Center for Pervasive Communications and Computing

We investigate the optimality of linear interference

alignment (allowing symbol extensions) for  3-user

$M_T\times M_R$ MIMO interference channel where $M_T$ and $M_R$

denote the number of antennas at each transmitter and each receiver,

respectively, and the \emph{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 $M_T, M_R$ values except

if $|M_T-M_R|=1$, $\min(M_T,M_R)\geq 2$. A partial proof of the conjecture is provided by Wang et al. for arbitrary $M_T, M_R$ values subject to a final numerical evaluation step that needs to be performed for each $M_T, M_R$ setting to complete the proof. The numerical evaluation step is also carried out explicitly by Wang et al. to settle the conjecture for all  $M_T, M_R$ values up to 10. For $|M_T-M_R|=1$, $\min(M_T,M_R)\geq 2$, 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 $M_T, M_R$ 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 \emph{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 $M_T, M_R$ values in the 3 user $M_T\times M_R$ MIMO interference channel with constant channel coefficients, except for the case $M_T=M_R=1$ where it is known that either time/frequency-varying channels or non-linear (e.g., rational alignment) schemes are required.