Allan Variance - Research History

Research History

The field of frequency stability has been studied for a long time, however it was found during the 1960s that there was a lack of coherent definitions. The NASA-IEEE Symposium on Short-Term Stability in 1964 was followed with the IEEE Proceedings publishing a special issue on Frequency Stability in its February 1966 issue.

The NASA-IEEE Symposium on Short-Term Stability in November 1964 brings together many fields and uses of short and long term stability with papers from many different contributors. The articles and panel discussions is interesting in that they concur on the existence of the frequency flicker noise and the wish for achieving a common definition for short and long term stability (even if the conference name only reflect the short-term stability intention).

The IEEE proceedings on Frequency Stability 1966 included a number of important papers including those of David Allan, James A. Barnes, L. S. Cutler and C. L. Searle and D. B. Leeson. These papers helped shape the field.

The classical M-sample variance of frequency was analysed by David Allan in along with a initial bias function. This paper tackles the issues of dead-time between measurements and analyses the case of M frequency samples (called N in the paper) and variance estimators. It provides the now standard α to µ mapping. It clearly builds on James Barnes work as detailed in his article in the same issue. The initial bias functions introduced assumes no dead-time, but the formulas presented includes dead-time calculations. The bias function assumes the use of the 2-sample variance as a base-case, since any other variants of M may be chosen and values may be transferred via the 2-sample variance to any other variance for of arbitrary M. Thus, the 2-sample variance was only implicitly used and not clearly stated as the preference even if the tools where provided. It however laid the foundation for using the 2-sample variance as the base case of comparison among other variants of the M-sample variance. The 2-sample variance case is a special case of the M-sample variance which produces an average of the frequency derivative.

The work on bias functions was significantly extended by James Barnes in in which the modern B₁ and B₂ bias functions was introduced. Curiously enough it refers to the M-sample variance as "Allan variance" while referencing to. With these modern bias functions full conversion among M-sample variance measures of variating M, T and τ values could used, by conversion through the 2-sample variance.

James Barnes and David Allan further extended the bias functions with the B₃ function in to handle the concatenated samples estimator bias. This was necessary to handle the new use of concatenated sample observations with dead time in between.

The IEEE Technical Committee on Frequency and Time within the IEEE Group on Instrumentation & Measurements provided a summary of the field in 1970 published as NBS Technical Notice 394. This paper could be considered first in a line of more educational and practical papers aiding the fellow engineers in grasping the field. In this paper the 2-sample variance with T = τ is being the recommended measurement and it is referred to as Allan variance (now without the quotes). The choice of such parametrisation allows good handling of some noise forms and to get comparable measurements, it is essentially the least common denominator with the aid of the bias functions B₁ and B₂.

An improved method for using sample statistics for frequency counters in frequency estimation or variance estimation was proposed by J.J. Snyder. The trick to get more effective degrees of freedom out of the available dataset was to use overlapping observation periods. This provides a square-root n improvement. It was included into the overlapping Allan variance estimator introduced in. The variable τ software processing was also included in. This development improved the classical Allan variance estimators likewise providing a direct inspiration going into the work on modified Allan variance.

The confidence interval and degrees of freedom analysis, along with the established estimators was presented in.