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I am learning about stochastic processes and I don't get one thing:

  • What is the advantage of calculating the PSD of a signal using the Wiener-Khinchin theorem $\Phi(\omega) =\mathcal{F}\{R_{xx}\}$ and calculating its PSD with $\Phi(\omega) =X(\omega) X^*(\omega)$.

I mean, if I record some random signal and I want to get its spectrum then I do $\mathcal{F}\{x(t)\}$.

  • What's wrong about using this way?
  • Why should I deal with autocorrelation and Wiener-Khinchin-Theorem?

I've created in MATLAB a cosine with random amplitude, so in my understanding this is a stochastic signal

t = 0:0.1:1000
x = rand() * cos(t) 
plot(abs(fft(xcorr(x,x))))
plot(abs(fft(x)).^2) 

Both plots seem to be very equal, I can't see the difference.

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Generally, one cannot compute the Fourier transform of any random process, because of integrability reasons. The PSD is one way to define it, based on autocorrelation.

And now, this mathematically sound definition of a periodogram is consistent in some way: if you have a realization of a random process, on a limited time interval, then its Fourier transform can be computed, and is consistent with the PSD, which you'd expect.

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    $\begingroup$ " you have a realization of a random process, on a limited time interval, then its Fourier transform can be computed, and is consistent with the PSD", it would be great if you can elaborate how consistent to PSD the FT is. Thanks $\endgroup$ – AlexTP Dec 10 '17 at 16:54
  • $\begingroup$ Sure, if you can wait for a few days $\endgroup$ – Laurent Duval Dec 10 '17 at 17:05
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    $\begingroup$ @AlexTP I don't know any theoretical results (it'd be great if Laurent can point to some). You can experiment with this, though: generate bandlimited white gaussian noise in Matlab, and find its FT. You'll see that it's not actually flat, as the theory predicts. But if you average the FT of many realizations of the noise, you'll see this average FT become flatter and flatter. $\endgroup$ – MBaz Dec 10 '17 at 17:52

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