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Chap12_English(1) - Chapter 12 Spectrum Estimation 12.1...

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Chapter 12 Spectrum Estimation 12.1 Ergodicity It is needed to estimate the statistical parameters of a random process from real data. Although most parameters to be estimated can be formulated as the expectation of a function of the random process ( ) t X , the problem lies in the fact that we need to implement the expectation via taking average (ensemble average) over the whole sample space. An example is given in the following. For a specific t , ( ) t X is a random variable. Let { } ( ) ( ) t E t η = X . We can estimate the mean of ( ) t X using n observed samples ( , ) i t ξ X of ( ) t X by 1 ˆ( ) ( , ) i i t t n η ξ = X . This is a point estimation. It is a consistent estimate of ( ) t η if its variance 0 t as n E . In this case, it is a good estimate for large n . But, we usually only have a sample of ( ) t X . Can we use it to estimate { } ( ) E t X ? If { } ( ) E t X depends on t , then this is impossible. If ( ) t X is a regular stationary process, then the time average will approach to { } ( ) E t X as the length t . This is the meaning of ergodicity. Mean-ergodic process Given a real stationary process ( ) t X , we want to estimate { } ( ) ( ) t E t η = X . We form the time average 1 ( ) 2 T T T t dt T - = η X . Since { } { } 1 ( ) 2 T T T E E t dt T η - = = η X , T η is an unbiased estimator of η . 1
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If the variance of T η , 2 0 T σ as T E , then T η η in the MS sense. We say that ( ) t X is mean-ergodic, i.e., ( ) t X is mean-ergodic if the time average T η h the ensemble average η , as T E . So, “mean-regodic” can be proved via calculating T σ and showing 0 T σ as T E . Ex. 12.1 Let c be an RV with mean c η and let ( ) t = X c . Then { } { } ( ) c E t E η η = = = X c . A specific sample ( ) t c = X is a line (constant). In this case, ( ) ( ) T ξ ξ = η c (i.e., a specific constant determined by ξ ). Obviously, ( ) T ξ η η . So ( ) t X is not mean- ergodic. Ex. 12.2 Given two mean-ergodic processes 1 ( ) t X and 2 ( ) t X , their means are 1 η and 2 η , respectively. We form 1 2 ( ) ( ) ( ) t t t = + X X cX , where c is a RV independent of 2 ( ) t X and 0 = c and 1 with probabilities 0.5 , respectively. Then { } { } { } { } { } 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 But for a specific sample, if ( ) 0 ξ = c , then 1 ( ) ( ) t t = X X and 1 T η η as T E . If ( ) 1 ξ = c , then 1 2 ( ) ( ) ( ) t t t = + X X X . In this case, 1 2 T η η = + η . Hence, ( ) t X is not mean-ergodic. To calculate 2 T σ of T η for an RP ( ) t X , we can form 1 ( ) ( ) 2 t T t T t d T α α + - = w X which is a moving average of ( ) t X . Obviously, (0) T = η w . ( ) t w is the output of 2
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an LTI system with input ( ) t X and impulse response 1 | | ( ) 2 . . 0 t T h t T o w E E E = So the autocovariance of ( ) t w can be expressed by 2 2 1 | | ( ) ( )(1 ) 2 2 T T C C d T T α τ τ α α - = - - ww where ( ) C τ is the autocovariance of ( ) t X and | | 1 ( ) ( ) 2 h t h t T α - = - . 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 0 1 ( )(1 ) 2 T C d T T α α α = - (12-4) Hence, ( ) t X is mean-ergodic iff 2 0 1 ( )(1 ) 0 2 T T C d T T α α α - 9 (12-5) T σ can also be used in the Tchebycheff’s inequality to define a confidence interval for the estimate T η of η . For instances, the probability that η lies in he range 10 T T σ η is greater than 0.99. So, if T is large enough to make T σ η = , then T η is a reliable estimate of η .
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