In order to better capture impulsive noises in dynamic systems, a state estimator in the presence of Cauchy distributed process and measurement noises has been studied in recent years. The Cauchy estimator is determined by expressing in closed form the characteristic function (CF) of the unnormalized conditional probability density functions of the states given measurement history. The CF is comprised of a sum of exponentials multiplied by a coefficient, both being nonlinear functions of the measurements and the spectral variable. In this paper, we uncover important properties of the exponential terms in the CF for the two-state Cauchy estimator. These properties can be used to simplify the estimator structure significantly.