A utilization of properties of the discrete-time Riccati equation in stochastic realization theory

The problem of generating families of wide-sense, stochastic realizations of a discrete-time stationary stochastic process is considered. To do this, it is known that a Riccati equation has to be solved. In this paper, the non-Riccati algorithm of Lindquist and Kailath is used to generate families of realizations, the state covariances of which are totally ordered. Finally, the property of constant directions which the discrete-time Riccati equation enjoys is utilized to obtain families of realizations, the state covariances of which have the same value in certain directions.