hw3soln

hw3soln - EE 670 : Homework #3 Solution Prof. Uf Tureli,...

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EE 670 : Homework #3 Solution Prof. Uf Tureli, Stevens Institute of Technology email/tel/fax: [email protected], 201.216.5603/8246 Note: Cover and Thomas, Q. 5.4,5.8,5.14,5.25 1. Question: Hufman coding. Consider the random variable: X = x 1 x 2 x 3 x 4 x 5 x 6 x 7 0 . 49 0 . 26 0 . 12 0 . 04 0 . 04 0 . 04 0 . 02 ! (a) Find a binary Hufman code ±or X . Solution : The Hufman tree ±or this distribution is: Codeword 1 x 1 0 . 49 0 . 49 0 . 49 0 . 49 0 . 49 0 . 51 1 00 x 2 0 . 26 0 . 26 0 . 26 0 . 26 0 . 26 0 . 49 011 x 3 0 . 12 0 . 12 0 . 12 0 . 13 0 . 25 01000 x 4 0 . 04 0 . 05 0 . 08 0 . 12 01001 x 5 0 . 04 0 . 04 0 . 05 01010 x 6 0 . 03 0 . 04 01011 x 7 0 . 02 (b) Find the expected codelength ±or this encoding. The expected length o± the codeword ±or the binary Hufman code E ( X )=2.02 bits. (c) Find a ternary Hufman code ±or X . Solution : Codeword 0 x 1 0 . 49 0 . 49 0 . 49 1 1 x 2 0 . 26 0 . 26 0 . 26 20 x 3 0 . 12 0 . 12 0 . 25 22 x 4 0 . 04 0 . 09 210 x 5 0 . 04 0 . 04 211 x 6 0 . 03 212 x 7 0 . 02 2. Question Simple optimum compression oF a Markov source. Consider the 3-state Markov process having transition matrix in Table 2: Thus, the probability that S 1 ±ollows S 3 is equal to zero. Design 3 codes C 1 , C 2 , C 3 (one ±or each state, S 1 , S 2 , S 3 ), each code mapping elements o± the set S i ’s into sequences o± 0’s and 1’s, such that this Markov process can be sent with maximal compression by the ±ollowing scheme: (a) Note the present symbol S i .

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This note was uploaded on 10/12/2009 for the course EE EE 670 taught by Professor Uftureli during the Spring '05 term at Stevens.

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hw3soln - EE 670 : Homework #3 Solution Prof. Uf Tureli,...

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