Optimal Filtering of Continuous-Time Stationary Processes by Means of the Backward Innovation Process

A new approach to linear least squares estimation of continuous-time (wide sense) stationary stochastic processes is presented. The basic idea is that the relevant estimates can be expressed not only in terms of the usual (forward) innovation process but also in terms of a backward innovation process. The functions determining the optimal filter as well as the error covariance functions are seen to satisfy some differential equations. As an important example the Kalman-Busy filter is considered. It is demonstrated that the optimal gain matrix can be determined from 2mn equations (where n is the dimension of the system and m of the output) rather than $\frac{1}{2}n(n + 1)$ as in the conventional theory. This is an advantage when, as is usually the case, $m \ll n$. These equations were first derived by Kailath, who used a different method. Also they are the continuous-time versions of some equations previously obtained (independently of Kailath) by the author.