On Achievable Degrees of Freedom for MIMO X Channels

In this paper, the achievable DoF of MIMO X channels for constant channel coefficients with Mt antennas at transmitter t and Nr antennas at receiver r (t, r = 1, 2) is studied. A spatial interference alignment and cancelation scheme is proposed to achieve the maximum DoF of the MIMO X channels. The scenario of M1 ≥ M2 ≥ N1 ≥ N2 is first considered and divided into 3 cases, 3N2 < M1+M2 < 2N1+N2 (Case A), M1 +M2 ≥ 2N1 +N2 (Case B), and M1 +M2 ≤ 3N2 (Case C). With the proposed scheme, it is shown that in Case A, the outer-bound M1+M2+N2 2 is achievable; in Case B, the achievable DoF equals the outer-bound N1 + N2 if M2 > N1, otherwise it is 1 2 or 1 less than the outer-bound; in Case C, the achievable DoF is equal to the outer-bound 2 3 (M1 + M2) if (3N2 − M1 − M2) mod 3 = 0, and it is 13 or 1 6 less than the outer-bound if (3N2 − M1 − M2) mod 3 = 1 or 2. In the scenario of Mt ≤ Nr , the exact symmetrical results of DoF can be obtained.