Unbiased Conversion of 3-D Bistatic Radar Measurements to Cartesian Position

Tracking with bistatic radar measurements is a challenging problem due to the nonlinear relationship between the radar measurements and the Cartesian coordinates, especially for long distances. This nonlinearity leads, for three-dimensional (3-D) bistatic radar, to a nonellipsoidal measurement uncertainty region in Cartesian coordinates, similar to a thin contact lens, that causes consistency problems for a tracking filter. A solution is proposed by developing an unbiased and statistically consistent conversion of the bistatic 3-D sine-space position measurements to Cartesian coordinates, based on second-order Taylor expansion. Such an approach was successfully used for monostatic radars but considered impractical for the bistatic case due to the difficulty to derive explicit conversion expressions, as done in this article. The recently developed 2-D approach for the bistatic case could not be successfully extended to 3-D and a different conversion was needed. It is shown that in contrast to the first-order conversion and the best nonlinear conversions (based on the cubature method), only the proposed conversion can cover the true uncertainty.