MA estimation in polynomial time

The parameter estimation of moving-average (MA) signals from second-order statistics was deemed for a long time to be a difficult nonlinear problem for which no computationally convenient and reliable solution was possible. We show how the problem of MA parameter estimation from sample covariances can be formulated as a semidefinite program that can be solved in a time that is a polynomial function of the MA order. Two methods are proposed that rely on two specific (over) parametrizations of the MA covariance sequence, whose use makes the minimization of a covariance fitting criterion a convex problem. The MA estimation algorithms proposed here are computationally fast, statistically accurate, and reliable. None of the previously available algorithms for MA estimation (methods based on higher-order statistics included) shares all these desirable properties. Our methods can also be used to obtain the optimal least squares approximant of an invalid (estimated) MA spectrum (that takes on negative values at some frequencies), which was another long-standing problem in the signal processing literature awaiting a satisfactory solution.