Indian Journal of Pure & Applied Physics
Vol. 45, December 2007, pp. 945-949
Determination of Allan deviation of Cesium atomic clock for lower
averaging time
P Banerjee, Arundhati Chatterjee & Suman
Time and Frequency Section, National Physical Laboratory, New Delhi 110 012
E-mail: [email protected]
Received 15 January 2007; revised 11 September 2007; accepted 8 October 2007
Absolute Allan deviation of the Cesium clock for averaging time (τ) of 5 days or more may be calculated from the
corresponding data of circular T published by Bureau International des Poids et Mesures (BIPM). For lower values of τ, the
Allan deviation may indirectly be found from the extrapolation of these values through τ-1/2 fit as recommended by CCTFWGMRA guidelines. Absolute Allan deviation may also be directly found out by inter-comparison of minimum three clocks
assuming that the noise in all clocks is fully uncorrelated. This paper analyses the values of Allan deviation determined by
the direct method keeping in mind the limitation of the measurement system. These values of Allan deviation tally well with
those found from the data of circular T.
Keywords: Cesium atomic clock, Allan deviation, Lower averaging time
1 Introduction
A stable signal generator in various fields such as
physics, radio communication, radar, space vehicle,
tracking, navigation, timing applications etc is the key
component. An oscillator has some nominal
frequency at which it operates. The frequency
stability of an oscillator is a term used to characterize
how small the frequency fluctuations are of the
oscillator signal. Frequency stability is the degree to
which an oscillating signal produces the same value
of frequency for any interval, throughout a specified
period of time. It is of interest to note that the
frequency stability actually refers to the frequency
instability. The frequency instability is the
spontaneous and/or environmentally caused frequency
change within a given time interval. Characterization
of phase and frequency instabilities of high-quality
time and frequency sources has been of great
importance and has drawn the attention of scientists
since mid sixties of last century1- 4. Given a set of data
of the fractional frequency or time fluctuations
between a pair of oscillators, it is useful to
characterize these fluctuations with reasonable and
tractable models of performance. The fluctuations that
are random in nature have been considered. These can
usually be best characterized statistically. If we
survey the literature, two major methods of specifying
the frequency stability of an oscillator may be
identified. One is time domain method namely the
Allan deviation [σ(τ)] of the frequency departure
averaged over time. Another method of characterizing
noise in the frequency fluctuations is by means of the
power spectral density [Sf(f)]. After the signing of
MRA, the determination of uncertainties of reference
frequency source to establish the calibration
measurement capabilities (CMC) has gained
momentum. The Allan deviation is normally used to
quantify the uncertainties of time and frequency
reference source against which time and frequency
devices are calibrated. So it becomes necessary for
practical requirement of calibration and testing to
determine the Allan variance or Allan deviation [σ(τ)]
of the source averaged over the time (τ) ranging from
few seconds to few tens of seconds.
Spectral power densities are theoretical concept
involving infinite duration process, infinite frequency
range and true average. In practice only finite duration
process are available. The spectrum analyzers have
non-zero bandwidth frequency window with lower
and upper frequency limit. A lower limit of 1 Hz and
upper limit of few kilohertz may be reached using
present-day technology of FFT analyzer. The spectral
power density may be used to find two sampled
variance σ2(τ) through the standard formulae5. But
the boundary condition of lower limit of bandwidth of
FFT does not allow us to find σ2(τ) for τ larger than
one or two seconds. It may, thus, be necessary to
attempt to find σ2y(τ) for average time τ of few
INDIAN J PURE & APPL PHYS, VOL. 45, DECEMBER 2007
946
seconds to few tens of seconds directly [i.e. in time
domain].
This paper attempts to address the problems of
determining the frequency stability in time domain for
lower averaging time. These problems have more
relevance particularly to a precision oscillator like
Rubidium clocks, Cesium clocks and Hydrogen
masers which may normally be used to maintain the
national time standard.
