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Estimation of Time Variant System Clock Period
for Wireless Sensor Network Applications
Hans-Peter Bernhard and Andreas Springer
Johannes Kepler University Linz
Institute for Communications Engineering and RF-Systems
Altenbergerstr. 69, 4040 Linz, Austria, Email: h.p.bernhard@ieee.org
Abstract—In industrial applications synchronized sensor nodes
are vital for many tasks like multi parameter sampling of
complex machinery behavior. The wirelessly connected nodes are
usually synchronized to a base station. Therefore, frequency or
period estimation of the reference clock is a key issue for all
connected tasks like sampling, localization or applying energy
optimized communication protocols. In wireless systems and
especially in harsh industrial environments, it is likely to miss
one or more synchronization events. The available data for clock
estimations is therefore sparse and periodogramm estimators,
at a complexity of at least O(Nlog N), are commonly used
for accurate clock estimation. We introduce a period estimator
for sparse clock signals with O(N)complexity. Furthermore,
we present an equation for the observation time necessary to
estimate the clock period at a certain quality. Mostly due to
temperature influences, the crystal frequency at the nodes are
varying. Our iterative period estimator follows those changes
with a given estimation accuracy. We analyze an equation, which
allows to calculate the accuracy of the estimator given a certain
change rate of time variant system clock. The work is concluded
with simulations considering sparse and time varying processes,
including a discussion of the application of the method.
Index Terms—fundamental frequency estimation, estimation
error, aliasing, pulse signals, point process, sparse process, low
complexity.
I. INTRODUCTION
For decades, sensors are part of the industrial equipment and
vital to monitor and control processes, gather data and sense
real-time machinery behavior. Requirements like low latency,
high reliability and synchronized operation or sampling are
essential in the industrial field. The number of monitoring
sensor nodes is also often very high for e.g. engine test beds
or similar measurement situations. These applications result
in many different requirements like synchronized sampling,
dependability, low or at least defined latency and high number
of sensors at low cost per individual sensors.
Moreover, if the sensors communicate wirelessly, this opens
a variety of new usage fields and flexibility at potentially low
cost. Based on energy autarky, low energy consumption per
node is an additional major challenge added to the list of
important requirements for wireless sensor networks (WSN).
Therefore, it is inevitable to focus on ultralow-power sensor
nodes providing a sample rate of e.g. 10 Hz with approxi-
mately 100 µW average power consumption. Moreover, the
power consumption contains also continuos wireless data
streaming with an average data rate of 0.4kbps. It enables the
application of battery-powered or energy harvesting wireless
sensor nodes. Compared to WSNs in home automation or
environmental monitoring, where small data traffic is expected,
industrial sensing applications are often characterized by a
higher data throughput in a dynamical changing, harsh, and
highly reflective environment. For data transmission in em-
bedded low-power sensor networks, a common approach is
the usage of established wireless standards such as ZigBee-
or other IEEE 802.15.4-based protocols (e.g., wirelessHART
and ISA100.11a [1]), Bluetooth low energy [2], ANT+ and
other proprietary solutions such as MiWi or SimpliciTI. Most
of them operate in the unlicensed 2.4-GHz industrial, scientific
and medical (ISM) band and are limited in their data transmis-
sion rate between 250 kbps and 2 Mbps. None of these systems
offer the possibility for an accurate distributed synchronized
data sampling without major modifications of the software
stack like done by Boggia et al. in [3] or the used hardware
[4]. The Chirp Spread Spectrum (CSS) communication scheme
described in [5] is also a method to introduce high accurate
synchronization near to physical (PHY) layer. In [6], a mea-
surement system is introduced, which features synchronized
sampling of sensor data in combination with a beacon driven
transmission protocol. Similar to [7], the synchronization is
derived from the time domain multiple access (TDMA) com-
munication protocol. Additionally, synchronous sensor nodes
help to save energy by reducing the active time of the node. If
perfectly synchronized, it is possible to wake up close to sam-
pling or transmission time so the sleeping periods of the nodes
can be increased to the maximum duration. During sleeping
period, the node operates its clock autonomously without any
synchronization. Therefore, the accuracy of the adjusted low
power clock is important for exact offline time measurement.
