sol3 - ECE 5620 Spring 2011 Homework #3: Solutions 1. 1 2....

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Unformatted text preview: ECE 5620 Spring 2011 Homework #3: Solutions 1. 1 2. 2 3. 4. 3 5. (c) The first channel has capacity 1- H ( p ) , while the second has capacity zero, so C = log 2 1- H ( p ) + 2 = log 2 1- H ( p ) + 1 . 4 6. 7. Take a sphere of radius r and consider the points in a shell within distance of the surface of the sphere. The volume of the overall sphere in n dimensions is given by C n r n for some constant C n . Likewise the volume of the smaller sphere with radius r- is given by C n ( r- ) n . The fraction of the volume of the radius- r sphere that is also in the shell is C n r n- C n ( r- ) n C n r n . Since this tends to one as n tends to infinity, we see that most of the points in the larger sphere are also contained in the shell. 8. (a) If = 1 / ( + 1) , then q ( n ) = (1- ) n n { , 1 , 2 ,... } is the geometric distribution over the nonnegative integers with mean . One can compute H ( q ) =- X n =0 (1- ) n log[ (1- ) n ] =- log - log(1...
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sol3 - ECE 5620 Spring 2011 Homework #3: Solutions 1. 1 2....

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