# 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. ${\ displaystyle \ sigma _ {y} ^ {2} (\ tau)}$

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 :

${\ displaystyle \ sigma _ {y} ^ {2} (\ tau) = {\ frac {1} {2}} \ langle (y_ {n + 1} -y_ {n}) ^ {2} \ rangle}$

With

• the duration of the sample period${\ displaystyle \ tau}$
• the normalized frequency deviation , averaged over the nth sample period:${\ displaystyle y_ {n}}$${\ displaystyle y_ {n} = \ left \ langle {\ delta \ nu \ over \ nu} \ right \ rangle _ {n}}$
• the frequency deviation  δν
• the frequency  ν .

For a clock, the time deviation  x n for the nth sample period is given by the sum of the previous frequency deviations:

${\ displaystyle x_ {n} = x_ {0} + \ tau \ sum _ {i = 0} ^ {n-1} y_ {i}}$

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

${\ displaystyle \ Rightarrow y_ {n} = {\ frac {1} {\ tau}} (x_ {n + 1} -x_ {n})}$

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

${\ displaystyle \ Rightarrow \ sigma _ {y} ^ {2} (\ tau) = {\ frac {1} {2 \ tau ^ {2}}} \ langle (x_ {n + 2} -2x_ {n + 1} + x_ {n}) ^ {2} \ rangle}$

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 .