We study downlink channel estimation in a frequency-division duplex (FDD) massive MIMO system from PMI-only feedback under a 5G NR-type limited-feedback architecture. In this architecture, the user selects a preferred codeword from a shared codebook based on the reduced-dimensional channel and only reports its index (known as the precoding matrix indicator, PMI) back to the base station. Therefore, the channel must be estimated from these highly quantized, nonlinear PMI observations. Based on a probabilistic perturbation model, a constrained maximum likelihood estimator (MLE) is proposed for this estimation problem, whose objective can also be interpreted as a relaxation of the hard empirical decision error. The Cramér--Rao bound is derived for the complex-valued model, with the global phase ambiguity handled via gauge-fixing. For the real-valued setting, a global excess-risk bound of order $O(1/\sqrt{T})$ is established, which is then refined to a sharp local rate of order $O(1/T)$ under suitable identifiability conditions. Numerical results show that the MLE asymptotically attains the Cramér--Rao bound and outperforms several baseline methods on both synthetic data and realistic FDD channels.

翻译：我们研究频分双工（FDD）大规模MIMO系统中基于仅PMI反馈的5G NR型有限反馈架构下的下行信道估计问题。在该架构中，用户根据降维信道从共享码本中选择优选码字，仅将其索引（称为预编码矩阵指示符，PMI）反馈给基站。因此，必须从这些高度量化、非线性的PMI观测值中估计信道。基于概率扰动模型，本文针对该估计问题提出了一种约束最大似然估计器（MLE），其目标函数也可解释为硬经验决策误差的松弛。针对复值模型推导了Cramér-Rao界，并通过规范固定处理全局相位模糊性。对于实值模型，建立了阶数为$O(1/\sqrt{T})$的全局过剩风险界，并在合适的可辨识性条件下将其细化为阶数为$O(1/T)$的尖锐局部速率。数值结果表明，MLE渐近达到Cramér-Rao界，并在合成数据和实际FDD信道中均优于若干基线方法。