Improved Time of Arrival Measurement Model for Non-Convex Optimization with Noisy Data

The quadratic system provided by the Time of Arrival technique can be solved analytical or by non-linear least squares minimization. In real environments the measurements are always corrupted by noise. This measurement noise effects the analytical solution more than non-linear optimization algorithms. On the other hand it is also true that local optimization tends to find the local minimum, instead of the global minimum. This article presents an approach how this risk can be significantly reduced in noisy environments. The main idea of our approach is to transform the local minimum to a saddle point, by increasing the number of dimensions. In addition to numerical tests we analytically prove the theorem and the criteria that no other local minima exists for non-trivial constellations.