5 - EE 478 Handout#15 Multiple User Information Theory...

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Unformatted text preview: EE 478 Handout #15 Multiple User Information Theory Thursday, October 23, 2008 Homework Set #5 Due: Thursday, October 30, 2008. 1. Duality between Gaussian broadcast and multiple access channels. Consider the following Gaussian broadcast and multiple access channels: The broadcast channel: At time i , Y 1 i = g 1 X i + Z 1 i and Y 2 i = g 2 X i + Z 2 i , where the Z 1 i and Z 2 i are i.i.d. ∼ N(0 , 1). Assume average power constraint P on X . The multiple access channel: At time i , Y i = g 1 X 1 i + g 2 X 2 i + Z i , where the Z i are i.i.d. ∼ N(0 , 1). Assume a sum average power constraint on each pair of codewords ( x n ( m 1 ) ,x n ( m 2 )): 1 n n X i =1 ( x 2 1 i + x 2 2 i ) ≤ P. (a) Provide expressions for the independent message capacity regions of these two channels in terms of C( x ) = 1 2 log(1 + x ). (b) Show that the two capacity regions are equal. (c) Consider any point ( R 1 ,R 2 ) on the boundary of the capacity region you found. Using random coding and successive cancelation decoding, argue that there exists a sequence of codes that achieve such a point for the given multiple access channel. Now, show that the same sequence of codes can be used to achieve the same point on the capacity region of the given broadcast channel.region of the given broadcast channel....
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5 - EE 478 Handout#15 Multiple User Information Theory...

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