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3 - EE 478 Multiple User Information Theory Handout#9...

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EE 478 Handout #9 Multiple User Information Theory Thursday, October 9, 2008 Homework Set #3 Due: Thursday, October 16, 2008. 1. Binary erasure MAC. Let X 1 Bern( p 1 ) and X 2 Bern( p 2 ) be independent. Show that max p 1 ,p 2 H ( X 1 + X 2 ) = 1 . 5 bits and is achieved when X 1 and X 2 are Bern(1 / 2). 2. Capacity of multiple access channels. (a) Consider the binary multiplier MAC example in the lecture notes, where X 1 , X 2 , and Y are binary, and Y = X 1 · X 2 . We established the capacity region using a time-sharing ar- gument. Show that the capacity region can also be expressed as the union of R ( X 1 , X 2 ) sets (with no time-sharing/convexification needed), specify the set of p ( x 1 ) p ( x 2 ) distri- butions on ( X 1 , X 2 ) that achieve the boundary of the region. (b) Find the capacity region for the modulo-2 additive MAC, where X 1 , X 2 , and Y are binary, and Y = X 1 + X 2 mod 2. Again show that the capacity region can be expressed as the union of R ( X 1 , X 2 ) sets and therefore time-sharing is not necessary. (c) The capacity regions of the above two examples and the Gaussian MAC can all be expressed as union of R ( X 1 , X 2 ) sets and no time-sharing is necessary. Is time-sharing ever necessary? Find the capacity of the push-to-talk MAC with binary inputs and output, and p (0 | 0 , 0) = p (1 | 0 , 1) = p (1 | 1 , 0) = 1 and p (0 | 1 , 1) = 0 . 5. Why is this channel called “push-to-talk”?
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