# Identifying Nonstationary Clock Noises in Navigation Systems

Authors: Lorenzo Galleani; Patrizia Tavella
Journal: International Journal of Navigation and Observation
 (2008-01-15)

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## Abstract

The stability of the atomic clocks on board the satellites of a navigation system should
remain constant with time. In reality there are numerous physical phenomena that make the
behavior of the clocks a function of time, and for this reason we have recently introduced the
dynamic Allan variance (DAVAR), a measure of the time-varying stability of an atomic clock. In
this paper we discuss the dynamic Allan variance for phase and frequency jumps, two common
nonstationarities of atomic clocks. The analysis of both numerical simulations and experimental
data proves that the dynamic Allan variance is an effective way of characterizing nonstationary
behaviors of atomic clocks.


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## Body

## 1. Introduction

Navigation is certainly one of the most effective
applications of atomic clocks. The exceptional stability of an atomic clock allows to reduce the localization error below one meter, as required, for example, by the Galileo system specifications. To guarantee andmaintain in
time a very high stability of the atomic clocks is therefore a fundamental
requirement of a navigation system. Unfortunately, the stability of an atomic
clock changes with time as a consequence of several different phenomena: sudden
and cyclic variations of temperature, aging of physical devices, sudden
breakdowns are among the main causes of nonstationarities.It is hence necessary to introduce a tool that allows
to represent the stability of an atomic clock as a function of time. We have
recently proposed thedynamic Allan variance, or DAVAR, a
quantity that measures the time-varying stability of a clock by sliding the
classical Allan variance on the data [1–4]. By using the dynamic Allan variance we are
classifying the typical nonstationarities of atomic clocks that operate on
board a satellite. The final goal is to identify the clock anomalies directly
from the DAVAR, which can reveal variations in the stability that cannot be
tracked with other methods [5]. In this way, proper warnings can be generated so that
the integrity of the clock and of the satellite signal can be monitored
continuously in time.In this paper, we consider two typical
nonstationarities of atomic clocks and we discuss the corresponding dynamic
Allan variance. We also analyze experimental data that show the anomalies
described.


## 2. The Dynamic Allan Variance

Time series from atomic  clocks are typically
represented by the phase deviation (we use bold symbols for stochastic quantities)x(t), or by the normalized frequency deviation y(t) [6]:(1)y(t)=dx(t)dt.The stability of a clock is
standardly defined through the Allan variance [7–10](2)σy2(τ)=12〈(y¯t+τ−y¯t)2〉,where τ is the
observation interval, the operator 〈⋅〉 stands for time
averaging, and y¯t is defined
as(3)y¯t(t)=1τ∫t−τty(t)dt=x(t)−x(t−τ)τ.In discrete-time we evaluate the
Allan variance with the following estimator:(4)σ^y2[k]=12k2τ021N−2k∑n=0N−2k−1(x[n+2k]−2x[n+k]+x[n])2,where N is the total
number of samples, k=τ/τ0 is an integer
number representing the discrete-time observation interval, and τ0 is the minimum
observation interval. To control the variance of the estimate that increases
with k, one typically takes k=1,2,…,kmax, where kmax=⌊N/3⌋, with N being the total
number of samples (the symbol ⌊⋅⌋ stands for the
integer part of the number).The dynamic Allan variance is defined
as(5)σy2[n,k]=12k2τ021N−2k×∑m=n−Nw/2+kn+Nw/2−k−1E[(x[m+k]−2x[m]+x[m−k])2],where n=t/τ0 is the discrete
time and Nw is the length
of the analysis window. In the definition we have used the expectation value E[⋅] because we
wanted σy2[n,k] to be a
deterministic quantity. In this way, we can study the properties of the DAVAR
without taking into account the random fluctuations that are present every time
that we consider one realization only of x[n].The DAVAR is in practice obtained by sliding the
estimatorσ^y2[k] of the Allan
variance on the data. The DAVAR at time n is made by the
Allan variance of the Nw samples
centered about n. When analyzing experimental data, we apply the
estimator(6)σ^y2[n,k]=12k2τ021N−2k×∑m=n−Nw/2+kn+Nw/2−k−1(x[m+k]−2x[m]+x[m−k])2,which is identical to (5) except
for the expectation value. We again take k=1,2,…,kmax, with kmax=⌊N/3⌋ (other choices
are possible). We also define the dynamic Allan deviationσy[n,k], or DADEV, as the square root of the DAVAR
(the DADEV estimator σ^y[n,k] is defined in
an identical way).There is a typical tradeoff in the computation of the dynamic
Allan variance. If the window is long, the variance of the estimate is small,
but the localization of events in time is poor. Conversely, a short window
guarantees an excellent localization of events, but has a poor variance
reduction. It is better to choose the window on a case-by-case basis, depending
on the type of data considered.


