Editing Allan variance (section)

==Bias functions==
The ''M''-sample variance, and the defined special case Allan variance, will experience [[systematic bias]] depending on different number of samples ''M'' and different relationship between ''T'' and ''τ''. In order to address these biases the bias-functions ''B''<sub>1</sub> and ''B''<sub>2</sub> has been defined<ref name=NBSTN375>Barnes, J. A.: [http://tf.boulder.nist.gov/general/pdf/11.pdf ''Tables of Bias Functions, ''B''<sub>1</sub> and ''B''<sub>2</sub>, for Variances Based On Finite Samples of Processes with Power Law Spectral Densities''], NBS Technical Note 375, 1969.</ref> and allows conversion between different ''M'' and ''T'' values.

These bias functions are not sufficient for handling the bias resulting from concatenating ''M'' samples to the ''Mτ''<sub>0</sub> observation time over the ''MT''<sub>0</sub> with the dead-time distributed among the ''M'' measurement blocks rather than at the end of the measurement. This rendered the need for the ''B''<sub>3</sub> bias.<ref name=NISTTN1318/>

The bias functions are evaluated for a particular μ value, so the α–μ mapping needs to be done for the dominant noise form as found using [[noise identification]]. Alternatively,<ref name=Allan1966/><ref name=NBSTN375/> the μ value of the dominant noise form may be inferred from the measurements using the bias functions.

===''B''<sub>1</sub> bias function===
The ''B''<sub>1</sub> bias function relates the ''M''-sample variance with the 2-sample variance, keeping the time between measurements ''T'' and time for each measurements ''τ'' constant. It is defined<ref name=NBSTN375/> as

:<math>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},</math>

where

:<math>r = \frac{T}{\tau}.</math>

The bias function becomes after analysis

:<math>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]}.</math>

===''B''<sub>2</sub> bias function===
The ''B''<sub>2</sub> 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<ref name=NBSTN375/> as

:<math>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},</math>

where

:<math>r = \frac{T}{\tau}.</math>

The bias function becomes after analysis

:<math>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)}.</math>

===''B''<sub>3</sub> bias function===
The ''B''<sub>3</sub> bias function relates the 2-sample variance for sample time ''MT''<sub>0</sub> and observation time ''Mτ''<sub>0</sub> with the 2-sample variance (Allan variance) and is defined<ref name=NISTTN1318>J. A. Barnes, D. W. Allan: [http://tf.boulder.nist.gov/general/pdf/878.pdf ''Variances Based on Data with Dead Time Between the Measurements''], NIST Technical Note 1318, 1990.</ref> as

:<math>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},</math>

where
:<math>T = M T_0,</math>
:<math>\tau = M \tau_0.</math>

The ''B''<sub>3</sub> bias function is useful to adjust non-overlapping and overlapping variable ''τ'' estimator values based on dead-time measurements of observation time ''τ''<sub>0</sub> and time between observations ''T''<sub>0</sub> to normal dead-time estimates.

The bias function becomes after analysis (for the ''N''&nbsp;=&nbsp;2 case)

: <math>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]},</math>

where

: <math>F(A) = 2A^{\mu+2} - (A + 1)^{\mu+2} - |A - 1|^{\mu+2}.</math>

===''τ'' bias function===
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

:<math>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}.</math>

The bias function becomes after analysis

:<math>B_\tau(\tau_1, \tau_2, \mu) = \left( \frac{\tau_2}{\tau_1} \right)^\mu.</math>

===Conversion between values===
In order to convert from one set of measurements to another the ''B''<sub>1</sub>, ''B''<sub>2</sub> and τ bias functions can be assembled. First the ''B''<sub>1</sub> function converts the (''N''<sub>1</sub>, ''T''<sub>1</sub>, ''τ''<sub>1</sub>) value into (2, ''T''<sub>1</sub>, ''τ''<sub>1</sub>), from which the ''B''<sub>2</sub> function converts into a (2, ''τ''<sub>1</sub>, ''τ''<sub>1</sub>) value, thus the Allan variance at ''τ''<sub>1</sub>. The Allan variance measure can be converted using the τ bias function from ''τ''<sub>1</sub> to ''τ''<sub>2</sub>, from which then the (2, ''T''<sub>2</sub>, ''τ''<sub>2</sub>) using ''B''<sub>2</sub> and then finally using ''B''<sub>1</sub> into the (''N''<sub>2</sub>, ''T''<sub>2</sub>, ''τ''<sub>2</sub>) variance. The complete conversion becomes

:<math>\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,</math>

where
:<math>r_1 = \frac{T_1}{r_1},</math>
:<math>r_2 = \frac{T_2}{r_2}.</math>

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

:<math>\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.</math>