Identifiability Analysis for Array Shape Self-Calibration Based on Hybrid Cramér-Rao Bound

In array shape self-calibration where the sensor position errors and source locations are both unknown, the identifiability of the unknowns is a fundamental problem. Previously, using an approximate hybrid Cramér-Rao bound (HCRB), it was found that under the assumption of small random errors, a nominally linear array is impossible to self-calibrate, but a nominally non-linear array is possible to self-calibrate with three noncollinear far-field sources. In this letter, both small and large random errors are considered, thus the crucial small-error approximation has to be dropped. An accurate HCRB is then derived using fully the prior information about the errors. The accurate HCRB proves that if it is tight, a perturbed nominally linear array is possible to self-calibrate. The larger the sensor position errors, the easier the self-calibration. This is important because of the wide application of linear array. Moreover, two noncollinear far-field sources are sufficient to self-calibrate an array of arbitrary nominal shape.