Downlink Channel Matrix Estimation from PMI-Only Feedback in FDD Systems: Maximum Likelihood and Sharp Excess Risk Bound

Abstract

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 Cram\'er--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 Cram\'er--Rao bound and outperforms several baseline methods on both synthetic data and realistic FDD channels.