An Hybrid Cramér-Rao Bound in Closed Form for Direction-of-Arrival Estimation by an “Acoustic Vector Sensor” With Gain-Phase Uncertainties

An “acoustic vector sensor” (also known as a “vector hydrophone” in underwater or sea-surface applications) is composed of three orthogonally oriented uni-axial particle-velocity sensors, plus a “pressure-sensor” (i.e., a microphone or a hydrophone) - all collocated in a point-like spatial geometry. (This collocated setup is versatile for direction finding, because its azimuth-elevation spatial response is independent of frequency.) This paper investigates how the acoustic vector sensor´s direction finding accuracy would be degraded by random deviations from its nominal gain response and/or phase response. Each type of deviation is statistically modeled herein as a random variable with a small variance, reasonably so for a well-built acoustic vector sensor. The resulting hybrid Cramér-Rao bound (HCRB) is derived exactly in open form for azimuth-elevation arrival-angle estimation, but also approximated to produce a closed form that is simple enough to yield qualitative observations. This closed-form hybrid Cramér-Rao lower bound´s tightness is illustrated by a new estimator developed in this paper.