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
Bartlett-window weighting to get
(7.22)


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 squared-magnitude FFTs) must be averaged together, as discussed in §6.12 below.
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Cyclic Autocorrelation