FUNDAMENTALS OF INFORMATION THEORY in .NET

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FUNDAMENTALS OF INFORMATION THEORY
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In this section, we give an overview on the background and mathematical concepts in information theory in order to establish Shannon s channel coding theorem. Concepts of ergodic and outage capacities will be followed. Finally, we give several examples of channel capacity in various channels. This will form the basis for the remaining chapters in the book.
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FUNDAMENTALS OF INFORMATION THEORY
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Entropy and Mutual Information
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The rst important concept in information theory is entropy, which is a measure of uncertainty in a random variable. De nition 1.2 (Entropy of Discrete Random Variable) The entropy of a discrete random variable X with probability mass function p(X) is given by H ( X ) = - p( x) log 2 ( p( x)) = -e [log 2 ( p( X ))]
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(1.51)
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where the expectation is taken over X. De nition 1.3 (Entropy of Continuous Random Variable) On the other hand, the entropy of a continuous random variable X with probability density function f(X) is given by H ( X ) = - f ( x) log 2 ( f ( x))dx
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(1.52)
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For simplicity, we shall assume discrete random variable unless otherwise speci ed. De nition 1.4 (Joint Entropy) The joint entropy of two random variables X1, X2 is de ned as H ( X 1 , X 2 ) = - p( x1 , x2 ) log 2 ( p( x1 , x2 )) = -e [log 2 ( p( X 1 , X 2 ))] (1.53)
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where the expectation is taken over (X1, X2). De nition 1.5 (Conditional Entropy) The conditional entropy of a random variable X2 given X1 is de ned as H ( X 2 X 1 ) = - p( x1 , x2 ) log 2 ( p( x2 x1 )) = -e [log 2 ( p( X 2 X 1 ))]
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(1.54)
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where the expectation is taken over (X1, X2). After introducing the de nitions of entropy, let s look at various properties of entropy. They are summarized as lemmas below. Please refer to the text by Cover and Thomas [30] for the proof. The rst lemma gives a lower bound on entropy. Lemma 1.2 (Lower Bound of Entropy) H (X ) 0 Equality holds if and only if there exists x0 X such that p(x0) = 1. (1.55)
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BASIC CONCEPTS IN WIRELESS COMMUNICATIONS
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Proof Directly obtained from Equation (1.51). The following lemma gives an upper bound on entropy for discrete and continuous random variable X. Lemma 1.3 (Upper Bound of Entropy) If X is a discrete random variable, then H ( X ) log 2 ( X ) (1.56)
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Equality holds if and only if p(X) = 1/|X|. On the other hand, if X is a continuous random variable, then H (X ) 1 2 log 2 (2pes X ) 2 (1.57)
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2 where sX = e[|X|2] and equality holds if and only if X is Gaussian distributed 2 with an arbitrary mean m and variance sX .
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From the lemmas presented-above, entropy can be interpreted as a measure of information because H(X) = 0 if there is no uncertainty about X. On the other hand, H(X) is maximized if X is equiprobable or X is Gaussian distributed. The chain rule of entropy is given by the following lemma. Lemma 1.4 (Chain Rule of Entropy) H (X1 , X 2 ) = H (X1 ) + H (X 2 X1 ) (1.58)
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On the other hand, as summarized in the lemma below, conditioning reduces entropy. Lemma 1.5 (Conditioning Reduces Entropy) H (X Y ) H (X ) (1.59)
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Equality holds if and only if p(XY) = p(X)p(Y) (X and Y are independent). Lemma 1.6 (Concavity of Entropy) H(X) is a concave function of p(X). Lemma 1.7 If X and Y are independent, then H(X + Y) H(X). Lemma 1.8 (Fano s Inequality) Given two random variables X and Y, let X = g(Y) be an estimate of X given Y. De ne the probability of error as