The choice of signal-process models in Kalman-Bucy filtering

Kalman and Bucy have shown how to obtain the linear least-squares estimate of a signal, given observations of the signal plus independent white noise, and given a lumped-parameter or state-variable model for the process. The filter producing the signal estimate produces it as a linear functional of an estimate of the state of the model; and although the variance in the error of the signal estimate is independent of that particular model out of the infinitely many possible assumed to generate the signal, the associated covariance of the estimation error in the system states is dependent on the choice of model. The paper establishes that there is one particular model yielding a smallest error-variance in a sense to be described, and that this model is causally invertible. In the particular case where the signal process is stationary and observed over a semi-infinite time interval, this means that the model has the minimum-phase property.