Probabilistic Source Localization Based on Time-of-Arrival Measurements

This article was motivated by the need to localize a sensor (signal source) in an Internet-of-Things (IoT) network to an area with a predetermined probability (credibility). The source is assumed to transmit a short time duration (burst) signal in a homogeneous line-of-sight environment. The source's signal is known to an array of spatially separated receivers, which can measure the times of arrival of the source's signal subject to Gaussian measurement errors. The time that the signal leaves the source (i.e., the transmit time) is, however, unknown. The authors develop a method for computing the exact a posteriori probability density functions (pdfs) of the coordinates of the source's in both 2-D and 3-D spaces. The obtained a posteriori pdfs incorporate arbitrary a priori densities, which makes them very useful in many practical scenarios. Unlike existing point-estimate methods, the probabilistic method does not simultaneously solve a set of equations so there is neither a lower nor upper limit on the number of receivers. Various examples are provided to demonstrate the superiority and usefulness of the proposed method. In particular, it is shown that the joint a posteriori pdf of the target's location is not always approximately jointly Gaussian, especially when the target is in close proximity to one of the gateways, or when the gateways are located in close proximity to each other.