Semidefinite Programming for Wireless Sensor Localization with Lognormal Shadowing

The problem of estimating the positions of the sensors in a wireless sensor network is commonly known as the wireless sensor localization problem and has been formulated as a relaxed semidefinite programming problem assuming inter-sensor distance measures corrupted by additive Gaussian noise. In this paper, we assume received signal strength measurements under a lognormal shadowing pathloss model and formulate the corresponding non-convex maximum likelihood distance estimator. We apply two different approximations of the objective, a Taylor approximation and a minimax approximation, and then relax the problem to a semidefinite program. The performance of the two approximations is analyzed and compared to the Cramer-Rao bound. Finally, we show that the localization performance is not appreciably reduced when the pathloss parameter is unknown.