Closed-Form and Near Closed-Form Solutions for TOA-Based Joint Source and Sensor Localization

We study the problem of joint source and sensor localization from time-of-arrival (TOA) measurements. For the case of nine sources and four sensors (or, by symmetry, vice versa), we derive a closed-form solution. For the cases of eight and seven sources, we derive near closed-form solutions depending on one or two parameters, respectively. The main aim of our method is to transform the original localization problem to other problems such that the number of unknown parameters is as small as possible. Although the constraints in the problems are multivariate and quadratic polynomial equations, by using a linear method of solving polynomial equations we can find closed-form solutions and near closed-form solutions if the number of unknown parameters is small and the number of constraints is large. Unlike off-the-shelf methods for solving polynomial systems, such as the Grobner basis method, our solution uses only low-degree monomials. Thus, unlike these other methods, our solution is more stable to noise, as verified using the Cramer-Rao lower bound.