A Closed-Form Pseudo-Linear Estimator for Bearing-Only Tracking With Signal Delay

In bearing-only tracking (BOT), due to the finite speed of signal propagation, the measured bearing signals are often subject to delay. If signal delays are ignored, the estimation of the target’s position and velocity will be biased, and this bias will not diminish as the number of measurements increases. To eliminate the estimation bias caused by signal delay, this article proposes an asymptotically unbiased two-stage pseudo-linear estimator (PLE). In the first stage, a PLE is employed without considering a signal delay to obtain the nominal solution for the target state. The bias of this solution is then analyzed, proving that the bias does not disappear with an increasing number of measurements. In the second stage, the impact of signal delay is taken into account, and an error vector is defined as the difference between the nominal solution and the true value of the target. First, the pseudo-linear equation for the error vector is established, and then, a closed-form weighted instrumental variable (WIV) estimator is introduced to estimate the error vector, with a proof of the asymptotic unbiasedness of the WIV estimator. The estimated error vector is added to the nominal solution, resulting in the final target state and thereby eliminating the estimation bias caused by measurement signal delay. The simulation results demonstrate that, in BOT scenarios for passive radar and passive sonar, the proposed method effectively eliminates the estimation bias due to signal delay, improves estimation accuracy, and achieves accuracy close to the Cramer-Rao lower bound (CRLB).