FREQUENCY CALIBRATIONS WITH CONVENTIONAL TIME
INTERVAL COUNTERS VIA GPS TRACEABILITY
Juan Jos
´
e Gonz
´
alez de la Rosa, I. Lloret
Univ. C
´
adiz. Research Group TIC168, EPSA. Av. Ram
´
on Puyol S/N. E-11202-Algeciras-C
´
adiz-Spain,
Carlos Garc
´
ıa Puntonet, J. M. G
´
orriz
Univ. Granada. Dept. ATC, ESII. Periodista Daniel Saucedo. E-18071-Granada-Spain
A. Moreno, M. Li
˜
n
´
an and V. Pallar
´
es
Univ. C
´
ordoba. Electronics Area, Campus Rabanales. A. Einstein C-2. E-14071-C
´
ordoba-Spain
Keywords:
AVAR, frequency calibrations, GPS receiver, MVAR, noise processes, traceable standard, type B uncertainty,
uncertainty calculations.
Abstract:
Calculation of the uncertainty in traceable frequency calibrations is detailed using low cost instruments, par-
tially characterized. Contributions to the standard uncertainty have been obtained under the assumption of
uniform probability density function of errors. Short term instability has been studied using non-classical sta-
tistics. A thorough study of the noise processes characterization is made with simulated data by means of our
variance estimators. The experiment is thought for frequencies close to 1 Hz.
1 INTRODUCTION
Time interval counters (TICs) and GPS receivers are
widely used in traceable frequency calibrations. A
transfer standard receives a signal that has a cesium
oscillator as source (Lombardi, 1996). This signal de-
livers a cesium derived frequency to the user, who is
benefited as not all laboratories can afford a cesium
(Lombardi, 1996). These instruments differ in speci-
fications and details regarding the time base, the main
gate and the counting assembly. Furthermore, manu-
facturers tend to omit the conditions under these spec-
ifications have been provided or measured.
The purpose of this paper is twofold. First we de-
tail the uncertainty calculations and the magnitudes
which contribute to the sensitivity coefficients in the
uncertainty propagation. Second, we show how to
deal with practical situations which involve incom-
plete specifications. Experimental results are ob-
tained under the assumption of white noise as the
main cause of short term instability, which is cor-
roborated later by means of the non-classical statis-
tics AVAR
1
and MVAR
2
. A prior analysis of noise
processes is made to show short term instability char-
acterization, by analyzing the slopes of the AVAR and
MVAR in the log-log curves. Noise time series have
1
Allan variance or two-sample Allan variance
2
Modified Allan variance
been simulated and estimators of the variances have
been programmed with the aim of having a thorough
vision of the time-domain slopes when compared to
former works: (Howe et al., 1999), (Allan, 1987),
(Rutman and Walls, 1991), (Vernotte, 1993), (Vig,
2001).
The paper is structured as follows: in Section 2
we review the oscillators independent noise processes
and the methods used to identify them; Section 3
shows the details concerning uncertainty calculations.
Experiments are drawn in Section 4, and conclusions
explained in Section 5.
2 CLASSICAL NOISE MODELS
2.1 Characterizing Instabilities
The instantaneous output voltage of an oscillator can
be expressed as:
v(t)=[V
o
+ ε(t)] sin [2πν
0
t + φ(t)] , (1)
where V
o
is the nominal peak voltage amplitude, ε(t)
is the deviation from the nominal amplitude, ν
0
is the
name-plate frequency, and φ(t) is the phase deviation
from the ideal phase 2πν
0
t. Changes in the peak value
of the signal is the amplitude instability. Fluctuations
in the zero crossings of the voltage is the phase insta-
bility. The so-called frequency instability is depicted
189
José González de la Rosa J., Lloret I., García Puntonet C., M. Górriz J., Moreno A., Liñán M. and Pallarés V. (2006).
FREQUENCY CALIBRATIONS WITH CONVENTIONAL TIME INTERVAL COUNTERS VIA GPS TRACEABILITY.
In Proceedings of the Eighth International Conference on Enterprise Information Systems - AIDSS, pages 189-195
DOI: 10.5220/0002442201890195
Copyright
c
SciTePress
by the fluctuations in the period of the voltage. The
situation was depicted in (Vig, 2001) and (de la Rosa
et al., 2005).
