On the degrees of freedom of the compound MISO broadcast channels with finite states

We consider compound multiple-antenna broadcast channels with M transmit antennas and K single-antenna receivers, where the channel of receiver r takes one of J_r complex vectors. We show that for any finite J_r, the degrees of freedom (DoF) of MK/M+K-1 is achievable. It is in contrary to the commonly believed conjecture that the DoF collapses to one for large J _r's. We also establish the optimality of this result, where J _r ≥ M, r = 1,⋯,K. The achievable scheme relies on using a number theoretic approach of interference alignment, and ignoring the possibility of cooperation among the transmit antennas. Moreover, we show that if J_r's for some of the users are relatively small, then both transmit cooperation and interference alignment are needed. Unlike the conventional schemes, here the transmit precoders are not full rank.