HW6_ES250

# HW6_ES250 - Harvard SEAS ES250 Information Theory Homework...

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Harvard SEAS ES250 – Information Theory Homework 6 (Due Date: Jan. 8 2007) 1. Consider the ordinary additive noise Gaussian channel with two correlated looks at X , i.e., Y = ( Y 1 , Y 2 ), where Y 1 = X + Z 1 Y 2 = X + Z 2 with a power constraint P on X , and ( Z 1 , Z 2 ) ∼ N 2 ( 0 , K ), where K = N N Find the capacity C for (a) ρ = 1 (b) ρ = 0 (c) ρ = - 1 2. Consider a pair of parallel Gaussian channels, i.e., Y 1 Y 2 = X 1 X 2 + Z 1 Z 2 where Z 1 Z 2 ∼ N 0 , σ 2 1 0 0 σ 2 2 ‚¶ , and there is a power constraint E [ X 2 1 + X 2 2 ] P . Assume that σ 2 1 > σ 2 2 . At what power does the channel stop behaving like a single channel with noise variance σ 2 2 , and begin behaving like a pair of channels, i.e., at what power does the worst channel become useful? 3. Consider the following channel: Throughout this problem we shall constrain the signal power E [ X ] = 0 , E [ X 2 ] = P, and the noise power E [ Z ] = 0 , E [ Z 2 ] = N, 1

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Harvard SEAS ES250 – Information Theory and assume that X and Z are independent. The channel capacity is given by I ( X ; X + Z ). Now for the game. The noise player chooses a distribution on Z to minimize I ( X ; X + Z ), while the signal player chooses a distribution on X to maximize I ( X ; X + Z ). Letting X * ∼ N (0 , P ), Z * ∼ N (0 , N ), show that Gaussian X * and Z *
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