S08.hw04 - ECE 320 Networks and Systems Spring 20072008...

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ECE 320 Networks and Systems Spring 2007–2008 Problem Set 4 Due March 3, 2008 1. This problem considers source encoding for a source which generates 4 different symbols. (a) The 4 symbols and their probabilities are shown in the following table. s i Pr( s i ) e .60 b .29 y .07 ! .04 Determine the Huffman code. What is the expected number of binary code symbols per 4-ary source symbol? (b) If the probabilities in Problem 1a were different, the following Huffman code might result: s i code word e 111 b 110 y 10 ! 0 (This code is not the answer to Problem 1a). Using this code, decode the message 110101110 2. The entropy ( H d ) of a probability mass function ( p k ) is defined by H d ( p ) = - n X k =1 p k log d p k where log d is logarithm to base d . Because the entropy describes the average length of a code word needed to encode realizations from the pmf p , it is important to determine what is the most unfavorable pmf p , i.e., the pmf that makes H d ( p ) as large as possible. Intuitively, you expect that it is hard to encode something with small average code word length when every possible input occurs frequently.
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