ch9 - Chapter 9 Random Processes ENCS6161 Probability and...

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Chapter 9 Random Processes ENCS6161 - Probability and Stochastic Processes Concordia University
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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 0 is fixed, X ( t 0 , w ) as a function of w , is a random variable. A random process is also called a stochastic process. ENCS6161 – p.1/47
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Definition of a Random Process Example : A random experment has two outcomes w ∈ { 0 , 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
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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 )] = integraldisplay -∞ xf X ( t ) ( x ) dx Autocorrelation function R X ( t 1 , t 2 ) = E [ X ( t 1 ) X ( t 2 )] = integraldisplay -∞ integraldisplay -∞ xyf X ( t 1 ) X ( t 2 ) ( x, y ) dxdy ENCS6161 – p.3/47
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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 ) radicalbig 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
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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
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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 integraldisplay 2 π 0 A cos( wt + θ ) = 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 integraldisplay 2 π 0 cos w ( t 1 - t 2 ) + cos[ w ( t 1 + t 2 ) + θ ] 2 = A 2 2 cos w ( t 1 - t 2 ) ENCS6161 – p.6/47
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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 ) = e - 1 2 ( x - m ) T K - 1 ( x - m ) (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
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Multiple Random Processes To specify joint random processes X ( t ) and Y ( t ) , we
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