Maximum likelihood positioning 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. Due to its asymptotic properties Maximum Likelihood estimation is discussed. Although the problem is quadratic, the proposed solution is based on solving a linear set of equations. For precise measurements we obtain explicitly the exact solution with a small number of operations. For noisy measurements the method provides an excellent initial point for the application of the Gerchberg-Saxton iterations. Proof of convergence is provided. The case of planar geometry is coached using complex numbers which reveals a strong relation to the celebrated problem of phase retrieval. We provide a compact, matrix form of the Cramér-Rao bound, small error analysis and evaluation of the computational load. Numerical examples are provided to corroborate the results.