Properties of the partial Cholesky factorization and application to reduced-rank adaptive beamforming

Abstract

Reduced-rank adaptive beamforming is a well established and efficient methodology, notably for disturbance covariance matrices which are the sum of a strong low-rank component (interference) and a scaled identity matrix (thermal noise). Eigenvalue or singular decomposition is often used to achieve rank reduction. In this paper, we study and analyze an alternative, namely a partial Cholesky factorization, as a means to retrieve interference subspace and to compute reduced-rank beamformers. First, we study the angles between the true subspace and that obtained from partial Cholesky factorization of the covariance matrix. Then, a statistical analysis is carried out in finite samples. Using properties of partitioned Wishart matrices, we provide a stochastic representation of the beamformer based on partial Cholesky factorization and of the corresponding signal to interference and noise ratio loss. We show that the latter follows approximately a beta distribution, similarly to the beamformer based on eigenvalue decomposition. Finally, numerical simulations are presented which indicate that a reduced-rank adaptive beamformer based on partial Cholesky factorization incurs almost no loss, and can even perform better in some scenarios than its eigenvalue or singular value-based counterpart.