Decomposition of Toeplitz Matrices via Convex Optimization

We point out that autocovariance functions of moving average processes of any given order m can be characterized via a linear matrix inequality (LMI). This LMI-condition can be used to decompose any Toeplitz autococovariance matrix into a sum of a singular Toeplitz covariance plus the autocovariance matrix of a moving average process of order m and of maximal variance. The decomposition is unique and subsumes the Pisarenko harmonic decomposition that corresponds to m=0. It can be used to account for mutual couplings between elements in linear antenna arrays or identify colored noise consistent with the covariance data. The same LMI-condition leads to an efficient computation of the least order of a MA-spectrum that agrees with covariance moments