Error Bounds for Localization with Noise Diversity

Abstract

In the context of acoustic monitoring, the location of a sound source can be passively estimated by exploiting time-of-arrival and time-difference-of-arrival measurements. To evaluate the fundamental hardness of a location estimator, the Cramer-Rao bound (CRB) has been used by many researchers. The CRB is computed by inverting the Fisher Information Matrix (FIM), which measures the amount of information carried by given distance measurements. The measurements are commonly expressed as actual distances plus white noise. However, the measurements do include extra noise types caused by time synchronization, acoustic sensing latency, and signal-to-noise ratio. Such noise can significantly affect the performance and depend highly on the sensing platforms such as Android smartphones. In this paper, we first remodel the acoustic-based distance measurements considering such additive errors. Then, we derive a new FIM with the new statistical ranging error models. As a result, we obtain new CRBs for both non-cooperative and cooperative localization schemes that provide better insight into the causality of the uncertainties. Theoretical analysis also proves that the proposed CRBs for localization become the old CRBs when the additional errors are ignored, which gives a robust check for the new CRBs. Thus, the new CRBs can serve as a benchmark for localization estimators with both new and old measurement models. The new CRBs also indicate that there is room to improve current localization schemes, however, it is a daunting challenge.