Gaussian Random Process Research Articles

An efficient space–time based simulation approach of wind velocity field with embedded conditional interpolation for unevenly spaced locations

The classical spectral representation method (SRM) has been extensively used in the simulation of multivariate stationary Gaussian random processes. Due to the application of fast Fourier transform (FFT), the simulation is usually efficient. However, for processes with a large number of simulation points, it becomes necessary to enhance the simulation efficiency. One example is the wind velocity field along a large-span bridge, where hundreds of wind velocity fluctuations are required. In the case of bridges built over a homogeneous terrain such as coastal area or flat plain, the wind velocity field can be modeled as a multivariate homogeneous random process i.e., the auto power spectral densities (PSDs) at evenly-spaced simulation points are same and the cross PSD is a function of separation distance between two simulation points. Furthermore, in some applications, additional simulation points need to be included to a set of uniformly distributed points in order to make the wind velocity field consistent to the structural dynamic analysis requirement.In this paper, a hybrid approach of space–time random-field based SRM and proper orthogonal decomposition (POD)-based interpolation is developed for simulating the above wind velocity process. In this approach, the random-field based SRM is used to simulate the multivariate homogeneous random process composed of a set of uniformly distributed simulation locations while POD-based interpolation is used to conditionally generate the wind velocities at a few unevenly distributed points using the previously simulated wind velocities. The idea of the former is based on transforming the simulation of the homogeneous random process into that of the corresponding space–time random field where the phase angle is assumed to be zero and the coherence function must be an even function in terms of separation distance. Through this procedure, customary requirement for spectral matrix decomposition is eliminated and application of two dimensional FFT can improve the simulation efficiency dramatically. The shortcomings of this method include a slight approximation regarding the simulated sample and the non-ergodicity for the correlation function. The numerical example of a homogeneous wind velocity field along a bridge deck shows that the proposed random field-based method is very efficient in terms of accuracy and efficiency when the number of simulation locations is large and the POD-based interpolation also has good performance.

Estimations of fatigue life and variability under random loading in aluminum Al-2024T351 using strip yield models from NASGRO

This paper is concerned with the application of the strip yield models implemented in NASGRO to estimate fatigue crack growth under random loading. The two different strip yield model options (constant constraint-loss (CCL) and variable constraint-loss (VCL)) implemented in the software are considered and compared. In addition, three crack grow rate relations, obtained from constant amplitude data, the NASGRO material database and the literature, were also considered and compared. The capacity of the models to estimate the fatigue life and variability is analyzed by comparing simulated results with experimental data of fatigue crack growth under different stationary Gaussian random load processes on compact, C(T), specimens of aluminum alloy 2024-T351. The analysis performed showed that both model options provide very similar good fatigue life predictions. The estimated fatigue lives are within ±13% of the test lives with the best option. The variability of the results due to the randomness of the load is also analyzed. In this case, the variable constraint-loss option provides better estimation than the constant constraint-loss option.

A kriging procedure for processes indexed by graphs

We provide a new kriging procedure of processes on graphs. Based on the construction of Gaussian random processes indexed by graphs, we extend to this framework the usual linear prediction method for spatial random fields, known as kriging. We provide the expression of the estimator of such a random field at unobserved locations as well as a control for the prediction error.

Canonical decomposition of fluctuation interferences using the delta function formalism

The paper deals with the discrete spectral-orthogonal decompositions of centered Gaussian random processes for two cases. In the first case, the process implementations are a sequence of pulses that are short in comparison with the observation time. The process decomposition was obtained as a generalized Fourier series on the basis of the delta function formalism, and the variances of the coefficients (random values) of this series were found as well. The resulting expressions complement Kotel'nikov's formula because they cover both the high-frequency and the low-frequency regions of the canonical-decomposition spectrum. In the second case, a random process is a superposition of narrow-band Gaussian random processes, and its implementations are characterized by oscillations. For such a process the canonical decomposition in terms of the Walsh functions was obtained on the basis of the generalized function formalism. Then this decomposition was re-decomposed in terms of trigonometric functions; it follows from the resulting series that the canonical decomposition spectrum is not uniform since a pedestal is formed in the constant component region.

