CCapacity Atlas
two-user-gaussian-mac

Two-user Gaussian multiple-access channel

Two power-constrained Gaussian users share one receiver, giving an exact pentagonal capacity region.

Multiple access Continuous alphabet Gaussian Additive noise Power constraint Capacity region Exact

Channel and question

Input
Independent inputs \(X_1,X_2\) with powers \(P_1,P_2\).
Output
\(Y=X_1+X_2+Z\)
Law
\(Z\sim\mathcal N(0,N)\) iid.
Quantity
Capacity region \(\mathcal C_{\mathrm{GMAC}}\), measured in rate pairs in bits per channel use.

Criterion. Vanishing average joint decoding error.

  • Messages are independent.
  • There is no feedback.

Parameters

\(P_1,P_2\)
User power constraints. Range: \(P_1,P_2\ge0\).
\(N\)
Noise variance. Range: \(N>0\).

Current status

\[\left\{\begin{array}{l}R_1\le\frac12\log_2(1+P_1/N),\\R_2\le\frac12\log_2(1+P_2/N),\\R_1+R_2\le\frac12\log_2(1+(P_1+P_2)/N)\end{array}\right\}\]
ResultRelationMethodYear
Inner Region\(R_i\le\frac12\log_2(1+P_i/N),\ R_1+R_2\le\frac12\log_2(1+(P_1+P_2)/N)\)Independent Gaussian codebooks and joint decoding.1971
Outer Region\(R_i\le\frac12\log_2(1+P_i/N),\ R_1+R_2\le\frac12\log_2(1+(P_1+P_2)/N)\)Gaussian entropy maximization in the MAC converse.1972

Lean formalization

Canonical statementNone

Version 1 · Lean. Gaussian random variables and multi-user regions 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. Rudolf Ahlswede (1971). Multi-way communication channels. Second International Symposium on Information Theory.
  2. Hsiao-Hwa Liao (1972). Multiple Access Channels. PhD thesis, University of Hawaii.
  3. Abbas El Gamal and Young-Han Kim (2011). Network Information Theory. Cambridge University Press. DOI 10.1017/CBO9781139030687.

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