Ergodic and outage performance of fading broadcast channels with 1-bit feedback

In this paper, the ergodic sum rate and the outage probability of a downlink single-antenna channel with K users are analyzed in the presence of Rayleigh flat fading, where limited channel state information (CSI) feedback is assumed. Specifically, only 1-bit feedback per fading block per user is available at the base station. We first study the ergodic sum rate of the 1-bit feedback scheme and consider the impact of feedback delay on the system. A closed-form expression for the achievable ergodic sum rate is presented as a function of the fading temporal correlation coefficient. It is proved that the sum rate scales as log log K, which is the same scaling law achieved by the optimal nondelayed full CSI feedback scheme. The sum-rate degradation due to outdated CSI is also evaluated in the asymptotic regimes of either large K or low SNR. The outage performance of the 1-bit feedback scheme for both instantaneous and outdated feedback is then investigated. Expressions for the outage probabilities are derived, along with the corresponding diversity-multiplexing tradeoffs (DMTs). It is shown that, with instantaneous feedback, a power allocation based on the feedback bits enables doubling of the DMT compared with the case with short-term power constraint in which a dynamic power allocation is not allowed. However, with outdated feedback, the advantage of power allocation is lost, and the DMT reverts to that achievable with no CSI feedback. Nevertheless, for finite SNR, improvement in terms of outage probability can still be obtained.