A uniform circular array of isotropic sensors that stochastically dislocate in three dimensions : the hybrid Cramér-Rao bound of direction-of-arrival estimation

Abstract

An array's constituent sensors could be spatially dislocated from their nominal positions. This paper investigates how such sensor dislocation would degrade a uniform circular array (UCA) of isotropic sensors (like pressure sensors) in their direction-finding precision. This paper analytically derives this direction finding's hybrid Cramér-Rao bound (HCRB) in a closed form that is expressed explicitly in terms of the sensors' dislocation parameters. In the open literature on UCA direction finding, this paper is the first to be three-dimensional in modeling the sensors' dislocation. Perhaps unexpectedly to some readers, sensor dislocation could improve and not necessarily degrade the HCRB; these opposing effects depend on the dislocation variances, the incident source's arrival angle, and the signal-to-noise power ratio—all analyzed rigorously in this paper. Interesting insights are thereby obtained: (a) The HCRB is enhanced for the impinging source's polar arrival angle as the sensors become more dislocated along the impinging wavefront due to aperture enlargement over the stochastic dislocation's probability space. (b) Likewise, the HCRB is improved for the azimuth arrival angle as the sensors become more dislocated on the circular array's plane, also due to aperture enlargement. (c) In contrast, sensor dislocation along the incident signal's propagation direction can only worsen the HRCBs due to nuisance-parameter effects in the Fisher information. (d) Sensor dislocation orthogonal to the array plane must degrade the HCRB for the azimuth arrival angle but could improve the HCRB for the polar arrival angle.