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ece6605-hwk1 - to see a sequence of independent equally...

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GEORGIA INSTITUTE OF TECHNOLOGY School of Electrical and Computer Engineering ECE 6605 Information Theory Assigned: Tuesday, Sep. 7, 2010 Due: Atlanta and Savannah students: Friday, Sep. 17, 2010 Due: Video students: Wed., September 22, 2010 Problem Set #1 Problem 1-5: Solve the questions 2.1 (from page 43), 2.4, 2.7 (part a only), 2.17 and 2.18 from Chapter 2 of the textbook (the second edition). Problem 6: In class, we considered the discrete memoryless source ( A, B, C, D ) with prob- abilities (1 / 2 , 1 / 4 , 1 / 8 , 1 / 8), with the binary source code: A 00 B 01 C 10 D 11 (a) With this code, calculate the probability of the channel seeing a “0” and a “1”. (b) See if you can rearrange the four codewords { 00 , 01 , 10 , 11 } differently, so that these probabilities are both 1/2. (c) Explain why the rearranged code you found in part (b) is still not much good, despite Shannon’s prescription, which says that the code should cause the channel
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Unformatted text preview: to see a sequence of independent, equally likely 0’s and 1’s. Problem 7: Let X , Y and Z be three discrete random variables. Using Jensen’s inequality, or otherwise (e.g., the chain rule), show that I ( X,Y ; Z ) ≥ I ( Y ; Z ). When does the equality hold? Problem 8: Let X 1 , X 2 be discrete random variables drawn according to probability mass functions p 1 ( · ) and p 2 ( · ) over the respective alphabets X 1 = { 1 , 2 ,... ,m } and X 2 = { m + 1 , 2 ,... ,n } . Let X = b X 1 with probability α X 2 with probability 1-α 1 Find H ( X ) in terms of H ( X 1 ) and H ( X 2 ) and α . Hint: de±ne a function of X as : γ = f ( X ) = b 1 when X = X 1 2 when X = X 2 and use H ( X,γ ). 2...
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