Ucsd Ece 255c Handout #12

Y1 if Y ′ 2 = Y2, e if Y ′ 2 = e. Then, Ỹ ′ 1 |{X = x} ∼ pY ′ 1 |X(ỹ ′ 1|x), i.e., Ỹ ′ 1 has the same conditional pmf as Y ′ 1 given X and X → Ỹ ′ 1 → Y ′ 2 form a Markov chain. Therefore, by definition, the erasure broadcast channel p(y 1, y ′ 2|x) is degraded (and hence less noisy as well as more capable). (b) Since it is degraded, by Theorem 5.2 in the text, the capacity region of the erasure broadcast channel p(y 1, y ′ 2|x) is the set of rate pairs (R1, R2) such that R1 ≤ I(X ;Y ′ 1 |U), R2 ≤ I(U ;Y ′ 2) for some pmf p(u, x). To express this region in terms of C , define binary random variables Ej for j = 1, 2 as