An optimal linear time-invariant estimator for certain types of nonstationary processes

A technique is developed whereby one can synthesize a causal, linear time-Invariant estimator that is optimal for restricted types of nonstationary processes. The technique is applicable to linear, time-invariant systems (driven by nonstationary state noise) for which scalar observations are made in the presence of additive nonstationary noise. Two-dimensional Fourier transforms are used to obtain an expression for the estimator´s mean square error. It is assumed that it is desirable to minimize the time integral of this expression. The calculation of this integral results in an expression which can be minimized by selecting an estimator depending in a prescribed way on the two-dimensional Fourier transforms of the state and observation noise. The resulting estimator is causal, linear, and time invariant. It is similar in some respects to the Wiener filter that can be derived under the assumptions of stationary state and observation noise processes. The estimator´s usefulness is limited by the requirement that the observations be scalar, and the nonstationary processes have Fourier transformable autocorrelation functions.