HW4_ES250 - Harvard SEAS ES250 Information Theory Homework...

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Harvard SEAS ES250 – Information Theory Homework 4 (Due Date: Nov. 13, 2007) 1. Consider a binary symmetric channel with Y i = X i Z i , where is mod 2 addition, and X i , Y i { 0 , 1 } . Suppose that { Z i } has constant marginal probabilities Pr( Z i = 1) = p and Pr( Z i = 0) = 1 - p , but that Z 1 , Z 2 , · · · , Z n are not necessarily independent. Let C = 1 - H ( p ). Show that max p ( x 1 ,x 2 , ··· ,x n ) I ( X 1 , X 2 , · · · , X n ; Y 1 , Y 2 , · · · , Y n ) nC Comment on the implications. 2. Consider the channel Y = X + Z (mod 13), where Z = 1 , with probability 1 3 2 , with probability 1 3 3 , with probability 1 3 and X ∈ { 0 , 1 , · · · , 12 } . (a) Find the capacity. (b) What is the maximizing p * ( x )? 3. Using two channels. (a) Consider two discrete memoryless channels ( X 1 , p ( y 1 | x 1 ) , Y 1 ) and ( X 2 , p ( y 2 | x 2 ) , Y 2 ) with capac- ities C 1 and C 2 respectively. A new channel ( X 1 × X 2 , p ( y 1 | x 1 ) × p ( y 2 | x 2 ) , Y 1 × Y 2 ) is formed in which x 1 ∈ X 1 and x 2 ∈ X 2 , are simultaneously sent, resulting in y 1 , y 2 . Find the capacity of this channel.
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