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Unformatted text preview: Stochastic Process 11/17/2006 Lecture 8 Random Sequence (1) NCTUEE Summary In this lecture, I will discuss: • Random Sequence • Stationarity • Widesense Stationary Random Sequence Notation We will use the following notation rules, unless otherwise noted, to represent symbols during this course. • Boldface upper case letter to represent MATRIX • Boldface lower case letter to represent vector • Superscript ( · ) T and ( · ) H to denote transpose and hermitian (conjugate transpose), respectively • Upper case italic letter to represent RANDOM VARIABLE 81 1 Random Sequence (1) In plain words, we can view a random sequence as follows: → A mathematical model of a probabilistic experiment that evolves in time → A random sequence can be considered as an evolution in time of random variables * The outcomes constitute a sequence of numerical values * The outcomes are measured in countable time instants, e.g. the time instants in the set T = { , 1 , 2 , ···} or T = {··· , 1 , , 1 , 2 , ···} . (2) For example, a random sequence can be used to model → the sequence of daily prices of a stock → the sequence of hourly traffic loads at a node of a network → the sequence of radar measurement of the position of an airplane → the sequence of failure times of a machine → the sequence of received and periodically sampled signal in a com munication link (3) Something of particular interests: → We tend to focus on the dependencies in the sequence. For example, how do future prices of a stock depend on past values? → We are often interested in longterm averages , involving the entire sequence of generated values. For example, what is the fraction of time that a machine is idle? → We sometimes wish to characterize the likelihood or frequency of certain boundary events. For example, what is the probability that within a given hour all circuits of some telephone system become simultaneously busy, or what is the frequency with which some buffer in a computer network overflows with data? 82 ] [ X n ] 1 [ X + n ] 1 [ X n n θ 1 n θ 1 + n θ (i) (ii) (iii) Figure 1 : The (i) filtering, (ii) smoothing, and (iii) prediction operations for timevarying unknown parameters θ n embedded in the random sequence X [ n ] for all n ....
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 Winter '10
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