A linear combination-based weighted least square approach for target localization with noisy range measurements

• The target localization is formulated as a weighted least square optimization problem.
• The coordinate of the sensor is expressed as a linear combination of virtual anchors.
• The originally non-convex optimization problem is converted to a convex form.
• The optimization is solved by linear least square method with high efficiency.
• A theorem is given to guarantee the accuracy of linear combination approximation.

Target localization based on range measurements from a set of anchors plays an important role in positioning systems and sensor networks. The localization is generally formulated as an optimization problem to tackle the noisy measurements. However, the objective is non-convex, and thus localization is difficult to solve in its original form. In this paper, a convex objective function is derived based on a linear combination scheme, within which the target position is expressed as a linear combination of positions of virtual anchors around its real position. In addition, the linear combination provides a highly accurate approximation for the computation of the distance from the anchors. Thus, the localization is formulated as a convex problem to find the optimal coefficients of the linear combination and is solved efficiently by the weighted linear least square method. As demonstrated by numerical experiments, the proposed approach, which achieves an approximately 35% improvement in accuracy and 98.5% shorter optimization time compared to the most accurate existing method, is very close to the Cramér–Rao lower bound (CRLB) while maintaining a quite high localization speed, and also works well with real measurement data.