On the Sum-Rate of Broadcast Channels with Outdated 1-Bit Feedback

In this paper, a single-input single-output (SISO) downlink channel with K users is analyzed in the presence of Rayleigh flat fading. A limited channel state information (CSI) feedback scheme is considered, where only an outdated 1-bit feedback per user is available at the base station for each fading block. A closed-form expression for the achievable ergodic sum- rate of the 1-bit feedback scheme is presented for any number of users, as a function of the fading temporal correlation coefficient, the threshold of the 1-bit CSI quantizer and the SNR. It is proved that the sum-rate scales with increasing number of users as log log K, which is the same scaling law achieved by the optimal non-delayed full CSI feedback scheme. In addition, the sum-rate degradation due to outdated CSI is evaluated in the asymptotic regimes of either large K or low SNR.