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Closed-form semi-blind separation of three sources from three real-valued instantaneous linear mixtures via quaternions

In the problem of blind source separation from instantaneous linear mixtures, a unitary transformation remains unknown after second-order spatial whitening. We present a novel approach for the identification of the orthogonal matrix in the real-valued three-signal scenario which, in contrast to existing procedures, operates in a single closed-form step, with no iterations. The new approach is based on intuitive geometrical notions and the theory of quaternions, and develops into a practical semi-blind method requiring certain prior knowledge on the source statistics. A simple numerical experiment illustrates the proposed algorithm.