Subspace alignment chains and the degrees of freedom of the three-user MIMO interference channel

We show that the 3 user M_T × M_R MIMO interference channel where each transmitter is equipped with M_T and each receiver is equipped with M_R antennas has min (M/2-1/k, N/2+1/k) degrees of freedom (DoF) per user normalized by time, frequency, and space dimensions, where N = max(M_T, M_R), M = min(M_T, M_R), k = [M/N-M]. While the information theoretic DoF outer bound is established for every M, N value, the achievability, relying only on linear interference alignment, is established in general subject to a normalization with respect to spatial-extensions, i.e., the scaling of the number of antennas at all nodes. In the absence of spatial extensions, we can also show through examples how essentially the same alignment scheme may be applied over time or frequency extensions. The central new insight to emerge from this work is the notion of subspace alignment chains as DoF bottlenecks. The subspace alignment chains are instrumental both in identifying the extra dimensions provided by a genie to a receiver for the DoF outer bound, as well as constructing the optimal interference alignment schemes. In addition, our results also settle the question of feasibility of linear interference alignment for the 3 user M_T × M_R MIMO interference channel, for all values of M_T, M_R.

I. Introduction

The number of degrees of freedom (DoF) of a communication network provides a lens into the most essential aspects of the communication problem. In this work we are interested primarily in interference channels (IC). The DoF of IC are known when all nodes are equipped with the same number of antennas for almost all channel realizations. However, if each node may have an arbitrary number of antennas, then a general DoF characterization is unavailable beyond the 2 user IC, due to the explosion of the number of parameters in considering arbitrary antenna configurations. However, as we show in this work, the problem involves fundamental challenges even when the number of parameters is restricted by symmetry. In particular, the difficulty is to deal with the new notion of “depth” of overlap between vector subspaces that come into play on the one hand, and of translating this notion into information theoretic bounds on the other. Specifically, we will explore the simplest setting for MIMO IC where the DoF are unknown.