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degraded-gaussian-broadcast-channel

Degraded Gaussian broadcast channel

A power-constrained Gaussian transmitter serves a strong and a weak receiver by superposition coding.

Broadcast Continuous alphabet Gaussian Degraded Power constraint Capacity region Exact

Channel and question

Input
\(X\in\mathbb R\) with power \(P\).
Output
\(Y_1=X+Z_1\), \(Y_2=X+Z_2\).
Law
Independent Gaussian noises have variances \(N_1\le N_2\).
Quantity
Capacity region \(\mathcal C\), measured in rate pairs in bits per channel use.

Criterion. Vanishing average error at both receivers.

  • Receiver 1 is stronger.
  • Each receiver requests a private message.

Current status

\[\bigcup_{0\le\alpha\le1}\!\left\{\begin{array}{l}R_1\le\frac12\log_2(1+\alpha P/N_1),\\R_2\le\frac12\log_2\!\left(1+\frac{(1-\alpha)P}{\alpha P+N_2}\right)\end{array}\right\}\]
ResultRelationMethodYear
Inner Region\(\mathcal R_{\mathrm{sup}}\subseteq\mathcal C\)Gaussian superposition coding.1973
Outer Region\(\mathcal C\subseteq\mathcal R_{\mathrm{sup}}\)Entropy-power converse for degraded Gaussian outputs.1974

Lean formalization

Canonical statementNone

Version 1 · Lean. Entropy power and Gaussian multi-user coding are not yet formalized.

Substantial proofs0 linked

No external Lean proof is registered. Proofs longer than roughly 50 lines or requiring problem-specific infrastructure should live in a dedicated repository and link back to this statement version.

References

  1. Peter P. Bergmans (1973). Random Coding Theorem for Broadcast Channels with Degraded Components. IEEE Transactions on Information Theory. DOI 10.1109/TIT.1973.1054980.
  2. Peter P. Bergmans (1974). A Simple Converse for Broadcast Channels with Additive White Gaussian Noise. IEEE Transactions on Information Theory. DOI 10.1109/TIT.1974.1055170.
  3. Abbas El Gamal and Young-Han Kim (2011). Network Information Theory. Cambridge University Press. DOI 10.1017/CBO9781139030687.

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