ece6605-hwk3

ece6605-hwk3 - X = 1 2 3 4 and that the probabilities of...

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GEORGIA INSTITUTE OF TECHNOLOGY School of Electrical and Computer Engineering ECE 6605 Information Theory Assigned: Monday, Sep. 27, 2010 Due: Atlanta and Savannah students: Wed, Oct. 6, 2010 Due: Video students: Wed, Oct. 13, 2010 Problem Set #3 Problem 1-2: Solve the questions 4.12 and 4.13 from Chapter 4 of the textbook (the second edition). Problem 3: Let assume the time is discretized to time slots or epochs. At each time epoch we throw a fair (unbiased) coin. Let X be the waiting time (i.e., the number of time epochs) for the ﬁrst heads to appear in successive ﬂips of a fair coin. Thus, the distribution of the waiting time is given by Pr { X = i } = (1 / 2) i . Let S n be the waiting time for the n th head to appear. Thus, we have: S 0 = 0 S n +1 = S n + X n +1 for n = 0 , 1 , 2 ,..., where X 1 ,X 2 ,X 3 ,... are iid according to the above distribution. (a) (10 points) Determine H ( S 1 ,S 2 ,...,S n ). (b) (10 points) Does the process S n have an entropy rate? If so, what is it? If not, why not? Problem 4: Suppose that

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Unformatted text preview: X = { 1 , 2 , 3 , 4 } and that the probabilities of the four possible outcomes are p = { 1 2 , 1 8 , 1 4 , 1 8 } (a) (2 points) Determine H ( X ). (b) (2 points) Let q = { 1 8 , 1 4 , 1 2 , 1 8 } be probabilities associated with a random variable Y also deﬁned on the set { 1 , 2 , 3 , 4 } . Compute H ( Y ). (c) (2 points) Find the relative entropy between p and q , (i.e., D ( p k q ). Also ﬁnd D ( q k p ). (d) (6 points) Find a Huﬀman code for X . (e) (2 points) Find the expected codeword length for the Huﬀman code. 1 (f) (6 points) Now suppose that q had been the true distribution, but the Huﬀman code was designed using p as in part d. Find the expected codeword length. What is the cost for not using the true distribution q to design the code? Problem 5: Solve the questions 4.28 from Chapter 4 of the textbook (the second edition). 2...
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ece6605-hwk3 - X = 1 2 3 4 and that the probabilities of...

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