2 Definition of Allan Deviation
One must also realize that any frequency
measurement involves two oscillators. It is impossible
to purely measure only one oscillator. In some
instances ,one oscillator may be enough better than
the other that the fluctuations measured may be
considered essentially those of the latter. However, in
general, because frequency measurements are always
dual, it is useful to define the normalized frequency
off set as:
ν −ν
y (t ) = 1 0
ν0
… (1)
y(t) is a dimensionless quantity and useful in
describing oscillator and clock performance. The time
deviation, x(t), of an oscillator is related to y(t) by:
t
x(t ) = ∫ y (t ) dt =
0
φ (t )
2πν 0
… (2)
Since it is impossible to measure instantaneous
frequency, any frequency or fractional frequency
measurement always involves some sample time,
some time window through which the oscillators are
observed; whether its a picosecond, a second, or a
day, there is always some sample time. So when
determining a fractional frequency, y(t), the time
deviation is measured say starting at some time t and
again at a later time, t + τ . The difference in these two
time deviations divided by τ gives the average
fractional frequency over that period:
y (t ) =
x(t +τ ) − x(t )
τ
… (3)
Tau, τ, may be called the averaging time.
The random frequency stability of an oscillator, in
time domain, may be estimated by several sample
variances. The most commonly used measure is the
two-sample standard deviation (also known as Allan
deviation) which is the square root of the two-sample
zero dead-time variance (also designated as Allan
variance). The normal Allan, or 2- sample variance is
defined as:
σ y2 (τ ) =
M −1
1
∑[ yi (t +τ ) − yi (t )]2
2( M −1) t =1
… (4)
where y(t) is the ith of M fractional frequency values
averaged over the measurement interval τ.
In terms of phase data, the Allan variance may be
calculated as:
σ y2 (τ ) =
1
2( N − 2)τ 2
N −2
×∑[ xi (t + 2τ ) − 2 xi (t +τ ) + xi (τ )]
… (5)
2
t =1
where x(t) is the ith of the N = M+1 phase values
spaced by the measurement interval τ.
The terms x(t+τ) and x(t) are proportional to
instantaneous phase (time) difference obtained from
the comparison between two clocks at date (t+τ) and t
and have the dimension of time. In any frequency and
phase measurement, one clock is compared with
respect to a reference clock. So, the measurement is a
relative one, so also Allan deviation. But the focus of
this paper relates to the absolute Allan deviation of an
oscillator.
3 Allan Deviation through Data of Circular T
International Bureau of Weights and Measures
(Bureau International des Poids et Mesures i.e. BIPM)
in Paris, has been coordinating the task of realizing
International Atomic Time (TAI) and Universal
Coordinated Time (UTC). To achieve this, the remote
clocks are compared through common-view GPS
method. These data are sent to BIPM through
internet/E-mail at regular interval of time. BIPM uses
all these data to generate a smooth time scale through
robust software, which is virtually the weighted
average of clocks of all participating laboratories.
This software generated time scale combining all the
received data is named as Universal Coordinated
Time6 (UTC). The software also generates the status
of the time scale of contributing laboratories with
respect to UTC through a circular T (published by
BIPM).
BANERJEE et al.: ALLAN DEVIATION OF CESIUM ATOMIC CLOCK
National Physical Laboratory, New Delhi, India
(NPLI) maintains the time scale of Indian Standard
Time (IST) with the help of a commercial Cesium
atomic clock (model HP5071A). The time scale
maintained by NPLI is designated as UTC(NPLI).
The status of time scale UTC (NPLI) is, thus, also
available through circular T which records (UTCUTC(NPLI)) at the interval of 5 days. Time scale
generated by UTC is for long term maintenance of
time scale. Long term normally refers to the averaging
time of more than one day. BIPM, thus, has decided
to generate data in Circular T every 5 days. The data
of circular T corresponding to UTC(NPLI) defines
[UTC(NPLI)-UTC] which is the local values of x(t).
UTC, being the average of more sixty atomic clocks,
may be considered to be a virtual noiseless oscillator.
Thus, it may be assumed that these x(t) values
correspond to the time differences (i.e. x(t)) with
respect to an ideal oscillator. So Allan deviation,
determined out of these data, may be assumed to be
the absolute Allan deviation of the Cesium clock,
not a relative one. So, these values of x(t)’s are used
in Eq. (2) to find σ1(τ) (implying absolute value of
σy(τ) for the Cesium clock #1) for averaging time τ of
5 days or more. These have been plotted in Fig. 1.
The status UTC (NPLI) is updated every 5 days in
Circular T. So Allan deviation for the averaging time
less than 5 days cannot be directly found out through
these data. But CCTF-WGMRA guideline7
recommends that τ- -1/2 fit may be applied to calculate
the values of Allan deviation for lower averaging
time. The Allan deviation for lower values of τ may
be obtained from the extraploted line as shown in
Fig. 1.
