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

SLIDE 1

1

• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics

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

• G. Ahmadi

ME 529 - Stochastics

Transformation of Stochastic Process Transformation of Stochastic Process

( ) ( ) [ ]

ξ ξ , , t X T t Y =

( )

ξ , t Y

( )

ξ , t X

T

• G. Ahmadi

ME 529 - Stochastics

System is deterministic if T operates on System is deterministic if T operates on t. t. if then if then System is stochastic if T operates on t System is stochastic if T operates on t and and ξ. ξ. responses to identical inputs differ. responses to identical inputs differ.

( ) ( )

2 1

, , ξ ξ t X t X =

( ) ( )

2 1

, , ξ ξ t Y t Y =

SLIDE 2

2

• G. Ahmadi

ME 529 - Stochastics

1 1st

st Order Density

Order Density Mean Mean

( ) ( ) [ ]

t X g t Y =

( )

( )

∑

′ =

j j X Y

g t x f t y f ; ;

( )

y g x

j j 1 −

=

( ) { } ( ) ( )

∫

+∞ ∞ −

= dx t x f x g t Y E

X

;

( ) ( ) { } ( ) ( ) ( )

∫ ∫

+∞ ∞ − +∞ ∞ −

=

2 1 2 1 2 1 2 1 2 1

, ; , dx dx t t x x f x g x g t Y t Y E

X

Autocorrelation Autocorrelation

• G. Ahmadi

ME 529 - Stochastics

Autocorrelation Autocorrelation Mean Mean Cross Correlation X & X’ Cross Correlation X & X’

( ) ( ) ( ) ε ε t X t X dt dX t X

ε

− + = = ′

→0

lim { } ( )

t dt d X E dt d dt dX E η = = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧

( ) ( ) ( ) { } ( ) ( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = ′ ′ =

′ ′ 2 2 1 1 2 1 2 1,

dt t dX dt t dX E t X t X E t t R

X X

( ) ( )

2 1 2 1 2 2 1

, , t t t t R t t R

XX X X

∂ ∂ ∂ =

′ ′

( ) ( )

2 1 2 2 1

, , t t R t t t R

XX X X

∂ ∂ =

′

• r
• r

Time Derivative Time Derivative

• G. Ahmadi

ME 529 - Stochastics

For X(t) stationary For X(t) stationary Mean Square Mean Square ( ) ( )

2 1 2 1,

t t R t t R

XX XX

− =

( ) ( )

τ τ τ d dR R

XX X X

− =

′ 2 1

t t − = τ

( ) ( )

2 2

τ τ τ d R d R

XX X X

− =

′ ′

( ) [ ]

{ }

( ) ( )

2 2 2

τ d R d R t X E

XX X X

− = = ′

′ ′

• G. Ahmadi

ME 529 - Stochastics

Mean of Y Mean of Y

( ) ( ) ( )

t X t Y a dt Y d a dt Y d a t Y L

n n n n n n t

= + + + =

− − − 1 1 1

...

( ) ( ) ( )

...

1 1

= = = =

− − n n

dt Y d dt dY Y

( ) ( ) { }

t Y E t

Y

= η ( ) ( ) ( )

t t a dt d a t L

X Y n Y n n Y t

η η η η = + + = ...

( ) ( ) ( )

...

1 1

= = = =

− − n Y n Y Y

dt d dt d η η η

Taking expected value of diff Taking expected value of diff eqn eqn and and I.C.’s I.C.’s

SLIDE 3

3

• G. Ahmadi

ME 529 - Stochastics

Cross Correlation Cross Correlation

• f Y and X
• f Y and X

( ) ( ) ( )

[ ]

2 2 1

2

t X t Y L t X

t

=

( ) ( )

2 1 2 1

, ,

2

t t R t t R L

XX XY t

= ( ) ( ) ( )

2 1 2 1 2 2 1

, , ... , t t R t t R a t t t R a

XX XY n XY n n

= + + ∂ ∂

( ) ( ) ( )

, ... , ,

1 2 1 1 2 1 1

= ∂ ∂ = = ∂ ∂ =

− − n XY n XY XY

t t R t t R t R

• r
• r

Multiply ICs by X(t Multiply ICs by X(t1

1) & taking expected value:

) & taking expected value:

• G. Ahmadi

ME 529 - Stochastics

Write X(t Write X(t1

1), multiply Y(t

), multiply Y(t2

2), take expected value

), take expected value ( ) ( )

2 1 2 1

, ,

1

t t R t t R L

XY YY t

=

( ) ( ) ( )

2 1 2 1 1 2 1

, , ... , t t R t t R a t t t R a

XY YY n YY n n

= + + ∂ ∂

( ) ( ) ( )

, ... , ,

1 1 2 1 1 2 2

= ∂ ∂ = = ∂ ∂ =

− − n YY n YY YY

t t R t t R t R

• r
• r

Note: Note: Y(t Y(t) is non ) is non-

• stationary

stationary

• G. Ahmadi

ME 529 - Stochastics

Concluding Remarks Concluding Remarks

• Memory

Memory-

• less Systems

less Systems

• Derivative of a Random Process

Derivative of a Random Process

• Statistics of

Statistics of Derivative Derivative

• Random Linear Differential

Random Linear Differential Equations Equations

• Response Mean and

Response Mean and Autocorrelation Autocorrelation

• G. Ahmadi

ME 529 - Stochastics