## 3. Analysis of Nonstationary Clock Noises

We now consider two typical nonstationary behaviors of
atomic clocks, namely, a phase and a frequency jump. Both cases are studied
using numerical simulations. The dynamic Allan variance is then applied to a
set of experimental data that show the same types of nonstationarity.


### 3.1. Case 1: Phase Jump

It is very common for clocks on board satellites to
experience jumps in the phase signal, which become spikes in the frequency
deviation, since frequency is defined as the derivative of phase (see (1)).
These frequency values are considered outliers, since they are numerically
distant from the rest of the data. Outliers should be removed in the
preprocessing of data, but since some of them could go unaltered through the
removal algorithm, it is of practical importance to understand what they look
like in the dynamic Allan variance domain.Therefore, we consider a white frequency noise to which
we add a delta function, and we numerically study the corresponding DAVAR. The
signal model is(7)y[n]=f[n]+cδ[n−n0],where f[n] is the usual
white Gaussian noise, c is an arbitrary
constant and n0 is the
discrete-time at which the delta function is located. The discrete-time delta
function is defined as(8)δ[n]={1,n=0,0,n≠0.In Figure 1, we show the
resulting frequency y[n], where n0=2500 and we have
taken c to be 30 times
the standard deviation of y[n]. In Figure 2, we show the estimated Allan deviation σ^y[k] of y[n], where we see the typical slope of a white frequency
noise, and we do not notice the delta function. In Figure 3, the estimated
dynamic Allan deviation σ^y[n,k] is represented.
We see that before and after the time instant n0, the DADEV surface is stationary, beside some obvious
fluctuations due to the variance of the estimate. In the stationary regions,
the slope indicates that locally y[n] is a white
frequency noise. Around n=n0 we instead see
a decrease in the stability (an increase in the dynamic Allan deviation
surface), which takes place at all observation intervals. This change in the
stability is due to the delta function in frequency, and is more intense as c increases. The
reason we see a change in stability for all observation intervals is that at
any k some of the
triplets x[m+k], x[m], x[m−k] used in the
DADEV computation include the delta function located at n=n0. Since the value of the delta function is much bigger
than the standard deviation of the stationary noise f[n], the corresponding triplets will be much bigger than
those that are located outside the nonstationary region. This fact implies that
the dynamic Allan deviation computed for the analysis times whose corresponding
window include the time instant n0 will be bigger
than the DADEV that is computed on the stationary regions alone, which is
precisely what we observe in Figure 3.Figure 1
White frquency noise plus a delta function (7).Figure 2
Allan deviation of the signal shown in Figure1.Figure 3
Dynamic Allan deviation of the signal shown in Figure1.


### 3.2. Case 2: Frequency Jump

Also frequency jumps can be detected in atomic clocks
on board satellites. A simple model for a frequency jump is given
by(9)y[n]={μ1+f[n],n≤n0,μ2+f[n],n>n0,where f[n] is a white
Gaussian noise with zero mean. The model of y[n] has been chosen
so that the mean value is(10)μ[n]=E[y[n]]={μ1,n≤n0,μ2,n>n0.This means that there is a sudden
variation in the mean of y[n], as shown in Figure 4. The estimated Allan deviation σ^y[k] is given in Figure 5. We see that the Allan deviation has the typical slopes of a white
frequency noise, and that there is no evident trace of the nonstationarity
going on in the clock noise. The reason is that the jump in the mean value of
the signal has been averaged out by the Allan variance. In Figure 6, we instead
see the estimate σ^y[n,k] of the dynamic
Allan deviation, computed with a window of Nw=200 samples. We
notice that for small values of the discrete observation interval k, the DADEV does not show the change in mean. The
reason is that for small k values the
frequency jump is present only in few of the triplets x[m+k], x[m], x[m−k] used in the
DADEV estimation. Most of the triplets are located in the stationary regions
before and after the discontinuity, and they are not influenced by the change
in the mean. For increasing values of k we see that the
stability steadily decreases. The reason is that for large k most of the
triplets are made by values located before and after the nonstationarity, that
will hence track the discontinuity in the mean.Figure 4
White frequency noise with a frequency jump (9).Figure 5
Allan deviation of the signal shown in Figure4.Figure 6
Dynamic Allan deviation of the signal shown in Figure4.