947
The above method has the following practical
difficulties.
1. It is necessary that the clock should be linked to
BIPM through GPS network to make the availability
of xi (t)’s values with respect to the virtually ideal
clock.
2. One should also have the feedback data from
circular T for several months to get sufficient number
of data to find σy(τ ).
4 Direct Method
Using the data of circular T, the determination
stability is quite time consuming. Further, it is
necessary to find the Allan deviation for lower
averaging time directly. One possible direct method is
to intercompare few clocks directly. In the direct
method, the phases of two Cesium clocks are
compared at a regular interval of time τ with the help
of a time interval counter (TIC). The measured data
may be assumed to correspond to xjk (t)’s of one clock
j with respect to the other clock k. With the help of
intercompared data, one may, thus, find out σjk(τ) (the
value of σy(τ) for the clock #j with respect to that of
clock #k) using Eq. (5). These values of Allan
deviation are, again, relative ones, not absolute ones.
But we may note that the noise in all the cesium
clocks is absolutely un-correlated. So one may write:
σjk2(τ) = σj2(τ) + σk2 (τ)
j, k=1,2,3…
(e.g. σj(τ) is absolute value of σy(τ) for clock #j)
The Eq.(6) may be used to find out independent
(absolute) value of the Allan deviation (i.e. σk (τ)) for
each of the clocks by intercomparing several clocks.
The Eq.(6) dictates that this method requires
minimum three clocks to find unique solutions for
σj(τ)’s. Thus, for three clocks, using all combinations
of Eq. (6), it is quite possible to find out Allan
deviation of each clock independently by using the
following relations:
1
2
… (7)
1
2
… (8)
1
2
… (9)
σ12 (τ ) = ((σ122 (τ ) +σ132 (τ ) −σ 232 (τ ))
σ 22 (τ ) = ((σ 232 (τ ) +σ 212 (τ ) −σ132 (τ ))
σ 32 (τ ) = ((σ 322 (τ ) +σ 312 (τ ) −σ122 (τ ))
Fig. 1Allan deviation of Cesium clock 1 at NPLI
… (6)
INDIAN J PURE & APPL PHYS, VOL. 45, DECEMBER 2007
948
NPLI has utilized three cesium clocks for this
purpose. A high resolution time interval counter (TIC)
to compare the phases of two clocks has also been
used as shown in Fig. 2. The selector switch helps in
selecting one chosen pair of Cesium clocks for
measurements. Linking of TIC to the computer
through RS232C port, a serial port, limits the speed to
one measurement every four seconds. So, minimum
value of τ (i.e. averaging time) has been four
seconds. Each of three sets of measurement has been
carried out for 6 hr continuously at the sampling rate
of four seconds. Each set of these measured values
(i.e. xjk (t)) are used in Eqs (5) to find σ12(τ), σ13(τ )
and σ23(τ). Using these values of σjk(τ)’s in Eqs (7−9),
σk’ s (i.e. for each of three Cesium atomic clocks)
have been found out. For example, with help of
Eq.(7) σ1 (τ) has been find out as shown in Table 1
and Fig. 2. As UTC(NPLI) is maintained through
Cesium Clock #1, so the emphasis has been given for
σ1(τ).
Fig. 2Experimental set up for determination of Allan deviation
by direct method
Table 1Calculation of absolute value of Allan deviation
of Cs #1
τ
σ12
σ 13
σ 32
σ1
4
8
16
32
64
128
256
512
1024
2048
1.05E-10
5.29E-11
2.85E-11
1.33E-11
7.95E-12
4.50E-12
2.62E-12
1.38E-12
9.58E-13
6.35E-13
4.13E-11
2.29E-11
1.30E-11
7.66E-12
5.02E-12
3.37E-12
2.43E-12
1.85E-12
1.12E-12
7.04E-13
4.34E-11
2.38E-11
1.37E-11
8.14E-12
5.22E-12
3.70E-12
2.27E-12
1.75E-12
8.56E-13
5.59E-13
7.36E-11
3.71E-11
1.99E-11
9.20E-12
5.53E-12
2.99E-12
1.95E-12
1.06E-12
8.48E-13
5.41E-13
5 Limitation of Measurement System
The universal counter (model HP53132A) has been
used in this measurement as TIC and this TIC has
RMS resolution of 300 ps8. So the uncertainty (1σ) of
frequency measurement through this counter is
∆y =
300×10−12
3τ
… (10)
This fact and a close look at Eqs (1−4), lead us to
conclude that the limiting values of Allan deviation
caused by the limitation of measurement ( Appendix
A) is:
∆σ1 = 3 ×∆y =
3×10−10
τ
… (11)
Eq. (11) implies that the system cannot measure the
stability whose value is actually lower than the
corresponding ∆σ1. In other words, if any value of
stability determined by the direct method becomes
lower than the uncertainty limit for that averaging
time, the stability value cannot be accepted. Thus, the
line indicating the limit of the measurement system
based on Eq. (11) has been shown in Fig. 1. So this
line implies that accepted values of Allan deviation
determined through the direct method should lie
above this limiting line.
