The relationship between spectrum and PEF

Knowledge of an autocorrelation function is equivalent to knowledge of a spectrum. The two are simply related by Fourier transform. A spectrum or an autocorrelation function encapsulates an important characteristic of a signal or an image. Generally the spectrum changes slowly from place to place although it could change rapidly. Of all the assumptions we could make to fill empty bins, one that people usually find easiest to agree with is that the spectrum should be the same in the empty-bin regions as where bins are filled. In practice we deal with neither the spectrum nor its autocorrelation but with a third object. This third object is the Prediction Error Filter (PEF), the filter in equation (10).

Take equation (10) for $\bold r$ and multiply it by the adjoint $\bold r'$ getting a quadratic form in the PEF coefficients. Minimizing this quadratic form determines the PEF. This quadratic form depends only on the autocorrelation of the original data y_t, not on the data y_t itself. Clearly the PEF is unchanged if the data has its polarity reversed or its time axis reversed. Indeed, we'll see here that knowledge of the PEF is equivalent to knowledge of the autocorrelation or the spectrum.