Polynomial bounds for the linear prediction coefficients

Abstract

The purpose of this paper is to introduce a new methodology for the study of the numerical behaviour of the Toeplitz system and the quantities involved in the linear prediction problem. For this reason, it is proved that the positive definiteness of the system matrix is equivalent to a set of constraints on the autocorrelation innovation, for which new explicit recursive formulae are given. Next, through these formulae, the minimum bounds of the absolute values of the linear prediction coefficients are computed, which are of the order of p sol;p 2 . However, it is proved that, by imposing proper restrictions on the autocorrelation values, a linear or a polynomial bound of a desired order for the LP coefficients can be obtained. Finally, using the previous analysis, the ill-conditioning of the Toeplitz system and the sensitivity of the determinants of the corresponding matrices are discussed.