On Optimality of Linear Interference Alignment for the Three-User MIMO Interference Channel with Constant Channel Coefficients

We investigate the optimality of linear interference alignment (allowing symbol extensions) for 3-user MT × MR MIMO interference channel where MT and MR denote the number of antennas at each transmitter and each receiver, respectively, and the channel coefficients are held constant. Recently, Wang et al. have conjectured that interference alignment based on linear beamforming using only proper Gaussian codebooks and possibly with symbol extensions, is sufficient to achieve the information theoretic DoF outer bound for all MT ,MR values except if |MT −MR| = 1, min(MT ,MR) ≥ 2. A partial proof of the conjecture is provided by Wang et al. for arbitrary MT ,MR values subject to a final numerical evaluation step that needs to be performed for each MT ,MR setting to complete the proof. The numerical evaluation step is also carried out explicitly by Wang et al. to settle the conjecture for all MT ,MR values up to 10. For |MT −MR| = 1, min(MT ,MR) ≥ 2, Wang et al. show that interference alignment schemes based on linear beamforming with proper Gaussian signaling and symbol extensions are not sufficient to achieve the information-theoretic DoF outer bonds. In contrast, in this note we show, for all MT ,MR values up to 10, that interference alignment schemes based on linear beamforming over symbol extensions are enough to achieve the information theoretic DoF outer bounds for constant channels, if asymmetric complex signaling is utilized. Based on this new insight, we conjecture that linear interference alignment is optimal for achieving the information theoretic DoF outer bounds for all MT ,MR values in the 3 user MT ×MR MIMO interference channel with constant channel coefficients, except for the case MT = MR = 1 where it is known that either time/frequency-varying channels or non-linear (e.g., rational alignment) schemes are required.