The clock rate in the energy efficient sleeping mode is gener-
ally low e.g. 32768Hz [8] and therefore, it introduces coarse
timing granularity, which has a high impact on synchronized
sampling and energy consumption. Consequently, some effort
is required to overcome the mentioned drawbacks in order
to gain high accurate offline synchronization. We use low
complex ∆Σ based period calculation [9] on the nodes to
maintain sub-tick-accurate synchronization. For all mentioned
tasks, we have to estimate the network or central clock
period accurately. The data source for the clock estimation
is given by beacon arrival time, TDMA frame signatures or
any other reliable synchronization event. The measurements
are noisy and, furthermore, some trigger events can be missed
randomly due to harsh environmental conditions. If in the
measurement sequence some measurements are missing, it is
called sparse. We use a point process model to describe such a
measurement sequence [10]. Most highly accurate frequency
estimation methods operate on a frequency domain representa-
tion of signals, e.g. Fourier transform, power spectral density
estimation or others. In the special case of identifying the
fundamental frequency, one of the most common methods is
the periodogram estimation [11]–[13]. Current research [10]
shows a variance of the estimation error according to O(N−3)
at a computational complexity of O(Nlog(N)), where Nis
the number of samples used in the estimation. As we focus
on periodic signals, recent research on frequency estimation of
cyclostationary signals is relevant. Cyclic spectrum estimation,
that can be used to estimate the fundamental frequency is
presented in [14] and [15]. But all of these methods do
not consider energy efficiency and time variance, which is
immanent in WSNs in industrial environments. Therefore, we
consider the wireless infrastructure in section II, discuss the
process model in section III and derive an estimator in section
IV. Addtionally, the article includes simulations in section
V, which show the applicability of the algorithms and final
conclusions VI summarize our results.
II. WIRELESS SE NS OR INFRASTRUCTURE
The high-level architecture of the WSN for the measurement
environment and its connection to an automation system (AS)
are shown in Fig. 1. Hundred and more sensors are arranged
in one centrally synchronized network managed by one base
station (BS) which consists, according to the background
structure of Transducer Electronic Data Sheets (TEDS), of a
wireless network processor (WNP) and a network application
processor (NCAP). TEDS is described in the ISO/IEC/IEEE
21451 – Networked Smart Transducer Interface Standard. An
detailed example is described in [16] and [9]. In typical
Fig. 1. System architecture
applications, the CPU and the node can be put into a sleep
mode for more than 90% of the time. The timing of the
protocol is maintained by the onboard real time clock (RTC).
Consequently, the clock frequency is reduced from 16 MHz
down to 32.768 kHz, resulting in a reduction of the averaged
sleep current from 400 µA down to 4-5 µA [8]. Hence, the
total power consumption of the node is reduced from 95%
to 99% for standard measurement tasks. Here, we have to
point out that the clock rate of the base station is kept at
16 MHz. The base station is mains-powered and with the
high rate clock, time durations can be set or measured with
0.0625 µs accuracy. The low clock frequency at the nodes has
a remarkable disadvantage on the timing resolution, which
is reduced from 0.0625 µs to 30.52 µs. Thus, a systematic
error is introduced to the generation of a periodic sampling
interval at the node and the TDMA schedule. Hence, the
superframe (SF) duration with 100 ms of the TDMA protocol
cannot be generated with sufficient accuracy by using the given
clock frequency. In worst case, the SF interval is 30.5 µs too
short or too long. If these inaccuracies are cumulated over 10
SF sleep periods, the total inaccuracy is ≈ ±0.3 ms. For a
typical TDMA time slot length of 0.3 ms to 1 ms, the node
has to listen to an additional synchronization beacon, which
is a comparable amount of energy as the transmission of the
data. Even using a crystal with a nominal clock frequency,
which is a multiple of the TDMA frame rate, would not solve
the problem caused by the usual manufacturing tolerances of
±150 ppm. Therefore, the clocks need to be adapted con-
tinuously to avoid being out of sync for longer periods of
radio silence. We proposed using a ∆Σ-converter approach to
generate sequences of SF with different lengths, so the average
SF length of the node accurately matches the TDMA schedule
of the base station [9]. However, this approach is based on
the assumption of an accurately known synchronization clock
period, needed to adapt sampling and all connected tasks
that rely on synchronicity. Thus, clock period estimation has
also to be done by low complex algorithms to avoid time
and memory consuming estimators. For that we propose a
radical simple approach with the same estimation accuracy
as frequency estimators in frequency domain and extend the
working regime to time variant frequency estimation without
increasing complexity.
