Linear degrees of freedom of the MIMO X-channel with delayed CSIT

We establish the sum degrees of freedom of the multiple-input multiple-output X-channel with delayed Channel State Information at the transmitters (CSIT), assuming linear coding strategies at the transmitters. The converse is based on developing a novel rank-ratio inequality that upper bounds the ratio between the dimensions of received linear subspaces at the two receivers. The achievability is based on a three-phase strategy that optimally exploits delayed CSIT in each phase.