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hw3278b

# hw3278b - UCSD ECE287B Prof Young-Han Kim Homework Set#3...

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UCSD ECE287B Handout #8 Prof. Young-Han Kim Tuesday, October 19, 2010 Homework Set #3 Due: Tuesday, October 26, 2010 1. Partialdecode–forwardforcausalrelaychannels. Considerthediscretememorylesscausalrelaychannel p ( y 2 | x 1 ) p ( y 3 | x 1 ,x 2 ,y 2 ). (a) By adapting partial decode–forward for the DM-RC to this case, show that the capacity is lower bounded as C 0 max p ( u,v,x 1 ) ,x 2 ( v,y 2 ) min braceleftbig I ( V,X 1 ; Y 3 ) , I ( U ; Y 2 | V )+ I ( X 1 ; Y 3 | U,V ) bracerightbig , (b) Suppose Y 2 = y 2 ( X 1 ). Using the partial decode–forward lower bound in part (a) and the cutset bound, show that the capacity of this semi-deterministic DM causal relay channel is C 0 = max p ( v,x 1 ) ,x 2 ( v,y 2 ) min braceleftbig I ( V,X 1 ; Y 3 ) , H ( Y 2 | V )+ I ( X 1 ; Y 3 | V,Y 2 ) bracerightbig . (c) Consider the DM noncausal relay channel with orthogonal sender components, where X 1 = ( X 1 ,X ′′ 1 ) and p ( y 2 | x 1 ) p ( y 3 | x 1 ,x 2 ,y 2 )= p ( y 2 | x ′′ 1 ) p ( y 3 | x 1 ,x 2 ). Show that the capacity is C 0 = max p ( x 1 | v ) p ( x ′′ 1 | v ) ,x 2 ( v,y 2 ) min braceleftbig I ( V,X 1 ; Y 3 ) , I ( X ′′ 1 ; Y 2 | V )+ I ( X 1
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