Stationary Gaussian Random Process Research Articles

Nonequilibrium fluctuation–dissipation relations for independent random rate processes with dynamical disorder

A class of rate processes with dynamical disorder is investigated based on the two following assumptions: (a) the system is composed of a random number of particles (or quasiparticles) which decay according to a first-order kinetic law; (b) the rate coefficient of the process is a random function of time with known stochastic properties. The formalism of characteristic functionals is used for the direct computation of the dynamical averages. The suggested approach is more general than the other approaches used in the literature: it is not limited to a particular type of stochastic process and can be applied to any type of random evolution of the rate coefficient. We derive an infinity of exact fluctuation–dissipation relations which establish connections among the moments of the survival function and the moments of the number of surviving particles. The analysis of these fluctuation–dissipation relations leads to the unexpected result that in the thermodynamic limit the fluctuations of the number of particles have an intermittent behavior. The moments are explicitly evaluated in two particular cases: (a) the random behavior of the rate coefficient is given by a non-Markovian process which can be embedded in a Markovian process by increasing the number of state variables and (b) the stochastic behavior of the rate coefficient is described by a stationary Gaussian random process which is generally non-Markovian. The method of curtailed characteristic functionals is used to recover the conventional description of dynamical disorder in terms of the Kubo–Zwanzig stochastic Liouville equations as a particular case of our general approach. The fluctuation–dissipation relations can be used for the study of fluctuations without making use of the whole mathematical formalism. To illustrate the efficiency of our method for the analysis of fluctuations we discuss three different physicochemical and biochemical problems. A first application is the kinetic study of the decay of positrons or positronium atoms thermalized in dense fluids: in this case the time dependence of the rate coefficient is described by a stationary Gaussian random function with an exponentially decaying correlation coefficient. A second application is an extension of Zwanzig’s model of ligand–protein interactions described in terms of the passage through a fluctuating bottle neck; we complete the Zwanzig’s analysis by studying the concentration fluctuations. The last example deals with jump rate processes described in terms of two independent random frequencies; this model is of interest in the study of dielectric or conformational relaxation in condensed matter and on the other hand gives an alternative approach to the problem of protein–ligand interactions. We evaluate the average survival function in several particular cases for which the jump dynamics is described by two activated processes with random energy barriers. Depending on the distributions of the energy barriers the average survival function is a simple exponential, a stretched exponential, or a statistical fractal of the inverse power law type. The possible applications of the method in the field of biological population dynamics are also investigated.

Computer-aided design of wide-range dc voltmeters based on two-parameter amplitude adaptation

A software package for computed-aided design of voltmeters is discussed assuming that the measured voltages are stationary or piecewise stationary Gaussian random processes. Examples are given of the design of a computer four-channel dc voltage acquisition module and a high-resolution analog-digital converter.

A Strain-Based Fatigue Reliability Analysis Method

A new fatigue reliability technique has been developed using a strain-based analysis. A probabilistic strain-life curve, where the variability in cycles to failure at constant strain range has been modeled with a three-parameter Weibull distribution, has been incorporated into the strain-based fatigue analysis. This formulation, which includes a notch strain analysis, rainflow cycle counting and damage accumulation according to Miner’s rule, is used to estimate fatigue life to crack initiation for notched components using smooth specimen laboratory data. Unlike other probabilistic fatigue models, the technique developed here does not include a distribution model for stress peaks such as the commonly-used stationary narrow band Gaussian random process assumption but rather uses strain histories directly. Using this model, techniques have been developed to estimate the number of cycles to failure at a specified reliability and to predict the reliability and failure rate at a specified time in the analysis.

Bottom friction beneath random waves

The effect of random waves on the bottom friction is studied by assuming that the wave motion is a stationary Gaussian narrow-band random process. The approach is also based on simple explicit friction coefficient formulas for sinusoidal waves. The probability distribution functions of the maximum bottom shear stress for laminar flow as well as smooth turbulent and rough turbulent flow are presented. The maximum bottom shear stress follows the Rayleigh distribution for laminar flow and the Weibull distribution for smooth turbulent and rough turbulent flow. Some characteristic statistical values of the maximum bottom shear stress for the three flow regimes are also given.

