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

Abstract

We establish the degrees of freedom of the two-user 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 that shows that, if two distributed transmitters employ linear strategies, the ratio of the dimensions of received linear subspaces at the two receivers cannot exceed 3 2 , due to delayed CSIT. As a byproduct, we also apply this general lemma to the three-user interference channel with delayed CSIT, thereby deriving a new upper bound of 9 7 on its linear degrees of freedom. This is the first bound that captures the impact of delayed CSIT on the degrees of freedom of this network, under the assumption of linear encoding strategies. I. INTRODUCTIONThe X-channel is a canonical setting for the information-theoretic study of interference management in wireless networks. This channel consists of two transmitters causing interference at two receivers, and each transmitter aims to communicate intended messages to both receivers. The question is: how can the transmitters optimally manage the interference and communicate their messages to the receivers? This problem has been studied extensively in the literature and various interference management techniques have been proposed. In particular, in [2] it is shown that, quite surprisingly, one can significantly improve upon conventional interference management schemes (e.g., orthogonalization) and achieve 4/3 degrees of freedom (DoF) by using interference alignment (IA) [3,4].However, in order to perfectly align the interference, the transmitters need to accurately know the current state of the channels, which is practically very challenging and may even be impossible (due to, for example, high mobility). Thus, a natural question would be: how can the transmitters optimally manage the interference with only delayed knowledge of the channel state information (i.e., delayed CSIT)?In the context of broadcast channel, Maddah-Ali and Tse in [5] have recently shown that delayed CSIT can still be very useful. In particular, for the multi-antenna broadcast channel with delayed CSIT, they developed an innovative transmission strategy that utilizes the past received signals to create signals of common interest to multiple receivers, hence significantly improving DoF by broadcasting them to the receivers. In a sense, these "signals of common interest" represent aligned interferences in the past receptions.Subsequently in [6][7][8][9], the impact of delayed CSIT has been explored for a variety of interference networks in which transmit antennas are now distributed at different locations. Unlike multi-antenna broadcast channels, in S. Lashgari is with the

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

“…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.…”

“…Satisfying (5) for all i, j ∈ {1, 2}, is also equivalent to satisfying the following conditions for all i, j ∈ {1, 2}, as proven in [14]:…”

“…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…”

“…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.…”