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Adaptive Period Estimation For Sparse Point Processes

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Hans-Peter Bernhard, Andreas Springer
Johannes Kepler University Linz,
Institute for Communications Engineering and RF-Systems
Altenbergerstr. 69, 4040 Linz, Austria, Email:
In this paper, adaptive period estimation for time varying
sparse point processes is addressed. Sparsity results from
signal loss, which reduces the number of samples available
for period estimation. We discuss bounds and minima of the
mean square error of fundamental period estimation suitable
in these situations. A ruleset is derived to determine the opti-
mum memory length which achieves the minimum estimation
error. The used low complex adaptive algorithm operates with
variable memory length Nto fit optimally for the recorded
time varying process. The algorithm is of complexity 3O(N),
in addition to that the overall complexity is reduced to 3O(1),
if a recursive implementation is applied. This algorithm is
the optimal implementation candidate to keep synchronicity
in industrial wireless sensor networks operating in harsh and
time varying environments.
Index TermsFrequency estimation, low complexity,
sparse process, synchronisation, industrial sensor networks
Many applications rely on period or frequency estimation
such as carrier frequency recovery in communication sys-
tems, vital sign monitoring or synchronization in wireless
sensor networks (WSNs) [1, 2, 3, 4]. Within networks, beacon
signals are sent out periodically from a master and received
by many communication partners. The time stamping with
the arrival time enables the estimation of the beacon period
and the synchronization the local clock. Sparsity is caused by
unavoidable occasional loss of communication links in harsh
environments. Hence, if beacons are lost and time-varying
clocks occur due to environmental influences, frequency esti-
mation becomes more complicated. Period estimation of such
This work has been supported in part by the Austrian Research Pro-
motion Agency (FFG) under grant number 853456 (FASAN: Flexible Au-
tonome Sensorik in industriellen ANwendungen) and by research from the
SCOTT project. SCOTT ( has received funding from
the Electronic Component Systems for European Leadership Joint Under-
taking under grant agreement No 737422. This Joint Undertaking receives
support from the European Union’s Horizon 2020 research and innovation
programme and Austria, Spain, Finland, Ireland, Sweden, Germany, Poland,
Portugal, Netherlands, Belgium, Norway.
time-varying processes has been considered in [5]. If Nis
the memory size of the estimator, the estimation mean square
error scales with O(N3)in situations where the signal is sta-
tionary [6] or time variation of the amplitude is much smaller
than measurement noise. We model measurements as sparse
periodic point process with additive phase noise in Sec. 2.
The period of a sparse point process is mostly assessed by
spectral estimation techniques [7, 8, 9]. One of the common
methods is the periodogram estimation [10, 11, 12] by con-
sidering stationary processes. Its computational complexity
is in the order of O(Nlog(N)) [13]. In [14, 15], funda-
mental frequency estimation of cyclo-stationary processes
was introduced, leading to similar results. To the best of our
knowledge, no method exists addressing the optimization of
period estimation for sparse time-varying point processes.
The optimization of the memory usage is discussed in Sec. 3,
leading to a design rule that depends on straightforwardly
measurable signal parameters. In Sec. 4.1, an adaptive pe-
riod estimation is presented with a computational complexity
3O(N). The proposed estimator is extremely simple and easy
to implement in digital hardware with limited computational
capabilities. Simulations in Sec. 3.3 and Sec. 4.2 support the
theoretic considerations and conclude the work.
Fig. 1. Time varying sparse periodic point process
Events like periodic beacons δt
bshould be detected by the re-
ceiver of a node in a sensor network. Whenever a beacon is
received, a time stamp y[n] = tis taken from the time vari-
able t.y[n]represents a periodic time series with period P,
written as y[n] = nP +e[n] + φ(1)
with random phase φand measurement noise e[n]=N(0, σe)
assumed to be Gaussian. This process is called a non-sparse
point process. A process is sparse if some events are missing
as depicted in Fig. 1. Additionally, if periods changes over
time, the process is also time variant and we have to replace
Pwith p[n]. In order to model time variance and sparsity, the
variable y[n]in (1) is replaced by the recursive formulation
y[n] = y[n1] + d[n]p[n] + e[n] + φ . (2)
A time discrete random variable d[n]with mean 1µd<
is used to model this behavior. The time variation of the
period pt[n]and the average period P0defines the individual
period p[n] = P0+pt[n](3)
with nN. We assume a periodic time variation as a start-
ing point before we extend the discussion to more complex
signals in section 3.2. The time variation is
p[n] = P0+pTsin(θn)(4)
with θ < π to fulfill the sampling theorem and pTis the peak
amplitude of the time variation.
