If I have the time history of the energy, Does it make sense to do the fourier transform of this energy? or if I want to see the energy in frequency the PSD is the only tool? and Why
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$\begingroup$ The field concerning itself with estimating the PSD is called "spectral estimation", and there's more ways than doing a Fourier transform of sampled data. You'll need to be Way more specific about where your data actually comes from, if possible with a mathematical model for the underlying phenomenon and the noise (but we can work together on that!), and a much more precise description of what you want to do with the PSD estimate. $\endgroup$– Marcus MüllerCommented Mar 28, 2022 at 13:41
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$\begingroup$ See DSP as toolset: you'll need to know what you want to build out of which material to choose the right tool. $\endgroup$– Marcus MüllerCommented Mar 28, 2022 at 13:41
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$\begingroup$ This really depends on what you want to do with the data and what the requirements of your application are. The spectrum of energy is a bit unusual. If your signal is a sine wave, the spectrum of the sine wave has a line at a single frequency. The spectrum of the energy (signal squared) has two lines: one at DC and one at twice the signal frequency. Is that what you want ? $\endgroup$– HilmarCommented Mar 28, 2022 at 14:18
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It could makes sense to do a Fourier Transform for the kinetic energy vs time (or of any parameter vs time) for evaluation of the power spectral density when the process is wide-sense-stationary, or at least stationary over the block of time used in the FFT (which specifically is the FT of the autocorrelation function and can be estimated from the magnitude squared of the DFT, normalized by the resolution bandwidth). JasonR further explains this here. For all other cases, the PSD does not exist and you will get a very different result for any shift in time or change in duration. (This is a motivation to instead use the STFT or wavelet transform to observe both time and frequency characteristics).
To assess if the process is sufficiently stationary, I recommend using the Allan Deviation which I detail in other posts such as here for that purpose.
answered Mar 28, 2022 at 15:15