HW4s[1] - ECE 534 Elements of Information Theory Fall 2010...

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ECE 534: Elements of Information Theory, Fall 2010 Homework 4 Solutions Problem 4.8 (Davide Basilio Bartolini) Since the sequence X comes from a discrete memoryless source, the samples X i are independent random variables and p 2 = 1 - p 1 . First, we need to compute the estimated duration for a sample, which is (due to the fact that the value of a symbol is equal to its duration): E [ T ] = E [ X i ] = p 1 + 2 p 2 = 2 - p 1 Then, we compute the entropy for a sample of the process: H ( X i ) = - ( p 1 log 2 p 1 + p 2 log 2 p 2 ) = - ( p 1 log 2 p 1 - (1 - p 1 ) log 2 (1 - p 1 )) So, the expression for the entropy per unit time is: H ( X ) = H ( X i ) E [ T ] = - ( p 1 log 2 p 1 - (1 - p 1 ) log 2 (1 - p 1 )) 2 - p 1 To find the value for p 1 that maximizes H ( X ), we can compute its partial derivative w.r.t. p 1 and find the value of p 1 that makes it zero (i.e. the value for which its numerator is zero): δ H ( X ) δp 1 = T ( δH ( X )) δp 1 - H ( X )( δT δp 1 ) T 2 = log e (1 - p 1 ) - 2 log e p 1 T 2 (using base e for the logs in entropies) log e (1 - p 1 ) - 2 log e p 1 = 0 1 - p 1 = p 2 1 p 2 1 + p 1 - 1 = 0 The last equation which gives the results for p 1 : - 1 2 ( 5 + 1) and 1 2 ( 5 - 1). Of the two found values, the second is a maximum for H ( X ) (moreover, the first solution is less than zero, so it cannot be a probability value) and the
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