Communications and Network, 2013, 5, 498-503
http://dx.doi.org/10.4236/cn.2013.53B2091 Published Online September 2013 (http://www.scirp.org/journal/cn)
On the Trade-off between Power Consumption and Time
Synchronization Q ualit y for Movin g Targets under
Large-Scale Fading Effe cts i n Wirel es s Sens or N etw orks
Pablo Briff1, Leonardo Rey Vega2, Ariel L utenberg1, Fabian Vargas3
1Embedded Systems Laboratory, Faculty of Engineering, University of Buenos Aires, Argentina
2Signal Processing and Communications Laboratory, Faculty of Engineering, University of Buenos Aires, Argentina
3Signals and Systems for Computing Group, Faculty of Engineering, Catholic University, PUCRS, Brazil
Received May 2013
ABSTRACT
In this work we find a lower bound on the energy required for synchronizing moving sensor nodes in a Wireless Sensor
Network (WSN) affected by large-scale fading, based on clock estimation techniques. The energy required for synchro-
nizing a WSN within a desired estimation error level is specified by both the transmit power and the required number of
messages. In this paper we extend our previous work introducing nodes’ movement and the average message delay in
the total energy, including a comprehensive analysis on how the distance between nodes impacts on the energy and
synchronization quality trade-off under la rge-scale fading effects.
Keywords: Wireless Sensor Networks; Clock Offset Estimation; Time Synchronization; Wireless Channel Fading;
Moving Targets
1. Introduction
With the advent of wireless technologies over the last
decade, Wireless Sensor Networks (WSN’s) are over-
taking wired networks in the field of sensing [1]. A WSN
typically consists of low cost battery-powered or self-
powered sensor nodes. Thus, energy management be-
comes a substantial matter in order to guarantee reasona -
ble sensors’ lifetime values. Time synchronization algo-
rithms aim to provide mechanisms for nodes to obtain an
estimate of their internal clocks with respect to the other
nodes, aiming to reach a consensus on the concept of
time among all the nodes. For each node i, its internal
clock
i
c
can be modeled as a linear equation with a
corres po nding skew
i
α
and offset
i
β
[2,3], namely
()
i ii
ct αβ = +
. In order to achieve a target synchroniza-
tion quality, parameter estimation techniques can be ap-
plied, being the estimation error

a function of the
estimator and the number of samples employed. Time
synchronization can be energy-consuming since it in-
volves wireless messages exchange, however, when
achieved, it allows significant energy savings through
network power management. In all, WSN synchroniza-
tion remains amongst the most challenging open topics in
WSN’s [4]. In our previous work [5] we had introduced
the existing tradeoff between time synchronization accu-
racy and synchronization energy, although we did not
contemplate nodes’ movement in our analysis. In this paper
we introduce this important feature under large-scale fad-
ing, a situation that is present in a realistic environment
where nodes change their relative distance within a
network.
2. Related Work
There is a number of clock synchronization techniques
that exploit the number of received messages from a given
sensor node to produce their clock estimation.. Examples
of these are Reference-Broadcast Synchronization (RBS)
[6], Timing-Sync Protocol for Sensor Networks (TPSN)
[7] and Pairwise Broadcast Synchronization (PBS) [8]. In
RBS, a reference node broadcasts reference beacons that
serve the nodes in the network to perform receiver-re-
ceiver pairwise synchronization. TPSN creates a hierar-
chical structure in which each node synchronizes to its
parent in a sender-receiver fashion. Yet, in PBS, a pair of
supernodes A and P exchange messages that are over-
heard by all nod es in the network, allowing each node to
construct their local estimate of the clock offset and skew
with respect to the supernodes based on reception time
stamps. Thus, for a given node B the quality of offset
estimation with respect to reference node P can be ob-

