On the estimation of structured covariance matrices

This paper discusses a method for estimating the covariance matrix of a multivariate stationary process w generated as the output of a given linear filter fed by a stationary process y. The estimated covariance matrix must satisfy two constraints: it must be positive semi-definite and it must be consistent with the fact that w is the output of the given linear filter. It turns out that these constraints force the estimated covariance to lie in the intersection of a cone with a linear space. While imposing only the first of the two constraints is rather straightforward, guaranteeing that both are satisfied is a non-trivial issue to which quite a bit of attention has already been devoted in the literature. Our approach extends the method for estimating the Toeplitz covariance matrix of order M of a process y based on the biased spectral estimator (Stoica & Moses, 1997). This extension is based on the characterization of the output covariance matrix in terms of the filter parameters and the sequence of covariance lags of the input process. After introducing our estimation method, we propose a comparison performance between this one and other methods proposed in the literature. Simulation results show that our approach constitutes a valid estimation procedure.