3s - EE 478 Handout #14 Multiple User Information Theory...

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Unformatted text preview: EE 478 Handout #14 Multiple User Information Theory October 20, 2008 Homework Set #3 Solutions 1. Solution: Let Y = X 1 + X 2 . Then P( Y = 0) = (1 p 1 )(1 p 2 ), P( Y = 1) = (1 p 1 ) p 2 + p 1 (1 p 2 ), and P( Y = 2) = p 1 p 2 , and H ( Y ) = summationdisplay y { , 1 , 2 } P( Y = y )log P( Y = y ) , = f ( p 1 ,p 2 ) . By differentiating with respect to p 1 and p 2 , and solving f p 1 = f p 2 = 0 , we get p 1 = p 2 = 0 . 5 which corresponds to H ( Y ) = 1 . 5. To verify that the entropy of Y is maximized at this point, we also need to investigate the boundary points. But if p i { , 1 } for i = 1 or i = 2, then H ( Y ) 1. 2. Solution: (a) The capacity region of this channel is the convex closure of the set of all ( R 1 ,R 2 ) such that R 1 H ( Y | X 1 ,Q ) , R 2 H ( Y | X 2 ,Q ) , R 1 + R 2 H ( Y | Q ) for some p ( q ) p ( x 1 | q ) p ( x 2 | q ). But since the RHS of each inequality is bounded by 1, the capacity region is the triangle region specified by R 1 + R 2 1. If we let X 1 Bern( p 1 ) and X 2 Bern( p 2 ), we see that the points on the boundary are achievable when p 1 p 2 = 1 p 1 p 2 , i.e., p 1 p 2 = 1 / 2. (b) The capacity region of this channel is again the convex closure of the set of all ( R 1 ,R 2 ) such that R 1 H ( X 1 | Q ) , R 2 H ( X 2 | Q ) , R 1 + R 2 H ( Y | Q ) for some p ( q ) p ( x 1 | q ) p ( x 2 | q ). But since the RHS of each inequality is bounded by 1, the capacity region is the triangle region specified by R 1 + R 2...
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3s - EE 478 Handout #14 Multiple User Information Theory...

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