Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 11I / c / Solution

Every row of the channel matrix is a permutation of

(1 - 2 α, α, α)

, and every column sum is one. It is therefore a weakly symmetric channel. The uniform input makes the output uniform, with entropy

lo g_{2} 3

, while the conditional output entropy is the entropy of one row:

H (Y ∣ X) = - (1 - 2 α) lo g_{2} (1 - 2 α) - 2 α lo g_{2} α .

The weakly symmetric channel capacity formula now gives the ternary symmetric channel capacity

C = lo g_{2} 3 + (1 - 2 α) lo g_{2} (1 - 2 α) + 2 α lo g_{2} α

bits per channel use, for

0 \leq α \leq 1/2

, with the convention

0 lo g_{2} 0 = 0

.

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