On the DoF of Two-Way MIMO Butterfly Networks

This paper studies the degrees of freedom (DoF) of two-way multiple-input multiple-output (MIMO) butterfly networks, a class of two-way four-unicast networks, that consists of four source/destination nodes and three relay nodes. We first give upper and lower bounds on the sum DoF of such network with any number of antennas in each node. We also study the DoF of the two-way MIMO butterfly network with caching at the relays. For the special case that each source/destination node has M antennas and each relay node has N antennas, we obtain the exact DoF for some special cases. Specifically, if N ≤ M, then without caching, DoF = 2N, which is equal to the DoF of a one-way MIMO butterfly network, and with relay caching, DoF = 4N. Moreover, if N > 2M, then with or without relay caching, DoF = 4M which doubles the DoF of a one-way MIMO butterfly network. Hence, for this MIMO butterfly network, when the number of relay antennas is large compared with the number of source/destination antennas, bidirectional transmission improves the DoF; and when the number of relay antennas is small, relay caching improves the DoF.