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Unformatted text preview: Chap.11 Spectral Representation 11.1 Factorization and Innovations ¨ A real WSS RP ( ) t X minimumphase system ( ) L s L output with input e white noise process ( ) t i h signal modeling i ( ) t X regulare Minimumphase system ( ) L s L causal e impulse response ( ) t l finite energy (i.e., causal and stable)e inverse system 1 ( ) ( ) s L s Γ = causal e ( ) t γ finite energy ( ) L s is minimumphase e ( ) L s and ( ) s Γ are analytic in the right splane, i.e., all poles of ( ) L s and ( ) s Γ are in the left splane. A RP e regular if it is linearly equivalent with a whitenoise process ( ) t i in the sense ( ) ( ) ( ) t t d γ α α α = i X e ( ) ( ) R τ δ τ = ii ( ) ( ) ( ) t t d α α α = X i l { } 2 2 ( ) ( ) E t t dt L = < X l ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) R R τ τ ρ τ δ τ τ τ τ τ = = = XX ii l l l l ( ) ( ) d τ α α α = + l l τ e ( ) ( ) R R τ τ = XX XX ( ) ( ) ( ) S s L s L s = 2 ( )  ( )  S L ϖ ϖ = ( ) L s L innovations filter of ( ) t X h 1 ( ) ( ) s L s Γ = whitening filter of ( ) t X h ( ) t i h innovations of ( ) t X ( ) L s C S Given a positive even function ( ) S ϖ a minimum phase system ( ) L s s.t . 2  ( )  ( ) L j S ϖ ϖ = PaleyWiener Conditione 2 log ( )  1 S d ϖ ϖ ϖ < + e ( ) S ϖ ( ) S ϖ line e BL( e e predictable) in generalbS for special casee ( ) L s L rational function of s L ( ) S ϖ 2 ϖ rational functionbS 2 * ( )  ( )  ( ) ( ) ( ) ( ) S L j L j L j L j L j ϖ ϖ ϖ ϖ ϖ ϖ = = = ∴ ( ) ( ) S S ϖ ϖ = 2 2 ( ) ( ) ( ) A S B ϖ ϖ ϖ = 2 2 ( ) ( ) ( ) A s S s B s = j s s = j s (e * j s L * j s for complex j s ) e ( ) S s L poles e zerose Fig. 112(a)z cw* poles e zerosbS L ( ) ( ) ( ) ( ) ( ) N s N s S s D s D s = ( ) ( ) L s L s = e ( ) ( ) N s D s C S L poles e zerose Fig. 112(b)e ∴ ( ) ( )/ ( ) L s N s D s = Ex. 111 2 2 ( ) N S ϖ α ϖ = + α ( ) L s Solutione 2 2 ( ) ( )( ) N N S s s s s α α α = = + ∴ ( ) N L s s α = + Ex. 112 2 4 2 49 25 ( ) 10 9 S ϖ ϖ ϖ ϖ + = + + 2 4 2 2 2 49 25 (7 5 )(7 5 ) (7 5 )(7 5 ) ( ) 10 9 (1 )(9 ) (1 )(1 )(3 )(3 ) s s s s s S s s s s s s s s s + + = = = + + + ∴ 7 5 ( ) (1 )(3 ) s L s s s + = + + Ex. 113 4 25 ( ) ( 1) S ϖ ϖ = + 4 2 2 25 25 ( ) ( 1) ( 2 1)( 2 1) S s s s s s s = = + + + + 2 5 ( ) ( 2 1) L s s s = + + Discretetime Processes A real WSS discretetime process [ ] n X is regular if ( ) S z can be written as 1 ( ) ( ) ( ) S z L z L z = 2 ( )  ( ) j j S e L e ϖ ϖ = where ( ) L z is a minimumphase systeme [ ] [ ] [ ] k n k n k γ = = i X h [ ] [ ] R m m δ = ii [ ] [ ] [ ] k n k n k L = = X i l [ ] { } [ ] 2 2 k E n k L = = < X l [ ] n X is linearly equivalent with a whitenoise process [ ] n i ( ) j S e ϖ of a real WSS process [ ] n X can be factorized as 1 ( ) ( ) ( ) S z L z L z = if it satisfies PaleyWiener Condition ln ( )  j S e d π ϖ π ϖ < General solution of ( ) j S e ϖ...
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 Fall '08
 SinHorngChen

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