Past exam of the communication Theory course of the University of Oxford / 2008 / 2 / c

The capacity of a discrete memoryless channel is $C={\max }_{P_X}I(X;Y)$, in bits per channel use. For the binary symmetric channel with crossover probability $p$, $H(Y\mid X)=h(p)$ and $H(Y)\le 1$. A uniform input makes the output uniform, attainingFor the binary erasure channel, let $E$ be its erasure indicator. This flag is visible in the output and independent of the input. If there is no erasure, the output reveals the input; otherwise it reveals nothing. Consequentlyagain attained by a uniform input. These formulas include crossover probabilities zero and one, and complete erasure.

Let $X$ be the original input, $U$ the output of the first channel, and $Y$ the final output. The independent erasure flag is observable from $Y$. The mutual information chain rule, conditioning on that flag, givesThe same uniform input maximizes $I(X;U)$ as in the binary symmetric channel. ThusThis scaling works because the erasure event is independent of both the input and the first channel noise.