Degrees of freedom of the rank-deficient MIMO X channel

The multiple-input multiple-output (MIMO) X channel with rank-deficient channels is considered. We characterize the total degrees of freedom (DoF) by defining an outer bound which is attained by the proposed precoding scheme. The total DoF are described in closed-form when all transmitters and receivers have M and N antennas, respectively. For an arbitrary number of antennas, the total DoF are obtained as solution of a linear programming problem. Results elucidate that, unlike point-to-point MIMO systems where DoF increase with the rank of the channel, the total DoF in a multipoint-to-multipoint MIMO system can increase up to a certain point if channels are rank-deficient. Hence, we corroborate the relevance of knowing the rank-deficiency in the MIMO X channel, a property already observed in the MIMO Interference channel.