# Chapter 13 - Chapter 13 Mean Square Estimation 13.1...

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Unformatted text preview: Chapter 13 Mean Square Estimation 13.1 Introduction H Z ( ) t S at a specific time t a in terms of ( ) ξ X h a b ξ () ( ) t X and ( ) t S h WSS) Linear estimation ) { } ˆ ˆ ( ) ( ) | ( ), ( ) ( ) b a t E t a b h d ξ ξ α α α = = S S X X (13-1) objective ) minimum of MS error defined below { } 2 2 ˆ ( ) ( ) ( ) ( ) ( ) b a P E t t E t h d α α α =- =- Q S S S X (13-2) extension of the orthogonality principle ) { } ( ) ( ) ( ) ( ) b a E t h d α α α ξ- = Q 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) ) { } ( ) ( ) ( ) ( ) (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 RP ) real and WSS}4 k cases (1) ( , ) t a b a a ˆ ( ) t S smoothing (2) ( , ) t a b a a ( ) ( ) t t = X S (no noise)) t b ˆ ( ) t S h forward predictor) t a < ˆ ( ) t S h backward predictor) case ) prediction (3) ( , ) t a b a a ( ) ( ) t t a X S h filtering and prediction 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 ) { } ( ( ) ( )) ( ) E t a t t λ +- = S S S ( ) (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 Special case) | | ( ) R Ae α τ τ- = 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 h prediction error ) ( ) t ξ- S h ξ orthogonal ) ( ) t λ + S h prediction ) ( ) t ξ- S h ξ ( ) t S process ) wide-sense Markov of order 1 estimate ( ) t λ + S h ( ) t S h ( ) t a S 1 2 ˆ ( ) ( ) ( ) t a t a t λ + = + S S S 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 Q (0) Ra = ( ) ( ) R R τ τ = - SS h ( ) ( ) R R τ τ = - S S ∴ 1 ( ) (0) R a R λ = 2 ( ) (0) R a R λ = ) { } 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 ) λ ( ) (0) R R λ ; ( ) (0) (0) (0) R R R R λ λ λ ξ⊃ k + = ; 1 1 a ; 2 a λ ; ˆ ( ) ( ) ( ) t t t λ λ + + S S S ; is the 1st approximation of Taylor series 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 Interpolation ) Fig. 13.1) estimate ( ) t λ + S for 0 T λ < < in terms of 2 1 N + samples ( ) t kT + S h N k N- ˆ ( ) ( ) N k k N t a t kT λ =- + = + S S T λ < < ( ( ) ( )) ( ) N k k N E t a t kT t nT λ =- +- + + = Q S S S | | n N a ) ( ) ( ) N k k N a R kT nT R nT λ =-- =- | | n N a ) 2 1 N + equations, ) 2 1 N + k a a ( ( ) ( )) ( ) (0) ( ) N N k k k N k N P E t a t kT t R a R kT λ λ λ =- =- = +- + + =-- S S S ˆ ( ) ( ) ( ) N t t t λ λ = +- + ε S S h output of filter ( ) N j jkT N k k N E e a e ϖλ ϖ ϖ + =- =- with input ( ) t S h P k {...
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## This note was uploaded on 07/21/2009 for the course CM EM5102 taught by Professor Sin-horngchen during the Fall '08 term at National Chiao Tung University.

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Chapter 13 - Chapter 13 Mean Square Estimation 13.1...

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