Transforms and Filters for Stochastic Processes (779449), страница 6
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We will considerthe simplest case only,whichisrelated to the Yule-Walker equations. Acomprehensive treatment of this subject would go far beyond the scope ofthis section.Recall that in Section 5.6.2 we showed that thecoefficients of a linear onestep predictor are identical to the parameters describing an autoregressiveprocess.
Hence the power spectral density may be estimated as(5.241)IThe coefficients b(n) in (5.241) are the predictor coefficients determined from142Chapter 5. Tkansforms and FiltersStochasticfor ProcessesHanningI......0-0.5...I0.5I""I."."..I.-0.5".'.'.'....l0.5Blackman'..0Normalized FrequencyHamming'l.-0.5Normalized FrequencyI''.0~.I.I""""II....0.5-0.5Normalized Frequency~.....00.5Normalized FrequencyFigure 5.6. Magnitude frequency responses of common window functions.the observed data, andaccording to (5.174):13;is the power of the white input8; = f Z Z(0)+(1) h.process estimated(5.242)If we apply the autocorrelation method to the estimation of the predictorcoefficients G(.), the estimated autocorrelation matrix has a Toeplitz structure, and the prediction filter is always minimum phase, just as when usingthe true correlation matrix R%%.For the covariance method this is not thecase.Finally, it shall beremarked that besides a forward prediction a backwardprediction may also be carried out.
By combining both predictors one canobtain an improved estimation of the power spectral density compared to(5.241). An example is the Burg method [19]..
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