Allan variance

The Allan variance (AVAR), also known as two-sample variance, is a measure of frequency stability in clocks, oscillators and amplifiers. It is named after David W. Allan and expressed mathematically as σ y 2 ( τ ) {\displaystyle \sigma _{y}^{2}(\tau )} . The Allan deviation (ADEV), also known as sigma-tau, is the square root of the Allan variance, σ y ( τ ) {\displaystyle \sigma _{y}(\tau )} . The M-sample variance is a measure of frequency stability using M samples, time T between measurements and observation time τ {\displaystyle \tau } . M-sample variance is expressed as

σ y 2 ( M , T , τ ) . {\displaystyle \sigma _{y}^{2}(M,T,\tau ).}

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 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 T = τ {\displaystyle T=\tau } is of greatest interest.

Background When investigating the stability of crystal oscillators and atomic clocks, it was found that they did not have a phase noise consisting only of white noise, but also of flicker frequency noise. These noise forms become a challenge for traditional statistical tools such as standard deviation, as the estimator will not converge. The noise is thus said to be divergent. Early efforts in analyzing the stability included both theoretical analysis and practical measurements. An important side consequence of having these types of noise was that, since the various methods of measurements did not agree with each other, the key aspect of repeatability of a measurement could not be achieved. This limits the possibility to compare sources and make meaningful specifications to require from suppliers. Essentially all forms of scientific and commercial uses were then limited to dedicated measurements, which hopefully would capture the need for that application. To address these problems, David Allan introduced the M-sample variance and (indirectly) the two-sample variance. While the two-sample variance did not completely allow all types of noise to be distinguished, it provided a means to meaningfully separate many noise-forms for time-series of phase or frequency measurements between two or more oscillators. Allan provided a method to convert between any M-sample variance to any N-sample variance via the common 2-sample variance, thus making all M-sample variances comparable. The conversion mechanism also proved that M-sample variance does not converge for large M, thus making them less useful. IEEE later identified the 2-sample variance as the preferred measure. An early concern was related to time- and frequency-measurement instruments that had a dead time between measurements. Such a series of measurements did not form a continuous observation of the signal and thus introduced a systematic bias into the measurement. Great care was spent in estimating these biases. The introduction of zero-dead-time counters removed the need, but the bias-analysis tools have proved useful. Another early aspect of concern was related to how the bandwidth of the measurement instrument would influence the measurement, such that it needed to be noted. It was later found that by algorithmically changing the observation τ {\displaystyle \tau } , only low τ {\displaystyle \tau } values would be affected, while higher values would be unaffected. The change of τ {\displaystyle \tau } is done by letting it be an integer multiple n {\displaystyle n} of the measurement timebase τ 0 {\displaystyle \tau _{0}} :

τ = n τ 0 . {\displaystyle \tau =n\tau _{0}.}

The physics of crystal oscillators were analyzed by D. B. Leeson, and the result is now referred to as Leeson's equation. The feedback in the oscillator will make the white noise and flicker noise of the feedback amplifier and crystal become the power-law noises of f − 2 {\displaystyle f^{-2}} white frequency noise and f − 3 {\displaystyle f^{-3}} flicker frequency noise respectively. These noise forms have the effect that the standard variance estimator does not converge when processing time-error samples. The mechanics of the feedback oscillators was unknown when the work on oscillator stability started, but was presented by Leeson at the same time as the set of statistical tools was made available by David W. Allan. For a more thorough presentation on the Leeson effect, see modern phase-noise literature.

Interpretation of value Allan variance 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, commonly denoted as τ, likewise 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. Allan deviation is widely used for plots (conventionally in log–log format) and presentation of numbers. It is preferred, as it gives the relative amplitude stability, allowing ease of comparison with other sources of errors. An Allan deviation of 1.3×10−9 at observation time 1 s (i.e. τ = 1 s) should be interpreted as there being an instability in frequency between two observations 1 second apart with a relative root mean square (RMS) value of 1.3×10−9. For a 10 MHz clock, this would be equivalent to 13 mHz RMS movement. If the phase stability of an oscillator is needed, then the time deviation variants should be consulted and used. One may convert the Allan variance and other time-domain variances into frequency-domain measures of time (phase) and frequency stability.

