Uncertainties in many practical systems, such as radar glint and sonar noise, have impulsive character and are better described by heavy-tailed non-Gaussian densities, for example, the Cauchy probability density function (pdf). The Cauchy pdf does not have a well defined mean and its second moment is infinite. Nonetheless, the conditional density of a Cauchy random variable, given a scalar linear measurement with an additive Cauchy noise, has a conditional mean and a finite conditional variance. In particular, for scalar discrete linear systems with additive process and measurement noises described by Cauchy pdfs, the unnormalized characteristic function of the conditional pdf is considered. It is expressed as a growing sum of terms that at each measurement update increases by one term, constructed from four new measurement-dependant parameters. The dynamics of these parameters is linear. These parameters are shown to decay, allowing an approximate finite dimensional recursion. From the first two differentials of the unnormalized characteristic functions of the conditional pdf evaluated at spectral value of zero, the conditional mean and variance are easily obtained. The effectiveness of this estimator is examined using a target interception example that also demonstrates its robustness when processing data with non-Cauchy noises.