# What is the difference between the PSD of a deterministic and stochastic signal?

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.

## 2 Answers

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.

• " 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 Dec 10 '17 at 16:54
• Sure, if you can wait for a few days Dec 10 '17 at 17:05
• @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.
– MBaz
Dec 10 '17 at 17:52

You need to be very careful about the differences between a "Power Spectral Density," a "Power Spectrum", a "Linear Spectral Density", and a "Linear Spectrum." Simply taking the Fourier transform of a signal does not give you power estimates, it estimates the complex amplitude of the signal at each frequency. Read Heinzel et al "Spectrum and spectral density estimation by the Discrete Fourier transform (DFT), including a comprehensive list of window functions and some new at-top windows" for a better explanation.