Degrees of freedom of the MIMO X channel

We provide achievability as well as converse results for the degrees of freedom region of a MIMO X channel, i.e., a system with two transmitters, two receivers, each equipped with multiple antennas, where independent messages need to be conveyed over fixed channels from each transmitter to each receiver. The inner and outerbounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message. If all nodes have equal number of antennas M > 1 and channel matrices are non-degenerate then the degrees of freedom eta(star)(X) = 4/3M. If the channels vary with time/frequency then the X channel with single antennas (M = 1) at all nodes has 4/3 degrees of freedom. Thus, the MIMO X channel has non-integer degrees of freedom when M is not a multiple of 3. Simple zero forcing without dirty paper encoding or successive decoding, suffices to achieve the 4/3 M degrees of freedom in all cases. The key idea for the achievability of the degrees of freedom is interference alignment - i.e., signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired. With equal number of antennas at all nodes, we also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio.