Array shape calibration using sources in unknown locations--Part II: Near-field sources and estimator implementation

Uncertainty concerning sensor locations can seriously degrade the ability of an array to estimate the location of radiating sources. Array calibration then becomes an important issue. This paper deals with array calibration using spectrally disjoint near-field sources whose locations are not known a priori. It calculates Cramer-Rao bounds on source location and array shape errors. It shows that, with mild restrictions on source-array geometry, the array shape errors can be made arbitrarily small by using a sufficient number of sufficiently strong calibrating sources. The required number of calibrating sources is five for a four-sensor array, four for a five-sensor array, and three for an array of six or more sensors. Accurate calibration is not possible for a three-sensor array. While calibration can establish array shape with great accuracy, it cannot resolve a rotational uncertainty in array orientation. This uncertainty translates into a residual error in source bearings, but not in source ranges. Thus, incremental errors in target range due to sensor location uncertainty can be reduced to any desired extent. The paper also proposes an actual calibration procedure and presents simulation results indicating that useful calibration can be obtained with calibrating sources of very moderate signal-to-noise ratio.