P. BRIFF ET AL.
499
tained as follows [8]:
22
() 1
22
11
ˆ
var()()
m
i
BP i
offset mm
ii
ii
D
mD D
σ
θ
=
= =
≥
−
∑
∑∑



(1)
From (1), the estimation quality (variance of the clock
offset estimator) depends on the number of received mes-
sages
m

and the time stamps differences
i
D
. Increas-
ing
m

will enhance estimation quality in detriment of
the energy consumed. While many authors expose this
energy-synchronization quality balance as a known open
topic (such as [9,10]), to the best of our knowledge, pre-
vious contributions in the field of clock synchronization
focus on the algorithmic aspect of the timing mechanism
with little concern on the energy and delay required to
attain such a goal. In [11], authors propose a mechanism
for synchronizing a WSN by means of constructive in-
terference in order to achieve low duty cycles in radio
operation, thus minimizing the energy spent, although
they do not mention the existing trade-off between ener-
gy and synchronization accuracy. In [12], the authors ex-
pose the trade-off of energy consumption and synchroni-
zation quality from a local sleep-time perspective, with-
out contemplating the energy spent in nodes interaction.
Also, [13] aims to minimize the energy for maximum
accuracy but from a local node’s perspective, with low
duty cycle and low frequency crystal oscillators hardware.
More recent works such as [11,14] approach the energy
efficiency problem from a protocol perspective, without
detailing the physical phenomena involved in wireless
channels. For example, [14] proposes a new algorithm,
the Recursive Time Synchronization Protocol (RTSP),
which aims to minimize the number of transmitted mes-
sages in a WSN, although the authors do not include in
their analysis either transmit or receive power in each
sensor node as part of the minimization problem. There-
fore, the tradeoff “power consumption-clock synchroni-
zation quality” is a critical issue for wireless embedded
systems that requires finding an optimal solution.
3. One-Way Message Clock Offset
Estimation Quality as a Function of
Transmit Power
3.1. Motivation
We will use one-way message mechanism as the starting
point for exposing the underlying issues associated with
clock synchronization by means of wireless messages.
The Flooding Time Synchronization Protocol [15] uses
this technique to e stimate the sender’s c l oc k offse t through
a linear regression of the received samples.
3.2. Model Statement
Transmit power and clock synchronization quality oper-
ate on different layers: the first one is a physical magni-
tude whereas the latter belongs to the application layer.
However, prior to estimation, physical layer reception oc-
curs with a given probability of failure as a function of
the transmit power S, given by the channel’s outage prob-
ability
out
P
, defined as the probability that the received
signal falls under a minimum acceptable threshold [16].
Let’s consider each node’s clock offset
θ
is estimated
with an unbiased estimator
ˆ
θ
and let
2
ˆ
θ
σ
be the va-
riance of the clock offset estimator. Consider sender node
A
sends
m
packets while receiver node
B
receives
[ ](1)
out
m EMmP== ⋅−

successful messages, where M
is a binomial random variable, i.e.
~ ( ,1)
out
M bmP−
.
The average delay per message can also be derived as a
function of the outage pro ba bility as follows:
1
M
Mout
T m
TmP δ = ⋅=
−

(2)
where M
T is the message transmission time. Thus, re-
ducing the outage probability will also enhance the syn-
chronization time. However, this must be balanced with
the application’s energy budget, since in order to reduce
out
P
, the transmit power
S
must be increased. Since the
estimation quality depends on the number of successfully
received packet
m

, the interesting relation
2
ˆ
()fS
θ
σ =
is sought. It will be necessary then to relate the estima-
tion quality’s dependence on the number of received
messages, namely
2
ˆ
()m
θ
σ

, and the number of received
messages dependence on the transmit power, i.e.
()mS

.
We will approach the synchronization problem from the
local perspective of a node that is synchronizing with a
neighbor, irrespective of the network size and topology.
The analysis presented in this work is not tied to a partic-
ular procedure but it repr ese nts a uni versal lower bound on
the “energy-sync hronization quality” trade-off.
3.3. Definitions
3.3.1. Estimators and Theoretical Limits
The trade-off studied in this work can be stated as an
estimation problem. Both expected value and variance of
the offset’s unbiased estimator are defined as shown be-
low:
ˆ
[]E θθ =
(3)
22 22
ˆˆ ˆˆˆ
[([]) ][() ]()EE Em
θθ
σθ θθθσ = −= −=
(4)
In order to formulate a general problem, the Cramer-
Rao lower bound [17] can be used for delimiting the best
performance an estimator can afford. Thus, the estima-
tion quality relates with the Fisher Information function
()I θ
as follows:

