Root-exchange property of constrained linear predictive models

In recent works, we have studied linear predictive models constrained by time-domain filters. In the present study, studied the one-dimensional case in more detail. Firstly, we obtain root-exchange properties between the roots of an all-pole model and corresponding constraints. Secondly, using the root-exchange property we can construct a novel matrix decomposition A^TRA^# = I, where R is a real positive definite symmetric Toeplitz matrix, superscript ^# signifies reversal of rows and I is the identity matrix. In addition, there exists also an inverse matrix decomposition C^TR^-1C^# = I, where C ∈ C is a Vandermonde matrix. Potential applications are discussed.