3.1. Sinusoidal time varying period
The MSE of the period estimator for sparse point process with
a sinusoidal time variation with frequency θis, referring to
21W(θ, µdN)W(θ, N )
3(1 + µdN)+σ2
if Nsamples are available. As an abbreviation W(θ,bµdNc)=
sin θbµdNc
2)is used. On average, the process has
µd1lost events and additive white Gaussian phase noise
d/2. According to [5], all three terms in (5) have an iden-
tifiable source. Firstly, the frequency estimation error of the
stationary process with phase noise is
Secondly, the error introduced by the sinusoidal time variance
of the frequency is
21W(θ, µdN)W(θ, N )
and finally, there is an additive upper bounded interpolation
error introduced by generated samples which are inserted in-
stead of lost samples
3(1 + µdN).(8)
All three MSE parts depend on process parameters µd,pT,θ
and one parameter of the estimator N. Therefore, only Ncan
be used to improve the MSE. Hence, we minimise the MSE
over Nby
which is solved by finding a solution for
= 0.(10)
Obviously, NN+and the first derivative exists only if
NR. Without losing generality, we can treat (5) as
continuous function in Rand remap N0to N+if the min-
ima is found. The derivative of MSEθis rather compli-
cated and it is not possible to find a closed form solution
for (10). To overcome this problem, we use approxima-
tions for the trigonometric functions by assuming Nθ 1.
Owing to the vanishing of first and second order approx-
imation, at least a fourth order approximation has to be
used. As a first step we use 2 sin(θµdN/2) sin(θN/2) =
cos(θ(µd1)N/2) cos(θ(µd+ 1)N/2) to circumvent the
trigonometric product and write
1sin(θµdN/2) sin(θN/2)
1cos(θ(µd1)N/2) cos(θ(µd+ 1)N/2)
2 sin2(θ/2)bµdNcN2
Consequently, we apply the fourth order approximation of the
cosine function cos(x) = 1 1
24 x4. After some trans-
formations we obtain
21W(θ, µdN)W(θ, N )
2 1 1
2 θ(µd+ 1)N
24 θ(µd+ 1)N
24!! 4
1152N4θ4(1 + µ2
and with (5), the approximation results in
1152N4θ4(1 + µ2
3(1 + µdN)+σ2
Finally, it is possible to solve (10) with this expression. It is
obvious that MSEiis approximately constant in Nand so its
derivative vanishes. Moreover, we can write a lower bound of
the MSE as
MSEθ[N] + MSEi[N] + MSEn[N]'
MSE[N]'MSEθ,n[N] = MSEθ[N] + MSEn[N].(14)
Thus, (10) leads to
d+ 12N3
= 0
MSE B,θ=2π
random time-varying
upper bound
sin time-varying
lower bound
µd=2 upper bound
d=1011 sin time-varying
pT=105lower bound
Fig. 2. Period estimation MSE of time-varying processes.
with the lower bound (14). An explicit expression for N0can
be found by solving (15) which yields
s864 σ2
d+ 1)2µd
N0has to be remapped by rounding Nθ=bN0e. This result
is confirmed by simulations depicted in Fig. 2 and described
in section 3.2. Nθis inversely proportional to θ4
7, therefore,
Nθis adapted to the time variance for achieving better esti-
mation quality. Consequently, the estimation algorithm uses
the optimized Nθto aquire the results presented in Fig. 2. The
algorithm itself is described in section 4.1.
3.2. White band-limited time varying period
According to (3), a random time variation is introduced with
pt[n]N(0, σp)and a band limitation of B= [B, B [. To
preserve power equivalence we set σp=pT
2. Within B,pt[n]
is white and therefore it implies an infinite number of fre-
quency components θwith individual MSEθ[N]. To empha-
size this we write MSE[N, θ]using (13). With Nθ 1, its
frequency dependence is proportional to θ4and for Nθ > 1
the MSE is limited with p2
2. The interpolation error is pro-
portional to θ2and therefore
MSE[N, θ]MSE[N, B]if θBB < .(17)
Hence, it is evident that MSE[N, B]is the worst case for the
estimation error. Consequently, (5) is also an upper bound
for the white band-limited time varying sparse process. The
MSE[N, θ]was derived based on a linear approximation,
therefore we proceed with the superposition assumption for
Nθ 1and use for the approximated mean MSE within B
θ[N] = 1
MSE[N, θ]
1152N4θ4(1 + µ2
1152 N4B
(1 + µ2
According to (15), (16) and the signal power equivalence, we
find for the optimal NBof the band limited time varying pe-
s864 σ2
d+ 1)2µd
3.3. Simulation results
Throughout this paper all simulations were done with 100
Monte Carlo simulations considering a time span of two pe-
riods of the time varying process realizations. In Fig. 2 the
MSE[N, θ]is depicted with θas parameter. Let us consider
the first simulation with θ=2π
250 . The simulation results are
lying within the derived upper and lower bound given by (14).
