Degrees of freedom of the three-user rank-deficient MIMO interference channel

We provide the degrees of freedom (DoF) characterization for the three-user M_T×M_R multiple-input-multiple-output (MIMO) interference channel (IC) with rank-deficient channel matrices, where each transmitter is equipped with M_T antennas and each receiver with M_R antennas, and the interfering channel matrices from each transmitter to the other two receivers are of ranks D₁ and D ₂, respectively. One important intermediate step for both the converse and achievability arguments is to convert the fully-connected rank-deficient channel into an equivalent partially-connected full-rank MIMO-IC by invertible linear transformations. As such, existing techniques developed for full-rank MIMO-IC can be incorporated to derive the DoF outer and inner bounds for the rank-deficient case. Our result shows that when the interfering links are weak in terms of the channel ranks, i.e., D_T+D_T≤ min(M_T, M_T), zero forcing is sufficient to achieve the optimal DoF. On the other hand, when D₁ + D₂ > (M _T, M_R, a combination of zero forcing and interference alignment is in general required for DoF optimality. The DoF characterization obtained in this paper unifies several existing results in the literature.