## 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)

*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 squared-magnitude FFTs) must be

*averaged together*, as discussed in §6.12 below.

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Cyclic Autocorrelation