Allan variance

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

:$sigma\_y^2(\; au)\; =\; frac\{1\}\{2\}\; langle(y\_\{n+1\}\; -\; y\_n)^2\; angle,$

where "y"_"n" is the normalized frequency departure, averaged over sample period "n", and $au$ is the time per sample period. The samples are taken with no dead-time between them.

:$y\_n\; =\; leftlangle\{delta\; u\; over\; u\}\; ight\; angle\_n,$

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, "x_n", at sampling period "n", is the sum of the preceding frequency errors, given by

:$x\_n\; =\; x\_0\; +\; ausum\_\{i=0\}^\{n-1\}\; y\_i.$

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

:$y\_n\; =\; frac\{1\}\{\; au\}(x\_\{n+1\}\; -\; x\_n),$

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

:$sigma\_y^2(\; au)\; =\; frac\{1\}\{2\; au^2\}\; langle(x\_\{n+2\}\; -\; 2\; x\_\{n+1\}\; +\; x\_n)^2\; angle.$

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, the moving Allan Variance, the Hadamard variance, the modified Hadamard Variance, the Picinbono Variance, the Sigma-Z Variance, and others. All these variances and their variants can be categorized into the same form of stability variances, mainly, as mean-square averages of the output of a finite-difference filter acting, not on the phase or frequency samples, but on their cummulative sums.

Development

The current review process by the IEEE for standard definitions of physical quantities for fundamental frequency and time metrology (Std 1139) has been going on for a long time. This standard covers the fundamental metrology for describing random instabilities of importance to frequency and time metrology. The primary mathematical method of analysis in Std 1139 is the two-sample variance also called the Allan variance. By “two-sample” variance the IEEE really means variance based on Finite Difference (FD), a method used in numerical solutions of differential equations. The need for FD methodology in precision oscillators stems from the facts that: a) the standard sample variance and its square root (standard deviation) commonly used and often associated with normal distributions diverges numerically for typical types of random noise affecting precision oscillators; and b) the instantaneous Frequency (limit of F as sample-interval goes to zero) is a physical impossibility to measure. These typical random noise types are obviously not normally distributed and generally follow an approximated Power Law type of distribution.

However, modern statistics have given analyst in the trade the tools to develop alternate analytical techniques based solely in statistical theory, mainly, distribution theory (generalized functions). With the advent of regression distributions and stochastic analysis, analysts now have the potential to derive second-moments (variance) from corresponding moment-generating functions then use the consequent confidence interval to provide complete solutions to instability risks. This is a developmental quantum-leap in analysis from ad-hoc applications of multidisciplinary branches of mathematics to the application of pure statistics for the analysis of non-deterministic (random) instabilities.

ee also

* Variance
* Variogram
* Metrology
* Network time protocol
* Precision Time Protocol
* Synchronization

References

1. "Uncertainty of Stability Variances Based on Finite Differences", C.A.Greenhall JPL-Caltech, W.J.Riley Symmetricon, Inc. 11-19-2003.

2. "IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology - Random Instabilities", IEEE STD 1139.

3. Stochastic Calculus and Theory of Distributions, "Lecture Notes on Complex Analysis", Dr Ivan F Wilde, published by Imperial College Press.

External links

* [http://www.allanstime.com/AllanVariance/ David W. Allan's Allan Variance Overview]
* [http://www.allanstime.com David W. Allan's official web site]
* [http://www.wriley.com/ Home page of Stable32] , a very popular clock stability analysis program, which includes many useful papers.
* [http://www.leapsecond.com/museum/manyadev.gifAllan deviation plots for a variety of high-quality oscillators.]