practiceFinalSol - EE 376A/Stat 376A Information Theory...

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EE 376A/Stat 376A Handout #32 Information Theory Thursday, March 12, 2009 Prof. T. Cover Solutions to Practice Final Examination (Note: When the solutions refer to particular homework problems, it is speaking with respect to homework problems that were assigned in the year in which this exam was given – not the ones assigned during the Winter 2008-2009 instantiation of EE376A.) 1. Graph. What is the entropy rate of a random walk on the star graph with a central hub node and n edges: Solution: Graph. Note that the stationary distribution is (1 / 2 n, . . . , 1 / 2 n bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright n , n/ 2 n ) and only the central hub node has non-zero conditional entropy. Hence, the entropy rate is 1 2 log n . 2. Optimal code when entropy is infinite. Let X be integer valued with H ( X ) = - i =1 p i log p i = . Thus the expected (binary) 1
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description length L = p i l i is infinite, even for the Shannon ideal codeword length l * i = log 1 p i . Show nonetheless that { l * i } is better than { l i } for any other instantaneous code in the sense that i =1 p i ( l i - l * i ) 0, for all { l i } satisfying the Kraft inequality. Solution: Optimal code when entropy is infinite. summationdisplay i =1 p i ( l i - l * i ) = summationdisplay p i log p i 2 - l i = summationdisplay p i log p i r i - log( summationdisplay 2 - l i ) 0 , where r i = 2 - l i 2 - l i and the last inequality follows from the nonnegativity of D ( p || r ) and Kraft inequality, 2 - l i 1. 3. Huffman code. (a) Find the binary Huffman code for p = { 7 20 , 4 20 , 4 20 , 3 20 , 2 20 } . (b) Guess the optimal (minimal expected description length) binary code for the integer valued random variable X , where Pr { X = i } = p i q, i = 0 , 1 , . . . , and p = . 6. Solution: Huffman code. (a) The Huffman tree for this distribution is Codeword 00 7 7 8 12 20 10 4 5 7 8 11 4 4 5 010 3 4 011 2 (b) Since p i - 1 q p i q k = i +1 p k q = p i +1 for all i , the Huffman tree is 2
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Codeword 0 q q . . . q q . . . q q 1 10 pq pq . . . pq pq . . . pq p 110 p 2 q p 2 q . . . p 2 q p 2 q . . . p 2 . . . 11 · · · 10 p i q p i q . . . p i q p i 11 · · · 110 p i +1 q p i +1 q . . . p i +1 .
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