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The Allan Variance

Clock algorithm in the generation of international time. Other metrology applications range from characterizing any precision clock or oscillator to the definition of the meter.


Page Index:
Brief Explanation | Equation | Detailed Explanation | Relevant Publications | Inspiration for | Related Sites


Brief Explanation

Allan variance equation:

Allan Variance: (sigma-sub-y)squared of tau = 1/2<(delta-y)squared>

where the variance is taken on the variable y. Each value of y in a set has been averaged over an interval J and the ys are taken in an adjacent series, i.e. no delay between the measurements of each. The brackets <> denote the expectation value. For a finite data set, it is taken as the average value of the quantity enclosed in the brackets. The )y denotes the first finite difference of the measures of y; i.e. if i denotes the ith measurement of y, then )y = yi+1 - yi. In total, each adjacent finite difference of y is squared and these then are averaged over the data set and divided by 2. The divide by two causes this variance to be equal to the classical variance if the ys are taken from a random and uncorrelated set; i.e. white noise.

The advantage of this variance over the classical variance is that it converges for most of the commonly encountered kinds of noise, whereas the classical variance does not always converge to a finite value. Flicker noise and random walk noise are two examples which commonly occur in clocks and in nature where the classical variance does not converge.

For time keeping, y is taken as the normalized rate of a clock. For example, if a wrist watch gained one second per day, then y = 1 second / 86400 seconds = 1.157e-5, and J is equal to 1 day. Notice that y is dimensionless.


For a More Detailed Explanation

Appendix A and B (pages 66-71) in HP application note 1289: "The Science of Timekeeping" are excellent tutorials for those who are beginning in their understanding of clock statistics and why the Allan variance.


Publications Relevant to the Allan Variance

See also
partial list of Independent References to the Allan Variance

See also
AVAR use in radio-astronomical instrumentation
- partial list of publications, compiled by Dr. Rudolf Schieder

starstarstar D.B. Sullivan, D.W. Allan, D.A. Howe, and F.L. Walls, Characterization of Clocks and Oscillators, NIST Tech Note 1337, 1990. (BIN: 868)

starstarstar D.W. Allan, H. Hellwig, P. Kartaschoff, J. Vanier, J. Vig, G.M.R. Winkler, and N.F. Yannoni, Standard Terminology for Fundamental Frequency and Time Metrology, Proceedings of the 42nd Annual Frequency Control Symposium, Baltimore, MD, June 1-4, 419-425, 1988. (BIN: 788)

starstarstar D.W. Allan, Statistics of Atomic Frequency Standard, Proceedings of the IEEE, 54, No. 2, 221-231, 1966. (BIN: 7)

starstarstar D.W. Allan and J.A. Barnes, A Modified "Allan Variance" with Increased Oscillator Characterization Ability, Proceedings of the 35th Annual Frequency Control Symposium,, 470-475, 1981. (BIN: 560) {click here for abstract}

starstarstar D.W. Allan, Should the Classical Variance Be Used as a Basic Measure in Standards Metrology?, IEEE Trans. on Instrumentation and Measurement, IM-36, 646-654, 1987. (BIN: 776)

starstarstar D.W. Allan, Time and Frequency (Time-Domain) Characterization, Estimation, and Prediction of Precision Clocks and Oscillators, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, UFFC-34, 647-654, 1987. (BIN: 752)

starstarstar David W. Allan, Wayne Dewey; Time-Domain Spectrum of GPS SA; Proceedings of 1993 Institute of Navigation ION GPS-93.

starstar D.W. Allan, M.A. Weiss and T.K. Peppler, In Search of the Best Clock, IEEE Transactions on Instrumentation and Measurement, 38, 624-630, 1989. (BIN: 815)

starstar D.W. Allan, Millisecond Pulsar Rivals Best Atomic Clock Stability, Proceedings of the 41st Annual Symposium on Frequency Control, Philadelphia, PA, 2-11, 1987. (BIN: 751)

star David W. Allan; The Impact of Precise Time in Our Lives: A Historical and Futuristic Perspective Surrounding GPS; 50th Anniversary Invited Talk at Institute of Navigation Annual Meeting, held in Colorado Springs, Colorado; 5-7 June 1995.

star D.W. Allan, Clock Characterization Tutorial, Proceedings of the 15th Annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting, 1983. (BIN: 662)

star F. Varnum, D.R. Brown, D.W. Allan, and T.K. Peppler, Comparison of Time Scales Generated with the NBS Ensembling Algorithm, Proceedings of the 19th Precise Time and Time Interval (PTTI) Meeting, 1987.

MORE . . . click here for list of publications relevant to the Allan Variance

Related Sites

List of Independent References to the Allan Variance
Inspiration for Allan Variance
AVAR Use in Radio-astronomical Instrumentation - Partial listing of Allan Variance-relevant papers compiled by Dr. Rudolf Schieder of the University of Cologne.
The Best Clock in the World ... and why we can't live without it (Discover Magazine's June 2000 Feature article on Boulder's Atomic Clock array)
Hamilton Technical Services [backup] - Software for frequency stability analysis. Stable32, Allan variance.
Time FAQ (from sci.astro news group) - Frequently Asked Questions about time
Today's Date and Time - across culture and through history - collection of time-keeping related links
Allan Variance Forum
Open Directory Project Listing on Time


Listing of other publications by David W. Allan

Page posted by Sterling D. Allan at least prior to March 11, 2003
Page last updated June 27, 2012
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