Identification of power spectral densities rely on measured second order statistics such as, e.g. covariance estimates. In the family of power spectra consistent with such an estimate a representative spectra is singled out; examples of such choices are the Maximum entropy spectrum and the Correlogram. Here, we choose a prior spectral density to represent a priori information, and the spectrum closest to the prior in the Itakura-Saito distance is selected. It is known that this can be seen as the limit case when the cross-entropy principle is applied to a gaussian process. This work provides a quantitative measure of how close a finite covariance sequence is to a spectral density in the Itakura-Saito distance. It is given by a convex optimization problem and by considering its dual the structure of the optimal spectrum is obtained. Furthermore, it is shown that strong duality holds and that a covariance matching coercive spectral density always exists. The methods presented here provides tools for discrimination between power spectrum, identification of power spectrum, and for incorporating given data in this process.
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