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Received: 1 March 2023 Revised: 30 June 2023 Accepted: 17 July 2023 IET Communications
DOI: 10.1049/cmu2.12662
ORIGINAL RESEARCH
Clock synchronization based on pulse with propagation delay
eliminated in wireless sensor networks
Wei Liu Ruijie Fan Yiyuan Mai Zehan Wan
The College of Electronic Science, National
University of Defense Technology, Changsha, China
Correspondence
Ruijie Fan, The College of Electronic Science,
National University of Defense Technology,
Changsha, China.
Email: fanruijiedut@sina.com
Abstract
Clock synchronization is indispensable for numerous applications of wireless sensor net-
works (WSNs). When no common reference clock is available, the nodes must employ
distributed synchronization techniques. This paper proposes, a distributed pulse-based
clock synchronisation approach, wherein the propagation delay is eliminated through signal
ping-pongs between neighbouring nodes. Such an approach can jointly estimate the clock
skew and offset without requiring any reference clock. The whole synchronization process
is completed at the physical (PHY) layer, effectively avoiding the random delay caused by
packet queuing and retransmission. Simulation results show that the proposed approach
can achieve higher synchronization accuracy compared with other existing methods.
1 INTRODUCTION
Wireless sensor networks (WSNs) have obtained a wide range
of applications recently. It is critical for nodes to operate coor-
dinately based on a common clock in those applications [1],
such as industrial control, target positioning and data fusion.
However, each node has its own oscillator to provide the local
clock, which often results in clock skew and offset. Conse-
quently, the clock synchronization technology is essential in
WSNs. In general, there are two main categories of clock syn-
chronization technologies for the current researches, namely:
timestamp-based and pulse-based. No matter which method
is used, delay is a key factor affecting clock synchronization
accuracy, including propagation delay and random delay, where
the propagation delay is generally considered to be fixed in the
current theoretical researches.
Timestamp-based clock synchronization relies on the
exchange of timestamps encoded in packets. Some clock syn-
chronization methods based on timestamp are presented in [2]
with no considering the delay existing. Then, [3]and[4] estimate
the fixed delay through two-way message interaction, which
improve the accuracy of clock parameter estimation. However,
all the algorithms in [2–4] need a reference clock, thus they
are usually adapted to fixed network structure and result in
poor flexibility.
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However, in distributed systems such as wireless sensor net-
works (WSNs), such a fixed reference clock is not available.
Later, some timestamp-based synchronization algorithms in
[5–8] are investigated without requiring any reference clock. In
[5]and[6], each node periodically broadcasts synchronization
messages and uses the messages received from neighbours to
estimate the clock parameters to complete global clock synchro-
nization. Unfortunately, the whole process ignores the existence
of propagation delay. Afterwards, [7] considers the impact of
propagation delay but assumes that the delay is known. As it is
a difficult problem to estimate the propagation delay without
reference clock, [8] proposes an improved clock synchroniza-
tion method through eliminating the influence of propagation
delay between neighbouring nodes. In summary, all the above
timestamp-based methods need to be performed at medium
access control (MAC) layer, hence the synchronization accuracy
will be affected by not only the fixed propagation delay but also
the uncertain random delay, and the latter (the uncertain random
delay) cannot be eliminated and cannot be estimated.
