Symbol-error rate optimized complex field network coding for wireless communications

In this paper, the complex field network coding (CFNC) over multi-user relay channel with non-orthogonal communications is studied. In order to improve the performance of CFNC coded relay channel, we propose to optimize the CFNC according to the symbol-error rate (SER) characteristics of the network. To achieve such an optimization, we first derive a bound for the SER of the coded relay channel, which uses maximum-likelihood detection both at the relay and destination node. Then, CFNC optimization is formulated as a convex program, which aims to minimize the SER-bound while considering the constraint on total transmit power and the network geometry. We next derive Karush-Kuhn-Tucker (KKT) conditions of optimal CFNC parameters. Due to the existence of nonlinearity in the simplified KKT conditions, a closed form solution for the optimal parameters is not possible. To overcome this difficulty, we also propose an approximate solution for the optimal CFNC, which utilizes an information theoretical result. After the SER-optimized CFNC parameters are determined, we investigate the average bit error rate (BER) of the network for various parameters. The performed numerical experiments reveal that SER-optimized CFNC could provide a BER improvement up-to 29 % over the conventional CFNC for a two-user relay channel.