Showing papers on "Gaussian published in 1969"

Computer Experiments With Fractional Gaussian Noises: Part 1, Averages and Variances

Abstract: Fractional Gaussian noises are a family of random processes such that the interdependence between values of the process at instants of time very distant from each other is small but nonnegligible. It has been shown by mathematical analysis that such interdependence has precisely the intensity required for a good mathematical model of long run hydrological and geophysical records. This analysis will now be illustrated, extended, and made practically usable, with the help of computer simulations. In this Part, we shall stress the shape of the sample functions and the relations between past and future averages.

Computer Experiments with Fractional Gaussian Noises: Part 2, Rescaled Ranges and Spectra

Abstract: Continuing the preceding paper, we shall report on computer experiments concerning the ‘rescaled range’ and the spectrum of fractional Gaussian noise

Computer Experiments with Fractional Gaussian Noises: Part 3, Mathematical Appendix

Abstract: The present ‘Appendix’ is devoted to mathematical considerations designed to fill in the innumerable logical gaps left in the preceding paper. Even though most proofs will be merely sketched or omitted, the notation remains heavy, and some readers may wish to skip to formulas 1 and 2, which define ‘Type 1 functions.’ Most of what follows is closely patterned after Mandelbrot and Van Ness [1968], which contains additional mathematical details and references to the few earlier articles, where fractional Gaussian noises were fleetingly considered as purely mathematical entities.

Analysis of nonlinear stochastic systems by means of the Fokker–Planck equation†

Abstract: Nonlinear systems disturbed by Gaussian white noises (or by signals obtained from Gaussian white noises) can sometimes be analysed by setting up and solving the Fokker–Planck equation for the probability density in state space. In the present paper a simplified derivation of the Fokker–Planck equation is given. The uniqueness of the steady-state solution is discussed. Steady-state solutions are obtained for certain classes of system. These solutions correspond to or slightly generalize the Maxwell–Boltzmann distribution which is well known in classical statistical mechanics

Optimized Gaussian Basis SCF Wavefunctions for First‐Row Atoms

Abstract: Optimum atomic Gaussian‐type‐function bases for self‐consistent‐field calculations are reported for B, C, N, O, and F, with from two to eight s functions and from one to four p functions. Orbital energies and coefficients are presented for the (3s1p), (5s2p), and (7s3p) bases for each atom, and the functions are shown to yield lower atomic energies than previously proposed Gaussian bases of the same size.

Small Gaussian Expansions of Atomic Orbitals

Abstract: Expansions of Clementi STO SCF AO's for some first‐row atoms with GTO's have been obtained by the method of least squares. Expansion lengths vary from two to five Gaussians for each AO. In addition, Gaussian sets with constraints that the exponential parameters be shared for 1s and 2s and 2s and 2p orbital expansions are reported. For all cases a full matrix least‐squares procedure was employed whereby the expansion coefficients and exponential parameters were simultaneously varied. All orbital expansions were constrained to a normalization condition. These expansions are useful for computing x‐ray scattering factors and can be used in quantum‐chemical studies.

Near Fields of Truncated-Gaussian Apertures*

Abstract: Near-field irradiance patterns have been calculated for plane-wave gaussian beams truncated by circular apertures. The Kirchhoff diffraction integral was integrated numerically, using the small-angle Fresnel approximation. The effect of truncation was important when the aperture radius was less than twice the spot size of the gaussian beam.

The minimum of a stationary Markov process superimposed on a U-shaped trend

Abstract: 1. This paper was motivated by some questions of Barnett and Lewis (1967) concerning extreme winter temperatures. The temperature during the winter can be hopefully regarded as generated by a stationary Gaussian process superimposed on a locally U-shaped trend. One is interested in statistical properties of the minimum of sample paths from such a process, and of their excursions below a given level. Equivalently one can consider paths from a stationary process crossing a curved boundary of the same form. Problems of this type are discussed by Cramer and Leadbetter (1967), extensively in the trend-free case and in less detail when a trend is present, following the method initiated by Rice (1945). While results on moments are easy to obtain, explicit results for the actual probability distributions are not usually available. However, in the important case when the level of values of interest is far below the mean, the asymptotic independence of up-crossing times makes it possible to derive simple approximate distributions. (See Cramer and Leadbetter (1967) page 256, Keilson (1966).) There is a dearth of particular examples of processes and trends for which the distributions of interest are known exactly. Such examples could give useful experience of the form of distribution to be expected in typical cases, and could serve as material on which to test out approximate methods. The object of the present paper is to provide an example of this kind. One process for which exact results are available in the trend-free case is the Ornstein-Uhlenbeck process, i.e., the stationary Gaussian Markov process X(t) generated by

Stochastic differential games with constrained state estimators

Abstract: Attention is given to stochastic differential games in which the two controllers have available only noise-corrupted output measurements. Consideration is restricted to the case in which the system is linear, the cost functional quadratic, and the noises corrupting the output measurements are independent, white, and Gaussian. A solution to this problem is presented under the constraint that each controller is limited to a linear dynamic system of fixed dimension for the generation of his estimate of the system state. The optimal controls are shown to satisfy a separation theorem, the optimal estimators are shown to be closely related to Kalman filters, and the various terms in the optimal cost are shown to be readily assignable to the appropriate contributing sources.

