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hw5sol

# hw5sol - EE 376B/Stat 376B Information Theory Prof T Cover...

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EE 376B/Stat 376B Handout #15 Information Theory Tuesday, May 16, 2006 Prof. T. Cover Solutions to Homework Set #5 1. Gaussian multiple access. A group of m users, each with power P , is using a Gaussian multiple access channel at capacity, so that m X i =1 R i = C mP N , (1) where C ( x ) = 1 2 log(1 + x ) and N is the receiver noise power. A new user of power P 0 wishes to join in. (a) At what rate can he send without disturbing the other users? (b) What should his power P 0 be so that the new users rate is equal to the combined communication rate C ( mP/N ) of all the other users? Solution: Gaussian multiple access. (a) If the new user is not to disturb other users, his message should be decodable at first. Therefore, R = C P 0 mP + N . (b) We need C P 0 mP + N = C mP N , or equivalently, P 0 = mP ( mP + N ) N . 2. Capacity of multiple access channels. Find the capacity region for each of the following multiple access channels: (a) Additive modulo 2 multiple access access channel. X 1 ∈ { 0 , 1 } , X 2 ∈ { 0 , 1 } , Y = X 1 X 2 . (b) Multiplicative multiple access channel. X 1 ∈ {- 1 , 1 } , X 2 ∈ {- 1 , 1 } , Y = X 1 · X 2 . 1

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Solution: Capacity of multiple access channels. (a) Additive modulo 2 multiple access channel. Quite clearly we cannot send at a total rate of more than 1 bit, since H ( Y ) 1. We can achieve a rate of 1 bit from sender 1 by setting X 2 = 0, and similarly we can send 1 bit/transmission from sender 2. By simple time sharing we can achieve the entire capacity region which is shown in Figure 1. - 6 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ R 1 R 2 0 C 2 = 1 C 1 = 1 Figure 1: Capacity region of additive modulo 2 MAC.
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• Spring '05
• TomCover
• Trigraph, Tier One, Scaled Composites, Scaled Composites White Knight, 2004 in spaceflight, Sub-orbital spaceflight

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hw5sol - EE 376B/Stat 376B Information Theory Prof T Cover...

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