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While reading up on oscillator stability, I noticed that authors characterize random walk noise (Brownian noise) as having a PSD of $S_y(f) = h_{-2} f^{-2} $ where $h_{-2}$ is some constant. This is in line with what Wikipedia has to say on the power spectrum of Brownian noise . However, random walk noise is a non-stationary process, and as far as I know PSD is only defined for stationary processes. For non-stationary processes, the PSD would also be time dependent. How does this align with the above definition for the PSD for random walk processes?

My question is essentially identical to this 3 year old question from cross validated, however this doesn't appear to have been answered satisfactorily.

EDIT: I have been made aware that my question is also similar to this one: Power Spectral Density of Brownian Motion despite non-stationary

user3120921
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  • Brownian noise is given by a Wiener process, which can be constructed as multiple independent stationary increments: https://en.wikipedia.org/wiki/Wiener_process#Characterisations_of_the_Wiener_process – Envidia May 10 '21 at 17:35
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    I see now that this question was already asked here: https://dsp.stackexchange.com/questions/45574/power-spectral-density-of-brownian-motion-despite-non-stationary. Although we can't compute the PSD down to DC due to its divergence, we can deduce the PSD everywhere else from knowing the PSD of a white noise process (constant in frequency), and that the integration of white noise is a random walk process, and the power response of integration goes as $1/f^2$. – Dan Boschen May 10 '21 at 17:48
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    @DanBoschen Ahh thank you I hadn’t seen that other thread until now. I am looking through some of the sources provided in that thread, and it seems that while it isn’t stationary, it’s a stationary increment processes and exhibits some similar properties of stationary processes. It is all a bit confusing, so I think some more reading from my side is in order. – user3120921 May 10 '21 at 18:11
  • Super- after your research if you have a better answer please feel free to add it to that question. – Dan Boschen May 10 '21 at 18:29
  • "PSD defined only for stationary processes" is a common misconception. The Wiener–Khinchin theorem states that for a WSS process, the PSD is the Fourier transform of the autocorrelation function. This is sometimes (wrongly) taken as the "definition" of PSD. The PSD is defined as $$S_y (f)= \lim_{T\to\infty}\frac{1}{T}|\hat{x}_T(f)|^2$$ where $\hat{x}_T(f)$ is the spectrum of the truncated signal $x_T(t)$ in a window of time $T$. – cjferes May 10 '21 at 19:05
  • You might be interested in this as well. The paper by Kasdin is well worth reading! – Ed V May 10 '21 at 19:33
  • @cjferes Can you add that as a comment (or even an answer) under the linked duplicate? I think it would be a valuable contribution. – Dan Boschen May 10 '21 at 22:07
  • Sure, will do right away. – cjferes May 11 '21 at 18:55

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