Abstract
This study focuses on the probabilistically constrained capacity of the Gaussian relay channel. Since the capacity itself remains unknown in general, we study the achievable epsilon-outage rates that are based on direct source-to-destination transmission, the decode-and-forward (DF) strategy, and the compress-and-forward (CF) strategy as well as two upper bounds that are based on the mutual information expressions of 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. We derive closed-form expressions for the outage rates of direct transmission and the DF lower bound. A closed-form expression for the probability of a successful transmission with CF is also provided. Since closed-form CSB probability terms for coherent source and relay transmission are difficult to obtain, we use a genie-aided and a loosened CSB formulation to obtain the upper bounds. Alternatively, we approximate the probability that the CSB exceeds a target rate assuming noncoherent transmission from the source and the relay to the destination. The numerical results verify that the CF lower bound meets the capacity if the relay is at the destination and that the DF scheme achieves the CSB for noncoherent transmission if the relay is close to the source.
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Notes
- 1.
In the considered relay channel, an outage occurs when the channel conditions are so poor that no error-free communication from the source to the destination is possible at the desired rate.
- 2.
Here, (6.37) defines the joint source and relay transmit covariance matrix of an equivalent system with i.i.d. unit variance source- and relay-to-destination channels and a source-to-relay channel with variance \({\sigma _{{\text {S}}{\text {R}}}^2}/{\sigma _{{\text {S}}{\text {D}}}^2}\).
- 3.
The state of \({h_{{\text {S}}{\text {R}}}}\) may be forwarded from the relay to the destination.
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Acknowledgments
The work of Lennart Gerdes and Andreas Gründinger was supported by the German Research Foundation (DFG) under Grant Jo 724/2-1.
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Gründinger, A., Gerdes, L., Joham, M., Utschick, W. (2016). Bounds on the Outage Constrained Capacity of the Gaussian Relay Channel. In: Utschick, W. (eds) Communications in Interference Limited Networks. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-22440-4_6
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DOI: https://doi.org/10.1007/978-3-319-22440-4_6
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