Conference PaperPDF Available

Error Characterization of Duty Cycle Estimation for Sampled Non-Band-Limited Pulse Signals With Finite Observation Period

Authors:

Abstract and Figures

In many applications the pulse duration of a periodic pulse signal is the parameter of interest. Thereby, the non-band-limited pulse signal is sampled during a finite observation period yielding to aliasing and windowing effects, respectively. In this work, the pulse duration estimation based on the mean value of the samples is considered, and a closed form expression of the mean squared estimation error (averaged over all possible time shifts) is derived. The resulting mean squared error expression depends on the observation period, the pulse period and the pulse duration. Analyzing the effect of these parameters shows that the mean squared error can be reduced (i) if the observation period is a multiple of the pulse period, (ii) if the pulse period is not a multiple of the sampling period, and (iii) if the total number of samples is a prime number. All results were validated with simulation results.
Content may be subject to copyright.
Error Characterization of Duty Cycle Estimation for
Sampled Non-Band-Limited Pulse Signals With
Finite Observation Period
Hans-Peter Bernhard, Bernhard Etzlinger 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 many applications the pulse duration of a peri-
odic pulse signal is the parameter of interest. Thereby, the non-
band-limited pulse signal is sampled during a finite observation
period yielding to aliasing and windowing effects, respectively.
In this work, the pulse duration estimation based on the mean
value of the samples is considered, and an exact expression of the
mean squared estimation error (averaged over all possible time
shifts) is derived. The resulting mean squared error expression
depends on the observation period, the pulse period and the pulse
duration. Analyzing the effect of these parameters shows that the
mean squared error can be reduced (i) if the observation period
is a multiple of the pulse period, (ii) if the pulse period is not
a multiple of the sampling period, and (iii) if the total number
of samples is a prime number. All results were validated with
simulation results.
Index Terms—sampling process, band width, signal reconstruc-
tion, sampling error, wireless sensor networks (WSN), synchro-
nization, localization, ultrasonic.
I. INTRODUCTION
Following the established sampling theory, band-limited
signals can be perfectly reconstructed from a set of samples
that are collected with a sampling frequency larger than the
double occupied bandwidth [1], [2]. However, this is only true
if the signal is sampled during its infinitely long duration.
In practical applications neither an infinite observation time
can be achieved, nor the underlying signals are band-limited
(e.g., discrete valued and continuous-time signals are not band-
limited).
Consider time-duration measurements in periodic signals.
For example, in ultrasonic reflection measurements for ranging
[3], [4], pulses are emitted periodically, and a device counts
the number of clock cycles between the emission of the pulse
and the reception of the reflection. This setting corresponds to
the sampling of a rectangular pulse signal, that is high between
emission and reception, and low until the next emission. The
pulse duration, i.e., the time that the pulse is on high, is used to
determine the range. Another example is round-trip time (RTT)
based clock synchronization for which clocks are modeled
with discrete events. The corresponding RTT measurements
are samples of a pulse function. In that scenario, the pulse
duration is related to the propagation delay and the clock offset
between two communication nodes [5], [6].
The two examples mentioned use a pulse shaped signal to
determine key system parameters. Thereby, both conditions of
the sampling theorem are violated, the bandwidth of the signal
is infinite and the observation period is limited. Although the
signals are considered as noise free, sampling and aliasing
distorts the signal reconstruction. In this work the question
is raised how accurately signal parameters can be estimated
from non band-limited signals with a finite number of samples.
Previous contributions have already derived upper bounds
on the squared error [7]. Here, for the function space of
sampled pulse signals, an exact formulation of the averaged
squared error on the estimation of the pulse duration is derived.
Moreover, the error analysis gives rise to the following design
suggestions in technical systems:
Predicting the estimation accuracy for a given number of
samples in a given observation period;
Selecting the number of pulse signal periods (i.e., the
total observation period) to achieve a desired estimation
accuracy.
Selecting the number of samples to achieve the minimum
estimation error for a given sampling period and obser-
vation period.
