Question

(b) For the BSC shown in Fig. Q2, establish the following relations: • (i) $H(Y/X) = \Omega(\alpha)$ • (ii) $I(X; Y) = \Omega(\alpha + p - 2\alpha p) - \Omega(\alpha)$ in which $\Omega(\ast)$ stands for the binary entropy function. Hence, deduce the mutual information $I(X; Y)$ when the channel noise is (i) small (ii) very large. (c) Calculate $I(X; Y)$ for $p = 0.5$ and $\alpha = 0.1$ and for $p = 0.5$ and $\alpha = 0.5$, and comment on the results.

          (b) For the BSC shown in Fig. Q2, establish the following relations:
• (i) $H(Y/X) = \Omega(\alpha)$
• (ii) $I(X; Y) = \Omega(\alpha + p - 2\alpha p) - \Omega(\alpha)$
in which $\Omega(\ast)$ stands for the binary entropy function. Hence, deduce the mutual
information $I(X; Y)$ when the channel noise is
(i) small
(ii) very large.
(c) Calculate $I(X; Y)$ for $p = 0.5$ and $\alpha = 0.1$ and for $p = 0.5$ and $\alpha = 0.5$, and comment on the
results.

(b) For the BSC shown in Fig. Q2, establish the following relations:
• (i) H(Y/X) = Ω(α)
• (ii) I(X; Y) = Ω(α + p - 2α p) - Ω(α)
in which Ω(∗) stands for the binary entropy function. Hence, deduce the mutual
information I(X; Y) when the channel noise is
(i) small
(ii) very large.
(c) Calculate I(X; Y) for p = 0.5 and α = 0.1 and for p = 0.5 and α = 0.5, and comment on the
results.

Added by Jacqueline B.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Ruirui Liu

09/08/2019

Step 1

Step 1: Calculate the conditional entropy H(Y|X) using the formula H(Y|X) = -ΣΣ p(y,x) log(p(y|x)). Show more…

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Please help me solve this, particularly the proving for small and very large channel noise. Please solve it accurately with correct appropriate steps, I need it. (b) For the BSC shown in Fig. Q2, establish the following relations: (i) H(Y/X) = Ω(α) (ii) I(X;Y) = Ω(α + p - 2αp) - Ω(α) in which Ω(**) stands for the binary entropy function. Hence, deduce the mutual information I(X;Y) when the channel noise is 0 (i) small (c) Calculate I(X;Y) for p = 0.5 and α = 0.1 and for p = 0.5 and α = 0.5, and comment on the results. 0 (ii) very large. (b) For the BSC shown in Fig. Q2, establish the following relations: (i) HY/X= (ii) I(X;Y=+p-2ap) -( in which (* stands for the binary entropy function. Hence deduce the mutual information I(X;Y) when the channel noise is (i) small 0 (ii) very large. results. p(x) = p 1 - α p(x) = 1 - p 1 - α Fig. Q2