Cramér-Rao Bounds for Complex-Valued Independent Component Extraction: Determined and Piecewise Determined Mixing Models

Blind source extraction (BSE) aims at recovering an unknown source signal of interest from the observation of instantaneous linear mixtures of the sources. This paper presents Cramér-Rao lower bounds (CRLB) for the complex-valued BSE problem based on the assumption that the target signal is independent of the other signals. The target source is assumed to be non-Gaussian or non-circular Gaussian while the other signals (background) are circular Gaussian or non-Gaussian. The results confirm some previous observations known for the real domain and yield new results for the complex domain. Also, the CRLB for independent component extraction (ICE) is shown to coincide with that for independent component analysis (ICA) when the non-Gaussianity of background is taken into account. Second, we extend the CRLB analysis to piecewise determined mixing models, where the observed signals are assumed to obey the determined mixing model within short blocks where the mixing matrices can be varying from block to block. This model has applications, for instance, when separating dynamic mixtures. Either the mixing vector or the separating vector corresponding to the target source is assumed to be constant across the blocks. The CRLBs for the parameters of these models bring new performance limits for the BSE problem.