In contrast, pulse-based synchronization methods are com-
pleted at the physical (PHY) layer, where the timing parameters
are estimated by processing the special sequence included in
pulses. Such methods can naturally avoid the uncertain ran-
dom delay caused by packet queuing and retransmission at MAC
layer. Therefore, in general, compared to timestamp-based
IET Commun. 2023;17:1955–1961. wileyonlinelibrary.com/iet-com 1955
1956 LIU ET AL.
methods, better synchronization accuracy can be achieved in
pulse-based methods, which has great engineering application
value, because the fixed propagation delay is estimable. Sim-
ilarly, synchronization methods based on pulse can also be
divided into two situations: with a reference clock and with-
out a reference clock. [9] estimates clock skew and offset of UE
regarding gNB clock as the reference, where the fixed propaga-
tion delay is directly obtained from time advance (TA) included
in 5G frames. As reported in [10], an universal pulse-based joint
estimation algorithm of clock skew and offset in WSNs is intro-
duced with the propagation delay estimated under a reference
clock. In addition, typical pulse-based synchronization meth-
ods without reference clock are presented in [11–13]. The clock
offset estimation methods are given in [11]and[12] with assum-
ing the ideal condition that takes neither propagation delay nor
clock skew into account. However, in practical systems, there
will inevitably be propagation delay and clock skew, and they
will have a crucial impact on the performance of clock synchro-
nization. Recently, a timing advance synchronization technique
is introduced in [13] considering the existence of propagation
delay and clock skew, where the impact of propagation delay
is eliminated by using a certain iterative method. However, the
clock skew is not solved so that the loss of synchronization
accuracy is inevitable. Therefore, it is meaningful to study the
pulse-based clock synchronization to estimate both clock skew
and offset jointly in the case of no reference clock, thus further
improving synchronization accuracy.
The main contributions of this article are summarized as
follows: firstly, compared to pulse-based method in [13], the
proposed method further solves the problem of clock skew
estimation and achieves better accuracy; secondly, compared to
timestamp-based method in [8], the proposed method employs
the physical layer pulse to avoid the random delay. In addi-
tion, compared to synchronization algorithms with a reference
clock, whether timestamp-based or pulse-based, the proposed
method is more suitable for distributed networks and has more
flexibility. In summary, this is the first time that a pulse-based
synchronization method is proposed to estimate the clock skew
and offset jointly, and at the same time resolve the influence of
propagation delay.
2SYSTEM MODEL
AWSNofLfully connected nodes indexed by i=
{1,…,L}without reference clock is considered here as illus-
trated in Figure 1. Each node has a local clock based on its
own oscillator which operates independently of other nodes
before synchronization.
The clock model of the ith node, which is an affine function
of the universal time t, is denoted as ti=
it+
i, where iand
idenote the clock frequency skew and phase offset of the ith
node, respectively. It is assumed that iis usually very close to
1, i(T0,T0)andT0is the clock period. The discrete logical
clock can be obtained by uniformly sampling the physical clock
at the time t=vT0. Therefore, the discrete clock model of the
FIGURE 1 Fully connected network structure.
ith node is [13]:
ti(v)=
ivT0+
i(1)
where vis a discrete-time index. We refer to ti(v)asthe
vth clock tick of the ith node. It is convenient to partition the
universal time axis into a sequence of non overlapping time
slots [vT0,(v+1)T0). In fact, idiffers from 1 by a very small
amount, and it is therefore reasonable to assume that each time
slot of the ith node contains a single clock tick tj(v)ofits
neighbouring node j(jand j≠i).
We define the time offset (TO) ∆tij(v)=ti(v)tj(v)atthe
vth clock tick between the ith node and the jth node. However,
the delay between two neighbouring nodes usually exist, thus
the time offset ∆tij(v) becomes:
∆tij(v)=ti(v)tj(v)+
ij (2)
where ij is the propagation delay and is supposedly fixed. Note
that the proposed synchronization process is completed at PHY
layer and there is no random delay caused by the upper layer.
Assume that at the vth clock tick, the ith node broadcasts a
time-shifted synchronization signal s(tti(v)) at time ti(v), and
s(t) is defined as:
s(t)=
2Nz1
n=0
z[n]xtnTz(3)
where Tzis the pulse spacing with 2NzTzT0;x(t)Ris
a normalized baseband pulse. In (3), z[n] is a synchronization
sequence of length 2Nzconstructed by two Zadof–Chu (ZC)
sequences of length Nzwith roots uand u, respectively, which
is recorded as [10]:
z[n]=
z+[n]=ej
Nzun2
,0≤n<Nz
z[n]=e
j
Nzu(nNz)2
,Nz≤n<2Nz
(4)
where uand Nzare coprime. Consider that the channel between
the ith node and its any neighbouring jth node is a single-path
LIU ET AL.1957
fading channel, the impulse response hij(t)is:
hij(t)=
ijt
ij(5)
where ij Cis the complex gain of the channel; ij >0isthe
propagation delay and ij T0is assumed here.
The jth node listens for the broadcasted synchronization sig-
nal during the receiving period [tj(v)T0
2,tj(v)+T0
2), centering
on its own clock tick tj(v). Suppose that the signal transmitted
from node iexists a carrier with the frequency fi,andletyij(t)
be the received signal from the ith node to the jth node, which
can be given as:
yij(t)=stti(v)ej2fithij(t)+nij(t)(6)
where the operator denotes convolution operation and nij(t)
is the additive white Gaussian noise. Combining with (5), (6)can
be rewritten as:
yij(t)=
ijstti(v)
ijej2fi(tij)+nij (t)(7)
Define that
fiis the estimated value of fiat the jth node with
the carrier frequency offset (CFO) ∆fij =fi
fi. When yij(t)
is down converted with frequency
fi, it is not difficult to get its
base-band form as y
ij(t)=
ijs(tti(v)
ij)e j2∆ fij(tij )+
nij(t).
