Bounds on the outage constrained capacity of the single-antenna Gaussian relay channel

We study the probabilistically constrained capacity of a Gaussian relay channel in terms of the corresponding outage constrained decode-and-forward (DF) achievable rate and the cut-set bound (CSB). The probabilistic outage constraint is necessary due to Rayleigh fading and the absence of channel state information (CSI) at the transmitting nodes. The DF outage probability can be calculated in closed form, which allows a direct optimization of the joint source and relay transmit strategy. In contrast, the CSB probability can only be calculated in closed form for noncoherent transmission from the source and relay to the destination. For coherent transmission, we propose two upper bounds based on a genie-aided transmission and Markov's inequality. The numerical results verify the quality of the bounds.