Geometry of the 3-user MIMO interference channel

TL;DR

This paper characterizes when interference alignment is feasible in a symmetric 3-user MIMO interference channel without time or frequency diversity, revealing the importance of alignment path length and geometric considerations.

Contribution

It provides a precise feasibility criterion for interference alignment based on geometric analysis and alignment path length, advancing understanding beyond simple equation counting.

Findings

Feasibility depends on a specific inequality involving M, N, d, and path length r.

Alignment path length is crucial for both proving feasibility and impossibility.

Counting equations alone is insufficient to determine feasibility.

Abstract

This paper studies vector space interference alignment for the three-user MIMO interference channel with no time or frequency diversity. The main result is a characterization of the feasibility of interference alignment in the symmetric case where all transmitters have M antennas and all receivers have N antennas. If N >= M and all users desire d transmit dimensions, then alignment is feasible if and only if (2r+1)d <= max(rN,(r+1)M) for all nonnegative integers r. The analogous result holds with M and N switched if M >= N. It turns out that, just as for the 3-user parallel interference channel \cite{BT09}, the length of alignment paths captures the essence of the problem. In fact, for each feasible value of M and N the maximum alignment path length dictates both the converse and achievability arguments. One of the implications of our feasibility criterion is that simply counting equations and comparing to the number of variables does not predict feasibility. Instead, a more careful investigation of the geometry of the alignment problem is required. The necessary condition obtained by counting equations is implied by our new feasibility criterion.