Characterization of polar to Cartesian coordinates transformation and its effect on target track quality

Abstract

Target tracking with radar and sonar is done in either spherical or rectangular coordinates. Often tracking is done in one reference frame while filtering, usually Kalman, is done in another reference frame. It is commonly assumed that the probability density functions can be treated the same in both reference frames. An extended Kalman filter is used under this assumption that the probability density function of the measurements after conversion can be adequately characterized by the mean and standard deviation. The transformations from spherical coordinates (see manuscript) to Cartesian coordinates (see manuscript) is a non-linear transformation, so the statistical characteristics of the measurement process noise are changed significantly by the transformation. Thus, the characteristics which tracking filters are designed to optimize with respect to are changed as well by these coordinate transformations. Typical engineering practice uses approximations rather than exact solutions. The objective of this paper is to provide means to analytically characterize the probability density functions of these coordinate transformations. We then investigate the impact of approximating the noise statistics of these transformed coordinates on track quality.