Editing Allan variance (section)

===Conventions===
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| 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.
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