Interference alignment with asymmetric complex signaling

It has been conjectured that complex Gaussian interference channels with constant channel coefficients have only one degree-of-freedom (DoF) regardless of the number of users. While several examples are known of constant channels that achieve more than 1 DoF, these special cases only span a subset of measure zero. In other words, for almost all channel coefficient values, it is not known if more than 1 DoF is achievable. In this paper, we settle this conjecture in the negative. We show that at least 1.2 DoF are achievable for all values of complex channel coefficients except for a subset of measure zero. To establish the achievability of 1.2 DoF we introduce the novel idea of asymmetric complex signaling - i.e., the inputs are chosen to be complex but not circularly symmetric. It is shown that unlike Gaussian point-to-point, multiple-access and broadcast channels where circularly symmetric complex Gaussian inputs are optimal, for interference channels optimal inputs are in general asymmetric. In addition, with this idea, we also show that 4/3 DoF can be achieved for 2 user X channel with constant, complex channel coefficients.