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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 and multiply it
by the adjoint 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.

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** Up:** PREDICTION-ERROR FILTER OUTPUT IS
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Stanford Exploration Project

4/27/2004