final_sol

# final_sol - SySc 645 Information Theory Final Exam...

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SySc 645 Information Theory: Final Exam. Solutions 1. (30 points) A Hamming code with block length 7 can be characterized by the matrix H = 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 Each legal input codeword x satisfies Hx = 0 (all arithmetic is understood to be mod 2). The 7 bits received can be represented by y = x + e where e is the error pattern. Assume that the bits are transmitted over a memoryless binary symmetric channel with the probability of error in each bit of which is much less than 1 2 . (a) How many codewords (legal input x ’s) are there? Answer: 16 (b) If y = 1110010 what is ˆ x ( y ), the best guess for x ? Answer: 1110000 (c) Assuming that you attempt to correct errors in received words, find P E , the probability that you will make an error in guessing the trans- mitted codeword: i. First write out an exact algebraic expression. Answer: P E = 1 - (1 - ) 7 - 7 (1 - ) 6 ii. Give an approximate numerical value for the expression assuming that = 10 - 9 . Answer: P E ( 7 2 ) 2 = 21 × 10 - 18 2. (20 points) Given the transition probabilities P Y | X

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