II. PRELIMINARIES
Sampling is the obvious starting point in all discrete signal
processing applications that have a relation to real world
processes. In the following, signals x(t)are considered that
are integrable in the sense of Lebesgue, i.e.,
L1(R) = x(t)Z
−∞
|x(t)|dt < .(1)
Sampling x(t)with a period Tyields the sampled signal
xs(n)nx(nT )nZT=1
2BB < o.(2)
To reconstruct x(t), the samples xs(n)are interpolated, i.e.,
xi(t) =
X
n=−∞
xs(n)sin(2πB(tnT ))
π(tnT ),(3)
is the interpolated signal. As commonly known [1], [2]
xi(t)x(t),(4)
if and only if x(t)is band-limited by B= [B, B [.
A. Signals of interest
In this work, the rectangular pulse function
x(t) =
X
l=−∞
rect tl P φ
D(5)
with t, P, φ R,0< D < P and lZare considered. The
signal period Pis considered to be a fixed parameter, and the
time shift φand the pulse duration Dare considered to be
uniformly distributed variables on the mentioned support. The
considered signal is a low-pass signal, and since it is periodic
in P, it is wide-sense cyclo stationary and ergodic.
Note that x(t)in (5) is discrete valued and hence not band-
limited. The condition (4) can not be achieved due to aliasing
induced by the sampling process.
B. Bounds on the Aliasing Error of Aribtrary Functions
The aliasing error is the difference between the original sig-
nal x(t)and the interpolated signal xi(t), i.e., |x(t)xi(t)|. In
[7] a upper bound is derived for sampling of one dimensional
low-pass and band-pass signals. The analysis was extended to
multidimensional signals in [8] yielding similar results. For
one-dimensional signals, the main finding is that the aliasing
error is bounded by the out-of-band signal contribution (cf.
[9], [10]) with
|x(t)xi(t)| ≤ 2Z
R\B
|X(f)|df , (6)
where X(f)is the spectral representation of x(t), and where
the band Bdepends on the sampling period in (2).
While the formulation in (6) provides an upper bound on
the aliasing error of arbitrary signals, this work focuses on
pulse-shaped signals as in (5). Hence, although Pand φare
unknown, the general shape (e.g., its discrete amplitude) is
known and can be interpreted as prior knowledge.
The aim of this work is to derive an exact formulation
of the reconstruction error similar to (6). Thereby, first the
frequency domain representation of the periodic infinite time-
continuous time signal (5) is derived, and then the frequency
representation of the sampled version of this signal. Finally,
a rectangular windowing function is included to account for a
finite observation period.
C. Time and Frequency Domain Pulse Signal Representation
The signal of (5) is an even function with a time shift φ
and can be represented by the Fourier coefficients for k1
and for k= 0, respectively,
ak=2
sin πkD
P,and a0=D
P.(7)
Hence, (5) can be rewritten as
x(t) = a0+
X
k=1
akcos 2πk
P(tφ)=
=D
P+
X
k=1
2
sin πkD
Pcos 2πk
P(tφ)(8)
and the corresponding frequency domain representation is
given by
X(f) = D
Pδ(2πf )+
X
k=−∞
2
sinπkD
Pδ2πf k2π
Pej2πkφ
P.(9)
D. Frequency Domain Representation of Sampled Pulse Sig-
nals
In this section the sampling of the non-time-limited signal
x(t)in (8) is considered. In contrast to [11], the considered
periodic pulse functions are unbounded in frequency domain.
Nevertheless, it is required that the pulse period Phas to be at
least twice the sampling period P2T. Due to sampling, all
out-of-band components of X(f)are folded into Baccording
to the Poisson sum formula [10], [12], [13]
Xs(f) = 1
T
X
m=−∞
Xfm1
T,(10)
where Xs(f)is the spectral representation of xs(n). Note that
Xs(f)is periodic in fwith a periodicity of 1/T . Hence, in
the following Xs(f)will only be considered for f∈ B with
B= [B, B [and B= 1/2T.
X(f)from (9) can be plugged into (10) yielding
Xs(f) = 1
T
X
m=−∞
D
Pδ2πf m2π
T+
1
T
X
m=−∞
X
k=−∞
2
sinπkD
Pδ2πf m2π
Tk2π
Pej2πkφ
P,
(11)
for f∈ B. The first delta-dirac function is non-equal to zero in
Bonly if f= 0 and m= 0. The second delta-dirac function
is non-equal to zero if
f=m1
T+k1
P,for f B .(12)
For each k, there exists exactly one mthat fulfills (12).
