Array shape calibration using sources in unknown locations--Part I: Far-field sources

This paper deals with source localization using a two-dimensional array of sensors whose locations are not known precisely. If only a single source is observed, uncertainties in sensor location increase errors in source bearing and range by an amount which is independent of signal-to-noise ratio and which can easily dominate over-all localization accuracy. Major performance gains could therefore result from successful calibration of array geometry. The paper derives Cramer-Rao bounds on calibration and source location accuracies achievable with far-field sources whose bearings are not initially known. The sources are assumed to radiate Gaussian noise and to be spectrally disjoint of each other. When the location of one sensor and the direction to a second sensor is known, three noncollinear sources are sufficient to calibrate sensor positions with errors which decrease to zero as calibrating source strength or time-bandwidth products tend to infinity. The sole exception to this statement is a nominally linear array for which such calibration is not possible. When one sensor location is known but no directional reference is available, three noncollinear sources can determine array shape, but there remains a residual error in angular orientation which is irremovable by the calibration procedure. When no sensor locations are known a priori, one adds to the residual error in rotation a translational component. In the far field, the latter should be unimportant. In addition to the asymptotic results, Cramer-Rao bounds are computed for finite signal-to-noise ratios and observation times. One finds that calibration permits significant reductions in localization errors for parameter values well within the practical range.