CCapacity Atlas
additive-white-gaussian-noise-channel

Real additive white Gaussian noise channel

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

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

Channel and question

Input
\(X\in\mathbb R\)
Output
\(Y\in\mathbb R\)
Law
\(Y=X+Z\), where \(Z\sim\mathcal N(0,N)\) independently across uses.
Quantity
Shannon capacity \(C_{\mathrm{AWGN}}(P,N)\), measured in bits per channel use.

Criterion. Average-error capacity without feedback. Resource averaging is fixed by the explicit model assumption below.

  • Average decoding error vanishes with blocklength.
  • For every message m, sum_t x_t(m)^2 <= n P. The constraint is block-average but holds for each codeword, not only after averaging over messages.

Parameters

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

Current status

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

A zero-mean Gaussian input of variance P achieves capacity.

Known results and bounds for Real additive white Gaussian noise channel
ResultRelationMethodYear
Lower\(C\ge\frac12\log_2(1+P/N)\)Gaussian random coding.1948
Upper\(C\le\frac12\log_2(1+P/N)\)Gaussian distributions maximize differential entropy at fixed variance.1948

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

Claims

  • Operational real AWGN capacity with maximum-codeword average power.
    operational-capacity · exact capacity · solved · Formally stated · v1
Lean declarations (1)

References

  1. Claude E. Shannon (1948). A Mathematical Theory of Communication. Bell System Technical Journal. DOI 10.1002/j.1538-7305.1948.tb01338.x.
  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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