A Message Passing Based Iterative Algorithm for Robust TOA Positioning in Impulsive Noise
Wenxin Xiong, Christian Schindelhauer, Hing Cheung So, Stefan J. Rupitsch
Abstract
In this contribution, we explore further possibilities for statistical robustification of the traditional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula> -space based time-of-arrival location estimator under impulsive noise conditions. We replace the non-robust <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula> loss by 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> counterpart with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1 \leq p < 2$</tex-math></inline-formula> , and devise an iteratively reweighted least squares (IRLS) type approach to tackle 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> -minimization formulation in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(N_{\text{IRLS}}L)$</tex-math></inline-formula> time. Here, the iteration number <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N_{\text{IRLS}}$</tex-math></inline-formula> is a constant typically of several tens and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L$</tex-math></inline-formula> represents the number of sensors. The key enabler for the rapid but reliable update of location estimate at each iteration, is the sum-product message passing implemented in an acyclic factor graph derived from the corresponding subproblem. Numerical results demonstrate the superiority of our algorithm over several existing statistical robustification methods in terms of computational simplicity and positioning accuracy in the presence of impulsive noise.