CCapacity Atlas
awgn-channel-with-feedback

AWGN channel with noiseless feedback

Noiseless output feedback improves reliability without changing AWGN capacity.

Point-to-point Continuous alphabet Gaussian Additive noise Feedback Power constraint Capacity Exact

Channel and question

Input
\(X_t\) may depend on the message and past outputs \(Y^{t-1}\).
Output
\(Y_t=X_t+Z_t\), with \(Y_t\) returned noiselessly to the encoder.
Law
\(Z_t\sim\mathcal N(0,N)\) iid.
Quantity
Feedback capacity \(C_{\mathrm{AWGN,fb}}\), measured in bits per channel use.

Criterion. Average-error capacity with causal feedback. Resource averaging is fixed by the explicit model assumption below.

  • Vanishing average decoding error.
  • E_{M,Z^n}[sum_t x_t(M,Y^{t-1})^2] <= n P, with M uniform and independent of the Gaussian noise. Each message-conditional energy is integrable. This is an expected block-power bound, not a pathwise bound on feedback realizations.

Parameters

\(P\)
Average input power. Range: \(P\ge0\).
\(N\)
Noise variance. Range: \(N>0\).

Current status

\[C_{\mathrm{AWGN,fb}}=\frac12\log_2\!\left(1+\frac PN\right)\]
Known results and bounds for AWGN channel with noiseless feedback
ResultRelationMethodYear
Lower\(C_{\mathrm{AWGN,fb}}\ge\frac12\log_2(1+P/N)\)Use a no-feedback Gaussian code.1956
Upper\(C_{\mathrm{AWGN,fb}}\le\frac12\log_2(1+P/N)\)Entropy maximization and the memoryless feedback converse.1956

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

Claims

  • Real AWGN feedback capacity under expected block-average power.
    operational-capacity · exact capacity · solved · Formally stated · v1
Lean declarations (1)

References

  1. Claude E. Shannon (1956). The Zero Error Capacity of a Noisy Channel. IRE Transactions on Information Theory. DOI 10.1109/TIT.1956.1056798.
  2. Thomas M. Cover and Joy A. Thomas (2006). Elements of Information Theory. Wiley, second edition. DOI 10.1002/047174882X.

Discussion

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