The two-state estimator for linear systems with additive measurement and process Cauchy noise

The conditional mean estimator for a two-state linear system with additive Cauchy measurement and process noises is developed. Although the Cauchy densities that model the process and measurement noise have an undefined first moment and an infinite second moment, the probability density function conditioned on linear noisy measurements does have a finite mean and variance. The conditional probability density function (cpdf) given the measurement history appears to be difficult to compute directly. However, the characteristic function of the unnormalized cpdf can be sequentially propagated through measurement updates and dynamic state propagation. A key step in processing a measurement lies in obtaining a closed form solution of an associated convolution integral. The solution obtained in this work is presented as a sum of terms all contributing to the structure of the characteristic function of the unnormalized cpdf, where the number of terms grows arithmetically with time. Many of these terms have a negligible contribution and can therefore be pruned. Once this characteristic is obtained, the mean and variance are easily computed from the first and second derivatives of the characteristic function, evaluated at the origin in the spectral variables´ domain. A two-state dynamic system example numerically demonstrates the Cauchy estimator´s performance compared with that of a Kalman filter for simulations using both Cauchy and Gaussian noises.