Input-to-state Covariances for Spectral Analysis: the Biased Estimate

In many practical applications second order moments are used for estimation of power spectrum. This can be cast as an inverse problem where we seek a spectrum consistent with a given state-covariance matrix. Such a solution exists if and only if the covariance matrix is positive and belong to a low dimensional subspace. The sample covariance estimate of such a matrix will typically fall outside this class. Several approaches have been proposed, determining a covariance matrix within the correct structure minimizing the distance to the estimated sample covariance. Here we take another approach, akin to the biased estimate in the classical autocovariance case, where the original estimate is guaranteed to belong to the correct class. The proposed procedure is computationally efficient compared to previous approximation approaches and numerical examples suggest increased reliability compared to the ordinary sample covariance.