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We show that the 3 user M_T x M_R MIMO interference channel hasd(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_Rvalue, the achievability is established in general subject to normalizationwith respect to spatial-extensions. Given spatial-extensions, the achievabilityrelies only on linear beamforming based interference alignment schemes with noneed for time/frequency extensions. In the absence of spatial extensions, weshow through examples how essentially the same scheme may be applied overtime/frequency extensions. The central new insight to emerge from this work isthe 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 orN being the bottleneck within each linear segment. The corner points of thesepiecewise linear segments correspond to A={1/2,2/3,3/4,...} andB={1/3,3/5,5/7,...}. The set A contains all values of M/N and only those forwhich there is redundancy in both M and N. The set B contains all values of M/Nand 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 interferencealignment 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 exceptionof the values M/N\in B, we show that for every M/N value there are propersystems that are not feasible.  Our results show that M/N\in A are the only values for which there is no DoFbenefit of joint processing among co-located antennas at the transmitters orreceivers.

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