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We present a frequency-domain technique based on PARAllel FACtor (PARAFAC) analysis that performs multichannel blind source separation (BSS) of convolutive speech mixtures. PARAFAC algorithms are combined with a dimensionality reduction step to significantly reduce computational complexity. The identifiability potential of PARAFAC is exploited to derive a BSS algorithm for the under-determined case (more speakers than microphones), combining PARAFAC analysis with time-varying Capon beamforming. Finally, a low-complexity adaptive version of the BSS algorithm is proposed that can track changes in the mixing environment. Extensive experiments with realistic and measured data corroborate our claims, including the under-determined case. Signal-to-interference ratio improvements of up to 6 dB are shown compared to state-of-the-art BSS algorithms, at an order of magnitude lower computational complexity. [ABSTRACT FROM PUBLISHER]