# Chapter 11 - Chap.11 Spectral Representation 11.1...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Chap.11 Spectral Representation 11.1 Factorization and Innovations ¨ A real WSS RP ( ) t X minimum-phase system ( ) L s L output with input e white noise process ( ) t i h signal modeling i ( ) t X regulare Minimum-phase 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 minimum-phase e ( ) L s and ( ) s Γ are analytic in the right s-plane, i.e., all poles of ( ) L s and ( ) s Γ are in the left s-plane. A RP e regular if it is linearly equivalent with a white-noise 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 = Paley-Wiener 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. 11-2(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. 11-2(b)e ∴ ( ) ( )/ ( ) L s N s D s = Ex. 11-1 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. 11-2 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. 11-3 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 = + + Discrete-time Processes A real WSS discrete-time 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 minimum-phase 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 white-noise 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 Paley-Wiener Condition |ln ( ) | j S e d π ϖ π ϖ- < General solution of ( ) j S e ϖ...
View Full Document

{[ snackBarMessage ]}

### Page1 / 24

Chapter 11 - Chap.11 Spectral Representation 11.1...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online