Unit content
Capacity of a discrete memoryless channel
For a discrete memoryless channel, the mutual information $I(X;Y)$ depends on both the channel transition probabilities and the distribution used for the input symbols.
Its capacity is
$$C=\max_{p(x)}I(X;Y)$$
bits per channel use.
The maximization chooses how frequently each allowed input symbol should be used.
A completely uninformative channel has capacity zero because its output is independent of its input. A noiseless channel with $M$ distinguishable symbols has capacity
$$\log_2M$$
bits per use, achieved by using the symbols uniformly.
For a binary symmetric channel with crossover probability $p$,
$$C=1-H_2(p),$$
where
$$H_2(p)=-p\log_2p-(1-p)\log_2(1-p)$$
is binary entropy. Capacity falls from one bit per use at $p=0$ to zero at $p=1/2$.