With these parameter settings, all three terms of the bound are
clearly identifiable. The first part for N= 3 . . . Nθis domi-
nated by the decay of the error proportional to O(N3)which
is represented by MSEn[N, θ]. The MSE decay reaches its
minimum at Nθ= 9 in accordance to the theoretically derived
minima. The value was observed by simulation and calcu-
lated with equation (16). The MSE minimum is increased by
the interpolation error MSEi[N, θ]which is noticeable by flat-
tening the peak of the minima. Beyond Nθthe MSEθ[N, θ]
dominates with its increase proportional to O(N4)and fi-
nally it saturates at p2
2. The second simulation in Fig. 2
with θ=2π
2000 shows the same behavior and a minimum at
2000 = 29.
Furthermore, we consider periodic point processes show-
ing a time varying period modeled by a white band limited
stochastic process. The MSE of period estimation is consid-
ered in Fig. 4. We depict the upper bound of the MSE[N, 2π
250 ]
as dashed curve. It represents the worst case as if the time
variation is assumed a single sinusoidal with power equiva-
lence to stochastic time variance. As expected, the simulated
MSE[N, θ]lays way below the worst case of the estimation
error. Moreover, there exists also a minimum for the estima-
tion error at NB, which is confirmed by the theoretical result
of (19). With B=2π
250 the value of NB= 17. These results
are supported by further simulations e.g. with frequency band
2000 as depicted using red curves in Fig. 2.
The time variance of the observed signal is based on sys-
tem parameters of the underlying process, pT,σd,µdand
B, hence the algorithm can be tuned using only the estima-
tor memory N. The parameter µdcan be easily measured as
ratio between the total number of samples and received events
of the sparse process. And, the parameter σdis derived from
the maximum prediction gain [16]. If the prediction horizon
is near zero, the prediction error is converging to the measure-
ment noise variance. If we consider the period estimation as
++ +
t>y[n]+Δ t
if event
else if
Fig. 3. Adaptive period estimation
MSE estimate bound σ2
d= 105
estimate bound σ2
d= 1010
estimate bound σ2
d= 1015
estimate bound σ2
d= 1020
estimate bound σ2
d= 1025
250 pT=105µd=2
Fig. 4. MSE for white time variance and different σ2
period prediction, the results of [16] can be applied straight
forward. To estimate σd, we are using an estimate with one
memory element and compare it with the following period
measurement. This is equivalent to the shortest possible pe-
riod prediction horizon and therefore the mean estimation er-
ror with one memory tap is the best guess we can get for the
measurement noise bσd/2used in (5). The variance of the time
varying process σPis calculated from the recorded samples
yd[n]. Finally, the bandwidth Bhas to be assumed from the
underlying physics or a band limiting input device.
4.1. The estimator
According to [17], the estimator is designed as depicted in
Fig. 3. It is estimating the period b
P[n]of repeating events as
shown in Fig. 1. Time stamps are stored in y[n]whenever a
trigger event occurs. The last bµdNctimestamps are stored to
calculate the difference yd[n]between the current timestamp
and the timestamp bµdNcsamples earlier. The difference is
squared y2
d[n]and averaged over Nsamples. In the follow-
ing, the scaled square root of the moving average is the es-
timate of the process period. Missing events are detected if
the time counter tgets larger than y[n] + ∆tmax. In case of
one or more missing events, new events are created by adding
an averaged sample difference yd[n]
bµdNcto the previous sam-
ple via a feedback loop. The ratio between the total sample
number (inserted and detected) and the number of detected
samples represents the parameter µdused for the calculation
of NB.σ2
dis estimated by using the first partial sum of the
moving average filter. Finally, σ2
Pis the variance of the dif-
ferences yd[n]. The optimized NBis calculated with (19) for
each estimated period. According to NBthe parameter Nis
updated. The computational complexity of the non-adaptive
algorithm was derived in [17] with O(N). The adaption pro-
cess is adding two-times O(N)to the total complexity based
on the two variance estimation tasks. For subsequent esti-
mates the algorithm can be altered to use the previous esti-
mates. Therefore, we reach an extremely efficient algorithm
having a constant complexity 3O(1) after the first initial cal-
culation of 3O(N).
4.2. The estimation results
The period estimation error of sparse periodic processes with
time varying period is depicted in Fig. 4. The figures show the
MSE of the presented estimator considering 5 different mea-
surement noise levels. For high measurement noise power the
MSE is decaying N3to the MSE level caused by the time
varying disturbance. This estimation error cannot be reduced
by increasing N. For lower measurement noise a minimum of
the estimation error for the memory size of NBexists, which
coincides with the theory. Finally, if the measurement noise is
below the interpolation error no minimum exists. In all three
cases the adaptive estimator is able to achieve the optimum
MSE based on its adaptation of N.
