On the Capacity Bounds of Non-Restricted Gaussian Amplify-and-Forward Two-Way Relay Channel

In this letter, we revisit the non-restricted half-duplex Gaussian amplify-and-forward two-way relay channel (AF-TWRC). The inner and outer bounds of its capacity region are derived by evaluating Shannon's inner and outer bounds for general two-way channels. We prove that, in contrast to the case of Gaussian two-way channel (GTWC) where the two bounds coincide thus establishing its capacity region, the two bounds of AF-TWRC do not coincide unless both terminals are noise-free, due to that the intermediate relay node introduces additional power constraint relating to the correlation of channel inputs from both terminals. This indicates that the capacity region of non-restricted Gaussian AF-TWRC, which was previously believed to be already established in a way similar to that of GTWC, still remains open. We further investigate a special case of non-restricted AF-TWRC where the relay is noise-free, and obtain its capacity region based on the tighter dependence balance bound.