By Kelbert M., Suhov Y.

ISBN-10: 0521139880

ISBN-13: 9780521139885

ISBN-10: 0521769353

ISBN-13: 9780521769358

This primary monograph introduces either the probabilistic and algebraic elements of knowledge concept and coding. It has advanced from the authors' years of expertise educating on the undergraduate point, together with a number of Cambridge Maths Tripos classes. The ebook presents suitable heritage fabric, a variety of labored examples and transparent recommendations to difficulties from genuine examination papers. it's a helpful instructing reduction for undergraduate and graduate scholars, or for researchers and engineers who are looking to grab the fundamental rules

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Show also that I(Xn : Xn+m ) is non-increasing in m, for m = 0, 1, 2, . . 36 Essentials of Information Theory Solution By the Markov property, Xn−1 and Xn+1 are conditionally independent, given Xn . Hence, h(Xn−1 , Xn+1 |Xn ) = h(Xn+1 |Xn ) + h(Xn−1 |Xn ) and I(Xn−1 : Xn+1 |Xn ) = 0. 21). 25 (An axiomatic definition of entropy) (a) Consider a probability distribution (p1 , . . , pm ) and an associated measure of uncertainty (entropy) such that h(p1 q1 , p1 q2 , . . , p1 qn , p2 , p3 , . . , pm ) = h(p1 , .

25 respectively (which is rather flattering to modern journalism). In case (c) the results were inconclusive; apparently the above assumptions are not appropriate in this case. ) Even more challenging is to compare different languages: which one is more appropriate for intercommunication? Also, it would be interesting to repeat the above experiment with the collected works of Tolstoy or Dostoyevsky. For illustration, we give below the original table by Samuel Morse (1791–1872), creator of the Morse code, providing the frequency count of different letters in telegraph English which is dominated by a relatively small number of common words.

24 Given random variables Y1 , Y2 , Y3 , define I(Y1 : Y2 |Y3 ) = h(Y1 |Y3 ) + h(Y2 |Y3 ) − h(Y1 ,Y2 |Y3 ). Now let the sequence Xn , n = 0, 1, . . be a DTMC. Show that I(Xn−1 : Xn+1 |Xn ) = 0 and hence I(Xn−1 : Xn+1 ) ≤ I(Xn : Xn+1 ). Show also that I(Xn : Xn+m ) is non-increasing in m, for m = 0, 1, 2, . . 36 Essentials of Information Theory Solution By the Markov property, Xn−1 and Xn+1 are conditionally independent, given Xn . Hence, h(Xn−1 , Xn+1 |Xn ) = h(Xn+1 |Xn ) + h(Xn−1 |Xn ) and I(Xn−1 : Xn+1 |Xn ) = 0.

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