Degrees of Freedom of the Circular Multirelay MIMO Interference Channel in IoT Networks

Abstract

In this paper, we study the degrees of freedom (DoF) of a new network information flow model named the circular multirelay multiple-input multiple-output interference channel (CMMI). In this model, there are two clusters and each of them contains three users. Each user equipped with $M$ antennas in one cluster intends to deliver data streams to another user in the same cluster in a circular one-way transmission via the common distributed $K~N$ -antenna relay nodes. The CMMI network model can be considered as a basic component to construct the complicated Internet of Things networks. By assuming linear processing at the users and the relays, we show that the original analysis of DoF comes down in finding solutions of some nonlinear matrix equations with rank constraints. Toward this end, by using linear precoding and post-processing techniques, we propose two different approaches to solve the nonlinear matrix equations based on different antenna configurations. We show that a DoF of $\text{max} \{ {\text{min} \{ M,({{\sqrt {6K} }}/{12}) \}{,}~\text{min} \{ {({M}/{3}),({KN}/{2})} \}} \}$ is achievable for ${\forall ({M}/{N}) \in ({0, + \infty })}$ . In addition, to assess the optimal DoF, the cut-set approach is used for deriving the DoF upper bound by innovatively separating certain users to form two-pair two-way relay channels. We show that the DoF of CMMI is upper bounded by ${\text{max} \{{\text{min} \{{M,({KN}/{3})} \}, \text{min} \{ {({2M}/{3)},({KN}/{2})} \}} \}}$ . By combining the achievable DoF and the upper bound, we finally show that the optimal DoF of CMMI can be achieved for ${({M}/{N}) {\in } [{0,({{\sqrt {6K} }}/{12})}] {\cup } [{({3K}/{2}), + \infty }),{\forall } {K \ge 1}}$ .