I'm working on generating noise signals $X(t)$ (with $t \in \left[0,T\right]$ with step size $\delta t$) with a prescribed power spectral density $S_{XX}(f)$ and I'm figuring out how well I am generating said noise; I need to know if it has the desired spectral density (in case someone notices, I posted about similar issues a while back. I've since then worked on a different topic and I am now returning to this).

However, estimating the spectral density of a signal is a tedious process. It involves a lot of estimation through periodograms and window functions and such, and it is therefore not so easy to quantify numerically if the spectrum not only has the desired trends but actually implements the desired parameters (in practice I will simply use a spectrum analyzer, but that is a different story).

The auto covariance function $C_{XX}(t') = \left<X(t) X(t+t') \right>$ on the other hand is very easy to calculate and subsequently fit. Moreover, the auto covariance function and the spectral density are related by a Fourier transform: $C_{XX}(t') = \int_0^\infty{S_{XX}(f)\cos{2\pi f}}$ (note that I'm using the cosine transform as this is customary in these settings, due to the evenness of the spectral density).

So my question is, how much do I learn about the spectral density by fitting the auto covariance? Say I am working with the Ornstein Uhlenbeck process, which has auto covariance $C_{XX}(t') = \frac{c \tau}{2}e^{-t'/\tau}$ and spectral density $S_{XX}(f) = \frac{2 c \tau^2}{1+(2\pi f \tau)^2}$. If I fit my numerically generated $X(t)$ to $C_{XX}$ to extract $c$ and $\tau$ and am able to do so with high accuracy, do I then know that my spectral density is also as desired?

Intuitively I would say yes, Fourier pairs contain the exact same information. The reason I am not sure if this is the case is due to the fact that I am unsure of how sampling rates and such come into this. Of course one cannot expect a spectral density to be accurate up to the GHz range if one has an $X(t)$ point every millisecond. But is this equally true for the auto covariance? In other words, if my autocovariance is accurate up to some specific lag time (which means I have a high enough sampling rate I suppose), is the spectral density then also accurate up to some specific frequency? And can I quantify this in some way?

What I mean by that last sentence is, can I tell up to (or down to) what frequencies I can trust my spectral density, given my auto covariance fit, $T$ and $\delta t$?

  • $\begingroup$ I know this question is old, but I wonder what you concluded regarding this issue. I'm in the same situation, I'm generating some noise and I want to test that it satisfies the prescriptions. $\endgroup$ – Pincopallino Aug 31 '18 at 14:09
  • $\begingroup$ In the end I ended up using periodograms, I believe with Welch's method, to directly look at the power spectral density instead of the autocovariance. $\endgroup$ – user129412 Sep 7 '18 at 7:51

The autocovariance and spectral density are Fourier transform pairs. Therefore, the autocovariance function and spectral density are equivalent.

Have a look to the definitions and examples for white noise in this presentation.

  • $\begingroup$ Thanks for the post! I agree, that does make sense. Note that I also wrote this, that intuitively I would say yes, Fourier pairs contain the exact same information. What I'm wondering about more is how the two correspond with regards to time and frequency components; for example it is not so easy to accurately put low frequency terms in a short time trace (with high sampling rate). I am not sure if that affects the autocovariance in the same way, trend wise. I understand that the information is equivalent, but does it affect the same part of the curve? $\endgroup$ – user129412 Apr 14 '16 at 19:14
  • $\begingroup$ To clarify a bit, say I have some exponentially decaying autocorrelation. Clearly the more important part here is the short timescales, for longer times it has just decayed. But on the other hand the spectral density is a lorentzian around zero, where a lot of the important information is contained at lower frequencies (and so intuitively at longer timescales?). That doesn't feel the same to me, and I hope this clarifies a bit what I mean by the above. In the sense that while the information might be the same, I am not sure if the details of the important features of the functions are the same $\endgroup$ – user129412 Apr 14 '16 at 19:16

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