# Real Interference Alignment: Exploiting the Potential of Single Antenna Systems

## TL;DR

This paper introduces a novel interference alignment scheme for single antenna systems that exploits fractional degrees of freedom, achieving optimal capacity in various multi-user channels using advanced number theory techniques.

## Contribution

It proposes a new coding scheme leveraging rational field properties and Diophantine approximation to optimize interference alignment in single antenna systems.

## Key findings

- Achieves total DOF of K/2 for K-user Gaussian Interference Channel.
- Determines total DOF of KM/(M+1) for cellular uplink channels.
- Establishes total DOF of KM/(K+M-1) for X channels.

## Abstract

In this paper, the available spatial Degrees-Of-Freedoms (DOF) in single antenna systems is exploited. A new coding scheme is proposed in which several data streams having fractional multiplexing gains are sent by transmitters and interfering streams are aligned at receivers. Viewed as a field over rational numbers, a received signal has infinite fractional DOFs, allowing simultaneous interference alignment of any finite number of signals at any finite number of receivers. The coding scheme is backed up by a recent result in the field of Diophantine approximation, which states that the convergence part of the Khintchine-Groshev theorem holds for points on non-degenerate manifolds. The proposed coding scheme is proved to be optimal for three communication channels, namely the Gaussian Interference Channel (GIC), the uplink channel in cellular systems, and the $X$ channel. It is proved that the total DOF of the $K$-user GIC is $\frac{K}{2}$ almost surely, i.e. each user enjoys half of its maximum DOF. Having $K$ cells and $M$ users within each cell in a cellular system, the total DOF of the uplink channel is proved to be $\frac{KM}{M+1}$. Finally, the total DOF of the $X$ channel with $K$ transmitters and $M$ receivers is shown to be $\frac{KM}{K+M-1}$.