Joint Approximate Covariance Diagonalization with Applications in MIMO\n Virtual Beam Design

Abstract

We study the problem of maximum-likelihood (ML) estimation of an approximate\ncommon eigenstructure, i.e. an approximate common eigenvectors set (CES), for\nan ensemble of covariance matrices given a collection of their associated i.i.d\nvector realizations. This problem has a direct application in multi-user MIMO\ncommunications, where the base station (BS) has access to instantaneous user\nchannel vectors through pilot transmission and attempts to perform joint\nmulti-user Downlink (DL) precoding. It is widely accepted that an efficient\nimplementation of this task hinges upon an appropriate design of a set of\ncommon "virtual beams", that captures the common eigenstructure among the user\nchannel covariances. In this paper, we propose a novel method for obtaining\nthis common eigenstructure by casting it as an ML estimation problem. We prove\nthat in the special case where the covariances are jointly diagonalizable, the\nglobal optimal solution of the proposed ML problem coincides with the common\neigenstructure. Then we propose a projected gradient descent (PGD) method to\nsolve the ML optimization problem over the manifold of unitary matrices and\nprove its convergence to a stationary point. Through exhaustive simulations, we\nillustrate that in the case of jointly diagonalizable covariances, our proposed\nmethod converges to the exact CES. Also, in the general case where the\ncovariances are not jointly diagonalizable, it yields a solution that\napproximately diagonalizes all covariances. Besides, the empirical results show\nthat our proposed method outperforms the well-known joint approximate\ndiagonalization of eigenmatrices (JADE) method in the literature.\n