Formulations

M-sample variance Given a time-series x ( t ) {\displaystyle x(t)} , for any positive real numbers T , τ {\displaystyle T,\tau } , define the real number sequence y ¯ i = x ( i T + τ ) − x ( i T ) τ i = 0 , 1 , 2 , . . . {\displaystyle {\bar {y}}_{i}={\frac {x(iT+\tau )-x(iT)}{\tau }}\quad i=0,1,2,...} Then the M {\displaystyle M} -sample variance is defined (here in a modernized notation form) as the Bessel-corrected variance of the sequence y ¯ 0 , . . . , y ¯ M − 1 {\displaystyle {\bar {y}}_{0},...,{\bar {y}}_{M-1}} : σ y 2 ( M , T , τ ) = M M − 1 ( 1 M ∑ i = 0 M − 1 y ¯ i 2 − [ 1 M ∑ i = 0 M − 1 y ¯ i ] 2 ) , {\displaystyle \sigma _{y}^{2}(M,T,\tau )={\frac {M}{M-1}}\left({\frac {1}{M}}\sum _{i=0}^{M-1}{\bar {y}}_{i}^{2}-\left[{\frac {1}{M}}\sum _{i=0}^{M-1}{\bar {y}}_{i}\right]^{2}\right),} The interpretation of the symbols is as follows:

t {\displaystyle t} is the reading on a reference clock (in arbitrary units).

x ( t ) {\displaystyle x(t)} is the reading of a clock we are testing (in arbitrary units), as a function of the reference clock's reading. It can also be interpreted as the average fractional frequency time series.

y ¯ n {\displaystyle {\bar {y}}_{n}} is the nth fractional frequency average over the observation time τ {\displaystyle \tau } .

M {\displaystyle M} is the number of clock reading intervals used in computing the M {\displaystyle M} -sample variance,

T {\displaystyle T} is the time between each frequency sample,

τ {\displaystyle \tau } is the time length of each frequency estimate, or the observation period. Dead-time can be accounted for by letting the time T {\displaystyle T} be different from that of τ {\displaystyle \tau } .

Allan variance The Allan variance is defined as

σ y 2 ( τ ) = ⟨ σ y 2 ( 2 , τ , τ ) ⟩ = 1 2 ⟨ ( y ¯ n + 1 − y ¯ n ) 2 ⟩ = 1 2 τ 2 ⟨ ( x n + 2 − 2 x n + 1 + x n ) 2 ⟩ {\displaystyle \sigma _{y}^{2}(\tau )=\left\langle \sigma _{y}^{2}(2,\tau ,\tau )\right\rangle ={\frac {1}{2}}\left\langle \left({\bar {y}}_{n+1}-{\bar {y}}_{n}\right)^{2}\right\rangle ={\frac {1}{2\tau ^{2}}}\left\langle \left(x_{n+2}-2x_{n+1}+x_{n}\right)^{2}\right\rangle }

where x n := x ( n τ ) {\displaystyle x_{n}:=x(n\tau )} and ⟨ ⋯ ⟩ {\displaystyle \langle \dotsm \rangle } denotes the expectation operator. The condition T = τ {\textstyle T=\tau } means the samples are taken with no dead-time between them.

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

σ y ( τ ) = σ y 2 ( τ ) . {\displaystyle \sigma _{y}(\tau )={\sqrt {\sigma _{y}^{2}(\tau )}}.}

Supporting definitions

Oscillator model The oscillator being analysed is assumed to follow the basic model of

V ( t ) = V 0 sin ⁡ ( Φ ( t ) ) . {\displaystyle V(t)=V_{0}\sin(\Phi (t)).}

The oscillator is assumed to have a nominal frequency of ν n {\displaystyle \nu _{\text{n}}} , given in cycles per second (SI unit: hertz). The nominal angular frequency ω n {\displaystyle \omega _{\text{n}}} (in radians per second) is given by

ω n = 2 π ν n . {\displaystyle \omega _{\text{n}}=2\pi \nu _{\text{n}}.}

The total phase can be separated into a perfectly cyclic component ω n t {\displaystyle \omega _{\text{n}}t} , along with a fluctuating component φ ( t ) {\displaystyle \varphi (t)} :

Φ ( t ) = ω n t + φ ( t ) = 2 π ν n t + φ ( t ) . {\displaystyle \Phi (t)=\omega _{\text{n}}t+\varphi (t)=2\pi \nu _{\text{n}}t+\varphi (t).}

