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Practical Bottom Line
Since we must use the DFT in practice, preferring an FFT for speed,
we typically compute the sample autocorrelation function for a
length
sequence
,
as follows:
 Choose the FFT size
to be a power of 2
providing at least
samples of zero padding
(
):

(7.21) 
 Perform a length
FFT to get
.
 Compute the squared magnitude
.
 Compute the inverse FFT to get
,
.
 Remove the bias, if desired, by inverting the implicit
Bartlettwindow weighting to get

(7.22) 
Often the sample mean (average value) of the
samples of
is
removed prior to taking an FFT. Some implementations also
detrend the data, which means removing any linear ``tilt'' in
the data.^{7.6}
It is important to note that the sample autocorrelation is itself a
stochastic process. To stably estimate a true autocorrelation
function, or its Fourier transform the power spectral density, many
sample autocorrelations (or squaredmagnitude FFTs) must be
averaged together, as discussed in §6.12 below.
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