Showing papers on "Gaussian published in 1968"

Diffusion by Continuous Movements

Abstract: The influence of velocity distribution on one‐particle dispersion is studied. The ensemble of particle releases is subdivided into subensembles characterized by particle velocity at the instant of release. Under certain simplifying assumptions, the motion of the particles of each subensemble is a systematic drift, combined with dispersal about the subensemble drift position. The displacement variance due to subensemble drift is dominant for small time t since particle release, whereas that due to subensemble is the more important at large t. The assumption that subensemble dispersal is Gaussian yields an estimate of the probability distribution density function θ(x, t) governing particle position which is correct both in the limits of t small and t large. The estimate is thus better than that given by the usual assumption that θ is Gaussian, and appears useful for t values less than the Lagrangian integral scale. The subensemble approach offers some new insights into the problem of the instantaneous structure of the individual plume.

On state estimation in switching environments

Abstract: Work concerned with the state estimation in linear discrete-time systems operating in Markov dependent switching environments is discussed. The disturbances influencing the system equations and the measurement equations are assumed to come from one of several Gaussian distributions with different means or variances. By defining the noise in this manner, periodic step changes in the inputs which cannot be feasibly measured for economic or technical reasons can be detected and corrected. These changes can be in the amplitudes of the inputs or in the variances of stochastic inputs. A Bayes estimation procedure is applied to the problem of filtering the observations of the system so that an estimate of the system state is obtained. Computer simulations for the optimal and suboptimal estimators are also presented.

Some Properties of Symmetric Stable Distributions

Abstract: This paper takes a few steps toward alleviating problems of data analysis that arise from the fact that elementary expressions for density and cumulative distribution functions (c.d.f.'s) for most stable distributions are unknown. In section 2 results of Bergstrom [3] are used to develop numerical approximations for the c.d.f.'s and the inverse functions of the c.d.f.'s of symmetric stable distributions. Tables of the c.d.f.'s and their inverse functions are presented for twelve values of the characteristic exponent. In section 3 the usefulness of the numerical c.d.f.'s and their inverse functions in estimating the parameters of stable distributions and testing linear models involving stable variables is discussed. Finally, section 4 presents a Monte Carlo study of truncated means as estimates of location. In every case but the Gaussian, some truncated mean is shown to have smaller sampling dispersion than the full mean.

Triggered Correlation

Abstract: This paper shows to what extent an average-response computer can be utilized for computing a cross-correlation function. This type of computer needs synchronization pulses, and the simplest methods of computation are those in which these pulses are directly derived from one of the signals (triggered correlation). The first method is to generate a synchronization pulse whenever the signal crosses a pre-set threshold in any direction. In this case, the computer output function is shown to be proportional to the true correlation function, for Gaussian signals. In a second method, synchronization pulses are produced when the signal crosses the threshold in a specified (e.g., positive) direction. Then the computer output is found to be contaminated by a systematic error, which, in turn, depends on the derivative of the correlation function. These two methods are described in detail, both with respect to the results and to the accuracy obtainable. Several other, less important, methods are only briefly described.

Limited memory optimal filtering

Abstract: Linear and nonlinear optimal filters with limited memory length are developed. The filter output is the conditional probability density function and, in the linear Gaussian case, is the conditional mean and covariance matrix where the conditioning is only on a fixed amount of most recent data. This is related to maximum-likelihood least-squares estimation. These filters have application in problems where standard filters diverge due to dynamical model errors. This is demonstrated via numerical simulations.

A comparison of different contractions for molecular calculations with gaussian-type functions

Abstract: The effect of various possible contractions of a gaussian basis set is investigated for atomic and molecular calculations. The gaussian basis set used consists of 11s-type functions and 7p-type functions. Atomic calculations for the atoms Li to F are reported with fourteen different contractions of the s orbitals. The effect of the same contractions has also been investigated for molecular calculations of LiH, BH, CH2, NH 2 − , H2O, and FH, together with the effect of the contraction for the p orbitals and for the s orbitals of the hydrogen atom. It is shown that the contraction in itself does not affect seriously the quality of a molecular calculation, but that a wrong choice of the contraction can produce a poor result.