Optimal transforms for multispectral and multilayer image coding

Multispectral images are composed of a series of images at differing optical wavelengths. Since these images can be quite large, they invite efficient source coding schemes for reducing storage and transmission requirements. Because multispectral images include a third (spectral) dimension with nonstationary behavior, these multilayer data sets require specialized coding techniques. The authors develop both a theory and specific methods for performing optimal transform coding of multispectral images. The theory is based on the assumption that a multispectral image may be modeled as a set of jointly stationary Gaussian random processes. Therefore, the methods may be applied to any multilayer data set which meets this assumption. Although the authors do not assume the autocorrelation has a separable form, they show that the optimal transform for coding has a partially separable structure. In particular, they prove that a coding scheme consisting of a frequency transform within each layer followed by a separate KL transform across the layers at each spatial frequency is asymptotically optimal as the block size becomes large. Two simplifications of this method are also shown to be asymptotically optimal if the data can be assumed to satisfy additional constraints. The proposed coding techniques are then implemented using subband filtering methods, and the various algorithms are tested on multispectral images to determine their relative performance characteristics.

Frictional Resistance Of Randomly Oscillating Surfaces

The frictional resistance of randomly oscillating surfaces is studied by assuming that the oscillatory motion is a stationary Gaussian narrow-band random process. The approach is also based on simple explicit frictional resistance formulas for harmonic oscillations. The probability distribution functions of the maximum surface shear stress for laminar flow as well as smooth turbulent and rough turbulent flow are presented. The maximum surface shear stress follows the Rayleigh distribution for laminar flow and the Weibull distribution for smooth turbulent and rough turbulent flow. Examples of the friction coefficients based on the mean values of the maximum surface shear stress for the three flow regimes are also given.

Response of an FM discriminator in the presence of noise and a cochannel interference

An exact expression for the output autocorrelation function of an FM discriminator is derived when the input of the FM discriminator is an FM signal, cochannel interference and Gaussian noise. The modulating baseband signals for the FM signal and the cochannel interference are assumed FDM signals whose statistics are modeled by a stationary Gaussian random process. The interaction among signal, interference and noise are determined by which the NPR (noise-power-ratio) can be calculated. >

A GAUSSIAN, NON-MARKOVIAN APPROACH TO LONG MEMORY WITH APPLICATION TO A STOCHASTIC THEORY OF LINE SHAPE

A new approach to long memory effects is suggested. The main assumption is that the regression of fluctuations occurs via a clustering mechanism. The regression of a scalar variable takes place through a succession of linear decay processes, resulting in a long time tail of the autocorrelation function. The number of decay events is a random variable obeying a self-similar probability distribution whose fractal exponent determines the long time behavior of fluctuation regression. The overall fluctuation dynamics is described by a stationary Gaussian and non-Markovian random process. The theory is applied to the stochastic theory of line shape. The relaxation function ϕ(t) can be exactly evaluated. We show that the memory effects lead to a narrowing of the relaxation function for large time. As the time t tends to infinity, ϕ(t) may be approximated by a ‘contracted’ exponential ϕ(t) ~ exp(-const. t2−H) where 1≥H>0 is the fractal exponent describing the clustering process.

Control of level crossings in stationary Gaussian random processes

A new optimal stochastic control problem that minimizes the intensity for a signal to upcross a level is solved by rewriting it as a one-parametric optimization problem over a set of LQG-problem solutions. The solution can sometimes be thought of as finding optimal weightings in an LQG problem. >

A MMSE Speckle Filter for Full Resolution SAR Polarimetric Data

The usual polarimetric speckle filters optimally combine the polarization channels into a single image [10] or only restore the radiometric information [8], i.e., only the four Ihh, Iυυ,Ιhυ and Iυh backscattered intensities (or the three Ihh, Iυυ, Ihυ intensities in the reciprocal case). The phase differences for one-look images are not restored. This implies a loss of information compared to the initial data, which contain five independent real parameters plus one absolute phase for the one-look scattering matrix format in the reciprocal case. In this paper we develop a fully polarimetric minimum mean square error (MMSE) filter under the multiplicative speckle noise model assumption for one-look images. This model assumes the data to be the product of the non-stationary "unspeckled" signal by statistically independent speckle noise, characterized by the covariance matrix of a stationary complex multivariate Gaussian random process, representing the sensor effects. For each pixel, one obtains on output of the filtering process a complex "unspeckled" scattering matrix and in addition, the complex degrees of coherence between the polarization channels. Furthermore, an adaptive window of sufficient size, defined by the coefficient of variation and various contrast ratios computed in an improved span image, allows the detection of structural features, ensuring reliable statistics for input parameters. Hence, speckle is highly reduced and the spatial resolution is not degraded. The extra information brought up by the filter (phase differences and magnitude of degrees of coherence) will be illustrated in this paper on NASA/JPL data.

