Toeplitz covariance estimation with applications to MUSIC

Abstract

We consider the maximum likelihood estimation of a covariance matrix that is known to be Toeplitz. The corresponding optimization problem is nonconvex, and to compute a local minimizer we propose an efficient implementation of Newton’s method. The cost per iteration of a straightforward implementation is of order O(n4), where n+1 is the dimension of the covariance matrix. By exploiting fast Fourier transforms to assemble the gradient and Hessian we reduce the iteration cost to order O(n3). Through extensive simulations we demonstrate that for direction-of-arrival estimation, the performance of MUSIC is significantly enhanced by using it together with the maximum likelihood Toeplitz covariance estimate. The simulations also demonstrate that our open-source implementation is orders of magnitude faster than two existing Toeplitz covariance estimators.