Allan variance

The Allan variance, named after W. David Allan, is a measure of the stability of clocks and oscillators; they also known as two-value variance. It is defined as half the average of the difference squares of two consecutive measurements of the frequency deviation.

The Allan variance depends on the temporal resolution of the data acquisition. It is thus a function of both the sample period, as well as the measured distribution, and is usually represented as a graph rather than as a single value.

A low Allan variance is a feature of a clock with high stability over the measured period.

The Allan variance is defined as

Wherein the normalized frequency deviation is averaged over the sampling period, n is the duration.

With ν the frequency and δν the frequency deviation. The average is formed where n over the sample period. A clock for the time differential is in the xn sample period n is given by the sum of the previous frequency deviations

This can be reversed to detect frequency deviations of the time differences:

This leads to the formula for the Allan variance as a time deviation:

Just as with deviation (standard deviation) and the Allan variance is defined as the square root deviation of the Allan variance.

The Allan variance is used as measure of the rate stability for a large number of partly exotic precision oscillators, such as frequency- stabilized laser. There are also some variants, especially the modified Allan variance, the total variance and the Hadamard variance. Another measure of the frequency stability is the phase noise.