Statistics of the derivatives of complex signal derived from Riesz transform and its application to pseudo-Stokes vector correlation for speckle displacement measurement.

As an improvement of the well-known intensity correlation used in conventional electronic speckle photography, we have proposed a new technique, to the best of our knowledge, for displacement measurement referred to as pseudo-Stokes vector correlation (PSVC). To provide a theoretical background for the superiority of the proposed PSVC technique, we study the statistical properties of the spatial derivatives of the complex signal representation generated from the Riesz transform. Under the assumption of a Gaussian random process, a theoretical analysis for the pseudo Stokes vector correlation has been provided. Based on these results, we show mathematically that PSVC has a performance advantage over conventional intensity-based correlation technique.

Statistical Peak to Average Power Ratio Bound

Central limit theorem, modern extreme value theory and the theory that the discrete orthogonal frequency division multiplexing signals converge weakly to a Gaussian random process are generally employed to derive the best approximation of peak-to-average power ratio distributions of discrete-time and continuous-time signals. In this paper, we arrive at a simple, rigorously justified, and accurate expression of the upper bound of the probability density function of the PAPR in context of orthogonal frequency division multiplexing systems. Since this bound has a complex expression not convenient to deal with in practice, we show also the demonstration of an accurate estimation of the proposed bound.

Fresnel diffraction by a square aperture with rough edge

In this paper, we study Fresnel diffraction of a square aperture with rough edge with Gaussian random process. The diffraction intensity formula depending on the random parameters is deduced by use of the scalar diffraction theory. The simulation calculations of the diffraction of the square aperture with different statistic parameters are performed. The diffraction intensity distributions exhibit elaborately the influence of the random parameters of rough edge on the diffraction of a square aperture. The results show that the larger the lateral correlation length and the larger roughness of the rough edge are, the more severe the modulation of the random spots is and the more unsymmetrical the diffraction distribution is. For the same random parameters, the modulation degree of the random spots changes with the propagation distance. Moreover, we also manufacture practically the random apertures with the help of the laser direct writing technology, and measure the diffraction distribution of the square aperture with random edge. The experimental results are consistent with the theoretic ones. These results will be instructively meaningful for the applications of the diffraction of the practical aperture.

Fatigue damage model comparison with formulated tri-modal spectrum loadings under stationary Gaussian random processes

A riser system experiences a tri-modal process. It is excited by combined loads composed of low frequency response due to riser tension, mid-range excitation frequency from waves and high frequency excitation due to vortex induced-vibration. This study proposes ideal tri-modal spectrum formulations for parametric study. The formulated spectrum is used to compare various fatigue damage models developed to predict fatigue damage for wide band spectrum through theoretical or empirical approaches. The most reasonable fatigue damage is calculated as a reference by inverse Fourier transform, rain-flow counting method and Palmgren–Miner rule. The results conclude that the Benasciutti–Tovo and JB models currently represent the most appropriate fatigue damage for tri-modal spectrum.

Characteristics of thunderstorms relevant to the wind loading of structures

"Wind and Ports" is a European project that has been carried out since 2009 to handle wind forecast in port areas through an integrated system made up of an extensive in-situ wind monitoring network, the numerical simulation of wind fields, the statistical analysis of wind climate, and algorithms for medium-term (1-3 days) and short term (0.5-2 hours) wind forecasting. The in-situ wind monitoring network, currently made up of 22 ultrasonic anemometers, provides a unique opportunity for detecting high resolution thunderstorm records and studying their dominant characteristics relevant to wind engineering with special concern for wind actions on structures. In such a framework, the wind velocity of thunderstorms is firstly decomposed into the sum of a slowly-varying mean part plus a residual fluctuation dealt with as a non-stationary random process. The fluctuation, in turn, is expressed as the product of its slowly-varying standard deviation by a reduced turbulence component dealt with as a rapidly-varying stationary Gaussian random process with zero mean and unit standard deviation. The extraction of the mean part of the wind velocity is carried out through a moving average filter, and the effect of the moving average period on the statistical properties of the decomposed signals is evaluated. Among other aspects, special attention is given to the thunderstorm duration, the turbulence intensity, the power spectral density and the integral length scale. Some noteworthy wind velocity ratios that play a crucial role in the thunderstorm loading and response of structures are also analyzed.

