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

  • Feedback is causal and noiseless.
  • Messages are independent.
  • For each user j, E_{M_1,M_2,Z^n}[sum_t x_{j,t}(M_j,Y^{t-1})^2] <= n P_j. The messages are independent and uniform. Every message-pair-conditional energy is integrable. The power bounds are separate and not pathwise.

Current status

\[\bigcup_{0\le\rho\le1}\!\left\{\begin{array}{l}R_1\le\frac12\log_2(1+P_1(1-\rho^2)/N),\\R_2\le\frac12\log_2(1+P_2(1-\rho^2)/N),\\R_1+R_2\le\frac12\log_2(1+(P_1+P_2+2\rho\sqrt{P_1P_2})/N)\end{array}\right\}\]
Known results and bounds for Two-user Gaussian multiple-access channel with feedback
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

Formal verification

Lean coverageFormally stated

New statements await mathematical review and proofs. Power convention: SeparateMessageNoiseAveragePowerAdmissible. No equivalence to another power convention is assumed.

Claims

  • Ozarow's capacity region under separate expected-power constraints and noiseless output feedback.
    operational-capacity · exact capacity · solved · Formally stated · v1
Lean declarations (1)

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.

Discussion

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