Path loss exponent estimation for wireless sensor … loss exponent estimation for wireless sensor...

Computer Networks 51 (2007) 2467–2483

www.elsevier.com/locate/comnet

Path loss exponent estimation for wireless sensornetwork localization

Guoqiang Mao a,*, Brian D.O. Anderson b, Barıs� Fidan b

a University of Sydney and National ICT Australia, Sydney, Australiab Australian National University and National ICT Australia, Canberra, Australia

Received 1 April 2006; received in revised form 3 September 2006; accepted 5 November 2006Available online 29 November 2006

Responsible Editor: S. Palazzo

Abstract

The wireless received signal strength (RSS) based localization techniques have attracted significant research interest fortheir simplicity. The RSS based localization techniques can be divided into two categories: the distance estimation basedand the RSS profiling based techniques. The path loss exponent (PLE) is a key parameter in the distance estimation basedlocalization algorithms, where distance is estimated from the RSS. The PLE measures the rate at which the RSS decreaseswith distance, and its value depends on the specific propagation environment. Existing techniques on PLE estimation relyon both RSS measurements and distance measurements in the same environment to calibrate the PLE. However, distancemeasurements can be difficult and expensive to obtain in some environments. In this paper we propose several techniquesfor online calibration of the PLE in wireless sensor networks without relying on distance measurements. We demonstratethat it is possible to estimate the PLE using only power measurements and the geometric constraints associated with pla-narity in a wireless sensor network. This may have a significant impact on distance-based wireless sensor networklocalization.� 2006 Elsevier B.V. All rights reserved.

Keywords: Sensor network; Path loss exponent; Cayley–Menger determinant; Data fusion

1. Introduction

Most wireless sensor network (WSN) applica-tions require knowing or measuring locations ofthousands of sensors accurately. In environmentalsensing applications such as bush fire surveillance,water quality monitoring and precision agriculture,

1389-1286/$ - see front matter � 2006 Elsevier B.V. All rights reserved

doi:10.1016/j.comnet.2006.11.007

* Corresponding author. Tel.: +61 2 93512962; fax: +61 293513847.

E-mail address: [email protected] (G. Mao).

for example, sensing data without knowing the sen-sor location is meaningless [1]. In addition, locationestimation may enable applications such as inven-tory management, intrusion detection, road trafficmonitoring, health monitoring, etc.

Sensor network localization estimates the loca-tions of sensors with unknown location informationwith regards to a few sensors with known locationinformation by using inter-sensor measurementssuch as distance and bearing measurements. Inter-sensor measurement techniques in WSN localization

.

mailto:[email protected]

2468 G. Mao et al. / Computer Networks 51 (2007) 2467–2483

can be broadly classified into three categories:received signal strength (RSS) measurements, angleof arrival (AOA) measurements, and propagationtime based measurements. Among them, RSS basedlocalization techniques have received considerableresearch interest [1–8]. Although RSS based localiza-tion techniques can only provide a coarse-grainedlocation estimate, they are attractive because of theirsimplicity. Received signal strength indicator (RSSI)has become a standard feature in most wirelessdevices, therefore RSS based localization techniquesrequire no additional hardware, and are unlikely tosignificantly impact local power consumption,sensor size and thus cost. RSS based localizationtechniques can be further divided into distance-estimation based [1,3–6] and RSS-profiling basedtechniques [2,7,9,10].

Existing distance-estimation based techniquesrely on a log-normal radio propagation model [11]to estimate inter-sensor distances from RSS mea-surements. The path loss exponent (PLE) is a keyparameter in the log-normal model. An accurateknowledge of the PLE is required in order to obtainan accurate estimate of the inter-sensor distancefrom the corresponding RSS measurement. Existingtechniques either consider the PLE is known a prioriby assuming the WSN environment is free space, orobtain the PLE through extensive channel measure-ment and modeling by measuring both RSS and dis-tances in the same environment of WSN prior tosystem deployment. However, the PLE is environ-ment dependent. Even in the same environment,the channel characteristics may change considerablyover a long period of time due to seasonal changesand weather changes [11]. Therefore, it is an over-simplification to assume a free space environment.On the other hand a priori channel measurementand modeling may not be possible for monitoringand surveillance applications in hostile or inaccessi-ble environments. In this paper, we proposeseveral techniques for online calibration of the pathloss exponent, which do not rely on distancemeasurements.

The RSS has been popularly modeled by a log-normal model [1,11–13]:

P ij ½dBm� � NðP ij ½dBm�; r2dBÞ; ð1Þ

P ij ½dBm� ¼ P 0 ½dBm� � 10� a� log10ðdij=d0Þ; ð2Þ

where Pij [dBm] is the received power at a receivingnode j from a transmitting node i in dB milliwatts,P ij ½dBm� is the mean power in dB milliwatts, r2

dB

is the variance of the shadowing, P0 [dBm] is the re-ceived power in dB milliwatts at a reference distanced0, and dij is the distance between nodes i and j. Inthis paper, we use the notation [dBm] to denote thatpower is measured in dB milliwatts. Otherwise, it ismeasured in milliwatts. The reference powerP0 [dBm] is calculated using the free space Friisequation or obtained through field measurementsat the reference distance d0 [11]. It is important tonote that the reference distance d0 should be in thefar field of the transmitter antenna so that thenear-field effect does not alter the reference power.In large coverage cellular systems, a 1 km referencedistance is commonly used, whereas in microcellularsystems, a much smaller distance such as 100 m or1 m is used [11]. In this paper, it is assumed that d0

and P0 [dBm] can be obtained from a prioricalibration of the wireless device and they are knownconstants. In many situations, Pij [dBm] 5 Pji [dBm]due to the asymmetric wireless signal propagationpath from node i to node j and from node j to nodei. However, in this paper, it is assumed thatPij [dBm] = Pji [dBm] for simplicity. In the case ofan asymmetric path, the average value of Pij [dBm]and Pji [dBm] is used [1]. It is also assumed that alldistances are normalized with respect to d0.

