Characterizing innovations realizations for random processes

Abstract

In this paper we are concerned with the theory of second order (linear) innovations for discrete random processes. We show that of existence of a finite dimensional linear filter realizing the mapping from a discrete random process to innovations is equivalent to a certain semiseparable structure of the covariance sequence of the process. We also show that existence of a finite dimensional realization (linear or nonlinear) of the mapping from a process to its innovations implies that the process have this semiseparable covariance sequence property. In particular, for a stationary random process, the spectral density function must be rational.

^†Work supported by the National Science Foundation through grant ECS82-05772

^†Work supported by the National Science Foundation through grant ECS82-05772

^‡Work supported by U.S. Air Force through grant AFOSR-80-0196

^†Work supported by the National Science Foundation through grant ECS82-05772

^†Work supported by the National Science Foundation through grant ECS82-05772

^‡Work supported by U.S. Air Force through grant AFOSR-80-0196

Notes

^†Work supported by the National Science Foundation through grant ECS82-05772

^†Work supported by the National Science Foundation through grant ECS82-05772

^‡Work supported by U.S. Air Force through grant AFOSR-80-0196