Editing Allan variance (section)

==Time and frequency filter properties==
In analysing the properties of Allan variance and friends, it has proven useful to consider the filter properties on the normalize frequency. Starting with the definition for Allan variance for

:<math>\sigma_y^2(\tau) = \frac{1}{2}\left\langle\left(\bar{y}_{i+1} - \bar{y}_i\right)^2\right\rangle,</math>

where

:<math>\bar{y}_i = \frac{1}{\tau} \int_0^\tau y(i\tau + t) \, dt.</math>

Replacing the time series of <math>y_i</math> with the Fourier-transformed variant <math>S_y(f)</math> the Allan variance can be expressed in the frequency domain as

:<math>\sigma_y^2(\tau) = \int_0^\infty S_y(f) \frac{2\sin^4\pi\tau f}{(\pi \tau f)^2} \, df.</math>

Thus the transfer function for Allan variance is

:<math>\left\vert H_A(f)\right\vert^2 = \frac{2\sin^4\pi \tau f}{(\pi \tau f)^2}.</math>