Degrees of Freedom Region of the MIMO <formula formulatype="inline"> <tex>$X$</tex></formula> Channel

We provide achievability as well as converse results for the degrees of freedom region of a multiple-input multiple-output (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 outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message. With $M=1$ antennas at each node, we find that the total (sum rate) degrees of freedom are bounded above and below as $1 \leq \eta _{X}^{\star} \leq {{ 4}\over { 3}}$. If $M>1$ and channel matrices are nondegenerate then the precise…