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Unformatted text preview: Chapter 9 Random Processes ENCS6161  Probability and Stochastic Processes Concordia University Definition of a Random Process Assume the we have a random experiment with outcomes w belonging to the sample set S . To each w ∈ S , we assign a time function X ( t,w ) , t ∈ I , where I is a time index set: discrete or continuous. X ( t,w ) is called a random process. If w is fixed, X ( t,w ) is a deterministic time function, and is called a realization, a sample path, or a sample function of the random process. If t = t is fixed, X ( t ,w ) as a function of w , is a random variable. A random process is also called a stochastic process. ENCS6161 – p.1/47 Definition of a Random Process Example : A random experment has two outcomes w ∈ { , 1 } . If we assign: X ( t, 0) = A cos t X ( t, 1) = A sin t where A is a constant. Then X ( t,w ) is a random process. Usually we drop w and write the random process as X ( t ) . ENCS6161 – p.2/47 Specifying a Random Process Joint distribution of time samples Let X 1 , ··· ,X n be the samples of X ( t,w ) obtained at t 1 , ··· ,t n , i.e. X i = X ( t i ,w ) , then we can use the joint CDF F X 1 ··· X n ( x 1 , ··· ,x n ) = P [ X 1 ≤ x 1 , ··· ,X n ≤ x n ] or the joint pdf f X 1 ··· X n ( x 1 , ··· ,x n ) to describe a random process partially. Mean function: m X ( t ) = E [ X ( t )] = i ∞∞ xf X ( t ) ( x ) dx Autocorrelation function R X ( t 1 ,t 2 ) = E [ X ( t 1 ) X ( t 2 )] = i ∞∞ i ∞∞ xyf X ( t 1 ) X ( t 2 ) ( x,y ) dxdy ENCS6161 – p.3/47 Specifying a Random Process Autocovariance function C X ( t 1 ,t 2 ) = E [( X ( t 1 )m X ( t 1 ))( X ( t 2 )m X ( t 2 ))] = R X ( t 1 ,t 2 )m X ( t 1 ) m X ( t 2 ) a special case: C X ( t,t ) = E [( X ( t )m X ( t )) 2 ] = V ar [ X ( t )] The correlation coefficient ρ X ( t 1 ,t 2 ) = C X ( t 1 ,t 2 ) r C X ( t 1 ,t 1 ) C X ( t 2 ,t 2 ) Mean and autocorrelation functions provide a partial description of a random process. Only in certain cases (Gaussian), they can provide a fully description. ENCS6161 – p.4/47 Specifying a Random Process Example : X ( t ) = A cos(2 πt ) , where A is a random variable. m X ( t ) = E [ A cos(2 πt )] = E [ A ] cos(2 πt ) R X ( t 1 ,t 2 ) = E [ A cos(2 πt 1 ) · A cos(2 πt 2 )] = E [ A 2 ] cos(2 πt 1 ) cos(2 πt 2 ) C X ( t 1 ,t 2 ) = R X ( t 1 ,t 2 )m X ( t 1 ) m X ( t 2 ) = ( E [ A 2 ]E [ A ] 2 ) cos(2 πt 1 ) cos(2 πt 2 ) = V ar ( A ) cos(2 πt 1 ) cos(2 πt 2 ) ENCS6161 – p.5/47 Specifying a Random Process Example : X ( t ) = A cos( wt + Θ) , where Θ is uniform in [0 , 2 π ] , A and w are constants. m X ( t ) = E [ A cos( wt + Θ)] = 1 2 π i 2 π A cos( wt + θ ) dθ = 0 C X ( t 1 ,t 2 ) = R X ( t 1 ,t 2 ) = A 2 E [cos( wt 1 + Θ) cos( wt 2 + Θ)] = A 2 2 π i 2 π cos w ( t 1t 2 ) + cos[ w ( t 1 + t 2 ) + θ ] 2 dθ = A 2 2 cos w ( t 1t 2 ) ENCS6161 – p.6/47 Gaussian Random Processes A random process X ( t ) is a Gaussian random process if for any n , the samples taken at t 1 ,t 2 , ··· ,t n are jointly Gaussian, i.e. if X 1 = X ( t 1 ) , ··· ,X n = X ( t n ) then f X 1 ··· X n ( x 1 , ··· ,x n ) = e1 2 ( xm ) T K1 ( xm ) (2 π ) n/ 2  K  1 / 2 where m = [ m X ( t 1 ) , ··· ,m X ( t n )] T and K = C X ( t 1 ,t 1 ) ··· C X ( t 1 ,t n ) ··· ··· ··· C X ( t n ,t 1 ) ··· C X ( t n ,t n ) ENCS6161 – p.7/47 Multiple Random Processes To specify joint random processes X ( t ) and Y ( t ) , we need to have the pdf of all samples of X ( t ) and Y ( t ) such as X ( t 1 ) , ··· ,X ( t i ) ,Y ( ´ t 1 ) , ··· ,Y ( ´ t j ) for all i and j and all choices of...
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This note was uploaded on 01/15/2011 for the course ECE 6161 taught by Professor Khkjk during the Winter '10 term at Concordia Canada.
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