III. PROC ES S MOD EL
A. Time invariant point process
The clock period estimation is based on received synchro-
nization events at the node. These events can be derived from
TDMA frames or separately sent beacons, depicted as pulses
a(t)in Fig. 2. If these events are detected by δa(t), the local
a
(
t
)
P
t
δa
(
t
)
y[4]
y[5]
y[3]
y[2]
y[1]
y[4]
Φ
t
Fig. 2. Ideal periodic generated events a(t)and the derived noisy point
process to show schematically a noise free event sequence and in parallel a
noisy sparse point process.
clock value nis used as timestamp for each event or trigger
time. It generates a time series with period Pand phase shift
φwhich is given by
y[n] = nP +e[n] + φ(1)
+
y
d
2
[n]
++ +
^
P[n]
y[n]
⏞
N
-
y
d
[n]
v[n]
yd[n]
μdN
T
T
T
T
T
T
+
t
sample:
t>y[n]+Δ t
max
⏞
μdN
if event
√
v[n]
μ
d
N
√
N
else if
Fig. 3. Iterative period estimator for
b
P[n]depicted using an unrolled iteration.
with measurement noise e[n] = N(0, σe), assumed to be
Gaussian. Equation (1) is representing a monotonously in-
creasing time series which can be used to estimate the fun-
damental frequency of the generating periodic events. The
processes is called a non-sparse point processes or simply a
point processes when all events a(t)are detected. A process is
called sparse if some events are missed as depicted in Fig. 2.
Hence, the variable nof (1) has to be replaced by a mapping
x[n]
y[n] = x[n]P+e[n] + φ, (2)
where x[n]6=n. A discrete time random variable d[n]with
mean 1≤µd<∞is used to model this behavior. d[n]
is assumed to be distributed geometrically, according to a
Poisson process or uniformly and it describes the number of
real events between sample x[n−1] and x[n]with
x[n] = x[n−1] + d[n] =
n−1
X
i=0
d[i].(3)
Therefore, the stationary sparse process can be written as
y[n] =
n−1
X
i=0
P d[i] + e[n] + φ. (4)
B. Time varying point process
The period of a network clock is generally not constant
because of temperature or other influences. As an example, the
period variation is in a range of ±100ppm for a temperature
range between −25◦C and +75◦C by applying a standard
crystal1. Hence, the period variation is modeled by
p(t) = P0+pT(t),(5)
where p(t)is the time varying period with a constant nominal
period P0and the variation of the period pT(t). For time
discrete sampling with a sampling period of P0, it has to be
guaranteed that the bandwidth Bpof p(t)is Bp<2
P0to avoid
aliasing. The period variation is written in time discrete form
as
p[n] = P0+pT[n](6)
1Quartz Crystal Specification 91SMX of IQD Frequency Products Ltd
with n∈N. Therefore, the sampled version of the time variant
point process is, with (1) and (6), a sum over all individual
periods
y[n] =
n−1
X
i=0
p[i] + e[n] + φ=nP0+
n−1
X
i=0
pT[i] + e[n] + φ. (7)
IV. THE E ST IM ATIOR
The estimator, as depicted in Fig. 3, is designed to determine
the period of repeating events. Each detected rising edge of
a(t)in Fig. 2 is referred as event. These events trigger the
sampling of the node time counter and the storing of the
value as y[n]. The last bµdNcevents are stored in a delay line
to calculate the difference yd[n]between the current sample
and the sample delayed by bµdNc. The squared sequence of
y2
d[n]is processed by a moving average filter of the length N.
The scaled square root of the moving average is the estimate
of the clock period. If the process is sparse, missing events
are inserted and calculated, by adding an averaged sample
difference yd[n]
bµdNcto the previous sample via a feedback loop.
A simple rule is used to decide whether or not a sample has
to be inserted. If the node time counter tis greater than
y[n]+∆tmax, then an additional sample is inserted. The
constant ∆tmax has to be defined and represents the maximum
period that can be measured or estimated.
