hw4sol - EE 376B/Stat 376B Information Theory Prof T Cover...

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EE 376B/Stat 376B Handout #13 Information Theory Tuesday, May 9, 2006 Prof. T. Cover Solutions to Homework Set #4 1. Multiple layer waterflling Let C ( x ) = 1 2 log(1 + x ) denote the channel capacity of a Gaussian channel with signal to noise ratio x . Show C µ P 1 N + C µ P 2 P 1 + N = C µ P 1 + P 2 N . This suggests that 2 independent users can send information as well as if they had pooled their power. Solution: Multiple layer waterflling C µ P 1 + P 2 N = 1 2 log µ 1 + P 1 + P 2 N = 1 2 log µ N + P 1 + P 2 N = 1 2 log µ N + P 1 + P 2 N + P 1 · N + P 1 N = 1 2 log µ N + P 1 + P 2 N + P 1 + 1 2 log µ N + P 1 N = C µ P 2 P 1 + N + C µ P N 1 2. Parallel channels and waterflling Consider a pair of parallel Gaussian channels, i.e., µ Y 1 Y 2 = µ X 1 X 2 + µ Z 1 Z 2 , where µ Z 1 Z 2 ∼ N µ 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 . 1
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(a) 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, ie., at what power does the worst channel become useful? (b) What is the capacity C ( P ) for large P ? Solution: Parallel channels and waterflling (a) By the result of Section 10.4, it follows that we will put all the signal power into the channel with less noise until the total power of noise + signal in that channel equals the noise power in the other channel. After that, we will split any additional power evenly between the two channels. Thus the combined channel begins to behave like a pair of parallel channels when the signal power is equal to the diFerence of the two noise powers, i.e., when P = σ 2 1 - σ 2 2 . (b) Let E ( X 2 1 ) = P 1 and E ( X 2 2 ) = P 2 . Therefore P = P 1 + P 2 . (1) ±rom water²lling we know P 2 = P 1 + σ 2 1 - σ 2 2 . (2) ±rom equations (1) and (2) we get P 1 = P - ( σ 2 1 - σ 2 2 ) 2 P 2 = P + ( σ 2 1 - σ 2 2 ) 2 . Hence C ( P ) = 1 2 log µ 1 + P - ( σ 2 1 - σ 2 2 ) 2 σ 2 1 + 1 2 log µ 1 + P + ( σ 2 1 - σ 2 2 ) 2 σ 2 2 2
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3. Vector channel Consider the 3 input 3 output Gaussian channel µ´ ¶³ - - ? X Y Z N 3 (0 , K ) where X, Y, Z R 3 , E || X || 2 = E ( X 2 1 + X 2 2 + X 2 3 ) P, and Z N 3 (0 , K ) . Find the capacity for K = 1 0 0 0 1 ρ 0 ρ 1 . Solution: Vector channel
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hw4sol - EE 376B/Stat 376B Information Theory Prof T Cover...

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