Reformulating the least-square source localization problem with contracted distances

In this paper a novel least-square (LS) formulation of the source localization problem is proposed. We prove that if the source lies within the convex-hull formed by the anchors, the source-to-anchor distance estimates d_i + ¿_i, ¿i are negative and the vector ¿ lies in the null subspace of the relative angle matrix ¿, then: 1) the associated least-square objective is a convex function, and 2) its global minimum coincides with the source´s true location. Consequently, the LS source localization problem can be cast as a null space problem (NSP), which proves mostly unaffected by to the most fundamental limitations of the classical LS source-localization problem, namely, sensitivity to noise and/or bias on the distance estimates and presence of local minima in the optimization objective. The results open an entirely new direction for the design of highly accurate and robust source localization algorithms, an example of which is provided.