Real Interference Alignment with Real Numbers

A novel coding scheme applicable in networks with single antenna nodes is proposed. This scheme converts a single antenna system to an equivalent Multiple Input Multiple Output (MIMO) system with fractional dimensions. Interference can be aligned along these dimensions and higher Multiplexing gains can be achieved. Tools from the field of Diophantine approximation in number theory are used to show that the proposed coding scheme in fact mimics the traditional schemes used in MIMO systems where each data stream is sent along a direction and alignment happens when several streams arrive at the same direction. Two types of constellation are proposed for the encoding part, namely the single layer constellation and the multi-layer constellation. Using the single layer constellation, the coding scheme is applied to the two-user $X$ channel and the three-user Gaussian Interference Channel (GIC). In case of the two-user $X$ channel, it is proved that the total Degrees-of-Freedom (DOF), i.e. 4/3, of the channel is achievable almost surely. This is the first example in which it is shown that a time invariant single antenna system does not fall short of achieving its total DOF. Using the multi-layer constellation, the coding scheme is applied to the symmetric three-user GIC. Achievable DOFs are derived for all channel gains. As a function of the channel gain, it is observed that the DOF is everywhere discontinuous.