Then, y
ij(t) is sampled at time instances kTsof the jth node
during the receiving period, where Tsis the sampling inter-
val and TsT0;k={K,…,1,0,1,…,K}is a discrete
time index and K=T0
2Ts. The sampling signal generated at the
vth clock tick can be expressed as:
y
ij(k)=y
ijtj(v)+kTs
=Aijstj(v)+kTsti(v)
ijej2∆ fijkTs+nij(k)(8)
where Aij =
ijej2∆ fij(tj(v)ij );nij(k) denotes the discrete
time noise.
3CLOCK SYNCHRONIZATION
PROTOCOL AND ESTIMATION
ALGORITHMS
To achieve clock synchronization in the WSN, any node ineeds
to obtain the estimated clock skew
iand phase offset
ibased
on pulse signals from its neighbours, and map its local clock ti(v)
to a certain virtual time
ti(v), that is [8]:
ti(v)=ti(v)
i
i
=i
i
vT0+i
i
i
(9)
Our goal is i
i
=j
j
and i
i
i
=j
j
j
with i≠j,
so that all nodes reach a consensus on such a virtual time.
FIGURE 2 Synchronization process.
The proposed clock synchronization approach is introduced
from three aspects in this section. Firstly, the whole synchro-
nization protocol is described. Then, the important time offset
parameter is estimated through correlation detection. Finally,
the clock skew and offset are estimated based on time offset
with propagation delay eliminated.
3.1 Synchronization protocol design
The specific synchronization protocol is performed from three
stages for Lfully connected nodes in WSNs, which is shown in
Figure 2. In the first stage, Lnodes broadcast the pulse signal in
turn until each node completes M(M2) broadcasts. At each
turn of broadcasting, any node j(j) will receive (L1)
signals from other nodes, and then can estimate (L1) time
offset parameters ∆tij(v), where iand i≠j. The first stage
is completed at PHY layer and the detailed estimation method
will be given in Part B. In the second stage, after Mturns of
broadcast finished, a node i0(i0) will be randomly selected
as the virtual centre when no differences exist among all nodes
of a distributed network, and other nodes send those estimated
time offsets to the virtual centre via packets. Thus, the virtual
centre can obtain all the ML(L1) estimated time offsets of
the whole network. Afterwards, the clock skew
iand phase off-
set
i(i) will be calculated centrally according to those time
offsets. The detailed process of estimating
iand
i(i)will
be presented in Part C.Inthethirdstage,the
iand
i(i
and i≠i0) will be distributed to other nodes from the virtual
centre, so that every node can obtain the estimated values of
its own clock skew and offset, and then compensate its logical
clock according to (9). Since the virtual centre need consume
extra energy in data computing and transmission processes, the
node with the highest remaining energy is usually chosen as the
virtual centre i0when considering the difference among nodes
in practical applications. Specially, the random delay, produced
by the packet transceiver and calculation in the second and third
stages, has no effect on the estimation of
iand
i,asthe
time offset parameters used in the packets have nothing to do
1958 LIU ET AL.
FIGURE 3 y
ij(k) is cross-correlated with s+(k)ands(k), respectively.
with the random delay. For clock synchronization in distributed
WSNs, each node only needs to exchange time information with
its neighbours to achieve global synchronization, so we only
consider information exchange between single hop nodes here.
It should be noted that the process of sending data packets in
the second and third stages is completely entrusted to the com-
munication protocols of the WSNs, just like ordinary business
communication data packets.