Nonstationary Stochastic Models of Earthquake Motions

Abstract: Eight strong-motion accelerograms recorded on firm ground and at moderate epicentral distances are studied for the purpose of developing a reasonable stochastic representation of the time history of ground accelerations during strong earthquakes. The results indicate that over durations which are of interest in structural response calculations, nonstationary random processes are needed for the description of the records. A nonstationary Gaussian filtered shot-noise process is examined for this purpose. The member functions of the mathematical model and their associated linear response spectra are compared with the corresponding information obtained from the recorded accelerograms. On this basis, a nonstationary Gaussian filtered shot-noise process with a second-order filter is proposed for stochastic simulation of strong-motion earthquakes.

Nonlinear filtering by approximation of the a posteriori density

Abstract: The problem of estimating from noisy measurement data the state of a dynamical system described by non-linear difference equations is considered. The measurement data have a non-linear relation with the state and are assumed to be available at discrete instants of time. A Bayesian approach to the problem is suggested in which the density function for the state conditioned upon the available measurement data is computed recursively. The evolution of the a posteriori density function cannot be described in a closed form for most systems; the class of linear systems with additive, white gaussian noise provides the major exception. Thus, the problem of non-linear filtering can be viewed as essentially a problem of approximating this density function. For linear systems with additive, white gaussian noise, the a posteriori density is gaussian. The results for linear systems are frequently applied to non-linear systems by introducing linear perturbation theory. Then, the linear equations and gaussian a posterio...

The Gaussian mode in optical resonators with a radial gain profile

Abstract: The dependence of the parameters of the Gaussian mode in laser resonators on the properties of the medium in the cavity is studied. Experimental verification of the theoretical results is presented. It is found that the modes in a high‐gain laser may differ widely from the usual free space resonator results. Also, resonator configurations which in free space are unstable may, with a suitable medium, support low‐loss Gaussian modes.

Response of the Bilinear Hysteretic System to Stationary Random Excitation

Abstract: Time‐average statistics of the response of the bilinear hysteretic system to an excitation with approximately white‐power spectral density and approximately Gaussian probability distribution are determined, using electronic‐analog techniques. Results are presented for the mean‐squared amplitude, the power spectral density, and the probability distribution of the response. The applicability of the Krylov‐Bogoliubov method of equivalent linearization to this problem is investigated by comparing predicted and experimentally measured values of the mean‐squared level of response.

The intensity-fluctuation distribution of Gaussian light

Abstract: We discuss the quantum-mechanical problem of intensity fluctuations in samples of Gaussian light and its connection with the corresponding classical scalar problem involving fluctuations of random noise power. We obtain an expression for the intensity-fluctuation distribution of light with a Lorentzian spectrum, and investigate analytically its asymptotic behaviour. Numerical techniques are used to evaluate the distribution for various intermediate linewidths, and some examples of the corresponding theoretical photon-counting distributions are also given.

Use of the Wiener—Hermite expansion for nearly normal turbulence

Abstract: The turbulence problem is formulated using the Wiener stochastic expansion. The expansion is useful for processes which are in some sense nearly normal, and can be used for non-linear non-Gaussian processes such as many turbulent fluid flows. Here we present the general formulation for statistically inhomogeneous and anistropic processes.The transfer term in the energy equation, or equivalently the third-order velocity correlation, forms a sensitive measure of the amount of non-Gaussianity present in real fluid flows. Experimental evidence shows that in many flows this component is small compared with the Gaussian part. It is shown that a homogeneous and isotropic flow which has but a small non-Gaussian part possesses a distribution at one point which is Gaussian to terms of second order. The experiments suggest that immediately behind a grid in a wind tunnel the flow is very nearly normal. The non-Gaussian part grows at a moderate rate, at least within the range of distance downstream (or decay time) available in the usual experiments. This growth is probably due to the relative increase in the amount of energy in the smallest eddies, which are non-normal.A necessary criterion for the validity of the zero-fourth-cumulant approximation is suggested: the transfer term in dimensionless form should be small. It is shown that calculations using the zero-fourth-cumulant approximation have given negative energy spectra when this condition is violated, probably for the reason that the process is no longer nearly Gaussian. However, even when this condition is fulfilled, it is shown that that approximation must be corrected.It is suggested that the present theory is valid for quite large times of decay if initial energy spectra are chosen which are not too far from the actual physical values for fluid in turbulent flow. Equations are given for the next-higher-order term in a nearly normal approximation. The expansion is also used in § 6 to describe turbulent mixing problems and is compared with the zero-fourth-cumulant approximation for these problems. Computational results are presented in § 7 and compared with experiments by Stewart and Townsend.

