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.