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It is a rather well-known fact, that measurement precision is limited by the 1/f noise of a signal [1]. One way to show this in a concise fashion, is to plot the Allan deviation of the signal. For sampling times roughly in excess of the 1/f noise corner frequency, the Allan deviation turns into a constant value, like in this figure from the above reference:

enter image description here

I have tried to study how the standard deviation evolves over time for 1/f noise signals. First, I created 1/f noise by defining the exact 1/f power spectrum with random phases and doing an inverse FFT: enter image description here

(Just for completeness: The amplitude spectrum is just a line $\frac{1}{\sqrt f}$, i.e. it has a value of $\frac{1}{\sqrt Hz}$ at 1 Hz. The sampling frequency of the signal was set to 12.5 Hz)

Then I was checking how the standard deviation and standard error of the mean (SEM) evolved depending on the measurement sample count $N$ and the sampling frequency of the individual data points $f_S$. I found that the standard deviation of the entire data vector was very nearly (maybe even exactly):

$$\sigma \approx \sqrt\frac{\ln N^2}{f_S}$$

The standard deviation slowly rises for large $N$, but the standard error of the mean falls steadily:

$$SEM \approx \sqrt\frac{\ln N^2}{f_S\cdot N}$$

I thought that the Allan deviation was sort of equivalent to the SEM. How can I solve this apparent discrepancy, that the Allan deviation suggests a precision limit in the 1/f noise signal, while the SEM keeps decreasing steadily for large $N$, almost in proportion to $\frac{1}{\sqrt N}$ like it would for white noise?

Coincidentally, I checked also brown noise, i.e. noise with a $\frac{1}{f^2}$ power spectrum, and for brown noise, the SEM indeed stays constant for increasing $N$, but the Allan deviation does not.

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For white noise (and only white noise) will the Allan Deviation (ADEV) be equivalent to the Standard Error of the mean. This is not the case for other noise types (and especially non-stationary signals where there is no actual valid mean!) which is the purpose of using ADEV in the first place.

I detail this further in this existing post which may further answer this question: https://dsp.stackexchange.com/a/87468/21048

It may also help to understand that the ADEV computation, for any given averaging interval $\tau$ has a high pass component as a moving difference (and ultimately has a bandpass frequency response as a filter to the noisy signal as the cascade of a Sinc filter (moving average) together with a comb filter (moving difference). ADEV specifically filters the signal as such and then computes the standard deviation of the resulting noise after that filter. My description here applies directly to "Overlap ADEV" or "OADEV" which converges more quickly to the same result of the original "block by block ADEV" and therefore is my preferred approach to computing ADEV typically. I describe both approaches here with the following graphic for OADEV copied again below:

Overlap ADEV

What is interesting is the area under the curve for the relevant frequency responses (A Sinc for a moving average, and the Sinc-Comb for ADEV) are the same! So if the noise is white, we will get the same result given the noise itself after processing will have the spectral shape of these filters (and the total variance is the integration of that power spectral density). But for any other spectral shape, the result will not necessarily be the same, as the OP has determined for 1/f noise specifically.

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  • $\begingroup$ So does that mean that the ADEV (and specifically the ADEV floor) is the statistically rigorous measure of precision that we can attribute to a signal? In contrast, the ever decreasing standard error of the mean provides a "false level of certainty" because the actual "mean" of the signal could be many many standard errors away from the measured mean? This is the source of my confusion. If the mean and its standard error are not guaranteed to converge on the actual mean, how are they used so widely, even in science 🤔 $\endgroup$
    – tobalt
    Jul 28, 2023 at 11:49
  • $\begingroup$ Dan, may I ask what tool you use to make these beautiful diagrams? $\endgroup$
    – Jdip
    Jul 28, 2023 at 12:26
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    $\begingroup$ @Jdip It's all with Powerpoint combined with graphics from plots I make in either Python or Octave. Most are plots I already have out of my course presentations. $\endgroup$ Jul 28, 2023 at 14:19
  • $\begingroup$ @tobalt ADEV provides a more consistent metric for non-stationary signals such as phase and frequency noise and gyroscope stability where this metric is commonly used. The mean and standard deviation does not converge for non-stationary processes. The "trick" that Allan conceived for the Clock World (where he is from and now part of my current work with Atomic Clock design) of transforming a non-stationary process to a stationary one is well known amongst statisticians. For example it is mentioned here where it is applied to financial data (without mentioning ADEV, but I have used it there): $\endgroup$ Jul 28, 2023 at 18:42
  • $\begingroup$ investopedia.com/articles/trading/07/stationary.asp Scroll down to the title "Trend and Difference Stationary". What Allan did was applied this to Clock Models and created the informative plot that shows what the resulting deviation is after the successive differences, as we vary the averaging interval over each block. Bottom line, with it we can consistently compare the stability of two different clocks (or gyroscopes, or other noise models that are potentially non-stationary). I suspect everything is non-stationary in the long term, but that is a bold statement. $\endgroup$ Jul 28, 2023 at 18:44

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