Design of Generalized Analog Network Coding for a Multiple-Access Relay Channel

In this paper, we propose a generalized analog network coding (GANC) scheme for a non-orthogonal multiple-access relay channel (MARC), where two sources transmit their information simultaneously to the destination with the help of a relay. In the GANC scheme, the relay receives interfered signals from the two sources and generates signals to be transmitted with a relay function. We focus on the design of the optimal relay function to achieve the minimum pair-wise error probability (PEP) of the system. Specifically, we first covert the relay function optimization problem to a transformation matrix (TM) design problem by presenting the received complex signals as signal matrices composed of real and imaginary parts. Then, we propose an optimization criteria, i.e., maximizing the minimal squared Euclidean distance (MMSED), to improve the PEP performance, since the PEP is determined by the Euclidean distance of the received constellation at destination. Next, we prove that the MMSED can be equivalently converted to a convex problem by introducing an intermediate matrix. We solve this convex problem by using the Lagrangian method and obtain the closed-form expression of the optimal TM. We further improve the PEP performance by optimizing transmission power of the two sources. Simulation results show that the proposed GANC scheme has a better PEP performance compared to other alternative schemes.