3.2 Time offset estimation based on
correlation detection
As shown in Figure 3,yij(k) is cross-correlated with two parts
of the synchronization signal s+(k)ands(k)atthevth clock
tick, respectively, to acquire ∆tij(v):
Rys±(l)=
kK
yij(k)s±(kl)(10)
where the superscript * denotes complex conjugation; y
ij(k)is
given in (8); the subscript ´
s denotes the symbol of the ZC roots
(i.e., +uor u)ands±(k) is obtained from (3)and(4):
s±(k)=
s+(k)=
Nz1
n=0
z[n]xkTsnTz
s(k)=
2Nz1
n=Nz
z[n]xkTsnTz(11)
If the noise term in (8) did not exist in the ideal case, the two
peak indexes of the cross-correlation functions in (10) should
be derived as:
lij+(v)=∆tij(v)
Ts
2Nz∆fij
u+
+
lij(v)=∆tij(v)
Ts
+2Nz∆fij
u+
(12)
where +=Nz1
2and =
Nz1
2+Nzdue to the property
of the sequences constructed by (4); 2Nz∆fij
uis the error caused
by CFO. As a result, the influence of CFO can be removed and
the time offset ∆tij(v) will be obtained from (12):
∆tij(v)=1
2lij+(v)+lij(v)NzTs(13)
Next, it is necessary to estimate lij±(v) in the actual case of
noise existing. The peak indexes are estimated by taking the
square of the cross-correlation function as the weight to aver-
age the index l. Then, the estimation expression of lij±(v)is:
lij±(v)=llRys±(l)
2
lRys±(l)
2(14)
Therefore, the estimated time offset is presented:
∆tij(v)=1
2
lij+(v)+
lij(v)NzTs(15)
3.3 Clock skew and offset estimation based
on time offset
Based on the above estimated time offset, iand iwill be
estimated by eliminating the propagation delay ij through
signal ping-pongs.
As shown in Figure 3, any two nodes iand jwill complete
a signal ping-pong at each turn of broadcast. At the mth turn
broadcast, m=1,2,…,M, we suppose that the ith node broad-
casts a synchronization signal at its local time ti(v), which is
received by the jth node at tj(v). Then according to (1)and(2),
the following equation is obtained:
∆t(m)
ij (v)=
ivT0+
i+
ij jvT0j(16)
Similarly, when the jth node broadcasts at tj(v)andtheith
node receives the signal at ti(v), we can get:
∆t(m)
ji (v)=
jvT0+j+ji
ivT0
i(17)
Supposing that a certain synchronization process starts at
the v0th clock tick and Lnodes broadcast according to the
order of node indexes, the clock tick indexes of the signal
ping-pongs between the ith and the jth node can be derived
as v=v0+(m1)L+i1andv=v0+(m1)L+j1.
LIU ET AL.1959
Meanwhile, the propagation delay between the ith node and the
jth node remains the same during the whole synchronization
process, that is, ji =
ij. Subtracting (17)from(16), we can
eliminate the propagation delay and obtain:
Ω(m)
ij ≜∆t(m)
ij (v)∆t(m)
ji (v)
=
(m)
ij i
(m)
ij j+2i2j
(18)
where (m)
ij ≜(2v0+2(m1)L+i+j2)T0and i≠j.
Then, C2
L=L(L1)
2equations are obtained from (18) for each
turn of broadcasting. After Mturns of broadcasting finished,
ML(L1)
2equations like (18) are obtained naturally. Since there
are 2Lunknown parameters i,iand i,M2 is required
in order to satisfy ML(L1)
22Lfor any Land to improve
the estimation accuracy as well. For convenience, (18) will be
rewritten as the matrix form:
F1H1
F2H2
FMHM
=
P1
P2
PM
(19)
where
Fm=
(m)
12 (m)
12 0…00
(m)
1L0…… 0(m)
1L
0(m)
23 (m)
23 0…0
0(m)
2L0…0(m)
2L
00 ……
(m)
(L1)L(m)
(L1)L
,
Hm=
220…00
20 ……02
0220…0
02 00…2
00 ……22
L(L1)
2×L
,
=11…
LT, = 12…
LT,
Pm=Ω(m)
12 …Ω
(m)
1LΩ(m)
23 ……Ω
(m)
(L1)LT
,
and m=1,2,…,M.
Besides, impose the constraint that L
i=1i=Land
L
i=1i=0, which means that the clock skew and phase off-
set of the Lnodes are close to 1 and 0, respectively. When the
constraint is added to (19), we can get
G
=P(20)
where
G=
F1H1
F2H2
FM
11×L0
01
1×L
,P=
P1
P2
PM
L
0
.
Then, ∆tij(v)inPis replaced with
∆tij(v)toget
P,and
iand
i
can be estimated via calculating G+
P, where the superscript +
denotes the pseudo inverse of the matrix. Finally,
iand
iare
substituted into (9) to estimate the virtual time of all nodes. The
flowchart of the proposed synchronization protocol is shown in
Figure 4.
