Best linear unbiased estimator approach for time-of-arrival based localisation

A common technique for source localisation is to utilise the time-of-arrival (TOA) measurements between the source and several spatially separated sensors. The TOA information defines a set of circular equations from which the source position can be calculated with the knowledge of the sensor positions. Apart from nonlinear optimisation, least squares calibration (LSC) and linear least squares (LLS) are two computationally simple positioning alternatives which reorganise the circular equations into a unique and non-unique set of linear equations, respectively. As the LSC and LLS algorithms employ standard least squares (LS), an obvious improvement is to utilise weighted LS estimation. In the paper, it is proved that the best linear unbiased estimator (BLUE) version of the LLS algorithm will give identical estimation performance as long as the linear equations correspond to the independent set. The equivalence of the BLUE-LLS approach and the BLUE variant of the LSC method is analysed. Simulation results are also included to show the comparative performance of the BLUE-LSC, BLUE-LLS, LSC, LLS and constrained weighted LSC methods with Crame-r-Rao lower bound.