Asymptotic Analysis of Precoding in Large-scale Multi-user Systems

This work provides a precise performance analysis of precoding in the asymptotic regime of the multi-user massive multiple-input multiple-output (MIMO) setting. Considering practical challenges, such precoders are typically formulated as non-linear, non-smooth optimization problems without closed-form solutions, often followed by non-linear post-processing operations. These aspects make it difficult to analyze the system behavior in multiple scenarios. To address these challenges, we utilize analytical tools based on the Gaussian Min-Max Theorem (GMT). For cases without post-processing, we employ the Convex Gaussian Min-Max Theorem (CGMT a rigorously established tool that enables the analysis of quantities of interest by connecting them to the optimal solutions of auxiliary models of the precoding problem. Notably, CGMT does not require closed-form solutions and facilitates inference of the distributions of these quantities. With the aid of CGMT, we uncover system behavior from multiple perspectives, offering deeper insights than existing works with similar motivations. To treat scenarios involving post-processing, we propose novel Gaussian Min-Max Theorems, with specific applications to one-bit quantization and thresholded precoders. The former had only been approximately characterized prior to our work, while the latter remained largely unexplored. Our newly developed theorems allow us to derive the same metrics of interest as in the non-post-processed cases, while also characterizing the effects of post-processing operations. These methodologies can further inspire new directions for addressing other post-processing schemes, potentially extending beyond the precoding context. The quantities of interest include the distributions of the transmit vector, transmit-receive distortion, and the asymptotic behavior of the received signal-to-distortion-and-noise ratio (SDNR) and bit error rate (BER), thus capturing system performance from all critical perspectives. We provide precise asymptotic predictions of these quantities?results that serve as valuable takeaways and reduce the need for extensive and laborious empirical experimentation.