A. Derivation of the estimator
Based on the point process (7), the sparse process is given
with (4) by
y[n] =
n−1
X
i=0
P[i]d[i] + e[n] + φ, (8)
where P[i]represents the average period of the last d[i]
periods. The estimator of Fig. 3 uses the time difference
between two events as
yd[n] = y[n]−y[n−N] =
N−1
X
i=0
P[n−i]d[n−i] + ed[n],(9)
where ed[n] = e[n]−e[n−N]. Without losing generality, (9)
can be written as
yd[n] =
N−1
X
i=0
P[n−i]d[n−i]+ed[n] = P[n]
N−1
X
i=0
d[n−i]+ed[n],
(10)
where P[n]corresponds to the period of a periodic process,
which is measured at discrete time nby using Nsamples of
the past. The mean of the sparse process is defined as µd=
1
NPN−1
i=0 d[n−i], and, thus, a simple product Nµdcan be
used in (10) and therefore,
yd[n] = P[n]Nµd+ed[n].(11)
This equation motivates to extend the data acquisition mem-
ory to a length of bµdNcwith equidistant samples for the
calculation of yd[n], as can be seen in Fig. 3. As already
mentioned, the process is sparse and not all periodic events
are sampled. The random gaps between observed events are
filled by sampling an interpolated value
ˆy[n+ 1] = y[n] + yd[n]
bµdNc,(12)
if the maximum period length y[n]+∆tmax has been exceeded.
∆tmax is determined by the smallest frequency of the search
window ∆fas ∆tmax =1
∆fmin . So far, up to equation (10), it
was assumed that y[n]and yn−bµdNcare non interpolated
real sampled values and therefore, it is obvious to have ed[n]in
(9) as resulting noise. With interpolated signals, the difference
yd[n]is
N−1
X
i=0
P[n]d[n−i] + ed[n] =
N−1
X
i=0 P[n]d[n−i] + ed[n]
N
(13)
and for sample distances d[n−i], interpolation is done by
adding a number of d[n−i]samples each with an individual
estimated period resulting from interpolation (11). Obviously,
there is an individual estimation error for each period.
eµd[n−i−k]
bµdNc=yd[n−i−k]
bµdNc−P[n],(14)
where yd[n−k−i]is the interpolated difference. By using
d[n−i]P[n] =
d[n−i]
X
k=1
P[n],(15)
we rewrite (13) with considering each individual interpolated
period
yd[n] =
N−1
X
i=0 P[n]d[n−i] + ed[n]
N=
N−1
X
i=0
d[n−i]−1
X
k=0 P[n] + eµd[n−i−k]
bµdNc.(16)
The error eµd[n−i−k]is an i.i.d. random variable and there-
fore, we can rewrite the sum without any loss of generality
as
N−1
X
i=0
P[n]d[n−i] +
N−1
X
i=0
ed[n]
N=
NµdP[n] +
N−1
X
i=0
d[n−i]−1
X
k=0
eµd[n−i−k]
bµdNc.(17)
Canceling out the equal parts of (17), the result is used to
write the second order moment as
N−1
X
i=0
σ2
d[n]
N2=
N−1
X
i=0
d[n−i]−1
X
k=0
σ2
µd[n−i−k]
bµdNc2(18)
Nσ2
d[n]
N2=bµdNcσ2
µd[n−i−k]
bµdNc2(19)
µdσ2
d[n] = σ2
µd[n−i−k],(20)
what means, that the error of the interpolated difference can
be assigned as eµd[i] = N(0,√µdσd).
To proceed, we introduce a vector notation
yd[n] = {yd[n], . . . , yd[n−N+ 1]}(21)
={P[n]dN
d[n]+eµd[n], . . . , P [n]dN
d[n−N+1]+eµd[n−N+1]},
with dN
d[n] = PN−1
i=0 d[n−i]as an intermediate step.
