Two methods for Toeplitz-plus-Hankel approximation to a data covariance matrix

Recently, fast algorithms have been developed for computing the optimal linear least squares prediction filters for nonstationary random processes (fields) whose covariances have (block) Toeplitz-Hankel form. If the covariance of the random process (field) must be estimated from the data, the following problem is presented: given a data covariance matrix, computer from the available data, find the Toeplitz-plus-Hankel matrix closest to this matrix in some sense. The authors give two procedures for computing the Toeplitz-plus-Hankel matrix that minimizes the Hilbert-Schmidt norm of the difference between the two matrices. The first approach projects the data covariance matrix onto the subspace of Toeplitz-plus-Hankel matrices, for which basis functions can be computed using a Gram-Schmidt orthonormalization. The second approach projects onto the subspace of symmetric Toeplitz plus skew-persymmetric Hankel matrices, resulting in a much simpler algorithm. The extension to block Toeplitz-plus-Hankel data covariance matrix approximation is also addressed