Asymptotic Analysis of the Amount of CSI Feedback in MIMO Broadcast Channels

In this paper, we consider a downlink communication system in which a base station (BS) equipped with M antennas and power constraint P communicates with N users each equipped with K receive antennas. It is assumed that the users have perfect channel state information (CSI) of their own channels, while the BS only knows the partial CSI provided by the receivers via a feedback channel. We study the fundamental limits on the amount of feedback required at the BS to achieve the sum-rate capacity of the system (when BS has perfect CSI for all users) in the asymptotic case of N →∞, considering various signal to noise ratio (SNR) regimes. The main results of this paper can be expressed as follows. 1) In the fixed-SNR regime (where the SNR does not scale with N) and low-SNR regime (where the SNR is much smaller than 1/In(N) ), to achieve the (1 - ε)-portion of the sum-rate capacity, the total amount of feedback should scale at least with (ε^-1). In the fixed-SNR regime, to reduce the gap between the achievable sum rate and the sum-rate capacity of the system (which is defined as the sum-rate gap) to zero, the amount of feedback should scale at least logarithmically with the sum-rate capacity, which is achievable by using the random beam-forming (RBF) scheme proposed by Sharif and Hassibi. In the low-SNR regime, we propose an opportunistic beam-forming (OBF) scheme, which is shown to be asymptotically feedback optimal. 2) In the high-SNR regime (where the SNR grows to infinity as N → ∞), the total amount of feedback depends on the number of receive antennas. In particular, to reduce the sum-rate gap to zero in the case of K <; M, the amount of feedback in the SNR regime of In(P)/In(N) >; 1/M - 1, should scale at least logarithmically with the SNR. In the case of K ≥ M , the amount of feedback does not need to scale with the SNR. Moreover, we show that RBF is asymptotically feedback optimal in the hig- -SNR regime.