4SIMULATION RESULTS
In this section, simulations are carried out to compare
the performance of the proposed approach with two typi-
cal timestamp-based [8] and pulse-based [13] synchronization
methods. In simulations, the discrete clock period is set to T0=
1 ms; clock skew iand phase offset iare constant for each
i=1,2,…,Land i[0.999,1.001], i[0.1,+0.1] ms;
the length of the ZC sequence is Nz=839; the pulse spacing
of the synchronization signal s(t)isTz=0.1µs; the sampling
interval Tsis also set to 0.1µs. The channel is assumed to
be a Rayleigh fading channel with the SNR fixed at 15 dB.
The propagation delay is set to 4 µs for all the three meth-
ods, but a special random delay is assumed existing with the
standard deviation 106.5for the method in [8]. Furthermore,
we use the standard deviation of virtual time in the whole
network, that is, =std{
t1(v),
t2(v),…,
tL(v)}to evaluate the
synchronization performance.
Figure 5describes the standard deviation of virtual time for
all nodes within 1 s after clock compensated when L=15
nodes are randomly generated and M=4 signal ping-pongs
are performed. Note that, the x-axis denotes the universal time,
zero scale indicates the clock instance when the parameter
estimation and clock compensation are completed. The logical
clock standard deviation of the three algorithms gradually
increases over time, because there are still slight deviations
in the logical clock skew between nodes after compensa-
tion, which is not completely consistent. The increase in
time will lead to the accumulation of synchronization errors.
However, the algorithm proposed can always achieve higher
1960 LIU ET AL.
FIGURE 4 The flowchart of the proposed synchronization protocol.
synchronization accuracy than the other two algorithms, and the
cumulative speed of synchronization errors is also slower. This
also indicates that under the same synchronization accuracy
requirements, the algorithm proposed can achieve longer syn-
chronization maintenance time, thereby reducing the frequency
of clock synchronization algorithm operation.
Figure 6demonstrates the relationship between synchroniza-
tion accuracy and the number of nodes when M=4signal
ping-pongs performed. As the number of nodes Lincreases,
the synchronization accuracy of the proposed algorithm does
not change significantly, whose virtual time standard deviation
is maintained around 108s. Otherwise, the synchronization
accuracy of [13] decreases gradually. And the standard devia-
tion of logical clocks in [8] gradually decreases from 106.48 to
106.91 s with the increase of L, indicating an improvement
in synchronization accuracy. Anyway, the proposed algorithm
always outperforms the above two algorithms for L50
at least.
The comparative results of synchronization accuracy with
the number of signal ping-pongs Mare shown in Figure 7
when L=20. With Mincreasing, the performance of both
the proposed approach and the algorithm in [8] are slightly
improved, nevertheless, the logical clock standard deviation in
FIGURE 5 Performance of different algorithms with the passage of time.
FIGURE 6 Performance of different algorithms with the number of
nodes L.
[13] has always been greater than 106s, and that of the algo-
rithm proposed can reach 107.7s, which is always lower than
the other two algorithms. In summary, the pulse-based method
in [13] avoids the influence of random delay, but it does not
estimate and compensate the clock skew, so that its perfor-
mance is the worst. The proposed algorithm not only avoids
the effect of random delay, but also eliminates the effect of
fixed propagation delay by signal ping-pongs and estimates the
clock skew and offset jointly. Therefore, under the same num-
ber of signal interactions, the proposed algorithm can have a
higher synchronization accuracy, which indicates that conver-
gence rate of the logic clock in the proposed algorithm is faster.
Under the same synchronization accuracy requirements, the
proposed algorithm requires less information interaction times.
Compared with other algorithms, it reduces the communication
overhead and is a lower cost clock synchronization scheme.
LIU ET AL.1961
FIGURE 7 Performance of different algorithms with the number of
signal ping-pongs M.
5CONCLUSION
Clock synchronization is significant in multiple applications of
WSNs. A pulse-based clock synchronization approach without
a reference clock is proposed in this paper. The approach elim-
inates the unknown propagation delay via signal ping-pongs
performing at PHY layer, and meanwhile avoids the random
delay caused by the upper layer. As a result, the proposed
approach achieves higher synchronization accuracy than the
recent typical methods. As far as we know, it is the first time
that both the clock skew and offset are estimated jointly at PHY
layer when existing no reference clock. In addition, introduc-
ing the multi hop mechanism for clock synchronization at the
physical layer in sparsely connected networks is valuable issue in
the further.