yd[n]is the sum of two vectors, the center vector c[n] =
{P[n]dN
d[n], . . . , P [n]dN
d[n−N+ 1]}and the error vec-
tor of a multivariate Gaussian random variable eµd[n] =
{eµd[n], . . . , eµd[n+−N+ 1]}. Therefore, this results in
yd[n] = c[n] + eµd[n].(22)
Hence, yd[n]represents a multivariate i.i.d. Gaussian ran-
dom variable with a diagonal covariance matrix, which is
shifted by the center vector c[n]. All diagonal elements of
the covariance matrix have the value σ2
dand therefore, it
is invariant to a rotation in the Ndimensional space. We
want to emphasize, that the variance of the N-dimensional
diagonal vector has the same value as the variance in each
of the Ndimensions. Hence, the center of the N-dimensional
observation is given by c[n]with an additive error in direction
of the vector c[n]
|c[n]|with eµd[n]c[n]
|c[n]|. We write the vector
components as
P[n]dN
d[n] = P[n]
N−1
X
i=0
d[n−i] = P[n]bNµdc.(23)
The length of the observed vector can be determined by using
the absolute value of the interpolated yd[n]as
|yd[n]|=|c[n]|+eµd[n]c[n]
|c[n]|
=v
u
u
tN
X
k=1bNµdc2P2[n] + eµd[n](24)
=P[n]bNµdc√N+eµd[n].(25)
Finally, we express v[n]in the estimator in Fig. 3 as
v[n] =
N−1
X
i=0
y2
d[n−i] = |yN
d[n]|2=P[n]bNµdc√N+eµd[n]2
(26)
and use the structure of Fig. 3, to estimate P[n]of the sparse
process. With (26), the estimator becomes
ˆ
P[n] = sN−1
P
i=0
y2
d[n−i]
bNµdc√N.(27)
The number of Nis a key parameter for the quality of
estimation. It was shown in [17] for periodic band limited
signals with B<2πK/P that nonuniform sampling with
N≥2K+ 1 number of samples is sufficient for perfect
reconstruction. Hence, it is necessary to have at least N≥3
samples based on the band limit of 2π/P where K= 1 for
the iterative frequency estimator.
B. Mean square error of the estimated period
1) Stationary sparse point process: The estimated period
ˆ
P[n]is given by (27) at discrete time n. If the process is
stationary for at least Nconsidered samples, the estimation
error can be deviated in the same way as the fundamental
frequency fP=1
Pof the point process presented in [18].
Considering (27), it was proven in [20] that, the mean square
error (MSE) of the period estimator for stationary sparse point
processes is
MSE =Eh(P[n]−ˆ
P[n])2i≥2σ2
e[n]
bµdNcN2(28)
for finite and infinite discrete time interval N.
2) Time variant sparse point process: We consider the time
varying processes given by (5) where the estimated parameter
p(t)is varied with temperature or other external influence.
Usually, a transfer function is derived to describe the effects
of changing input to the resulting output ˆ
P[n]. But as we
have a nonlinear element in the estimation algorithm, it is not
possible to derive a standard transfer function. The estimator
can be interpreted as nonlinear Wiener-Hammerstein system
[19] consisting of two linear blocks and a static nonlinearity
in between. Therefore, the difference yd[n]is given by straight
forward application of (9)
yd[n] = y[n]−y[n− bµdNc](29)
which is the input of the squarer. The squarer is interpreted
as second order Volterra operator H2(yd[n]) according to
[19]. This is followed by the second linear system which is
the moving average filter. To overcome the missing spectral
description of a Wiener-Hammerstein model, we investigate
the linear models separated from H2(yd[n]) and focus on
prototypical time variant processes which change the period
duration according to a sinusoidal function in (5). Let
p[n] = P0+ ˆpTsin(θn)(30)
with θ < π to fulfill the sampling theorem. Hence, the
averaged period P[n]of (10) is
P[n] = 1
bµdNc
bµdN−1c
X
k=0
p[n−k](31)
=P0+ˆpT
bµdNc
bµdN−1c
X
k=0
sin(θ(n−k)) (32)
which is used as reference for the estimator considering a
sinusoidal type of time varying period. The resulting in-
12345678
10-9
10-8
10-7
10-6
10-5
10-4
Average of the measurement loss µd
MSE
θP= 200s
θP= 200s and θP= 314.5s
θP= 200s and
3 additive sinusoids each θP>200s
θP= 200s and
5 additive sinusoids each θP>200s
Lower bound θP= 200s
Fig. 4. Dependence of estimation quality (MSE) and the average number of
lost measurements.
terpolation error, is after some straightforward mathematical
manipulations,
Ee2
di[n]≥E"1
2ˆpTθbµdNc
2−i2#(33)
≥4
105θ2ˆp2
Tµ5
d7N2−14N+ 8N. (34)
Following Fig. 3, we have to use the moving average filter for
the squared signal part and apply the square root which yields
with (26) to
pvs[n]≥√NbµdNcP0+
ˆpTsin θn−bµdNc+N
2+ 1W(θ, µdN)W(θ, N )
√N,
(35)
where W(θ, y) = sin(θy
2)
sin( θ
2). The estimated value ˆ
P[n]is
calculated by using pvs[n]according to (27). The MSE is
based on the difference P[n]−ˆ
P[n]and therefore, the group
delay of the estimator has to be canceled out by delaying the
signal in (35) by bµdNc+N
2−1. Thus, the MSE of the estimator
is covering the sparse, time variant and the stationary case by
MSE ≥ˆp2
T
21−W(θ, µdN)W(θ, N )
bµdNcN2
+4
105θ2ˆp2
Tµ3
d7−14 1
N+ 8 1
N2+2σ2
e[n]
bµdNcN2.(36)
A detailed prove is given in [20].
