Optimal Analog Precoder Design for Hybrid Beamforming is Possible

Hybrid beamforming is a cost-effective solution for millimeter wave massive multiple-input-multiple-output communications, which highly reduces the number of required radio frequency chains by jointly using a digital precoder and an analog precoder. The digital precoder can be optimized by using the classical least squares method. However, due to the discrete resolution of analog phase shifters, the analog precoder design is a high-dimensional integer programming problem, for which the exsiting methods can only provide suboptimal solutions. In this article, we first show that optimal analog precoder design is possible in typical antenna settings. Specifically, we show that this high-dimensional problem can be divided into multiple low-dimensional subproblems in parallel and there exists an analytical lower bound for each subproblem that can be efficiently utilized to obtain the optimal solution by using the branch and bound method. Numerical results show that the proposed hybrid precoder with an optimal analog precoder can achieve 98
$\%$
performance of the fully digital solution in a typical antenna setting.