Chapter 11 - Chapter 11 Spectral Representation Teacher:...

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Unformatted text preview: Chapter 11 Spectral Representation Teacher: Office: 805 Tel: ext. 31822 Email: schen@mail.nctu.edu.tw 2 11.1 Factorization and Innovations x & real WSS RP ( ) t X minimum-phase system ( ) L s L output with input e white noise process ( ) t i i signal modeling i h ( ) t X i i regulare Minimum-phase system ( ) L s L causal e impulse response ( ) t finite energy (i.e., causal and stable)e inverse system 1 ( ) ( ) s L s = causal e ( ) t finite energy 3 ( ) L s is minimum-phase ( ) 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. 4 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 { } 2 2 ( ) ( ) E t t dt = < X ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) R R = =- =- XX ii ( ) ( ) d = + e ( ) ( ) R R =- XX XX 5 ( ) ( ) ( ) S s L s L s =- 2 ( ) | ( )| S L = ( ) L s L innovations filter of ( ) t X i 1 ( ) ( ) s L s = whitening filter of ( ) t X i ( ) t i i innovations of ( ) t X 6 h& ( ) L s bA Given a positive even function ( ) S a minimum-phase system ( ) L s s.t. 2 | ( ) | ( ) L j S = h Paley-Wiener Conditione 2 |log ( )| 1 S d - < + e ( ) S ( ) S line e BL( predictable) 7 h& in general for special casee e ( ) L s L rational function of sL ( ) S 2 rational functione e 2 * ( ) | ( )| ( ) ( ) ( ) ( ) S L j L j L j L j L j = = = - ( ) ( ) S S = - e 2 2 ( ) ( ) ( ) A S B = 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. 11-2(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. 11-2(b)e ( ) ( )/ ( ) L s N s D s = 9 10 Ex. 11-1 2 2 ( ) N S = + ( ) L s Solutione 2 2 ( ) ( )( ) N N S s s s s = =-- + ( ) N L s s = + 11 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 + = + + 12 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 = + + 13 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 14 [ ] [ ] [ ] k n k n k = = - i X i [ ] [ ] R m m = ii [ ] [ ] [ ] k n k n k = = - X i [ ] { } [ ] 2 2 k E n k = = < X 15 h& [ ] n X is linearly equivalent with a white-noise process...
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Chapter 11 - Chapter 11 Spectral Representation Teacher:...

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