# What happens if you use the Fourier transform of the autocorrelation of a non-WSS process to compute power spectral density?

The Wiener–Khinchin theorem states that the power spectral density of a wide-sense stationary stochastic process can be obtained through the Fourier transform of the autocorrelation of the signal i.e. $$S(\omega) = \int_{-\infty}^{\infty} R(\tau) e^{-j \omega \tau} d\tau$$

I would like to find out what happens when I use this theorem for non-WSS signals?

To be specific, I want to assess the validity, for non-WSS, of a derivation which uses this theorem to arrive to the conclusion that when two signals with phase noise are multiplied, the output phase noise $$S_{\phi_{IF}}$$ is related to the input phase noise $$S_{\phi}$$ by $$S_{\phi_{IF}}(f) = S_{\phi}(f) 4 \sin^2 (\frac{\alpha f}{2})$$

I am aware of the more generally applicable approach for computing PSD through Fourier decomposition but it's a little difficult to apply to the above mentioned derivation.

Thank you.

• For non-WSS signal, $R(\tau)$ does not exist. As the premise is not true, the theorem is not applicable. See en.wikipedia.org/wiki/Theorem Dec 1, 2022 at 13:50
• The autocorrelation function of any process (whether WSS or not) is defined as $R_x(t_1, t_2) = E[X(t_1)X(t_2)]$ and is a function of two variables. It is just that for WSS processes, $R_x(t_1, t_2)$ happens to depend only on $t_2-t_1$, and not upon the individual values of $t_1$ and $t_2$. And so, setting $t_1=t, t_2 = t+\tau, t_2-t_1=\tau$, we arrive at the usual expression $R_X(\tau)=E[X(t)X(t+\tau)]$ for WSS processes. So, please tell us what you mean by "the Fourier transform" of $R_x(t_1, t_2)$ of a nonWSS process. How is it defined? -1 pending edits. Dec 1, 2022 at 14:42
• Thank you for the responses. @DilipSarwate rather than having a formal definition, it is a signal I have captured with a radar receiver which I know to be non-stationary according to literature on phase-locked loop synthesisers. I have definitely computed its autocorrelation using MATLAB/Python although I have gone through theory that says the autocorrelation of a non-WSS signal doesn't exist. Dec 2, 2022 at 5:57
• The signal is likely stationary up to a certain capture length in time (in the short time frame)-- I assume you are dealing with phase noise effects which become non stationary as we approach the carrier (as would happen over long time intervals). If you include the specifics on the testing and waveforms you are working with, we may be able to help further. Dec 2, 2022 at 14:46
• Thanks @DanBoschen. I will gather up a bit more information and as soon as I have enough I'll post a comment. It's been a bit of a struggle to formally define the problem as the synthesiser, I've read, has all sorts of weird things stationary, cyclo-stationary, non-stationary, Alan variance etc. components. All I know at the moment is that the dominant phase noise sources are the reference clock and VCO which have power law phase noises i.e. $1/f^\alpha$. Dec 2, 2022 at 16:40