Allan variance

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The Allan variance (AVAR), also known as two-sample variance is a measure of frequency stability in clocks, oscillators and amplifiers. It is named after David W. Allan. It is expressed mathematically as

The Allan deviation (ADEV), is the square root of Allan variance. It is also known as sigma-tau, and is expressed mathematically as

The M-sample variance, is a measure of frequency stability using M samples, time T between measures and observation time τ. M-sample is expressed as

The Allan variance is intended to estimate stability due to noise processes and not that of systematic errors or imperfections such as frequency drift or temperature effects. The Allan variance and Allan deviation describe frequency stability, i.e. the stability in frequency. See also the section entitled "Interpretation of value" below.

There are also different adaptations or alterations of Allan variance. Notably the modified Allan variance, the total variance, and the Hadamard variance. There also exist time stability variants such as time deviation or time variance. Allan variance and its variants have proved useful outside of the scope of timekeeping and are a set of improved statistical tools to use whenever the noise processes are not unconditionally stable, but a derivative will be.

The M-sample variance is of historic importance as well as important background but has essentially been replaced by its special case of 2-sample variance with T = τ now being called Allan variance. It remains important since it allows dead time in measurements and bias functions allows conversion into Allan variance values.