Weighted sum degrees of freedom of the asymmetric MIMO y channel with common and private messages

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 Mj antennas, 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{3αN/2, 2(N-M3) + 3αM3/2, 2(2N-M2-M3) + 3α(M2+M3-N)/2, (6N-2(M1+M2+M3))+ 3α(M1+M2+M3-2N)/2, 2M2+2M3, M1+M2+M3}, and finally, for α > 2, the weighted sum DoF is equal to min{3αN/2, α(2N +M3)/2, 2(N-3M3/2) + 2αM3, α(N +M2 +M3)/2, N+M2-3M3+2αM3, α(M1 +M2 +M3)/2, 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. © 2017 IEEE.