Near to Toeplitz structure and efficient schemes for linear modeling

The purpose of this paper is to give reduced complexity algorithms in order to compute linear models in situations of non-stationnary signals. The case of impulse response identification is examined as well as the case of pure AR-modeling. Both problems lead to a linear system which must be solved to give the unknown AR-model or the unknown impulse response. Furthermore the linear systems that occur in both cases have an identical left hand side member characterized by a symmetric matrix having the supplementary property of being the product of two non-symmetric Toeplitz matrices. The "near to Toeplitz" structure of this matrix permits the fast solution of the system with specially adapted algorithms to this situation. Many properties of the near to Toeplitz matrices are outlined as the key formulas for the derivation of all fast schemes. Order, time and order-time recursive methods are derived using a unified theoretical framework. The methodology used here is similar to that used by Morf et al. The algorithms presented are new and can be converted to efficient and automated computer packages.