Time error The time-error function x(t) is the difference between expected nominal time and actual normal time:

x ( t ) = φ ( t ) 2 π ν n = Φ ( t ) 2 π ν n − t = T ( t ) − t . {\displaystyle x(t)={\frac {\varphi (t)}{2\pi \nu _{\text{n}}}}={\frac {\Phi (t)}{2\pi \nu _{\text{n}}}}-t=T(t)-t.}

For measured values a time-error series TE(t) is defined from the reference time function Tref(t) as

T E ( t ) = T ( t ) − T ref ( t ) . {\displaystyle TE(t)=T(t)-T_{\text{ref}}(t).}

Frequency function The frequency function ν ( t ) {\displaystyle \nu (t)} is the frequency over time, defined as

ν ( t ) = 1 2 π d Φ ( t ) d t . {\displaystyle \nu (t)={\frac {1}{2\pi }}{\frac {d\Phi (t)}{dt}}.}

Fractional frequency The fractional frequency y(t) is the normalized difference between the frequency ν ( t ) {\displaystyle \nu (t)} and the nominal frequency ν n {\displaystyle \nu _{\text{n}}} :

y ( t ) = ν ( t ) − ν n ν n = ν ( t ) ν n − 1. {\displaystyle y(t)={\frac {\nu (t)-\nu _{\text{n}}}{\nu _{\text{n}}}}={\frac {\nu (t)}{\nu _{\text{n}}}}-1.}

Average fractional frequency The average fractional frequency is defined as

y ¯ ( t , τ ) = 1 τ ∫ 0 τ y ( t + t v ) d t v , {\displaystyle {\bar {y}}(t,\tau )={\frac {1}{\tau }}\int _{0}^{\tau }y(t+t_{v})\,dt_{v},}

where the average is taken over observation time τ, the y(t) is the fractional-frequency error at time t, and τ is the observation time. Since y(t) is the derivative of x(t), we can without loss of generality rewrite it as

y ¯ ( t , τ ) = x ( t + τ ) − x ( t ) τ . {\displaystyle {\bar {y}}(t,\tau )={\frac {x(t+\tau )-x(t)}{\tau }}.}

Estimators This definition is based on the statistical expected value, integrating over infinite time. The real-world situation does not allow for such time-series, in which case a statistical estimator needs to be used in its place. A number of different estimators will be presented and discussed.

Conventions

Fixed τ estimators A first simple estimator would be to directly translate the definition into

σ y 2 ( τ , M ) = AVAR ⁡ ( τ , M ) = 1 2 ( M − 1 ) ∑ i = 0 M − 2 ( y ¯ i + 1 − y ¯ i ) 2 , {\displaystyle \sigma _{y}^{2}(\tau ,M)=\operatorname {AVAR} (\tau ,M)={\frac {1}{2(M-1)}}\sum _{i=0}^{M-2}({\bar {y}}_{i+1}-{\bar {y}}_{i})^{2},}

or for the time series:

σ y 2 ( τ , N ) = AVAR ⁡ ( τ , N ) = 1 2 τ 2 ( N − 2 ) ∑ i = 0 N − 3 ( x i + 2 − 2 x i + 1 + x i ) 2 . {\displaystyle \sigma _{y}^{2}(\tau ,N)=\operatorname {AVAR} (\tau ,N)={\frac {1}{2\tau ^{2}(N-2)}}\sum _{i=0}^{N-3}(x_{i+2}-2x_{i+1}+x_{i})^{2}.}

These formulas, however, only provide the calculation for the τ = τ0 case. To calculate for a different value of τ, a new time-series needs to be provided.

Non-overlapped variable τ estimators Taking the time-series and skipping past n − 1 samples, a new (shorter) time-series would occur with τ0 as the time between the adjacent samples, for which the Allan variance could be calculated with the simple estimators. These could be modified to introduce the new variable n such that no new time-series would have to be generated, but rather the original time series could be reused for various values of n. The estimators become

