Smoothing of linear scalar systems with cauchy distributed noises

The smoothing problem for a scalar linear system with additive Cauchy noises is to determine the mean of the history of the state given the measurement history over a defined time interval. Since the state history is structured as a vector, the conditional mean for additive Cauchy noises is best determined from a characteristic function approach. The derived smoothing algorithm recursively and analytically updates the characteristic function of the state history given the increasing measurement history, thereby increasing the dimension of the state history vector and the associated spectral vector, until the final stage is reached. The resulting characteristic function is then used to efficiently determine the smoothed conditional mean state history and the associated estimation error variance matrix.