c c X X t t t t dt dt n n n n 0 0 - - PowerPoint PPT Presentation

SLIDE 1

1

• G. Ahmadi

ME 639-Turbulence

• G. Ahmadi

ME 639-Turbulence

Outline

• Expansion of a function
• Orthonormal set
• Expansion of a random function
• K-L Expansion for periodic and non-

periodic functions

• Response of linear system
• K-L expansion for Brownian motion
• G. Ahmadi

ME 639-Turbulence

Let n(t) be an orthonormal set

   



  t c t X

n n

   



  t c t X

n n

   



 

T n n

dt t t X c

   



 

T n n

dt t t X c

   

nm T * m n

dt t t    



   

nm T * m n

dt t t    



• G. Ahmadi

ME 639-Turbulence

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

     

1 n n T 2 n 2 1 xx

t t t , t R    



     

1 n n T 2 n 2 1 xx

t t t , t R    



SLIDE 2

2

• G. Ahmadi

ME 639-Turbulence

K-L Expansion converges in mean-square sense:

 

n 2 n

c E  

 

n 2 n

c E  

 

  nm

2 n * m n

c E c c E  

 

  nm

2 n * m n

c E c c E  

   

t c t X E

2 n n n

                  

   

t c t X E

2 n n n

                  

• G. Ahmadi

ME 639-Turbulence

  

  

n 2 n n xx

t , t R 

 

  

n 2 n n xx

t , t R

  

   

n 2 * n 1 n n 2 1 xx

) t ( ) t ( t , t R 

 

   

n 2 * n 1 n n 2 1 xx

) t ( ) t ( t , t R

Autocorrelation

• G. Ahmadi

ME 639-Turbulence

Stationary and Periodic Processes

 

2 1 xx xx

t t R R  

 

2 1 xx xx

t t R R  

 

t in n

e T 1 t



   

t in n

e T 1 t



  T 2    T 2   

  

   



t in n

• e

T c t x  

   



t in n

• e

T c t x

 

n 2 n

c E  

 

n 2 n

c E  

Stationary Periodic

• G. Ahmadi

ME 639-Turbulence

 

 



    

 

2 1

t t in n 2 1 xx

e T 1 t , t R 



 



    

 

2 1

t t in n 2 1 xx

e T 1 t , t R

   



  

      

n xx

n T 1 S  

 



  

      

n xx

n T 1 S

 

 



  

 

n 2

T 1 t X E

 

 



  

 

n 2

T 1 t X E Correlation and Spectrum

Correlation Spectrum

SLIDE 3

3

• G. Ahmadi

ME 639-Turbulence

Expansion n() = White Noise

Stationary Non-Periodic Processes      



   

    d S n e t X

t i

     



   

    d S n e t X

t i

       

2 1 2 1 n

n E       

       

2 1 2 1 n

n E       

Correlation

 

   



     

    d s e t t R

1 2 t

t i 2 1 xx



   



     

    d s e t t R

1 2 t

t i 2 1 xx

• G. Ahmadi

ME 639-Turbulence

Linear System n() = White Noise

Response of a Linear System

   

t n t X L t 

   

2 1 2 1 nn

t t S 2 t , t R    

     



    

t

d n t h t X

   

t t h L t  

Response Impulse Response

• G. Ahmadi

ME 639-Turbulence

         



    

t 2 t 2 xx t

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

         



    

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

   

         

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

2 2 2 xx t

   

     

t dt t t , t R

T

• 2

2 2 xx

  



     

t dt t t , t R

T

• 2

2 2 xx

  



     

t L dt t t t h S 2

t T 2 2 2

     



     

t L dt t t t h S 2

t T 2 2 2

     



• G. Ahmadi

ME 639-Turbulence

       

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

T 2 2 2 t t

         





       

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

T 2 2 2 t t

         





  

i

   

i

 

  

| t L

T t i t

 



  

| t L

T t i t

 



1 N ,..., 1 , i   1 N ,..., 1 , i  

SLIDE 4

4

• G. Ahmadi

ME 639-Turbulence

• G. Ahmadi

ME 639-Turbulence

Concluding Remarks

• Expansion of a function
• Orthonormal set
• Expansion of a random function
• K-L Expansion for periodic and non-

periodic functions

• Response of linear system
• K-L expansion for Brownian motion
• G. Ahmadi

ME 639-Turbulence