Exact Solution of an Approximate Weighted Least Squares Estimate of Energy-Based Source Localization in Sensor Networks

In this paper, we consider an approximate weighted least squares (AWLS) estimate for the energy-based localization in wireless sensor networks, which is a quartic programming and is nonconvex in general. Based on a matrix rank-one decomposition (ROD) technique, we prove that this model can be transformed into a convex constrained programming model, and the proof is constructive. Furthermore, by a theoretical analysis of the semidefinite programming (SDP) model proposed by Wang, we present an easy verifiable condition that characterizes when the SDP model admits no gap with the primal AWLS model. By using this condition, we design a new algorithm that combines the SDP algorithm with the ROD technique. The new algorithm can solve the AWLS model exactly in theory and overcomes the shortcomings of the SDP algorithm. The numerical results show that, at low noise levels, the performance of the proposed algorithm is very close to Cramer-Rao bound accuracy, while the SDP method is not; at high noise levels, the performance of the proposed algorithm is similar to the SDP method while the SDP method is regarded best in the corresponding reference. Numerical simulations indicate that the proposed new algorithm performs better than other existing methods.