Exact solutions of time difference of arrival source localisation based on semi-definite programming and Lagrange multiplier: complexity and performance analysis

In this study, the authors investigate the problem of source localisation based on the time difference of arrival (TDOA) in a group of sensors. Aiming to minimise the squared range-difference errors, the problem leads to a quadratically constrained quadratic programme. It is well known that this approach results in a non-convex optimisation problem. By proposing a relaxation technique, they show that the optimisation problem would be transformed to a convex one which can be solved by semi-definite programming (SDP) and Lagrange multiplier methods. Moreover, these methods offer the exact solution of the original problem and the affirmation of its uniqueness. In contrast to other complicated state-of-the-art SDP algorithms presented in the TDOA localisation literature, the authors methods are derived in a few straightforward reformulations and insightful steps; thus, there are no confusing and unjustifiable changes in the main optimisation problem. Furthermore, complexity analysis and a new approach for performance analysis, which show the merit of their methods, are introduced. Simulations and numerical results demonstrate that the positioning estimators resulted from the proposed algorithms outperform existing SDP-based methods presented so far.