P. BRIFF ET AL.
500
2
ˆ
1
()I
θ
σθ ≥
(5)
with the Fisher Information’s expression as shown below
[17]:
2
2
(, )ln (, )Im Efm θθ
θ

∂
−
∂



(6)
where
f
is the likelihood function of the parameter
θ
.
3.3.2. Communication Channel Model
For wireless channels, the received to transmit power
ratio
/
R
SS
is dictated by [16]:
0
()10log10 log
RdB
Sd
dB K
Sd
γψ =−−
(7)
where
K
is a constant that models the antenna gain,
0
d
a reference distance,
γ
the path loss exponent,
d
the distance between transmitter and receiver nodes, and
10log
dB
ψψ =
, being
ψ
a random variable that models
large-scale (shadowing) effects. A communication is de-
fined to be successful, i.e. the receiver can process the
transmitted message, when the received Signal-to-Noise
Ratio (SNR)
s
γ
satisfies
0s
γγ >
, being
0
γ
the min-
imum acceptable SNR by the receiver [16]. We will co n-
sider that the wireless channel is memory-less and time-
invariant, meaning that each channel use will be inde-
pendent and uncorrelated from each other, i.e., they will
undergo independent and identically distributed (i.i.d.)
fading effects [16], which means that two subsequent
messages sent over the wireless channel will present in-
dependent and uncorrelated impairments.
3.4. Problem To Solve: Energy Optimization
The number of successfully received packets
m

is re-
lated to the transmit power
S
as shown below:
[1( )]
out
mm PS= ⋅−

(8)
The main challenge is to find the transmit power
S
that satisfies the following condition:
2
ˆ
min() s.t. ()Sm
θ
σ <

(9)
Equation (9) seeks the minimum transmit power
S
that guarantees the necessary amount of received mes-
sages
m

so that the clock offset estimation error
2
ˆ
θ
σ
is
less than a desired level

. For Cramer-Rao efficient
estimators, i.e. estimators that attain equality in (5), the
following inequality can be stated:
2
ˆ
1
(, )Im
θ
σθ = <

(10)
where
I
is the Fisher Information of the estimated pa-
rameter
θ
as a function of the received samples
m

.
Thus, the problem can be stated as follows:
1
Find: min() s.t. (,)S Im θ > 

(11)
Equation (11) seeks the minimum transmit power S for
achieving a desired estimation error

on the clock off-
set
θ
by successfully receiving
m

messages after trans-
mitting m messages. In order to account for energy opti-
mization, both transmitter and receiver energy must be
minimized; the first one depends on the transmit power S
and the number of transmitted messages m, whereas the
latter is determined by the total time the receiver circuit
is powered-on. Since the transmitter sends m messages
and the average message delay is
δ
, the receiver must
be turned on for at least
m δ ⋅
to successfully receive
m

messages. Thus, the total energy function for a pair
of nodes
(, )ij
, where node i is transmitting messages
to node
j
, can be expressed as follows:
() () ()
(1 )
iji j
M
out
ETotalE TxERx
SmT Sm
Sm P
ηδ
δη
= +
=⋅⋅ +⋅⋅⋅
=⋅⋅⋅+−
(12)
where
/1
M out
TP δ = −
as per (2) and
/
Rx
SS η 
represents the ratio between the receive power and the
transmit power, which typically falls in the range 0.5 ~
0.8 for commercial transceivers [18]. The term
(1)( ,1)
out
P η ηη +−∈ +
in (12) has a smooth variation
with
S
for which it does not strongly contribute to the
overall variation as the rest of the unknowns S, m and
δ
do, i.e. it is sufficient to minimize the product of all S, m
and
δ
to find the minimum energy working point. Hence,
let
(, ,)ASmS m δδ =⋅⋅
(13)
be a representative measure of the total energy required
for synchronizing a pair of nodes. Thus, the objective is
to minimize the
(, ,)ASm δ
function for large-scale fad-
ing effects. This will be the main motivation throughout
the rest of this work.
3.5. Gaussian Observations of t he Clock Offset
As per (6), the Fisher Information function requires a
likelihood function to be applied. Considering the case of
Gaussian distributed likelihood functions, for
m