Defining the round modulo function [14] with
:R×(R\ {0})R,(r, p)7→ rp:= rr
p+ 0.5p ,
(13)
where b·c denotes the rounding to the next lower integer,
allows to rewrite (12) as a function of konly, with
fk=m1
T+k1
P1
T=k1
P1
T.(14)
Finally, (11) can be rewritten to
Xs(f) = D
P T δ(2πf)
+
X
k=−∞\0
2
kπT sin πkD
Pδ(2π(ffk)) ej2πfkφ.(15)
The sampled time-domain signal for arbitrary φis finally
obtained analogue to (8) with
xs(n) = D
P T +
X
k=1
2
kπT sin πkD
Pcos (2πfk(nT φ)) .
(16)
The sampled signal representation of xs(n)in (16) includes
all aliasing components due to sampling. It will be the starting
point for considering a limited observation period in the
following section.
E. Finite Observation Period
To include a finite observation period N T in the frequency
domain representation of Xs(f)in (15), we consider a
non-causal rectangular window yielding
Xs(f) =
bN1
2c
X
n=bN
2c
xs(n)ej2πf nT ,(17)
with xs(n)from (16) for arbitrary φ. Because of its linearity,
we can transform each akweighted term of the Fourier series
of (16) separately
Xs(f) =
X
k=0
akXfk
s(f).(18)
The finite discrete Fourier transform of each cosine function
of (16) can be written by
Xfk
s(f) =
bN1
2c
X
n=bN
2c
cos(2πfk(nT φ)) ej2πf nT
=ej(2πfkφ)sin(πN T (ffk))
2 sin(πT (ffk)) ,(19)
where the simplification from the first to the second line ap-
plies the generalized formula of the geometric series. Finally,
the frequency domain representation for f∈ B of the sampled
pulse signal with finite observation period NT is
Xs(f) = D
P T
sin(πN T (f))
sin(πT (f))
+
X
k=−∞\0
ej(2πfkφ)sin πkD
Psin(πN T (ffk))
kπT sin(π T (ffk)) .
(20)
III. ESTIMATION ERRO R OF T HE PU LS E DUR ATION
The pulse duration Ddetermines the mean value of the
periodic time-continous signal x(t)in (5), and appears in
X(f= 0) = D/P in (9). A natural choice of estimating
Dis using the mean value1of the sampled signal. Hence,
the estimation error can be characterized by the influence of
aliasing and windowing on Xs(0). l’Hospital’s rule yields
Xs(0) = D
P T N
+
X
k=−∞\0
ej(2πfkφ)sin πkD
Psin(πN T fk)
T kπ sin(πT fk).(21)
1Based on the binary nature of the observed process this estimator is
equivalent to an implementation of a nonlinear edge detector .
The estimator of the pulse duration can be written by
b
D(D, φ;N , P ) = P T
NXs(0).(22)
A. Mean Square Estimation Error
Considering (22) and (21), the estimation error eDas a
function of the random variables D,φand the fixed parameters
N,P, is given by
eD(D, φ;N , P ) = Db
D(D, φ;N , P )(23)
=P
N
X
k=−∞\0
ej(2πfkφ)sin πkD
Psin(πN T fk)
sin(πT fk)
=
X
k=−∞\0
ckej(2πfkφ),(24)
with
ck=P
Nsin πkD
Psin(πN T fk)
sin(πT fk).(25)
Note that (24) has the structure of a discrete Fourier transform
with the coefficients ck. Note that ckonly depends on T fk.
