Editing Allan variance (section)

==Background==
When investigating the stability of [[crystal oscillator]]s and [[atomic clock]]s, it was found that they did not have a [[phase noise]] consisting only of [[white noise]], but also of [[flicker noise|flicker frequency noise]]. These noise forms become a challenge for traditional statistical tools such as [[standard deviation]], as the estimator will not converge. The noise is thus said to be divergent. Early efforts in analysing the stability included both theoretical analysis and practical measurements.<ref name="Cutler1966">{{Citation |last1=Cutler |first1=L. S. |last2=Searle |first2=C. L. |url=http://wwwusers.ts.infn.it/~milotti/Didattica/Segnali/Cutler&Searle_1966.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://wwwusers.ts.infn.it/~milotti/Didattica/Segnali/Cutler&Searle_1966.pdf |archive-date=2022-10-09 |url-status=live |title=Some Aspects of the Theory and Measurements of Frequency Fluctuations in Frequency Standards |journal=Proceedings of the IEEE |volume=54 |number=2 |date=February 1966 |pages=136–154 |doi=10.1109/proc.1966.4627}}</ref><ref name="Leeson1966">{{Citation|last=Leeson |first=D. B |title=A simple Model of Feedback Oscillator Noise Spectrum |url=http://ccnet.stanford.edu/cgi-bin/course.cgi?cc=ee246&action=handout_download&handout_id=ID113350669026291 |pages=329–330 |journal=Proceedings of the IEEE |volume=54 |number=2 |date=February 1966 |access-date=20 September 2012 |url-status=dead |archive-url=https://web.archive.org/web/20140201231407/http://ccnet.stanford.edu/cgi-bin/course.cgi?cc=ee246&action=handout_download&handout_id=ID113350669026291 |archive-date=1 February 2014 |doi=10.1109/proc.1966.4682}}</ref>

An important side consequence of having these types of noise was that, since the various methods of measurements did not agree with each other, the key aspect of repeatability of a measurement could not be achieved. This limits the possibility to compare sources and make meaningful specifications to require from suppliers.  Essentially all forms of scientific and commercial uses were then limited to dedicated measurements, which hopefully would capture the need for that application.

To address these problems, David Allan introduced the ''M''-sample variance and (indirectly) the two-sample variance.<ref name="Allan1966">Allan, D.  [http://tf.nist.gov/general/pdf/7.pdf ''Statistics of Atomic Frequency Standards''], pages 221–230. Proceedings of the IEEE, Vol. 54, No 2, February 1966.</ref> While the two-sample variance did not completely allow all types of noise to be distinguished, it provided a means to meaningfully separate many noise-forms for time-series of phase or frequency measurements between two or more oscillators. Allan provided a method to convert between any ''M''-sample variance to any ''N''-sample variance via the common 2-sample variance, thus making all ''M''-sample variances comparable. The conversion mechanism also proved that ''M''-sample variance does not converge for large ''M'', thus making them less useful. IEEE later identified the 2-sample variance as the preferred measure.<ref name="IEEE1139">{{cite journal | doi = 10.1109/IEEESTD.1999.90575 | journal=IEEE STD 1139-1999| isbn=978-0-7381-1753-9 | year=1999 | title=IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology-Random Instabilities }}</ref>

An early concern was related to time- and frequency-measurement instruments that had a [[dead time]] between measurements. Such a series of measurements did not form a continuous observation of the signal and thus introduced a [[systematic bias]] into the measurement. Great care was spent in estimating these biases. The introduction of zero-dead-time counters removed the need, but the bias-analysis tools have proved useful.

Another early aspect of concern was related to how the [[Bandwidth (signal processing)|bandwidth]] of the measurement instrument would influence the measurement, such that it needed to be noted. It was later found that by algorithmically changing the observation <math>\tau</math>, only low <math>\tau</math> values would be affected, while higher values would be unaffected. The change of <math>\tau</math> is done by letting it be an integer multiple <math>n</math> of the measurement [[timebase]] <math>\tau_0</math>:

:<math>\tau = n \tau_0.</math>

The physics of [[crystal oscillator]]s were analyzed by D. B. Leeson,<ref name=Leeson1966/> and the result is now referred to as [[Leeson's equation]]. The feedback in the [[oscillator]] will make the [[white noise]] and [[flicker noise]] of the feedback amplifier and crystal become the [[power-law noise]]s of <math>f^{-2}</math> white frequency noise and <math>f^{-3}</math> flicker frequency noise respectively. These noise forms have the effect that the [[standard variance]] estimator does not converge when processing time-error samples. The mechanics of the feedback oscillators was unknown when the work on oscillator stability started, but was presented by Leeson at the same time as the set of statistical tools was made available by [[David W. Allan]]. For a more thorough presentation on the [[Leeson effect]], see modern phase-noise literature.<ref name="Rubiola2009">{{Citation |last=Rubiola |first=Enrico |title=Phase Noise and Frequency Stability in Oscillators |publisher=Cambridge university press |isbn=978-0-521-88677-2 |year=2008}}</ref>