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hw4soln - EE 670 Homework#4 Solution Prof Uf Tureli Stevens...

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EE 670 : Homework #4 Solution Prof. Uf Tureli, Stevens Institute of Technology email/tel/fax: [email protected], 201.216.5603/8246 Note: Cover and Thomas, Q. 8.6,8.12,9.1,9.3 1. Question: Using two channels at once. Consider two discrete memoryless channels ( X 1 , p ( y 1 | x 1 ) , Y 1 ) and ( X 2 , p ( y 2 | x 2 ) , Y 2 ) with capacities C 1 , and C 2 respectively. A new channel ( X 1 × X 2 , p ( Y 1 | x 1 ) × p ( y 2 | x 2 ) , Y 1 times Y 2 ) is formed in which x 1 ∈ X 1 and x 2 ∈ X 2 are simultaneously sent, resulting in y 1 , y 2 . Find the capacity of this channel. Solution: Suppose we are given two channels, ( X 1 , p ( y 1 | x 1 ) , Y 1 ) and ( X 2 , p ( y 2 | x 2 ) , Y 2 ) which we can use simultaneously. We can define the product channel as the channel, ( X 1 × X 2 , p ( y 1 , y 2 | x 1 , x 2 ) = p ( y 1 | x 1 ) p ( y 2 | x 2 ) , Y 1 × Y 2 ) To find the capacity of the product channel, we must find the distribution of p ( x 1 , x 2 ) on the input alphabet X 1 × X 2 that maximizes I ( X 1 , X 2 ; Y 1 , Y 2 ). Since the joint distribution: p ( x 1 , x 2 , y 1 , y 2 ) = p ( x 1 , x 2 ) p ( Y 1 | x 1 ) p ( y 2 | x 1 ) , Y 1 X 1 X 2 Y 2
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