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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. Resource averaging is fixed by the explicit model assumption below.

  • Receiver 1 is stronger.
  • Each receiver requests a private message.
  • For every pair (m_1,m_2), sum_t x_t(m_1,m_2)^2 <= n P. The constraint is not merely an expectation over the message pair.

Current status

\[\bigcup_{0\le\alpha\le1}\!\left\{\begin{array}{l}R_1\le\frac12\log_2(1+\alpha P/N_1),\\R_2\le\frac12\log_2\!\left(1+\frac{(1-\alpha)P}{\alpha P+N_2}\right)\end{array}\right\}\]
Known results and bounds for Degraded Gaussian broadcast channel
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

Formal verification

Lean coverageFormally stated

Concrete operational definitions and admitted research statements are present. Existing proofs are preserved. New statements require mathematical review and proof completion. Power convention: MessagePairPowerAdmissible. No equivalence to another power convention is assumed.

Claims

  • The scalar Gaussian broadcast power-splitting capacity region, stronger receiver first.
    operational-capacity · exact capacity · solved · Formally stated · v1
Lean declarations (1)

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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