Discrete optimal linear smoothing for systems with uncertain observations

The smoothing filter and smoothing error covariance matrix equations are developed for discrete linear systems whose observations may contain noise alone, where only the probability of occurrence of such cases is known to the estimator. An example of such a system arises in trajectory tracking, where the signal is first detected and then is processed by the estimator for tracking purposes. The results apply to any detection decision process, however, any such decision is associated with a false alarm probability, which is the probability that the detected signal contains only noise. The present results together with the earlier work of Nahi on prediction and filtering give a complete treatment of the discrete linear estimation problem for systems characterized by uncertain observations. These results, of course, reduce to well-known formulations for the classical estimation problem in the case where the observation is always assumed to contain the signal to be estimated.