Editing Allan variance (section)

==Supporting definitions==

===Oscillator model===

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

: <math>V(t) = V_0 \sin (\Phi(t)).</math>

The oscillator is assumed to have a nominal frequency of <math>\nu_\text{n}</math>, given in cycles per second (SI unit: [[hertz]]). The nominal [[angular frequency]] <math>\omega_\text{n}</math> (in radians per second) is given by

: <math>\omega_\text{n} = 2\pi \nu_\text{n}.</math>

The total phase can be separated into a perfectly cyclic component <math>\omega_\text{n} t</math>, along with a fluctuating component <math>\varphi(t)</math>:

: <math>\Phi(t) = \omega_\text{n}t + \varphi(t) = 2\pi \nu_\text{n}t + \varphi(t).</math>

===Time error===
The time-error function ''x''(''t'') is the difference between expected nominal time and actual normal time:

: <math>x(t) = \frac{\varphi(t)}{2\pi \nu_\text{n}} = \frac{\Phi(t)}{2\pi \nu_\text{n}} - t = T(t) - t.</math>

For measured values a time-error series TE(''t'') is defined from the reference time function ''T''{{sub|ref}}(''t'') as

: <math>TE(t) = T(t) - T_\text{ref}(t).</math>

===Frequency function===
The frequency function <math>\nu(t)</math> is the frequency over time, defined as

: <math>\nu(t) = \frac{1}{2\pi} \frac{d\Phi(t)}{dt}.</math>

===Fractional frequency===
The fractional frequency ''y''(''t'') is the normalized difference between the frequency <math>\nu(t)</math> and the nominal frequency <math>\nu_\text{n}</math>:

:<math>y(t) = \frac{\nu(t) - \nu_\text{n}}{\nu_\text{n}} = \frac{\nu(t)}{\nu_\text{n}} - 1.</math>

===Average fractional frequency===
The average fractional frequency is defined as

:<math>\bar{y}(t, \tau) = \frac{1}{\tau} \int_0^\tau y(t + t_v) \, dt_v,</math>

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

:<math>\bar{y}(t, \tau) = \frac{x(t + \tau) - x(t)}{\tau}.</math>