Derivation of a Kalman-type filter for linear systems with pointwise delay in signal noise

Abstract

Recently, it was demonstrated that a modification of the Kalman-filtering model with a pointwise delay of the signal noise could improve communication with considerably distanced spacecraft. However, a complete and correct derivation of the equations of the Kalman-type filter for this case has not yet been provided. In this paper, we close this gap. The method of derivation is based on a passage from the distributed delay to pointwise by means of a delta function. The derived equations constitute a system of first-order partial differential equations with the initial and boundary conditions.