In this paper, we consider the one-bit precoding problem for the multiuser downlink massive multiple-input multiple-output (MIMO) system with phase shift keying (PSK) modulation. We focus on the celebrated constructive interference (CI)-based problem formulation. We first establish the NP-hardness of the problem (even in the single-user case), which reveals the intrinsic difficulty of globally solving the problem. Then, we propose a novel negative $\ell_1$ penalty model for the considered problem, which penalizes the one-bit constraint into the objective by a negative $\ell_1$-norm term, and show the equivalence between (global and local) solutions of the original problem and the penalty problem when the penalty parameter is sufficiently large. We further transform the penalty model into an equivalent min-max problem and propose an efficient alternating proximal/projection gradient descent ascent (APGDA) algorithm for solving it, which performs a proximal gradient decent over one block of variables and a projection gradient ascent over the other block of variables alternately. The APGDA algorithm enjoys a low per-iteration complexity and is guaranteed to converge to a stationary point of the min-max problem and a local minimizer of the penalty problem. To further reduce the computational cost, we also propose a low-complexity implementation of the APGDA algorithm, where the values of the variables will be fixed in later iterations once they satisfy the one-bit constraint. Numerical results show that, compared to the state-of-the-art CI-based algorithms, both of the proposed algorithms generally achieve better bit-error-rate (BER) performance with lower computational cost.

翻译：本文研究采用相移键控（PSK）调制的多用户下行大规模多输入多输出（MIMO）系统中的单比特预编码问题，重点关注基于构造性干扰（CI）的经典问题建模。首先证明该问题的NP难解性（即使是在单用户情形下），揭示了全局求解该问题的内在困难。继而针对所考虑问题提出新颖的负$\ell_1$惩罚模型——通过在目标函数中添加负$\ell_1$范数项来惩罚单比特约束，并证明当惩罚参数足够大时，原始问题与惩罚问题的（全局和局部）解具有等价性。进一步将惩罚模型转化为等价的最小-最大问题，提出高效的交替近端/投影梯度下降上升（APGDA）算法进行求解：该算法交替执行一个变量块的近端梯度下降与另一个变量块的投影梯度上升。APGDA算法具有低单次迭代复杂度，且能保证收敛至最小-最大问题的驻点及惩罚问题的局部极小点。为降低计算开销，进一步提出APGDA算法的低复杂度实现——在后续迭代中，一旦变量满足单比特约束即将其固定。数值结果表明，与现有的CI基算法相比，所提两类算法通常能以更低计算代价获得更优的比特误码率（BER）性能。