Capacity bounds for full-duplex two-way relay channel with feedback

We consider a two-way relay channel (TRC) in which two terminals exchange their messages with the help of a relay between them. The two terminals transmit messages to the relay through the Multiple Access Channel (MAC) and the relay transmits messages to the two terminals through the Broadcast Channel (BC). We assume that the MAC and the BC do not interfere with each other, and each terminal receives signals only from the relay but not the other terminal. All the nodes are assumed to be full-duplex, which means that they can transmit and receive information at the same time. We present with detailed proofs an outer bound on the capacity region of each of the discrete memoryless TRC with feedback and the Gaussian TRC with feedback. We show that the outer bound for the discrete memoryless TRC with feedback is loose for some TRC.