RSS-based Multiple Sources Localization with Unknown Log-normal Shadow Fading

Multi-source localization based on received signal strength (RSS) has drawn great interest in wireless sensor networks. However, the shadow fading term caused by obstacles cannot be separated from the received signal, which leads to severe error in location estimate. In this paper, we approximate the log-normal sum distribution through Fenton-Wilkinson method to formulate a non-convex maximum likelihood (ML) estimator with unknown shadow fading factor. In order to overcome the difficulty in solving the non-convex problem, we propose a novel algorithm to estimate the locations of sources. Specifically, the region is divided into $N$ grids firstly, and the multi-source localization is converted into a sparse recovery problem so that we can obtain the sparse solution. Then we utilize the K-means clustering method to obtain the rough locations of the off-grid sources as the initial feasible point of the ML estimator. Finally, an iterative refinement of the estimated locations is proposed by dynamic updating of the localization dictionary. The proposed algorithm can efficiently approach a superior local optimal solution of the ML estimator. It is shown from the simulation results that the proposed method has a promising localization performance and improves the robustness for multi-source localization in unknown shadow fading environments. Moreover, the proposed method provides a better computational complexity from $O(K^3N^3)$ to $O(N^3)$.