The parameter a is called the path loss exponent,which measures the rate at which the received signalstrength decreases with distance. The value of adepends on the specific propagation environment.In this paper, it is assumed that a is an unknownconstant, which needs to be determined. For a largewireless sensor network spanning a wide area withdifferent environmental conditions, the area can bedivided into smaller regions and a can be consideredas a constant in each region. Path loss exponentestimation plays an important role in distance-basedwireless sensor network localization, where distanceis estimated from the received signal strengthmeasurements [14]. Path loss exponent estimationis also useful for other purposes like sensor networkdimensioning. As the value of the path loss expo-nent depends on the environment in which a sensornetwork is deployed, existing techniques on pathloss exponent estimation rely on both RSS measure-ments and distance measurements (obtained using amethod other than from the RSS, e.g., by using aruler) in the same environment to calibrate thepath loss exponent [1,13]. RSS measurements arereadily available. However, distance measurementscan be difficult and expensive to obtain in someenvironments. Moreover, the reliance on distance

G. Mao et al. / Computer Networks 51 (2007) 2467–2483 2469

measurements impedes deployment in unknownenvironments.

In this paper, we propose several techniques foronline calibration of the path loss exponent, whichdo not rely on distance measurements. The pro-posed technique also does not assume knowledgeof the quantity r2

dB in Eq. (1). We demonstrate thatit is possible to estimate the PLE using only powermeasurements and the geometric constraints associ-ated with planarity in a sensor network. This mayhave a significant impact on distance-based wirelesssensor network localization. For ease of explana-tion, this paper focuses on path loss exponent esti-mation in 2D. However the proposed techniquemay also be extended to 3D.

The rest of the paper is organized as follows. InSection 2, we present a technique which estimatesthe path loss exponent based on known probabilitydistribution of inter-sensor distances. In Section 3,we present the principle of the path loss exponentestimation using the Cayley–Menger determinant[15,16]. Sections 4 and 5 present the proposed algo-rithm and an improvement of it using data fusion.Both algorithms do not rely on knowledge of dis-tance-related information. The validity of thesealgorithms is established using simulation and realmeasurement data. Finally, conclusions and sugges-tions for further work are summarized in Section 6.

2. Path loss exponent estimation based on known

probability distribution of inter-sensor distances

In this section, we consider the estimation of thepath loss exponent assuming that the probabilitydistribution of distance between neighboring sen-sors is known. In some applications, it is possibleto know a priori the probability distribution of dis-tance between neighboring sensors. For example,Stoleru and Stankovic [17] consider that in someapplications like precision agriculture, sensors willbe deployed approximately on the grid points withsome error. In that case, the approximate probabil-ity distribution of distance between neighboringsensors can be obtained by assuming a certain distri-bution of the error, e.g., Gaussian distribution.Some other researchers also consider the deploy-ment of sensors near grid points [18,19]. In addition,it is possible to obtain an accurate knowledge of theprobability distribution of inter-sensor distancesfrom the distribution of wireless sensors [20–22].In particular, it was shown in [20] that under certainconditions, the inter-sensor distance distributions

are very similar when sensors are distributed in theregion following a uniform distribution or a Gauss-ian distribution.

If the distribution has a simple form allowingstraightforward computation, the method ofmoments estimator [23, Section 12.6] can be usedto obtain the path loss exponent estimate. Forexample, if the distance between neighboring nodesis uniformly distributed in the range [a,b], usingEqs. (1) and (2) the expected value of the measuredpower can be obtained:

EðP ij ½dBm�Þ

¼Z b

aP 0 ½dBm� � 10� a� log10 xdx; ð3Þ

¼ P 0 ½dBm�ðb� aÞ � 10� aln 10

�xþ x ln x½ �ba: ð4Þ

An estimate of a can be obtained by equating the ex-pected value of Pij [dBm] with the sample mean ofthe measured received power. In this paper, we as-sume that all sensors radiate at the same power levelfor simplicity, i.e., P0 [dBm] is the same for all sen-sors. However, the derivation in this paper does notrely on this assumption but rather the weakerassumption that the receiver knows the exact valueof the radiated power of the transmitter.

A more generic technique, based on the sameidea as the quantile–quantile (q–q) plot [24], canbe used if the distance distribution has a complexform which is not suitable for computation. Theq–q plot is a graphical technique for determiningwhether two data sets come from populations witha common distribution. Using an approach similarto the q–q plot, a set of points k1,k2, . . .,kN uni-formly distributed in the interval (0, 1) is selected.The quantile points of the distance distributiond1,d2, . . .,dN and the measured received powerP1 [dBm],P2 [dBm], . . .,PN [dBm] corresponding tok1,k2, . . .,kN can then be determined, where

di ¼ argfPrðd P diÞ ¼ kig; ð5Þand

P i ½dBm� ¼ argfPrðP ½dBm� 6 P i ½dBm�Þ ¼ kig:ð6Þ

An estimate of a can be obtained by minimizing thefunction f ðaÞ:

f ðaÞ ¼XN

i¼1

ðP i ½dBm� � P 0 ½dBm� þ 10� alog10diÞ2:

ð7Þ

0 200 400 600 800 1000 1200 1400 1600 1800 200010–4

10–3

10–2

10–1

Number of power measurements

Mea

n sq

uare

err

or o

f the

est

imat

ed p

ath

loss

exp

onen

t

Fig. 1. Variation of the mean square error of a with the numberof power measurements.

1p

2p

3p

4p

1212 ,dP

1414 , dP

1313, dP

2323 , dP

3434 , dP

2424 , dP

Fig. 2. A fully-connected planar quadrilateral in sensor networks.


Differentiating f ðaÞ and setting it to be zero, the va-lue of a can be obtained:

a ¼ �PN

i¼1ðP i ½dBm� � P 0 ½dBm�Þlog10diPNi¼110ðlog10diÞ2

: ð8Þ

Fig. 1 shows the variation of the mean squareerror of the estimated path loss exponent a withthe number of power measurements. The meansquare error is obtained by running the simulation100 times with the same number of power measure-ments but using different random seed. The param-eters used in the simulation are: the true value of ais 2.3; rdB = 3.92; P0 [dBm] = �37.4603 dBm; d0 =1 m. The distances dij are uniformly distributed inthe interval [1,18]. These parameters are drawnfrom real measurement data reported in [1]. Asshown in the figure, this method is very accurateand a converges to the true value very quickly.When the number of power measurements is greaterthan 200, the mean square error drops below 0.003.

