Weighted Sum Degrees of Freedom of the Asymmetric MIMO Y Channel with Common and Private Messages

摘要

This paper investigates the weighted sum degrees of freedom (DoF) of the MIMO Y channel that consists of three users, where the j-th user is equipped with M_jantennas, and a relay equipped with N antennas. In this network, each user conveys two private messages to the other two users in addition to a common message directed to both of them. As there is no direct link between the users, communication occurs through the relay. We define a weighted sum DoF metric that integrates all the network messages and weights the common message by a factor of α. Then, we study the weighted sum DoF maximization problem for α ≥ 0. Specifically, we show that the weighted sum DoF of the network, with M₁≥ M₂≥ M₃, is given by min{2N, 2M₂+ 2M₃, M₁+ M₂+ M₃}, for 0 ≤ α ≤ 4/3. While, for 4/3 <; α ≤ 2, the weighted sum DoF is equal to min{3α/2 N, 2(N - M₃) + 3α/2 M₃, 2(2N - M₂- M₃) + 3α/2 (M₂+ M₃- N), (6N - 2(M₁+ M₂+M₃))+ 3α/2 (M₁+ M₂+ M₃-2N), 2M₂+ 2M₃, M₁+ M₂+ M₃}, and finally, for α > 2, the weighted sum DoF is equal to min{3α/2 N, α/2 (2N + M₃), 2(N - 3/2M₃) + 2αM₃, α/2 (N + M₂+ M₃), N + M₂- 3M₃+ 2αM₃, α/2 (M₁+ M₂+ M₃), M₁+ M₂- 3M₃+ 2αM₃, 2(M₂- M₃) + 2αM₃}. Achievability of the sum DoF is shown by using signal space alignment for network coding in the uplink phase, and zero-forcing precoding in the downlink phase.