  Multiple-antenna broadcast channels with $M$ transmit antennas and $K$
single-antenna receivers is considered, where the channel of receiver $r$ takes
one of the $J_r$ finite values. It is assumed that the channel states of each
receiver are randomly selected from $\mathds{R}^{M\times 1}$ (or from
$\mathds{C}^{M\times 1}$). It is shown that no matter what $J_r$ is, the
degrees of freedom (DoF) of $\frac{MK}{M+K-1}$ is achievable. The achievable
scheme relies on the idea of interference alignment at receivers, without
exploiting the possibility of cooperation among transmit antennas. It is proven
that if $J_r \geq M$, $r=1,...,K$, this scheme achieves the optimal DoF. This
results implies that when the uncertainty of the base station about the channel
realization is considerable, the system loses the gain of cooperation. However,
it still benefits from the gain of interference alignment. In fact, in this
case, the compound broadcast channel is treated as a compound X channel.
  Moreover, it is shown that when the base station knows the channel states of
some of the receivers, a combination of transmit cooperation and interference
alignment would achieve the optimal DoF.
  Like time-invariant $K$-user interference channels, the naive vector-space
approaches of interference management seem insufficient to achieve the optimal
DoF of this channel. In this paper, we use the Number-Theory approach of
alignment, recently developed by Motahari et al.[1]. We extend the approach of
[1] to complex channels as well, therefore all the results that we present are
valid for both real and complex channels.
