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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 ME 639-Turbulence G. Ahmadi

  1. 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 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi Let  n (t) be an orthonormal set In the expansion, the coefficients c n become uncorrelated (orthogonal) random               X X t t c c t t variables if and only if  n (t) are the eigen- n n n n functions of the following Fredholm’s         integral equation: T T       c c X X t t t t dt dt n n n n 0 0             T T           R R t t , , t t t t t t         T T           xx xx 1 1 2 2 n n 2 2 n n n n 1 1 * * t t t t dt dt 0 0 n n m m nm nm 0 0 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 1

  2.        nm  nm     Autocorrelation 2 2 2 2         * * E E c c c c E E c c E E c c n n m m n n n n n n     R           * * R t t , , t t ( ( t t ) ) ( ( t t ) ) xx xx 1 1 2 2 n n n n 1 1 n n 2 2 K-L Expansion converges in mean-square sense: n n         R   2 2         2 2                       R t t , , t t     E E   X X t t c c t t   0 0 n n n n xx xx n n n n             n n n n ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi Correlation and Spectrum Stationary and Periodic Processes           1 1 R R R R t t t t R           Stationary         in in t t t t Correlation R t t , , t t e e 0 0 1 1 2 2 xx xx xx xx 1 1 2 2 xx xx 1 1 2 2 n n T T            1 1 2 2        in in t t     1 1 t t e e S         Periodic   0 0               Spectrum n n S n n 0 0 T T T T xx xx n n 0 0 T T           x       c c 2 2               in in t t 1 1 n n E E c c       x t t e e o o     2 2 E E X X t t n n n n T T n n     T T     ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 2

  3. Response of a Linear System Stationary Non-Periodic Processes      L t X t n t                           Linear System i i t t Expansion X X t t e e n n S S d d             n(  ) = White Noise R t , t 2 S t t nn 1 2 0 1 2                 n(  ) = White Noise               E E n n 1 n 1 n 2 2 1 1 2 2       t       Response X t h t n d 0       xx           Correlation               i i t t 2 t 2 t     R R t t t t e e s s d d 1 1   Impulse Response L t h t t xx 1 1 2 2     ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi                                   t t T T                           L L R R t t , , t t L L h h t t E E n n X X t t d d L L L L t t 2 2 S S t t t t t t dt dt t t xx xx 2 2 t t 2 2 0 0   t t t t 0 0 2 2 2 2 2 2 0 0                                       L L R R t t , , t t E E n n t t X X t t 2 2 S S h h t t t t 2 2 S S t t t t xx xx 2 2 2 2 0 0 2 2 0 0             T T                  R R t t , , t t t t dt dt t t    i i 0 0 0 0 xx xx 2 2 2 2 2 2 o o     i i 0 0 , , 1 1 ,..., ,..., N N 1 1                       T T                 i i 2 2 S S h h t t t t t t dt dt L L t t L L t t | | 0 0   0 0 2 2 2 2 2 2 t t t t t t T T 0 0 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 3

  4. 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 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 4

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