The basic idea of least-squares fitting
is that the residual is orthogonal to the fitting functions.
Applied to the PE filter, this idea means
that the output of a PE filter is orthogonal to lagged inputs.
The orthogonality applies only for lags in the past because
prediction knows only the past while it aims to the future.
What we want to show is different,
namely, that the output is uncorrelated with itself
(as opposed to the input) for lags in both directions;
hence the output spectrum is white.
We are given a signal yt and
filter it by
We found by setting to zero
We interpret this to mean that the residual is orthogonal
to the fitting function,
or the present PE filter output is orthogonal to its past inputs,
or one side of the crosscorrelation vanishes.
Taking an unlimited number of time lags and filter coefficients,
the crosscorrelation vanishes not only for but for larger values, say where and s>0.
In other words,
the future PE filter outputs are orthogonal to present and past inputs:
Recall that if
for any a1 and a2.
So for any we have
and for any linear combination we have
Therefore, substituting from ((3)), we get
which is an autocorrelation function
and must be symmetric. Thus,
Since the autocorrelation of the prediction-error output
is an impulse, its spectrum is white.
Stanford Exploration Project