Diffraction of Gaussian Laser Beams by a Semi-Infinite Plane

Abstract: Theoretical and experimental aspects of the diffraction of gaussian laser beams by the straight edge bounding an opaque plane are investigated. Theoretical analysis is based upon the Kirchhoff scalar wave theory in the Fresnel limit, assuming an incident electromagnetic field having spatial amplitude and phase variation appropriate to a fundamental-mode gaussian beam. Experimental observation consisting of irradiance as a function of position is in good agreement with this theory. Both theoretical and experimental results are found to depend strongly on gaussian-beam parameters.

Gaussian Lobe Function Expansions of Hartree–Fock Solutions for the Second‐Row Atoms

Abstract: Gaussian expansions of ground‐state Hartree–Fock wavefunctions for the atoms sodium through argon are reported. The functions presented are designed primarily for use in molecular computations and consist only of functions of the form exp(− ar2); angular properties of real p orbitals are achieved by displacement of basis functions along coordinate axes. Variable exponents and coefficients are determined by maximization of the overlap of Gaussian orbitals with exponential Hartree–Fock expansions followed by self‐consistent‐field minimization of atomic energies.

Gaussian-Orbital Basis Sets for the First-Row Transition-Metal Atoms

Abstract: Contracted Gaussian-type function basis sets have been produced for the first-row transition-metal atoms Sc through Cu for use in molecular SCF calculations. The basis functions consist of 15 s-type, eight p-type, and five d-type Gaussian primitives contracted to a four s-type, two p-type, and one d-type basis set and are exponent and coefficient optimized after contraction. It is shown that this basis set is of single-zeta quality for the core functions and of double-zeta quality for the valence-electron functions. Relaxation of the degree of contraction results in a significant lowering of the total energy.

Gaussian Quadratures for the Integrals ∞ 0 exp(-x 2 )f(x)dx and b 0 exp(-x 2 )f(x)dx

Abstract: Gaussian quadratures are developed for the evaluation of the integrals given in the title. The weights and abscissae for the semi-infinite integral are given for two through fifteen points with fifteen places. For b = 1, the weights and abscissae are given for two through ten points with fifteen places.

Forced Model Equation for Turbulence

Abstract: An artificial random force is introduced into Burgers' model equation for turbulence. This forced model equation is solved numerically as an initial‐value problem. Both the driving force and initial velocity field are assumed Gaussian and are generated by a white noise process. Many statistical properties of this model for turbulence are studied. By adjusting the external force, the turbulence can reach an equilibrium state. The velocity correlation function and energy spectrum are calculated for the equilibrium turbulence. It is found that the energy spectrum falls off as the inverse second power of the wavenumber. The velocity correlation function is similar to the result obtained in real turbulence experiments. With Gaussian random driving force and Gaussian initial velocity field, it is found that the velocity field remains very nearly Gaussian by comparing the fourth‐order velocity correlation with the quasinormal assumption. Although the process remains very nearly Gaussian, it is found that the projection of the process on the initial white noise process becomes smaller and smaller. This is to be expected, since the dynamic system is escaping from the original random base.

An algorithm for the decomposition of a distribution into Gaussian components.

Abstract: An algorithm for the decomposition of a mixture of Gaussian components is given, partially in Algol-60 language, and experiences with its use on a digital computer are described. The algorithm contains three steps: (1) finding the mean of each component by the aid of a Fourier transform of the given density function and the method of decreasing the standard deviations (Medgyessi [1961]); (2) determining the standard deviations and frequencies of the components using the continued fraction approximation of the error function; and (3) testing results by the Kolmogorov-Smirnov method. Illustrative examples are given.

Floating Spherical Gaussian Orbital Model of Molecular Structure. VI. Double-Gaussian Modification

Abstract: The floating spherical gaussian orbital model is modified to improve molecular energies and geometry by using a linear combination of two concentric spherical Gaussian orbitals (a double Gaussian) as the localized orbital. The double‐Gaussian model was applied to hydrogen, the first‐row atom hydrides, and a series of hydrocarbons. Bond lengths are predicted within 5.8% and bond angles within 6.6% for all the cases investigated. Molecular energies are generally 96% of Hartree–Fock values, and bond angles are improved.