### 3.3. Experimental data

We now analyze a set of experimental data coming from
a Rubidium clock undergoing tests for space flight certification. In Figure7,
we show a section of the frequency data y(t). We notice a frequency jump located approximately at t1=1.6104 seconds. After
this sudden variation, y(t) gradually
recovers a mean value close to the one that it had before the nonstationarity.
There is also a spike in frequency located roughly at t2=3.9104 seconds, which
indicates that a jump in the phase x(t) has taken place
at the same time instant.Figure 7
Frequency data of a Rubidium clock.In Figure8, we see the Allan deviation σ^y[k] of y(t), which shows the typical slopes of a Rubidium clock
and does not point out the presence of nonstationary behaviors. In Figure 9, we
instead represent the dynamic Allan deviation. Around t=t1 we notice that
the DADEV surface increases for large τ values, which
implies the presence of a step change in the mean of the frequency y(t), as discussed in Section 3.2. Also, around t=t2 we spot an
increase in the DADEV for all τ values, which
means that there is a spike in frequency or, equivalently, that there is a jump
in the phase data x(t), as previously discussed in Section 3.1. Outside the
regions around t1 and t2 the dynamic
Allan deviation is mostly stationary and it is in accordance with the slope of
a Rubidium clock.Figure 8
Allan deviation of the signal shown in Figure7.Figure 9
Dynamic Allan deviation of the signal shown in Figure7.It is therefore possible to characterize the stability
of the Rubidium clock by directly observing the dynamic Allan variance surface.


## 3.1. Case 1: Phase Jump

It is very common for clocks on board satellites to
experience jumps in the phase signal, which become spikes in the frequency
deviation, since frequency is defined as the derivative of phase (see (1)).
These frequency values are considered outliers, since they are numerically
distant from the rest of the data. Outliers should be removed in the
preprocessing of data, but since some of them could go unaltered through the
removal algorithm, it is of practical importance to understand what they look
like in the dynamic Allan variance domain.Therefore, we consider a white frequency noise to which
we add a delta function, and we numerically study the corresponding DAVAR. The
signal model is(7)y[n]=f[n]+cδ[n−n0],where f[n] is the usual
white Gaussian noise, c is an arbitrary
constant and n0 is the
discrete-time at which the delta function is located. The discrete-time delta
function is defined as(8)δ[n]={1,n=0,0,n≠0.In Figure 1, we show the
resulting frequency y[n], where n0=2500 and we have
taken c to be 30 times
the standard deviation of y[n]. In Figure 2, we show the estimated Allan deviation σ^y[k] of y[n], where we see the typical slope of a white frequency
noise, and we do not notice the delta function. In Figure 3, the estimated
dynamic Allan deviation σ^y[n,k] is represented.
We see that before and after the time instant n0, the DADEV surface is stationary, beside some obvious
fluctuations due to the variance of the estimate. In the stationary regions,
the slope indicates that locally y[n] is a white
frequency noise. Around n=n0 we instead see
a decrease in the stability (an increase in the dynamic Allan deviation
surface), which takes place at all observation intervals. This change in the
stability is due to the delta function in frequency, and is more intense as c increases. The
reason we see a change in stability for all observation intervals is that at
any k some of the
triplets x[m+k], x[m], x[m−k] used in the
DADEV computation include the delta function located at n=n0. Since the value of the delta function is much bigger
than the standard deviation of the stationary noise f[n], the corresponding triplets will be much bigger than
those that are located outside the nonstationary region. This fact implies that
the dynamic Allan deviation computed for the analysis times whose corresponding
window include the time instant n0 will be bigger
than the DADEV that is computed on the stationary regions alone, which is
precisely what we observe in Figure 3.Figure 1
White frquency noise plus a delta function (7).Figure 2
Allan deviation of the signal shown in Figure1.Figure 3
Dynamic Allan deviation of the signal shown in Figure1.