σ y 2 ( n τ 0 , M ) = AVAR ⁡ ( n τ 0 , M ) = 1 2 M − 1 n ∑ i = 0 M − 1 n − 1 ( y ¯ n i + n − y ¯ n i ) 2 {\displaystyle \sigma _{y}^{2}(n\tau _{0},M)=\operatorname {AVAR} (n\tau _{0},M)={\frac {1}{2{\frac {M-1}{n}}}}\sum _{i=0}^{{\frac {M-1}{n}}-1}\left({\bar {y}}_{ni+n}-{\bar {y}}_{ni}\right)^{2}}

with n ≤ M − 1 2 {\displaystyle n\leq {\frac {M-1}{2}}} , and for the time series:

σ y 2 ( n τ 0 , N ) = AVAR ⁡ ( n τ 0 , N ) = 1 2 n 2 τ 0 2 ( N − 1 n − 1 ) ∑ i = 0 N − 1 n − 2 ( x n i + 2 n − 2 x n i + n + x n i ) 2 {\displaystyle \sigma _{y}^{2}(n\tau _{0},N)=\operatorname {AVAR} (n\tau _{0},N)={\frac {1}{2n^{2}\tau _{0}^{2}\left({\frac {N-1}{n}}-1\right)}}\sum _{i=0}^{{\frac {N-1}{n}}-2}\left(x_{ni+2n}-2x_{ni+n}+x_{ni}\right)^{2}}

with n ≤ N − 1 2 {\displaystyle n\leq {\frac {N-1}{2}}} . These estimators have a significant drawback in that they will drop a significant amount of sample data, as only 1/n of the available samples is being used.

Overlapped variable τ estimators A technique presented by J. J. Snyder provided an improved tool, as measurements were overlapped in n overlapped series out of the original series. The overlapping Allan variance estimator was introduced by Howe, Allan and Barnes. This can be shown to be equivalent to averaging the time or normalized frequency samples in blocks of n samples prior to processing. The resulting predictor becomes

σ y 2 ( n τ 0 , M ) = AVAR ⁡ ( n τ 0 , M ) = 1 2 n 2 ( M − 2 n + 1 ) ∑ j = 0 M − 2 n ( ∑ i = j j + n − 1 y i + n − y i ) 2 = 1 2 ( M − 2 n + 1 ) ∑ j = 0 M − 2 n ( y ¯ j + n − y ¯ j ) 2 , {\displaystyle {\begin{aligned}\sigma _{y}^{2}(n\tau _{0},M)&=\operatorname {AVAR} (n\tau _{0},M)={\frac {1}{2n^{2}(M-2n+1)}}\sum _{j=0}^{M-2n}\left(\sum _{i=j}^{j+n-1}y_{i+n}-y_{i}\right)^{2}\\[5pt]&={\frac {1}{2(M-2n+1)}}\sum _{j=0}^{M-2n}\left({\bar {y}}_{j+n}-{\bar {y}}_{j}\right)^{2},\end{aligned}}}

or for the time series:

σ y 2 ( n τ 0 , N ) = AVAR ⁡ ( n τ 0 , N ) = 1 2 n 2 τ 0 2 ( N − 2 n ) ∑ i = 0 N − 2 n − 1 ( x i + 2 n − 2 x i + n + x i ) 2 . {\displaystyle \sigma _{y}^{2}(n\tau _{0},N)=\operatorname {AVAR} (n\tau _{0},N)={\frac {1}{2n^{2}\tau _{0}^{2}(N-2n)}}\sum _{i=0}^{N-2n-1}(x_{i+2n}-2x_{i+n}+x_{i})^{2}.}

The overlapping estimators have far superior performance over the non-overlapping estimators, as n rises and the time-series is of moderate length. The overlapped estimators have been accepted as the preferred Allan variance estimators in IEEE, ITU-T and ETSI standards for comparable measurements such as needed for telecommunication qualification.

Modified Allan variance In order to address the inability to separate white phase modulation from flicker phase modulation using traditional Allan variance estimators, an algorithmic filtering reduces the bandwidth by n. This filtering provides a modification to the definition and estimators and it now identifies as a separate class of variance called modified Allan variance. The modified Allan variance measure is a frequency stability measure, just as is the Allan variance.

Time stability estimators A time stability (σx) statistical measure, which is often called the time deviation (TDEV), can be calculated from the modified Allan deviation (MDEV). The TDEV is based on the MDEV instead of the original Allan deviation, because the MDEV can discriminate between white and flicker phase modulation (PM). The following is the time variance estimation based on the modified Allan variance:

σ x 2 ( τ ) = τ 2 3