On fatigue crack growth under random loading

A probabilistic analysis of the fatigue crack growth, fatigue life and reliability of a structural or mechanical component is presented on the basis of fracture mechanics and theory of random processes. The material resistance to fatigue crack growth and the time-history of the stress are assumed to be random. Analytical expressions are obtained for the special case in which the random stress is a stationary narrow-band Gaussian random process, and a randomized Paris-Erdogan law is applicable. As an example, the analytical method is applied to a plate with a central crack, and the results are compared with those obtained from digital Monte Carlo simulations.

System for generating sequences of phased gust or taxi loadings

A method is presented for Monte Carlo generation of phased load conditions for multiple loading durability testing. These load conditions are consistent with the gust and taxi environmental-dynamic model used in aircraft design. Time histories of individual loads from sequences of load conditions are consistent with the load sequences used in standard single load point durability and damage tolerance testing and analysis. Truncation can be applied to a sequence of load conditions so that only the most severe are retained. The truncation process is consistent with the mission analysis approach to design and to durability and damage tolerance. The load conditions can be modified using an induced autocorrelation approach to give variation in the number of zero crossings of the load time histories. OAD sequences generated for durability and damage tolerance analysis are usually applied at a single point. However, the stresses in a component of structure often arise from combinations of multiple loads. Selection of a single application point and direction is often a difficult task. This is particularly true for aircraft structures flying in turbulence. In that case, loads arise from complex combinations of ex- ternal and internal forces. At McDonnell Douglas the full-scale durability test of the C-17 aircraft requires the simultaneous application of test loads which are properly phased by some rational definition of phasing. The following criteria was developed for satisfying this requirement: 1) Time histories for each individual load (e.g., wing root normal bending) must be consistent with load sequences used in durability and damage tolerance testing and analysis. 2) The phasing of the simultaneously applied loads must be consistent with the dynamic gust and taxi models used to design the aircraft. A method is presented here that meets this criteria. The method generates load conditions, that is, multiple loads for simultaneous application at several points on an aircraft struc- ture. The multiple loads are correlated in conformance with the mathematical model of the gust environment and the structure. In this model the magnitude, frequency response, and correlation of loads are based on a flexible linear aircraft response to a forcing function having the characteristics of a stationary Gaussian random process (see Fig. 1). The specifics of the mathematical model of an aircraft flying in turbulence have been summarized by Hoblit in Ref. 1. This method as implemented at Douglas Aircraft Corpo- ration is called the Phased Loading Sequence Generator Sys- tem (PLSGS). This system will be used to generate gust and taxi loadings for the C-17 full-scale durability test. For the C-17 application the taxi forcing function (the runway) was treated as a stationary Gaussian random process, and the

Three-dimensional stochastic response of offshore towers to random sea waves

In this paper, a three-dimensional model for dynamic response analysis of jacket-type offshore platforms under random sea waves is established. The sea waves are considered to be zero mean, stationary and ergodic Gaussian random processes specified by the P-M wave height spectrum. Linear wave theory and the generalized Morrison equation are used to calculate the wave forces on the structure. The fluid-structure interaction and the couplings of the motion in different directions at each node are considered. The fluid drag forces are linearized by stochastic linearization techniques. The mean-square responses are obtained by Gauss-Legendre numerical integration. Numerical results for a platform with a height of 65 m to symmetrical and unsymmetrical loads are presented and contrast with the corresponding results of one- and two-dimensional models. Some more detail and practical description, such as the response in the vertical direction and the unsymmetrically loaded behaviour, are given.

Effect of Small Stress Ranges on Fatigue Crack Propagation under Random Loading

This is a part of the serial studies on the experimental method to infer fatigue life of a structural member under service loadings. When a load profile can be approximated by a narrow band stationary Gaussian random process, it has been pointed out that random fatigue tests are easily done with a personal computer by simulating the envelope of the process by FFT. However, it may not be realistic to repeat a time and cost consuming test to get the probability distribution function of fatigue life. In this study a method is proposed to make the total experimental hours as short as possible by eliminating small amplitude cycles, that is, an experimental procedure is established to see the effect of the elimination and determine the cut-off-level beforehand.