Using the round modulo definition (13),
T fk=kT
PkT
P+ 0.5=kT
Pv(k)
with v(k) = kT
P+ 0.5Z. The coefficients ckin (25) can
be simplified to
ck=P
Nsin πkD
P(1)(N1)v(k)sin(πN k T
P)
sin(πk T
P).(26)
To characterize the error independent of φ, the mean square
error with respect to φis evaluated. Due to the ergodic and
wide-sense cyclo-stationary signal in (5), the expectation can
be evaluated with Φ = mP and Φ→ ∞ to
e2
D(N) = E{|eD(D, φ;N , P )|2}
= lim
Φ→∞
1
Φ
Φ
2
Z
Φ
2
|eD(D, φ;N , P )|2dφ
= lim
Φ→∞
1
Φ
Φ/2
Z
Φ/2
X
k=−∞\0
ckej(2πfkφ)
2
=
X
k=−∞\0
|ck|2(27)
=
X
k=−∞\0
P2
N2sin2πkD
Psin2(πN k T
P)
()2sin2(π k T
P),(28)
where the simplification from the third to the fourth line is
possible since the integration limits tend to infinity. The mean
squared error on the estimation of the pulse duration D, i.e.,
e2
D(N)is exactly characterized by (28). It is averaged over all
possible time shifts φ, and it depends on the pulse period P,
the number of samples N, and the sampling period T. In the
following, special cases on the relation of the parameters P,
Nand Tare discussed.
0 1 5 10 15 20
101
100
101
NT
P
e2
D(N).e2
Fig. 1. Mean square error of duty cycle estimation depending on the
observation length NT . The value T= 25/501 = 0.0499 ... was selected
to guarantee that mP 6=NT and nT 6=P. The error is decaying slowly
to 0.1 of the quantization error at an observation interval of about 20 signal
periods. The () indicate the simulation results.
IV. DISCUSSION ON THE PARAMETER RELATION
By observing (28), one finds in the squared sine in the
enumerator the relation N T /P , and in the squared sine in
the denominator T /P . If NT =mP with mN+, i.e., the
observation period is a multiple of the pulse period, some
coefficients in (28) will be zero. If the pulse period is a
multiple of the sampling period, i.e., P=nT with nN+,
the enumerator and the denominator of some coefficients will
tend to zero. In the following, four different cases on the
relation of NT to mP and of nT to Pare discussed.
The discussion on the parameter selection plays a key role in
applications, e.g. the number of repeated pulses for ultrasonic
distance measures and the sample frequency used to measure
the time to incoming reflections. Both parameters are easy to
determine and we can achieve higher estimation performance
by optimized parameter relations.
In the following analysis, the pulse period is normalized to
one second, i.e. P= 1 s, and the mean squared estimation
error is normalized to the quantization error induced by T,
i.e. e2
D(N)/e2with e2=T2/12 [15]. The normalized error
is compared with simulation results using 4000 realizations.
A. Parameter: mP 6=NT and nT 6=P
For the given parameter setting, Fig. 1 depicts the normal-
ized error following the derived analytical expression (28),
indicated by the solid line, and simulation results, indicated
by the () markers. Both results show the same behavior.
Local minima can be observed whenever the NT is close to
a multiple of the pulse period P. In contrast, local maxima
can be observed in between. An evident conclusion is to use
NT =mP as observation length to have the lowest estimation
error. This case will be considered in the following section.
B. Parameter: mP =NT and nT 6=P
Considering NT =mP , the coefficients ckin (28) can be
rewritten to
|ck|2=T2
m2sin2πkmD
NT sin2(πkm)
k2π2sin2(πk m
N)(29)
where the numerator is equal to zero. It is obvious, that also
the denominator is zero for some kZ, i.e., if
k=vN
m,(30)
0 1 5 10 15 20
102
100
NT
P
e2
D(N).e2
Fig. 2. Plot of the mean square error of duty cycle estimation over NT with
T= 5/107 = 0.04672 .... The values for mP =NT and nT 6=Pare
indicated by (o). The error is decaying to 0.01 of the quantization error at an
observation interval of about 5 signal periods due to the prime number N.
The () indicate the simulation results.
with vN+. For those kZ, the coefficients ckwill not be
equal to zero. Note that if Nis a prime number, vmust be
a multiple of mand only a minimum number of coefficients
ckexists for which numerator and denominator tend to zero.
Hence, for Nis a prime number, the minimum mean squared
estimation error e2
D(N)is expected.
Applying de l’Hoptials rule to the non-zero ckcoefficients,
i.e. for kT mP , yields
lim
kT mP |ck|2=P2
N2sin2πkD
PN2
k2π2(31)
which results for k=i mN in
e2
D(N)N=mP
T
=
X
i=1
2T2
(i mπ)2sin2πi mD
T.(32)
In Fig. 2, the analytical estimation error (solid line) and the
simulated estimation error () for mP 6=N T , and the special
cases for mP =NT (o) are depicted. Interestingly, the
minimum value of the mean square error, i.e., for mP =NT ,
is constant also for multiples of fN,fN+. It can be
concluded that (30) holds also for fmultiples of Nand leads
to the same set of ck. Therefore, increasing the observation
period does not improve the estimation accuracy, because there
is no dependence on Nin (32).
