Allan Plot of an Oscillator with Deterministic Perturbations

​
white frequency noise amplitude
0.05
oscillation amplitude
0.012
oscillation period (s)
100
drift (1/s)
5.×
-6
10
The Allan deviation
σ
y
(τ)
is a statistical measure of the relative frequency instability of an oscillator on a sampling period
τ
. This Demonstration simulates an oscillator having white frequency noise plus deterministic perturbations. The upper graph shows the time series of relative frequency fluctuations
y(t)=(ν(t)-
ν
0
)/
ν
0
and the lower graph shows the corresponding Allan plot
σ
y
(τ)
. You can modify the noise amplitude, add a drift or an oscillation, and observe the resulting change in Allan deviation. Each of the three perturbations contributes to the Allan deviation as follows: the white frequency noise gives rise to
σ
y
(τ)∝
-1/2
τ
, the linear drift
y(t)=dt
adds a contribution
σ
y
(τ)=
-1/2
2
dτ
(green curve), and the oscillation
y(t)=bcos(2πt/T)
results in
σ
y
(τ)=b
2
sin(πτ/T)
(πτ/T)
(red curve).

Details

Snapshot 1: oscillator with white frequency noise
Snapshot 2: oscillator with white frequency noise plus a deterministic oscillation
Snapshot 3: oscillator with white frequency noise plus a linear drift
The Allan deviation
σ
y
(τ)
is the square root of the Allan variance defined by:
2
σ
y
(τ)=
1
2

2

τ
y
n+1
-
τ
y
n


,
where
τ
y
n
is the
th
n
relative frequency fluctuations average over the integration time
τ
.

References

[1] C. Audoin and B. Guinot, The Measurement of Time, Cambridge: Cambridge University Press, 2001.
[2] J. Vanier and C. Audoin, The Quantum Physics of Atomic Frequency Standards, Philadelphia: Hilger, 1989.

Permanent Citation

Gianni Di Domenico
​
​"Allan Plot of an Oscillator with Deterministic Perturbations"​
​http://demonstrations.wolfram.com/AllanPlotOfAnOscillatorWithDeterministicPerturbations/​
​Wolfram Demonstrations Project​
​Published: March 7, 2011