Simulation of stationary stochastic processes

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 specifically, 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 ¿ greater than or equal to the `switch-on¿ time of the system. The advantage of the `spectral-factorisation¿ 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.