On the approximate stochastic realisation problem

The approximate stochastic realization problem is considered. This is to find a linear discrete time model, driven by white noise, whose output has a covariance which is approximately equal to a given covariance sequence and the error lies within some specified bounds. A parameterization of the covariance of an ARMA (autoregressive moving average) process in terms of the poles and the residues of the partial fraction expansion of the model transfer function is presented. Applying the Nelder-Mead simplex algorithm for nonlinear optimization, the above parameters are estimated in such a way that the error between the model output covariance and the given sequence satisfies the tolerance requirements