Evaluation of the integrals of oscillating functions by interpolation at nodes of gaussian quadratures

Abstract: The direct computation of these integrals by the Newton-Cotes quadrature formulas is not very efficient. Vhen such formulas are applied the integrand, in this case f(x) exp (iox), is approximated by polynomials. Therefore, in order to obtain a good approximation it is necessary for the integration step to be smaller than the typical segment of variation of the integrand, which is of the order of 2R/O.

Wind Shear and Reflectivity Gradient Effects on Doppler Radar Spectra

Abstract: Other investigators have discussed the effects of wind and reflectivity gradients across the radar beam on Doppler measurements, but have either estimated their magnitude from a simple approximation or set them aside as negligible. This paper deals with the component of the shear vector along the beam. Exact solutions and simple approximations for both the mean and variance of the Doppler spectrum are derived for two types of reflectivity gradients combined with a linear velocity, gradient. In the case of an exponential reflectivity, gradient it is found that the “effective” beam (i.e., the reflectivity-weighed two-way illumination pattern) remains Gaussian with identical beamwidth to the real beam, but its mean is shifted to an angle ϕm on the high-reflectivity side of the actual beam. With a linear velocity profile in the ϕ direction, the approximate solution shows that the mean Doppler velocity, is then shifted to the scatterer velocity found at ϕm. This approximation is shown to be valid for ...

Statistics of heterodyction of Gaussian light

Abstract: Discusses the statistical properties of the intensity fluctuations of light fields consisting of an incoherent Gaussian component heterodyned with a single-frequency coherent beam. The second moment of the intensity- fluctuation distribution is obtained in an analytic form for a Lorentzian incoherent spectrum for arbitrary values of the bandwidth and frequency displacement from the coherent mode. The analysis includes as special cases many of the exact results obtained in statistical optics over the last decade.

Two-Pole Approximation for the Plasma Dispersion Function

Abstract: A simple algebraic approximation to the Hilbert transform of the Gaussian, valid over most of the complex plane, is given.

The uncorrelated output components of a nonlinearity

Abstract: Using his characteristic-function approach, Rice (1945) obtained a double series for the autocorrelation function of a sinusoidal signal and Gaussian noise after passage through a memoryless nonlinearity. It is shown here that the output of the nonlinearity can be expressed as the sum of uncorrelated terms whose auto-correlation functions are the terms of Rice's double series. Such a decomposition of the output is shown to be generally possible if and only if the bivariate probability density functions of the input signal and the input noise can both be expressed in the diagonal form studied by Barrett and Lampard (1955), though not necessarily involving polynomials, as they can in the sinusoidal and Gaussian cases. In addition, a more direct and meaningful equation is found for the coefficients in Rice's double series.

Optimal filtering for Gauss—Markov noise†

Abstract: The optimal continuous-filtering problem for the caso of linear dynamics, linear measurements, and gaussian whito disturbance and measurement noise has been Solved by Kalman and Buey. In this study, their rosults are generalized for the caso where measurement noise is a Gauss—Markov process, but without the technique of state augmentation, as has already been done. Proof is presented that the optimal continuous-filtering problem can be solved by simply replacing the observation vector with a derived observation vector and an initial condition. When the derived observation vector is used, coloured noise is eliminated and only the standard Kalman filter problem, easily solvable, remains.