C. Parameter: mP =NT and nT =P
In some practical scenarios (e.g., the ultrasonic range mea-
surement in [3]), the pulse period is equal to a multiple of the
sampling period, and the observation period is a multiple of
the pulse period. Reformulating mP =NT to N=mP /T ,
i.e. Nis a multiple of the pulse period due to T=P/n, and
using (30) yields k=vP/T N+. As (31) holds also for
nT Pit can be simplified to cknot depending on Nand
used in (27) to
e2
D(N) =
X
v=1
2T2
π2v2sin2πvD
T.(33)
In Fig. 3, the (o) marker indicate the analytical results for
mP =NT and nT =P. The solid line and the () marks
in between indicate the analytical results and the simulation
results, respectively, for mP 6=NT and nT =P. It can
be observed, that for nT =P, setting mP =NT yields
0 1 5 10 15 20
100.5
101
NT
P
e2
D(N).e2
Fig. 3. Plot of the mean square error of duty cycle estimation over NT with
T= 0.01. The values for mP =NT (o), mP 6=N T (-) and nT =Pare
plotted. The error is not decaying beyond 2 times the standard quantization
error. The () indicate the simulation results.
the minimum estimation error. However, this error is constant
regardless how many pulse periods are observed, and it is 200
times larger than the error obtained by setting nT 6=P.
D. Parameter: mP 6=NT and nT =P
In Fig. 3 it could be observed that this parameter setting
yields the highest mean squared error of all compared settings.
It remains to find an analytical expression of the error. For
the special setting of mP 6=NT and nT =P, the sum
in (28) consists of two types of factors: those that can be
directly computed, i.e., k6=vP /T ; and those that require the
application of de l’Hospital’s rule. The latter is for k=vP/T .
Finally, the analytical formulation of the mean squared error
is
e2
D(N) =
X
v=1
2T2
π2v2sin2πvD
T+
X
k=−∞\0k6=vP
T
P2
N2sin2πkD
Psin2(πN k T
P)
()2sin2(π k T
P).(34)
The comparison of the analytical and the numerical results in
Fig. 3 indicate the correctness of (34).
V. CONCLUSION
For sampled periodic pulse signals with finite observation
period, an exact formulation of the mean squared error for
estimating the pulse duration was derived. The resulting error
formulation depends on the observation time, the pulse period
and the pulse duration. As certain relations of these parameters
prevent a direct evaluation of the mean squared error, exact
expressions for these specific settings were presented using de
l’Hospital’s rule. The error formulations were validated with
simulation results. A discussion on the parameter relations
revealed that a minimum mean squared error is achieved if
a full number of pulse periods is covered by the observation
time, and if the number of used samples is equal to a
prime number. For such system settings, an increase of the
observation time can not improve the estimation accuracy.
Moreover, it could be seen that technical systems, where the
pulse period is a multiple sampling period have significant
inferior performance. Both results are counterintuitive and may
have an important impact to design systems.
ACKNOWLEDGMENT
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.
REFERENCES
[1] E. T. Whittaker, “On the functions which are represented by the
expansions of the interpolation-theory,Proceedings of the Royal Society
of Edinburgh, vol. 35, pp. 181–194, 1 1915.
[2] C. Shannon, “Communication in the presence of noise (reprint of classic
paper),” Proceedings of the IEEE, vol. 86, no. 2, pp. 447–457, Feb 1998.
[3] M. Scherhaufl, R. Pfeil, M. Pichler, and A. Berger, “A novel ultrasonic
indoor localization system with simultaneous estimation of position and
velocity,” in Wireless Sensors and Sensor Networks (WiSNet), 2012 IEEE
Topical Conference on, Jan 2012, pp. 21–24.
[4] L. Smith, B. Bomar, and B. Whitehead, “Measuring the level of liquid in
a partially-filled pipe via the ultrasonic pulse-echo method using acoustic
modeling,” in Systems Engineering (ICSEng), 2011 21st International
Conference on, Aug 2011, pp. 292–296.
