Stability Measures for Frequency Sources - Understanding GPS: Principles and Applications - page 667

B.3 Measures of Stability

Two basic approaches can be taken to analyze the stability of an oscillator: a fre-
quency domain approach and a time domain approach. One can map from one to
the other. The time domain approach is more commonly used for stability analysis.
The interest in oscillators and the common measurement of their stability
became such an item of interest that the IEEE Standards Committee 14 developed a
standard in the 1980s. With this standard in place, oscillator stability evaluations
could be performed on a common basis using standard definitions and evaluation
techniques. The latest revision of this standard was published in 1999 [1].

B.3.1 Allan Variance
One common measure of oscillator stability based on the instantaneous fractional
frequency deviation is the Allan variance,

στ
y

2

(), defined by

[ ]

1
2

() ( )

2

2

στ
y

= Ey − y

k + 1 k

where:

φ( t +− τ) φ( t )

k k

y =

2 πν τ

0

τ = sampling interval

E is the expected value operator. In theory, E is an infinite sum of elements, but
in practice the sum is limited to a large but finite number.
The square root of the Allan variance is referred to as the Allan deviation .

B.3.2 Hadamard Variance
The Allan variance works well for cesium-based AFS with no linear drift effects. It is
also often used to characterize the stability of quartz crystal oscillators. Rubidium-
based AFS have a significant linear drift above the random noise, which degrades
the fidelity of the Allan variance and thus does not provide an accurate measure of
stability. The linear drift can be removed by a separate processing step, but an alter-
nate measure of stability has been defined which overcomes this inherent limitation
of the Allan variance. This measure is referred to as the Hadamard variance, which
removes any linear drift and is thus not effected by linear drift. Thus, the Hadamard
variance is a good measure of stability for rubidium AFS.
The Hadamard variance, H στ

2

(), is defined by

[ ]

1
2

() ( )

2

2

στ = Ey − 2 y + y

H y k + 2 k + 1 k

As in the Allan variance, E is the expected value operator. In theory, E is an infi-
nite sum of elements, but in practice the sum is limited to a large but finite number.