The Geometry of Far-Field Passive Source Localization With TDOA and FDOA

Passive localization of acoustic or radio-frequency sources is often performed using time-difference-of-arrival (TDOA) measurements and/or frequency-difference-of-arrival (FDOA) measurements. TDOA localization has been thoroughly studied, but FDOA less so. This is largely because the TDOA level surfaces are hyperboloids, which are well understood, whereas the FDOA level curves and surfaces are much more complicated. This article addresses the case of known sensor positions and velocities and a stationary source. This article shows examples of the FDOA level curves and surfaces, and shows that they simplify dramatically in the far field, i.e., when the source is much farther from the origin than the sensors. The far-field behavior is of two types, depending on whether the sensor velocities are equal or unequal. The far-field behavior gives insight into conditions needed for far-field TDOA-FDOA localization and FDOA-only localization. This article includes a characterization of feasible far-field TDOA and unequal-velocity FDOA data.