Massive multiple-input multiple-output (MIMO) systems employing one-bit digital-to-analog converters offer a hardware-efficient solution for wireless communications. However, the one-bit constraint poses significant challenges for precoding design, as it transforms the problem into a discrete and nonconvex optimization task. In this paper, we investigate a widely adopted ``convex-relaxation-then-quantization" approach for nonlinear symbol-level one-bit precoding. Specifically, we first solve a convex relaxation of the discrete minimum mean square error precoding problem, and then quantize the solution to satisfy the one-bit constraint. Focusing on a real-valued system with an independently and identically distributed (i.i.d.) Gaussian channel, we develop a novel analytical framework based on approximate message passing (AMP) to characterize the high-dimensional asymptotic performance of the considered scheme. The key technical ingredient is an auxiliary AMP iteration that dedicatedly incorporates the nonlinear quantization function into the state evolution analysis. With the proposed framework, we derive a closed-form expression for the symbol error probability (SEP) at the receiver side in the large-system limit, which provides a quantitative characterization of how model and system parameters affect the SEP performance. Our empirical results suggest that the $\ell_\infty^2$ regularizer, when paired with an optimally chosen regularization parameter, achieves optimal SEP performance within a broad class of convex regularization functions. As a first step towards a theoretical justification, we prove the optimality of the $\ell_\infty^2$ regularizer within the mixed $\ell_\infty^2$-$\ell_2^2$ regularization functions.

翻译：采用单比特数模转换器的大规模多输入多输出系统为无线通信提供了一种硬件高效的解决方案。然而，单比特约束给预编码设计带来了显著挑战，因为它将问题转化为离散非凸优化任务。本文研究了一种广泛采用的"凸松弛-量化"方法用于非线性符号级单比特预编码。具体而言，我们首先求解离散最小均方误差预编码问题的凸松弛，然后对解进行量化以满足单比特约束。针对具有独立同分布高斯信道的实值系统，我们开发了一种基于近似消息传递的全新分析框架，用以刻画所考虑方案的高维渐近性能。关键技术要素是辅助AMP迭代，该迭代专门将非线性量化函数纳入状态演化分析中。借助所提出的框架，我们推导出大系统极限下接收端符号错误概率的闭式表达式，从而定量刻画模型和系统参数如何影响SEP性能。实证结果表明，当与优化选择的正则化参数配对时，$\ell_\infty^2$正则化器在广泛凸正则化函数类别中实现了最优SEP性能。作为理论证明的第一步，我们在混合$\ell_\infty^2$-$\ell_2^2$正则化函数中证明了$\ell_\infty^2$正则化器的最优性。