The short-term stability measures most frequently
found on oscillator specification sheets is the two-
sample deviation, also called Allan deviation, σ
2
y
(τ)
(Howe et al., 1999), (Vig, 2001).
Classical variance in non-stationary noise
processes doesn’t converge to concrete values.
It diverges for some noise processes (de la Rosa
et al., 2005). This is the reason whereby non-classical
statistics are used to characterize short term instabil-
ity. AVAR and MVAR have proven their adequacy
in characterizing frequency phase and instabilities.
These easy-to-compute variances converge for all
noise processes observed in precision frequency
sources, have a straightforward relationship to power
law spectral density of noise processes, and are faster
and more accurate than the FFT (Lesage and Ayi,
1984).
The estimates of AVAR and MVAR for a given cali-
bration time τ for a m-data series of phase differences,
x, are given by equations 2 and 3, (Greenhall, 1988):
AV AR ≡ σ
2
y
(τ,m)=
1
2(m − 1)
m
j=2
y
j
− y
j−1
2
=
1
2τ
2
(m − 1)
m
j=2
∆
2
τ
x(jτ)
2
(2)
MVAR ≡
1
2τ
2
∆
2
τ
x
2
, (3)
where the bar over x denotes the average in the
time interval τ (averaging time), and ∆
2
τ
x = x
i+2
−
2x
i+1
+ x
i
, is the so called second difference of
x. The fractional frequency deviation is the relative
phase difference in an interval τ. It is defined by equa-
tion 4:
y =
1
τ
t
t−τ
y(s)ds =
x(t) − x(t − τ)
τ
=
∆
τ
x(t)
τ
.
(4)
Non-classical statistics estimators, defined above, in
equations 2 and 3, for non-stationary series charac-
terization, give an average dispersion of the fractional
frequency deviation due to the noise processes cou-
pled to the oscillator. As a consequence time do-
main instability (two-sample variance) is related to
the noise spectral density via (Rutman and Walls,
1991):
σ
2
y
(τ)=
2
(πν
0
τ)
2
f
h
0
S
φ
(f)sin
4
(πfτ)df , (5)
where ν
0
is the carrier frequency and f is the Fourier
frequency (the variable), and f
h
is the band-width of
the measurement system. S
φ
(f) is the spectral den-
sity of phase deviations, which is in turn related to
the spectral density of fractional frequency deviations
by(Rutman and Walls, 1991):
S
φ
(f)=
ν
2
0
f
2
S
y
(f), (6)
The classical power-law noise model is a sum of the
five common spectral densities. The model can be de-
scribed by the one-sided phase spectral density S
φ
(f)
via (IEE, 1988), (Greenhall, 1988):
S
φ
(f)=
ν
2
0
f
2
2
α=−2
h
α
f
α
= ν
2
0
4
β=0
h
β
f
β
, (7)
for 0 ≤ f ≤ f
h
. Where, again, f
h
is the high-
frequency cut-off of the measurement system (the
band-width); h
α
and h
β
are constants which rep-
resent, respectively, the independent characteristic
models of oscillator frequency and phase noise (Al-
lan, 1987), (IEE, 1988), (Greenhall, 1988).
For integer values (the most common case) we have
the following approximate expression:
σ
y
(τ) ∼ τ
µ/2
, (8)
where µ = −α − 1, for −3 ≤ α ≤ 1; and µ ≈−2
for α ≥ 1. In the case of the modified Allan variance,
the time-domain instability can be approximated via:
Modσ
y
(τ) ∼ τ
µ
(9)
Hereinafter we use expressions 8 and 9 for analyzing
noise in these work.
2.2 Time Domain Stability
Characterization Curves
Equations 8 and 9 are used to make the graphical
representation of σ
y
(τ) vs. τ , and lets us infer the
noise processes which causes frequency instability by
means of measuring the slope in a log-log graph (Rut-
man and Walls, 1991). These functional characteris-
tics of the independent processes are widely used in
modelling frequency instability of oscillators. Table
1 shows the experimental criteria adopted in the main
references. In the second column or MVAR we have
picked up two different criteria according to the ref-
erences (Rutman and Walls, 1991) and (Lesage and
Ayi, 1984), respectively. We have kept the notation in
the works (Rutman and Walls, 1991) and (Lesage and
Ayi, 1984) for µ/2 and µ
, respectively.
