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

  • Messages are independent.
  • There is no feedback.
  • For user j and every local message m_j, sum_t x_{j,t}(m_j)^2 <= n P_j. There is no averaging over messages and no feedback.

Parameters

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

Current status

\[\left\{\begin{array}{l}R_1\le\frac12\log_2(1+P_1/N),\\R_2\le\frac12\log_2(1+P_2/N),\\R_1+R_2\le\frac12\log_2(1+(P_1+P_2)/N)\end{array}\right\}\]
Known results and bounds for Two-user Gaussian multiple-access channel
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

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: SeparateCodewordPowerAdmissible. No equivalence to another power convention is assumed.

Claims

  • The Gaussian MAC capacity region under separate maximum-codeword power constraints.
    operational-capacity · exact capacity · solved · Formally stated · v1
Lean declarations (1)

References

  1. Rudolf Ahlswede (1973). Multi-way communication channels. Proceedings of the Second International Symposium on Information Theory (September 1971), Akadémiai Kiadó, pp. 23–52.
  2. Henry Herng-Jiunn 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.

Discussion

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Related problems

Two-user Gaussian multiple-access channel with feedback

Ozarow's feedback scheme and converse determine the full two-user Gaussian MAC feedback region.

Multiple access Continuous alphabet Gaussian Additive noise Feedback Power constraint Capacity region Exact
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Two independent senders communicate to one receiver through a finite memoryless channel.

Multiple access Finite alphabet Discrete memoryless Capacity region Exact Single-letter characterization
Solved \(\bigcup_{p(q)p(x_1|q)p(x_2|q)}\!\left\{\begin{array}{l}R_1,R_2\ge0,\\R_1\le I(X_1;Y|X_2,Q),\\R_2\le I(X_2;Y|X_1,Q),\\R_1+R_2\le I(X_1,X_2;Y|Q)\end{array}\right\}\)