The Extended “Sequentially Drilled” Joint Congruence Transformation and Its Application in Gaussian Independent Vector Analysis
Independent vector analysis (IVA) has emerged in recent years as an extension of independent component analysis (ICA) into multiple sets of mixtures, where the sources in each set are independent, but may depend on sources in the other sets. In a semiblind IVA framework, information regarding the probability distributions of the sources may be available, giving rise to maximum likelihood (ML) separation. In recent work, we have shown that under the multivariate Gaussian model, with arbitrary temporal covariance matrices (stationary or non-stationary) of the sources, ML separation requires the solution of a “Sequentially Drilled” Joint Congruence (SeDJoCo) transformation of a set of matrices, which is reminiscent of (but different from) classical joint diagonalization. In this paper, we extend our results to the IVA problem, showing how the ML solution for the Gaussian model (with arbitrary covariance and cross-covariance matrices) takes the form of an extended SeDJoCo solution. We formulate the extended problem, derive a condition for the existence of a solution, and propose two iterative solution algorithms. Additionally, we derive the induced Cramér–Rao lower bound (iCRLB) on the resulting interference-to-source ratios (ISR), and demonstrate by simulation how the ML separation obtained by solving the extended SeDJoCo problem attains the iCRLB (asymptotically), as opposed to other separation approaches, which cannot exploit prior knowledge regarding the sources distributions.