# Allan variance

The **Allan variance** , named after David W. Allan , also **two-** value **variance** , represents a measure of the frequency stability of clocks and oscillators : a low Allan variance is a characteristic of a clock with high stability over the measured period.

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

The Allan variance is defined as half of the mean of the squares of the differences between two consecutive measured values of the normalized frequency deviation :

With

- the duration of the sample period
- the normalized frequency deviation , averaged over the
*nth*sample period:- the frequency deviation
*δν* - the frequency
*ν*.

- the frequency deviation

For a clock, the time deviation *x _{n}* for the

*nth*sample period is given by the sum of the previous frequency deviations:

This can be reversed to determine frequency deviations from time deviations:

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

The Allan variance is used as a measure of the frequency stability for a large number of partly exotic precision oscillators, e.g. B. frequency stabilized laser used. There are also some variants, above all the modified Allan variance, the total variance and the Hadamard variance .

Analogous to the standard deviation and variance , the **Allan deviation is** defined as the square root of the Allan variance.

Another measure of frequency stability is phase noise .

## See also

## Web links

- David W. Allan's Allan Variance Overview
- David W. Allan's official web site
- Home page of Stable32 , a program for analyzing the time stability of clocks
- Allan deviation plots for a range of oscillators

## Individual evidence

- ^ WP Robins: Phase Noise in Signal Sources: Theory and Applications . IET, 1984, pp. 184 ( limited preview in Google Book search).