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Unformatted text preview: 1 Chapter 12 Spectrum Estimation Teacher: Office: 805 Tel: ext. 31822 Email: schen@mail.nctu.edu.tw 2 12.1 Ergodicity M & * data 1 RP 1 statistical parameters1 1 parameters L function of ( ) t X expectation1 L expectation 1 L 5 sample space 1 average (ensemble average)1 1 For a specific t E ( ) t X is an RV1 { } ( ) ( ) t E t = X observed n samples ( , ) i t X of ( ) t X mean of ( ) t X by 1 ( ) ( , ) i i t t n = X 1 point estimation1 consistent estimate of ( ) t n variance n E H 5 estimate1 3 &* sample of ( ) t X { } ( ) E t X h { } ( ) E t X depends on t 5 E h ( ) t X regular stationary process1 time average 1 1 1 { } ( ) E t X as length h Ergodicity 1 4 Meanergodic process Given a real stationary process ( ) t X 5 { } ( ) ( ) t E t = X form time average 1 ( ) 2 T T T t dt T = X 5 { } { } 1 ( ) 2 T T T E E t dt T  = = X T unbiased estimator of h variance of T 2 T as T T in the MS sense1 ( ) t X meanergodic1 i.e.1 ( ) t X is meanergodic if T ensemble average as T meanregodic 1 T T as T 6 Ex. 12.1 Let c is an RV with mean c and ( ) t = X c { } { } ( ) c E t E = = = X c . For a specific sample ( ) t c = X (constant)1 ( ) ( ) T = c (i.e.} 5 determined by )1 ( ) T ( ) t X meanergodic 7 Ex. 12.2 Given two meanergodic processes 1 ( ) t X 2 ( ) t X mean 1 1 1 2 form 1 2 ( ) ( ) ( ) t t t = + X X cX c RV independent of 2 ( ) t X = c 11 probabilities 1 0.51 { } { } { } { } { } 1 2 1 2 1 2 ( ) ( ) ( ) ( ) 0.5 E t E t E t E E t = + = + = + X X cX c X 1 for a specific sample1 if ( ) = c 1 ( ) ( ) t t = X X 1 T as T 1 ( ) 1 = c 1 2 ( ) ( ) ( ) t t t = + X X X 1 2 T = + ( ) X t E meanergodic 8 h& 2 T of T for an RP ( ) t X C form 1 ( ) ( ) 2 t T t T t d T + = w X 1 moving average of ( ) t X (0) T = w ( ) t w ( ) t X LTI system with impulse response 1 1   ( ) 2 0 o.w. t T h t T = 9 2 2 1   ( ) ( )(1 ) 2 2 T T C C d T T  =   ww 1 ( ) t w 1 autocovariance 1 ( ) C 1 ( ) t X 1 autocovariance and   1 ( ) ( ) 2 t h t h t T =  10 2 ( (0)) (0) T Var C = = ww w 1 ( ) ( ) C C = 2 2 2 1   ( )(1 ) 2 2 T T T C d T T  = 2 1 ( )(1 ) 2 T C d T T = (124) 1 ( ) t X is meanergodic iff 2 1 ( )(1 ) 2 T T C d T T  (125) 11 h& T & Tchebycheffs inequality 1 1 confidence interval for the estimate T of 1 10 T T 1 1 0.991 T E T T 1 1 1 1 12 Ex. 123 If   ( ) c C qe  = 2 2 2 1 (1 ) (1 ) 2 2 cT T c T q q e e d T T cT cT  = = 2 T as T ( ) t X is meanergodic If 1 T c E E then 2 T q cT ( 1 2 cT e ) 13...
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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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