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Unformatted text preview: 1 Chapter 13 Mean Square Estimation Teacher: SinHorng Chen Office: Engineering Bld. #4, Room 805 Tel: ext. 31822 Email: [email protected] 2 13.1 Introduction We want to estimate ( ) t S at a specific time t in terms of ( ) ξ X for a b ξ ≤ ≤ . Here we assume that both ( ) t X and ( ) t S are WSS. The general form of linear estimation can be expressed by { } ˆ ˆ ( ) ( )  ( ), ( ) ( ) b a t E t a b h d ξ ξ α α α = ≤ ≤ = ∫ S S X X (131) The objective is to minimize the mean square (MS) error defined below { } 2 2 ˆ ( ) ( ) ( ) ( ) ( ) b a P E t t E t h d α α α = = ∫ S S S X (132) 3 Based on the extension of the Orthogonality Principle , we have { } ( ) ( ) ( ) ( ) b a E t h d α α α ξ  = ∫ S X X a b ξ ≤ ≤ (133) ⇒ ( , ) ( ) ( , ) b a R t h R d ξ α α ξ α = ∫ SX XX a b ξ ≤ ≤ (134) In this case, the prediction error is given by { } ( ) ( ) ( ) ( ) (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 In the following discussions, we assume RPs are real and WSS. We consider several cases: (1) ( , ) t a b ∈ . In this case, ˆ ( ) t S is called smoothing . (2) ( , ) t a b ∉ and ( ) ( ) t t = X S (i.e., no noise). This is a prediction case. For t b , ˆ ( ) t S is a forward predictor; while for t a < , ˆ ( ) t S is a backward predictor. (3) ( , ) t a b ∉ and ( ) ( ) t t ≠ X S . This is a filtering and prediction case. 5 Prediction of ( ) t λ + S in terms of ( ) t S : The linear predictor is expressed by { } ˆ ˆ ( ) ( )  ( ) ( ) t E t t a t λ λ + = + = S S S S . From { } ( ( ) ( )) ( ) E t a t t λ +  = S S S , we obtain ( ) (0) R a R λ = . The prediction error is { } 2 ( ) ( ( ) ( )) ( ) (0) ( ) (0) (0) R P E t a t t R aR R R λ λ λ λ = +  + =  =  S S S Simple Illustrations 6 Special case1 If   ( ) R Ae α τ τ = , then a e αλ = . In this case, { } ( ) ( ( ) ( )) ( ) 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 The prediction error of ( ) t λ + S is orthogonal to ( ) t ξ S for ξ . So, the prediction of ( ) t λ + S using ( ) t ξ S for ξ ≥ is equivalent to that of using ( ) t S only. The process is called widesense Markov of order 1. 7 Estimate ( ) t λ + S using ( ) t S and ( ) t ′ S . The estimator is 1 2 ˆ ( ) ( ) ( ) t a t a t λ ′ + = + S S S From ˆ ( ) ( ) ( ), ( ) t t t t λ λ ′ +  + ⊥ S S S S , we have 1 2 ( ) (0) (0) 0 R a R a R λ ′  = S S 1 2 ( ) (0) (0) 0 R a R a R λ ′ ′ ′ ′  = SS SS S S 1 (0) R ′ = , ( ) ( ) R R τ τ ′ ′ =  SS and ( ) ( ) R R τ τ ′ ′ ′′ =  S S ∴ 1 ( ) (0) R a R λ = and 2 ( ) (0) R a R λ ′ = ′′ 8 and { } 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 If λ is small, then ( ) (0)...
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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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