Cooperative localization in hybrid infrared/visible light networks: Theoretical limits and distributed algorithms

Light emitting diode (LED) based visible light positioning (VLP) networks can provide accurate location information in environments where the global positioning system (GPS) suffers from severe signal degradation and/or cannot achieve high precision, such as indoor scenarios. In this manuscript, we propose to employ cooperative localization for hybrid infrared/visible light networks that involve multiple LED transmitters having known locations (e.g., on the ceiling) and visible light communication (VLC) units equipped with both LEDs and photodetectors (PDs) for the purpose of cooperation. In the considered scenario, downlink transmissions from LEDs on the ceiling to VLC units occur via visible light signals, while the infrared spectrum is utilized for device-to-device communications among VLC units. First, we derive the Cramer-Rao lower bound (CRLB) and the maximum likelihood estimator (MLE) for the localization of VLC units in the proposed cooperative scenario. To tackle the nonconvex structure of the MLE, we adopt a set-theoretic approach by formulating the problem of cooperative localization as a quasiconvex feasibility problem, where the aim is to find a point inside the intersection of convex constraint sets constructed as the sublevel sets of quasiconvex functions resulting from the Lambertian formula. Next, we devise two feasibility-seeking algorithms based on iterative gradient projections to solve the feasibility problem. Both algorithms are amenable to distributed implementation, thereby avoiding high-complexity centralized approaches. Capitalizing on the concept of quasi-Fej'er convergent sequences, we carry out a formal convergence analysis to prove that the proposed algorithms converge to a solution of the feasibility problem in the consistent case. Numerical examples illustrate the improvements in localization performance achieved via cooperation among VLC units and evidence the convergence of the proposed algorithms to true VLC unit locations in both the consistent and inconsistent cases. IEEE