Genie Tree and Degrees of Freedom of the Symmetric MIMO Interfering Broadcast Channel

Abstract: In this paper, we study the information theoretic maximal degrees of freedom (DoF) for the symmetric multi-input-multi-output (MIMO) interfering broadcast channel (IBC) with arbitrary antenna configurations. For the G-cell K-user MXN MIMO-IBC network, we find that the information theoretic maximal DoF per user are related to three DoF bounds: 1) the decomposition DoF bound d^{Decom}=MN/(M+KN), a lower-bound of linear interference alignment (IA) with infinite time/frequency extensions (called asymptotic IA); 2) the proper DoF bound d^{Proper}=(M+N)/(GK+1), an upper-bound of linear IA without time/frequency extensions (called linear IA); and 3) the quantity DoF bound d^{Quan}, a zigzag piecewise linear function of M and N. The whole region of M/N can be divided into two regions, Region I and Region II. Specifically, for most configurations in Region I, the information theoretic maximal DoF are the decomposition DoF bound and can be achieved by the asymptotic IA. For all configurations in Region II, the information theoretic maximal DoF are the quantity DoF bound and can be achieved by the linear IA. To obtain the tight upper-bound, we propose a unified way to construct genies, where the genies help each base station or user resolve the maximal number of interference. According to the feature that the designed genies with the same dimension can derive identical DoF upper-bound, we convert the information theoretic DoF upper-bound problem into a linear algebra problem and obtain the closed-form DoF upper-bound expression. Moreover, we develop a non-iterative linear IA transceiver to achieve the DoF upper-bound for the networks with antenna configurations in Region II, which means that the derived DoF upper-bound is tightest. The basic principles to derive the DoF upper-bound and design the linear IA transceiver to achieve the DoF upper-bound can be extended into general asymmetric networks.