Gaus-
sian i.i.d observations of
θ
, the joint probability distri-
bution function is expressed as:
2
2 /22
1
() 1
( ,)exp
(2 )2
mi
mj
VV
fm
θθ
θπσ σ
=

−
= −


∑



(14)
where
2
V
σ
is the variance of the perturbations that im-
pair the measurements around the real value of the para-
meter
θ
to be estimated. Operating with (6), (11) and
(14), we obtai n :

P. BRIFF ET AL.
501
2
1
(, )
V
m
Im
θσ = >



(1 5)
3.6. Large-Scale Effects: Path Loss and
Shadowing
Large-scale fading represents the average signal power
attenuation or path loss over large areas, a phenomenon
affected by prominent terrain contours (billboards, clump
of buildings, etc.) between the transmitter and receiver
[19]; still, for indoor applications, this phenomenon is
also present for distances smaller than 10 meters [16].
Under path loss and shadowing, the outage probability
out
P
is defined as the probability that the received power
falls below a given outage threshold
Rx
S
expressed in
dBm as found below [16]:
2
0
11
( )experfc
22
22
[10log10 log(/)]
()
()1(())
dB
z
Rx
out
xz
Q zdx
S SKdd
zS
P SQzS
ψ
π
γ
σ
+∞  
−=
 


−+−
=
= −
∫

(16)
with the unknow n transmit power
S
expressed in dBm.
Parameters
K
,
d
,
0
d
,
γ
,
d
defined in (7) are as-
sumed known. In this scenario, the random variable
dB
ψ
assumes a Gaussian distribution with zero mean and va-
riance
dB
ψ
σ
(assumed kno wn). Involving (8) , (11), (15)
and (16), it can be seen that for a desired estimation pre-
cision

, the transmit power
S
must fulfill the follow-
ing:
2
0
[10log10 log(/)]
dB
Rx V
S SKdd
Qm
ψ
γσ
σ

−+ −>


⋅


(17)
Equation (17) shows that for decreasing estimation er-
ror

, either power
S
or number of transmitted mes-
sages m must be increased accordingly. Since Q increas-
es with increasing S, the minimum transmit power
min
S
will be found on the limit of equality in (17). It is then
convenient to rewrite this equation into a function as fol-
lows:
]
2
1
()
( ,,)0
dB
min V
min
Kd S
BS mdQm
ψ
σ
σ

−
= −= 

⋅



(18)
with
10
( )10log10log(/)
Rx
Kd SKdd γ =−+
(19)
For nodes moving at a relative velocity
V
, the dis-
tance
d
between them at time
t
is determined by:
0
()ddtdV t== +⋅
(2 0)
where
0
d
is the initial distance between the nodes ex-
changing messages. Equation (20) is a scalar expression
due to the fact that we study the fundamentals of the
energy trade-off problem with two nodes communicatin g
with each other; for the general case of N nodes, the ex-
pression can be better represented by a vector equation.
Furthermore, V can be either positive or negative; in case
of negative relative velocity (nodes’ approaching each
other) we will consider that th e nodes’ instantaneous dis-
tance d fulfills
()
min
dt dt>∀
in order to remain in the
large-scale effects scenario. The minimum distance
min
d
for large-scale effects is typically 10 m for indoor appli-
cations and 100 m for outdoor applications.
3.7. Large-Scale Effects: Towards Energy
Minimization
From (18), the number of transmitted messages m is de-
termi ne d by:
2
1()
dB
V
min
min
mKd S
Q
ψ
σ
σ
=

 
−
⋅



(21)
After combining (2) and (16), the delay
δ
adopt s the
following expression under large -scale effects:
1
()
dB
M
min
min
T
Kd S
Q
ψ
δ
σ
=
−