[5] B. Etzlinger, N. Palaoro, and A. Springer, “Synchronization and delay
estimation with sub-tick resolution,” in Proc. Asilomar Conf. Sig., Syst.,
Comput., Pacific Grove, CA, Nov. 2015.
[6] H.-P. Bernhard, A. Berger, and A. Springer, “Analysis of delta-sigma-
synchronization in wireless sensor nodes,” in Industrial Informatics
(INDIN), 2015 IEEE 13th International Conference on, July 2015, pp.
914–918.
[7] L. J. Brown, Jr., “On the error in reconstructing a non-bandlimited
function by means of the bandpass sampling theorem,” Journal of
Mathematical Analysis and Applications, vol. 18, pp. pp. 75–84, 1967.
[8] J. R. Higgins, Numerical Functional Analysis and Optimization, vol. 12,
no. 3-4, pp. 327–337, 1991.
[9] P. L. Butzer and R. L. Stens, “Sampling theory for not necessarily
band-limited functions: A historical overview,SIAM Review, vol. 34,
no. 1, pp. pp. 40–53, 1992. [Online]. Available: http://www.jstor.org/
stable/2132784
[10] J. Brown, J.L., “Estimation of energy aliasing error for nonbandlimited
signals,” Multidimensional Systems and Signal Processing, vol. 15, no. 1,
pp. 51–56, 2004.
[11] E. Matusiak and Y. Eldar, “Sub-nyquist sampling of short pulses,Signal
Processing, IEEE Transactions on, vol. 60, no. 3, pp. 1134–1148, March
2012.
[12] H. G. Feichtinger and K. Gr¨
ochenig, “Error analysis in regular and
irregular sampling theory,Appl. Anal, vol. 50, pp. 167–189, 1992.
[13] A. Y. Olenko and T. K. Pog`
any, “Time shifted aliasing error upper
bounds for truncated sampling cardinal series,” Journal of Mathematical
Analysis and Applications, vol. 324, no. 1, pp. 262 – 280, 2006.
[14] “IEEE standard for floating-point arithmetic - redline,” IEEE Std 754-
2008 (Revision of IEEE Std 754-1985) - Redline, pp. 1–82, Aug 2008.
[15] A. Oppenheim and R. Schafer, Discrete-Time Signal Processing. Pear-
son Education, 2011.
... The estimation is corrupted by jitter (3) and aliasing [14]. Therefore, the MSE of the RTT estimator is derived as function of D, P, T and N . ...
... Hence, it is given as the expectation of the squared error Because n[l] are independent random variables similar to those of (2) we can reuse the previous results for the error contribution of the jitter. In [14] the error analysis of the duration estimation is performed and we can apply these results to the PPE as ...
... If k ∈ Z \ 0 and N is a finite number, still infinite contributions at multiples of N exist, where a fraction of terms in (21) result in a division zero by zero. This was analyzed further in [14]. There, we have shown that e 2 D can be rewritten as ...
Article
Timestamp free clock synchronization in a masterslave network, i.e., synchronization where no timestamps are exchanged between the nodes, is considered. For highly accurate synchronization, a novel discrete-valued clock model is introduced. It is based on the observation that clocks are discrete counters in digital wireless radios. Considering this model, it is shown that the round-trip time (RTT) measurements follow specific pulse or step shaped functions. The estimated parameters of these RTT functions are used to determine the clock parameters (clock skew and phase) and the propagation delay. Numerical analysis illustrate that when RTT measurements are collected using discrete-valued clocks, the proposed estimation schemes outperform estimators derived from the continuous clock model, which is used in state-of-the-art methods.Moreover, the presented scheme performs similar to recently presented discrete-valued clock approaches with more stringent hardware assumption. The correctness of the proposed models is validated through hardware experiments.
Research
Full-text available
(work accepted at Asilomar 2015 conference) We consider the problem of clock synchronization between a pair of nodes using round-trip communication where only the initiating node collects time stamps, and where no time stamps are exchanged. These time stamps are the current timer values at the event of transmission or reception of a signal. Traditionally, the timer is modeled by a continuous function, i.e. as affine clock in case of asynchronism in clock skew and clock phase. In this work we deviate from the traditional model and propose a discrete-value clock model, which is motivated by observations from real hardware. Employing the clock model allows us to find a new signaling model for round-trip time measurements, which enables us to simultaneously estimate clock skew, clock phase and propagation delay with a resolution below one clock tick period. Hence, we set a first step towards enabling high timing resolution with limited clock hardware.
