6 Atomic Clock Background
has an influence on how we make sense of the world. Of course, the ultimate test is success in predicting and manipulating the real world, but we must remember that there is a degree of arbitrariness.
From empirical usage, we prefer to treat the frequency error, ∆ν, as a random stationary process, such that for eacht, ∆ν(t) is a random variable whose joint distribution across different times depends only on the time difference, ∆t.
Lastly, to aid comparisons, it is useful to define the fractional/relative (or normalized) frequency noise as,
y(t)≡ ∆ν ν0
.
It is this quantity that is treated as our fundamental random variable and becomes the basis of the Allan variance [1].
6.4.1 Statistics
Now that we have decided to focus ony(t), let us discuss how to quantify the frequency noise of our model oscillator and how to collect statistics from real oscillators.
Interval Averaging
We begin by looking at the average frequency over some interval (tk, tk +τ). The averaging process begins at timetkand lastsτ long. Experimentally this is implemented by counting the number of oscillations within the interval and dividing by the length of the interval. Hence in terms of the model,
ν(t)t
k,τ =ν0+ 1 τ
Z tk+τ tk
∆ν(t)dt.
We can write the average frequency in terms of the fractional frequency noise.
ν(t)t
k,τ =ν0
1 + 1 τ
Z tk+τ tk
∆ν(t) ν0
dt
(6.3)
=ν0
1 + 1 τ
Z tk+τ tk
y(t)dt
(6.4)
=ν0 1 +ytk,τ
, (6.5)
80
6.4 Modelling and Randomness where we have introducedyt
k,τ = 1τRtk+τ
tk y(t)dt, the average fractional frequency noise over the interval.
Ensemble Averaging and Variance
With y(t) as our random variable we may impose assumptions, our first being a zero mean‡, for each t,
hy(t)i= 0.
Here the average is an ensemble average, over infinitely many runs of the experiment.
Theτ averaging of the random variabley(t) still leavesytk,τ a random variable. But the zero mean carries through, regardless of the interval length τ, because of the ensemble average,
ytk,τ
= 0.
With the mean of the noise eliminated, we next introduce the variance of the noise. It is the variance that turns out to be our main window into the workings of noise. Indeed from signal processing theory, the variance of a signal is very important and is called by analogy to many physical situations, the expected power of the signal.
Thus, the variance ofytk,τ is
σ2[ytk,τ] = y2tk,τ
,
where we have used the zero mean to simplify the standard definition of the variance.
For a given tk and τ and having taken the ensemble average, the variance is a single number. It is the true mean of the square of the average of the fractional frequency noise over (tk, tk+τ).
For a stationary process, as we are assuming, σ2[ytk,τ] is actually independent of the starting time tk. Hence the convention to define a new function as follows,
I2(τ)≡σ2[yt0,τ].
‡In general, if there is some simple systematic drift, it could still be handled by first removing the drift before analysis.
6 Atomic Clock Background
Taking the square-root yields
I(τ) = q
σ2[yt0,τ],
which, in words, is the true root mean square of theτ-averaged fractional noise.
If we take the limitτ →0, we get I(τ)→p
hyi, the instantaneous root mean square of y(t). If we take the other extreme τ → ∞, we get I(τ)→0, which is just another way of saying (assuming ergodicity) that the y(t) has zero mean. That is the τ-averaging is over a long enough interval that the noise fluctuations wash away to zero before being squared and ensemble averaged.
Allan Variance
In trying to measureI(τ) for real world oscillators we have to contend with finite sample sizes. Since variance estimators can be biased or unbiased§ and the variance of the variance can behave differently for different sample sizes; and since it is important to be able to compare different frequency sources, a convention needs to be established.
We have already assumed stationarity and if we add the ergodicity assumption, then we are able to takeτ-long averages, one after another, without “restarting” the experiment tot0#. During oneτ-long run we are feverishly counting the number of oscillations after which we produce only one sample ofytk,τ, so we need to specify how many τ runs, the gap between runs and the manner of combining them to arrive at a variance estimate.
The IEEE [6] has recommended the following variance estimator:
σb2y(τ)≡ 1
2(yt1,τ −yt0,τ)2, t1 =t0+τ.
Notice that the second interval starts immediately after the first interval with zero, so called, dead time. This estimator is itself a random variable, and its expectation is what completes our quest for a time-domain measure of frequency stability, called the Allan
§Sometimes, whether an estimator is biased even depends on the underlying noise distribution.
#Restarting is in some sense not even theoretically possible because it would be restarting the universe to the same conditions. Then again, the scientific method presupposes the ability to prepare an identical closed system afresh. However, when we are dealing with noise we are in general assuming that we are working with an open system where we have no control over the environment. The environment cannot be reset, so we rather assume ergodicity and see how far it takes us.
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6.4 Modelling and Randomness
variance,
σy2(τ)≡ σby2(τ)
.
Notice that in terms of notationk, the expectation of the estimator “removes” the hat and tells us something about the underlying variance. Indeed, this estimator was chosen so that its expectation is unbiased, with respect to white noise,
σy2(τ)white=I2(τ).
For other types of noise the Allan variance is in general biased, σy2(τ) = 2(I2(τ)−I2(2τ))6=I2(τ).
Ironically, this bias is actually helpful for some noise models. For example, it is useful when the true variance I2 is unbounded and yet the Allan variance (with the infinities
‘cancelling’) is finite! In this way the Allan variance can still be used to help identify and characterize these cases.
Ultimately, because we can’t take an infinite expectation in practice, we are forced to approximate, as best as possible, the ensemble average in the Allan variance. So even though one estimate calls for only two τ-averaged runs, we still require many estimates in order that we may build up enough statistics to accurately approximate the ensemble average,
\σ2y(τ)(m) =\ bσy2(τ)
(m) = 1
2(m−1)
m−2
X
i=0
(yti+1,τ −yti,τ)2, ti+1=ti+τ.
Them−2 term is there, because we are counting from zero, and them−1 term appears because we can only get m−1 estimates of the Allan variance from m τ-long runs∗∗. This mean isalso its own separate estimator (and a random variable) whose (empirical) variance may be used to plot the error bars in our estimate of the Allan variance. Thus
kThe bar over theyis absent from this widely used notation. We use it to remind us of theτ-averaging ofy. The bar is left out, since, withτ being specified, the averaging process is implied anyway and our bar notation is redundant (though still a useful visual aid).
∗∗It has nothing to do with making the mean unbiased as in the variance case. The natural definition of the mean is automatically unbiased.
6 Atomic Clock Background
at this level, we are calculating a realization of an estimator of the Allan variance (which is the true/full expectation of a two-sample estimator of the variance of theτ-averaged fractional frequency noise).