Talk:Allan variance

Stable32
As of fairly recently Stable32 is freely available from the IEEE UFFC website, at least in executable binary form for Windows. (https://ieee-uffc.org/frequency-control/frequency-control-software/stable32/). Stable32 is widely used in time/frequency so this link might be worth adding.

variable names
there is missing a concise explanation of variable names:

with the first equation should be explained what y, T and tau are. x appears without any explanation ... later appears the nebulous remark that y is the derivative of x - but what is x ?

the oscillator model has a variable name for the amplitude (big) V (13th letter of alpabet).

The name of frequency is (small) v (same letter).

This is even for a quality C article not sufficient.

--Janschween (talk) 14:52, 31 October 2010 (UTC)

explanation for high-school-level
Could someone add an explanation for high-school-level people as well? —Preceding unsigned comment added by 82.136.198.205 (talk) 13:15, 19 January 2008 (UTC)

Could the nature of the "frequency deviation" be explained? Michael Hardy 21:52, 2 Feb 2005 (UTC)
 * I took a stab at it.--Srleffler 05:58, 29 November 2005 (UTC)

Note: This article was originally based on text derived from a public domain entry in Federal Standard 1037C. -- The Anome 00:38, 29 November 2005 (UTC)

I have just done a general overhaul of the article. There is a few bits and pieces that certainly could be improved. I intend to add and refined the references. I think the field of Bias terms could be improved. A few nice graphs could be added, so I will attend to that somewhat later. Cfmd (talk) 15:35, 10 April 2010 (UTC)

More restructuring, revamping the definitions, adding M-sample variance, getting the definition correct, more hints to the reader being new to the field. The bias functions, error bars and edf is the main fields missing from the current article for it to cover the field sufficiently well. References needs to be worked on. The old formulations was sample-series oriented, while true definitions needs to be continuous time based and then confined into the sample-series world as a special case, which has taken some effort. Cfmd (talk) 01:58, 12 April 2010 (UTC)

Resolved the issue relating to measurement bandwidth, since it is obvious from the frequency domain representation that all forms of noise depends on the measurement bandwidth, but the formulas only let the WPM and FPM noise forms depend on it. Digging up the original references for those formulas was fruitful as it clearly state the assumption in the beginning of the Appendix A and then detailing the derivation in Appendix B. By adding the references and the assumption text and formula it becomes clear that the tabulated formulas is not exact. Additional clarification was added to the bandwidth limitation text to better bring out the point. Now I feel much more confident that the stated problem and solution is correct, while it certainly could be improved further. Also happy to bring in some more original references in order to better complete the reference aspect of the article. Cfmd (talk) 23:53, 18 April 2010 (UTC)

Added section on bias functions. This is by far completed, but provides a starting point. It also makes it bleeding obvious that the alpha-to-mu mapping that I have been avoiding needs to be presented as the bias functions uses the mu value which is not being referred to anywhere else. The bias calculators needs to be included. The classical bias reference is added.

In addition has a new section been added to provide some research history. While it at first glace may seem to duplicate the background section, it is intended to focus on the research while the background is more to help orient a new reader to what this is all about. Bias functions is to almost no use for the new reader where as they are of much more importance in the research history scope.

Once the bias functions is done, the error bars, confidence intervals and edf analysis needs to be addressed.

Another aspect to cover is the current lack of graphics. There is a wealth of graphical plots to aid the process of understanding in the traditional literature, but so far this article has none. I think this article would benefit from a few graphs, but I am resisting the obvious temptation to "borrow" from others (with associated Copyright issues).

Considering adding a section of educational and practical resources giving some guidance to the wealth of material available. Different articles and books distinguish themselves in explaining different aspects and giving short presentations on that may help the interested in finding the right material for deeper studies. Cfmd (talk) 00:22, 21 April 2010 (UTC)

The bias section is essentially complete including the B1, B2 and B3 bias functions as well as the guide to conversion between measurements. Only minor additions such as supporting references for tau bias and conversion of values.

The main areas needing attention is the alpha-mu mapping as well as confidence intervals and edf. Cfmd (talk) 16:45, 25 April 2010 (UTC)

The removal of huge part of introduction section done 2010-05-05T06:21:46 by 210.212.130.71 was contra-productive as it removed key material from the introduction. The introduction could certainly be improved, but removing those section such that the article name, common alternate definition etc. being removed does not help the reader to orient him/her-self. I have made a manual undo to restore the content. Please provide comments here instead of removing it. Cfmd (talk) 19:04, 5 May 2010 (UTC)

The alpha-mu mapping, confidence interval and effective degrees of freedom has been addressed.