V. SIMULATIONS AND DISC US SI ON O F TH E RE SU LTS
The simulation is set up according to an engine test bed
instrumentation, where the time variance is mainly based
on temperature variations of the measurement equipment.
We assume a worst case scenario of temperature differences
causing 10 ppm clock period differences. The time variant
clock is analyzed with respect to parameter change periods
of 10 s to 200 s. Additive measurement noise is considered
as an additive Gaussian noise process according to N(0, σe).
The measurement losses are considered in a range of 0 to 20
measurements in a row, modeled by a uniformly distributed
random variable showing a mean from 1 to 10 at maximum.
With Monte Carlo simulation, 1000 realizations of the pro-
cesses were created to gain statistical randomized experiments.
The first analysis considers the dependence of estimation
quality (MSE) and the average number of lost measurements.
The approximation, based on (36), shows an increasing trend
proportional to µdin the log-log plot of Fig. 4. This behavior
is confirmed by the simulation, if the average of the sparse
process µd≤5(see Fig. 4). For high numbers of lost
measurements in a block (µd>5) the linear approximation
of W(θ, bµdNc)ubµdNcfails and the error increases.
Additionally, Fig. 4 shows that complex time variant behavior
does not influence the estimation quality. We used time variant
disturbances modeled with one, two, three and five additive
sinusoids. The sinusoids have similar amplitudes, and, as can
be seen from Fig. 4, it does not alter the estimation quality.
This behavior is introduced by the time variant error decay,
proportional to θ2, concluded from (36).
Fig. 5 shows a comparison between lower bound of the
MSE and the MSE of simulations for different length of
estimation samples N. If we consider 3 processes with a
time variation of Pθ= 50 s,100 s,200 s respectively and a
measurement noise σ2
e= 10−12, it can be seen that, for
small numbers of N, the MSE decays with increasing N. This
behavior is based on pure measurement noise and the decay
is according to relation (28) of the stationary case. MSE of
simulations and lower bound of the MSE follow this decay
similarly, also for different time variation frequencies. For high
Nfrom 50 to 200 the error due to time variance dominates the
MSE, and it rises with increasing N. The first part of (36) is
explaining this increase. In the transient region from decaying
to increasing MSE, the MSE is not perfectly approximated
by theory, because of some simplifications necessary to derive
a closed form approximation of the MSE. Nevertheless, this
behavior is similar to all considered periodic time variant
processes. The dominating parameter is the period Pθwhich
scales the MSE of both the simulation and the lower bound. It
is very important to learn from Fig. 5 that the time variation
has a great impact on the MSE. Obviously, if the time variance
has a period of 50s, a moving average filter with a memory of
µdN= 20 b= 2 s fails to follow the time variation closely with
a neglectable error. This fact leads to the paradox situation that
the MSE is not increasing, if the number of estimator samples
is increased. Moreover, considering Fig. 5, there is a minimum
of the MSE. This minimum can be used to select an optimized
memory N.
In Fig. 7, the next step in analyzing sparse or lossy mea-
surements is done by different µdbetween 1.0 and 5 and a
time varying period of 50 s. If measurement noise is less than
the error caused by the time variation, we see no dependence
on the measurement noise. Contrary, if the time variant error is
less than the measurement noise, the situation can be analyzed
by the right most part of Fig. 7. In this range the MSE is
101102
10−16
10−15
10−14
10−13
10−12
10−11
10−10
Number of considered samples N
MSE
Simulation Low. bound Pθ=500 samples
Simulation Low. bound Pθ=1000 samples
Simulation Low. bound Pθ=2000 samples
Fig. 5. The MSE of period estimation for the time variant reference clock
period with increasing Nfrom 2 to 200 with µd= 2.
10−1
10−5
10−10
10−15
10−15
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
Measurement noise σ2
e
MSE
Simulation Low. bound Pθ=10s
Simulation Low. bound Pθ=50s
Simulation Low. bound Pθ=100s
Simulation Low. bound Pθ=200s
Fig. 6. The MSE of period estimation for the time variant reference clock
period with µd= 2 of the lossy (at average every second measurement)
observation with respect to additive measurement noise. The time variance is
considered with periods of 10, 50, 100 and 200 seconds.
proportional to σ2
eresulting from (28).
