An asymptotically unbiased estimator for bearings-only and Doppler-bearing target motion analysis

Bearings-only (BO) and Doppler-bearing (DB) target motion analysis (TMA) attempt to obtain a target trajectory based on bearings and on Doppler and bearing measurements, respectively, from an observer to the target. The BO-TMA and DB-TMA problems are nontrivial because the measurement equations are nonlinearly related to the target location parameters. The pseudolinear formulation provides a linear estimator solution, but the resulting location estimate is biased. The instrumental variable method and the numerical maximum likelihood approach can eliminate the bias. Their convergence behavior, however, is not easy to control. This paper proposes an asymptotically unbiased estimator of the tracking problem. The proposed method applies least squares minimization on the pseudolinear equations with a quadratic constraint on the unknown parameters. The resulting estimator is shown to be solving the generalized eigenvalue problem. The proposed solution does not require initial guesses and does not have convergence problems. Sequential forms of the proposed algorithms for both BO-TMA and DB-TMA are derived. The sequential algorithms improve the estimation accuracy as a new measurement arrives and do not require generalized eigenvalue decomposition for solution update. The proposed estimator achieves the Cramer-Rao Lower Bound (CRLB) asymptotically for Gaussian noise before the thresholding effect occurs.