Capacity of the Gaussian relay channel with independent receiver noises
Consider the memoryless real channel $$Y_{r,t} = a X_{s,t} + Z_{r,t}, \qquad Y_{d,t} = b X_{s,t} + c X_{r,t} + Z_{d,t},$$ where $a, b, c > 0$ and all noise variables are independent $N(0, 1)$. A source encodes a uniformly distributed message $M$ into $X_{s,1}, \dots, X_{s,N}$, without feedback; a relay transmits $X_{r,t} = f_t(Y_r^{t-1})$ by strictly causal rules; and the destination estimates $M$ from $Y_d^N = (Y_{d,1}, \dots, Y_{d,N})$. The inputs satisfy the average block-power constraints $N^{-1} \sum_t \mathbb{E} X_{s,t}^2 \le P_s$ and $N^{-1} \sum_t \mathbb{E} X_{r,t}^2 \le P_r$, with $P_s, P_r > 0$. The capacity $C(a, b, c, P_s, P_r)$ is the supremum of the rates $(\log_2 |\mathcal{M}|)/N$ achievable with error probability tending to zero as $N \to \infty$, where $\mathcal{M}$ is the message set. Cover and El Gamal established the cutset upper bound and the lower bounds later called decode–forward, partial decode–forward and compress–forward, and the cutset bound is attained for degraded relay channels (1979); with independent noises as here, the channel is neither physically degraded nor reversely degraded. El Gamal, Gohari and Nair proved an upper bound that is strictly smaller than the cutset bound for all positive gains (2022).
Problem. Determine $C(a, b, c, P_s, P_r)$ for all $a, b, c, P_s, P_r > 0$.
Source: Van der Meulen (1971) introduced the relay channel; this Gaussian model and its capacity question are as in El Gamal and Kim (2011), Chapter 16. Stated by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- IT — Information Theory
- Secondary
- PR
- Keywords
- AIM Open Applied Problems, channel capacity, Gaussian channels, relay channels
References
@article{vandermeulen1971,
author = {van der Meulen, Edward C.},
title = {Three-terminal communication channels},
journal = {Adv. in Appl. Probab.},
volume = {3},
number = {1},
year = {1971},
pages = {120--154},
doi = {10.2307/1426331},
}
@incollection{elgamal2011relay,
author = {El Gamal, Abbas and Kim, Young-Han},
title = {Relay channels},
booktitle = {Network Information Theory},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {2011},
pages = {382--426},
doi = {10.1017/CBO9781139030687.018},
}
@article{cover1979,
author = {Cover, Thomas M. and El Gamal, Abbas},
title = {Capacity theorems for the relay channel},
journal = {IEEE Trans. Inform. Theory},
volume = {25},
number = {5},
year = {1979},
pages = {572--584},
doi = {10.1109/TIT.1979.1056084},
}
@article{elgamal2022,
author = {El Gamal, Abbas and Gohari, Amin and Nair, Chandra},
title = {A strengthened cutset upper bound on the capacity of the relay channel and applications},
journal = {IEEE Trans. Inform. Theory},
volume = {68},
number = {8},
year = {2022},
pages = {5013--5043},
doi = {10.1109/TIT.2022.3163647},
eprint = {2101.11139},
archivePrefix = {arXiv},
}
@misc{aim-gaussian-relay-capacity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Capacity of the Gaussian relay channel with independent receiver noises}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/136-gaussian-relay-capacity.md},
note = {Entry 136 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00523, "Capacity of the Gaussian relay channel with independent receiver noises", clopen.solutions.