Rate-splitting multiple access (RSMA) is a general multiple access scheme for downlink multi-antenna systems embracing both classical spatial division multiple access and more recent non-orthogonal multiple access. Finding a linear precoding strategy that maximizes the sum spectral efficiency of RSMA is a challenging yet significant problem. In this paper, we put forth a novel precoder design framework that jointly finds the linear precoders for the common and private messages for RSMA. Our approach is first to approximate the non-smooth minimum function part in the sum spectral efficiency of RSMA using a LogSumExp technique. Then, we reformulate the sum spectral efficiency maximization problem as a form of the log-sum of Rayleigh quotients to convert it into a tractable non-convex optimization problem. By interpreting the first-order optimality condition of the reformulated problem as an eigenvector-dependent nonlinear eigenvalue problem, we reveal that a leading eigenvector is a local optimal solution. To find the leading eigenvector, we propose a computationally efficient algorithm inspired by a power iteration method. Simulation results show that the proposed RSMA transmission strategy provides significant improvement in the sum spectral efficiency compared to the state-of-the-art RSMA transmission methods, while requiring considerably less computational complexity.
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