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MA (Moving Average) model for Power Spectrum Estimation Sarbjeet Singh NITTTR- Chandigarh
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Introduction to MA model Parametric method based model for the power spectrum estimation. Parametric methods are based on modeling the data sequence x(n) as the output of a linear system characterized by a system function of the form In model based approach the spectrum estimation procedure consists of two steps.
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Spectrum Estimation procedural steps For a given data sequence x(n), where n ranges from 0 to N-1.1. The parameters {ak} and {bk} are to be estimated.2. From the estimated parameters {ak} and {bk} power spectrum estimation is computed according to the equations based on stationary random process. ---(1) This equation represents power spectrum density of data, where is the power density spectrum of the input sequence.
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Power density spectrum estimation To estimate power density spectrum we assume input sequence w(n) is a zero mean white noise sequence with autocorrelation. ---(2) where is the variance i.e. The power density spectrum of the observed data is ---(3) where H(f) is the frequency response of the model.
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Moving Average process MA(q) The Moving Average process of order q is denoted as MA(q). The MA(q) model for the observed data, the autocorrelation is related to MA parameters {bk} by With this background established, we describe power spectrum estimation for the MA(q) model.
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Power spectrum estimation Calculating coefficients and power spectrum estimation. The parameters in a MA(q) model are related to the statistical autocorrelation given by equation ---(4) However, the coefficients {dm} are related to the MA parameters by the expression
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Power spectrum estimation (cont.) Clearly then, And the power spectrum for the MA process is It is apparent from the expressions that we do not have to solve for the MA parameters {bk} to estimate the power spectrum. The estimates of the autocorrelation for |m| <=q. From such estimates we compute the estimated MA power spectrum.
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Estimated MA power spectrum The estimated power spectrum obtained is ---(5) which is identical to the classical (non parametric) power spectrum estimate. There is an alternative method for determining {bk} based on a high order AR approximation to the MA process. The MA(q) process be modeled by an AR(p) model.
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Alternative method for determiningcoefficients {bk} Alternative method for determining {bk} is based on a high order AR approximation to the MA process. Let the MA(q) process be modeled by an AR(p) model, where p>>q. then B(z)=1/A(z), or equivalently, B(z)A(z)=1. Thus, the parameters {bk} and {ak} are related by a convolution sum
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Alternative method for determining {bk}(cont.) Here are the parameters obtained by fitting the data to an AR(p) model. Although this set of equations can be easily solved for {b k} , better fit is obtained by using a least square error criterion. That is, we form the squared error ---(6)
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Minimizing squared error The least square error is minimized by selecting the MA(q) parameters {bk}. The result of this minimization is ---(7) Where the elements of Raa and raa are given as where
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Minimizing squared error (cont.) raa is given as where This least square method for determining the parameters of the MA(q) model is attributed to Durbin(1959). It has been shown by Kay(1988) that this estimation method is approximately the maximum likelihood under the assumption.
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Order q of the MA model The order q of the MA model may be determined empirically by several methods. For example, the AIC (Akaike Information Criteria) for MA models has the same form as for AR models, Where is an estimate of the variance of the white noise. Another approach, proposed by Chow(1927), is to filter the data with the inverse MA(q) filter and test the filtered output for whiteness.
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REFERENCES Digital Signal Processing: “Principles, Algorithms & Applications” by Proakis & Manolakis, 3rd edition.