FDD CSI Feedback under Finite Downlink Training: A Rate-Distortion Perspective

Abstract

This paper establishes the theoretical limits of channel state information (CSI) feedback in frequency-division duplexing (FDD) multi-antenna orthogonal frequency-division multiplexing (OFDM) systems under finite-length training with Gaussian pilots. The user employs minimum mean-squared error (MMSE) channel estimation followed by asymptotically optimal uplink feedback. Specifically, we derive a general rate-distortion function (RDF) of the overall CSI feedback system. We then provide both non-asymptotic bounds and asymptotic scaling for the RDF under arbitrary downlink signal-to-noise ratio (SNR) when the number of training symbols exceeds the antenna dimension. A key observation is that, with sufficient training, the overall RDF converges to the direct RDF corresponding to the case where the user has full access to the downlink CSI. More importantly, we demonstrate that even at a fixed downlink SNR, the convergence rate is inversely proportional to the training length. The simulation results show that our bounds are tight, and under very limited training, the deviation between the overall RDF and the direct RDF is substantial.