Chapter 13 - 1 Chapter 13 Mean Square Estimation Teacher ¨...

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Unformatted text preview: 1 Chapter 13 Mean Square Estimation Teacher: ¨ d ª Office: 805 Tel: ext. 31822 Email: [email protected] 2 13.1 Introduction ¨ ¶ D … “ ( ) t S at a specific time t a in terms of ( ) ξ X X a b ξ ≤ ≤ (1 ( ) t X and ( ) t S X WSS) Linear estimation 1 { } ˆ ˆ ( ) ( ) | ( ), ( ) ( ) b a t E t a b h d ξ ξ α α α = ≤ ≤ = ∫ S S X X (13-1) objective 1 minimum of MS error defined below { } 2 2 ˆ ( ) ( ) ( ) ( ) ( ) b a P E t t E t h d α α α =- =- ∫ S S S X (13-2) 3 h extension of the orthogonality principle 1 { } ( ) ( ) ( ) ( ) b a E t h d α α α ξ - = ∫ S X X a b ξ ≤ ≤ (13-3) ⇒ ( , ) ( ) ( , ) b a R t h R d ξ α α ξ α = ∫ SX XX a b ξ ≤ ≤ (13-4) 1 { } ( ) ( ) ( ) ( ) (0) ( ) ( , ) b b a a P E t h d t R h R t d α α α α α α =- =- ∫ ∫ SS SX S X S 4 h& RP 1 real and WSSk cases (1) ( , ) t a b ∈ ˆ ( ) t S 1 smoothing (2) ( , ) t a b ∉ ( ) ( ) t t = X S (no noise)1 t b ˆ ( ) t S X forward predictor1 t a < ˆ ( ) t S X backward predictor1 case 1 prediction (3) ( , ) t a b ∉ ( ) ( ) t t ≠ X S X filtering and prediction 5 Simple Illustrations Prediction of ( ) t λ + S in terms of ( ) t S { } ˆ ˆ ( ) ( ) | ( ) ( ) t E t t a t λ λ + = + = S S S S 1 { } ( ( ) ( )) ( ) E t a t t λ +- = S S S 1 ( ) (0) R a R λ = { } 2 ( ) ( ( ) ( )) ( ) (0) ( ) (0) (0) R P E t a t t R aR R R λ λ λ λ = +- + =- =- S S S 6 Special case1 | | ( ) R Ae α τ τ- = 1 a e αλ- = { } ( ) ( ( ) ( )) ( ) 0 ( ) ( ) E t a t t R aR Ae Ae e α λ ξ αλ αξ λ ξ ξ λ ξ ξ- +-- +-- = +- =- = S S S ⇒ ( ) t λ + S X prediction error 1 ( ) t ξ- S X ξ orthogonal ⇒ ( ) t λ + S X prediction 1 ( ) t ξ- S X ξ ≥ ( ) t S X X h process 1 wide-sense Markov of order 1 7 estimate ( ) t λ + S X ( ) t S X ( ) t ′ S 1 2 ˆ ( ) ( ) ( ) t a t a t λ ′ + = + S S S h ˆ ( ) ( ) ( ), ( ) t t t t λ λ ′ +- + ⊥ S S S S ⇒ 1 2 ( ) (0) (0) R a R a R λ ′-- = S S 1 2 ( ) (0) (0) R a R a R λ ′ ′ ′ ′-- = SS SS S S 8 1 (0) R ′ = ( ) ( ) R R τ τ ′ ′ = - SS X ( ) ( ) R R τ τ ′ ′ ′′ = - S S ∴ 1 ( ) (0) R a R λ = 2 ( ) (0) R a R λ ′ = ′′ 1 { } 1 2 1 2 ( ( ) ( ) ( )) ( ) (0) ( ) ( ) P E t a t a t t R a R a R λ λ λ λ ′ = +-- + ′ =- + S S S S 1 λ ( ) (0) R R λ ( ) (0) (0) (0) R R R R λ λ λ ′ ′ ′′ ′′ + = ⇒ 1 1 a a a 2 a λ ˆ ( ) ( ) ( ) t t t λ λ ′ + + S S S X is the 1st approximation of Taylor series 9 Filtering { } ˆ ˆ ( ) ( ) | ( ) ( ) t E t t a t = = S S X X { } ( ( ) ( )) ( ) E t a t t- = S X X ⇒ (0) (0) R a R = SX XX { } ( ( ) ( )) ( ) (0) (0) P E t a t t R aR =- =- SS SX S X S 10 Interpolation 1 Fig. 13.11 estimate ( ) t λ + S for 0 T λ < < in terms of 2 1 N + samples ( ) t kT + S X N k N- ≤ ≤ ˆ ( ) ( ) N k k N t a t kT λ =- + = + ∑ S S T λ < < 11 Fig. 13-1 12 ( ( ) (...
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Chapter 13 - 1 Chapter 13 Mean Square Estimation Teacher ¨...

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