The five noise processes have been modelled and
VAR and MVAR have been calculated. Hereinafter
we show the simulation results of the time-series and
their associated VAR and MVAR graphs. From this
simulations we adopt the criteria depicted in the sec-
ond column of MVAR in table 1. Figures 1-5 show
ICEIS 2006 - ARTIFICIAL INTELLIGENCE AND DECISION SUPPORT SYSTEMS
190
Table 1: Noise processes characterized by the time and fre-
quency domain slopes. Up to bottom: random walk fre-
quency modulation, flicker frequency modulation, white
frequency modulation, flicker phase modulation, white
phase modulation.
AVA R M VAR
S
y
(f) S
φ
(f) σ
y
(τ) ∼|τ|
µ
2
σ
y
(τ) ∼|τ|
µ
αβ= α − 2
µ
2
µ
−2 −40.51(0.5)
−1 −3 0 0 (0)
0 −2 −0.5 −1(−0.5)
1 −1 −1 −2(−1)
20 −1 −3(−1.5)
the results. Each sequence contains 4096 points for a
time resolution of τ =10
−4
s. Allan deviation curves
have been depicted for averaging times τ = n × τ
0
,
with n ∈ [1, 500].
0.02 0.04 0.06 0.08 0.1
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Time (s)
Amplitude (V)
White phase modulation
10
-4
10
-3
10
-2
10
-2
10
-1
10
0
10
1
10
2
log(tau)
log(sigma)
AVAR and MVAR; beta=0
Figure 1: Characterization of a noise process corresponding
to β =0.
In practice, two or more noise processes simulta-
neously affect clocks performance. In this cases in-
stability of the device under test is explained away
through the behaviour of the upper enveloping curve.
If the individual variance curves cross each other, it
is possible to see the slope changes in the variance
curve, for a time-series which includes several types
of noise(Vernotte, 1993). This situation is shown in
figures 6 and 7.
In figure 6, the individual variance curves cross. So
the enveloping curve characterizes the short-term in-
stability. By the contrary, in figure 7 the β =0noise
processes has a variance greater than the β = −4 per-
turbation. In this case the enveloping curve is the first
(upper) AVAR curve.
0.02 0.04 0.06 0.08 0.1
-2
-1
0
1
2
3
x 10
-3
Time (s)
Amplitude (V)
Flicker phase mod.
10
-4
10
-3
10
-2
10
-2
10
-1
10
0
log(tau)
log(sigma)
AVAR and MVAR; beta=-1
Figure 2: Characterization of a noise process corresponding
to β = −1.
0.1 0.2 0.3 0.4
-6
-4
-2
0
2
4
6
8
x 10
-4
Time (s)
Amplitude (V)
White freq. mod.
10
-4
10
-3
10
-2
10
-2
10
-1
log(tau)
log(sigma)
AVAR and MVAR; beta=-2
Figure 3: Characterization of a noise process corresponding
to β = −2.
3 UNCERTAINTY PROPAGATION
USING A REFERENCE SIGNAL
OF 1 PPS
3.1 Sensitivity Coefficients in the
Measurement System
In calibration we usually deal with a measurand, Z,
which is the particular quantity subject to the mea-
surement and is considered as the output of the mea-
surement system. This quantity depends upon a set
of input random variables X
i
according to a func-
tional relationship given by a function f, representing
the procedure of the measurement and the method of
evaluation (Force, 1999):
Z = f(X
1
,X
2
,...,X
N
) (10)
FREQUENCY CALIBRATIONS WITH CONVENTIONAL TIME INTERVAL COUNTERS VIA GPS TRACEABILITY
191
0.1 0.2 0.3 0.4
-3
-2
-1
0
1
2
x 10
-4
Time (s)
Amplitude (V)
Flicker freq. mod.
10
-4
10
-3
10
-2
10
-2.4
10
-2.3
10
-2.2
log(tau)
log(sigma)
AVAR and MVAR; beta=-3
Figure 4: Characterization of a noise process corresponding
to β = −3.
0.1 0.2 0.3 0.4
-1
0
1
x 10
-4
Time (s)
Amplitude (V)
Random walk freq. mod.
10
-4
10
-3
10
-2
10
-3
log(tau)
log(sigma)
AVAR and MVAR; beta=-4
Figure 5: Characterization of a noise process corresponding
to β = −4.