The result shown in Fig. 1 is obtained based onthe assumption that the probability distribution ofdistance between neighboring sensors is knownaccurately. This may not be the case in real applica-tions. Further work needs to be done on investigat-ing the sensitivity of the aforementioned techniquewith respect to inaccurate knowledge of the proba-bility distribution of inter-sensor distances.

3. General principle of path loss exponent estimation

using the Cayley–Menger determinant

In the last section, we presented a path loss esti-mation technique based on the knowledge of the

probability distribution of inter-sensor distance. Itcan calibrate the path loss exponent accurately.However, the proposed technique relies on theknowledge of distance distribution, which can beunrealistic to obtain in some applications. This dis-advantage of the proposed technique motivates usto find a more generic technique for path loss expo-nent estimation without relying on knowledge of thedistance distribution, let alone the distance itself.The techniques we present in the following sectionsalso do not assume knowledge of the quantity r2

dB inEq. (1).

Consider a sensor with three neighbors and theneighbors of that sensor are also neighbors of eachother. Two sensors i and j are neighbors if Pij is non-zero. The sensor and its neighbors can be repre-sented by a full-connected planar quadrilateralshown in Fig. 2. The tuple (Pij,dij) represents themeasured power and the true distance between nodei and node j, respectively. Pij and dij are relatedthrough Eqs. (1) and (2). The path loss exponentestimation problem can be formulated as the simul-taneous estimation of the true distances d12, d13, d14,d23, d24, d34 and a given the power measurementsP12, P13, P14, P23, P24, P34. Using the maximumlikelihood estimator (MLE), the likelihood functioncan be obtained as

Lðd12;d13;d14;d23;d24;d34;aÞ

¼ 1

ðffiffiffiffiffiffi2pp

rdBÞ6Y

16i<j64

� exp �ðP ij ½dBm� � P 0 ½dBm� þ 10a log10dijÞ2

r2dB

!:

ð9Þ

3000

3500

to th

e re

gion


Maximizing L is equivalent to minimizing g(d12,d13,d14,d23,d24,d34,a), where

gðd12; d13; d14; d23; d24; d34; aÞ¼

X16i<j64

ðP ij ½dBm� � P 0 ½dBm� þ 10a log10dijÞ2:

ð10Þ

As the number of power measurements is smallerthan the number of parameters to be estimated,minimizing g gives us a set of simple equations. Inthese equations Pij and P0 are in decimal units,not in dB units, so that P ij ¼ 10P ij ½dBm�=10.

P ij ¼ P 0 � d�aij ; 1 6 i < j 6 4: ð11Þ

Another constraint that is required to solve theabove equations can be found from the geometricconstraint on a fully-connected planar quadrilateralusing the Cayley–Menger determinant [15,16]. TheCayley–Menger determinant of a quadrilateral isgiven by:

Dðp1; p2; p3; p4Þ ¼

0 d212 d2

13 d214 1

d212 0 d2

23 d224 1

d213 d2

23 0 d234 1

d214 d2

24 d234 0 1

1 1 1 1 0

��

��ð12Þ

A classical result on the Cayley–Menger determi-nant is given by the following theorem.

Theorem 1 (Theorem 112.1 in [16]). Consider an n-

tuple of points p1,. . .,pn in m-dimensional space with

n P m + 1. The rank of the Cayley–Menger matrix

M(p1, . . ., pn) (defined analogously to the right side ofEq. (12) but without the determinant operation) is at

most m + 1.

0 2 4 6 8 10 12 140

500

1000

1500

2000

2500

Estimated path loss exponent

Num

ber

of e

stim

ated

pat

h lo

ss e

xpon

ent f

allin

g in

Fig. 3. Histogram of the estimated path loss exponent usingCayley–Menger determinant. The figure is obtained using 10,000quadrilaterals whose vertices are uniformly distributed in asquare region of 15 · 15.

A direct application of the theorem leads to that,in 2D

Dðp1; p2; p3; p4Þ ¼ 0: ð13ÞCombining Eqs. (13) and (11), a nonlinear equationfor a can be obtained:

hðaÞ ¼

0 C�2=a12 C�2=a

13 C�2=a14 1

C�2=a12 0 C�2=a

23 C�2=a24 1

C�2=a13 C�2=a

23 0 C�2=a34 1

C�2=a14 C�2=a

24 C�2=a34 0 1

1 1 1 1 0

��

��¼ 0;

ð14Þ

where Cij = Pij/P0,1 6 i < j 6 4, and the Cij are allknown. An analytical solution to Eq. (14) is difficultto find. However, Eq. (14) can be convenientlysolved using the bracketing and bisection numericaltechnique [25, Section 9.1]. The correctness of thenumerical technique has been validated by remov-ing the noise term from Eq. (1), the numerical tech-nique can give the exact value of a. A gradualincrease in the standard deviation of the noise termresults in a larger and larger estimation error.

Unfortunately, the estimate of a obtained usingthis method shows a strong bias, i.e., the bias

Ba ¼ EðaÞ � a ð15Þ

is certainly nonzero and not a small fraction of a.Fig. 3 shows a histogram of the estimated path lossexponent. The same parameters used in the last sec-tion have been used here. The figure is obtainedfrom 10,000 different quadrilaterals whose verticesare uniformly distributed in a square region of15 · 15. It should be noted that in the presence ofnoise in power measurements, Eq. (14) may havenonunique solutions even in the typical range of afor some quadrilaterals. In that case, we simply dis-card that quadrilateral and do not use it in the cal-culation to avoid any ambiguity. In the varioussimulations shown in this paper, the number ofquadrilaterals discarded accounts for about 15%of the total number of quadrilaterals. For the exam-ple shown in the figure, the true value of a is 2.3 andthe estimated value of a has a bias of 2.08. The


simulation shows that a obtained using Eq. (14) hasa strong bias which cannot be ignored.

Bias removal requires an analysis on EðaÞ. How-ever, a direct analysis of EðaÞ is difficult because ofthe difficulty in obtaining an explicit analyticalexpression of a from Eq. (14). Therefore, we resortto numerical experiments to evaluate EðaÞ. Fig. 4shows the relationship between the bias of a (i.e.,

Fig. 4. Relationship between the bias of a, the standard deviation of noirdB.

