Maximum-likelihood position estimation of network nodes using range measurements

Given a network of stations with incomplete and possibly imprecise inter-station range measurements, it is required to find the relative positions of the stations. The authors show that for a planar geometry the problem can be couched using complex numbers. It then becomes evident that location estimation is equivalent to the celebrated problem of phase retrieval. Although the equations are quadratic, the proposed solution is based on solving a set of linear equations. For precise measurements, the exact solution is obtained with a small number of operations. For noisy measurements, the method provides an excellent initial point for the application of the Gerchberg-Saxton iterations that are usually associated with phase retrieval. Proof of convergence is provided for the iterations. Small error analysis of the algorithm proves that it is statistically efficient and therefore for small measurement errors achieves the Cramér-Rao lower bound. The authors provide a compact, matrix form expression for the Cramér-Rao bound and evaluation of the computational load. Numerical examples are provided to corroborate the results.