The auto-correlation functions could possibly be added for completeness, as these are being used to establish the bias functions among other things. They will be helpful for completeness but most of the daily use should be covered by now. Cfmd (talk) 01:12, 9 May 2010 (UTC)

Copyright problem
This article has been revised as part of a large-scale clean-up project of multiple article copyright infringement. Earlier text must not be restored, unless it can be verified to be free of infringement. For legal reasons, Wikipedia cannot accept copyrighted text or images borrowed from other web sites or printed material; such additions must be deleted. Contributors may use sources as a source of information, but not as a source of sentences or phrases. Accordingly, the material may be rewritten, but only if it does not infringe on the copyright of the original or plagiarize from that source. Please see our guideline on non-free text for how to properly implement limited quotations of copyrighted text. Wikipedia takes copyright violations very seriously. --r.e.b. (talk) 03:05, 24 December 2009 (UTC)


 * Considering that the article now has undergone a major overhaul, which has essentially removed the old structure by freshly written material, I think that this copyright notice may be dropped as there will be less incentive to revert the removal. I will have a look at the old text and see if there is any meaningful contents not already covered. Cfmd (talk) 01:58, 12 April 2010 (UTC)

Sample indexing convention
The casual reader will notice that the convention being used here for numbering and indexing phase and average frequency samples differs from most other texts. The traditional convention numbers time error samples x_i with i from 1 to N and average fractional frequency samples y_i from 1 to M. The mapping functions lets


 * $$x_i = x(i*T)$$

which looks OK at first. But then considering that the average fractional frequency integration goes from 0 to tau will let the first sample cover the time from x(T) to x(T+tau) rather than starting at x(0) up to x(tau) which one would believe it to be and in fact what most implementations would do. IEEE 1139 has the constant x_0 added, but to no help since you expect the first average to cover x(x_0) to x(x_0 + tau). To circumvent this issue four different approaches can be used

1) Offset the mapping function with T such that it becomes


 * $$x_i = x((i-1)T)$$

which will solve the problem, but leaving the explanation of the somewhat unexplained -1 term.

2) Offset the difference function in the Allan variance such that it uses the backward frequency derivate.

This would break the relationship to the M-variance function which is not a derivate but becomes that in the special case of doing a 2-sample variance.

3) Offset the average fractional frequency function to use back-ward integration.


 * $$\bar{y}_i = \frac{1}{\tau}\int\limits_{-T}^{-T+\tau}y(t+t_v) \, dt_v = \frac{x(iT-T+\tau) - x(iT-T)}{\tau}$$

which again is inelegant. Notice how the continous x(t) function is needed as the T vs tau subtle detail can't be expressed in the indexed sample variant, another inelegance in presentation.

4) Use a straight mapping and offset the predictor to run from 0 rather than 1.

From these choices solution 4 has been selected as M-sample variance with dead-time can be made to colapase nicely into the Allan variance when M=2 and T=tau.

This convention has proven useful when implementing in base-0 indexed languages such as C.

One should also consider what happens for different taus, either by evaluating for different taus or by concatenating samples to create an extended measurement. In all these case we want the starting point to remain the same, 0 (or some offset x_0) for all these cases. Using an offset like T or tau would create a shift which needs to be cancelled correctly.

The used convention and traditional convention does not deliver different results, but the used convention is hopefully more consistent and useful over the range of variances that is included in this series of articles. I believe that the convention chosen is for best consistency and presentation solution available, at the cost of not being exactly transcribed from the referenced papers or IEEE 1139. Cfmd (talk) 22:56, 27 April 2010 (UTC)

Introduction
I moved some material around to make the introduction easier to understand. The equations were hampering the readability of the main theme of the introduction. So, I moved the equations to the end of the paragraphs. There is one paragarph that was so technical that it was opaque, to anyone who doesn't know anything about this topic. I moved that paragarph to a newly created section entitled "Technical considerations". Hopefully this helps, because the introduction needed improvements. It still needs more imporvements. Ciao. Steve Quinn (formerly Ti-30X) (talk) 05:22, 6 May 2010 (UTC)


 * I agree that there was room for improvements there.


 * The edit that made "sigma-tau" into the greek letters of that appear was also incorrect, as that term is always used in latin form. Your movement of that into formula form is further incorrect. Thus, I removed it.


 * The section name "Technical considerations" is a misnomer. I was already contemplating on moving that paragraph to a separate section, but naming it "Technical considerations" is not helpful, as the sections "Measurement issues" present one out of many sections you would expect under that title. I changed it to "Interpretation of value" as it is closer to what it attempts to achieve, namely describe what kind of layman-meaning the values has.


 * The movement of the M-sample variance presentation I think may be inappropriate. I will consider moving it back. Also, the locations of the new sections may not have been the best. Both M-sample variance and Interpretation of values may have better be put after the Background section, which is there to present the "Why do we do this" aspect.