Fig. 5. Relationship between EðaÞ, the standard deviation of noise in po

Ba), the standard deviation of noise in power mea-surement rdB, and a. Fig. 5 shows the correspondingrelationship between EðaÞ, rdB, and a. Both figuresare obtained using 3000 quadrilaterals whose verti-ces are uniformly distributed in a square of15 · 15. The power measurements are obtainedusing Eqs. (1) and (2). Fig. 6 shows the relationshipbetween ra, rdB, and a. The parameter ra is the

se in power measurement rdB and a. Ba depends on a as well as on

wer measurement rdB and a. EðaÞ depends on a as well as on rdB.

Fig. 6. Relationship between the standard deviation of a, the standard deviation of noise in power measurement rdB and a. ra depends onrdB, and to a small degree on a as well. At larger values of rdB, ra is almost independent of a. The dependence of ra on a is more significantwith smaller values of rdB.

2

3

4

5

6

7

8

9

10

σ =4

σ =2σ =3


square root of the sample variance of a. With a littlebit abuse of terminology, we call ra the standarddeviation of a.

Fig. 7 shows the relationship between EðaÞ andrdB at specific values of a = 2.0, a = 3.0 anda = 4.0. Fig. 8 shows the relationship between ra

and rdB at specific values of a = 2.0, a = 3.0 anda = 4.0.

Further simulations show that the relationshipsbetween EðaÞ, rdB and a, and between ra, rdB and

0 5 10 15 20 25 300

5

10

15

20

25

Standard Deviation of Noise in Power (dB)

Mea

n V

alue

of t

he E

stim

ated

Pat

h Lo

ss E

xpon

ent

=2

=3

=4

Fig. 7. Relationship between EðaÞ and the standard deviation ofnoise in power measurement rdB at specific values of a = 2.0,a = 3.0 and a = 4.0.

0 5 10 15 20 25 300

1

Standard Deviation of Noise in Power (dB)

Fig. 8. Relationship between the standard deviation of a and thestandard deviation of noise in power measurement rdB at specificvalues of a = 2.0, a = 3.0 and a = 4.0.

a, are almost independent of the distribution of thevertices of the quadrilaterals, and independent ofthe shape of the area in which the vertices of thequadrilaterals are located. These are shown in Figs. 9and 10. Fig. 9 shows the relationship between EðaÞ,rdB and a for quadrilaterals whose vertices arelocated in areas of a variety of different shapes anddistributed in the area following different distribu-tions. This relationship between EðaÞ, rdB and a is

Fig. 9. Relationship between the bias of EðaÞ, rdB and a. The relationship between EðaÞ, rdB and a is almost independent of thedistribution of the vertices of the quadrilaterals and independent of the shape of the area in which the vertices of the quadrilaterals arelocated. (a) Generated from 3000 quadrilaterals whose vertices are uniformly distributed in a ‘‘L’’ shaped area. (b) Generated from 3000quadrilaterals whose vertices are uniformly distributed in a rectangular area of 10 · 20. (c) Generated from 3000 quadrilaterals whosevertices are distributed in a square area of 15 · 15 following a truncated Gaussian distribution with a zero mean and a standard variationof 5. (d) Generated from 3000 quadrilaterals whose vertices are located in a ring. The inner radius of the ring is 5 and the outer radius is 10.The coordinates of the vertices are generated by first selecting a number uniformly distributed in [5,10] and then selecting a numberuniformly distributed in [0,2p]. The two numbers are used as the polar coordinate of the vertex to obtain the rectangular coordinate.


explored in Section 4 to approximately correct thebias in EðaÞ. Fig. 10 shows the relationship betweenra, rdB and a. This relationship between ra, rdB anda is further explored in Section 5.

4. Path loss exponent estimation based on pattern

matching

Based on the observation shown in Fig. 9 that therelationship between EðaÞ, rdB and a is independentof the distribution of the vertices of various quadri-laterals and the shape of the area in which verticesof the quadrilaterals are located, a pattern matchingtechnique can be used to estimate the path lossexponent using the power measurements only.

Specifically, via a priori simulation, a data basecan be established where each entry in the data base

is arranged in the form ðai; r2dB;j;EðaÞai;r2

dB;jÞ. The

symbol EðaÞai;r2dB;j

is used to emphasize dependenceof EðaÞ on the path loss exponent ai and the vari-ance of noise in power measurements r2

dB;j. Thisdata base can be obtained using a large number ofquadrilaterals whose vertices, say, are uniformlydistributed in an area. The corresponding powermeasurements are obtained using Eqs. (1) and (2).The distance between adjacent r2

dB is the same, i.e.,

r2dB;jþ1 � r2

dB;j ¼ Dr2dB: ð16Þ

The distance between adjacent a is also the same,i.e.,

aiþ1 � ai ¼ Da: ð17ÞThe parameters Dr2

dB and Da are empirically chosenconstants, which do not change with i and j.

Fig. 10. Relationship between ra, rdB and a. The relationship between between ra, rdB and a is almost independent of the distribution ofthe vertices of the quadrilaterals and independent of the shape of the area in which the vertices of the quadrilaterals are located. Sub-figures (a)–(d) are obtained under the same conditions as those in Fig. 9.


Given this data base, the estimation of a can beobtained using the following procedure for patternmatching:

1. Identify a set of fully connected quadrilateralsin the wireless sensor network for furthercomputation.

2. Add a random Gaussian noise with varianceDr2

dB into each power measurement. The poweris measured in dB milliwatts unit.

3. For each individual quadrilateral, solve for ausing Eq. (14). An EðaÞr;1 corresponding toDr2

dB can be obtained as the average value of aobtained from each individual quadrilateral.Here the subscript r is used to mark the differencewith the corresponding value in the database.

4. Repeat steps 2 and 3 using different noise vari-ance values MDr2

dB, where M = 0,1,2, . . .,m.Here, M = 0 corresponds to the original dataset without the additionally introduced noise. Aseries of tuples ð0;EðaÞr;0Þ, ðDr2

dB; EðaÞr;1Þ; . . . ;ðmDr2

dB;EðaÞr;mÞ can then be obtained.

