The net benefit of delayed finite-rate feedback in the MISO broadcast channel

A novel approach recently proposed by Maddah-Ali and Tse (the MAT scheme) achieves significant degrees of freedom (DoF) gain in the multi-input single-output (MISO) broadcast channel (BC) even with channel state information feedback that is completely stale, i.e., uncorrelated with the current channel state. However, their result does not consider the cost of the feedback, which can be quite large with moderate to high mobility/Doppler. In this paper, we introduce a new “net” degrees of freedom metric which is defined as the prelog capacity term remaining after subtracting off the feedback DoF consumed. We closely examine the MAT scheme and compare its maximum net DoF gain to single user transmission (which always achieves 1 DoF) and zero-forcing precoding (which achieves up to K DoF). In particular, assuming the channel coherence time is N symbol periods and the feedback delay is Nfd, we show that when N < (1+o(1))K logK (short coherence time), single user transmission performs best; whereas for N > (1+o(1))(Nfd+K/log K)(1-log−1 K)−1 (long coherence time), zero-forcing precoding outperforms the other two. The MAT scheme is optimal for intermediate coherence times, which for practical parameter choices is indeed quite a large and significant range, even accounting for the feedback cost.