Forming Pseudo-MIMO by Embedding Infinite Rational Dimensions Along a Single Real Line: Removing Barriers in Achieving the DOFs of Single Antenna Systems

The K-user single-antenna Gaussian Interference Channel (GIC) is considered, where the channel coefficients are NOT necessarily time-variant or frequency selective. It is proved that the total Degrees-Of-Freedom (DOF) of this channel is K 2 almost surly, i.e. each user enjoys half of its maximum DOF. Indeed, we prove that the static time-invariant interference channels are rich enough which allow simultaneous interference alignment at all receivers. To derive this result, we show that single-antenna interference channels can be treated as pseudo multiple-antenna systems with infinitely-many antennas, as many as rationally-independent irrational numbers. Such machinery enables us to prove that the real or complex M ×M Multiple Input Multiple Output (MIMO) GIC achieves its total DOF, i.e., MK 2 , M ≥ 1. The pseudo multiple-antenna systems are developed based on a recent result in the field of Diophantie approximation which states that the convergence part of the Khintchine-Groshev theorem holds for points on non-degenerate manifolds.