Editing Allan variance (section)

{{Short description|Measure of frequency stability in clocks and oscillators}}
{{Use dmy dates|date=June 2023}}
[[File:AllanDeviation.svg|thumb|right|upright=1.25|A clock is most easily tested by comparing it with a ''far more accurate'' reference clock. During an interval of time ''τ'', as measured by the reference clock, the clock under test advances by ''τy'', where ''y'' is the average (relative) clock frequency over that interval. If we measure two consecutive intervals as shown, we can get a value of {{nowrap|(''y'' − ''{{prime|y}}'')<sup>2</sup>}}—a smaller value indicates a more stable and precise clock. If we repeat this procedure many times, the average value of {{nowrap|(''y'' − ''{{prime|y}}'')<sup>2</sup>}} is equal to twice the Allan variance (or Allan deviation squared) for observation time ''τ''.]]

The '''Allan variance''' ('''AVAR'''), also known as '''two-sample variance''', is a measure of [[frequency stability]] in [[clock]]s, [[oscillator]]s and [[amplifier]]s. It is named after [[David W. Allan]] and expressed mathematically as <math>\sigma_y^2(\tau)</math>.
The '''Allan deviation''' ('''ADEV'''), also known as '''sigma-tau''', is the square root of the Allan variance, <math>\sigma_y(\tau)</math>.

The ''M-sample variance'' is a measure of frequency stability using ''M'' samples, time ''T'' between measurements and observation time <math>\tau</math>.  ''M''-sample variance is expressed as

:<math>\sigma_y^2(M, T, \tau).</math>

The Allan variance is intended to estimate stability due to noise processes and not that of systematic errors or imperfections such as frequency drift or temperature effects. The Allan variance and Allan deviation describe frequency stability. See also the section [[#Interpretation of value|Interpretation of value]] below.

There are also different adaptations or alterations of Allan variance,  notably the [[modified Allan variance]] MAVAR or MVAR, the [[total variance]], and the [[Hadamard variance]]. There also exist time-stability variants such as [[time deviation]] (TDEV) or time variance (TVAR). Allan variance and its variants have proven useful outside the scope of [[timekeeping]] and are a set of improved statistical tools to use whenever the noise processes are not unconditionally stable, thus a derivative exists.

The general ''M''-sample variance remains important, since it allows [[dead time]] in measurements, and bias functions allow conversion into Allan variance values.  Nevertheless, for most applications the special case of 2-sample, or "Allan variance" with <math>T = \tau</math> is of greatest interest.

[[File:AllanDeviationExample.gif|thumb|right|upright=1.25|Example plot of the Allan deviation of a clock. At very short observation time ''τ'', the Allan deviation is high due to noise. At longer ''τ'', it decreases because the noise averages out. At still longer ''τ'', the Allan deviation starts increasing again, suggesting that the clock frequency is gradually drifting due to temperature changes, aging of components, or other such factors. The error bars increase with ''τ'' simply because it is time-consuming to get a lot of data points for large ''τ''.]]

[[File:5 regimes of Allan variance as a function of averaging time.png|thumb|upright=1.25|Diagram of Allan deviation as a function of averaging time, showing the 5 typical regimes.<ref>[https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication1065.pdf NIST Special Publication 1065, Handbook of Frequency Stability Analysis. July 2008]</ref>

1. white/flicker phase-modulation noise (PM): At the highest frequency, phase noise dominates. This corresponds to <math>\sigma(\tau) \propto \tau^{-1}</math>. However, White PM has <math>S[f] = f^3</math> but Flicker PM has <math>S[f] = f^2</math>. The Allan variance plot does not distinguish them. It requires [[modified Allan variance]] plot to distinguish them.
2. White frequency-modulation noise (FM): at a lower frequency, white noise in frequency dominates. This corresponds to <math>\sigma(\tau) \propto \tau^{-1/2}, S[f] = f^0</math> 
3. Flicker FM: <math>\sigma(\tau) \propto \tau^0, S[f] \propto f^{-1}</math>. This is also called "pink noise".
4. Random Walk FM: <math>\sigma(\tau) \propto \tau^{+1/2}, S[f] \propto f^{-2}</math>. This is also called "brown noise" or "brownian noise". In this regime, the frequency of the system executes a random walk. In other words, <math>df/dt</math> becomes a white noise.
5. Frequency drift: <math>\sigma(\tau) \propto \tau^{+1}, S[f] \propto f^{-3}</math>. In this regime, the frequency of the system executes a pink noise walk. In other words, <math>df/dt</math> becomes a pink noise.]]