Channel and question
- Input
- Source input \(X\) and relay input \(X_r\).
- Output
- Relay observation \(Y_r\) and destination output \(Y\).
- Law
- The channel factors as \(p(y_r|x,x_r)p(y|y_r,x_r)\).
- Quantity
- Relay-channel capacity \(C\), measured in bits per channel use.
Criterion. Average-error source-to-destination capacity.
- The relay acts causally.
- Average decoding error vanishes.
Current status
\[C=\max_{p(x,x_r)}\min\{I(X;Y_r|X_r),\ I(X,X_r;Y)\}\]
| Result | Relation | Method | Year |
|---|---|---|---|
| Lower | \(C\ge\max\min\{I(X;Y_r|X_r),I(X,X_r;Y)\}\) | The relay fully decodes and cooperates with the source. | 1979 |
| Upper | \(C\le\max\min\{I(X;Y_r|X_r),I(X,X_r;Y)\}\) | Degradedness reduces the cut-set observation term. | 1979 |
Lean formalization
Canonical statementNone
Version 1 · Lean. This theorem should live in a dedicated external proof repository once causal relay codes are shared.
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
- Thomas M. Cover and Abbas El Gamal (1979). Capacity Theorems for the Relay Channel. IEEE Transactions on Information Theory. DOI 10.1109/TIT.1979.1056084.
- Abbas El Gamal and Young-Han Kim (2011). Network Information Theory. Cambridge University Press. DOI 10.1017/CBO9781139030687.
Discussion
Thread key: capacityatlas:physically-degraded-relay-channel