Large System Analysis of Linear Precoding in Correlated MISO Broadcast Channels under Limited Feedback

In this paper, we study the sum rate performance of zero-forcing (ZF) and regularized ZF (RZF) precoding in large MISO broadcast channels under the assumptions of imperfect channel state information at the transmitter and per-user channel transmit correlation. Our analysis assumes that the number of transmit antennas $M$ and the number of single-antenna users $K$ are large and of the same order of magnitude. We derive deterministic approximations of the empirical signal-to-interference plus noise ratio (SINR) at the receivers, which are almost surely exact as $M,K\to\infty$. In the course of this derivation, the assumed channel model requires the development of a novel deterministic equivalent of the empirical Stieltjes transform of the eigenvalue distribution of certain kinds of large dimensional random matrices. The deterministic SINR approximations enable us to solve various practical optimization problems. Under sum rate maximization, we derive (i) the optimal regularization term of the RZF precoder, (ii) for ZF the optimal number of users and (iii) for ZF and RZF the optimal amount of feedback in large FDD/TDD multi-user systems. Numerical simulations suggest that the deterministic approximations are accurate even for small $M,K$.