Algebraic Complete Solution for Joint Source and Sensor Localization Using Time of Flight Measurements

Joint source and sensor localization is a challenging problem, nevertheless it has numerous potential applications. This paper takes time of flight measurements and develops an approximate and yet reasonably accurate algebraic solution to the problem by converting it to the identification of an upper triangular linear transformation matrix in the localization space. The proposed solution is in closed-form and not iterative, and it reduces the solution evaluation to the positive root of a polynomial of degree 3 or less. It is among the first in solving the problem when the source and/or sensor positions can be embedded in a lower dimensional space than the localization space, leading to a complete and robust solution. The solution derivation establishes naturally the conditions for the solvability of the localization problem. The resulting solution is sufficiently close to the optimum estimate and gives better results than those from the literature. Iterative refinement of the proposed solution on the source and sensor positions provides the Cramér-Rao Lower Bound accuracy under Gaussian noise with sufficient number of sources and sensors. Performance evaluation using the data from two experiments under real environment confirms the reliability and promising behavior of the proposed solutions.