 Solution

ID: past-exam-of-the-communication-theory-course-of-the-university-of-oxford/2008/2/c/ii/solution

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, gives

I (X; Y) = (1 - ϵ) I (X; U) + ϵ \cdot 0.

The same uniform input maximizes

I (X; U)

C = (1 - ϵ) (1 - h (p)) .

This scaling works because the erasure event is independent of both the input and the first channel noise.

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