Noisy feedback linear precoding: A Bayesian Cramér-Rao bound

Transmitter precoding strategies in broadcast systems generally assume perfect knowledge of channel state information (CSI) at the transmitter. In this paper, we study linear precoding with arbitrary error in the CSI from an estimation theoretic point of view. We derive a Bayesian Cramer-Rao type bound on the sum mean squared error (SMSE) achievable at the receiver for any linear precoding scheme for arbitrary feedback noise and channel fading. We next specialize this result to power constrained precoders. It is shown that the regularity conditions of the bound may be significantly weakened for power constrained precoders. Interestingly, we obtain a bound whose validity depends on rather weak conditions of continuity and differentiability on the joint distribution of the channel and feedback. We demonstrate the bound by applying it to Gaussian, Nakagami-m and Weibull fading models, with the assumption of feedback corrupted by additive Gaussian noise.