CCapacity Atlas
gaussian-fading-receiver-state

Gaussian fading channel with receiver-only state information

Iid real fading gains known only to the receiver determine ergodic capacity under a fixed power budget.

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

Channel and question

Input
Real symbols with a maximum-codeword block-average power limit P.
Output
The receiver observes each fading gain H and the corresponding real output Y.
Law
Y=HX+Z, with iid gains of a fixed Borel probability law and independent iid N(0,N) noise.
Quantity
Gaussian fading channel with receiver-only state information 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.

  • P >= 0 and N > 0.
  • The encoder has no gain information. The receiver sees the complete gain and output words.
  • The expected value of log2(1+H^2 P/N) is finite.
  • This is ergodic capacity, not outage capacity. Decoder decision regions are measurable.
  • For every message m, sum_t x_t(m)^2 <= n P. Inputs do not depend on the fading gains. There is no gain-dependent power adaptation or averaging of the power constraint over gains.

Current status

\[C=\mathbb E[\tfrac12\log_2(1+H^2P/N)]\]
Known results and bounds for Gaussian fading channel with receiver-only state information
ResultRelationMethodYear
Exact\(C=\mathbb E[\tfrac12\log_2(1+H^2P/N)]\)Receiver-state channel coding and a Gaussian conditional-entropy converse.1997

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

Claims

  • Iid ergodic real fading known only to the receiver, with finite expected information.
    operational-capacity · exact capacity · solved · Formally stated · v1
Lean declarations (1)

References

  1. Andrea J. Goldsmith and Pravin P. Varaiya (1997). Capacity of Fading Channels with Channel Side Information. IEEE Transactions on Information Theory. DOI 10.1109/18.641562.

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