Covariance and Time-Scale Methods for Blind Separation of Delayed Sources

An extension of the blind source separation technique based on the second-order blind identification (SOBI) approach is presented to separate mixtures of delayed sources. When the delay is small such that the first-order Taylor approximation holds, the delayed mixture is transformed as the mixture of the original sources and their derivatives. Two algorithms are proposed for the rotation step that recovers the extended source vector (original sources and the corresponding derivatives). The first approach is based on the odd symmetry of the derivative of the autocorrelation function; and the second method identifies the locations of single auto terms in the optimized time-scale plane. A simulation analysis was conducted to evaluate the performance of the proposed algorithms. The results showed that the proposed methods substantially improved the performance of SOBI and its extension in the time-scale plane when the sources presented delays in the mixtures. In addition, the proposed algorithms were applied representatively to experimental multichannel surface electromyographic signals to identify motor unit action potential trains from the interference signal. The performance of the proposed methods was superior to previous methods also in this representative application. In conclusion, extensions of the SOBI approach of source separation have been proposed for the case of sources being delayed in the mixtures. These techniques were proven superior to previous approaches.