5. Search the database and find the values of i and j

such that:

fi; jg ¼ arg minfi;jgXm

N¼0

ðEðaÞr;N � EðaÞai ;r2dB;jþNÞ2:

ð18Þ

The parameter m has to be a large number, sayP500, in order to obtain a reasonably accurateestimate of a, which is robust against therandomness in the data. As will be shown inthe next section, the accuracy of the path lossexponent estimation can be increased by choos-ing a larger value of m and smaller values ofDr2

dB and Da.6. Finally, an improved estimate of a approximately

correcting for the bias can be obtained as

a ¼ ai: ð19Þ

Similarly, an estimate of rdB can be obtained as

rdB ¼ rdB;j: ð20Þ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dB Estimation Error

CD

F o

f the

σdB

Est

imat

ion

Err

or

Empirical CDF

σ

Fig. 12. The empirical CDF of error in estimating rdB using thepattern matching technique. The vertices of the quadrilaterals areuniformly distributed in a rectangular area of 10 · 20.


4.1. Simulation validation

In this section, we shall validate the proposedtechnique using both simulations and real measure-ments. The database is established by simulationsusing 3000 quadrilaterals whose vertices are uni-formly distributed in a square region of 15 · 15.The measured power is generated using Eqs. (1)and (2). Dr2

dB is set to be 1 and the distance betweenadjacent ai is set to be 0.1, i.e., Da = 0.1. Another setof 3000 quadrilaterals whose vertices are uniformlydistributed in a rectangular area of 10 · 20 are usedto establish the performance of the proposed algo-rithm. 700 points are used in the search, i.e.,m = 700 in Eq. (18). The simulation is repeated byvarying the value of a from 2 to 4 and varying thevalue of rdB from 5 to 10, which are the typicalranges of the two parameters [11]. Fig. 11 showsthe empirical cumulative distribution function(CDF) of error in estimating a and Fig. 12 showsthe the empirical CDF of error in estimating r.

As shown in Fig. 11, the error in estimating a iscontained in the region [�0.3,0.1]. The mean esti-mation error is �0.1175 and the error variance is0.0115. Fig. 12 shows that the error in estimatingrdB is contained in the region [0,1]. The mean esti-mation error is 0.4827 and the error variance is0.0263. The estimation error is attributable to therandomness in the data but can be reduced by usingsmaller values of Dr2

dB and Da at the expense ofincreased computational load. For example, oursimulation shows that by using m = 2800,Da = 0.05 and Dr2

dB ¼ 0:25, the maximum error in

–0.35 –0.3 –0.25 –0.2 –0.15 –0.1 –0.05 0 0.05 0.1 0.150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Path Loss Exponent Estimation Error

CD

F o

f the

Pat

h Lo

ss E

xpon

ent E

stim

atio

n E

rror Empirical CDF

Fig. 11. The empirical CDF of error in estimating a using thepattern matching technique. The vertices of the quadrilaterals areuniformly distributed in a rectangular area of 10 · 20.

estimating a reduces to 0.2. The mean error reducesto �0.0603 and the error variance reduces to 0.0056.

Simulations using quadrilaterals whose verticesare distributed in an area of a different shape and/or following a different distribution show similarperformance. Figs. 13 and 14 show the simulationresults using a set of 3000 quadrilaterals whose ver-tices are distributed in a square region of 15 · 15following a truncated two-dimensional Gaussiandistribution. The mean of the Gaussian distributionis at the center of the square region and the standarddeviation of the Gaussian distribution is 5. Both fig-ures show similar performance as those in Figs. 11and 12 except that the estimation error of a has avalue of �0.4 at a couple of points in Fig. 13. The

–0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F o

f the

Pat

h Lo

ss E

xpon

ent E

stim

atio

n E

rror Empirical CDF

Fig. 13. The empirical CDF of error in estimating a using thepattern matching technique. The vertices of the quadrilaterals aredistributed in a square region of 15 · 15 following a truncatedtwo-dimensional Gaussian distribution.

–0.25 –0.2 –0.15 –0.1 –0.05 0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F o

f the

Pat

h Lo

ss E

xpon

ent E

stim

atio

n E

rror Empirical CDF

Fig. 15. The empirical CDF of error in estimating a using thepattern matching technique. The vertices of the quadrilaterals aredrawn randomly from nodes distributed in a rectangular regionof 10 · 15 following a Poisson distribution with a uniform nodedensity of 100/(10 · 15).

–0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4 0.5 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dB Estimation Error

CD

F o

f the

σdB

Est

imat

ion

Err

or

Empirical CDF

σ

Fig. 16. The empirical CDF of error in estimating rdB using thepattern matching technique. The vertices of the quadrilaterals aredrawn randomly from nodes distributed in a rectangular regionof 10 · 15 following a Poisson distribution with a uniform nodedensity of 100/(10 · 15).

–0.6 –0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dB Estimation Error

CD

F o

f the

σdB

Est

imat

ion

Err

or

Empirical CDF

σ

Fig. 14. The empirical CDF of error in estimating rdB using thepattern matching technique. The vertices of the quadrilaterals aredistributed in a square region of 15 · 15 following a truncatedtwo-dimensional Gaussian distribution.


mean error in estimating a is �0.1785. The estima-tion error of rdB is better than that in Fig. 12 andthe mean estimation error is �0.1252. Figs. 15 and16 show the simulation results using a set of 3000quadrilaterals whose vertices are distributed in arectangular region of 10 · 15 following a Poissondistribution. The mean error in estimating a is�0.0113 and the mean estimation error of rdB is0.1021.

4.2. A comparison with a MLE of the PLE using

both power and distance measurements

To further evaluate the performance of the pro-posed technique, we apply the proposed PLE esti-

mation technique to real measurement data andcompare the PLE estimate obtained using the pro-posed technique with that in [1] which obtains amaximum likelihood estimate of the PLE using bothpower measurements and distance measurements.The measurement data and the deployment ofsensors can also be found at http://www.eecs.umi-ch.edu/~hero/localize/. The wireless sensor networkin [1] consists of 44 fully connected nodes, whichmake up 135,751 quadrilaterals. It is both computa-tionally intensive and unnecessary to compute a foreach quadrilateral. Here, we randomly choose10,000 quadrilaterals for computation. The refer-ence power P0(d0) [dBm] is calculated using the freespace Friis equation at a reference distance d0 = 1m

