Chapter 12 - 1 Chapter 12 Spectrum Estimation Teacher:...

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 Mean-ergodic 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 mean-ergodic1 i.e.1 ( ) t X is mean-ergodic if T ensemble average as T mean-regodic 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 mean-ergodic 7 Ex. 12.2 Given two mean-ergodic 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 mean-ergodic 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 =- (12-4) 1 ( ) t X is mean-ergodic iff 2 1 ( )(1 ) 2 T T C d T T - (12-5) 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. 12-3 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 mean-ergodic If 1 T c E E then 2 T q cT ( 1 2 cT e- ) 13...
View Full Document

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.

Page1 / 139

Chapter 12 - 1 Chapter 12 Spectrum Estimation Teacher:...

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online