Notes on Allan Variance
O. B. Alam
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853
(Dated: July 10, 2014)
I.
INTRODUCTION
The Allan variance is a statistical tool that is used
to characterize an oscillator’s white frequency noise and
flicker frequency (1/f ) noise. It allows for the separation
of multiple noise-forms in time-domain measurements of
frequency and phase. This cannot be achieved with the
standard variance, whose estimator function fails to converge for data sets with f −2 and f −3 noise. Statistically,
an oscillator with a lower Allan variance has a smaller
spread of frequency data points over a given observation
time. A smaller variance implies greater frequency stability over that time period.
II.
MATHEMATICAL DEFINITION
III.
PHYSICAL INTERPRETATION
If an oscillator has an Allan deviation σy at observation time τ, then it has a root-mean-square frequency
instability of σy . For example, a 10 MHz oscillator with
σy = 5.6 · 10−9 and τ = 1 second has 56 mHz RMS noise.
IV.
IN THE FREQUENCY DOMAIN
Given a normalized frequency f, and power spectral
density Sy (f ), the Allan variance in the frequency domain is given by
σy2 (τ )
Z
=2
Sy (f )
0
The instantaneous frequency under study is represented by the formula
1 dφ
+ f0
(1)
2π dt
where φ is the phase of the signal and f0 is the oscillator’s
intrinsic nominal frequency. The fractional frequency error is defined as
∆f
f (t) − f0
1 dφ
y(t) ≡
.
(2)
=
=
f0
f0
ω0 dt
f (t) =
where ω0 is the angular nominal frequency. Now, for
some observation period τ, the Allan variance is given by
E
1D
2
σy2 (τ ) ≡
(ȳn+1 − ȳn )
(3)
2
where the bar designates the average fractional frequency
error over the observation time τ :
Z
1 τ
y(t + nτ ) dt.
(4)
ȳn =
τ 0
In effect, there are three steps in (3) to compute the Allan
deviation:
1. Take the average fractional frequency errors ȳk in
the observation time intervals kτ.
2. Square the difference between two consecutive averages.
3. Take one half the overall ensemble average by computing the summation of the squares of the differences, and dividing by the total sample size.
The result is the Allan variance of the frequency, which is
traditionally expressed in its square-root form, the Allan
deviation σy , and plotted in a log-log scale against τ .
fH
sin4 (πf t)
df
(πf t)2
where fH is the system cutoff frequency.
(5)