Probability and Random Processes for Electrical Engineering

Probability and Random Processes for Electrical Engineering

Optimal Linearization of Morison‐Type Wave Loading

The linearization of Morison-type wave loading is to replace \Iu\N|\N\Iu\N|\N by \Icu\N, where \Iu\N is a zero mean, stationary Gaussian random process and \Ic\N is a constant. The coefficient \Ic\N is traditionally determined by a use of the statistical least-square error linearization; that is minimizing \IE\N[(\Iu\N|\N\Iu\N|\N-\Icu\N)²]. In this study, the optimal linearization refers to a coefficient \Ic\N, chosen in such a way that the statistics of structural response calculated from the linearized force will be identical to that calculated from the originally nonlinear force. To accomplish this, the optimal linearization must not only consider the ocean wave properties, but incorporate the structural properties. While most offshore structural engineers believe that the least-square linearization may cause significantly nonconservative error only when the fundamental structural frequency is nearly three times the peak wave frequency, this study shows that worse errors may occur for other frequency ratios.

Comparison theorems and asymptotic behavior of correlation estimators in spaces of continuous functions. II

This paper is the second part of [12]. Using the comparison theorems which were proved in the first part, the asymptotic normality of the estimator — in a model of a series of several samples — of the correlation function of a stationary Gaussian random process in spaces of continuous functions with weights is established. A method for constructing functional confidence intervals for an unknown correlation function in these spaces is described.

Asymptotic error probability expressions for multihypothesis testing using multisensor data

Existing upper bounds to the error probability in testing between m>2 hypotheses, and H. Chernoff's (1952) asymptotically correct error probability expression for m=2 hypotheses, as the number of observations n to infinity , are discussed. The multidimensional version of Chernoff's bound and its relationship to large deviation theory is presented. Large deviation theory is used to develop new bounds. The new bounds are asymptotically exact, in the sense that as n to infinity , they converge to the correct asymptotic rate, which is guaranteed to be the optimum one by the large deviation theorem. Necessary and sufficient conditions are determined so that asymptotic convergence of the error rates to zero is sustained in the presence of mismatch, which occurs when inaccurate versions of the true probability density functions are utilized in the maximum-likelihood decision rule. The conditions are expressed in terms of informational divergence distances, for Markov chain data and Gaussian multivariate stationary random processes. The results for multisensor data are generalized. >

Seismic reliability analysis of flat-plate structures

This paper presents an extension of the reliability analysis method for a plane frame structure to a flat-plate structure. The flat-plate structure is modelled as a frame structure by using the effective beam approach proposed by Moehle and Diebold. The condensation technique in TABS77 is further utilized to transform the frame structure into a stick model. The earthquake ground acceleration is represented by a segment of stationary Gaussian random process with mean zero and a Kanai-Tajimi power spectrum. The power spectrum together with the dynamic characteristics of structure is utilized in the random vibration analysis to derive the probabilistic structural response. The limit analysis procedure proposed by Aoyama is modified to determine the failure mechanism and the corresponding storey shear capacity of the flat-plate structure. The reliability of the structure is measured in terms of the largest storey limit state probability. The fragility curve of the structure is generated by evaluating the limit state probabilities at various levels of peak ground acceleration. From the hazard curve for a site and the fragility curve for a structure, the annual limit state probability of the structure can also be evaluated. A five-storey flat-plate structure assumed to be located in New York City is used to illustrate the reliability analysis method.

A separable cross-entropy approach to power spectral estimation

An approach to power spectrum estimation that is based on a separable cross-entropy modeling procedure is presented. The authors start with a model of a multichannel, multidimensional, stationary Gaussian random process that is sampled on a nonuniform grid. An approximate separable model in which selected frequency samples of the process are modeled as independent random variables, is then fitted to it. Two cross-entropy-like criteria are used to select optimal separable approximations. One of them yields a spectral estimation algorithm that is a generalized version of Capon's maximum-likelihood method, and the other is similar to classical windowing methods. They discuss different strategies for designing bandpass filters for use with the cross-entropy approach.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>