Allan variance

Overview
The Allan variance, named after David W. Allan, is a measurement of stability in clocks and oscillators. It is also known as the two-sample variance. It is defined as one half of the time average of the squares of the differences between successive readings of the frequency deviation sampled over the sampling period. The Allan variance depends on the time period used between samples: therefore it is a function of the sample period, as well as the distribution being measured, and is displayed as a graph rather than a single number. A low Allan variance is a characteristic of a clock with good stability over the measured period.

The Allan variance is given by


 * $$\sigma_y^2(\tau) = \frac{1}{2} \langle(y_{n+1} - y_n)^2\rangle,$$

where yn is the normalized frequency departure, averaged over sample period n, and $$\tau$$ is the time per sample period. The samples are taken with no dead-time between them.


 * $$y_n = \left\langle{\delta\nu \over \nu}\right\rangle_n,$$

where ν is the frequency, δν is the frequency error, and the average is taken over sampling period n. For a clock, the time error, xn, at sampling period n, is the sum of the preceding frequency errors, given by


 * $$x_n = x_0 + \sum_{i=0}^{n-1} y_i.$$

This can be reversed to compute frequency error from time error measurements


 * $$y_n = x_{n+1} - x_n,$$

which leads to the equation for Allan variance in terms of time errors:


 * $$\sigma_y^2(\tau) = \frac{1}{2} \langle(x_{n+2} - 2 x_{n+1} + x_n)^2\rangle.$$

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

Allan variance is used as a measure of frequency stability in a variety of exotic precision oscillators, such as frequency-stabilized lasers over a period of a second or more. Short term stability (under a second) is typically expressed as phase noise. The Allan variance is also used to characterize the bias stability of gyroscopes, including fiber optic gyroscopes and MEMS gyroscopes. There are also a number of variants, notably the modified Allan variance, the total variance, and the Hadamard variance.