5s - EE 478 Handout #20 Multiple User Information Theory...

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Unformatted text preview: EE 478 Handout #20 Multiple User Information Theory November 6, 2008 Homework Set #5 Solutions 1. Solution: (a) From the lecture notes, the capacity region C BC ( g 1 ,g 2 ,P ) of the Gaussian broadcast channel is given by the set of rate pairs ( R 1 ,R 2 ) such that R 1 C( S 1 ) , R 2 C parenleftbigg (1- ) S 2 S 2 + 1 parenrightbigg , for some 0 1, where S 1 = g 2 1 P and S 2 = g 2 2 P . For the multiple-access channel assume P 1 = P and P 2 = (1- ) P . The capacity region C MAC ( P ) is given by the set of rate pairs ( R 1 ,R 2 ) such that R 1 C( S 1 ) , R 2 C((1- ) S 2 ) , R 1 + R 2 C( S 1 + (1- ) S 2 ) , for some 0 1. (b) We show that the corner points of each region is in the other region. Consider ( R 1 ,R 2 ) = (C( S 1 ) , C parenleftBig (1 ) S 2 S 2 +1 parenrightBig ) C BC ( g 1 ,g 2 ,P ). For = ( S 2 + 1) S 2 + 1 , we have C parenleftbigg (1- ) S 2 S 2 + 1 parenrightbigg = C((1- ) S 2 ) . Note that the defined satisfies , therefore R 1 = C( S 1 ) < C( S 1 ). Finally, it suffices to show that C( S 1 ) + C parenleftbigg (1- ) S 2 S 2 + 1 parenrightbigg C( S 1 + (1- ) S 2 ) . Note that C( S 1 ) + C parenleftbigg (1- ) S 2 S 2 + 1 parenrightbigg = 1 2 log (1 + S 2 )( S 1 + 1) S 2 + 1 , and C( S 1 + (1- ) S 2 ) = 1 2 log( ( S 2 + 1) S 1 S 2 + 1 + (1- ) S 2 S 2 + 1 + 1) = C( S 1 ) + C parenleftbigg (1- ) S 2 S 2 + 1 parenrightbigg . 1 Therefore, C BC ( g 1 ,g 2 ,P ) C MAC ( g 1 ,g 2 ,P ). To prove the other direction, let ( R 1 ,R 2 ) = (C( S 1 ) , C( S 1 +(1- ) S 2 )- C( S 1 )) be a corner point of the MAC region. Note that C( S 1 + (1- ) S 2 )- C( S 1 ) = 1 2 log parenleftbigg S 1 + (1- ) S 2 + 1 S 1 + 1 parenrightbigg = 1 2 log(1 + (1- ) S 2 S 1 + 1 ) = C( (1- ) S 2 S 1 + 1 ) . Therefore, ( R 1 ,R 2 ) C BC . The other possibility for a corner point of the MAC region is ( R 1 ,R 2 ) = (C( S 1 + (1- ) S 2 )- C((1- ) S 2 ) , C((1- ) S 2 )). Note that C( S 1 + (1- ) S 2 )- C((1- ) S 2 ) = 1 2 log parenleftbigg S 1 + (1- ) S 2 + 1 (1- ) S 2 + 1 parenrightbigg = C( S 1 (1- ) S 2 + 1 ) C( S 1 ) . Hence, C MAC ( g 1 ,g 2 ,P ) C BC ( g 1 ,g 2 ,P ) as well. (c) Consider the point ( R 1 ,R 2 ) = (C( S 1 ) , C( S 1 + (1- ) S 2 )- C( S 1 )) = parenleftbigg C( S 1 ) , C( (1- ) S 2 S 1 + 1 ) parenrightbigg , on the boundary of the multiple-access region. The rate pair ( R 1 ,R 2 ) can be achieved using successive cancelation by first decoding the message of user 2 assuming the received sequence of user 1 as noise and then decoding message of user 1 after subtracting off the received sequence of user 2. However, R 1 = C( S 1 ) , R 2 = C parenleftbigg (1- ) S 2 S 1 + 1 parenrightbigg , is also on the boundary of the the broadcast region. Hence the same codebook (with proper power scaling) can be used for the broadcast channel and achieves the same point...
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This note was uploaded on 10/24/2011 for the course ELECTRICAL ECE 571 taught by Professor Kelly during the Spring '11 term at University of Illinois, Urbana Champaign.

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5s - EE 478 Handout #20 Multiple User Information Theory...

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