By Raymond W. Yeung (auth.)

A First path in details idea is an updated advent to details idea. as well as the classical issues mentioned, it offers the 1st entire therapy of the speculation of I-Measure, community coding thought, Shannon and non-Shannon kind details inequalities, and a relation among entropy and workforce idea. ITIP, a software program package deal for proving details inequalities, can be incorporated. With quite a few examples, illustrations, and unique difficulties, this e-book is great as a textbook or reference ebook for a senior or graduate point direction at the topic, in addition to a reference for researchers in comparable fields.

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**Example text**

The entropy rate HXt exists, and it is equal to H'x . 54 that H'x always exists for a stationary source {Xd, in order to prove the theorem, we only have to prove that H x = H'x . 200) from (2. 201) The theorem is proved . 0 In this theorem , we have proved that the entropy rate of a random source {Xd exists under the fairly general assumption that {Xd is stationary. However, the entropy rate of a stationary source {Xk} may not carry any physical meaning unless {X k } is also ergodic . 4. 36 A FIRST COURSEIN INFORMATION THEORY PROBLEMS I.

94). 28 that this is the case if and only if p(x) = q(x ) for all x ESp . 94) holds with equality. Finally, since L x q(x) = 1 and q(x) 2: 0 for all x , p(x) = q(x) for all x E Sp implies q(x) = 0 for all x ~ Sp, and therefore p = q. The theorem is proved. 0 We now prove a very useful consequence of the divergence inequality called the log-sum inequality. 31 ( LOG- SUM INEQ UALITY ) For positi ve numbers a I, a2, ... and nonn egati ve numbers bi , b2 , . . 98) A FIRST COURSE IN INFORMATIONTHEORY 22 with the convention that log ~ = constant for all i.

Since an -t a as n -t 00, for every e > 0, there exists N(f) such that Ian - c] < f for all n > N(f) . L lai - al + E. (2. I98) The first term tends to 0 as n --+ 00 . Therefore, for any E > 0, by taking n to be sufficiently large, we can make Ibn - al < 2E. Hence bn --+ a as n --+ 00 , proving the lemma . 0 We now prove that H'x is an alternative definition/interpretation of the entropy rate of {Xd when {Xd is stationary. 56 For a stationary source {Xk} . the entropy rate HXt exists, and it is equal to H'x .