Allan variance

Given a time-series ${\displaystyle x(t)}$, for any positive real numbers ${\displaystyle T,\tau }$, define the real number sequence

$${\displaystyle {\bar {y}}_{i}={\frac {x(iT+\tau )-x(iT)}{\tau }}\quad i=0,1,2,...}$$

Then the ${\displaystyle M}$-sample variance is defined[4] (here in a modernized notation form) as the Bessel-corrected variance of the sequence ${\displaystyle {\bar {y}}_{0},...,{\bar {y}}_{M-1}}$:

$${\displaystyle \sigma _{y}^{2}(M,T,\tau )={\frac {M}{M-1}}\left({\frac {1}{M}}\sum _{i=0}^{M-1}{\bar {y}}_{i}^{2}-\left[{\frac {1}{M}}\sum _{i=0}^{M-1}{\bar {y}}_{i}\right]^{2}\right),}$$

The interpretation of the symbols is as follows:

• ${\displaystyle t}$ is the reading on a reference clock (in arbitrary units).
• ${\displaystyle x(t)}$ 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.
• ${\displaystyle {\bar {y}}_{n}}$ is the nth fractional frequency average over the observation time ${\displaystyle \tau }$.
• ${\displaystyle M}$ is the number of clock reading intervals used in computing the ${\displaystyle M}$-sample variance,
• ${\displaystyle T}$ is the time between each frequency sample,
• ${\displaystyle \tau }$ is the time length of each frequency estimate, or the observation period.

Dead-time can be accounted for by letting the time ${\displaystyle T}$ be different from that of ${\displaystyle \tau }$.

The Allan variance is defined as

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

where ${\displaystyle \langle \dotsm \rangle }$ denotes the expectation operator.

The condition ${\textstyle T=\tau }$ means the samples are taken with no dead-time between them.

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

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

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

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

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

${\displaystyle \omega _{\text{n}}=2\pi \nu _{\text{n}}.}$

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

${\displaystyle \Phi (t)=\omega _{\text{n}}t+\varphi (t)=2\pi \nu _{\text{n}}t+\varphi (t).}$

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

${\displaystyle x(t)={\frac {\varphi (t)}{2\pi \nu _{\text{n}}}}={\frac {\Phi (t)}{2\pi \nu _{\text{n}}}}-t=T(t)-t.}$

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

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

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

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

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

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

The average fractional frequency is defined as

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

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

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

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

${\displaystyle \sigma _{y}^{2}(\tau ,M)=\operatorname {AVAR} (\tau ,M)={\frac {1}{2(M-1)}}\sum _{i=0}^{M-2}({\bar {y}}_{i+1}-{\bar {y}}_{i})^{2},}$

or for the time series:

${\displaystyle \sigma _{y}^{2}(\tau ,N)=\operatorname {AVAR} (\tau ,N)={\frac {1}{2\tau ^{2}(N-2)}}\sum _{i=0}^{N-3}(x_{i+2}-2x_{i+1}+x_{i})^{2}.}$

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.

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

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

with ${\displaystyle n\leq M-1}$,

and for the time series:

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

with ${\displaystyle n\leq {\frac {N-1}{2}}}$.

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.

A technique presented by J. J. Snyder[8] 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.[9] 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

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

or for the time series:

${\displaystyle \sigma _{y}^{2}(n\tau _{0},N)=\operatorname {AVAR} (n\tau _{0},N)={\frac {1}{2n^{2}\tau _{0}^{2}(N-2n)}}\sum _{i=0}^{N-2n-1}(x_{i+2n}-2x_{i+n}+x_{i})^{2}.}$

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,[5] ITU-T[10] and ETSI[11] standards for comparable measurements such as needed for telecommunication qualification.

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.

A time stability (σ_x) 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:

${\displaystyle \sigma _{x}^{2}(\tau )={\frac {\tau ^{2}}{3}}{\bmod {\sigma }}_{y}^{2}(\tau ),}$

and similarly for modified Allan deviation to time deviation:

${\displaystyle \sigma _{x}(\tau )={\frac {\tau }{\sqrt {3}}}{\bmod {\sigma }}_{y}(\tau ).}$

The TDEV is normalized so that it is equal to the classical deviation for white PM for time constant τ = τ₀. To understand the normalization scale factor between the statistical measures, the following is the relevant statistical rule: For independent random variables X and Y, the variance (σ_z²) of a sum or difference (z = x − y) is the sum square of their variances (σ_z² = σ_x² + σ_y²). The variance of the sum or difference (y = x_2τ − x_τ) of two independent samples of a random variable is twice the variance of the random variable (σ_y² = 2σ_x²). The MDEV is the second difference of independent phase measurements (x) that have a variance (σ_x²). Since the calculation is the double difference, which requires three independent phase measurements (x_2τ − 2x_τ + x), the modified Allan variance (MVAR) is three times the variances of the phase measurements.

