Achieving Asymptotic Efficient Performance for Squared Range and Squared Range Difference Localizations

The estimation of a source location using directly the range or range difference measurements is difficult and requires numerical solution, which is caused by the highly non-linear relationship between the measurements and the unknown. We can obtain a computationally efficient and non-iterative algebraic solution by squaring the measurements first before solving for the unknown. However, a recent study has shown that such a solution is suboptimum in reaching the CRLB performance and the localization accuracy could be significantly worse in some localization geometries. This paper demonstrates that when range weighting factors are introduced to the squared measurements, the resulting solution will be able to reach the CRLB accuracy. Both the squared range and squared range difference cases are considered, and the mean-square error (MSE) and the bias of the resulting solutions are derived. The asymptotic efficiency of the proposed cost functions are proven theoretically and validated by simulations. The effects of range weighting factors on the localization performance under different sensor number, noise correlation, and localization geometry are examined. Introducing range weightings to the squared range measurements increases the bias but it is negligible in the MSE. Having range weightings in the squared range difference measurements improves both the MSE and bias.