Diophantine Approach to Blind Interference Alignment of Homogeneous K-user 2x1 MISO Broadcast Channels

TL;DR

This paper investigates the conditions under which a K-user 2x1 MISO broadcast channel can achieve optimal degrees of freedom using blind interference alignment, by translating the problem into solving Diophantine equations.

Contribution

It introduces a Diophantine equation framework to determine the feasibility of BIA in K-user channels and derives conditions on user offsets for BIA feasibility.

Findings

Derived necessary and sufficient conditions for BIA feasibility based on Diophantine equations.

Proved that with independent, uniformly distributed offsets, 11 users suffice to find 3 BIA-feasible users with 95% certainty.

Established a link between user fading offsets and the ability to achieve maximal DoF in homogeneous channels.

Abstract

Although the sufficient condition for a blindly interference-aligned (BIA) 2-user 2x1 broadcast channel (BC) in homogeneous fading to achieve its maximal 4/3 DoF is well understood, its counterpart for the general K-user 2x1 MISO BC in homogeneous block fading to achieve the corresponding 2k/(2+K-1) (DoF) remains unsolved and is, thus, the focus of this paper. An interference channel is said BIA-feasible if it achieves its maximal DoF only via BIA. In this paper, we cast this general feasibility problem in the framework of finding integer solutions for a system of linear Diophantine equations. By assuming independent user links each of the same coherence time and by studying the solvability of the Diophantine system, we derive the sufficient and necessary conditions on the K users' fading block offsets to ensure the BIA feasibility of the K-user BC. If the K offsets are independent and uniformly distributed over a coherence block, we can further prove that 11 users are enough for one to find, with certainty of 95%, 3 users among them to form a BIA-feasible 3-user 2x1 BC.