Joint blind source separation from second-order statistics: Necessary and sufficient identifiability conditions

This paper considers the problem of joint blind source separation (J-BSS), which appears in many practical problems such as blind deconvolution or functional magnetic resonance imaging (fMRI). In particular, we establish the necessary and sufficient conditions for the solution of the J-BSS problem by exclusively exploiting the second-order statistics (SOS) of the observations. The identifiability analysis is based on the idea of equivalently distributed sets of latent variables, that is, latent variables with covariance matrices related by means of a diagonal matrix. Interestingly, the identifiability analysis also allows us to introduce a measure of the identifiability degree based on Kullback-Leibler projections. This measure is clearly correlated with the performance of practical SOS-based J-BSS algorithms, which is illustrated by means of numerical examples.