## 3.2. Case 2: Frequency Jump

Also frequency jumps can be detected in atomic clocks
on board satellites. A simple model for a frequency jump is given
by(9)y[n]={μ1+f[n],n≤n0,μ2+f[n],n>n0,where f[n] is a white
Gaussian noise with zero mean. The model of y[n] has been chosen
so that the mean value is(10)μ[n]=E[y[n]]={μ1,n≤n0,μ2,n>n0.This means that there is a sudden
variation in the mean of y[n], as shown in Figure 4. The estimated Allan deviation σ^y[k] is given in Figure 5. We see that the Allan deviation has the typical slopes of a white
frequency noise, and that there is no evident trace of the nonstationarity
going on in the clock noise. The reason is that the jump in the mean value of
the signal has been averaged out by the Allan variance. In Figure 6, we instead
see the estimate σ^y[n,k] of the dynamic
Allan deviation, computed with a window of Nw=200 samples. We
notice that for small values of the discrete observation interval k, the DADEV does not show the change in mean. The
reason is that for small k values the
frequency jump is present only in few of the triplets x[m+k], x[m], x[m−k] used in the
DADEV estimation. Most of the triplets are located in the stationary regions
before and after the discontinuity, and they are not influenced by the change
in the mean. For increasing values of k we see that the
stability steadily decreases. The reason is that for large k most of the
triplets are made by values located before and after the nonstationarity, that
will hence track the discontinuity in the mean.Figure 4
White frequency noise with a frequency jump (9).Figure 5
Allan deviation of the signal shown in Figure4.Figure 6
Dynamic Allan deviation of the signal shown in Figure4.


## 3.3. Experimental data

We now analyze a set of experimental data coming from
a Rubidium clock undergoing tests for space flight certification. In Figure7,
we show a section of the frequency data y(t). We notice a frequency jump located approximately at t1=1.6104 seconds. After
this sudden variation, y(t) gradually
recovers a mean value close to the one that it had before the nonstationarity.
There is also a spike in frequency located roughly at t2=3.9104 seconds, which
indicates that a jump in the phase x(t) has taken place
at the same time instant.Figure 7
Frequency data of a Rubidium clock.In Figure8, we see the Allan deviation σ^y[k] of y(t), which shows the typical slopes of a Rubidium clock
and does not point out the presence of nonstationary behaviors. In Figure 9, we
instead represent the dynamic Allan deviation. Around t=t1 we notice that
the DADEV surface increases for large τ values, which
implies the presence of a step change in the mean of the frequency y(t), as discussed in Section 3.2. Also, around t=t2 we spot an
increase in the DADEV for all τ values, which
means that there is a spike in frequency or, equivalently, that there is a jump
in the phase data x(t), as previously discussed in Section 3.1. Outside the
regions around t1 and t2 the dynamic
Allan deviation is mostly stationary and it is in accordance with the slope of
a Rubidium clock.Figure 8
Allan deviation of the signal shown in Figure7.Figure 9
Dynamic Allan deviation of the signal shown in Figure7.It is therefore possible to characterize the stability
of the Rubidium clock by directly observing the dynamic Allan variance surface.


## 4. Conclusion

Navigation requires atomic clocks on board the
satellites to have a very high stability, and to maintain it with time. Since
in reality there are several physical phenomena that produce variations in the
clock behavior, it is fundamental to understand how its stability changes with
time. For this reason we have proposed the dynamic Allan variance, or DAVAR, a
quantity that is able to characterize the nonstationary behaviors of atomic
clocks. In this paper, we have analyzed two typical nonstationarities that
affect atomic clocks on board a satellite, namely, phase and frequency jumps.
Numerical simulations demonstrate that the DAVAR correctly represents these
anomalous behaviors. We have also validated our method with experimental data,
proving that it is possible to understand the nonstationarities of a clock by directly
inspecting the DAVAR surface. This means that it is possible to design anomaly
detection methods directly in the dynamic Allan variance domain (a free Matlab
implementation of the DAVAR can be found atwww.ien.it/tf/ts/clock_behavior.shtml) [5].



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*Source: 524317-2008-04-06.xml*