lectr19 - 19 Series Representation of Stochastic Processes...

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1 19. Series Representation of Stochastic Processes Given information about a stochastic process X ( t ) in can this continuous information be represented in terms of a countable set of random variables whose relative importance decrease under some arrangement? To appreciate this question it is best to start with the notion of a Mean-Square periodic process . A stochastic process X ( t ) is said to be mean square (M.S) periodic, if for some T > 0 i.e Suppose X ( t ) is a W.S.S process. Then Proof : suppose X ( t ) is M.S. periodic. Then , 0 T t (19-1) . all for 0 ] ) ( ) ( [ 2 t t X T t X E = + ( ) ( ) with 1 for all . X t X t T probability t = + ) ( PILLAI ( ) is mean-square perodic ( ) is periodic in the ordinary sense, where X t R τ * ( ) [ ( ) ( )] R E X t X t T τ = +
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2 But from Schwarz’ inequality Thus the left side equals or i.e., i.e., X ( t ) is mean square periodic. . period with periodic is ) ( T R τ 2 2 2 * 1 2 2 1 2 2 0 [ ( ){ ( ) ( )} ] [ ( ) ] [ ( ) ( ) ] E X t X t T X t E X t E X t T X t + + ±²²²²³²²²²´ * * 1 2 1 2 2 1 2 1 [ ( ) ( )] [ ( ) ( )] ( ) ( ) E X t X t T E X t X t R t t T R t t + = + = Then periodic. is ) ( Suppose ) ( τ R 0 ) ( ) ( ) 0 ( 2 ] | ) ( ) ( [| * 2 = = + τ τ τ R R R t X t X E ( ) ( ) for any R T R τ τ τ + = . 0 ] ) ( ) ( [ 2 = + t X T t X E (19-2) PILLAI (19-3) * 1 2 2 [ ( ){ ( ) ( )} ] 0 E X t X t T X t + =
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3 Thus if X ( t ) is mean square periodic, then is periodic and let represent its Fourier series expansion. Here In a similar manner define Notice that are random variables, and 0 0 1 ( ) . T jn n R e d T ω τ γ τ τ = 0 0 1 ( ) T jk t k c X t e dt T ω = +∞ −∞ = k c k , (19-5) PILLAI (19-6) ) ( τ R 0 0 2 ( ) , jn n R e T ω τ π τ γ ω +∞ −∞ = = (19-4) 0 1 0 2 0 1 0 2 0 2 1 0 1 * * 1 1 2 2 2 0 0 2 1 1 2 2 0 0 ( ) ( ) 2 1 2 1 1 0 0 1 [ ] [ ( ) ( ) ] 1 ( ) 1 1 [ ( ) ( )] m T T jk t jm t k m T T jk t jm t T T jm t t j m k t E c c E X t e dt X t e dt T R t t e e dt dt T R t t e d t t e dt T T τ ω ω ω ω ω ω τ τ γ = = = ∫ ∫ µ¶· ±³´ ±³´ ±²²²²²²³²²²²²²´
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4 i.e., form a sequence of uncorrelated random variables, and, further, consider the partial sum We shall show that in the mean square sense as i.e., Proof: But 0 1 , 1 ( ) * 1 0 0, [ ] { } 0 . m k T m j m k t k m m T k m E c c e dt k m ω δ γ γ > = = = ±²²²³²²²´ (19-7) +∞ = −∞ = n n n c } { 0 ( ) . N jk t N k K N X t c e ω =− = ¸ ) ( ) ( ~ t X t X N = 2 2 * 2 [ ( ) ( ) ] [ ( ) ] 2Re[ ( ( ) ( )] [ ( ) ]. N N N E X t X t E X t E X t X t E X t = + ¸ ¸ ¸ (19-8) . N . as 0 ] ) ( ~ ) ( [ 2 N t X t X E N (19-9) (19-10) PILLAI
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5 0 0 0 2 * * ( ) * 0 ( ) 0 [ ( ) ] (0) , [ ( ) ( )] [ ( )] 1 [ ( ) ( ) ] 1 [ ( ) ( )] . k k k N jk t N k k N N T jk t k N N N T jk t k k N k N E X t R E X t X t E c e X t E X e X t d T R t e d t T ω ω α ω α γ γ α α α α γ +∞ =−∞ =− =− =− =− = = = = = = ¸ ±²²²²²³²²²²²´ PILLAI (19-12) Similarly i.e., 0 0 2 ( ) ( ) * * 2 [ ( ) ] [ [ ] . [ ( ) ( ) ] 2( ) 0 as N j k m t j k m t N k m k m k k m k m k N N N k k k k N E X t E c c e E c c e E X t X t N ω ω γ γ γ =− +∞ =−∞ =− = = = = → ∞ ∑∑ ∑∑ ¸ ¸ (19-13) 0 ( ) , .
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