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I am currently studying oscillator stability and have come across the Allan variance. I gather that it was developed as an alternative to the standard variance as it doesn't necessarily converge for power law noise. However, I have a hard time seeing how this niche tool differentiates itself (or how it is superior) to other more conventional statistical tools like the autocorrelation or PSD.

Does anyone have any insight?

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My current work involves the design details of atomic clocks where we use the Allan Variance and Allan Deviation (ADEV) extensively. The primary point is that it can be used for non-stationary processes (which frequency noise is). For non-stationary signals where the autocorrelation or power spectral density can't be used with consistency, the Allan Deviation will converge to a consistent metric for comparison of the frequency accuracy of two different clocks (and generally for the comparison of the noise in non-stationary signals).

Non-stationary signals will often have a shorter time duration over which the signal can be sufficiently assumed to be stationary. Oscillator phase noise is an excellent example of this: for larger frequency offsets (typically 1 Hz and above for lower frequency oscillators such as 10MHz) a power spectral density (PSD) can be computed and given as the oscillator’s phase noise performance. The issue is this PSD cannot be computed consistently at lower frequency offsets as the non—stationary contributions start to dominate and this is where the ADEV computation will shine: in computing the longer term frequency accuracy of the clock over time durations where non-stationary contributions dominate (1/f noise, drift). ADEV is computed as an rms error given by the difference between the average frequency accuracy over interval tau seconds compared to the average frequency accuracy tau seconds ago. The rms of this error is computed over a very long duration (this error difference is a stationary signal and thus the standard deviation can be consistently computed).

One specific utility for me that the ADEV has provided well beyond the world of clocks is to determine the time constraints under which a random process can be sufficiently assumed to be stationary: if the random process is stationary we can continue to average to get a better estimate of the underlying mean— therefore we can use the ADEV to determine the optimum averaging time after which no further improvement can be gained (and through further averaging our estimate can get worst). This applies to everything from channel estimation to financial markets, wherever we may be observing past results to make an estimate of an underlying process.

I explain further details of the Allan Deviation and its potential use and application well beyond oscillator stability in these other posts:

How to interpret Allan Deviation plot for gyroscope?

What determines the accuracy of the phase result in a DFT bin?

Averaging data from 2 sensors

Is it possible to add some kind of noise to a non stationary (having unit root) signals to make them stationary?

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  • $\begingroup$ Thank you for the detailed answer. It along with your other posts were very helpful. I didn’t realize ADEV could be so helpful outside of the clock world. I am still a bit confused regarding the mathematical details as to why ADEV will converge for non-stationary processes. Could you possibly elaborate (or provide a different sources) on why this is? $\endgroup$ – user3120921 May 10 at 12:41
  • $\begingroup$ Further, I gather that ADEV can be directly related to a signal’s frequency PSD (ADEV and PSD plots are essentially mirrored), and still have a hard time seeing how it differentiates itself. $\endgroup$ – user3120921 May 10 at 12:49
  • $\begingroup$ I have a simple explanation: consider a waveform from a drifting non-stationary random-walk process (has no mean; the longer we look the more the mean drifts)- now consider a moving difference of that drifting waveform- that difference will be bound and stationary. May be best to try this experimentally and see first hand: create a RW by accumulating the output of a zero-mean white noise random variable (randn()) and observe that output and the moving difference of that output. ADEV functionally works with a moving difference and returns results vs that distance ( $\endgroup$ – Dan Boschen May 10 at 12:50
  • $\begingroup$ ADEV and PSD plots are not at all mirrored; they are completely different. In order to get one sample on the ADEV plot (at a particular tau) you need to integrate the phase noise power spectral density after weighting it by a filter shape equivalent to the differencing and averaging operation that the ADEV performs. $\endgroup$ – Dan Boschen May 10 at 12:54
  • $\begingroup$ PSD (Phase Noise) is useful to observe clock performance at higher offset frequencies (1 Hz and above, or higher for higher frequency clocks) but can’t be used at all for lower offset frequencies as that is where the signal is non-stationary so where we would instead use ADEV (for tau at 1 second and above). We can predict ADEV for lower taus from the PSD but we can not predict the PSD from the ADEV at lower taus given what I described how the two are related. $\endgroup$ – Dan Boschen May 10 at 12:57

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