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

We consider a K-user multiple input multiple output (MIMO) Y channel consisting of K(≥ 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
≥ K-1 and N ≥ (K-1)
²
for even K, and M
_i
≥ K-1 and N ≥ 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.