Inertial Sensors and Their Applications - Signal Processing Systems - page 75

Fig. 2 An example Allan
variance plot

obey the principle of “a chain only being as strong as its weakest link”. Errors
such as interference, noise, and instability could be eliminated through chopping
or dynamic amplification and division applied to individual sensors.

2.1.2 Allan Variance

Named after Dr. David W. Allan, the Allan variance [ 1 ] is a quantity to characterize
the stability of oscillator systems. Although originally developed for frequency
standards, the Allan variance is widely used to characterize the performance of
inertial sensors; it reveals the contributions of uncorrelated and random walk type
error processes on the measurement noise. The Allan variance

A is a function of

σ

the averaging time τ , computed as

N 1

i = 1 ( y τ ( i + 1 ) − y τ ( i ))

−

1

2

A

( τ )=

σ (1)

2 ( N − 1 )

where the data y have been partitioned into N disjoint bins of length is

the average value of the i th such bin. The square root of Allan variance is known as
the Allan deviation, which is in accordance with common statistical terminology.
Usually, the Allan variance function is visualized as a log-log graph; an example
is shown in Fig. 2 . Generally, the Allan variance curve is U-shaped. At short averag-
ing times, uncorrelated noise dominates the output. The variance of independent
and identically distributed data is inversely proportional to the averaging time,
which causes a negative slope to the Allan variance at short averaging times. As
the averaging time increases, after some point, 1

τ ,and y τ ( i )

f noise starts to dominate over
uncorrelated noise and the curve levels off—the Allan variance of 1

/

f noise is
constant [ 30 ] . Eventually, the curve starts to increase due to rate random walk.
Based on the Allan variance plot, it is possible to quantify certain characteristics
of the sensor noise. The variance of white noise can be estimated as the Allan

/