Further developments have produced improved estimation methods for the same stability measure, the variance/deviation of frequency, but these are known by separate names such as the Hadamard variance, modified Hadamard variance, the total variance, modified total variance and the Theo variance. These distinguish themselves in better use of statistics for improved confidence bounds or ability to handle linear frequency drift.

The confidence interval can be established using chi-squared distribution by using the distribution of the sample variance:[5][9]

${\displaystyle \chi ^{2}={\frac {{\text{df}}\,s^{2}}{\sigma ^{2}}},}$

where s² is the sample variance of our estimate, σ² is the true variance value, df is the degrees of freedom for the estimator, and χ² is the degrees of freedom for a certain probability. For a 90% probability, covering the range from the 5% to the 95% range on the probability curve, the upper and lower limits can be found using the inequality

${\displaystyle \chi ^{2}(0.05)\leq {\frac {{\text{df}}\,s^{2}}{\sigma ^{2}}}\leq \chi ^{2}(0.95),}$

which after rearrangement for the true variance becomes

${\displaystyle {\frac {{\text{df}}\,s^{2}}{\chi ^{2}(0.95)}}\leq \sigma ^{2}\leq {\frac {{\text{df}}\,s^{2}}{\chi ^{2}(0.05)}}.}$

The degrees of freedom represents the number of free variables capable of contributing to the estimate. Depending on the estimator and noise type, the effective degrees of freedom varies. Estimator formulas depending on N and n has been found empirically:[9]

Allan variance degrees of freedom

Noise type	degrees of freedom
white phase modulation (WPM)	${\displaystyle {\text{df}}\cong {\frac {(N+1)(N-2n)}{2(N-n)}}}$
flicker phase modulation (FPM)	${\displaystyle {\text{df}}\cong \exp \left[\left(\ln {\frac {N-1}{2n}}\ln {\frac {(2n+1)(N-1)}{4}}\right)^{-1/2}\right]}$
white frequency modulation (WFM)	${\displaystyle {\text{df}}\cong \left[{\frac {3(N-1)}{2n}}-{\frac {2(N-2)}{N}}\right]{\frac {4n^{2}}{4n^{2}+5}}}$
flicker frequency modulation (FFM)	${\displaystyle {\text{df}}\cong {\begin{cases}{\frac {2(N-2)}{2.3N-4.9}}&n=1\\{\frac {5N^{2}}{4n(N+3n)}}&n\geq 2\end{cases}}}$
random-walk frequency modulation (RWFM)	${\displaystyle {\text{df}}\cong {\frac {N-2}{n}}{\frac {(N-1)^{2}-3n(N-1)+4n^{2}}{(N-3)^{2}}}}$

The detailed mapping of a phase modulation of the form

${\displaystyle S_{x}(f)={\frac {1}{4\pi ^{2}}}h_{\alpha }f^{\alpha -2}={\frac {1}{4\pi ^{2}}}h_{\alpha }f^{\beta },}$

where

${\displaystyle \beta \equiv \alpha -2,}$

or frequency modulation of the form

${\displaystyle S_{y}(f)=h_{\alpha }f^{\alpha }}$

into the Allan variance of the form

${\displaystyle \sigma _{y}^{2}(\tau )=K_{\alpha }h_{\alpha }\tau ^{\mu }}$

can be significantly simplified by providing a mapping between α and μ. A mapping between α and K_α is also presented for convenience:[5]

Allan variance α–μ mapping

α	β	μ	Kα
−2	−4	1	${\displaystyle {\frac {2\pi ^{2}}{3}}}$
−1	−3	0	${\displaystyle 2\ln 2}$
0	−2	−1	${\displaystyle {\frac {1}{2}}}$
1	−1	−2	${\displaystyle {\frac {3[\gamma +\ln(2\pi f_{H}\tau )]-\ln 2}{4\pi ^{2}}}}$
2	0	−2	${\displaystyle {\frac {3f_{H}}{4\pi ^{2}}}}$