On Adler space‐time extremes during ocean storms

AbstractThe paper concerns the statistical properties of extreme ocean waves in the space‐time domain. In this regard, a solution for the exceedance probability of the maximum crest height during a sea state over a certain area is obtained. The approach is based on the Adler's solution for the extremal probability for Gaussian random processes in a multidimensional domain. The method is able to include the effects of spatial variability of three‐dimensional sea waves on short‐term prediction, both over an assigned area XY and in a given direction. Next, the storm‐term predictions in the space‐time are investigated. For this purpose, the exceedance probability of during an ocean storm over an assigned area is derived. This solution gives a generalization to the space‐time of the Borgman's time‐based model for nonstationary processes. The validity of the model is assessed from wave data of two buoys of the NOOA‐NDBC network located along the Pacific and the Atlantic U.S. coasts. The results show that the size of the spatial domain remarkably influences the expected maximum crest height during a sea storm. Indeed, the exceedance probabilities of the maximum crest height during an ocean storm over a certain area significantly deviate from the classical Borgman's model in time for increasing area. Then, to account for the nonlinear contributions on crest height, the proposed model is exploited jointly with the Forristall's distribution for nonlinear crest amplitudes in a given sea state. Finally, Monte Carlo simulations of a sea storm are performed showing a very good agreement with theoretical results.

Spectral diffusion model of heavy inertial particles in a random velocity field of the continuous medium

Based on the spectral expansion of Euler correlation of the carrier medium the authors have obtained a closed system of functional equations for the Lagrange spectra of heavy inertial particles and the velocity fluctuations of the carrier medium on the particle trajectory. To split the fourth moments the approximation of quasinormality and velocity fluctuations of particles is performed by a random Gaussian process. The approximate self-consistent method is proposed for solving the resulting system of functional equations. The spectrum of Euler correlations of medium velocity fluctuations is modeled by Saffman and Karman distributions. The influence of the spatial microstructure of turbulence, the particles inertia and velocity slip on the intensity of chaotic motion and the coefficient of turbulent diffusion of dispersed particles has been studied.

Closed-form solutions for the evolutionary frequency response function of linear systems subjected to separable or non-separable non-stationary stochastic excitations

In the seismic engineering, in order to reproduce the typical characteristics of real earthquakes ground-motion time history, the so-called uniformly modulated and the fully non-stationary random processes have been introduced. The first process is constructed as the product of a stationary zero-mean Gaussian random process by a deterministic function of time; for this reason it is also called separable non-stationary stochastic process. However, this process catches only the time-varying intensity of the accelerograms. To consider simultaneously both the amplitude and frequency changes, time–frequency varying deterministic functions have been introduced in the characterization of the input process. The latter process is referred as fully or non-separable non-stationary stochastic process.The evolutionary frequency response function plays a central role in the evaluation of the statistics of the response of linear structural systems subjected to both separable or non-separable stochastic excitations. In fact, by means of this function, it is possible to evaluate in explicit form the evolutionary power spectral density of the response and consequently the non-geometric spectral moments, which are required in the prediction of the safety of structural systems subjected to non-stationary random excitations.In this paper a method to evaluate in closed-form the evolutionary frequency response function of classically damped linear structural systems subjected to both separable and non-separable non-stationary excitations is presented. In order to evidence the flexibility of the proposed procedure the evolutionary frequency response function is evaluated by very handy explicit closed-form solutions for the most adopted time varying and time–frequency varying modulating functions.

