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 = N. It turns out that, just as for the 3-user parallel interference channel 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.
We show that the 3 user M_T x M_R MIMO interference channel has d(M,N)=min(M (2-1 k),N (2+1 k)) degrees of freedom (DoF) normalized by time, frequency, and space dimensions, where M=min(M_T,M_R), N=max(M_T,M_R), k=ceil M (N-M) . While the DoF outer bound is established for every M_T, M_R value, the achievability is established in general subject to normalization with respect to spatial-extensions. Given spatial-extensions, the achievability relies only on linear beamforming based interference alignment schemes with no need for time frequency extensions. In the absence of spatial extensions, we show through examples how essentially the same scheme may be applied over time frequency extensions. The central new insight to emerge from this work is the notion of subspace alignment chains as DoF bottlenecks. The DoF value d(M,N) is a piecewise linear function of M,N, with either M or N being the bottleneck within each linear segment. The corner points of these piecewise linear segments correspond to A= 1 2,2 3,3 4,... and B= 1 3,3 5,5 7,... . The set A contains all values of M N and only those for which there is redundancy in both M and N. The set B contains all values of M N and only those for which there is no redundancy in either M or N. Our results settle the feasibility of linear interference alignment, introduced by , for the 3 user M_T x M_R MIMO interference channel, completely for all values of M_T, M_R. Specifically, the linear interference alignment problem (M_T x M_R, d)^3 (as defined in previous work by ) is feasible if and only if d<=floor d(M,N) . With and only with the exception of the values M N B, we show that for every M N value there are proper systems that are not feasible. Our results show that M N A are the only values for which there is no DoF benefit of joint processing among co-located antennas at the transmitters or receivers.