Improvements on cauchy estimation for linear scalar systems

Recently, a new estimation paradigm was presented for scalar discrete linear systems entailing additive process and measurement noises that have Cauchy probability density functions (pdf). For systems with Gaussian noises, the Kalman filter has been the main estimation paradigm. However, many practical system uncertainties that have impulsive character, such as radar glint, are better described by stable non-Gaussian densities, for example, the Cauchy pdf. Although the Cauchy pdf does not have a well defined mean and has an infinite second moment, the conditional density of a Cauchy random variable, given its linear measurements with an additive Cauchy noise, has a conditional mean and a finite conditional variance, both being functions of the measurement. Based on this fact, a sequential estimator was derived analytically. This recursive Cauchy conditional mean estimator has parameters that are generated by linear difference equations with stochastic coefficients. Analytically, the number of filter parameters grows constantly. In the current paper we prove that the majority of those parameters decay with time, thus allowing an accurate low dimensional approximation of the estimator that provides computational efficiency. Moreover, the original estimator structure is simplified by separating the deterministic and stochastic parts of the problem. In simulations, the performance of the Cauchy estimator is superior to a Kalman in the presence of Cauchy noise, whereas the Cauchy estimator deteriorates only slightly compared to the Kalman filter in the presence of Gaussian noise.