인공 신경망을 이용한 광대역 과정의 피로 손상 모델 개발

For the frequency-domain spectral fatigue analysis, the probability density function of stress range needs to be estimated based on the stress spectrum only, which is a frequency domain representation of the response. The probability distribution of the stress range of the narrow-band spectrum is known to follow the Rayleigh distribution, however the PDF of wide-band spectrum is difficult to define with clarity due to the complicated fluctuation pattern of spectrum. In this paper, efforts have been made to figure out the links between the probability density function of stress range to the structural response of wide-band Gaussian random process. An artificial neural network scheme, known as one of the most powerful system identification methods, was used to identify the multivariate functional relationship between the idealized wide-band spectrums and resulting probability density functions. To achieve this, the spectrums were idealized as a superposition of two triangles with arbitrary location, height and width, targeting to comprise wide-band spectrum, and the probability density functions were represented by the linear combination of equally spaced Gaussian basis functions. To train the network under supervision, varieties of different wide-band spectrums were assumed and the converged probability density function of the stress range was derived using the rainflow counting method and all these data sets were fed into the three layer perceptron model. This nonlinear least square problem was solved using Levenberg-Marquardt algorithm with regularization term included. It was proven that the network trained using the given data set could reproduce the probability density function of arbitrary wide-band spectrum of two triangles with great success.

Long-haul quasi-single-mode transmissions using few-mode fiber in presence of multi-path interference.

We study long-haul Quasi-Single-mode (QSM) systems in which signals are transmitted in the fundamental modes of a few-mode fiber (FMF) while keeping other system components such as amplifiers and receivers are kept single-moded. The large-effective-area nature of the FMF fundamental modes improves system nonlinear tolerance in the expense of mode coupling along FMF transmissions which induces multi-path interference (MPI) and needs to be compensated. We analytically investigate 6-spatial-polarization mode QSM transmission systems in presence of MPI and show that in the weak coupling regime, the QSM channel is a Gaussian random process in frequency. MPI compensation filters are derived and performance penalties due to MPI and signal loss from higher-order modes are characterized. We also experimentally demonstrate 256 Gb/s polarization multiplexed (PM)-16-QAM QSM transmissions over a record distance of 2600 km with 100-km span using decision directed least mean square (DD-LMS) algorithm for MPI compensation.

Bayesian Sequential Parameter Estimation by Cognitive Radar With Multiantenna Arrays

In this paper we consider the problem of Bayesian sequential parameter estimation of extended targets for cognitive radar with multi-antenna arrays using adaptive waveforms. The target is modeled as a complex Gaussian random process. Using iterative waveform transmission, the cognitive radar estimates the target's characteristic parameters and updates its probabilistic model based on new measurements. The adaptive waveform is designed by minimizing the conditional entropy from the posterior density of the model parameters. We analyze the performance of the developed Bayesian sequential estimation algorithm and derive expressions for the signal-to-noise ratio gain, the asymptotical posterior Cramer Rao bound, and the mutual information gain. The analysis and numerical simulations demonstrate that the adaptive sequential Bayesian estimator yields accelerated convergence of the estimate towards its true value and a smaller estimation error compared with the conventional Bayesian estimator that uses fixed waveform transmission under Gaussian or non-Gaussian noise.

New Robust Estimators of Correlation and Weighted Basis Pursuit

The autocorrelation function is a commonly used tool in statistical time series analysis. In this paper we examine the robustness of two estimators of correlation based on memoryless nonlinear functions of the observations, the Median-of-Ratios Estimator (MRE) and the Phase-Phase Correlator (PPC), which are applicable to complex-valued Gaussian random processes. We show that they are robust to impulsive noise from a bias perspective at the expense of statistical efficiency at the Gaussian distribution. Additionally, we develop iterative versions of these estimators named the IMRE and IPPC, realizing an improved bias performance over their non-iterative counterparts and the well-known robust Schweppe-type Generalized M-estimator utilizing a Huber cost function (SHGM). We use robust Mahalanobis distances generated by the IMRE and IPPC to improve the performance of impulsive noise suppression through the use of weighted basis pursuit methods. These estimation and data-cleaning techniques are applied to both synthetic and actual collected data using an Ettus Research USRP to perform robust spectral estimation on signals from the digital television band in the presence of additive impulsive noise. Finally, the analysis reveals that if the time series is highly correlated and the contamination rate is low, the MRE outperforms the PPC estimator from a variance and bias perspective. However, the PPC is the estimator of choice when the correlation is lower and is better suited to time critical applications due to lower computation costs.

