and ellipsoids

Abstract: A new approach to the scattering of electromagnetic radiation by dielectric scatterers and application of it to the case of scattering by homogeneous spheroidal and ellipsoidal raindrops is presented. We transform the (singular) integral equation for the scattering into an integral equation for the Fourier transform of the internal field, which has a nonsingular kernel. This equation is solved by reducing it by quadrature into a set of algebraic equations. The scattering amplitude so obtained is shown to satisfy the Schwinger variational principle, and the method is thus both numerically stable and known to be convergent. We present sample calculations for spheres, for spheroids, and for ellipsoids.

Scale Invariant Optical Transform

Abstract: The scale invariance of the Mellin transform and its optical synthesis in real-time are discussed. An initial off-line demonstration of an optical Mellin transform is presented. A scale invariant correlation and a combined Fourier-Mellin transform that is both scale and shift invariant are discussed. Applications of these transforms in optical data processing and optical pattern recognition are emphasized.

Numerical Solutions of the Response of a Two-Dimensional Earth to AN Oscillating Magnetic Dipole Source

Abstract: A finite difference formulation is developed for computing the frequency domain electromagnetic fields due to a point source in the presence of two‐dimensional conductivity structures. Computing costs are minimized by reducing the full three‐dimensional problem to a series of two‐dimensional problems. This is accomplished by Fourier transforming the problem into the x-wavenumber (kx) domain; here the x-direction is parallel to the structural strike. In the kx domain, two coupled partial differential equations for H⁁x(kx,y,z) and E⁁x(kx,y,z) are obtained. These equations resemble those of two coupled transmission sheets. For a requisite number of kx values these equations are solved by the finite difference method on a rectangular grid on the y-z plane. Application of the inverse Fourier transform to the solutions thus obtained gives the electric and magnetic fields in the space domain. The formulation is general; complex two‐dimensional structures containing either magnetic or electric dipole sources can ...

Nitrogen-15 nuclear magnetic resonance spectrum of alumichrome. Detection by a double resonance Fourier transform technique.

Abstract: Alumichrome, the Al/sup 3 +/ analogue of ferrichrome, a ferric cyclohexapeptide, was enriched in /sup 15/N by growing the fungus Ustilago sphaerogena in a medium containing 99.5% /sup 15/N-ammonium acetate as the sole nitrogen source. The magnitude of the amide proton--nitrogen scalar coupling constant, obtained directly from the /sup 1/H NMR spectrum is correlated with the angular deviation from planarity at the peptide bond. The /sup 15/N amide resonanceees were measured indirectly by heteronuclear double resonance methods. The amide /sup 1/H NMR spectrum was obtained by Fourier transform at 220 MHz while the /sup 15/N resonances were irradiated selectively with low power near 22.3 MHz. The chemical shifts of the amide /sup 15/N resonances are sensitive to conformation and the responses to temperature and solvent changes provide excellent criteria for determining the spatial configuration of each -NH--CO-group within the peptide structure. An extension of the Fourier transform double resonance method, Fourier internuclear difference spectroscopy (FINDS), was tested successfully on the amide region of the unenriched peptide thus permitting detection of /sup 15/N resonances at natural abundance. The results suggest a large potential for the method.