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Biased Sample
Autocorrelation
The sample autocorrelation defined in (5.2) 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

(6.5) 
Thus,
can be seen as a
Bartlettwindowed sample
autocorrelation:

(6.6) 
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.5.1 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).
Previous: Sample Power Spectral DensityNext: Smoothed Power Spectral Density
About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at
Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See
http://ccrma.stanford.edu/~jos/ for details.