6 Conclusions
Values of σ1(τ) of Cesium atomic clock calculated
by the direct method for τ higher than 100’s of
seconds have been found to lie above the limit line
(Fig. 2) and thus, they can be accepted to be reliable
ones (i.e. for τ-values of 256 s, 512 s, 1024 s and
2048 s ). But for averaging time lower than 100 s,
values of σ1(τ)’s have been found to lie almost on
the limit line and thus, these values cannot be reliable.
The reliably calculated values of Allan deviations (i.e.
for τ higher than 100 s) may be used for extrapolation
through τ−1/2 law to find Allan deviations for lower
values of τ. It is of interest to note that τ -1/2 line fit by
circular T method almost coincides with that by direct
method confirming the reliability of the measurement
by direct method. For example, for τ of 100s, Allan
deviation is 2.58×10-12 and 2.04×10-12 by direct
method and circular T method ,respectively.
So, the direct method may be used to find the
absolute Allan deviation for lower values of τ. This
BANERJEE et al.: ALLAN DEVIATION OF CESIUM ATOMIC CLOCK
2
does not require the linking of clocks to BIPM and
thus, one need not wait for a long time to get sufficient
data from BIPM. But the limitation of the measurement
system has to be kept in mind. For example, to get
Allan deviation for averaging time lower than 100 s
directly (i.e. without extrapolation) particularly for
precise clocks like atomic clocks, one needs to have a
TIC with an rms resolution better than 300 ps.
3
4
5
6
7
References
8
1
949
Barnes J A, Chi A R, Cutler L S, Healey D J, Leeson D B,
McGunigal T E, Mullen J A, Smith W L, Sydnor R, Vessot R
F & Winkler G M R, IEEE Trans Instr Meas, IM-20 (1971)
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Cutler L S & Searle C L, IEEE, 54 (1983) 136.
Rutman J, Proc. IEEE, 66 9 (1978) 48.
Kartaschoff P, Frequency and Time, Academic Press, 1978.
Annual report of the BIPM Time Section, 18 (2005) 89.
Uncertainty extrapolation for T&F CMC entries, CCTF
WGMRA Guideline 3 (Rev. 20021210).
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Appendix A
Uncertainty of Allan deviation determined through direct
method
Using Eq. (1) for Allan deviation and noting that y(t+τ) and y(t)
are correlated, one may write :
∆σ12
σ12
≈
∆σ13
σ13
≈
∆σ 23
σ 23
≈
∆σ y
σy
… (3A)
Thus, with the help of Eqs(3A) and (1A), (2A) reduces to :
2
 ∆σ y   ∆y 

 = 
 σ y   y 
2
… (1A)
∆ represents the uncertainty of the corresponding parameter and
∆y is, thus, the uncertainty of the frequency measurement through
TIC.
It may be noted that all σ’s in Eq. (4) are partially uncorrelated.
So it follows:
2
2
2
2
 ∆σ 
 ∆σ  
 ∆σ    ∆σ 
2  1  = 2  12  + 2  13  + 2  23  
 σ1    σ12 
 σ13 
 σ 23  
For the practical purpose, one may assume:
… (2A)
2
 ∆σ 1 
 ∆y 

 = 3 
σ
 y
 1 
2
… (4A)
Eq. (4A) simplifies to:
∆σ1 = 3
σ1
y
Assuming,
∆y
σ1
∆σ1 = 3∆y
y
… (5A)
≈1 , one may write:
… (6A)