By Raymond W. Yeung
This e-book features a thorough dialogue of the classical subject matters in info concept including the 1st complete remedy of community coding, an issue first emerged less than info thought within the mid 1990's that has now subtle into coding thought, desktop networks, instant communications, complexity concept, cryptography, graph conception, and so forth. With a lot of examples, illustrations, and unique difficulties, this ebook is superb as a textbook or reference ebook for a senior or graduate point direction at the topic, in addition to a reference for researchers in similar fields.
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Extra info for Information Theory and Network Coding
The variational distance between p and q is defined as |p(x) − q(x)|. 68) x∈X 5 The variational distance is also referred to as the L1 distance in mathematics. 3 Continuity of Shannon’s Information Measures for Fixed Finite Alphabets 19 For a fixed finite alphabet X , let PX be the set of all distributions on X . 69) H(p) = − x∈Sp where Sp denotes the support of p and Sp ⊂ X . 71) or equivalently, lim H(p ) = H lim p p →p p →p = H(p), where the convergence p → p is in variational distance. , l(a) is a continuous extension of a log a.
17. For random variables X and Y , the mutual information between X and Y is defined as I(X; Y ) = p(x, y) log x,y 3 p(X, Y ) p(x, y) = E log . p(x)p(y) p(X)p(Y ) See Problem 5 at the end of the chapter. 53) 16 2 Information Measures Remark I(X; Y ) is symmetrical in X and Y . 18. , I(X; X) = H(X). Proof. This can be seen by considering p(X) p(X)2 = −E log p(X) = H(X). 56) The proposition is proved. Remark The entropy of X is sometimes called the self-information of X. 19. 8). The proof of this proposition is left as an exercise.
136) i=1 n ≤ i=1 where the inequality follows because we have proved in the last theorem that conditioning does not increase entropy. 137) for 1 ≤ i ≤ n. From the last theorem, this is equivalent to Xi being independent of X1 , X2 , · · · , Xi−1 for each i. 138) = p(p(x1 , x2 , · · · , xn−2 )p(xn−1 )p(xn ) .. , X1 , X2 , · · · , Xn are mutually independent. , X1 , X2 , · · · , Xn are mutually independent. 40. 147) with equality if and only if X → Y → Z forms a Markov chain. Proof. By the chain rule for mutual information, we have I(X; Y, Z) = I(X; Y ) + I(X; Z|Y ) ≥ I(X; Y ).
Information Theory and Network Coding by Raymond W. Yeung