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The sample autocorrelation defined in (6.6) is not quite
the same as the autocorrelation function for infinitely long
discretetime sequences defined in §2.3.6,
viz.,
where the signal
is assumed to be of finite support
(nonzero over a finite range of samples), and
is the DTFT
of
. The advantage of the definition of
is that
there is a simple
Fourier theorem associated with it. The disadvantage is that
it is biased as an estimate of the statistical autocorrelation.
The bias can be removed, however, since

(7.15) 
Thus,
can be seen as a Bartlettwindowed sample
autocorrelation:

(7.16) 
It is common in practice to retain the implicit Bartlett
(triangular) weighting in the sample autocorrelation. It merely
corresponds to smoothing of the power spectrum (or
crossspectrum) with the
kernel, and smoothing is necessary
anyway for statistical stability. It also downweights the less
reliable largelag estimates, weighting each lag by the number of
lagged products that were summed, which seems natural.
The left column of Fig.6.1 in fact shows the Bartlettbiased
sample autocorrelation. When the bias is removed, the autocorrelation
appears noisier at higher lags (near the endpoints of the plot).
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