CCapacity Atlas
two-user-gaussian-mac-with-feedback

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

Channel and question

Input
At time \(t\), each encoder uses its message and common past outputs \(Y^{t-1}\).
Output
\(Y=X_1+X_2+Z\), returned noiselessly to both encoders.
Law
\(Z\sim\mathcal N(0,N)\) iid, with powers \(P_1,P_2\).
Quantity
Feedback capacity region \(\mathcal C_{\mathrm{GMAC,fb}}\), measured in rate pairs in bits per channel use.

Criterion. Vanishing average joint decoding error.

  • Feedback is causal and noiseless.
  • Messages are independent.

Current status

\[\bigcup_{0\le\rho\le1}\!\left\{\begin{array}{l}R_1\le\frac12\log_2(1+P_1(1-\rho^2)/N),\\R_2\le\frac12\log_2(1+P_2(1-\rho^2)/N),\\R_1+R_2\le\frac12\log_2(1+(P_1+P_2+2\rho\sqrt{P_1P_2})/N)\end{array}\right\}\]
ResultRelationMethodYear
Inner Region\(\mathcal R_{\mathrm{Ozarow}}\subseteq\mathcal C_{\mathrm{fb}}\)Linear feedback creates controlled correlation between users.1984
Outer Region\(\mathcal C_{\mathrm{fb}}\subseteq\mathcal R_{\mathrm{Ozarow}}\)A dependence constraint and Gaussian extremality.1984

Lean formalization

Canonical statementNone

Version 1 · Lean. The exact feedback region is a substantial external formalization target.

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. Lawrence H. Ozarow (1984). The Capacity of the White Gaussian Multiple Access Channel with Feedback. IEEE Transactions on Information Theory. DOI 10.1109/TIT.1984.1056935.
  2. Abbas El Gamal and Young-Han Kim (2011). Network Information Theory. Cambridge University Press. DOI 10.1017/CBO9781139030687.

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