An estimate of the measurand, denoted by z, is ob-
tained from equation 10 using input estimates x
i
:
z = f(x
1
,x
2
,...,x
N
) (11)
The standard uncertainty associated with that estimate
u(z), depends on the particular uncertainties of the
input quantities u(x
i
). For uncorrelated inputs the
square of the standard uncertainty of the output es-
timate is given by:
u
2
(z)=
N
i=1
u
2
i
(z), (12)
where the individual contributions in equation 12 are
obtained through the sensitivity coefficients c
i
via:
u
i
(z)=c
i
u(x
i
),c
i
=
∂f
∂X
i
x
i
(13)
10
-4
10
-3
10
-2
10
-4
10
-3
10
-2
log(tau)
log(sigma)
AVAR
10
-4
10
-3
10
-2
10
-6
10
-5
10
-4
10
-3
10
-2
log(tau)
MVAR
Figure 6: Noise processes corresponding to β =0and β =
−4. Situation of changing slope.
10
-4
10
-3
10
-2
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
log(tau)
log(sigma)
AVAR
10
-4
10
-3
10
-2
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
log(tau)
log(sigma)
MVAR
Figure 7: Noise processes corresponding to β =0and β =
−4. The upper noise process is the enveloping curve.
3.2 Types of Uncertainty for the
Input Estimates
The Type A evaluation of standard uncertainty is the
method which considers the statistical analysis of a
series of observations. The standard uncertainty is the
experimental standard deviation of the mean, which
in turn results from a regression analysis. By the
contrary, the Type B method is based on scientific
knowledge (Force, 1999). The standard uncertainty
of one input estimate u(x
i
), evaluated via the Type
B method, comprises all the information related to
the variability of the measurand X
i
. This variability
can fall into the following six categories, described in
(de la Rosa et al., 2005).
Insight and general knowledge are the sources of
information for a Type B evaluation of standard un-
certainty. In this paper no probability distribution is
provided in the data sheets for the quantities X
i
. Only
ICEIS 2006 - ARTIFICIAL INTELLIGENCE AND DECISION SUPPORT SYSTEMS
192
upper and lower limits can be estimated for the values
of the quantities in the manufacturer’s specifications.
So a rectangular probability distribution is a reason-
able description of one’s inadequate knowledge about
an input quantity in absence of any other information
apart from its limits of variability.
3.3 The Measurand in Traceable
Frequency Characterization
In traceable frequency calibrations the expression for
the measurand f
meas
is given by:
f
meas
=
f
REF
1 ± f
REF
∆x
τ
f
REF
,
¯
∆x,τ
(14)
where f
REF
is the reference (1 pps), ∆x represents
the phase shift between the source under test and the
reference, and τ is the averaging time or the cali-
bration period of the measurement system. Expres-
sion 14 is evaluated in the averaged phase shift dur-
ing the calibration period. For a zero phase shift or
an infinity averaging time, we have the ideal case
(f
meas
=f
REF
).
Using equations 12, 13 and 14, the uncertainty of
the frequency is obtained from equation 15:
u
2
(f
meas
)=
1 − f
REF
¯
∆x
τ
−4
×
u
2
(f
REF
)+u
2
(∆x)+u
2
(τ)
(15)
Sensitivity coefficients in expressions 12 and 13 de-
termine the contributions of the type B uncertainty,
which is associated to the instrument specifications.
4 EXPERIMENTAL RESULTS
4.1 Uncertainty Calculations
A high resolution function generator is chosen as de-
vice under test. It is set up to deliver a 1.1 Hz TTL
signal. The experimental arrangement is depicted in
figure 8. The measurement system comprises a TIC
3
,
a GPS receiver and the frequency source under test.
These instruments have been connected via GPIB to
the computer. Data points are captured every 1 s.
Figure 9 shows the signals involved in the measure-
ment process. Each measurement cycle corresponds
to 1 s. The bottom graph corresponds to the in-
stantaneous phase-deviation series, which comprises
m = 898 data. These data are the result of filter-
ing the spiky time-series of phase differences, and are
3
Time Interval Counter
GPIB
GPS Receiver
TIC
UTC 1 pps
CH
A
CHB
10 MHz
Ext. Ref.
Input
oscillato
r
50 Ω output
1.1 Hz
Figure 8: Experimental arrangement.
used to perform the calibration. These data are sup-
posed to be corrupted by white noise, with a rectangu-
lar probability density function. This is corroborated
later by means of AVAR and MVAR.
100 200 300 400 500 600 700 800 900 1000
0.2
0.4
0.6
0.8
From GPIB (sec.)
