T Fredholms integral equation: Fredholms integral equation: = - - PowerPoint PPT Presentation

SLIDE 1

1

• G. Ahmadi

ME 639-Turbulence

• G. Ahmadi

ME 639-Turbulence

Outline Outline

• Expansion of a function

Expansion of a function

• Orthonormal

Orthonormal set set

• Expansion of a random function

Expansion of a random function

• K

K-

• L Expansion for periodic and non

L Expansion for periodic and non-

• periodic functions

periodic functions

• Response of linear system

Response of linear system

• K

K-

• L expansion for Brownian motion

L expansion for Brownian motion

• G. Ahmadi

ME 639-Turbulence

Let Let ϕ ϕn

n(t) be an

(t) be an orthonormal

• rthonormal set

set

( ) ( )

∑

ϕ = t c t X

n n

( ) ( )

∫

ϕ =

T n n

dt t t X c

( ) ( )

nm T * m n

dt t t δ = ϕ ϕ

∫

• G. Ahmadi

ME 639-Turbulence

In the expansion, the coefficients cn become uncorrelated (orthogonal) random variables if and only if ϕn(t) are the eigen-functions of the following Fredholm’s integral equation: In the expansion, the coefficients In the expansion, the coefficients c

cn

n

become uncorrelated (orthogonal) become uncorrelated (orthogonal) random variables if and only if random variables if and only if ϕ

ϕn

n(t)

(t) are

are the the eigen eigen-

• functions of the following

functions of the following Fredholm’s Fredholm’s integral equation: integral equation:

( ) ( ) ( )

1 n n T 2 n 2 1 xx

t t t , t R ϕ λ = ϕ

∫

SLIDE 2

2

• G. Ahmadi

ME 639-Turbulence

K K-

• L Expansion converges in mean

L Expansion converges in mean-

• square sense:

square sense:

{ }

n 2 n

c E λ =

{ }

{ } nm

2 n * m n

c E c c E δ =

( ) ( )

t c t X E

2 n n n

= ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ϕ −∑

• G. Ahmadi

ME 639-Turbulence

( ) ∑

ϕ λ =

n 2 n n xx

t , t R

( ) ∑

ϕ ϕ λ =

n 2 * n 1 n n 2 1 xx

) t ( ) t ( t , t R

Autocorrelation Autocorrelation

• G. Ahmadi

ME 639-Turbulence

Stationary and Periodic Processes Stationary and Periodic Processes

( )

2 1 xx xx

t t R R − =

( )

t in n

e T 1 t

ω

= ϕ T 2 π = ω

( ) ∑

+∞ ∞ − ω

=

t in n

• e

T c t x

{ }

n 2 n

c E λ =

Stationary Stationary Periodic Periodic

• G. Ahmadi

ME 639-Turbulence

( )

( )

∑

+∞ ∞ − − ω

λ =

2 1

t t in n 2 1 xx

e T 1 t , t R

( ) ( )

∑

+∞ ∞ −

ω − ω δ λ = ω

n xx

n T 1 S

( )

{ }

∑

+∞ ∞ −

λ =

n 2

T 1 t X E Correlation and Spectrum Correlation and Spectrum

Correlation Correlation Spectrum Spectrum

SLIDE 3

3

• G. Ahmadi

ME 639-Turbulence

Expansion Expansion n( n(ω ω) = ) = White Noise White Noise

Stationary Non Stationary Non-

• Periodic Processes

Periodic Processes ( ) ( ) ( )

∫

+∞ ∞ − ω

ω ω ω = d S n e t X

t i

( ) ( ) { } ( )

2 1 2 1 n

n E ω − ω δ = ω ω

Correlation Correlation

( )

( ) ( )

∫

+∞ ∞ − − ω −

ω ω = − d s e t t R

1 2 t

t i 2 1 xx

• G. Ahmadi

ME 639-Turbulence

Linear System Linear System n( n(ω ω) = ) = White Noise White Noise

Response of a Linear System Response of a Linear System

( ) ( )

t n t X L t =

( ) ( )

2 1 2 1 nn

t t S 2 t , t R − δ π =

( ) ( ) ( )

∫

τ τ τ − =

t

d n t h t X

( ) ( )

t t h L t δ =

Response Response Impulse Response Impulse Response

• G. Ahmadi

ME 639-Turbulence

( ) ( ) ( ) ( ) { }

∫

τ τ τ − =

t 2 t 2 xx t

d t X n E t h L t , t R L

( ) ( ) ( ) { } ( )

t t h S 2 t X t n E t , t R L

2 2 2 xx t

− π = =

( ) ( ) ( )

t dt t t , t R

T

• 2

2 2 xx

λϕ = ϕ

∫

( ) ( ) ( )

t L dt t t t h S 2

t T 2 2 2

φ λ = ϕ − π

∫

• G. Ahmadi

ME 639-Turbulence

( ) ( ) ( ) ( )

t S 2 dt t t t S 2 t L L

T 2 2 2 t t

ϕ π = φ − δ π = φ λ

∫

−

( )( )

i

= ϕ

( )( )

| t L

T t i t

= ϕ

=

1 N ,..., 1 , i − =

SLIDE 4

4

• G. Ahmadi

ME 639-Turbulence

• G. Ahmadi

ME 639-Turbulence

Concluding Remarks Concluding Remarks

• Expansion of a function

Expansion of a function

• Orthonormal

Orthonormal set set

• Expansion of a random function

Expansion of a random function

• K

K-

• L Expansion for periodic and non

L Expansion for periodic and non-

• periodic functions

periodic functions

• Response of linear system

Response of linear system

• K

K-

• L expansion for Brownian motion

L expansion for Brownian motion

• G. Ahmadi

ME 639-Turbulence