[ ] ( ) ( ) t. t. = Y t T X t , , ( ) ( ) ( - PowerPoint PPT Presentation

Outline Outline � Memory � Memory- -less Systems less Systems � Derivative of Random Processes � Derivative of Random Processes � Mean and Autocorrelation � Mean and Autocorrelation � Random Linear Differential � Random Linear Differential Equations Equations � Evaluation of Mean and � Evaluation of Mean and autocorrelation of Response autocorrelation of Response ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Transformation of Stochastic Process Transformation of Stochastic Process System is deterministic if T operates on System is deterministic if T operates on [ ] ( ) ( ) t. t. ξ = ξ Y t T X t , , ( ) ( ) ( ) ( ) ξ = ξ Y t Y t ξ = ξ if then if then , , X t X t , , 1 2 1 2 System is stochastic if T operates on t System is stochastic if T operates on t ξ. and ξ. ( ) ( ) and ξ ξ Y t X t , , T responses to identical inputs differ. responses to identical inputs differ. ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 1

[ ] ( ) ( ) ( ) ( ) = + ε − dX X t X t Y t g X t ( ) ′ = = X t Time Derivative Time Derivative lim ε dt → 0 ε ( ) f x t ⎧ ⎫ dX d d ; Mean Mean { } ( ) ( ) ∑ ( ) = = η st Order Density = X j E ⎨ ⎬ E X t f y t = − 1 st Order Density x g 1 y 1 ; ⎩ dt ⎭ dt dt Y ′ g j j j ( ) ( ) ⎧ ⎫ Autocorrelation Autocorrelation dX t dX t ( ) { ( ) ( ) } { ( ) } +∞ ( ) ( ) ′ ′ R t t = E X t X t = E ∫ ⎨ 1 2 ⎬ 1 , = E Y t g x f x t dx ′ ′ X X Mean Mean ; 2 1 2 dt dt ⎩ ⎭ X 1 2 − ∞ ( ) ∂ 2 R t t ( ) , or or = XX Autocorrelation Autocorrelation R t t 1 2 , ′ ′ X X ∂ ∂ 1 2 t t 1 2 { ( ) ( ) } +∞ +∞ ( ) ( ) ( ) ∫ ∫ = E Y t Y t g x g x f x x t t dx dx ∂ , ; , ( ) ( ) = X Cross Correlation X & X’ R t t R t t Cross Correlation X & X’ 1 2 1 2 1 2 1 2 1 2 , , − ∞ − ∞ ′ X X XX 1 2 ∂ t 1 2 2 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi ( ) ( ) = − − R t t R t t n n d Y d 1 Y 1 , For X(t) stationary ( ) ( ) ( ) For X(t) stationary XX XX = + + + = 2 1 2 L Y t a a a Y t X t ... − t n n n 1 n − 0 dt dt 1 ( ) τ dR ( ) ( ) ( ) τ = − τ = t − − R XX t dY d n 1 Y ( ) 0 0 = = = = ′ Y X X 0 ... 0 τ d 1 2 n − dt dt 1 ( ) { ( ) } η = Mean of Y t E Y t Mean of Y Y ( ) τ d 2 R ( ) τ = − XX R ′ ′ X X Taking expected value of diff eqn Taking expected value of diff eqn and and I.C.’s I.C.’s τ d 2 ( ) { } ( ) ( ) d 2 R [ ] ( ) ( ) 0 η n − η η d d 1 ′ d n ( ) 0 0 = = − XX Mean Square E X t 2 R ( ) ( ) ( ) Mean Square η = Y = = Y = 0 η = + + η = η L t a Y a t t 0 ... 0 X ′ X ′ ... τ Y − d 2 dt dt n t Y n n Y X 1 dt 0 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 2

[ ] ( ) ( ) ( ) Cross Correlation = Write X(t 1 ), multiply Y(t 2 ), take expected value Cross Correlation X t L Y t X t Write X(t 1 ), multiply Y(t 2 ), take expected value t 1 2 2 2 of Y and X ( ) ( ) of Y and X = L R t t R t t ( ) ( ) , , = t XY YY L R t t R t t 1 2 1 2 , , 1 t XY XX 1 2 1 2 2 ( ) ∂ n R t t , ( ) ( ) ( ) + + = a YY a R t t R t t ∂ or 1 2 n or R t t ... , , , ( ) ( ) or or n ∂ YY XY + + = n 0 1 2 1 2 a XY a R t t R t t t 1 2 ... , , n XY XX ∂ n 0 1 2 1 2 t 1 2 ( ) ( ) ∂ ∂ n − R t 1 R t ( ) 0 , 0 , = YY = = YY = R t 2 2 0 , ... 0 Multiply ICs by X(t 1 Multiply ICs by X(t 1 ) & taking expected value: ) & taking expected value: YY ∂ ∂ − 2 t t n 1 1 1 ( ) ( ) − ∂ ∂ n R t 1 R t ( ) , 0 , 0 = = = = R t XY XY 1 1 , 0 ... 0 Note: Y(t Y(t) is non ) is non- -stationary stationary Note: XY ∂ − 1 t ∂ t n 1 2 2 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Concluding Remarks Concluding Remarks � � Memory- -less Systems less Systems Memory � � Derivative of a Random Process Derivative of a Random Process � � Statistics of Derivative Derivative Statistics of � � Random Linear Differential Random Linear Differential Equations Equations � � Response Mean and Response Mean and Autocorrelation Autocorrelation ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 3