Comparison of linearly and quadratically modified spectral estimates of Gaussian signals

Abstract: Although the use of quadratically modified periodograms as spectral estimators for the Gaussian case is well established in the literature, the linearly modified form has grown in popular use even when the data involved can be reasonably presumed to be Gaussian. The mean and variance of the two forms of estimators are calculated as well as the equivalent duration of the data window. It is shown that variance is always greater or the equivalent duration of the sample data length is always less for the linear case when the results are normalized with respect to the effective bandwidth. In addition, a generalized form of the Hanning window is considered.

A non-gaussian turbulence simulation

Abstract: : A comparison of the statistical properties of low altitude atmospheric turbulence and the characteristics of presently used simulation techniques shows that these techniques do not satisfactorily account for the non-Gaussian nature of turbulence. A non-Gaussian turbulence simulation, intended to be used in conjunction with piloted flight simulators, is developed. The simulation produces three simultaneous random processes which represent the three orthogonal gust components. The probability distribution of each component is characterized by a modified Bessel function. The rms intensity and scale length of each component are independent parameters. A general method of introducing cross spectra between components is demonstrated. The multiplication of independent random processes is used to generate each of the gust components. Gaussian white noise generators, analog multipliers, and linear filters are used throughout the simulation. A complete analog circuit diagram is presented.

Self-Consistent Molecular-Orbital Methods. III. Comparison of Gaussian Expansion and PDDO Methods Using Minimal STO Basis Sets

Abstract: Self‐consistent‐field molecular‐orbital calculations over a minimal basis set of Slater‐type atomic orbitals are presented for a set of organic molecules and positive ions containing up to eight first‐row atoms. The necessary molecular integrals are calculated by two previously introduced schemes: the Gaussian expansion (STO–KG) method and the projection of diatomic‐differential‐overlap (PDDO) method. Atomization energies, electric dipole moments, density matrices, optimum STO ζ exponents, and computation times are compared for the PDDO, STO‐3G, and STO‐4G methods, the latter of which has previously been shown to closely reproduce the full STO results. Relative to the STO‐4G values, the PDDO method leads to errors of up to 0.22 a.u. in the atomization energy, 0.16D for the dipole moment, and 0.05 for the optimum ζ exponents. The corresponding limits for the STO‐3G method are 0.06 a.u., 0.07D, and 0.02. Two electron integrals are evaluated at rates of 125–175, 25–140, and 10–70 integrals per second for the...

Gaussian expansions of minimal STO basis for calculations in molecular quantum mechanics

Abstract: By means of minimal basis SCF calculations for HF, H2O, NH3 and CH4 different expansions of Slater orbitals (STO) in terms of Gaussian orbitals (GTO) are tested in order to find an appropriate compromise between sufficient accuracy of the results and reasonable computing times. The least squares fit of the GTO expansion to STO does not appear to have any advantages over the expansion based on a variational procedure. It turns out that for hydrogens an expansion of the 1s orbital in terms of three GTO is quite sufficient, whereas for first row atoms an expansion of the 1s orbital in terms of three to five GTO, of the 2s orbital in terms of two GTO and of the 2p orbitals in terms of three GTO seems to be adequate.

Stress-optical coefficient of cis-1,4-polybutadiene and cis-1,4-polyisoprene networks. Measurements on cis-1,4-polybutadiene networks and theoretical interpretation

Abstract: Stress, strain, and birefringence measurements have been carried out on swollen and unswollen networks of ′cis-1,4-polybutadiene polymers. Neither stress-strain nor birefringence-strain relations of unswollen specimens obey the Gaussian network theory, but both can be fitted by the Mooney-Rivlin equation. On the contrary, data on specimens swollen in tetralin, decalin, benzene, and carbon tetrachloride strictly obey the Gaussian network theory. Existing methods for evaluating the temperature coefficient of the unperturbed dimensions, d In 〈r2〉/dT, from the stress-temperature relation are applied to the present data and discussed in some detail. It is concluded that reliable values of d In 〈r2〉/dT are not obtainable from data on unswollen samples because of the pronounced non-Gaussian effect. The value 7.5 A3 for the optical anisotropy A3 (an alternative to the stress-optical coefficient) for unswollen specimens is markedly larger than values (5.8 A3 on the average) for swollen specimens. This is interpreted as due to the shortrange orientational order among polymer segments. The quantities 〈r2〉, ΔΓ, and their temperature coefficients are calculated for both cis-1,4-polybutadiene and cis-1,4-polyisoprene chains, on the basis of the rotational isomeric state approximation for bond rotations. Values of ΔΓ for cis-1,4-polybutadiene calculated using Clement and Bothorel's set of anisotropic bond polarizabilities are in good agreement with observed values for swollen specimens. Those for cis-1,4-polyisoprene obtained using the same set of anisotropic bond polarizabilities are somewhat smaller than observed values for unswollen specimens. This departure is in the direction expected from the behavior of ΔΓ upon swelling (i.e., a decrease in ΔΓ upon swelling).