Simulations without lost measurements but time variance
conclude our analysis. Fig. 8 shows impressively, that com-
pared to Fig. 6, losing measurement data is significantly
changing the MSE. Addtionally, the lower bound and the MSE
of simulations are nearly identical.
A. Numerical Example
Finally, we apply our results by using an example published
in [9]. Concluding from [9], the necessary MSE is 10−14
to reach sufficient synchronization quality. The measurement
noise is about σ2
e= 1.6·10−10. With those prerequisits, the
number Nof the estimator is calculated. For non sparse and
statinonary samples (28) is resulting to
N=&3
r2σ2
e
MSE '=&3
r1.6·10−10
10−14 '= 26.(37)
10−1
10−5
10−10
10−15
10−5
10−8
10−11
10−14
Measurement noise σ2
e
MSE
Simulation Low. bound µd= 1.0no lost events
Simulation Low. bound µd= 2
Simulation Low. bound µd= 3
Simulation Low. bound µd= 4
Fig. 7. MSE of period estimation for time variant signals with lossy
observations and additive measurement noise. The period of the time variance
is 50 s.
10−1
10−5
10−10
10−15
10−16
10−15
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
Measurement noise σ2
e
MSE
Simulation Lower bound Pθ=10s
Simulation Lower bound Pθ=50s
Simulation Lower bound Pθ=100s
Simulation Lower bound Pθ=150s
Simulation Lower bound Pθ=200s
Fig. 8. The MSE of period estimation for the time variant reference
clock period with non sparse (lossless) observation with respect to additive
measurement noise. The time variance is considered with periods of 10, 50,
100, 150 and 200 seconds.
The considered nodes are using a crystal with ±10ppm in
the defined temperature range. Therefore, the time variation is
according to (36) with ˆpT= 10−5
W(θ, N ) = Nv
u
u
t 1−s2·MSE
ˆp2
T!= 26r1−√2·10−2.
This equation was solved numerically and yields θ≈0.008.
Due to superframes of 0.1s, the shortest allowed full scale
variation period is 12.5s, whilst keeping the nodes syn-
chronous. Furthermore, we consider sparse data with losing
samples so that µd= 2. If we analyze Fig. 5, we see that
for N= 26 the time variation with Pθ= 50 s will fulfill the
required error of 10−14. Hence, a shortest period of 50s for
full scale temperature variations is allowed, and we do not
lose synchronicity of the nodes.
VI. CONCLUSIONS
We introduced a highly efficient estimator for the fundamen-
tal period of network clocks. Such an estimate, with low MSE,
is needed to apply an efficient synchronization of wireless
sensor networks. The complexity of the presented algorithm
is O(N)with an accuracy of O(N−3)for stationary data,
which coincides with the MSE for frequency based estimation
as shown in [10], [21]. Therefore, the presented estimator
has a similar performance as estimators based on frequency
domain analysis, but with less complexity. Thus, we meet
the energy requirements for ultralow-power wireless nodes.
Moreover, an approximation of the MSE for this estimator was
introduced. The approximation even applies to time variant
systems when the clock rate of the base station, sensor node or
both changes. Based on theory and simulation, we showed that
increasing the number of samples to estimate the fundamental
period of time variant processes, can reduce estimation quality.
We introduce, how the number of samples can be optimized
with respect to measurement noise and change rate of the
time varying process. The closed form equation for the MSE
even holds if the data is sparse, which means losing samples
needed for the estimation. Hence, it is possible to quantify
the reliability of the estimate, if the average number of lost
samples is known. The complex properties of the estimator are
illustrated by simulations to show the influence of memory,
time varying clocks, measurement noise and lost samples.
The presented framework, estimator and MSE approximation,
allow to support the development of synchronized sensor nets
in harsh environments.
ACKNOWLEDGMENT
This work has been supported in part by the Austrian Re-
search Promotion Agency (FFG) under grant number 853456
FASAN and also supported in part by the research from
DEWI project (www.dewi-project.eu) leading to these results
has received funding from the ARTEMIS Joint Undertaking
under grant agreement no621353 and the Austrian Research
Promotion Agency (FFG) under grant no. 842547.
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