Simulation of stationary stochastic processes

Abstract

A ‘spectral-factorisation’ procedure involving the solution of a Riccati matrix differential equation is considered to determine systems which, with white-noise input signals, may be used in the simulation of stochastic processes having prescribed stationary covariances. More spe$itically, the specification of a system is made so that the covariance of the system output is a prescribed stationary covariance R(t – ~) for all t and -r greater than or equal to the ‘switch-on’ time of the system. The advantage of the ‘spectralfactorisation’ procedure described compared with those previously given is that, assuming an initial-state mean of zero, a suitable initial-state covariance is calculated as an intermediate result in the procedure. The calculation of an appropriate initial-state covariance is of interest since, if zero initial conditions are used in an attempted simulation, an undesirable time lapse may be necessary for the output covariance to be acceptable as a simulation of the prescribed stationary covariance. For the case when the system is given or is determined using alternative procedures to those described in the paper, the initial-state covariance is calculated from the solution of a linear matrix equation. The problem considered in the paper is the simulation of stationary stochastic processes with prescribed covariances using linear, finite-dimensional, time-invariant systems with white-noise input. Of particular interest is the selection of an initial-state covariance, so that the covariance of the outputs will be indistinguishable from that observed over the same time interval for the hypothetical limiting case as the initial time approaches – m. Systems which may be used in the simulation of stationary stochastic processes with prescribed covariances may be determined from any of a number of spectral-factorisation procedures, ] z,* With regard to the initial conditions, Cttrrent practice is to set these to zero and ignore the outputs for a period corresponding to a few time constants of the system. The inadequacy of this procedure has been recognised.3 In the paper two results are presented. The first is a method for selecting an initial-state covariance for a given system, so that the application of white noise at the input results in outputs that may be considered, after the switch-on, as sample functions of a stationary stochastic process; this is the best possible real-time simulation for a stationary stochastic process. All that is required in order to obtain the result is the solution of a linear matrix equation. The second result of the paper is a spectral factorisation of a specified covariance matrix using theorems from Anderson. t The procedure gives a system having a stable transferfunction matrix with a stable inverse (often required in certain optimisation problems), together with the initial-state covariance; the advantage of the particular approach presented is that all the information necessary for the simulation is given in one procedure. The key step in the procedure is the solution of a quadratic matrix equation which satisfies certain constraints. This solution, which is unique, may be found using algebraic means similar to those of Reference 4 or by determining the steadystate solution of a Riccati matrix differential equation. t The method avoids the need to carry out any of the procedures in References 1, 2 or *, which prove very complex in cases where the covariance is a matrix rather than scalar.