Signals in the measurement system
100 200 300 400 500 600 700 800 900 1000
-80
-60
-40
-20
0
Phase shift (sec.)
100 200 300 400 500 600 700 800 900
0.1
0.12
0.14
0.16
0.18
x(i) (sec.)
100 200 300 400 500 600 700 800
0.0906
0.0908
0.091
0.0912
Measurement cycles
Filtered x(i), (sec.)
Figure 9: Signals in the measurement chain. From top to
bottom: original data from the TIC and the GPIB interface,
accumulated phase shift, spiky phase differences, filtered
phase differences.
Table 2 summarizes the results of the type B evalu-
ation of the standard uncertainty. It has been reported
under the assumption of a rectangular (uniform) prob-
ability distribution of the magnitudes X
i
(see the fac-
tor
√
3 in the particular uncertainties). The rightmost
column has been rounded according to the resolution
of the TIC.
The expression for the standard uncertainty is ob-
tained from equation 16:
u
2
(z)=2×
N
i=1
u
2
i
(z)+VAR, (16)
where the double factor is due to the fact that we
are measuring phase differences. Type A uncertainty
(VAR) have been included, resulting 2 ×10
−4
s. The
expanded uncertainty of the measurement is stated as
the standard uncertainty multiplied by the coverage
factor k=2, which for a normal distribution attributed
to the measurand corresponds to a coverage probabil-
ity of approximately 0.95. The reported result of the
measurement is f
meas
=1.0974 ± 0.0004 Hz, for a
total measurement time of 898 s.
FREQUENCY CALIBRATIONS WITH CONVENTIONAL TIME INTERVAL COUNTERS VIA GPS TRACEABILITY
193
Table 2: Sources of the type B uncertainty assuming white
noise (TIC HM8122). Top to bottom: X
1
(±1 ext. clock
from GPS receiver), X
2
(Time base error from GPS clock’s
accuracy), X
3
(Jitter), X
4
(Systematic error), X
5
(Reso-
lution from GPS receiver HM8125), X
6
(Accuracy), X
7
(Jitter), X
8
(Averaging time of the measurement system:
u
2
(x
8
)=u
2
(x
6
)+u
2
(x
7
)). Units in [ns].
Value Std. uncertainty Contribution
u(x
i
) u
i
(z)=c
i
× u(x
i
)
100
100
√
3
70
100
100
√
3
70
5
5
√
3
4
< 4
4
√
3
3
100
50
√
3
4
100
100
√
3
70
5
5
√
3
4
66 0.5
4.2 Testing for White Noise
The ratio of the classical variance (VAR) to the Al-
lan variance (AVAR) provides a primary test for white
noise. This quantity (0.672) is less than 1+1/
√
m
1.033; thus it is probably safe to assume that the data
set is dominated by white noise, and the classical sta-
tistical approach can safely be used. Failure of the
test does not necessarily indicate the presence of non-
white noise (Fluke, 1994). A slope test (based in
AVAR and MVAR curves) has been developed to con-
firm the presence of white noise. AVAR and MVAR
curves are depicted in figure 10.
10
0
10
1
10
2
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
log(tau)
log(VAR), log(MVAR)
Figure 10: AVAR (upper) and MVAR (lower) log-log
curves. The final calibration period is τ = 500 × τ
0
for
τ
0
=1s.
Measures of the slopes over the log-log graphs in
figure 10 offer the results -1 and -1.5 for log(AV AR)
vs. log(τ ), and log(MVAR) vs. log(τ), respec-
tively; which indicate that a white phase modulation
process is coupled to the frequency source under test
(see table 1).
5 CONCLUSION
Frequency calibrations using incomplete data sheets
can be performed by means of the white noise hypoth-
esis. This conveys the idea of using uniform probabil-
ity distributions for which classical variances are eas-
ily computed. Since the sensibility coefficients in the
expression of the uncertainty of the measurement are
computed under this assumption, it has to be corrob-
orated later. Two tests have been revised and applied
successfully. The numerical (first) test is in turned
corroborated by the slope test. Sources of Type B un-
certainty have been calculated considering the white
noise assumption.
ACKNOWLEDGEMENTS
The authors would like to thank the Spanish Min-
istry of Education and Science for funding the project
DPI2003-00878 which involves noise processes mod-
elling and time-frequency calibration.
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