An Efficient Iterative Least Square Method for Indoor Visible Light Positioning Under Shot Noise

Abstract

In this paper, we develop a set of effective algorithms for performing efficient and accurate visible light positioning (VLP) in the presence of shot noise, which is an important component in the received optical signal yet has been largely neglected in prior works. In particular, we formulate the positioning problem as a maximum log-likelihood optimization problem, which is nonconvex so that the standard numerical algorithm such as gradient descent (GD) and stochastic gradient descent (SGD) may not be able to find the global solution. To address this, we propose a novel least-square (LS) solver that can find a sub-optimal solution to the aforementioned non-convex optimization problem. Based on the LS solver, a set of more effective algorithms can be developed to further enhance the optimality of the solution. Specifically, we consider (1) combining the LS solver with GD, giving rise to the GD-LS algorithm; and (2) applying the LS solver in an iterative manner, giving rise to the iterative LS algorithm, which is a novel and efficient positioning algorithm. Moreover, we also provide a closed-form lower bound on the positioning error based on the Cramér-Rao lower bounds (CRLB). Numerical simulation shows that the proposed GD-LS and iterative LS algorithms cannot only achieve high positioning accuracy, but also enjoy low computation complexity: the average positioning accuracy of LS-GD is 0.009 m using computation time 0.046 s, and the iterative LS algorithm can achieve average positioning accuracy 0.023 m with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1.94\times 10^{-4}$</tex-math></inline-formula> s computation time, which outperform GD and SGD method.