Editing Allan variance (section)

==Research history==
The field of frequency stability has been studied for a long time. However, during the 1960s it was found that coherent definitions were lacking. A NASA-IEEE Symposium on Short-Term Stability in November 1964<ref name="NASA1964">NASA: [https://wayback.archive-it.org/all/20100206151245/http://hdl.handle.net/2060/19660001092] ''Short-Term Frequency Stability'', NASA-IEEE symposium on Short Term Frequency Stability Goddard Space Flight Center 23–24 November 1964, NASA Special Publication 80.</ref> resulted in the special February 1966 issue of the IEEE Proceedings on Frequency Stability.

The NASA-IEEE Symposium brought together many fields and uses of short- and long-term stability, with papers from many different contributors. The articles and panel discussions concur on the existence of the frequency flicker noise and the wish to achieve a common definition for both short-term and long-term stability.

Important papers, including those of David Allan,<ref name=Allan1966/> James A. Barnes,<ref name=Barnes1966>Barnes, J. A.: [http://tf.boulder.nist.gov/general/pdf/6.pdf ''Atomic Timekeeping and the Statistics of Precision Signal Generators''], IEEE Proceedings on Frequency Stability, Vol 54 No 2, pages 207–220, 1966.</ref> L. S. Cutler and C. L. Searle<ref name=Cutler1966/> and D. B. Leeson,<ref name=Leeson1966/> appeared in the IEEE Proceedings on Frequency Stability and helped shape the field.

David Allan's article analyses the classical ''M''-sample variance of frequency, tackling the issue of dead-time between measurements along with an initial bias function.<ref name=Allan1966/> Although Allan's initial bias function assumes no dead-time, his formulas do include dead-time calculations. His article analyses the case of M frequency samples (called N in the article) and variance estimators. It provides the now standard α–μ mapping, clearly building on James Barnes' work<ref name=Barnes1966/> in the same issue.

The 2-sample variance case is a special case of the ''M''-sample variance, which produces an average of the frequency derivative.  Allan implicitly uses the 2-sample variance as a base case, since for arbitrary chosen ''M'', values may be transferred via the 2-sample variance to the ''M''-sample variance.  No preference was clearly stated for the 2-sample variance, even if the tools were provided. However, this article laid the foundation for using the 2-sample variance as a way of comparing other ''M''-sample variances.

James Barnes significantly extended the work on bias functions,<ref name=NBSTN375/> introducing the modern ''B''<sub>1</sub> and ''B''<sub>2</sub> bias functions. Curiously enough, it refers to the ''M''-sample variance as "Allan variance", while referring to Allan's article "Statistics of Atomic Frequency Standards".<ref name=Allan1966/> With these modern bias functions, full conversion among ''M''-sample variance measures of various ''M'', ''T'' and ''τ'' values could be performed, by conversion through the 2-sample variance.

James Barnes and David Allan further extended the bias functions with the ''B''<sub>3</sub> function<ref name=NISTTN1318/> to handle the concatenated samples estimator bias. This was necessary to handle the new use of concatenated sample observations with dead-time in between.

In 1970, the IEEE Technical Committee on Frequency and Time, within the IEEE Group on Instrumentation & Measurements, provided a summary of the field, published as NBS Technical Notice 394.<ref name=NBSTN394/> This paper was first in a line of more educational and practical papers helping fellow engineers grasp the field. This paper recommended the 2-sample variance with ''T'' = ''τ'', referring to it as '''Allan variance''' (now without the quotes). The choice of such parametrisation allows good handling of some noise forms and getting comparable measurements; it is essentially the least common denominator with the aid of the bias functions ''B''<sub>1</sub> and ''B''<sub>2</sub>.

J. J. Snyder proposed an improved method for frequency or variance estimation, using sample statistics for frequency counters.<ref name=Snyder1981/> To get more effective degrees of freedom out of the available dataset, the trick is to use overlapping observation periods. This provides a {{sqrt|''n''}} improvement, and was incorporated in the '''overlapping Allan variance estimator'''.<ref name=Howe1981/> Variable-τ software processing was also incorporated.<ref name=Howe1981/> This development improved the classical Allan variance estimators, likewise providing a direct inspiration for the work on [[modified Allan variance]].

Howe, Allan and Barnes presented the analysis of confidence intervals, degrees of freedom, and the established estimators.<ref name=Howe1981/>