A menagerie of processes

The problem of digital modeling of random processes with given either correlation function or spectral density of the process is considered. These functions of a random process are interconnected by the Wiener–Khinchin theorem. The solution of one function can be used to solve another. The development of a mathematical representation of a stationary random process with a given correlation function based on the Karhunen-Loeve transformation, which is most often used to decorrelate the original process in order to describe it more concisely (data compression problem), has been completed. It is proposed to use the Karhunen–Loev transformation to impart the required correlation properties to the original uncorrelated random process by inverting (converting) this transformation. The form of the required transformation for a discrete (in time) representation of input and output processes of various lengths and methods for ensuring the required modeling accuracy are substantiated. A procedure for obtaining a correlation function from a given spectral density of a simulated random process is presented. An experimental study of the proposed method was carried out in the course of computer simulation in the Mathcad package which simplified the solution of the required computational problems. The initial random process was obtained as a sequence of independent (and, therefore, uncorrelated) random numbers, and the output process, as a result of the transformation, was obtained in the work. The calculated approximate correlation function is compared with the given one and the error variance is determined. The results of modeling random processes with given correlation functions and a homogeneous Markov process with a given transition probability are given as well as an example of the transition from a given spectral density of a random process to its correlation function. The results obtained confirm the effectiveness and feasibility of the developed modeling methods which will allow them to be used in computer research and design of various systems.

Importance sampling for reliability assessment of dynamic systems under general random process excitation

3 - Time varying signals

These notes provide an introduction to three aspects of mechanics of materials with random and multiscale microstructures: stochastic tools, scale effects in constitutive response, and computational methods. The overriding objective in the first Section (Basic Probability Tools) is to give an account of tools of probability theory and random (or so-called stochastic) processes — admittedly a vast subject area per se — at a level sufficient to describe non-deterministic, spatial material fluctuations. An attempt is made here to present random processes as objects parametrized by space coordinates, rather than time. The latter viewpoint is classical throughout most of the literature on random processes, but ours allows a natural generalization to random fields. Indeed, from that standpoint, each and every material microstructure may be considered a random field, although, as pointed out in the chapter by D. Jeulin, a different modeling strategy — based, say, on mathematical morphology — may be more suitable. The presentation covers scalar and vector processes and fields, all illustrated by measurements of paper mechanical properties. This introduces basic concepts of wide-sense stationarity and ergodicity, and sets the base for analyses of random continua.

The chapter contains elements of theory of random processes required for time series analysis and for understanding its results. The time series discussed in this book belong to the class of stationary random processes, which means that their statistical properties averaged over an ensemble of realizations of the process do not depend upon the time origin. If averaging over an ensemble of realizations is equivalent to averaging over time of any single realization, the process is ergodic. If a stationary process is Gaussian, it is also ergodic. In our research, we mostly have just one sample realization (one time series) of finite length and by extending the properties estimated from a single time series to the entire process under the study, we assume that the process is ergodic. The most important characteristics of any time series include the covariance function and spectral density, which describe the time series properties in the time and frequency domains, respectively. The two stationary discrete random processes typical for climatology and other disciplines are white noise and Markov chains. Examples of the true and estimated correlation functions and spectral densities including their sampling variability are briefly discussed for simulated time series of length \( N = 100 \).

Characterizing the relation between weight structure and input/output statistics is fundamental for understanding the computational capabilities of neural circuits. In this work, I study the problem of storing associations between analog signals in the presence of correlations, using methods from statistical mechanics. I characterize the typical learning performance in terms of the power spectrum of random input and output processes. I show that optimal synaptic weight configurations reach a capacity of 0.5 for any fraction of excitatory to inhibitory weights and have a peculiar synaptic distribution with a finite fraction of silent synapses. I further provide a link between typical learning performance and principal components analysis in single cases. These results may shed light on the synaptic profile of brain circuits, such as cerebellar structures, that are thought to engage in processing time-dependent signals and performing on-line prediction.

This paper presents an approach for the analysis of gated Gaussian impulsive noise and its effects in M-ary Quadrature Amplitude Modulation (M-QAM) schemes. In the approach, both amplitude variation and noisy pulse duration can be characterized as an amplitude modulation of the impulsive component by a discrete (binary or m-ary) random process. New exact expressions are obtained for the probability density function, autocorrelation function and power spectral density of the noise as well as for the bit error probability of M-QAM considering the maximum {\em a posteriori} probability optimum receiver. An important aspect of the proposed approach is the fact that the discrete random process incorporates the main parameters of the impulsive noise, such as amplitude, duration, instants in which the noise is added and time intervals between instants in which the noise is added.

Estimation of nonlinear communication channel has long been an important problem in digital communications. Being able to identify the nonlinear characteristics of the channels can help in the design of the nonlinear equalizer. In this paper, we propose a novel method for estimation of cubically nonlinear channels using random multisine signals as the input. By analyzing the higher-order moment spectra of random multisine input signals, a computationally efficient algorithm for identifying the Volterra kernels of the nonlinear channel is derived. In addition, we show that, with properly designed random multisines as driving signals, the channel estimate obtained by the proposed method is equal to the optimum minimum mean square error solution. The goodness of the proposed method is also demonstrated by computer simulation.

A mathematical model for discrete-time buffer systems with correlated output process

A class of discrete second-order random processes, described by linear time-invariant state-space models, is investigated. Equivalent realizations with reduced number of noise inputs are presented. In contrast to the innovations approach these realizations have time-invariant parameters.

Riemann and Weierstrass walks revisited

The performance of various discrete adaptive delta modulation (ADM) systems in the presence of random signal input is analyzed. The ADM systems considered are (a) Winkler system, (b) Abate (1967) system, (c) Bosworth and Candy (1968) system, and (d) Tazaki system (1977). It is assumed that the input random signal has a slope which varies randomly having Gaussian, Rayleigh and exponential probability density function (PDF) and has rectangular and triangular power spectral density (PSD). For an input random signal with specific PSD and PDF of its slope, the SNR has been calculated and plotted against the signal power. A performance evaluation has been made and summarized for the four systems above. >

Publication in the conference proceedings of EUSIPCO, Tampere, Finland, 2000

In many engineering problems, the power spectra of random signals have to be measured. In this paper, a method for estimating the unknown parameters in the power spectral density functions of discrete random processes is presented. The spectral density functions are represented by a ratio of two real polynomials in z. It is assumed that only a finite record of the random process is observed and no knowledge about its probability distribution is required. An estimation procedure is developed. It is shown that the denominator parameter estimates can be first obtained. Then the numerator parameters are estimated by filtering the poles from the spectrum. A four parameter example is given to demonstrate the method.

We investigate the waiting time statistics of solar flares both in a flare-productive active region (AR 12673) of the solar cycle 24 and in a three-dimensional magnetohydrodynamic (MHD) simulated AR. The statistical models of a discrete random process and a continuous memory-dependent process are applied to interpret the waiting time distributions (WTDs) of solar flares. Our results indicate that the occurrence of a solar flare in an individual AR maintains a certain amount of memory, and probably arises from MHD turbulence rather than from intermittent avalanches in a self-organized criticality system. It differs from the occurrence of ‘super flares’ when treating the star/Sun as a single non-linear system.