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

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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 = causal e ( ) t finite energy ( ) L s is minimum-phase e ( ) L s and ( ) s are analytic in the right s-plane, i.e., all poles of ( ) L s and ( ) s 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 =- i X e ( ) ( ) R = ii ( ) ( ) ( ) t t d =- X i l { } 2 2 ( ) ( ) E t t dt L = < X l ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) R R = =- =- XX ii l l l l ( ) ( ) d = + l l e ( ) ( ) R R =- XX XX ( ) ( ) ( ) S s L s L s =- 2 ( ) | ( ) | S L = ( ) L s L innovations filter of ( ) t X h 1 ( ) ( ) s L s = whitening filter of ( ) t X h ( ) t i h innovations of ( ) t X ( ) L s C S Given a positive even function ( ) S a minimum- phase system ( ) L s s.t . 2 | ( ) | ( ) L j S = Paley-Wiener Conditione 2 |log ( ) | 1 S d - < + e ( ) S ( ) S line e BL( e e predictable) in generalbS for special casee ( ) L s L rational function of s L ( ) S 2 rational functionbS 2 * ( ) | ( ) | ( ) ( ) ( ) ( ) S L j L j L j L j L j = = =- ( ) ( ) S S =- 2 2 ( ) ( ) ( ) A S B = 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 = + ( ) L s Solutione 2 2 ( ) ( )( ) N N S s s s s = =-- + ( ) N L s s = + Ex. 11-2 2 4 2 49 25 ( ) 10 9 S + = + + 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 = + 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 = where ( ) L z is a minimum-phase systeme [ ] [ ] [ ] k n k n k = =- i X h [ ] [ ] R m m = 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 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 - < General solution of ( ) j S e...
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Chapter 11 - Chap.11 Spectral Representation 11.1...

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