The serial/matrix technique applied to the analysis of linear systems with stationary random inputs

A method is presented for the determination of the output autocorrelation functions and output mean square values of time-invariant linear systems subjected to stationary random inputs with a range of possible input autocorrelation functions. The factorized transfer function of the system is regarded as representing that of a chain of filters and the input autocorrelation function is assumed to be a linear combination of certain basic autocorrelation functions. The coefficients of these basic functions form an input coefficient vector. The output autocorrelation function is then shown to be obtainable by successive matrix multiplications of this coefficient vector by autocorrelation response matrices representing the individual filters. The output mean square value is obtained by setting τ = 0 in the output autocorrelation function. The autocorrelation response matrices for standard filter elements are tabulated for a basic set of commonly occurring input autocorrelation functions. It is demonstrated that the method is easily adapted to cope with ideal low-pass and band-pass filtering.