We presented an adaptive period estimator for sparse periodic
time varying point processes. The main result of this contribu-
tion is an algorithm designed to follow the changes of a time
varying process by adapting its memory length Nto achieve
a minimum MSE in period estimation. The algorithm is ex-
tremely low complex with 3O(N)and can be implemented
recursively which reduces the complexity to 3O(1). The es-
timator is applicable in e.g. low power sensing devices which
are exposed to temperature variations. These environmen-
tal changes introduce time varying behavior of the crystals
and the presented algorithm is able to follow the frequency
changes with a minimum MSE.
[1] H.-P. Bernhard, A. Berger, and A. Springer, “Tim-
ing synchronization of low power wireless sensor nodes
with largely differing clock frequencies and variable
synchronization intervals, in 2015 IEEE 20th Con-
ference on Emerging Technologies Factory Automation
(ETFA), Luxemburg, Sept 2015, pp. 1–7.
[2] E. Conte, A. Filippi, and S. Tomasin, “ML period esti-
mation with application to vital sign monitoring,” IEEE
Sig. Process. Letters, vol. 17, no. 11, pp. 905–908, 2010.
[3] L. Tavares Bruscato, T. Heimfarth, and E. Pignaton de
Freitas, “Enhancing time synchronization support in
wireless sensor networks, Sensors, vol. 17, no. 12,
[4] M. Xu, W. Xu, T. Han, and Z. Lin, “Energy-efficient
time synchronization in wireless sensor networks via
temperature-aware compensation, ACM Trans. Sen.
Netw., vol. 12, no. 2, pp. 12:1–12:29, Apr. 2016.
[5] H.-P. Bernhard, B. Etzlinger, and A. Springer, “Period
estimation with linear complexity of sparse time vary-
ing point processes,” in 2017 51th Asilomar Conference
on Signals, Systems and Computers, Asilomar, Pacific
Grove, Nov 2017, pp. 1–5.
[6] I. Vaughan L. Clarkson, Approximate maximum-
likelihood period estimation from sparse, noisy timing
data,” IEEE Transactions on Signal Processing, vol. 56,
no. 5, pp. 1779–1787, May 2008.
[7] H. Ye, Z. Liu, and W. Jiang, “Efficient maximum-
likelihood period estimation from incomplete timing
data,” in International Conference on Automatic Control
and Artificial Intelligence (ACAI 2012), March 2012,
pp. 959–962.
[8] P. Stoica, J. Li, and H. He, “Spectral analysis of nonuni-
formly sampled data: A new approach versus the pe-
riodogram,” IEEE Transactions on Signal Processing,
vol. 57, no. 3, pp. 843–858, March 2009.
[9] J. K. Nielsen, M. G. Christensen, and S. H. Jensen, “De-
fault bayesian estimation of the fundamental frequency,”
IEEE Transactions on Audio, Speech, and Language
Processing, vol. 21, no. 3, pp. 598–610, March 2013.
[10] E. J. Hannan, “The estimation of frequency, Journal of
Applied Probability, vol. 10, no. 3, pp. 510–519, 1973.
[11] B. G. Quinn, “Recent advances in rapid frequency esti-
mation,” Digital Signal Processing, vol. 19, no. 6, pp.
942 – 948, 2009, DASP’06 - Defense Applications of
Signal Processing.
[12] B. G. Quinn, R. G. McKilliam, and I. V. L. Clark-
son, “Maximizing the periodogram,” in IEEE Global
Telecommunications Conference, Nov 2008, pp. 1–5.
[13] R. G. McKilliam, I. V. L. Clarkson, and B. G. Quinn,
“Fast sparse period estimation, IEEE Signal Processing
Letters, vol. 22, no. 1, pp. 62–66, Jan 2015.
[14] A. Napolitano, Asymptotic normality of cyclic au-
tocorrelation estimate with estimated cycle frequency,”
in 23rd European Signal Processing Conference (EU-
SIPCO 2015), Aug 2015, pp. 1481–1485.
[15] A. Napolitano, “On cyclic spectrum estimation with es-
timated cycle frequency,” in 24rd European Signal Pro-
cessing Conference (EUSIPCO 2016), Budapest, Hun-
gary, Aug 2016, pp. 160–164.
[16] H.-P. Bernhard, “A tight upper bound on the gain of
linear and nonlinear predictors for stationary stochastic
processes,” IEEE Transactions on Signal Processing,
vol. 46, no. 11, pp. 2909–2917, Nov 1998.
[17] H.-P. Bernhard and A. Springer, “Linear complex itera-
tive frequency estimation of sparse and non-sparse pulse
and point processes,” in 2017 25th European Signal
Processing Conference (EUSIPCO 2017), Kos, Greece,
Aug. 2017, pp. 1150–1154.
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