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