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We study the problem of maximum-likelihood (ML) estimation of an approximatecommon eigenstructure, i.e. an approximate common eigenvectors set (CES), foran ensemble of covariance matrices given a collection of their associated i.i.dvector realizations. This problem has a direct application in multi-user MIMOcommunications, where the base station (BS) has access to instantaneous userchannel vectors through pilot transmission and attempts to perform jointmulti-user Downlink (DL) precoding. It is widely accepted that an efficientimplementation of this task hinges upon an appropriate design of a set ofcommon "virtual beams", that captures the common eigenstructure among the userchannel covariances. In this paper, we propose a novel method for obtainingthis common eigenstructure by casting it as an ML estimation problem. We provethat in the special case where the covariances are jointly diagonalizable, theglobal optimal solution of the proposed ML problem coincides with the commoneigenstructure. Then we propose a projected gradient descent (PGD) method tosolve the ML optimization problem over the manifold of unitary matrices andprove its convergence to a stationary point. Through exhaustive simulations, weillustrate that in the case of jointly diagonalizable covariances, our proposedmethod converges to the exact CES. Also, in the general case where thecovariances are not jointly diagonalizable, it yields a solution thatapproximately diagonalizes all covariances. Besides, the empirical results showthat our proposed method outperforms the well-known joint approximatediagonalization of eigenmatrices (JADE) method in the literature.

Joint Approximate Covariance Diagonalization with Applications in MIMO Virtual Beam Design