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Outline Outline � Independent Increment Processes � Independent Increment Processes � Cross � Cross- -Correlation & Cross Correlation & Cross- - Covariance Covariance � Strict � Strict- -Sense Stationary Process Sense Stationary Process � Jointly SSS Processes � Jointly SSS Processes � Wild � Wild- -Sense Stationary Process Sense Stationary Process ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Cross- -Correlation Correlation If increments X(t 2 ) - - X(t X(t 1 ) and X(t 4 ) - - X(t X(t 3 ) Cross If increments X(t 2 ) 1 ) and X(t 4 ) 3 ) ( ) { ( ) ( ) } ( ) of process X(t) are uncorrelated (or of process X(t) are uncorrelated (or = = R t , t E X t Y t R t , t 2 ≤ ≤ t independent) for any t 1 independent) for any t 1 < t < t 2 t 3 3 < t < t 4 4 , then , then XY 1 2 1 2 YX 2 1 X(t) is a process with uncorrelated (or ) is a process with uncorrelated (or X(t Cross- Cross -Covariance Covariance independent) increments. independent) increments. ( ) { [ ( ) ( ) ] ( ) [ ( ) ] } = − η − η C t , t E X t t Y t t XY 1 2 1 X 1 2 Y 2 Poisson, Weiner Poisson, Weiner are independent are independent ( ) ( ) ( ) = − η η R t , t t t increment increment XY 1 2 X 1 Y 2 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 1

( ) = R XY t , 2 t 0 Orthogonal Processes Orthogonal Processes Strict- -Sense Stationary (SSS) Process Sense Stationary (SSS) Process Strict 1 Statistics not affected by shift in time origin. Statistics not affected by shift in time origin. ( ) = C XY t , 2 t 0 Uncorrelated Processes Uncorrelated Processes ( ) ( ) 1 = + τ + τ f x ,..., x , t ,..., t f x ,..., x , t ,..., t 1 n 1 n 1 n 1 n Independent Processes – – group of random group of random Independent Processes First order density is First order density is variables X(t 1 ), …, X(t X(t n ) are independent of the variables X(t 1 ), …, n ) are independent of the independent of time independent of time group Y(t 1 ’), …, Y(t m ’): group Y(t 1 ’), …, Y(t m ’): ( ) ( ) ( ) ( ) = + τ = f x ; t f x ; t f x ; t f x ( ) ( ) ( ) = f x ,..., x , y ,..., y f x ,..., x f y ,..., y n m n m 1 1 1 1 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Similarly: Similarly: Jointly SSS Processes Jointly SSS Processes Second order density depends on time difference Second order density depends on time difference Joint statistics of X(t Joint statistics of X(t) & ) & Y(t Y(t) are the ) are the ( ) ( ) same as X(t + τ τ ), Y(t + ), Y(t + τ τ ) = − ) same as X(t + f x , x ; t , t f x , x ; t t 1 2 1 2 1 2 1 2 Statistics of Stationary processes Statistics of Stationary processes ( ) ( ) = − f x , y ; t , t f x , y , t t { } { ( ) } ( ) = η = = σ = 2 2 XY 1 2 XY 1 2 E X t const E X t const { ( ) ( ) } ( ) = − ( ) { ( ) ( ) } ( ) = = − E X t Y t R t t R t 1 , t E X t X t R t t 2 1 2 1 2 1 2 XY 1 2 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 2

Wide- -Sense Stationary (WSS) Process Sense Stationary (WSS) Process Wide Concluding Remarks Concluding Remarks � Cross � Cross- -Correlation & Cross Correlation & Cross- - If mean is independent of time and If mean is independent of time and autocorrelation depends on τ τ = t = t 1 – t t 2 autocorrelation depends on 1 – 2 Covariance Covariance { ( ) } { ( ) ( ) } ( ) = η = + τ = τ � Strict � E X t const E X t Y t R Strict- -Sense Stationary Process Sense Stationary Process XY � Jointly SSS Processes � Jointly SSS Processes Jointly WSS Processes X(t) & Y(t) Jointly WSS Processes X(t) & Y(t) � Wild � If both X(t If both X(t) & ) & Y(t Y(t) are ) are WSS, and cross WSS, and cross- - Wild- -Sense Stationary Process Sense Stationary Process correlation depends on time difference correlation depends on time difference { ( ) ( ) } ( ) + τ = τ E X t X t R ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 3