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# 7.22 - EE 376A Information Theory Prof T Cover Handout#28...

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EE 376A Handout #28 Information Theory Tuesday, March 3, 2009 Prof. T. Cover Solutions to Homework Set #7 1. Postprocessing the output. One is given a communication channel with transition probabilities p ( y | x ) and channel capacity C = max p ( x ) I ( X ; Y ) . A helpful statistician postprocesses the output by forming ˜ Y = g ( Y ) , yielding a channel p y | x ). He claims that this will strictly improve the capacity. (a) Show that he is wrong. (b) Under what conditions does he not strictly decrease the capacity? Solution: Postprocessing the output. (a) The statistician calculates ˜ Y = g ( Y ). Since X Y ˜ Y forms a Markov chain, we can apply the data processing inequality. Hence for every distribution on x , I ( X ; Y ) I ( X ; ˜ Y ) . Let ˜ p ( x ) be the distribution on x that maximizes I ( X ; ˜ Y ). Then C = max p ( x ) I ( X ; Y ) I ( X ; Y ) p ( x )=˜ p ( x ) I ( X ; ˜ Y ) p ( x )=˜ p ( x ) = max p ( x ) I ( X ; ˜ Y ) = ˜ C. Thus, the helpful suggestion is wrong and processing the output does not increase capacity. (b) We have equality (no decrease in capacity) in the above sequence of inequalities only if we have equality in data processing inequality, i.e., for the distribution that maximizes I ( X ; ˜ Y ), we have X ˜ Y Y forming a Markov chain. Thus, ˜ Y should be a sufficient statistic. 2. Noisy typewriter. Consider a 26-key typewriter. 1

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(a) If pushing a key results in printing the associated letter, what is the capacity C in bits? (b) Now suppose that pushing a key results in printing that letter or the next (with equal probability). Thus A A or B , and Z Z or A. What is the capacity? (c) What is the highest rate code with block length one that you can find that achieves zero probability of error for the channel in part (b) . Solution: Noisy typewriter. (a) If the typewriter prints out whatever key is struck, then the output Y is the same as the input X and C = max I ( X ; Y ) = max H ( X ) = log 26 , (1) attained by a uniform distribution over the letters. (b) In this case, the output is either equal to the input (with probability 1 2 ) or equal to the next letter ( with probability 1 2 ). Hence H ( Y | X ) = log 2 independent of the distribution of X , and hence C = max I ( X ; Y ) = max H ( Y ) log 2 = log 26 log 2 = log 13 , (2) which is attained for a uniform distribution over the output, which in turn is attained by a uniform distribution on the input. (c) A simple zero error block length one code is the one that uses every alternate letter, say A,C,E,. . . ,W,Y. In this case, none of the codewords will be confused, since A will produce either A or B, C will produce C or D, etc. The rate of this code, R = log(# codewords) Block length = log 13 1 = log 13 . (3) In this case, we can achieve capacity with a simple code with zero error. Note that the uniform distribution over the output is attained also by this input distribution.
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7.22 - EE 376A Information Theory Prof T Cover Handout#28...

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