An algebraic description of realizations of partial covariance sequences

The solutions of the partial realization problem have to satisfy a finite number of interpolation conditions at 1. The minimal degree of an interpolating deterministic system is called the algebraic degree or McMillan degree of the partial covariance sequence and is easy to compute. The solutions of the partial stochastic realization problem have to satisfy the same interpolation conditions and have to fulfill a positive realness constraint. The minimal degree of a stochastic realization is called the positive degree. The interpolating deterministic solutions can be parameterized by the KimuraGeorgiou parameterization. In the literature, the solutions of the partial stochastic realization problem are then described by checking the positive realness constraints for each interpolating deterministic system. In this paper, an alternative parameterization for the deterministic solutions of the interpolation problem is presented. Both the solutions of the partial and partial stochastic realization problem are analyzed in this alternative parameterization. Based on the structure of the parameterization, a lower bound for the positive degree is obtained.