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

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 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi ϕ n Let ϕ Let n (t) be an (t) be an orthonormal orthonormal set set In the expansion, the coefficients c n In the expansion, the coefficients In the expansion, the coefficients c c n n become uncorrelated (orthogonal) become uncorrelated (orthogonal) become uncorrelated (orthogonal) ( ) ( ) ∑ = ϕ X t c t random variables if and only if ϕ n (t) are random variables if and only if ϕ ϕ n (t) are are random variables if and only if n (t) n n the eigen-functions of the following the eigen eigen- -functions of the following functions of the following the ( ) ( ) Fredholm’s integral equation: ∫ T Fredholm’s integral equation: Fredholm’s integral equation: = ϕ c X t t dt n n 0 ( ) ( ) ( ) T ∫ ϕ = λ ϕ R t , t t t ( ) ( ) T ∫ ϕ ϕ = δ xx 1 2 n 2 n n 1 * t t dt 0 n m nm 0 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 1

{ } { } nm { } Autocorrelation Autocorrelation 2 = 2 δ = λ * E c c E c E c n m n n n ) ∑ ( = λ ϕ ϕ * R t , t ( t ) ( t ) xx 1 2 n n 1 n 2 K- -L Expansion converges in mean L Expansion converges in mean- -square sense: square sense: K n ) ∑ ⎧ ⎫ ( 2 ⎡ ⎤ ⎪ ⎪ = λ ϕ 2 ( ) ( ) − ∑ ϕ = R t , t ⎨ ⎬ E X t c t 0 ⎢ ⎥ n n xx n n ⎪ ⎣ ⎦ ⎪ ⎩ ⎭ n n ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi Correlation and Spectrum Stationary and Periodic Processes Correlation and Spectrum Stationary and Periodic Processes ( ) = − +∞ R R t t ( ) 1 ∑ ( ) ω − = λ Stationary Stationary in t t Correlation R t , t e Correlation 0 1 2 xx xx 1 2 xx 1 2 n T − ∞ π 1 ( ) 2 ϕ = ω +∞ ω = in t ( ) 1 ( ) t e ∑ Periodic 0 Periodic ω = λ δ ω − ω Spectrum Spectrum n S n 0 T T xx n 0 T − ∞ { } +∞ ( ) ∑ c 2 = λ { } ω +∞ = in t 1 n E c ( ) ∑ x t e o = λ 2 E X t n n T n − ∞ T − ∞ ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 2

Response of a Linear System Stationary Non- -Periodic Processes Periodic Processes Response of a Linear System Stationary Non ( ) ( ) = +∞ L t X t n t ( ) ( ) ( ) ∫ = ω ω ω ω i t Linear System Linear System Expansion X t e n S d Expansion − ∞ ( ) ( ) = π δ − ω ) = n( ω R t , t 2 S t t n( ) = White Noise White Noise nn 1 2 0 1 2 { ( ) ( ) } ( ) n( ω ω ) = ω ω = δ ω − ω E n 1 n n( ) = White Noise White Noise 2 1 2 ( ) ( ) ( ) ∫ t = − τ τ τ Response Response X t h t n d 0 ) ( ) ( ) +∞ ( ∫ − ω − Correlation − = ω ω Correlation i t 2 t ( ) ( ) R t t e s d 1 = δ Impulse Response L t h t t xx 1 2 Impulse Response − ∞ ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi ( ) ( ) { ( ) ( ) } ( ) ( ) ( ) t ∫ T = − τ τ τ ∫ λ φ = π δ − φ L R t , t L h t E n X t d L L t 2 S t t t dt t xx 2 t 2 0 − t t 0 2 2 2 0 ( ) { ( ) ( ) } ( ) ( ) = = π − = π ϕ L R t , t E n t X t 2 S h t t 2 S t t xx 2 2 0 2 0 ( ) ( ) ( ) ∫ T ϕ = λϕ ( ) ( ) R t , t t dt t ϕ = i xx 2 2 2 0 0 o = − i 0 , 1 ,..., N 1 ( ) ( ) ( ) ( ) ( ) T ∫ π − ϕ = λ φ ϕ = i 2 S h t t t dt L t L t | 0 = 0 2 2 2 t t t T 0 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 3

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