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Unformatted text preview: Stochastic Process 12/15/2006 Lecture 11 Advanced Topics in Random Processes NCTUEE Summary In this lecture, I will discuss: • MeanSquare Stochastic Integral • Mean Ergodic Property • Karhunen–Lo` eve Expansion 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 111 1 MeanSquare Stochastic Integral (1) Integration of a random process X ( t ) over a finite interval ( T 1 ,T 2 ) . — Difficulty arises since a random process is a 2variable function — Analogous to integration of a 1variable deterministic function of time, we first consider a Riemann sum I n , n X i =1 X ( t i )Δ t i , where ( t 1 ,t 2 , ··· ,t n ) are n points that partition ( T 1 ,T 2 ) into n intervals each with width Δ t i , i = 1 , ··· ,n. — Since X ( t i ) for fixed t i is a random variable. Clearly, I n , n = 1 , 2 , ··· is a sequence of random variables (random sequence) — If the meansquare convergence of I n exists , then the converged random variable I is called the meansquare integral of X ( t ) over the interval ( T 1 ,T 2 ) . Mathematically, we have lim n →∞ E £  I I n  2 / = 0 and denote I = Z T 2 T 1 X ( t ) dt. (2) We can check the existence using the Cauchy criterion. Specifically, the integral I (i.e. the convergence of I n ) exists if and only if lim m,n →∞ E [  I n I m  2 ] = 0 . 112 2 Mean Ergodic Property (1) Physical motivation Suppose we want to know the average temperature at Hsinchu on the day of 12/14 over the past 100 years. But we don’t have that many records at hand. How sure are we that we measure the average temp. of 12/14 in the year 2006, and say the average over the past 100 years is approximately the same? That is: does time average of only one realization tend to be the en semble average in some sense? (2) Mathematical definition A widesense stationary random process X ( t ) is ergodic in the mean if the timeaverage of X ( t ) defined by ˆ M ( T ) , 1 2 T Z T T X ( t ) dt converges in the meansquare sense to the ensemble average E [ X ( t )] = μ X . That is ˆ M ( T ) → μ X (m.s.) as T → ∞ ....
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This note was uploaded on 11/28/2010 for the course EE 301 taught by Professor Gfung during the Winter '10 term at National Chiao Tung University.
 Winter '10
 GFung

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