A New DoF Upper Bound and Its Achievability for $K$-User MIMO Y Channels

This work is to study the degrees of freedom (DoF) for the $K$-user MIMO Y channel. Previously, two transmission frameworks have been proposed for the DoF analysis when $N \geq 2M$, where $M$ and $N$ denote the number of antennas at each source node and the relay node respectively. The first method is named as signal group based alignment proposed by Hua et al. in [1]. The second is named as signal pattern approach introduced by Wang et al. in [2]. But both of them only studied certain antenna configurations. The maximum achievable DoF in the general case still remains unknown. In this work, we first derive a new upper bound of the DoF using the genie-aided approach. Then, we propose a more general transmission framework, generalized signal alignment (GSA), and show that the previous two methods are both special cases of GSA. With GSA, we prove that the new DoF upper bound is achievable when $\frac{N}{M} \in \left(0,2+\frac{4}{K(K-1)}\right] \cup \left[K-2, +\infty\right)$. The DoF analysis in this paper provides a major step forward towards the fundamental capacity limit of the $K$-user MIMO Y channel. It also offers a new approach of integrating interference alignment with physical layer network coding.