Error bounds for the white Gaussian and other very noisy memoryless channels with generalized decision regions

Abstract: For a class of generalized decision strategies, which afford the possibility of erasure or variable-size list decoding, asymptotically tight upper and lower error bounds are obtained for orthogonal signals in additive white Gaussian noise channels. Under the hypothesis that a unique signal set is asymptotically optimal for the entire class of strategies, these bounds are shown to hold for the optimal set in both the white Gaussian channel and the class of input-discrete very noisy memoryless channels.

An ionospheric channel simulator

Abstract: An ionospheric channel simulator with characteristics based on an experimentally verified channel model is described. In the channel model, the input (transmitted) signal feeds a delay line that delivers undistorted delayed versions of the input signal to a limited number of adjustable taps with delays corresponding to the relative propagation times of typical ionospheric modes. Each delayed signal is modulated in amplitude and phase by an independent baseband complex bivariate Gaussian random tap-gain function with a zero mean value and quad­rature components with equal RMS values that produce Rayleigh fading. Each tapgain function has a spectrum that, in general, is the sum of two Gaussian functions of frequency, one for each magnetoionic component, which have independently adjustable amplitudes, frequency shifts, and two-sigma frequency spreads. The delayed and modulated signals are summed to form the output (received) signal. Detailed specifications are presented for the channel simulator, and descriptions are given of the analog and digital circuit techniques used.

LCAO–MO–SCF Calculations Using Gaussian Basis Functions. III. Determination of Geometry by SCF Calculations, CF2

Abstract: Results are presented of an LCAO–MO–SCF investigation of the geometry of the CF2 molecule. It is shown that a well‐chosen 582p Gaussian set is sufficient to determine reliably the geometry of triatomic molecules. A Walsh‐type orbital diagram is given which shows a number of features which differ from the predictions of Walsh. The bonding in CF2 is discussed in terms of the Walsh diagram and the population analysis results of a calculation using a more extensive 985p basis set.

Optimum Control of Non-Gaussian Linear Stochastic Systems with Inaccessible State Variables

Abstract: This article presents a new result in the optimum control of linear systems with respect to a quadratic performance criterion. It is assumed that the system is subject to additive random disturbance and that some state variables cannot be measured or can only be measured with additive noise. It is well known that when the disturbances and noise are Gaussian random variables, the optimum controller is a certain linear function of the mean of the posteriori distribution of state variables. It is shown here that this result holds without qualification.

The optimum linear modulator for a Gaussian source used with a Gaussian channel

Abstract: The optimum linear modulator and demodulator which provide transmission of a gaussian vector source through an additive gaussian vector channel are derived in this paper. The measure of performance that is used is the transmission distortion, which is defined here as the mean square error between the source output and the decoder output. It is assumed that the source and channel are mutually independent but that correlations can exist among the components of each. The performance of the best linear system is then compared with the distortion shown by Shannon to be theoretically obtainable when no functional constraint is imposed at the modulator other than an energy constraint. Although the precise form of this optimum modulator is not known for general gaussian vector sources and channels, it is known to be nonlinear and to require arbitrarily long coding block lengths. However, it is a commonly held notion that when the source and channel dimensionalities are equal the optimum modulator is linear and requires a block length of only one. It is shown here that this belief is incorrect except in very particular situations which are described. Some relations between the optimum linear modulator-demodulator pair and Shannon's test channel are discussed, and an example is included which shows that the nonoptimality of linear devices can be quite small.

Energy Detection of a Random Process in Colored Gaussian Noise

Abstract: A likelihood receiver for a Gaussian random signal process in colored Gaussian noise is realized with a quadratic form of a finite-duration sample of the input process. Such a receiver may be called a "filtered energy detector." The output statistic is compared with a threshold and if the threshold is exceeded, a signal is said to be present. False alarm and detection probabilities may be estimated if tabulated distributions can be fitted to the actual distributions of the test statistic which are unknown. Gamma distributions were fitted to the conditional probability densities of the output statistic by equating means and variances, formulas for which are derived assuming a large observation interval. A numerical example is given for the case in which the noise and signal processes have spectral densities of the same shape or are flat. The optimum filter turns out to be a band-limited noise whitener. The factors governing false alarm and detection probabilities are the filter bandwidth, the sample duration, and the signal level compared to the noise. Two sets of receiver operating characteristic curves are presented to complete the example.