Amplify-and-modulo for Gaussian two-way relay channel

We consider a two-way relay channel (TWRC) in which two terminals exchange 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. A transmission scheme for the Gaussian TWRC is said to be analog-relaying if the relay does not need any codebook for encoding. The simplest analog-relaying scheme is amplify-and-forward (AF), under which the relay amplifies the received codeword and forwards the resultant codeword to the two terminals. In this paper, we propose a new analog-relaying scheme called amplify-and-modulo (AM) based on lattice operations. AM is a slight modification of AF. Under AM, the relay first amplifies the received codeword followed by reducing the power of the amplified codeword using the modulo-lattice operation, and then forwards the resultant codeword to the two terminals. After receiving the codeword transmitted by the relay, each terminal subtracts its own information before decoding. We prove an achievable rate region for AM, and obtain a necessary and sufficient condition under which AM outperforms AF. In addition, we show by graph that AM can achieve a strictly higher equal-rate than AF and another existing analog-relaying scheme together under some scenario.