AUTHOR CONTRIBUTIONS
Wei Liu: Methodology, supervision, writing - review and edit-
ing. Ruijie Fan: Software, writing - original draft. Yiyuan
Mai: Supervision. Zehan Wan: Supervision, writing - review
and editing.
CONFLICT OF INTEREST STATEMENT
The authors declare no conflict of interest.
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
ORCID
Wei Liu https://orcid.org/0000-0001-5180-6563
Ruijie Fan https://orcid.org/0000-0002- 9236-6485
Zehan Wan https://orcid.org/0000-0001-7703-2500
REFERENCES
1. Simeone, O., Spagnolini, U., Bar Ness, Y., Strogatz, S.H.: Distributed syn-
chronization in wireless networks. IEEE Signal Process Mag. 25(5), 81–97
(2008)
2. Bosamiya, N.S., Jinwala, D.C.: Analysis of time synchronization proto-
cols in wireless sensor networks. In: 2009 International Conference on
Advances in Recent Technologies in Communication and Computing, pp.
39–41. IEEE, Piscataway, NJ (2009)
3. Mei, L., Yik Chung, W.: On clock synchronization algorithms for wireless
sensor networks under unknown delay. IEEE Trans. Veh. Technol. 59(1),
182–190 (2010)
4. Wang, H., Shao, L., Li, M., Wang, B., Wang, P.: Estimation of clock skew
for time synchronization based on two-way message exchange mechanism
in industrial wireless sensor networks. IEEE Trans. Ind. Inf. 14(11), 4755–
4765 (2018)
5. Schenato, L., Fiorentin, F.: Average timesynch: A consensus-based pro-
tocol for clock synchronization in wireless sensor networks. Automatica
47(9), 1878–1886 (2011)
6. Jianping, H., Peng, C., Ling, S., Jiming, C., Youxian, S.: Time synchroniza-
tion in wsns: A maximum-value-based consensus approach. IEEE Trans.
Autom. Control 59(3), 660–675 (2014)
7. Shi, F., Tuo, X., Ran, L., Ren, Z., Yang, S.X.: Fast convergence time syn-
chronization in wireless sensor networks based on average consensus.
IEEE Trans. Ind. Inf. 16(2), 1120–1129 (2020)
8. Yang, R., Jiang, Y.: Consensus-based clock synchronization for wide area
networks. In: 2021 IEEE Wireless Communications and Networking
Conference (WCNC), pp. 1–6. IEEE, Piscataway, NJ (2021)
9. Chai, Z., Liu, W., Lai, K., Lei, J.: Clock synchronization based on
physical-layer pulse and timestamp free mechanism in 5g. In: 2021
13th International Conference on Wireless Communications and Signal
Processing (WCSP), pp. 1–5. IEEE, Piscataway, NJ (2021)
10. Fan, R., Liu, W., Li, M., Chai, Z.: Clock offset and skew estimation based on
correlation detection with one-way dissemination in wireless sensor net-
works. In: 2022 10th International Workshop on Signal Design and Its
Applications in Communications (IWSDA), pp. 1–5. IEEE, Piscataway,
NJ (2022)
11. Roche, D.T.L., Champagne, B., Psaromiligkos, I., Pelletier, B.: On the
use of distributed synchronization in 5g device-to-device networks. In:
2017 IEEE International Conference on Communications (ICC), pp. 1–7.
IEEE, Piscataway, NJ (2017)
12. Antonieta Alvarez, M., Spagnolini, U.: Distributed time and carrier fre-
quency synchronization for dense wireless networks. IEEE Trans. Signal
Inf. Process. Networks. 4(4), 683–696 (2018)
13. K aratalay, O., Psaromiligkos, I., Champagne, B., Pelletier, B.: A dis-
tributed pulse-based synchronization protocol for half-duplex D2D
communications. IEEE Open J. Commun. Soc. 2, 245–261 (2021)
How to cite this article: Liu, W., Fan, R., Mai, Y., Wan,
Z.: Clock synchronization based on pulse with
propagation delay eliminated in wireless sensor
networks. IET Commun. 17, 1955–1961 (2023).
https://doi.org/10.1049/cmu2.12662
