Capacity of the two-receiver Gaussian broadcast channel with feedback
Fix $P, N_1, N_2 > 0$. A real memoryless broadcast channel has outputs $$Y_{k,t} = X_t + Z_{k,t}, \qquad k \in \{1, 2\},$$ at time $t$, where all $Z_{k,t}$ are mutually independent, $Z_{k,t} \sim \mathcal{N}(0, N_k)$, and the noises are independent of the messages. The transmitter knows two independent uniform messages $W_k \in \{1, \dots, M_k\}$, with $W_k$ intended only for receiver $k$. It receives noiseless feedback of both past output sequences and may use arbitrary measurable encoders $$X_t = f_t(W_1, W_2, Y_1^{t-1}, Y_2^{t-1}), \qquad \frac{1}{n} \sum_{t=1}^{n} \mathbb{E}[X_t^2] \le P.$$ Receiver $k$ estimates its message from $Y_k^n$ alone. Let $\mathcal{C}_{\mathrm{fb}}(P, N_1, N_2)$ be the closure of the set of nonnegative rate pairs $(R_1, R_2)$ achieved by a sequence of such codes with $\liminf_{n \to \infty} n^{-1} \log_2 M_k \ge R_k$ for $k = 1, 2$ and $$\Pr\{\widehat{W}_1 \ne W_1 \ \text{or} \ \widehat{W}_2 \ne W_2\} \to 0.$$
Without feedback the capacity region depends only on the marginal channels. El Gamal proved that feedback does not enlarge the capacity region of the physically degraded Gaussian broadcast channel, and Ozarow and Leung-Yan-Cheong showed that with independent noises, as here, it does. Gastpar, Lapidoth, Steinberg and Wigger proved that the sum-rate capacity $C_\Sigma$ with feedback satisfies $$\lim_{P \to \infty} \Bigl[ C_\Sigma - \tfrac{1}{2} \log_2\Bigl(1 + \frac{P (N_1 + N_2)}{N_1 N_2}\Bigr) \Bigr] = 0,$$ the subtracted term being the full-cooperation bound.
Problem. Determine $\mathcal{C}_{\mathrm{fb}}(P, N_1, N_2)$ for every fixed $P, N_1, N_2 > 0$, with matching achievability and converse bounds. The encoders are not restricted to linear feedback schemes.
Source: Ozarow and Leung-Yan-Cheong (1984), who gave an achievable region and an outer bound; Gastpar, Lapidoth, Steinberg and Wigger (2014), §I, note that the exact capacity region with feedback is unknown. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- IT — Information Theory
- Secondary
- PR
- Keywords
- AIM Open Applied Problems, broadcast channels, channel capacity, Gaussian channels
References
@article{ozarow1984,
author = {Ozarow, L. H. and Leung-Yan-Cheong, S. K.},
title = {An achievable region and outer bound for the {G}aussian broadcast channel with feedback},
journal = {IEEE Trans. Inform. Theory},
volume = {30},
number = {4},
year = {1984},
pages = {667--671},
doi = {10.1109/TIT.1984.1056923},
}
@article{gastpar2014,
author = {Gastpar, Michael and Lapidoth, Amos and Steinberg, Yossef and Wigger, Mich{\`e}le},
title = {Coding schemes and asymptotic capacity for the {G}aussian broadcast and interference channels with feedback},
journal = {IEEE Trans. Inform. Theory},
volume = {60},
number = {1},
year = {2014},
pages = {54--71},
doi = {10.1109/TIT.2013.2287531},
eprint = {1003.6082},
archivePrefix = {arXiv},
}
@article{elgamal1981,
author = {El Gamal, Abbas},
title = {The capacity of the physically degraded {G}aussian broadcast channel with feedback},
journal = {IEEE Trans. Inform. Theory},
volume = {27},
number = {4},
year = {1981},
pages = {508--511},
doi = {10.1109/TIT.1981.1056372},
}
@misc{aim-gaussian-broadcast-feedback,
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 two-receiver Gaussian broadcast channel with feedback}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/299-gaussian-broadcast-feedback.md},
note = {Entry 299 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-00522, "Capacity of the two-receiver Gaussian broadcast channel with feedback", clopen.solutions.