CCapacity Atlas
amplitude-constrained-gaussian-channel

Amplitude-constrained scalar Gaussian channel

A scalar Gaussian channel with a hard amplitude constraint has an optimizing input with finite support.

Point-to-point Continuous alphabet Gaussian Power constraint Capacity Single-letter characterization

Channel and question

Input
A real input satisfying |X| <= A at every coordinate of every codeword.
Output
A real output corrupted by independent Gaussian noise.
Law
Y=X+Z with iid Z distributed as N(0,N).
Quantity
Amplitude-constrained scalar Gaussian channel capacity \(C\), measured in bits per channel use.

Criterion. Vanishing average block error, with the constraints specified in the model. Resource averaging is fixed by the explicit model assumption below.

  • A >= 0 and N > 0.
  • The constraint is peak amplitude, not a replacement average-power constraint.
  • All decoder decision regions are measurable. Average decoding error vanishes.
  • For every message m and time t, |x_t(m)| <= A. This is a pathwise peak-amplitude bound, not an average-power constraint.

Current status

\[C=\max_{\operatorname{supp}(P_X)\subseteq[-A,A]} I(X;X+Z)\]

Conditions. An optimal input has finite support. Also C <= (1/2) log2(1+A^2/N).

Known results and bounds for Amplitude-constrained scalar Gaussian channel
ResultRelationMethodYear
Exact\(C=\max_{\operatorname{supp}(P_X)\subseteq[-A,A]} I(X;X+Z)\)Smith optimality conditions, finite-support attainment and a channel coding theorem.1971
Upper\(C\le\tfrac12\log_2(1+A^2/N)\)A peak amplitude bound implies the block power bound P=A^2.1971

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

Claims

  • Amplitude-constrained AWGN capacity is attained by a finite-support input law.
    operational-capacity · exact capacity · solved · Formally stated · v1
  • The peak-amplitude operational capacity equals the compact-input information supremum.
    capacity-formula · exact capacity · solved · Formally stated · v1
  • A finite-support input attains the operational peak-amplitude capacity.
    finite-support-attainment · structural · solved · Formally stated · v1
  • Peak-amplitude capacity is bounded above by the relaxed average-power formula.
    power-relaxation-converse · converse · solved · Formally stated · v1
Lean declarations (4)

References

  1. Joel G. Smith (1971). The Information Capacity of Amplitude- and Variance-Constrained Scalar Gaussian Channels. Information and Control. DOI 10.1016/S0019-9958(71)90346-9.

Discussion

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