Clutter Removal in Sub-Nyquist Radar

Recently, a new approach to sub-Nyquist sampling and processing in pulse-Doppler radar was introduced, based on the Xampling approach to reduced-rate sampling combined with Doppler focusing performed on the low-rate samples. This method imposes no restrictions on the transmitter, reduces both the sam- pling and processing rates and exhibits linear signal-to-noise ratio improvement with the number of pulses. Here we extend previous work on sub-Nyquist pulse-Doppler radar by incorporating a clutter removal algorithm operating on the sub-Nyquist samples, allowing to reject clutter modeled as a colored Gaussian random process. In particular, we show how to adapt standard clutter rejection techniques to work directly on the sub-Nyquist samples, allowing to preserve high detection rates even in the presence of strong clutter, while avoiding the need to first interpolate the samples to the Nyquist grid.

Characterisation of random Gaussian and non-Gaussian stress processes in terms of extreme responses

In the field of military land vehicles, random vibration processes generated by all-terrain wheeled vehicles in motion are not classical stochastic processes with a stationary and Gaussian nature. Non-stationarity of processes induced by the variability of the vehicle speed does not form a major difficulty because the designer can have good control over the vehicle speed by characterising the histogram of instantaneous speed of the vehicle during an operational situation. Beyond this non- stationarity problem, the hard point clearly lies in the fact that the random processes are not Gaussian and are generated mainly by the non-linear behaviour of the undercarriage and the strong occurrence of shocks generated by roughness of the terrain. This non-Gaussian nature is expressed particularly by very high flattening levels that can affect the design of structures under extreme stresses conventionally acquired by spectral approaches, inherent to Gaussian processes and based essentially on spectral moments of stress processes. Due to these technical considerations, techniques for characterisation of random excitation processes generated by this type of carrier need to be changed, by proposing innovative characterisation methods based on time domain approaches as described in the body of the text rather than spectral domain approaches.

A Review of Spectral Methods for Variable Amplitude Fatigue Prediction and New Results

A review of the available methods for estimating fatigue damage from variable amplitude loading is presented. The dependence of fatigue damage accumulation on power spectral density (psd) is investigated for stationary Gaussian random processes. Improvements to the Rayleigh approximation are examined by comparison to rainflow analysis of time histories simulated from psd functions representative of simple theoretical and real world applications. Spectral methods investigated include those by Wirsching and Light, Ortiz and Chen, the Dirlik formula, and the Single-Moment method, among other recent methods. Guidelines and limitations are given for using spectral methods to increase confidence in the damage estimate.

POLYNOMIAL-CHAOS-BASED KRIGING

Computer simulation has become the standard tool in many engineering fields for designing and optimizing systems, as well as for assessing their reliability. To cope with demanding analysis such as optimization and reliability, surrogate models (a.k.a meta-models) have been increasingly investigated in the last decade. Polynomial Chaos Expansions (PCE) and Kriging are two popular non-intrusive meta-modelling techniques. PCE surrogates the computational model with a series of orthonormal polynomials in the input variables where polynomials are chosen in coherency with the probability distributions of those input variables. On the other hand, Kriging assumes that the computer model behaves as a realization of a Gaussian random process whose parameters are estimated from the available computer runs, i.e. input vectors and response values. These two techniques have been developed more or less in parallel so far with little interaction between the researchers in the two fields. In this paper, PC-Kriging is derived as a new non-intrusive meta-modeling approach combining PCE and Kriging. A sparse set of orthonormal polynomials (PCE) approximates the global behavior of the computational model whereas Kriging manages the local variability of the model output. An adaptive algorithm similar to the least angle regression algorithm determines the optimal sparse set of polynomials. PC-Kriging is validated on various benchmark analytical functions which are easy to sample for reference results. From the numerical investigations it is concluded that PC-Kriging performs better than or at least as good as the two distinct meta-modeling techniques. A larger gain in accuracy is obtained when the experimental design has a limited size, which is an asset when dealing with demanding computational models.