11.7.3 Image reconstruction  11.7 The Maximum-Entropy Approach  11.7.1 Fundamentals

11.7.2 Application to spectral analysis

The application of these ideas to spectral analysis was developed by Burg (1967). Part of the problem he addressed was the effect on the spectrum of contributions with wavelength or period beyond the size of the record. In the method outlined in Chapter 8, such contributions are not included when the variance spectrum is calculated. Because of the connection (8.26) between the variance spectrum and the autocovariance function, the estimated spectral density at any frequency is in truth affected by values of the autocovariance function at all lags. The equivalence between the FFT-based spectral method of Chapter 8 and the spectral approach via the autocovariance function shows that neglecting the long-wavelength or long-period components is equivalent to assuming that the autocovariance function is zero for all lags longer than the time series. Unless it is known that this is true, this is an assumption that can introduce error into the result. In fact, if the experimental values for the observed portion of the autocovariance function do not taper smoothly to zero at the maximum lag, the jump in value assumed at that point introduces unwanted side lobes into the result. Furthermore, in an effort to treat these problems, window functions are introduced that have the effect of tapering the autocovariance spectrum to zero at the longest lag. This distorts the data still further from their actual values, and reduces the resolution possible (by in effect smoothing together nearby estimates of the spectral density), to gain the benefits of smoothing the resulting spectrum and reducing sidelobe contributions. Burg pointed out that this approach then proceeds on the basis of a spectrum that could not have produced the observations, when a better approach is to select a "best" spectrum from among the many spectra that could have produced the observed spectrum.

The ambiguity among these possible spectra arises because they have nonzero contributions from wavelengths or periods outside the range of observations. The Burg approach is to select from among these the result that conforms to the maximum-entropy solution; in effect, this selects a solution that matches the data exactly but extrapolates the autocovariance function beyond the range of observations in a way that, in the maximum-entropy sense, makes the least restrictive assumption about the form of that extrapolation.^11.6

Press et al. (1992) give the details of the maximum-entropy solution and provide a computer routine that can be used to find that solution. They also provide useful guidance for an analyst wanting to use that routine.

   
 11.7.3 Image reconstruction  11.7 The Maximum-Entropy Approach  11.7.1 Fundamentals