Abstract
Let z(t) denote a random signal process, its observation process y(t) be given by y(t)_??_z(t)+v(t), and z(t) denote the linear least squares estimate of z(t) based on {y(τ);τ≤t}, where v(t) is a white process which is uncorrelated with z(t). It was first shown by Kailath in a heuristic manner that the innovation process ν(t) of y(t) defined by ν(t)_??_y(t)-z(t) is a white process with the same correlation operator as that of v and has the same information as the original observation process y as far as linear estimates are concerned. The complete proofs of these properties requires a rigorous treatment of the white process because v(t) appearing in the observation process is not an ordinary random process. One way to treat the white process is to employ a stochastic integral, which enables one to discuss a problem involving a white process without defining the white process explicitly. Another way is to define it in the framework of generalized random processes.
In 1972, Kailath gave a rigorous proof of the properties of the innovation process using stochastic integrals. In the same year, the proof using generalized random processes was given by H. Inaba. In this paper, we discuss the linear filtering problem for random fields and show that even for the case of random fields the corresponding innovation process has the same properties as those of the innovation process for stochastic processes.