and P0(d0) [dBm] = �37.4663 dBm [1]. The PLEestimated using the proposed technique is 2.2. Usingboth power measurements and distance measure-ments (946 distance measurements and 946 powermeasurements from 44 fully connected nodes), amaximum likelihood estimate of the PLE is shownto be 2.3022 [1]. Assuming the PLE estimateobtained using both power measurements and dis-tance measurements represents the true value ofPLE, the PLE estimate obtained using the proposedtechnique which does not use distance measure-ments has an estimation error of �0.1022. The truevalue of rdB (i.e., the value obtained using bothpower measurements and measured distances) is3.92 and the estimation error is �1.50. The slightlylarger error in estimating rdB when using real mea-surement data may be attributable to a deviation of

http://www.eecs.umich.edu/~hero/localize/

http://www.eecs.umich.edu/~hero/localize/

0 10 20 30 40 50 60 70 80 90 10010–3

10–2

10–1

100

Number of distance measurements

Mea

n sq

uare

err

or o

f est

imat

e

dB=5

dB=6

dB=7

dB=8

dB=9

dB=10

σ

σ

σ

σ

σ

σ

Fig. 17. Variation of the mean square error of the estimated PLEwith rdB and number of distance measurements. The PLE isobtained from a maximum likelihood estimator using both powerand distance measurements.


the density of the noise in the power measurementsfrom a Gaussian distribution.

As a further comparison of the proposed tech-nique with the maximum likelihood estimator usingboth power and distance measurements, we analyzethe number of distance measurements required bythe maximum likelihood estimator to make a PLEestimate with a similar accuracy as the proposed tech-nique. Given k distance measurements d1,d2, . . .,dk

(recall that all distances are normalized with regardsto known distance d0) and the corresponding powermeasurements P1 [dBm],P2 [dBm], . . .,Pk [dBm], itcan be readily shown from Eqs. (1) and (2) that amaximum likelihood estimate of the PLE is:

am ¼�Pk

i¼1ðP i ½dBm� � P 0 ½dBm�Þlog10di

10Pk

i¼1ðlog10diÞ2: ð21Þ

According to Eqs. (1) and (2), Pi [dBm] is related todi by:

P i ½dBm� ¼ P 0 ½dBm� � 10alog10di þ ni; ð22Þwhere the sequence {ni} is a stationary zero meanwhite Gaussian sequence of random variables withstandard deviation rdB. Replacing Pi [dBm] in Eq.(21) with Eq. (22), it can be shown that

am ¼ a�Pk

i¼1nilog10di

10Pk

i¼1ðlog10diÞ2; ð23Þ

and

Eðam � aÞ2 ¼ r2dB

100Pk

i¼1ðlog10diÞ2: ð24Þ

Eq. (24) shows that the mean square error of theestimated PLE depends on the distribution of dis-tances. Considering a simple scenario of k distancemeasurements uniformly distributed between 1 and10, i.e., di = 1 + 9i/k, Eq. (24) becomes

Eðam � aÞ2 ¼ r2dB

100Pk

i¼1ðlog10ð1þ 9i=kÞÞ2: ð25Þ

Fig. 17 shows the variation of the mean square errorof the estimated PLE with rdB and number of dis-tance measurements. Using the performance of theproposed technique shown in Fig. 11 for example,the mean square error of the estimated PLE, i.e.,the sum of the square of the mean estimation errorand the error variance, is 0.025. When rdB is equalto 5, the maximum likelihood estimator using bothpower and distance measurements needs 18 distancemeasurements to achieve a similar accuracy as theproposed technique. A larger number of distance

measurements is required at a larger value of rdB

and the converse.In addition to not requiring distance measure-

ments, another advantage of the proposed tech-nique is it makes the PLE estimation possible inhostile, dangerous or inaccessible environments.Note that even in the same environment, the chan-nel characteristics may change considerably over along period of time due to seasonal changes andweather changes [11], the proposed technique alsoshow advantage for implementation in theseenvironments.

4.3. A discussion on the computational complexity

and the impact of the PLE estimation error

In this section, we have a brief discussion of thecomputational complexity of the proposed tech-nique and the impact of the PLE estimation erroron sensor network localization.

To evaluate the computational complexity of theproposed algorithm, further simulations are per-formed to estimate the number of quadrilateralsrequired in order to obtain a stable performance.The simulation results are shown in Fig. 18.Fig. 18 shows that a minimum of 1000 quadrilater-als are required in order for the proposed algorithmto achieve a stable performance.

In the proposed algorithm, the data baserequired for pattern matching can be establishedvia a priori simulation. The memory space requiredfor storing the data base is in the order of hundredsof kilo-bytes which is not a major constraint forthe implementation of the proposed algorithm.

–5 0 5 10—2

0

2

4

6

8

10

12

14Non–anchorsAnchors

Fig. 19. True positions of anchors and non-anchor nodes in thesimulation.

0 500 1000 1500 2000 2500 3000–0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Number of quadrilaterals

Err

or m

etric

Maximum estimation errorMean estimation errorVariance of estimation error

Fig. 18. Variation of the maximum estimation error of a, meanestimation error and error variance with number of quadrilateralsused in the simulation. Each point shown in the figure is theaverage value of ten simulations using different random seed. Thevertices of the quadrilaterals are drawn randomly from nodesdistributed in a rectangular region of 10 · 15 following a Poissondistribution with a uniform node density of 100/(10 · 15).


However the online computation involving 1000quadrilaterals and the pattern matching may pres-ent a major challenge for distributed implementa-tion of the proposed algorithm. Therefore theproposed algorithm in its present form is not suit-able for implementation in a distributed environ-ment. In a centralized environment, the proposedalgorithm can be implemented in a central stationor in a cluster head which has more energy andcomputation power than an ordinary sensor node.It remains a future research topic to design a distrib-uted algorithm in which each sensor does the com-putation for the quadrilateral it resides in and animproved estimate of PLE is obtained by fusingthe computation results of individual sensors.

