CCapacity Atlas
awgn-channel-with-feedback

AWGN channel with noiseless feedback

Noiseless feedback dramatically improves reliability schemes but leaves the ordinary AWGN capacity unchanged.

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.

  • Average power \(P\).
  • Vanishing average decoding error.

Parameters

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

Current status

\[C_{\mathrm{AWGN,fb}}=\frac12\log_2\!\left(1+\frac PN\right)\]
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

Lean formalization

Canonical statementNone

Version 1 · Lean. Continuous feedback codes are outside the current finite Lean core.

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. 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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Related problems

Finite discrete memoryless channel with noiseless feedback

Causal noiseless output feedback changes coding strategies and reliability but not ordinary DMC capacity.

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Solved \(C_{\mathrm{fb}}(W)=C(W)=\max_{P_X}I(X;Y)\)
Finite-group additive-noise channel

A symbol in a finite group is corrupted by independent additive noise with a known distribution.

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Gaussian dirty-paper channel

Additive Gaussian interference known noncausally to the encoder causes no capacity loss.

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Real additive white Gaussian noise channel AWGN

The power-constrained real Gaussian channel has a closed-form capacity attained by a Gaussian input.

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Solved \(C_{\mathrm{AWGN}}(P,N)=\frac12\log_2\!\left(1+\frac PN\right)\)