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

Linear Degrees of Freedom of the MIMO X-Channel With Delayed CSIT

“…Lemma 2 generalizes the rank-ratio inequality given in Lemma 1 of [13,14] to all antenna configurations: for the case of single antennas (i.e.,…”

“…We restrict ourselves to linear coding strategies as defined in [13][14][15], in which DoF simply represents the dimension of the linear subspace of transmitted signals. More specifically, consider a communication scheme with blocklength T , in which Transmitter j wishes to transmit a vector u ij ∈ C m (T ) ij of m (T ) ij ∈ N information symbols to Receiver i.…”

“…Our sum DoF converse generalizes the one given in [13,14] to the multiple antenna setting, and requires establishing a general rank-ratio inequality for the ratio between dimensions of received signal subspaces at the two receivers. Our rank-ratio inequality results from establishing two bounds.…”

“…However, the DoF-optimal transmission strategy for the general MIMO XC with delayed CSIT is still unknown, except for the case where all nodes have a single antenna, for which a novel converse was recently derived in [13,14] that shows the sum DoF optimality of the scheme proposed in [10] under the restriction of linear encoding strategies. The key insight of [13,14] relied on establishing a rank-ratio inequality that shows that, if two distributed single antenna transmitters employ linear strategies, the ratio of the dimensions of received linear subspaces at two arbitrary receivers cannot exceed…”

“…The key insight of [13,14] relied on establishing a rank-ratio inequality that shows that, if two distributed single antenna transmitters employ linear strategies, the ratio of the dimensions of received linear subspaces at two arbitrary receivers cannot exceed…”