Matrix Manifold Precoder Design for Massive MIMO Downlink

We investigate the weighted sum-rate (WSR) maximization linear precoder design under total power constraint (TPC) for massive MIMO downlink with matrix manifold optimization. Particularly, we prove that the precoders under TPC are on a Riemannian submanifold, and transform the constrained problem in Euclidean space to the unconstrained one on manifold. In accordance with this, Riemannian design methods using Riemannian steepest descent and Riemannian conjugate gradient are provided to design the WSR-maximization precoders under TPC. Riemannian methods are free of the inverse of large dimensional matrix, posing significant computational savings and potentially allowing to avoid ill numerical behavior in algorithms. Complexity analysis and performance simulations demonstrate the advantages of the proposed precoder design.