Levinson-type extensions for non-Toeplitz systems

It is shown Levinson´s basic principle for the solution of normal equations which are of Toeplitz form may be extended to the case where these equations do not possess this specific symmetry. The method is illustrated by application to various examples which are chosen so that the coefficient matrix possesses various symmetries. Specifically, the solution of the normal equations when the associated matrix is the doubly symmetric non-Toeplitz covariance matrix is considered. Next, the solution of extended Yule-Walker equations where the coefficient matrix is Toeplitz, but nonsymmetric is obtained. Finally, the approach is illustrated by considering the determination of the prediction error operator when the normal equations are of symmetric Toeplitz form