  Spectral and energy efficiency in 3-way relay channels are studied in this
paper. First, achievable sum rate expressions for 3-way relay channels are
derived for different relaying protocols. Moreover, an outer bound for the
capacity of the 3-way relay channel is presented. Next, leveraging the derived
achievable sum rate expressions, two algorithms for joint power allocation at
the users and at the relay are designed so as to maximize the system energy
efficiency. Numerical results are provided to corroborate and provide insight
on the theoretical findings.
