Editing Allan variance (section)

==Power-law noise==
The Allan variance will treat various [[power-law noise]] types differently, conveniently allowing them to be identified and their strength estimated. As a convention, the measurement system width (high corner frequency) is denoted ''f''<sub>''H''</sub>.
{| class="wikitable"
|+ Allan variance power-law response
|-
!Power-law noise type
!Phase noise slope
!Frequency noise slope
!Power coefficient
!Phase noise<br /> <math>S_x(f)</math>
!Allan variance<br /> <math>\sigma_y^2(\tau)</math>
!Allan deviation<br /> <math>\sigma_y(\tau)</math>
|-
|white phase modulation (WPM)
|<math>f^0=1</math>
|<math>f^2</math>
|<math>h_2</math>
|<math>\frac{1}{(2\pi)^2}h_2</math>
|<math>\frac{3 f_H}{4\pi^2\tau^2}h_2</math>
|<math>\frac{\sqrt{3 f_H}}{2\pi\tau}\sqrt{h_2}</math>
|-
|flicker phase modulation (FPM)
|<math>f^{-1}</math>
|<math>f^1=f</math>
|<math>h_1</math>
|<math>\frac{1}{(2\pi)^2f}h_1</math>
|<math>\frac{3[\gamma+\ln(2\pi f_H\tau)]-\ln 2}{4\pi^2\tau^2}h_1</math>
|<math>\frac{\sqrt{3[\gamma+\ln(2\pi f_H\tau)]-\ln 2}}{2\pi\tau}\sqrt{h_1}</math>
|-
|white frequency modulation (WFM)
|<math>f^{-2}</math>
|<math>f^0=1</math>
|<math>h_0</math>
|<math>\frac{1}{(2\pi)^2f^2}h_0</math>
|<math>\frac{1}{2\tau}h_0</math>
|<math>\frac{1}{\sqrt{2\tau}}\sqrt{h_0}</math>
|-
|flicker frequency modulation (FFM)
|<math>f^{-3}</math>
|<math>f^{-1}</math>
|<math>h_{-1}</math>
|<math>\frac{1}{(2\pi)^2f^3}h_{-1}</math>
|<math>2\ln(2)h_{-1}</math>
|<math>\sqrt{2\ln(2)}\sqrt{h_{-1}}</math>
|-
|random walk frequency modulation (RWFM)
|<math>f^{-4}</math>
|<math>f^{-2}</math>
|<math>h_{-2}</math>
|<math>\frac{1}{(2\pi)^2f^4}h_{-2}</math>
|<math>\frac{2\pi^2\tau}{3}h_{-2}</math>
|<math>\frac{\pi\sqrt{2\tau}}{\sqrt{3}}\sqrt{h_{-2}}</math>
|}

As found in<ref name="NBSTN394">J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. Sydnor, R. F. C. Vessot, G. M. R. Winkler: ''[https://tf.nist.gov/general/tn1337/Tn146.PDF Characterization of Frequency Stability]'', NBS Technical Note 394, 1970.</ref><ref>J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, Jr., W. L. Smith, R. L. Sydnor, R. F. C. Vessot, G. M. R. Winkler: ''[https://tf.nist.gov/general/tn1337/Tn146.PDF Characterization of Frequency Stability]'', IEEE Transactions on Instruments and Measurements 20, pp. 105–120, 1971.</ref> and in modern forms.<ref name=Bregni2002>Bregni, Stefano: [https://books.google.com/books?id=APEBaL4WHNoC ''Synchronisation of digital telecommunication networks''], Wiley 2002, {{ISBN|0-471-61550-1}}.</ref><ref name=NISTSP1065>NIST SP 1065: [https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=50505 ''Handbook of Frequency Stability Analysis''] .</ref>

The Allan variance is unable to distinguish between WPM and FPM, but is able to resolve the other power-law noise types. In order to distinguish WPM and FPM, the [[modified Allan variance]] needs to be employed.

The above formulas assume that

:<math>\tau \gg \frac{1}{2\pi f_H},</math>

and thus that the bandwidth of the observation time is much lower than the instruments bandwidth. When this condition is not met, all noise forms depend on the instrument's bandwidth.

===''α''–''μ'' mapping===
The detailed mapping of a phase modulation of the form

:<math>S_x(f) = \frac{1}{4\pi^2} h_\alpha f^{\alpha - 2} = \frac{1}{4\pi^2} h_\alpha f^\beta,</math>

where

:<math>\beta \equiv \alpha - 2,</math>

or frequency modulation of the form

:<math>S_y(f) = h_\alpha f^\alpha</math>

into the Allan variance of the form

:<math>\sigma_y^2(\tau) = K_\alpha h_\alpha \tau^\mu</math>

can be significantly simplified by providing a mapping between ''α'' and ''μ''. A mapping between ''α'' and ''K''<sub>''α''</sub> is also presented for convenience:<ref name=IEEE1139/>

:{| class="wikitable"
|+ Allan variance ''α''–''μ'' mapping
|-
!''α''
!''β''
!''μ''
!''K''<sub>''α''</sub>
|-
| −2
| −4
| 1
|<math>\frac{2\pi^2}{3}</math>
|-
| −1
| −3
| 0
|<math>2\ln 2</math>
|-
| 0
| −2
| −1
|<math>\frac{1}{2}</math>
|-
| 1
| −1
| −2
|<math>\frac{3[\gamma+\ln(2\pi f_H\tau)]-\ln 2}{4\pi^2}</math>
|-
| 2
| 0
| −2
|<math>\frac{3f_H}{4\pi^2}</math>
|}

===General conversion from phase noise===
A signal with spectral phase noise <math>S_\varphi</math> with units rad<sup>2</sup>/Hz can be converted to Allan Variance by<ref name=NISTSP1065/>

: <math>\sigma^2_y(\tau) = \frac{2}{\nu_0^2} \int^{f_b}_0 S_\varphi(f) \frac{\sin^4(\pi \tau f)}{(\pi \tau)^2} \, df.</math>