Allan variance, $\sigma^2[ \tau ]$, or its square root (Allan deviation, $\sigma[ \tau ]$) is a quantity (as function of parameter $\tau$) which is said to be a measure of (or related to) "stability of clocks".

For a recent example cmp. "First accuracy evaluation of NIST-F2" (T. P. Heavner et al.), especially

"Figure 1. The Total deviation (TOTDEV) of NIST-F2".

Clearly, this quantity is referring to one clock itself; e.g. "the NIST-F2" in the article.

However, the Wikipedia page points out that for "practical measurements":

"All measurements of Allan variance will in effect be the comparison of two different clocks [...] a reference clock and a device under test (DUT)."

Likewise, the article on the NIST-F2 states that

"The measurement was made using a commercial hydrogen maser as a reference."

With two clocks being involved, there's necessarily concern about clock drift; indeed:

"[...] drift will contribute to [the raw] output result. When measuring a real system, the linear drift or other drift mechanism may need to be estimated and removed from the time-series prior to calculating the Allan variance."

My question:

Can you please give an expression (as explicitly as reasonably achievable here)
of this mentioned "(raw) output result before post-processing",
in terms of

  • the total phase $\Phi_a[ t ]$ of the "device under test",

  • the total phase $\Phi_b[ t ]$ of the "reference",

  • and other parameters as needed, such as perhaps the nominal angular frequencies, $\omega_a$ and $\omega_b$, of the device under test and of the reference, respectively ?


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