Editing Allan variance (section)

==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===
{{bulleted list
| The number of frequency samples in a fractional-frequency series is denoted by ''M''.
| The number of time error samples in a time-error series is denoted by ''N''.

The relation between the number of fractional-frequency samples and time-error series is fixed in the relationship
: <math>N = M + 1.</math>
| For [[#Time error|time-error]] sample series, ''x''<sub>''i''</sub> denotes the ''i''-th sample of the continuous time function ''x''(''t'') as given by

: <math>x_i = x(iT),</math>

where ''T'' is the time between measurements. For Allan variance, the time being used has ''T'' set to the observation time ''τ''.

The [[#Time error|time-error]] sample series let ''N'' denote the number of samples (''x''<sub>0</sub>...''x''<sub>''N''−1</sub>) in the series. The traditional convention uses index 1 through ''N''.
| For [[#Average fractional frequency|average fractional-frequency]] sample series, <math>\bar{y}_i</math> denotes the ''i''th sample of the average continuous fractional-frequency function ''y''(''t'') as given by

: <math>\bar{y}_i = \bar{y}(Ti, \tau),</math>

which gives

:<math>\bar{y}_i = \frac{1}{\tau} \int_0^\tau y(iT + t_v) \, dt_v = \frac{x(iT + \tau) - x(iT)}{\tau}.</math>

For the Allan variance assumption of ''T'' being ''τ'' it becomes

:<math>\bar{y}_i = \frac{x_{i+1} - x_i}{\tau}.</math>

The [[#Average fractional frequency|average fractional-frequency]] sample series lets ''M'' denote the number of samples (<math>\bar{y}_0 \ldots \bar{y}_{M-1}</math>) in the series. The traditional convention uses index 1 through ''M''.

As a shorthand, [[#Average fractional frequency|average fractional frequency]] is often written without the average bar over it. However, this is formally incorrect, as the [[#Fractional frequency|fractional frequency]] and [[#Average fractional frequency|average fractional frequency]] are two different functions. A measurement instrument able to produce frequency estimates with no dead-time will actually deliver a frequency-average time series, which only needs to be converted into [[#Average fractional frequency|average fractional frequency]] and may then be used directly.
| It is further a convention to let ''τ'' denote the nominal time difference between adjacent phase or frequency samples. A time series taken for one time difference ''τ''<sub>0</sub> can be used to generate Allan variance for any ''τ'' being an integer multiple of ''τ''<sub>0</sub>, in which case ''τ'' = ''nτ''<sub>0</sub> are being used, and ''n'' becomes a variable for the estimator.
| The time between measurements is denoted by ''T'', which is the sum of observation time ''τ'' and dead-time.
}}

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

:<math>\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,</math>

or for the time series:

:<math>\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.</math>

These formulas, however, only provide the calculation for the ''τ'' = ''τ''<sub>0</sub> 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''&nbsp;−&nbsp;1 samples, a new (shorter) time-series would occur with ''τ''<sub>0</sub> 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

:<math>\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</math>

with <math>n \le \frac{M - 1}{2}</math>,

and for the time series:

:<math>\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</math>

with <math>n \le \frac{N - 1}{2}</math>.

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<ref name=Snyder1981>Snyder, J. J.: ''An ultra-high resolution frequency meter'', pages 464–469, Frequency Control Symposium #35, 1981.</ref> 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.<ref name=Howe1981/> 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

:<math>
\begin{align}
\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{align}
</math>

or for the time series:

:<math>\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.</math>

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,<ref name=IEEE1139/> ITU-T<ref name=itutg810>ITU-T Rec. G.810: [http://www.itu.int/rec/dologin_pub.asp?lang=e&id=T-REC-G.810-199608-I!!PDF-E&type=items ''Definitions and terminology for synchronization and networks''], ITU-T Rec. G.810 (08/96).</ref> and ETSI<ref name=ETSIEN3004610101>ETSI EN 300 462-1-1: [http://www.etsi.org/deliver/etsi_en/300400_300499/3004620701/01.01.01_20/en_3004620701v010101c.pdf ''Definitions and terminology for synchronisation networks''], ETSI EN 300 462-1-1 V1.1.1 (1998–05).</ref> 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 (σ<sub>''x''</sub>) 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:

:<math>\sigma_x^2(\tau) = \frac{\tau^2}{3}\bmod\sigma_y^2(\tau),</math>

and similarly for modified Allan deviation to [[time deviation]]:

:<math>\sigma_x(\tau) = \frac{\tau}{\sqrt{3}}\bmod\sigma_y(\tau).</math>

The TDEV is normalized so that it is equal to the classical deviation for white PM for time constant ''τ'' =&nbsp;''τ''<sub>0</sub>.  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<sub>''z''</sub><sup>2</sup>) of a sum or difference (''z'' = ''x'' − ''y'') is the sum square of their variances (σ<sub>''z''</sub><sup>2</sup> = σ<sub>''x''</sub><sup>2</sup> + σ<sub>''y''</sub><sup>2</sup>).  The variance of the sum or difference (''y'' = ''x''<sub>2''τ''</sub> − ''x''<sub>''τ''</sub>) of two independent samples of a random variable is twice the variance of the random variable (σ<sub>''y''</sub><sup>2</sup> = 2σ<sub>''x''</sub><sup>2</sup>).  The MDEV is the second difference of independent phase measurements (''x'') that have a variance (σ<sub>''x''</sub><sup>2</sup>). Since the calculation is the double difference, which requires three independent phase measurements (''x''<sub>2''τ''</sub> − 2''x''<sub>''τ''</sub> + ''x''), the modified Allan variance (MVAR) is three times the variances of the phase measurements.

===Other estimators===
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.