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Unformatted text preview: 1 Chapter 13 Mean Square Estimation Teacher: d Office: 805 Tel: ext. 31822 Email: schen@mail.nctu.edu.tw 2 13.1 Introduction D ( ) t S at a specific time t a in terms of ( ) X X a b (1 ( ) t X and ( ) t S X WSS) Linear estimation 1 { } ( ) ( )  ( ), ( ) ( ) b a t E t a b h d = = S S X X (131) objective 1 minimum of MS error defined below { } 2 2 ( ) ( ) ( ) ( ) ( ) b a P E t t E t h d = = S S S X (132) 3 h extension of the orthogonality principle 1 { } ( ) ( ) ( ) ( ) b a E t h d  = S X X a b (133) ( , ) ( ) ( , ) b a R t h R d = SX XX a b (134) 1 { } ( ) ( ) ( ) ( ) (0) ( ) ( , ) b b a a P E t h d t R h R t d = = SS SX S X S 4 h& RP 1 real and WSSk cases (1) ( , ) t a b ( ) t S 1 smoothing (2) ( , ) t a b ( ) ( ) t t = X S (no noise)1 t b ( ) t S X forward predictor1 t a < ( ) 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 + S in terms of ( ) t S { } ( ) ( )  ( ) ( ) t E t t a t + = + = S S S S 1 { } ( ( ) ( )) ( ) E t a t t + = S S S 1 ( ) (0) R a R = { } 2 ( ) ( ( ) ( )) ( ) (0) ( ) (0) (0) R P E t a t t R aR R R = + + = = S S S 6 Special case1   ( ) R Ae  = 1 a e  = { } ( ) ( ( ) ( )) ( ) 0 ( ) ( ) E t a t t R aR Ae Ae e  + + = + = = S S S ( ) t + S X prediction error 1 ( ) t  S X orthogonal ( ) t + S X prediction 1 ( ) t  S X ( ) t S X X h process 1 widesense Markov of order 1 7 estimate ( ) t + S X ( ) t S X ( ) t S 1 2 ( ) ( ) ( ) t a t a t + = + S S S h ( ) ( ) ( ), ( ) t t t t + + S S S S 1 2 ( ) (0) (0) R a R a R  = S S 1 2 ( ) (0) (0) R a R a R  = SS SS S S 8 1 (0) R = ( ) ( ) R R =  SS X ( ) ( ) R R =  S S 1 ( ) (0) R a R = 2 ( ) (0) R a R = 1 { } 1 2 1 2 ( ( ) ( ) ( )) ( ) (0) ( ) ( ) P E t a t a t t R a R a R = + + = + S S S S 1 ( ) (0) R R ( ) (0) (0) (0) R R R R + = 1 1 a a a 2 a ( ) ( ) ( ) t t t + + S S S X is the 1st approximation of Taylor series 9 Filtering { } ( ) ( )  ( ) ( ) 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 + S for 0 T < < in terms of 2 1 N + samples ( ) t kT + S X N k N ( ) ( ) N k k N t a t kT = + = + S S T < < 11 Fig. 131 12 ( ( ) (...
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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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