Achievable Degrees of Freedom for K-User MIMO Y Channels Using Signal Group Based Alignment

Hua Mu, Jitendra K. Tugnait

Abstract: We consider a $K$-user multiple input multiple output (MIMO) Y channel consisting of $K(\geq 3)$ users and a relay. Each user has $K - 1$ independent messages for all the other $K - 1$ users. Degrees of freedom (DoF) of such channels is not known in general but it is known that the DoF of $K(K - 1)/2$ is achievable for a network operating in a half-duplex mode by using signal space alignment for network coding during both the multiple access phase and the broadcast phase. In this paper, a novel signal group based alignment scheme is proposed, which divides all $K(K - 1)$ signals into $l$ groups where $l = K$ or $K - 1$. Then, the signals in each group are aligned into a smaller subspace at the relay. If the $i$-th user is equipped with $M_{i}$ antennas and the relay is equipped with $N$ antennas where all antennas are used for both transmitting and receiving, we prove that when $M_{i} = K - 1$, $N = (K - 1)^{2}$ for even $K$ and $M_{i} = K - 1$, $N = K(K - 2)$ for odd $K$, the optimal total DoF of this $K$-user MIMO Y channel is $K(K - 1)/2$. As a consequence, to achieve the total DoF of $K(K - 1)/2$, the requirements on $M_{i}$ and $N$ are $M_{i} \geq K - 1$ and $N \geq (K - 1)^{2}$ for even $K$, and $M_{i} \geq K - 1$ and $N \geq K(K - 2)$ for odd $K$. In our proposed approach, we significantly decrease the minimum $M_{i}$ at the expense of higher $N$ for a given number of users $K$ and achievable DoF of $K(K - 1)/2$, compared to an existing approach. This signal group alignment concept also motivates other signal grouping methods, which provide a tradeoff between number of antennas at end users and the relay. Also, for the $K$-user Y channel where all end users have a single antenna and the relay node has $N$ antennas, it is shown that the DoF of $\min\{K/2, (N + 1)/2\}$ is achievable.