On the modeling of discrete time Auto-Regressive representations

It is well known [2], [6], that given the discrete-time AutoRegressive representation A(σ)β(k) = 0; where σ denotes the shift forward operator and A(σ) a polynomial matrix, we can always construct the forward-backward behavior of this system, by using the finite and infinite elementary divisor structure of A(σ). The main theme of this work is to study the inverse problem: given a specific forward-backward behavior, find a family of polynomial matrices A(σ), such that the system A(σ)β(k) = 0 has exactly the prescribed behavior. As we shall see, the problem can be reduced either to a linear system equation problem or to an interpolation problem.