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# ex4_sol - Introduction to Information Theory(67548...

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Introduction to Information Theory (67548) January 12, 2009 Assignment 4: Gaussian Channel and Differential Entropy Lecturer: Prof. Michael Werman Due: Sunday, January 25, 2009 Note: Unless specified otherwise, all entropies and logarithms should be taken with base 2 . Problem 1 Differential Entropy 1. If Y = aX + c , we have that f Y ( y ) = 1 | a | f X ( y - c a ) , where f X ( · ) , f Y ( · ) are the density functions of X, Y respectively. Therefore, h ( Y ) = - y = -∞ f Y ( y ) log( f Y ( y )) dy = - y = -∞ 1 | a | f X y - c a log 1 | a | f X y - c a dy = - y = -∞ 1 | a | f X y - c a log f X y - c a dy - y = -∞ 1 | a | f X y - c a log 1 | a | dy = - z = -∞ f X ( z ) log( f X ( z )) dz - z = -∞ f X ( z ) log 1 | a | dz = H ( X ) + log( | a | ) , where the shift from the third to the fourth line was performed with a change of variables z = ( y - c ) /a . So we see that shifting a random variable X by a constant c does not change its differential entropy, while multipliying by a constant a changes its differential entropy by log( | a | ). 2. The chain rule for differential entropy simply follows from the definitions (try it!), and so is the resulting conclusion. Problem 2 Discrete Input, Continuous Output Recall that I ( X ; Y ) = h ( Y ) - h ( Y | X ). Given the value of X , Y is uniformly distributed in an interval of length 2. Therefore, h ( Y |

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ex4_sol - Introduction to Information Theory(67548...

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