recover time history from power spectral data...

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.

I appreciate if anyone can help or make a hint.

[dpb]

No can do w/o both magnitude AND phase the latter of which the PSD throws away...

Answer 1

[Sandip Kumar]

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);

Thanks, Sandip, MathWorks