The most important property of a prediction-error filter is that its output tends to a white spectrum. No matter what the input to this filter, its output tends to whiteness as the number of the coefficients tends to infinity. Thus, the PE filter adapts itself to the input by absorbing all its color. If the input is already white, the aj coefficients vanish. The PE filter is frustrated because with a white input it can predict nothing, so the output is the same as the input. Thus, if we were to cascade one PE filter after another, we would find that only the first filter does anything. If the input is a sinusoid, it is exactly predictable by a three-term recurrence relation, and all the color is absorbed by a three-term PE filter (see exercises). The power of a PE filter is that a short filter can often extinguish, and thereby represent the information in, a long filter.
That the output spectrum of a PE filter is white is very useful. Imagine the reverberation of the soil layer, highly variable from place to place, as the resonance between the surface and deeper consolidated rocks varies rapidly with surface location as a result of geologically recent fluvial activity. The spectral color of this erratic variation on surface-recorded seismograms is compensated for by a PE filter. Of course, we do not want PE-filtered seismograms to be white, but once they all have the same spectrum, it is easy to postfilter them to any desired spectrum.
Because the PE filter has an output spectrum that is white, the filter itself has a spectrum that is inverse to the input. Indeed, an effective mechanism of spectral estimation, developed by John P. Burg and described in FGDP, is to compute a PE filter and look at the inverse of its spectrum.
Another interesting property of the PE filter is that it is minimum phase. The best proofs of this property are found in FGDP. These proofs assume uniform weighting functions.