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Unformatted text preview: Chapter 11 Spectral Representation Teacher: ø ò ª Office: 805 Tel: ext. 31822 Email: [email protected] 2 11.1 Factorization and Innovations x ¡ & ” “¢ real WSS RP ( ) t X minimumphase system ( ) L s L output with input e white noise process ( ) t i i signal modeling i h ( ) t X i i regulare Minimumphase system ( ) L s L causal e impulse response ( ) t finite energy (i.e., causal and stable)e inverse system 1 ( ) ( ) s L s Γ = causal e ( ) t γ finite energy 3 ( ) L s is minimumphase ⇒ ( ) 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. 4 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 { } 2 2 ( ) ( ) E t t dt ∞ = < ∞ ∫ X ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) R R τ τ ρ τ δ τ τ τ τ τ = = = XX ii ( ) ( ) d τ α α α ∞ = + ∫ τ ≥ e ( ) ( ) R R τ τ = XX XX 5 ( ) ( ) ( ) S s L s L s = 2 ( )  ( ) S L ϖ ϖ = ( ) L s L innovations filter of ( ) t X i 1 ( ) ( ) s L s Γ = whitening filter of ( ) t X i ( ) t i i innovations of ( ) t X 6 h& ( ) L s bA Given a positive even function ( ) S ϖ a minimumphase system ( ) L s s.t. 2  ( )  ( ) L j S ϖ ϖ = h PaleyWiener Conditione 2 log ( ) 1 S d ϖ ϖ ϖ ∞∞ < ∞ + ∫ e ( ) S ϖ ( ) S ϖ line e BL( ⇒ predictable) 7 h& in general for special casee e ( ) L s L rational function of sL ( ) S ϖ 2 ϖ rational functione e 2 * ( )  ( ) ( ) ( ) ( ) ( ) S L j L j L j L j L j ϖ ϖ ϖ ϖ ϖ ϖ = = =  ∴ ( ) ( ) S S ϖ ϖ =  e 2 2 ( ) ( ) ( ) A S B ϖ ϖ ϖ = ⇒ 2 2 ( ) ( ) ( ) A s S s B s = 8 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)e e e poles e zeros L ( ) ( ) ( ) ( ) ( ) N s N s S s D s D s = ( ) ( ) L s L s = e ( ) ( ) N s D s L poles e zerose e Fig. 112(b)e ∴ ( ) ( )/ ( ) L s N s D s = 9 10 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 = + 11 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 + = + + 12 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 = + + 13 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 14 [ ] [ ] [ ] k n k n k γ ∞ = =  ∑ i X i [ ] [ ] R m m δ = ii [ ] [ ] [ ] k n k n k ∞ = =  ∑ X i [ ] { } [ ] 2 2 k E n k ∞ = = < ∞ ∑ X 15 h& [ ] n X is linearly equivalent with a whitenoise process...
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This note was uploaded on 07/21/2009 for the course CM EM5102 taught by Professor Sinhorngchen during the Fall '08 term at National Chiao Tung University.
 Fall '08
 SinHorngChen

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