A Two-Stage Iteratively Reweighted Least Squares Estimator for Robust Elliptic Localization in the Presence of Impulsive Noise

As a range-based localization approach, elliptic localization based on bistatic ranges (BRs) has wide range of applications in multistatic systems. This paper addresses the problem of robust elliptic localization using the minimum <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{p}$</tex-math></inline-formula>-norm criterion in scenarios affected by impulsive <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula>-stable noise. To solve the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ \ell _{p}$</tex-math></inline-formula>-norm formulation, we develop an asymptotically efficient two-stage iteratively reweighted least squares (TSIRLS) estimator via incorporating pseudolinear estimation and multi-stage weighted least squares estimation into the framework of iteratively reweighted least squares (IRLS) methods. The estimator consists of two cascaded <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ \ell _{p}$</tex-math></inline-formula>-norm minimization estimators. To obtain a closed-form solution at each iteration, we transform nonlinear BR measurement equations into pseudolinear forms through introducing auxiliary variables, thus resulting in an iteratively reweighted pseudolinear least squares (IRPLS) estimator in the first stage. The dependencies of unknown parameters and the estimate error derived from the IRPLS are utilized in the second stage to enhance the performance of target location estimate. The analytical derivation demonstrates that, under the assumptions of a sufficiently large number of measurements and small noise, the final position estimate obtained from the TSIRLS achieves the theoretical covariance of the general <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ \ell _{p}$</tex-math></inline-formula>-norm minimization estimation. Extensive numerical simulations highlight the advantages of TSIRLS over existing least squares and robust estimators in terms of positioning performance and runtime. The TSIRLS is also observed to generate approximately unbiased estimates with mean square errors that closely approach the Cramér-Rao lower bound.