A simple smoother: closed-form solution and analytical results

A closed-form solid-state solution for a simple one-dimensional smoother is presented. The target position-velocity dynamics are modeled as a linear discrete Gauss-Markov process: the discrete position measurements are assumed to be independent and identically normally distributed. Analytical results derived from estimating optimum steady-state position and velocity are given as well as the improvement provided by smoothing relative to pure filtering estimates. Analytical results, including appropriate limiting cases, show that the ratios of the RMS smoothing to the RMS filtering error are from 1/2 to 1 in position and from 1/2 to 1/√2 in velocity. As a numerical example a highly maneuvering target characterized by an acceleration width of 30 m/s² is considered