Scaling law of the sum-rate for multi-antenna broadcast channels with deterministic or selective binary feedback

The sum-capacity of the multi-antenna Gaussian broadcast channel is known to be achieved by Dirty Paper Coding techniques, that require full channel state information at the base station. It has been recently shown that a sum-rate having the same scaling law of the sum-capacity with respect to the number of users n for a fixed signal to noise ratio (i.e., M log log n where M is the number of transmitting antennas) can be achieved by using reduced feedback (or equivalently reduced channel state information at the transmitter). In particular, it has been proved that n real and n integer numbers are enough to guarantee the optimal scaling law. In this paper, the optimal scaling law of the sum-rate is shown to be achievable with an even smaller amount of feedback and, more precisely, with 1) n log₂ (M + 1) bits, if a deterministic feedback scheme is employed; 2) an average number of feedback bits that scales as M log₂ M log n with the number of users n, if a selective (random) feedback scheme is employed.