On Secrecy Rate and Optimal Power Allocation of the Full-Duplex Amplify-and-Forward Relay Wire-Tap Channel

We present the secrecy rate of a relay wire-tap channel in which a source node communicates securely to a destination node in the presence of an eavesdropper using an amplify-and-forward (AF) relay operating in full-duplex (FD) mode. We explicitly account for the residual self-interference due to FD transmission and compute the optimal power allocation (PA) that maximizes the secrecy rate under both individual and joint power constraints of the source and the relay nodes. For slowly varying fading channels, we show that the optimal PA problem is quasiconcave and, hence, determine the globally optimal solution. Applying the method of dominant balance to analyze the capacity and PA schemes in different high-power regimes, we demonstrate that full PA at the relay is only necessary when the power at the relay is sufficiently small compared to the power at the source. Our results show that FD relaying achieves a significantly higher secrecy rate than half-duplex (HD) relaying. We extend the results to ergodic fading channels where the channel gains are assumed to be available at the receivers but not the transmitters. To this end, we first calculate the expectation of linear functions of exponentially distributed random variables using the exponential integral function. This method allows us to obtain a closed-form expression for the ergodic secrecy rate. The bisection method can then be applied to find the optimal PA scheme. Numerical results also reveal the superiority of FD over HD relaying in channels with ergodic fading.