The impact of the PLE estimation error on theestimation of individual inter-sensor distances canbe readily evaluated. The answer to the moreinvolved question of the impact of the PLE estima-tion error on sensor network localization is actuallylinked to a fundamental question: what is theimpact of error in distance measurements on theperformance of distance-based localization algo-rithm? Numerous localization algorithms haveshown via simulations that the impact of distancemeasurement error possibly depends on a numberof factors which include the distribution of anchors(i.e, sensor nodes with known positions) and non-anchor (i.e., sensor nodes whose positions are tobe estimated) nodes, node degree (or equivalentlynode density and sensing range) and the shape ofthe area in which sensors are deployed [14]. We

are yet to develop an accurate knowledge in thearea. In this paper, instead of giving a comprehen-sive evaluation of the impact of PLE estimationerror on sensor network localization, we give anillustration of the possible impact of the PLE esti-mation error using the measurement data in [1].The wireless sensor network in [1] consists of 44fully connected nodes. The four sensors located atthe four corners are chosen to be the anchor nodesand the rest are the non-anchor nodes. Fig. 19shows the placement of these sensors. The true valueof PLE is 2.3022 [1]. An error term varying from�0.4 to 0.4 is added to the true value to evaluatethe impact of the PLE estimation error. The sumof the true PLE and the error is termed as theestimated PLE, i.e., a. The sensor network localiza-tion is formulated as the following optimizationproblem:

fxi;m<i6ng ¼ arg minfxi;m<i6ng

�Xn

i¼mþ1

Xj2Ni

ðP ij � 10alog10kxi � xjkÞ2;

ð26Þ

where anchor nodes are numbered from 1 to m,non-anchor nodes are numbered from m+1 to n,Ni denotes the set of neighbors of node i and kÆk de-notes the Euclidean norm. xi and xi are the esti-mated position and the true position of the ithsensor respectively and for anchor nodes xi ¼ xi.The simulated annealing algorithm [26] is used tosolve the problem and obtain the estimated posi-tions of non-anchor nodes. Fig. 20 shows thevariation of the sensor network position estimationerror with the PLE estimation error. As shown in

–0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.40

5

10

15

20

25

Path loss exponent estimation error

Sen

sor

netw

ork

posi

tion

estim

atio

n er

ror

Fig. 20. Variation of the sensor network position estimationerror with the PLE estimation error. The sensor network positionestimation error shown in the figure is 1

n�m

Pni¼mþ1kxi � xik2, where

xi and xi are the estimated position and the true position of the ithsensor, respectively, sensors 1 to m are anchors, and sensorsm + 1 to n are non-anchors. Each point shown in the figure is theaverage value of 10 simulations using different random seed.

0.1

0.2

0.3

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0.5

0.6

0.7

0.8

0.9

1

of t

he P

ath

Loss

Exp

onen

t Est

imat

ion

Err

or Empirical CDF


the figure, the sensor network localization appears tobe more robust against an overestimation of the PLEthan an underestimation. Actually, with an overesti-mation of the PLE, we find a slight improvement inthe sensor network localization performance. Aplausible explanation of the simulation result is:

• First, the MLE of inter-sensor distance in Eq.(11) is a biased estimate [14] and it tends to over-estimate the distance even when the PLE is accu-rate. A slightly overestimate of the PLE may helpto balance this effect.

• Second, this may also be attributable to theplacement of anchors and the high node degreeof anchors. Given a set of power measurements,an overestimate of the PLE will result in smallerinter-sensor distance estimates. Therefore a‘‘shrinkage’’ of the estimated sensor positionsmay occur. However because the anchors areplaced at the four corners of the map and theyhave a high node degree, they constrain theshrinking effect caused by an overestimation ofthe PLE. This constraint formed by placing theanchors at the corners becomes less effective withan underestimation of the PLE.

–1.2 –1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.40


CD

F

Fig. 21. The empirical CDF of error in estimating a using thepattern matching technique and the relationship between ra, rdB

and a. The vertices of the quadrilaterals are uniformly distributedin a rectangular area of 10 · 20.

5. Further improvement on the proposed technique

using data fusion

In Section 4, we presented a technique whichapproximately corrects the bias of a using the rela-

tionship between EðaÞ, rdB and a. Unsurprisinglya similar technique can also be developed, whichbegins by establishing a database similar to that inSection 4 but describing the relations between ra

(instead of EðaÞ), rdB and a, and uses the patternmatching technique to correct the bias of a basedon the relationship between ra, rdB and a. Fig. 21shows the error in estimating a. The mean estima-tion error is �0.2597 and the error variance is0.1295. The same parameters as used in Section 4are used here.

As shown in the figure, the error in estimating a issignificantly worse than that shown in Section 4.This result is expected because in comparison withEðaÞ, the value of ra is less dependent on the valueof a, as suggested in Figs. 7 and 8. However the rela-tionship between ra, rdB and a does provide extrainformation concerning a, which can be exploitedto improve a obtained in Section 4 by using datafusion [27].

A popular approach in data fusion is to linearlycombine the estimated parameters. Denote the esti-mated a using the relationship between ra, rdB and aby as. Denote the estimated a using the relationshipbetween EðaÞ, rdB and a by ab. The dual estimatedvector is defined as

~a ¼ ½asab�T: ð27Þ

The estimated fusion calculation is given by

a ¼WT a!; ð28Þ

–0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F o

f the

Pat

h Lo

ss E

xpon

ent E

stim

atio

n E

rror

Empirical CDF

Fig. 22. Error in estimating a using data fusion. The vertices ofthe quadrilaterals are uniformly distributed in a rectangular areaof 10 · 20.


where W is a vector in our specific application. Inthe more general case of simultaneous estimationof multiple parameters, W is a full rank weight ma-trix. A typical data fusion approach is to calculateW such that the sum of the diagonal elements forthe error covariance matrix of the fused estimateis minimized. The optimal W, which gives the lowesterror covariance is given by [27]:

W ¼ C�1ATðAC�1ATÞ�1: ð29Þ

where

C ¼ Eas � a

ab � a

� �as � a

ab � a

� �T( )

; ð30Þ

and

A ¼ ½11� ð31Þ

in our specific application. a is the correspondingtrue value. It should be noted that the optimalityof data fusion in Eq. (29) relies on the assumptionsthat covariance is the dominant factor in estimationerror in comparison with the bias, and that the esti-mation errors have a Gaussian distribution [27].These requirements make the optimality of this datafusion method difficult to prove. However thismethod is generally robust in the sense that givenan accurate error covariance matrix, it always re-sults in a fused estimator with a lower covariancethan the individual estimators [28].

To establish the performance improvement ofdata fusion approach, we use the path loss exponentestimation from the quadrilaterals, whose verticesare distributed in a square region of 15 · 15 follow-ing a truncated two-dimensional Gaussian distribu-tion (as shown in Fig. 13), to obtain the errorcovariance matrix C and the weight vector W. Theobtained weight vector is W = [�0.1718,1.1718]T.A negative value of �0.1718 indicates that the errorin as and the error in ab are positively correlated.Using this weight vector, we linearly combine thepath loss exponent estimates obtained using therelationship between ra, rdB and a, and usingthe relationship between EðaÞ, rdB and a (as shownin Eq. (28)).

