Question
Hello having the power spectral density data (like what we get from pwelch) is it possible to retrieve the original time history. I have obtained the spectrum using pwelch, have done some modifications directly on the spectrum and now want to find the time history data which can build such a spectrum, actually going the reverse path from spectrum to the original signal.
Expert Answer
John Williams
PhD Expert
Answered Aug 23, 2026
The mapping from Power Spectrum back to time history is not unique. By definition, power spectrum has no phase information in there. Here is a simple example that two time series arrives at the same power spectrum. A filtering method could get a time series that generates the desired power spectrum, however there is no guarantee that it is the original time series.
t = 0:0.01:0.99; x1 = sin(2*pi*10*t); x2 = sin(2*pi*10*t+pi/6); P1 = periodogram(x1,[],100); P2 = periodogram(x2,[],100); max(abs(P1-P2)) figure; subplot(211); periodogram(x1); subplot(212); periodogram(x2);
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