# Chapter 12 - Chapter 12 Spectrum Estimation 12.1...

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Unformatted text preview: Chapter 12 Spectrum Estimation 12.1 Ergodicity & CS K data a RP a statistical parametersdp 5 a parametersdp function of ( ) t X h expectationd p 5 expectationd p 5 sample space a average (ensemble average)a a For a specific t E ( ) t X is an RVa { } ( ) ( ) t E t η = X t a observed n samples ( , ) i t ξ X of ( ) t X h mean of ( ) t X by 1 ˆ( ) ( , ) i i t t n η ξ = X a point estimation a consistent estimate of ( ) t η n E E variance a a n dp 5 estimatea L dp 5 sample of ( ) t X { } ( ) E t X h { } ( ) E t X depends on t dp 5 E ( ) t X h regular stationary processa time average a { } ( ) E t X as length a Ergodicity a Mean-ergodic process Given a real stationary process ( ) t X dp 5 { } ( ) ( ) t E t η = X h form time average 1 ( ) 2 T T T t dt T- = η X { } { } 1 ( ) 2 T T T E E t dt T η- = = η X h ∴ T η h unbiased estimator of η variance of T η 2 T σ as T E E T η η in the MS sensea 1 ( ) t X h mean-ergodica i.e.a ( ) t X is mean-ergodic if T η h ensemble average η as T E ∴ mean-regodic a T σ T σ as T E Ex. 12.1 Let c be an RV with mean c η and ( ) t = X c h { } { } ( ) c E t E η η = = = X c . For a specific sample ( ) t c = X 1 (constant)a a ( ) ( ) T ξ ξ = η c (i.e.D1 5 determined by ξ )a ( ) T ξ η η h ∴ ( ) t X h mean-ergodic Ex. 12.2 Given two mean-ergodic processes 1 ( ) t X h 2 ( ) t X h mean a 1 η 2 η form 1 2 ( ) ( ) ( ) t t t = + X X cX h c h RV independent of 2 ( ) t X h = c h 1a probabilities a 0.5 a { } { } { } { } { } 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 h for a specific samplea if ( ) ξ = c h 1 ( ) ( ) t t = X X h 1 T η η as T E E E ( ) 1 ξ = c h 1 2 ( ) ( ) ( ) t t t = + X X X h 1 2 T η η = + η h ( ) t X h mean- ergodic 2 T σ of T η for an RP ( ) t X D1 form 1 ( ) ( ) 2 t T t T t d T α α +- = w X h moving average of ( ) t X h (0) T = η w h h ( ) t w h ( ) t X h LTI system with impulse response a 2 1 | | ( ) 2 0 o.w. t T h t T E E E = ∴ 2 2 1 | | ( ) ( )(1 ) 2 2 T T C C d T T α τ τ α α- =-- ww a ( ) t w autocovariance a ( ) C τ ( ) t X autocovariance and 1 | | (1 ) ( ) ( ) 2 2 t h t h t T T- =- 2 ( (0)) (0) T Var C σ = = ww w Q ( ) ( ) C C α α =- ∴ 2 2 2 1 | | ( )(1 ) 2 2 T T T C d T T α σ α α- =- 2 1 ( )(1 ) 2 T C d T T α α α =- (12-4) a ( ) t X is mean-ergodic iff 2 1 ( )(1 ) 2 T T C d T T α α α- V (12-5) T σ Tchebycheff’s inequality a confidence interval for the estimate T η of η η 10 T T σ η É 5 0.99a ∴ T E E T σ η = T η η ← 5 Ex. 12-3 If | | ( ) c C qe τ τ- = 2 2 2 1 (1 ) (1 ) 2 2 cT T c T q q e e d T T cT cT τ τ σ τ--- =- =- ∴ 2 T σ as T E a ( ) t X is mean-ergodic If 1 T c ?...
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Chapter 12 - Chapter 12 Spectrum Estimation 12.1...

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