Achievable rate regions of multi-way relay channel with direct links

Rate regions of a multi-way relay channel with direct links (MWRC-DLs), where K users exchange their messages via a relay terminal and all users can overhear each other directly, are studied in this study. Under the assumption that a restricted encoder is employed at each user, the cut-set outer bound on the capacity region is derived first. Then, achievable rate regions of the MWRC-DLs with decode-and-forward (DF) and compress-and-forward (CF) strategies are characterised. Meanwhile, the explicit expressions of the outer bound and the achievable rate regions for the Gaussian MWRC-DLs are also derived. It is shown that the rate regions of the DF and CF strategies for the two-way relay channel and the multiple-access relay channel can be obtained from those of the MWRC-DLs. To give more insights on the two strategies of the MWRC-DLs system, the common rates of a symmetric Gaussian network are analysed. It is shown that the CF strategy achieves common rates within 1/2(K - 1) bits of the capacity when the relay´s power is at least (K - 1) times as large as the user power. Numerical examples are also provided to verify the theoretical analysis.