MA (Moving Average) model for Power Spectrum Estimation

Sarbjeet SinghSarbjeet Singh

NITTTR- ChandigarhNITTTR- Chandigarh

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.

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.

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.

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.

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

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.

Power spectrum estimation (cont.)

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.

Alternative method for determining coefficients {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

Here are the parameters obtained by fitting the data to an AR(p) model.

Although this set of equations can be easily solved for {bk} , better fit is obtained by using a least square error criterion. That is, we form the squared error

---(6)

Alternative method for determining {bk} (cont.)

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

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.

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.

REFERENCES

Digital Signal Processing: “Principles, Algorithms & Applications” by Proakis & Manolakis, 3rd edition.

QUERIES

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