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 Xt R τ * ( ) [ ( ) ( )] RE 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 22 * 12 2 1 2 2 0 [() {( ) () } ] [()][( ) ()] EXt Xt T T +− ±²²²²³²²²²´ ** 2 1 2 1 [() ( ) ] [() () ] ( ) ( ) EXt X t T Rt t T t += += − Then periodic. is ) ( Suppose ) ( τ R 0 ) ( ) ( ) 0 ( 2 ] | ) ( ) ( [| * 2 = = τ τ R R R t X t X E ( ) ( ) for any RTR ττ τ ⇒+ = . 0 ] ) ( ) ( [ 2 = + t X T t X E (19-2) PILLAI (19-3) * 2 ) } ]0 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 Re d T ωτ γτ τ = 0 0 1 T jk t k cX t e d t T ω = +∞ −∞ = k c k , (19-5) PILLAI (19-6) ) ( τ R 0 0 2 ( ) , jn n T π τγ ω +∞ −∞ == (19-4) 01 02 02 1 0 1 ** 112 2 2 0 0 21 1 2 2 0 1 0 1 [] [ ( ) ( ) ] 1 11 [( ) ( ) ] m TT jk t jm t km jk t jm t jm t t j m k t Ecc E Xt e d t X t e d t T Rt t e e d td t T dt t e d t τ ωω ττ γ −− = =− =− − ∫∫ ±²³ ´µ¶ ´······µ······¶
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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 01 , 1 () * 1 0 0, [] { } 0 . mk T m jm k t km m T Ecc e d t ω δ γ γ −− >= == ±²²²³²²²´ (19-7) +∞ = −∞ = n n n c } { 0 . N jk t Nk KN Xt c e ω =− = µ ) ( ) ( ~ t X t X N = 2 2 * 2 [( ) ( ) ] [( ) ]2 R e [( ( ) ( ) ] ) ] . NN N EXt X t EX tX t EX t −= − + µµ µ (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 Nk kN N T jk t NN T jk t k EXt R EX tX t E ce X t EXe X t d T Rt e dt T ω ωα γ γ αα γ +∞ =−∞ =− −− == = = =− − = ∑∑ ± ²³³³³³´³³³³³µ PILLAI (19-12) Similarly i.e., 00 2 2 ) ][ [ ] . [ ( ) ( ) ] 2( ) 0 as N j k mt j k m k m k km k N N k kk N E X t E cce Ecc e N ωω γ γγ +∞ == = ⇒− = −→ ± ± (19-13) 0 ( ) , . jk t k k Xt t ω +∞ −∞< <+∞ (19-14) and
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6 Thus mean square periodic processes can be represented in the form of a series as in (19-14). The stochastic information is contained in the random variables Further these random variables are uncorrelated and their variances This follows by noticing that from (19-14) Thus if the power P of the stochastic process is finite, then the positive sequence converges, and hence This implies that the random variables in (19-14) are of relatively less importance as and a finite approximation of the series in (19-14) is indeed meaningful.
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This note was uploaded on 10/01/2009 for the course ECE 2521 taught by Professor Jacobs during the Fall '09 term at Pittsburgh.

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lectr19 - 19. Series Representation of Stochastic Processes...

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