Achievable rate for a multi-source relaying system

In this work we determine the achievable rate in a multi-source relaying system with Gaussian phase-fading channels. In our system, M sources simultaneously transmit their messages to a common destination in M separate frequency bands with the help of a single relay (an M − 1 − 1 system). The achievable rates of both a separate processing scheme at the relay, and a network coding scheme at the relay, are considered. For the separate processing scheme, we propose an new constrained water-filling algorithm which determines the power allocation at the relay in order to obtain the achievable rate. For the network coding scheme we derive the achievable rate based on the use of a new Galois field rate-splitting theorem, and discuss why power allocation at the relay in this scheme can be set using a traditional water-filling algorithm. We show how our network coding scheme will always obtain higher achievable rates relative to those obtained from a separate processing scheme.