Determining Power Spectral Density from Autocorrelation Function and Time-Series Data

So I think I've confused myself a bit here. I know that taking the FFT of time-series data (wind velocities, for example) yields a power spectral density. However, I just read that taking the Fourier Transform of an autocorrelation function also yields the power spectral density. So, if one were to take the inverse Fourier Transform of power spectral density, would you get the original time series data (with the mean subtracted out) or would you get the autocorrelation function?

• You can compute the autocorrelation in an efficient way by using the FFT: (1) compute the dft of $x$, i. e. $F=FFT(x)$, (2) compute the power spectrum $S=F*F^*$ and (3) transform it back, to get the autocorrelation $R=IFFT(S)$ – Vertex Mar 27 '15 at 18:54
• @AndrewM. your question has been answered directly, please mark it as such. – Antoine Bassoul Apr 27 '15 at 11:03

Taking the FFT of a time-series does not give you the power spectral density (PSD). What you can do is take the squared magnitude of the FFT to get an estimate of the PSD. And the latter is equivalent to taking the Fourier transform of the (deterministic) autocorrelation function of the data. See also this answer to a related question.

So the inverse Fourier transform of the PSD is the autocorrelation function.

• @AntoineBassoul: I understand now what you meant (re-reading the original question), and I added the last sentence above. This should be really clear now I hope. Thanks. – Matt L. Apr 27 '15 at 14:53