One binary symbol is transmitted perfectly while the other can flip in only one direction.
\(X\in\{0,1\}\)
\(Y\in\{0,1\}\)
\(P(Y=0\mid X=0)=1\) and \(P(Y=0\mid X=1)=p\).
| Symbol | Meaning | Range |
|---|---|---|
| \(p\) | Probability that input 1 is received as 0. | \(0\le p\le1\) |
Conditions. \(0<p<1\), with the endpoint values \(C_Z(0)=1\) and \(C_Z(1)=0\) obtained by continuity.
Unlike a symmetric channel, the capacity-achieving input is generally not uniform.
| Type | Claim | Method | Source |
|---|---|---|---|
| lower | \(C_Z(p)\ge\log_2\!\left(1+(1-p)p^{p/(1-p)}\right)\) | Evaluate mutual information at the optimizing Bernoulli input distribution. | 1948 [1] [2] |
| upper | \(C_Z(p)\le\log_2\!\left(1+(1-p)p^{p/(1-p)}\right)\) | The discrete-memoryless channel converse followed by a one-variable maximization. | 1948 [1] [2] |
The asymmetric transition kernel is formalized. The closed-form mutual-information maximization is not.
Each input bit is independently deleted without an erasure marker. The exact capacity is unknown for every nontrivial deletion probability.
Each bit is independently flipped with probability p. This is the canonical finite noisy channel.
A relay assists communication from a source to a destination. Decode-forward and the cut-set bound do not coincide in general.