  This paper presents a computationally efficient approach to blind source
separation (BSS) of audio signals, applicable even when there are more sources
than microphones (i.e., the underdetermined case). When there are as many
sources as microphones (i.e., the determined case), BSS can be performed
computationally efficiently by independent component analysis (ICA).
Unfortunately, however, ICA is basically inapplicable to the underdetermined
case. Another BSS approach using the multichannel Wiener filter (MWF) is
applicable even to this case, and encompasses full-rank spatial covariance
analysis (FCA) and multichannel non-negative matrix factorization (MNMF).
However, these methods require massive numbers of matrix inversions to design
the MWF, and are thus computationally inefficient. To overcome this drawback,
we exploit the well-known property of diagonal matrices that matrix inversion
amounts to mere inversion of the diagonal elements and can thus be performed
computationally efficiently. This makes it possible to drastically reduce the
computational cost of the above matrix inversions based on a joint
diagonalization (JD) idea, leading to computationally efficient BSS.
Specifically, we restrict the N spatial covariance matrices (SCMs) of all N
sources to a class of (exactly) jointly diagonalizable matrices. Based on this
approach, we present FastFCA, a computationally efficient extension of FCA. We
also present a unified framework for underdetermined and determined audio BSS,
which highlights a theoretical connection between FastFCA and other methods.
Moreover, we reveal that FastFCA can be regarded as a regularized version of
approximate joint diagonalization (AJD).
