On the DoF Region of the MIMO Gaussian Two-User Interference Channel With an Instantaneous Relay

Abstract

This paper studies inner and outer bounds on the degrees of freedom (DoF) region of the multi-antenna two-user Gaussian interference channel with an instantaneous relay (IR) or relay without delay. It is assumed that the two transmitters and the two receivers have $\mathop {\mathrm {\mathsf {M}}}\nolimits$ antennas, while the IR receives through $\mathop {\mathrm {\mathsf {N_{r}}}}\nolimits$ antennas and transmits through $\mathop {\mathrm {\mathsf {N_{t}}}}\nolimits$ antennas. In the proposed achievable scheme, which generalizes a known one for the case $\mathop {\mathrm {\mathsf {M}}}\nolimits = \mathop {\mathrm {\mathsf {N_{r}}}}\nolimits = \mathop {\mathrm {\mathsf {N_{t}}}}\nolimits$ to any $(\mathop {\mathrm {\mathsf {M}}}\nolimits, \mathop {\mathrm {\mathsf {N_{r}}}}\nolimits, \mathop {\mathrm {\mathsf {N_{t}}}}\nolimits)$ , the IR performs memoryless linear operations on its received signal so as to neutralize interference at the receivers, and the beamforming matrices used by the IR and the transmitters are jointly designed. This joint design strictly outperforms known achievable schemes. Two outer bounds are derived. An information theoretic outer bound is obtained by giving the receivers or the IR genie side information, so that the DoF region of the resulting enhanced channel is known; this converse is valid for any type of processing at the IR and shows the optimality of the proposed achievable scheme for some $(\mathop {\mathrm {\mathsf {M}}}\nolimits, \mathop {\mathrm {\mathsf {N_{r}}}}\nolimits, \mathop {\mathrm {\mathsf {N_{t}}}}\nolimits)$ . A linear processing outer bound is obtained when the IR is restricted to performs linear operations, without any memoryless restriction, on its received signal and shows the optimality of the proposed achievable scheme among all linear processing schemes at the IR. As a result of independent interest, the DoF region of the classical multi-antenna two-user Gaussian interference channel without relay when the channel matrices can have any structure is also derived, which generalized available DoF region results that were derived under certain assumptions on the structure of the channel matrices.