Cramér-Rao Bound for Circular and Noncircular Complex Independent Component Analysis

Despite an increased interest in complex independent component analysis (ICA) during the last two decades, a closed form expression for the Cramér-Rao bound (CRB) for the demixing matrix is not known yet. In this paper, we fill this gap by deriving a closed-form expression for the CRB of the demixing matrix for instantaneous noncircular complex ICA. It contains the CRB for circular complex ICA and noncircular complex Gaussian ICA as two special cases. We also study the CRB numerically for the family of noncircular complex generalized Gaussian distributions and compare it to simulation results of two ICA estimators. Furthermore, we show how to extend the CRB to the case where the source signals are not temporally independent and identically distributed.