LCAO–MO–SCF Calculations Using Gaussian Basis Functions. II. BeH2

Abstract: Results of extensive LCAO–MO–SCF calculations on BeH2 utilizing Gaussian basis functions are presented. It is established that BeH2 is linear in its ground state, and the equilibrium bond distance has been determined. Force constants and normal frequencies were obtained. Utilizing an accurate estimate of the correlation energy, the dissociation energy of BeH2 was determined to within the experimental error of the National Bureau of Standards measurement.

Power transfer efficiency between focused circular antennas with Gaussian illumination in Fresnel region

Abstract: The power transfer efficiency between two circular aperture antennas in the Fresnel region is obtained analytically and explicitly. They are focused at each other and illuminated as a simple Gaussian taper. Two apertures also have generally different sizes and unequal illuminations. It is shown that the Gaussian distribution is close to the optimum, which is derived from the theory of the confocal optical resonators. The effect of the deviation of the illumination from the optimum upon the efficiency is numerically evaluated. The result shows that the reduction of the efficiency for a constant deviation increases as the Fresnel number decreases. The dissimilarity of both aperture illuminations also reduces the efficiency. The qualitative trend is such that the larger the dissimilarity, the smaller the efficiency. When the order of dissimilarity is small or the taper of the aperture is large, the efficiency can be approximately predicted by assuming equal illumination with the geometrical mean of the two illuminations. A simplified approximate efficiency which can be applied to the greatly tapered illumination is also derived.

The rate distortion function of a Gaussian process with a weighted square error criterion (Corresp.)

Abstract: Rate distortion function for Gaussian random process with weighted mean square error distortion measure criterion

Physical nature of the chemical bond

Abstract: The effect of such variables as size of basis set, number of S and P type functions in the basis set, size of initial exponent in the basis set, size of multiplicative factor in the geometrical progression of exponents, and interatomic distance has been examined by a self-consistent-field calculation with Gaussian orbitals on HeH+. A quasi-optimized wave function has been obtained by allowing only the initial exponent and multiplicative factor to vary in the optimization process.

Fluctuation Distribution of Gaussian Beam Propagating through a Random Medium

Abstract: The mean-square fluctuations of log amplitude and phase are analytically obtained for gaussian light-beam propagating through a randomly inhomogeneous medium with gaussian covariance of the refractive-index fluctuations. The beam used is radiated from an extended source with a circularly symmetric gaussian amplitude distribution and a curved wavefront (phase variation) which characterize the beam shape. The dependence of the fluctuation distribution upon the beam shape and the scale of the medium turbulence is discussed.

Formaldehyde Molecule in a Gaussian Basis. A Self‐Consistent Field Calculation

Abstract: Accurate LCAO–MO–SCF calculations have been carried out for the formaldehyde molecule using (73/2) and (95/3) Gaussian basis sets. The energy parameters, molecular orbitals, dipole moments, and population analyses are reported. The results are compared to a previous calculation with a minimum Slater basis and to experiment.

Gaussian Beams from GaAs Junction Lasers

Abstract: Gaussian beams have been obtained from a GaAs junction laser operating in conjunction with a microscope objective and a slit. The laser mirror illumination, as viewed in the image plane of the objective, is asymmetric perpendicular to the junction plane and consists of a primary intensity maximum along with several weaker secondary peaks. The slit, placed in the image plane, suppresses the secondary peaks thereby eliminating their effects in the diffraction field. The isolated primary maximum is approximately Gaussian shaped and the radiation emanating from the slit retains its Gaussian profile as it propagates into the far‐field region.

Evolution of a Gaussian Wavepacket

Abstract: An alternative derivation is given for the evolution in time of a Gaussian wavepacket representing a free particle. This is based on explicit evaluation of the action of the evolution operator on the initial wavefunction, rather than on expansions in free-particle eigenstates. Both the cases of a free particle initially at rest and in uniform linear motion are considered.