A rank ratio inequality and the linear degrees of freedom of X-channel with delayed CSIT

We establish the degrees of freedom of the X-channel with delayed channel knowledge at transmitters (i.e., delayed CSIT), assuming linear coding strategies at the transmitters. We derive a new upper bound and characterize the linear degrees of freedom of this network to be 6/5. The converse builds upon our development of a general lemma for wireless networks consisting of two transmitters and two receivers, with delayed CSIT. The lemma states that, if the two transmitters employ linear strategies, then the ratio of the rank of received signal subspaces at the two receivers cannot exceed 3/2, due to delayed CSIT. This lemma can also be applied to any arbitrary network, in which a receiver decodes its desired message in the presence of two interferers.