On Feasibility of Interference Alignment in MIMO Interference Networks

TL;DR

This paper investigates the conditions under which interference alignment is feasible in MIMO interference channels, linking the problem to algebraic geometry and polynomial system solvability, and providing new theoretical insights.

Contribution

It introduces a novel algebraic geometric framework to determine interference alignment feasibility, especially relating proper systems to solvability in MIMO networks.

Findings

Feasibility linked to polynomial system solvability.

Proper systems are generally feasible under certain conditions.

Multi-beam cases require additional bounds for feasibility assessment.

Abstract

We explore the feasibility of interference alignment in signal vector space -- based only on beamforming -- for K-user MIMO interference channels. Our main contribution is to relate the feasibility issue to the problem of determining the solvability of a multivariate polynomial system, considered extensively in algebraic geometry. It is well known, e.g. from Bezout's theorem, that generic polynomial systems are solvable if and only if the number of equations does not exceed the number of variables. Following this intuition, we classify signal space interference alignment problems as either proper or improper based on the number of equations and variables. Rigorous connections between feasible and proper systems are made through Bernshtein's theorem for the case where each transmitter uses only one beamforming vector. The multi-beam case introduces dependencies among the coefficients of a polynomial system so that the system is no longer generic in the sense required by both theorems. In this case, we show that the connection between feasible and proper systems can be further strengthened (since the equivalency between feasible and proper systems does not always hold) by including standard information theoretic outer bounds in the feasibility analysis.