Performance bound for robust array calibration in radio interferometry

Robust calibration of next-generation radio-interferometers, as the square kilometer array (SKA) for instance, is a crucial prepro-cessing step for sky imaging. Recently, several robust calibration es-timators based on the use of well known strong sources in the field of view (FOV) have been proposed in the literature. For that, usually a compound-Gaussian (CG) noise is considered since it can take into account the presence of outliers, such that unknown weak sources in the FOV, and remains mathematically tractable. Specifically, the CG model is a zero-mean Gaussian process with a random variance, usually called texture. Performance bounds provide the lowest mean squared error (MSE) that an unbiased estimator can hope to reach. To the best of our knowledge, there is a lack of practical bounds dedicated to the robust calibration of future radio interferometers as far as they can be used as design tools for a given FOV. In this work, the hybrid Cramér-Rao bound (HCRB) is derived for a propagation model based on the Jones-matrix formalism corrupted by a CG noise for several texture distributions (K-distribution, Student's t, Cauchy, and Inverse-Gaussian compound Gaussian distribution (IG-CG)). In this standard physical model, the waveform propagation model is parametrized by a set of deterministic physical parameters of interest (complex gains, phases, DOA, etc.). In this work, we show that the HCRB w.r.t. the physical deterministic parameters is in fact given by the modified CRB (MCRB). The MCRB is easy to derive since only the first-order moment of the inverse of the texture variable is needed.