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^{}
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 *y*_{t} and
filter it by

| |
(11) |

We found by setting to zero
:
| |
(12) |

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:
| |
(13) |

Recall that if
and
, then
for any *a*_{1} and *a*_{2}.
So for any we have
| |
(14) |

and for any linear combination we have
| |
(15) |

Therefore, substituting from (11), we get
| |
(16) |

which is an **autocorrelation** function
and must be symmetric. Thus,
| |
(17) |

Since the autocorrelation of the prediction-error output
is an impulse, its spectrum is **white**.
This has many interesting philosophical implications,
as we will see next.

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Stanford Exploration Project

10/21/1998