 * When looking at the intro, moving a few sentences from the Allan variance and Allan deviation paragraphs to form new paragraphs in the Interpretation section makes the clean-up separation better. Those sentences where there help explain the values to the layman, and where needing a new location. I think that the readability of the intro becomes better and that the layman would have better use of the assisting material in one location. Considering the now much sparser intro, reintroducing the M-sample variance there makes better sense, but I have not done that so far. Cfmd (talk) 10:44, 6 May 2010 (UTC)


 * It sounds like you have some good ideas, and taken some positive action regarding this article. I say, do whatever you think is best. Steve Quinn (formerly Ti-30X) (talk) 21:49, 6 May 2010 (UTC)

- vs. –
User:Michael Hardy and User:Materialscientist makes changes to or from - and –. This location is here to have them agree on which is the correct variant and then we will use only that format convention. My goal is to see the oscillation go away. Cfmd (talk) 17:57, 8 May 2010 (UTC)
 * Upon cursory look, I haven't seen oscillations, but Michael rather fixed what I missed. Hyphens are not to be used for page ranges, minus signs, separation of sentence parts, etc. Materialscientist (talk) 00:17, 9 May 2010 (UTC)

It looks to me as if what "Materialscientist" did was to change one kind of en-dash to another, both of which look the same to the reader. He didn't change en-dashes to hyphens, as far as I can see.

en-dashes are used in ranges of pages, ranges of years, months, etc., things named after two people (e.g. the Cauchy–Riemann equations). Michael Hardy (talk) 01:16, 9 May 2010 (UTC)

Measurement class and importance
In lack of finding another way to get assessment, I simply try to set the class and importance of the article. I beleive it has matured significantly, and to the best of my knowledge it should meet the C grade at least, possibly B grade. If not, I want to learn what is missing and correct it.

As for importance, within the measurement field of time and frequency, Allan variance is a key concept. For that field, tagging the importance as mid is an understatement, where as for others outside of that field it may seem specialized. However, considering that more units directly or indirectly reverts back to the SI second, the tools to assess the stability of time becomes necessary for those fields as well, and indeed we have seen the use of Allan variance for other measurements than pure time, such as in accelerometers. Cfmd (talk) 01:01, 16 May 2010 (UTC)

Modified Allan estimators section
I reverted the removal of the section on modified Allan estimators. Please do not remove it without discussion here. There is a line of estimators achieving the modified variant of the Allan variance, but they line up to the same scale. The section is intended to explain the benefits, and is the part which goes into most detail where as the hints about it exists in more places. Cfmd (talk) 12:14, 7 June 2012 (UTC)

Allan Variance Power Law Response (table)
There seems to be a rather opaque normalisation in the table that gives equations for Allan variance/deviation in terms of Sf. That is to say: I would expect Allan deviation to be inversely proportional to the carrier frequency, but the carrier frequency does not appear anywhere in the table. It could of course be hidden in the parameter h (h-1, h-2, etc.) but this is not locally apparent. Please clarify or correct.

ThanksPhysicistQuery (talk) 18:20, 30 November 2015 (UTC)

Sorry for the delay in answering. Considering that both forms relates to normalized form for the same carrier, it is a static component in this conversion. This table exists in this form in a wide range of publications. Cfmd (talk) 20:33, 26 May 2018 (UTC)

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Effective degrees of freedom
There appear to be some typographic errors in these equations in the original source, which includes a PDF annotation referring to an erratum that I could not track down. Google scanned a series of books containing a reproduction of the original article, but that scan seems to cut off right before the page where the appropriately-numbered erratum would appear.

As a general principle, we ought to expect df to scale linearly with N when n is held constant. My intuition for offering this guideline is as follows. The Allan variance estimator takes (in effect) a second-order finite difference of the phase time-series. Among the five noise models, the one with the strongest long-time correlation behavior, the frequency random walk, is the double-integral of white noise. All of the other noise models have less spectral constant at low frequencies, and hence weaker long-time correlations. Taking a second-order finite difference of a double integral of white noise gives you back more-or-less white noise. So Allan variance samples taken far apart (compared to n) under any of the five noise models will be more-or-less independent. Thus, when N is much greater than n, the number of effective degrees of freedom had better scale linearly with N.

This general principle is violated by the expressions given in the article (as of Jan. 23, 2020) for FPM and FFM in the n=1 case.

In the FPM expression, the original source omits the inverse-square-root operation entirely. They write exp(ln * ln). That is clearly wrong; df would then scale like exp(ln(N)^2) for large N. The inverse square root does not make sense, either, scaling like exp(1/ln(N)). I have not found any source that corroborates the inverse-square-root version. The correction that makes sense is a non-inverted square root. Perhaps the original authors took the geometric mean of two fits on two different log-log plots? In any event, with the exponent changed from (-1/2) to (+1/2), the expression gains sensible asymptotic behavior for large N: unless I am mistaken, it runs like (N-1) √((2n+1)/(8n)).

As for the n=1 FFM case: The original authors cite, in an earlier paragraph, the work of a number of other authors (Lesage, Audoin, and Yoshimura). Those equations are numbered (6.5). The authors then cast some doubt upon them. Nevertheless, it appears that they intended for the n=1 FFM expression in their results, (6.6), to be a copy of that in (6.5); however, the copy-editor seems to have missed that the (N-2) expression in the numerator must be squared. Not only does this resolve the conflict between (6.5) and (6.6), it also recovers sensible asymptotic behavior for large N.

Mumble mumble original research-- I can't make the edit without locating a better source, but I definitely want to leave this breadcrumb here for the next person who gets confused. Good luck! — Preceding unsigned comment added by Piannucci (talk • contribs) 10:30, 23 January 2020 (UTC)