Channel and question
- Input
- \(X\in\mathbb R\), with average power \(P\).
- Output
- \(Y=X+S+Z\).
- Law
- Gaussian state \(S\) is known noncausally to the encoder; \(Z\sim\mathcal N(0,N)\) is independent.
- Quantity
- Dirty-paper capacity \(C_{\mathrm{DPC}}\), measured in bits per channel use.
Criterion. Average-error capacity.
- The decoder does not know S.
- State and noise are iid Gaussian.
Current status
\[C_{\mathrm{DPC}}=\frac12\log_2\!\left(1+\frac PN\right)\]
The value is independent of the state variance.
| Result | Relation | Method | Year |
|---|---|---|---|
| Lower | \(C_{\mathrm{DPC}}\ge\frac12\log_2(1+P/N)\) | Gaussian Gel'fand-Pinsker coding pre-cancels the known interference. | 1983 |
| Upper | \(C_{\mathrm{DPC}}\le\frac12\log_2(1+P/N)\) | Reveal the interference to the decoder and apply the AWGN converse. | 1983 |
Lean formalization
Canonical statementNone
Version 1 · Lean. A faithful proof needs Gaussian conditional distributions and the Gel'fand-Pinsker theorem.
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
- Max H. M. Costa (1983). Writing on Dirty Paper. IEEE Transactions on Information Theory. DOI 10.1109/TIT.1983.1056659.
Discussion
Thread key: capacityatlas:costa-dirty-paper-channel