A “Sequentially Drilled” Joint Congruence (SeDJoCo) Transformation With Applications in Blind Source Separation and Multiuser MIMO Systems

We consider a particular form of the classical approximate joint diagonalization (AJD) problem, which we call a “sequentially drilled” joint congruence (SeDJoCo) transformation. The problem consists of a set of symmetric real-valued (or Hermitian-symmetric complex-valued) target-matrices. The number of matrices in the set equals their dimension, and the joint diagonality criterion requires that in each transformed (“diagonalized”) target-matrix, all off-diagonal elements on one specific row and column (corresponding to the matrix-index in the set) be exactly zeros, yet does not care about the other (diagonal or off-diagonal) elements. The motivation for this form arises in (at least) two different contexts: maximum likelihood blind (or semiblind) source separation and coordinated beamforming for multiple-input multiple-output (MIMO) broadcast channels. We prove that SeDJoCo always has a solution when the target-matrices are positive-definite . We also propose two possible iterative solution algorithms, based on defining and optimizing two different criteria functions, using Newton's method for the first function and successive Jacobi-like transformations for the second. The algorithms' convergence behavior and the attainable performance in the two contexts above are demonstrated in simulation experiments.