Linear Filtering for Wide Band Noise Driven Observation Systems

Abstract

In this paper, a linear filtering problem for the observation system driven by the sum of white and wide band noises and the signal system driven by white noise is considered. It is assumed that the signal noise is independent of the observation noises. Two filtering results are proved. The first result assumes that the observation white and wide band noises are correlated and present a complete set of equations for the optimal filter. This result is non-invariant in the sense that it depends on the relaxing function, generating the wide band noise, that is not available in real applications. To remove the non-invariance, the problem is handled under additional condition on independence of observation noises, and the equations of the optimal filter are modified to this case. The obtained result is invariant in the sense that it is independent of the relaxing function but depends on the autocovariance function of the wide band noise that is available in real applications. This filtering result is applied to linear quadratic Gaussian control problem. Application of these filtering results to tracking satellites and digital signal processing are discussed.