A signal with spectral phase noise ${\displaystyle S_{\varphi }}$ with units rad²/Hz can be converted to Allan Variance by[15]

${\displaystyle \sigma _{y}^{2}(\tau )={\frac {2}{\nu _{0}^{2}}}\int _{0}^{f_{b}}S_{\varphi }(f){\frac {\sin ^{4}(\pi \tau f)}{(\pi \tau )^{2}}}\,df.}$

The B₁ bias function relates the M-sample variance with the 2-sample variance (Allan variance), keeping the time between measurements T and time for each measurements τ constant. It is defined[16] as

${\displaystyle B_{1}(N,r,\mu )={\frac {\left\langle \sigma _{y}^{2}(N,T,\tau )\right\rangle }{\left\langle \sigma _{y}^{2}(2,T,\tau )\right\rangle }},}$

where

${\displaystyle r={\frac {T}{\tau }}.}$

The bias function becomes after analysis

${\displaystyle B_{1}(N,r,\mu )={\frac {1+\sum _{n=1}^{N-1}{\frac {N-n}{N(N-1)}}\left[2(rn)^{\mu +2}-(rn+1)^{\mu +2}-|rn-1|^{\mu +2}\right]}{1+{\frac {1}{2}}\left[2r^{\mu +2}-(r+1)^{\mu +2}-|r-1|^{\mu +2}\right]}}.}$

The B₂ bias function relates the 2-sample variance for sample time T with the 2-sample variance (Allan variance), keeping the number of samples N = 2 and the observation time τ constant. It is defined[16] as

${\displaystyle B_{2}(r,\mu )={\frac {\left\langle \sigma _{y}^{2}(2,T,\tau )\right\rangle }{\left\langle \sigma _{y}^{2}(2,\tau ,\tau )\right\rangle }},}$

where

${\displaystyle r={\frac {T}{\tau }}.}$

The bias function becomes after analysis

${\displaystyle B_{2}(r,\mu )={\frac {1+{\frac {1}{2}}\left[2r^{\mu +2}-(r+1)^{\mu +2}-|r-1|^{\mu +2}\right]}{2\left(1-2^{\mu }\right)}}.}$

The B₃ bias function relates the 2-sample variance for sample time MT₀ and observation time Mτ₀ with the 2-sample variance (Allan variance) and is defined[17] as

${\displaystyle B_{3}(N,M,r,\mu )={\frac {\left\langle \sigma _{y}^{2}(N,M,T,\tau )\right\rangle }{\left\langle \sigma _{y}^{2}(N,T,\tau )\right\rangle }},}$

where

${\displaystyle T=MT_{0},}$

${\displaystyle \tau =M\tau _{0}.}$

The B₃ bias function is useful to adjust non-overlapping and overlapping variable τ estimator values based on dead-time measurements of observation time τ₀ and time between observations T₀ to normal dead-time estimates.

The bias function becomes after analysis (for the N = 2 case)

${\displaystyle B_{3}(2,M,r,\mu )={\frac {2M+MF(Mr)-\sum _{n=1}^{M-1}(M-n)\left[2F(nr)-F{\big (}(M+n)r{\big )}+F{\big (}(M-n)r{\big )}\right]}{M^{\mu +2}[F(r)+2]}},}$

where

${\displaystyle F(A)=2A^{\mu +2}-(A+1)^{\mu +2}-|A-1|^{\mu +2}.}$

While formally not formulated, it has been indirectly inferred as a consequence of the α–µ mapping. When comparing two Allan variance measure for different τ, assuming same dominant noise in the form of same µ coefficient, a bias can be defined as

${\displaystyle B_{\tau }(\tau _{1},\tau _{2},\mu )={\frac {\left\langle \sigma _{y}^{2}(2,\tau _{2},\tau _{2})\right\rangle }{\left\langle \sigma _{y}^{2}(2,\tau _{1},\tau _{1})\right\rangle }}.}$

The bias function becomes after analysis

${\displaystyle B_{\tau }(\tau _{1},\tau _{2},\mu )=\left({\frac {\tau _{2}}{\tau _{1}}}\right)^{\mu }.}$

In order to convert from one set of measurements to another the B₁, B₂ and τ bias functions can be assembled. First the B₁ function converts the (N₁, T₁, τ₁) value into (2, T₁, τ₁), from which the B₂ function converts into a (2, τ₁, τ₁) value, thus the Allan variance at τ₁. The Allan variance measure can be converted using the τ bias function from τ₁ to τ₂, from which then the (2, T₂, τ₂) using B₂ and then finally using B₁ into the (N₂, T₂, τ₂) variance. The complete conversion becomes

