Efficient construction of canonical ladder forms for vector autoregressive processes

The paper treats the problem of constructing canonical ladder realizations for vector autoregressive (AR) processes specified by their characteristic matrix polynomials. The difficulty of this problem is rooted in the fact that the backward matrix polynomial corresponding to a given vector AR process is a nontrivial function of the forward matrix polynomial. The construction calls for solving a discrete Lyapunov equation in block-controller form. Two efficient procedures for solving this equation are presented, both requiring a number of operations that is proportional to at most the square of the model order. Applications of the new procedures to stability, tests, simulation of AR processes, and model reduction are described.