Fig. 22 shows the path loss exponent estimationerror for quadrilaterals, whose vertices are uni-formly distributed in a rectangular area of 10 · 20.An overall improvement has been achieved in com-parison with the results shown in Fig. 11, althoughat some points the estimation error becomes larger,which can be possibly explained by the statistical

nature of data fusion. The mean estimation errorreduces from the original �0.1175 to �0.0930. Theerror variance remains essentially the same.

6. Conclusions and further work

In this paper, we presented some techniques foronline calibration of path loss exponent in wirelesssensor networks without relying on distance mea-surements. Specifically, techniques were proposedwhich are based on different assumptions aboutknowledge of distance information.

The first technique assumes that the probabilitydistribution of distance between neighboringsensors is known. Then an algorithm similar tothe quantile–quantile plot was proposed, whichcan estimate the path loss exponent accurately usinga small number of received power measure-ments. However, this assumption of knowing thedistance distribution can be unrealistic in someapplications. This has motivated us to find a moregeneric technique without using any distance infor-mation.

Then we presented a technique based on the Cay-ley–Menger determinant, which estimates the pathloss exponent using only power measurements andthe geometric constraints associated with planarityin a wireless sensor network. The technique can givean accurate estimate of a when there is no noise inpower measurements, but it has a large bias in thepresence of noise. A pattern matching techniqueapproximately correcting the bias is proposed basedon the empirical observation that the relation-ship between EðaÞ, rdB and a is independent of the


distribution of the vertices of various quadrilateralsand the shape of the area in which vertices of thequadrilaterals are located. We also presented animprovement of the earlier technique using datafusion. The proposed algorithms may have signifi-cant impact on distance-based wireless sensor net-work localization, where distance is estimatedfrom the received signal strength measurements.

In this paper, we observed the empirical law thatthe relationship between EðaÞ, rdB and a is indepen-dent of the distribution of the vertices of the quad-rilaterals and is also independent of the shape of thearea in which the vertices of the quadrilaterals arelocated. It is desirable to obtain an analyticalexpression of the relationship between EðaÞ, rdB

and a. This is the direction of our future research.Furthermore, the proposed algorithm relies on

the log-normal propagation model in Eqs. (1) and(2) in the sense that the maximum likelihood estima-tor shown in Eq. (11) may have a different formwhen the received signal strength has a differentmodel. Although the log-normal propagation modelis a popular model for wireless networks, there areenvironments in which the log-normal propagationmodel is not the best model [11,12]. In that case, atechnique needs to be developed to select the bestmodel and choose the best estimator for distanceto replace Eq. (11) accordingly. Therefore how todevelop an algorithm for environments in whichthe log-normal propagation model does not applyis also a future research topic.

Acknowledgements

This research is supported by National ICTAustralia. National ICT Australia is funded by theAustralian Government’s Department of Commu-nica- tions, Information Technology and the Artsand the Australian Research Council through theBacking Australia’s Ability initiative and the ICTCentre of Excellence Program.

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Guoqiang Mao received the Bachelordegree in electrical engineering, Masterdegree in engineering and Ph.D. in tele-communications engineering in 1995,1998 and 2002 respectively. After grad-uation from PhD, he worked in ‘‘Intel-ligent Pixel Incorporation’’ as a SeniorResearch Engineer for one year. Hejoined the School of Electrical andInformation Engineering, the Universityof Sydney in December 2002 where he is

a Senior Lecturer now. His research interests include wirelesssensor networks, wireless localization techniques, network QoS,
telecommunications traffic measurement, analysis and modeling,and network performance analysis.
Brian D.O. Anderson was born in Syd-ney, Australia, and received his under-graduate education at the University ofSydney, with majors in pure mathemat-ics and electrical engineering. He subse-quently obtained a Ph.D. degree inelectrical engineering from StanfordUniversity. Following completion of hiseducation, he worked in industry in Sil-icon Valley and served as a facultymember in the Department of Electrical

Engineering at Stanford. He was Professor of Electrical Engi-

neering at the University of Newcastle, Australia from 1967 until1981 and is now a Distinguished Professor at the AustralianNational University and Chief Scientist of National ICT Aus-tralia Ltd. His interests are in control and signal processing. He isa Fellow of the IEEE, Royal Society London, Australian Acad-emy of Science, Australian Academy of Technological Sciencesand Engineering, Honorary Fellow of the Institution of Engi-neers, Australia, and Foreign Associate of the US NationalAcademy of Engineering. He holds doctorates (honoris causa)from the Universite Catholique de Louvain, Belgium, SwissFederal Institute of Technology, Zurich, Universities of Sydney,Melbourne, New South Wales and Newcastle. He served a termas President of the International Federation of Automatic Con-trol from 1990 to 1993 and as President of the AustralianAcademy of Science between 1998 and 2002. His awards includethe IEEE Control Systems Award of 1997, the 2001 IEEE JamesH Mulligan, Jr Education Medal, and the Guillemin-CauerAward, IEEE Circuits and Systems Society in 1992 and 2001, theBode Prize of the IEEE Control System Society in 1992 and theSenior Prize of the IEEE Transactions on Acoustics, Speech andSignal Processing in 1986.

Barıs� Fidan received the B.S. degrees inelectrical engineering and mathematicsfrom Middle East Technical University,Turkey in 1996, the M.S. degree in elec-trical engineering from Bilkent Univer-sity, Turkey in 1998, and the Ph.D.degree in Electrical Engineering-Systemsat the University of Southern California,Los Angeles, USA in 2003. After work-ing as a postdoctoral research fellow atthe University of Southern California for

one year, he joined the Systems Engineering and Complex Sys-tems Program of National ICT Australia and the Research
School of Information Sciences and Engineering of the Austra-lian National University, Canberra, Australia in 2005, where he iscurrently a researcher. His research interests include autonomousformations, sensor networks, adaptive and nonlinear control,switching and hybrid systems, mechatronics, and various controlapplications including high performance and hypersonic flightcontrol, semiconductor manufacturing process control, and disk-drive servo systems.

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