${\displaystyle \left\langle \sigma _{y}^{2}(N_{2},T_{2},\tau _{2})\right\rangle =\left({\frac {\tau _{2}}{\tau _{1}}}\right)^{\mu }\left[{\frac {B_{1}(N_{2},r_{2},\mu )B_{2}(r_{2},\mu )}{B_{1}(N_{1},r_{1},\mu )B_{2}(r_{1},\mu )}}\right]\left\langle \sigma _{y}^{2}(N_{1},T_{1},\tau _{1})\right\rangle ,}$

where

${\displaystyle r_{1}={\frac {T_{1}}{r_{1}}},}$

${\displaystyle r_{2}={\frac {T_{2}}{r_{2}}}.}$

Similarly, for concatenated measurements using M sections, the logical extension becomes

${\displaystyle \left\langle \sigma _{y}^{2}(N_{2},M_{2},T_{2},\tau _{2})\right\rangle =\left({\frac {\tau _{2}}{\tau _{1}}}\right)^{\mu }\left[{\frac {B_{3}(N_{2},M_{2},r_{2},\mu )B_{1}(N_{2},r_{2},\mu )B_{2}(r_{2},\mu )}{B_{3}(N_{1},M_{1},r_{1},\mu )B_{1}(N_{1},r_{1},\mu )B_{2}(r_{1},\mu )}}\right]\left\langle \sigma _{y}^{2}(N_{1},M_{1},T_{1},\tau _{1})\right\rangle .}$

A measurement system is expected to have a bandwidth at or below that of the Nyquist rate, as described within the Shannon–Hartley theorem. As can be seen in the power-law noise formulas, the white and flicker noise modulations both depends on the upper corner frequency ${\displaystyle f_{H}}$ (these systems is assumed to be low-pass filtered only). Considering the frequency filter property, it can be clearly seen that low-frequency noise has greater impact on the result. For relatively flat phase-modulation noise types (e.g. WPM and FPM), the filtering has relevance, whereas for noise types with greater slope the upper frequency limit becomes of less importance, assuming that the measurement system bandwidth is wide relative the ${\displaystyle \tau }$ as given by

${\displaystyle \tau \gg {\frac {1}{2\pi f_{H}}}.}$

When this assumption is not met, the effective bandwidth ${\displaystyle f_{H}}$ needs to be notated alongside the measurement. The interested should consult NBS TN394.[12]

If, however, one adjust the bandwidth of the estimator by using integer multiples of the sample time ${\displaystyle n\tau _{0}}$, then the system bandwidth impact can be reduced to insignificant levels. For telecommunication needs, such methods have been required in order to ensure comparability of measurements and allow some freedom for vendors to do different implementations. The ITU-T Rec. G.813[18] for the TDEV measurement.

It can be recommended that the first ${\displaystyle \tau _{0}}$ multiples be ignored, such that the majority of the detected noise is well within the passband of the measurement systems bandwidth.

Further developments on the Allan variance was performed to let the hardware bandwidth be reduced by software means. This development of a software bandwidth allowed addressing the remaining noise, and the method is now referred to modified Allan variance. This bandwidth reduction technique should not be confused with the enhanced variant of modified Allan variance, which also changes a smoothing filter bandwidth.

Many measurement instruments of time and frequency have the stages of arming time, time-base time, processing time and may then re-trigger the arming. The arming time is from the time the arming is triggered to when the start event occurs on the start channel. The time-base then ensures that minimal amount of time goes prior to accepting an event on the stop channel as the stop event. The number of events and time elapsed between the start event and stop event is recorded and presented during the processing time. When the processing occurs (also known as the dwell time), the instrument is usually unable to do another measurement. After the processing has occurred, an instrument in continuous mode triggers the arm circuit again. The time between the stop event and the following start event becomes dead time, during which the signal is not being observed. Such dead time introduces systematic measurement biases, which needs to be compensated for in order to get proper results. For such measurement systems will the time T denote the time between the adjacent start events (and thus measurements), while ${\displaystyle \tau }$ denote the time-base length, i.e. the nominal length between the start and stop event of any measurement.

Dead-time effects on measurements have such an impact on the produced result that much study of the field have been done in order to quantify its properties properly. The introduction of zero-dead-time counters removed the need for this analysis. A zero-dead-time counter has the property that the stop event of one measurement is also being used as the start event of the following event. Such counters create a series of event and time timestamp pairs, one for each channel spaced by the time-base. Such measurements have also proved useful in order forms of time-series analysis.

