handout_week10_b[1]

# handout_week10_b[1] - Stochastic Signals and Systems Random...

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Stochastic Signals and Systems Random Processes Virginia Tech Fall 2008 Mean and Correlation Functions As in the case of rvs, the moment functions play an important role in practical applications. These functions, however, only partially describe a random process. The mean function m X ( t ) of a random process X ( t ) is defined by m X ( t ) = E [ X ( t )] = Z -∞ xf X ( t ) ( x ) dx , where f X ( t ) ( x ) is the pdf of X ( t ) . In general, m X ( t ) is a function of time. The variance function of X ( t ) is defined by σ 2 X ( t ) = E h ( X ( t ) - m X ( t )) 2 i = Z -∞ ( x - m X ( t )) 2 f X ( t ) ( x ) dx

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Mean and Correlation Functions The autocorrelation function R X ( t 1 , t 2 ) of a random process X ( t ) is defined as R X ( t 1 , t 2 ) = E [ X ( t 1 ) X ( t 2 )] = Z -∞ Z -∞ xyf X ( t 1 ) , X ( t 2 ) ( x , y ) dxdy , where f X ( t 1 ) , X ( t 2 ) ( x , y ) is the second order pdf of X ( t ) . The autocovariance function C X ( t 1 , t 2 ) of a random process X ( t ) is defined as 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 ) The correlation coefficient function ρ X ( t 1 , t 2 ) of X ( t ) is
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