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Optical Networks: A Practical Perspective - Part 79

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Optical Networks: A Practical Perspective - Part 79. This book describes a revolution within a revolution, the opening up of the capacity of the now-familiar optical fiber to carry more messages, handle a wider variety of transmission types, and provide improved reliabilities and ease of use. In many places where fiber has been installed simply as a better form of copper, even the gigabit capacities that result have not proved adequate to keep up with the demand. The inborn human voracity for more and more bandwidth, plus the growing realization that there are other flexibilities to be had by imaginative use of the fiber, have led people. | 750 Multilayer Thin-Film Filters The three-cavity filter is described by the sequence G HL 5HLL HL nHLL HL nHLL HL 5HG. Again the values G 1.52 1.46 and riH 2.3 were used. _ References Kni76 z. Knittl. optics of Thin Films. John Wiley New York 1976. RWv93 s. Ramo J. R. Whinnery and T. van Duzer. Fields and Waves in Communication Electronics. John Wiley New York 1993. appendix Random Variables and Processes In many places in the book we use random variables and random processes to model noise polarization and network traffic. Understanding the statistical nature of these parameters is essential in predicting the performance of communication systems. Random Variables A random variable X is characterized by a probability distribution function Fx x P X x . The derivative of F x is the probability density function dFx x fxM . ax Note that ZOO fx x dx 1. -00 In many cases we will be interested in obtaining the expectation or ensemble average associated with this probability function. The expectation of a function g x is defined as g X fx x g x dx. 751 752 Random Variables and Processes The mean of X is defined to be X xfx x dx J-00 and the mean square second moment of X is E X2 x2fx x dx. J 00 The variance of X is defined as a2x E X2 - E X 2. In many cases we are interested in determining the statistical properties of two or more random variables that are not independent of each other. The joint probability distribution function of two random variables X and Y is defined as Fx Y x y P X x Y y . Sometimes we are given some information about one of the random variables and must estimate the distribution of the other. The conditional distribution of X given Y is denoted as Fx r x y P X x T y . An important relation between these distributions is given by Bayes theorem . Fx y x y Fx y x y . Fy y H.1.1 Gaussian Distribution A random variable X is said to follow a Gaussian distribution if its probability density function fx x e x lP _ 00 x 00. V2 cr Here z is the mean and a2 .