This paper investigates the secrecy rate of a relay network consisting of four nodes: a source node, a destination node, an eavesdropper, and a trusted relay operating in full-duplex (FD) mode. Attention is paid to ergodic Rayleigh fading channels where the channel gains change randomly over time and they are available at the receivers but not the transmitters. To deal with fading, we first use a simple technique to calculate the expectation of an exponentially distributed random variable. Using this result, the ergodic secrecy rates of the considered FD relay channel are then established in closed-form for both cases of amplify-and-forward (AF) and decode-and-forward (DF) relaying. Numerical results show that the proposed method provides a simple yet effective way to compute the ergodic secrecy rate without the need of lengthy Monte Carlo simulations. The established closed-forms can also be exploited to determine the optimal power allocation schemes among the source and relay nodes for further secrecy enhancements.
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