Measurements being performed with dead time can be corrected using the bias function B₁, B₂ and B₃. Thus, dead time as such is not prohibiting the access to the Allan variance, but it makes it more problematic. The dead time must be known, such that the time between samples T can be established.

Studying the effect on the confidence intervals that the length N of the sample series have, and the effect of the variable τ parameter n the confidence intervals may become very large, since the effective degree of freedom may become small for some combination of N and n for the dominant noise form (for that τ).

The effect may be that the estimated value may be much smaller or much greater than the real value, which may lead to false conclusions of the result.

It is recommended that the confidence interval is plotted along with the data, such that the reader of the plot is able to be aware of the statistical uncertainty of the values.

It is recommended that the length of the sample sequence, i.e. the number of samples N is kept high to ensure that confidence interval is small over the τ range of interest.

It is recommended that the τ range as swept by the τ₀ multiplier n is limited in the upper end relative N, such that the read of the plot is not being confused by highly unstable estimator values.

It is recommended that estimators providing better degrees of freedom values be used in replacement of the Allan variance estimators or as complementing them where they outperform the Allan variance estimators. Among those the total variance and Theo variance estimators should be considered.

A large number of conversion constants, bias corrections and confidence intervals depends on the dominant noise type. For proper interpretation shall the dominant noise type for the particular τ of interest be identified through noise identification. Failing to identify the dominant noise type will produce biased values. Some of these biases may be of several order of magnitude, so it may be of large significance.

Systematic effects on the signal is only partly cancelled. Phase and frequency offset is cancelled, but linear drift or other high-degree forms of polynomial phase curves will not be cancelled and thus form a measurement limitation. Curve fitting and removal of systematic offset could be employed. Often removal of linear drift can be sufficient. Use of linear-drift estimators such as the Hadamard variance could also be employed. A linear drift removal could be employed using a moment-based estimator.

Traditional instruments provided only the measurement of single events or event pairs. The introduction of the improved statistical tool of overlapping measurements by J. J. Snyder[8] allowed much improved resolution in frequency readouts, breaking the traditional digits/time-base balance. While such methods is useful for their intended purpose, using such smoothed measurements for Allan variance calculations would give a false impression of high resolution,[19][20][21] but for longer τ the effect is gradually removed, and the lower-τ region of the measurement has biased values. This bias is providing lower values than it should, so it is an overoptimistic (assuming that low numbers is what one wishes) bias, reducing the usability of the measurement rather than improving it. Such smart algorithms can usually be disabled or otherwise circumvented by using time-stamp mode, which is much preferred if available.

All measurements of Allan variance will in effect be the comparison of two different clocks. Consider a reference clock and a device under test (DUT), and both having a common nominal frequency of 10 MHz. A time-interval counter is being used to measure the time between the rising edge of the reference (channel A) and the rising edge of the device under test.

In order to provide evenly spaced measurements, the reference clock will be divided down to form the measurement rate, triggering the time-interval counter (ARM input). This rate can be 1 Hz (using the 1 PPS output of a reference clock), but other rates like 10 Hz and 100 Hz can also be used. The speed of which the time-interval counter can complete the measurement, output the result and prepare itself for the next arm will limit the trigger frequency.

A computer is then useful to record the series of time differences being observed.

The recorded time-series require post-processing to unwrap the wrapped phase, such that a continuous phase error is being provided. If necessary, logging and measurement mistakes should also be fixed. Drift estimation and drift removal should be performed, the drift mechanism needs to be identified and understood for the sources. Drift limitations in measurements can be severe, so letting the oscillators become stabilized, by long enough time being powered on, is necessary.

The Allan variance can then be calculated using the estimators given, and for practical purposes the overlapping estimator should be used due to its superior use of data over the non-overlapping estimator. Other estimators such as total or Theo variance estimators could also be used if bias corrections is applied such that they provide Allan variance-compatible results.

To form the classical plots, the Allan deviation (square root of Allan variance) is plotted in log–log format against the observation interval τ.

The time-interval counter is typically an off-the-shelf counter commercially available. Limiting factors involve single-shot resolution, trigger jitter, speed of measurements and stability of reference clock. The computer collection and post-processing can be done using existing commercial or public-domain software. Highly advanced solutions exists, which will provide measurement and computation in one box.