(2 2)
By substituting (21) and (22) into (13), and expressing
min
S
in dBm, the minimization problem is stated as:
0.1 2
21
10 0
()
min
dB
S
MV
min min
T
SKd S
Q
ψ
σ
σ


 ⋅
∂=

∂
−

⋅





(23)
A solution to (23) was shown in [5], leading to:
2
1
1
() 1
2 exp2
() 0
0.23 2
dB
dB dB
min
min
Kd S
Kd S
Q
ψ
ψψ
σ
σπσ


−

⋅−



 
−
−= 


(24)
which can be graphically solved to find the optimal
min
S
value, provided that
1
()
min
S Kd≥
. Equations (21) and
(24) represent an energy-efficient solution to the target
estimation error

under the effect of large-scale fading.
4. Simulation Results
This section exposes the simulations results for typical
WSN parameters as referenced in [16] under one-way
message exchange. Figure 1 shows the dependence of
the number of transmitted messages m and the required

P. BRIFF ET AL.
502
energy
(, ,)ASm δ
with transmit power S under the in-
fluence of large-scale fading for different values of the
estimation quality

, for a fixed distance d between
transmitter and receiver nodes. Figure 2 shows the de-
pendence of m with S and d under large-scale fading. It
can be seen that for S fixed, as d grows, m has to be in-
creased accordingly.
Since under large-scale effects the dominating factor
in signal fading is the distance d between pairs, the nodes’
relative velocity is not an independent variable in the energy
graphics. However, it is always possible to define a fixed
time window, e.g.
1ts∆=
, substitute d with its defini-
tion in (20), and analyze the variation of the energy mi-
nima as dict a ted by (24) . He nce, although t he re i s a unique
relation between d and V, it is more accurate to analyze
the variations the energy minima with d under this scena-
rio. The figures in this section have the following simula-
tion parameters:
24
0
,1,80,7.0146 10
/110, , 3.71,1s
dB
V Rx
M
SdBm K
d ddBT
ψ
σ
σγ
−
= =
=
=−⋅
= ==
.
Figure 1. Number of transmitted messages m and Energy required A(S, m, d) as a function of transmit power S for large-scale
fading, for different clock offset estimation qualities e using one-way messages.
Figure 2. Number of transmitted messages m as a function of transmit power S and nodes’ separation distance d for
large-scale fading, for different clock offset estimation qualities e using one-way me ssages.

P. BRIFF ET AL.
503
5. Summary and Discussions
Time synchronization for a WSN can be achieved by
means of parameter estimation techniques which require
a number of messages to be transmitted from a sender to
a receiver node. The minimum amount of total energy
required for achieving a desired estimation quality

is
represented by the product of the transmit power S, num-
ber of messages m and the average message delay d. By
introducing the concept of outage probability of the wire-
less channel for large-scale fading, a minimization prob-
lem can be stated for the total energy function. The reso-
lution of the entire system finds the energy-optimal work-
ing point which represents a lower bound for the estima-
tion quality. We have also analyzed the effect of distance
and relative movement between sensor nodes and pre-
sented a set of equations describing their impact on the
system’s parameters, which constitutes an interesting and
realistic problem to real-world applications. The general
results obtained in this work have been applied to the
particular case of Gaussian p erturbations for the Cramer-
Rao efficient, unbiased offset estimator
ˆ
θ
. For other
estimators that do not fulf ill these conditions, the estima-
tion error
2
ˆ
θ
σ
shall be used instead of the Fisher Infor-
mation function in order to compute the theoretical limits
for that particular case. As part of our future work, we
are working on the small-scale effects counterpart when
moving targets are synchronizing in a WSN.
Finally, unlike under large-scale effects where the dis-
tance between no des plays a predominant role in the ener-
gy minimization problem, under small-scale effects the
relative velocity between nodes is a determining factor in
the synchronization accuracy, due to the Doppler spread
effect [19]. Thus, our aim is to complete a comprehen-
sive analysis of “energy-synchronization quality” trade-
off under both fadi ng sc enari os as part of our fut ure work.
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