We investigate the weighted sum-rate (WSR) maximization linear precoder design for massive MIMO downlink and propose a unified matrix manifold optimization framework applicable to total power constraint (TPC), per-user power constraint (PUPC) and per-antenna power constraint (PAPC). Particularly, we prove that the precoders under TPC, PUPC and PAPC are on different Riemannian submanifolds, and transform the constrained problems in Euclidean space to unconstrained ones on manifolds. In accordance with this, we derive Riemannian ingredients including orthogonal projection, Riemannian gradient, Riemannian Hessian, retraction and vector transport, which are needed for precoder design in matrix manifold framework. Then, Riemannian design methods using Riemannian steepest descent, Riemannian conjugate gradient and Riemannian trust region are provided to design the WSR-maximization precoders under TPC, PUPC or PAPC. Riemannian methods are free of the inverse of large dimensional matrix, which is of great significance for practice. Complexity analysis and performance simulations demonstrate the advantages of the proposed precoder design.

翻译：本文研究了用于大规模MIMO下行链路的加权和速率（WSR）最大化线性预编码器设计，并提出了一个统一的矩阵流形优化框架，适用于总功率约束（TPC）、每用户功率约束（PUPC）和每天线功率约束（PAPC）。特别地，我们证明了在TPC、PUPC和PAPC下的预编码器位于不同的黎曼子流形上，并将欧几里得空间中的约束问题转化为流形上的无约束问题。据此，我们推导了矩阵流形框架中预编码器设计所需的黎曼要素，包括正交投影、黎曼梯度、黎曼海森矩阵、收缩和向量传输。随后，提供了使用黎曼最速下降法、黎曼共轭梯度法和黎曼信赖域法的黎曼设计方法，以在TPC、PUPC或PAPC约束下设计WSR最大化预编码器。黎曼方法无需对大型矩阵求逆，这对实际应用具有重要意义。复杂度分析和性能仿真证明了所提出的预编码器设计的优势。