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# Chapter 12 - Chapter 12 Spectrum Estimation Te r ache...

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1 Chapter 12 Spectrum Estimation Teacher: Ø ª Office: 805 Tel: ext. 31822 Email: [email protected]

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2 12.1 Ergodicity M ¥ ° » *±± data 1 RP 1 statistical parameters 1 1 parameters L function of ( ) t X η expectation 1 L expectation 1 L 5 sample space 1 average (ensemble average) 1 1 For a specific t E ( ) t X is an RV 1 { } ( ) ( ) t E t η = X observed n samples ( , ) i t ξ X of ( ) t X η mean of ( ) t X by 1 ˆ ( ) ( , ) i i t t n η ξ = X 1 point estimation 1 consistent estimate of ( ) t η n → ∞ variance 0 n E H 5 estimate 1
3 ¨ ° » sample of ( ) t X { } ( ) E t X η h { } ( ) E t X depends on t 5 E h ( ) t X η regular stationary process1 time average 1 1 1 { } ( ) E t X as length → ∞ h Ergodicity 1

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4 Mean-ergodic process Given a real stationary process ( ) t X 5 { } ( ) ( ) t E t η = X η η form time average 1 ( ) 2 T T T t dt T - = η X
5 { } { } 1 ( ) 2 T T T E E t dt T η - = = η X η T η η unbiased estimator of η h variance of T η 2 0 T σ as T → ∞ T η η in the MS sense 1 ( ) t X η mean-ergodic 1 i.e. 1 ( ) t X is mean-ergodic if T η ensemble average η as T → ∞ mean-regodic 1 T σ 0 T σ as T → ∞

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6 Ex. 12.1 Let c is an RV with mean c η and ( ) t = X c η { } { } ( ) c E t E η η = = = X c . For a specific sample ( ) t c = X η η (constant) 1 ( ) ( ) T ξ ξ = η c (i.e. } 5 determined by ξ ) 1 ( ) T ξ η η η ( ) t X η mean-ergodic
7 Ex. 12.2 Given two mean-ergodic processes 1 ( ) t X η 2 ( ) t X η mean 1 1 1 η 2 η form 1 2 ( ) ( ) ( ) t t t = + X X cX η c η RV independent of 2 ( ) t X η 0 = c η 11 probabilities 1 0.5 1 { } { } { } { } { } 1 2 1 2 1 2 ( ) ( ) ( ) ( ) 0.5 E t E t E t E E t η η η = + = + = + X X cX c X 1 for a specific sample 1 if ( ) 0 ξ = c η 1 ( ) ( ) t t = X X η 1 T η η as T → ∞ 1 ( ) 1 ξ = c η 1 2 ( ) ( ) ( ) t t t = + X X X η 1 2 T η η = + η η ( ) X t E mean-ergodic

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8 2 T σ of T η for an RP ( ) t X C form 1 ( ) ( ) 2 t T t T t d T α α + - = w X 1 moving average of ( ) t X η (0) T = η w η η ( ) t w η ( ) t X η 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 α τ τ α α - = - - ww 1 ( ) t w 1 autocovariance 1 ( ) C τ 1 ( ) t X 1 autocovariance and | | 1 ( ) ( ) 2 t h t h t T - = -

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10 2 ( (0)) (0) T Var C σ = = ww w 1 ( ) ( ) C C α α = - 2 2 2 1 | | ( )(1 ) 2 2 T T T C d T T α σ α α - = - 2 0 1 ( )(1 ) 2 T C d T T α α α = - (12-4) 1 ( ) t X is mean-ergodic iff 2 0 1 ( )(1 ) 0 2 T T C d T T α α α →∞ - → (12-5)
11 T σ ± Tchebycheff’s inequality 1 1 confidence interval for the estimate T η of η η 1 10 T T σ ± η 1 1 0.99 1 T E T σ η T η 1 η 1 1 1

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12 Ex. 12-3 If | | ( ) c C qe τ τ - = 2 2 2 0 1 (1 ) (1 ) 2 2 cT T c T q q e e d T T cT cT τ τ σ τ - - - = - = - 2 0 T σ as T → ∞ ( ) t X is mean-ergodic If 1 T c E E then 2 T q cT σ ( 1 2 0 cT e - )
13 From (12-5) 1 ergodicity of a process 1 ( ) C τ τ behavior 1 h If ( ) 0 C τ = for a τ ( ) t X η a -dependent 1 T a E E 2 0 0 1 1 ( )(1 ) ( ) (0) 2 a a T a C d C d C T T T T τ σ τ τ τ τ = - < 0 T →∞ → ( 1 | ( ) | (0) C C τ < ) ( ) t X is mean-ergodic

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14 ± for large τ ( ) t τ + X η ( ) t X η nearly uncorrelated 1 i.e. 1 ( ) 0 C τ as τ →∞ 2 2 0 1 ( ) T T C d T σ τ τ 0 1 ( ) C d T τ τ (0) c C T τ = 0 T →∞ → 1 c τ correlation time of ( ) t X . So ( ) t
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Chapter 12 - Chapter 12 Spectrum Estimation Te r ache...

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