Let's say we have a realization (time-series data) of a random process. I take fft() of that to get its frequency domain representation. Then how can I estimate the PSD of this random process by having this frequency domain representation?

Note 1: It is known that the PSD is the fft() of auto-correlation function. But here we don't have access to the auto correlation function, we only have one realization.

Note 2: I want the PSD estimation approach from the 'frequency domain' representation, not from the time-domain data.

Note 3: My question is not about the relationship between system transfer function and power spectral density (i.e. it is not about the PSD of filter input and output).

My proposed answer (not sure if it is right):

Calculate the signal power in a 1 Hz bandwidth as a function of frequency. The PSD can be formed by taking the ratio of the signal power in a 1 Hz bandwidth at a specified frequency offset, $f_m$, to the signal peak amplitude at frequency $f_0$, something like that.

But what if there is no peak or multiple peaks in the the frequency domain representation?

  • $\begingroup$ I don't see any explanation of getting PSD from frequency representation in that post. The only comment about that approach is the following which is very general and vague: ["Spectrogram using short-time Fourier transform: Now you're crossing into a different domain. This is used for time-varying spectra. That is, one whose spectrum changes with time. This opens up a whole other can of worms, and there are just as many methods as you have listed for time-frequency analysis. This is certainly the cheapest, which is why its so frequently used."] So I don't think this question is a duplicate. $\endgroup$ – shampar May 7 '19 at 15:30