On the bias of pseudolinear estimators for time-of-arrival based localization

Closed-form pseudolinear estimators are computationally attractive alternatives to iterative nonlinear techniques. For time-of-arrival (TOA) based localization, several pseudolinear estimators have been proposed such as the least squares calibration (LSC) estimator, the linear least squares (LLS) estimator, and their best linear unbiased estimator (BLUE) variants (namely, the BLUE-LSC and BLUE-LLS estimators). Despite their stable performance and low computational complexity, these pseudolinear estimators suffer from bias problems due to the nonzero mean of the pseudolinear noise vectors. In this paper, we present a bias analysis for the TOA-based pseudolinear estimators. Based on the bias analysis we develop bias compensation methods that lead to new bias-compensated versions of the LSC, LLS, BLUE-LSC and BLUE-LLS estimators. The superior performance of the proposed bias-compensated estimators is demonstrated via numerical simulations. The new estimators are observed to exhibit negligible estimation bias even at high measurement noise levels.