Article
Full-text available
Shannon’s sampling theorem is one of the most powerful results in signal analysis. The aim of this overview is to show that one of its roots is a basic paper of de la Vallée Poussin of 1908. The historical development of sampling theory from 1908 to the present, especially the matter dealing with not necessarily band-limited functions (which includes the duration-limited case actually studied in 1908), is sketched. Emphasis is put on the study of error estimates, as well as on the delicate pointwise behavior of sampling sums at discontinuity points of the signal to be reconstructed.
Article
Full-text available
Time shifted aliasing error upper bound extremals for the sampling reconstruction procedure are fully characterized. Sharp upper bounds are found on the aliasing error of truncated cardinal series and the corresponding extremals are described for entire functions from certain specific Lp, p>1, classes. Analogous results are obtained in multidimensional regular sampling. Truncation error analysis is provided in all cases considered. Moreover, sharpness of bounding inequalities, convergence rates and various sufficient conditions are discussed.
Conference Paper
We analyse the use of a ΔΣ-modulator in the nodes of a wireless sensor network, which is a new method to achieve long term synchronization. We consider star topology WSNs (Wireless Sensor Networks) with a central base station and address timing synchronization using low frequency realtime clocks. The WSN uses a beacon driven TDMA-protocol for bidirectional node/base communication. Between the beacons, which are sent by the base station, lie the superframe time intervals to handle data transmission from node to base. The ΔΣ-modulator is used to generate — at average — the accurate superframe duration for any rational number of clock ticks, by generating a sequence of superframes with different time durations, but each consisting of integer multiples of clock ticks. We discuss the synchronization accuracy based on the internal arithmetic of the ΔΣ-modulator and show by theory a relation between synchronization accuracy and word length of the internal arithmetic. Additionally the fractional part of a crystal clock module is responsible for variations in the synchronization quality. We present an equation that allows us to interpret measurements showing periodic variations of synchronization quality.
Conference Paper
We consider the problem of clock synchronization between a pair of nodes using round-trip communication where only the initiating node collects time stamps, and where no time stamps are exchanged. These time stamps are the current timer values at the event of transmission or reception of a signal. Traditionally, the timer is modeled by a continuous function, i.e. as affine clock in case of asynchronism in clock skew and clock phase. In this work we deviate from the traditional model and propose a discrete-valued clock model, which is motivated by observations from real hardware. Employing the clock model allows us to find a new signaling model for round-trip time measurements, which enables us to simultaneously estimate clock skew, clock phase and propagation delay with a resolution below one clock tick period. Hence, we set a first step towards enabling high timing resolution with limited clock hardware.
Patent
A method for signal processing includes accepting an analog signal, which consists of a sequence of pulses confined to a finite time interval. The analog signal is sampled at a sampling rate that is lower than a Nyquist rate of the analog signal and with samples taken at sample times that are independent of respective pulse shapes of the pulses and respective time positions of the pulses in the time interval. The sampled analog signal is processed.
Conference Paper
We present a novel ultrasonic (US) indoor localization system based on the time-of-arrival (TOA) principle realized with radio frequency (RF) synchronization. The US signal is assembled as a Gold sequence of chirp signals to guarantee a high signal-to-noise ratio (SNR) and, at the same time, provide the position estimates of multiple mobile transmitters (MTs). By exploiting the TOA in the time domain and the Doppler shift in the frequency domain, position and velocity of the MT can be determined simultaneously. The concept is proven by means of both simulated and measured data and performs excellently even in a harsh industrial environment.
Article
Bounds are given on the error incurred when a real valued function with multi-dimensional domain is approximately reconstructed from samples by the multi-dimensional Whittaker-Shannon theorem. The functions considered are band-limited to a thin spherical shell. To obtain these bounds, an aliasing error bound, and a bound for Fourier transforms due to Boas and Kac, are given multi-dimensional treatments; the inequalities involved are both shown to be sharp by exhibiting extremals.