soln_ex2_makeup - ECE320 Solutions to Makeup Second...

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ECE320 Solutions to Makeup Second Examination Spring 2006 Cornell University T.L.Fine 1. Consider a source with five symbols [a,b,c,d,e] having correspond- ing probabilities of being chosen of p = [1 , 1 , 2 , 4 , 8] / 16. Consider a prefix code for this set of symbols with the corresponding codewords being [0000 , 0001 , 001 , 01 , 1]. (a)(2pts) What is the expected codeword length? EL = X i l i p i = (4 + 4 + 2 · 3 + 4 · 2 + 1 · 8) / 16 = 15 8 . (b)(3pts) Calculate the source entropy H in bits. In this case, all the probabilities are powers of 1 / 2 and therefore we can exactly evaluate logarithms to base 2, H = - X i p i log 2 ( p i ) = (1 · 4 + 1 · 4 + 2 · 3 + 4 · 2 + 8 · 1) / 16 = 15 8 . (c)(4pts) Construct a Huffman code for this source, evaluate its expected length, and compare it to the entropy found in (b). The code given in (a) is a prefix code whose expected length equals the source entropy. Hence, it is as short as is possible. The code of (a) is a Huffman code, although such codes are not unique. (d)(6pts) Consider another set of codewords [00 , 01 , 10 , 11 , 1] for the sym- bols listed in (a). Evaluate the expected codeword length and compare it to the expected length found in (c). Comment on the comparison. EL
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soln_ex2_makeup - ECE320 Solutions to Makeup Second...

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