Error exponents for two-hop Gaussian multiple source-destination relay channels

We investigate the two-hop multiple source-destination relay channels using the rate splitting transmission scheme where each source can split its message into private and public parts. We determine the system error probability via integrated exponent function under amplify-and-forward (AF) and decode-and-forward (DF) relay strategies. The most significant difference between AF and DF system error probability evaluations lies in a minimized cut-set bound of transmission rate under the DF strategy. There are many cases among transmission rate intervals for different system parameters (e.g. transmit power) and it is extremely complex to derive the system error probability for the DF strategy. We obtain a relatively simple result, which unifies various cases by a few expressions. Numerical results show that the system error probability decreases with the increase of the public message. Moreover, in order to draw deep insight into the reliability requirement of each source node in this network, we provide the error exponent region (EER) for different source nodes to show the trade-off of error probability among source nodes.