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Subspace Alignment Chains and the Degrees of Freedom of the Three-User MIMO Interference Channel

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 Cenk et al., 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 Cenk et al.) is feasible if and only if d<=floor{d(M,N)}. With and only with the exception of the values M/N\in B, we show that for every M/N value there are proper systems that are not feasible. Our results show that M/N\in A are the only values for which there is no DoF benefit of joint processing among co-located antennas at the transmitters or receivers.