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4(DRAFT) 1

Collision Tolerant Packet Scheduling forUnderwater Acoustic Localization

↑Hamid Ramezani*,Student Member, IEEE, †Fatemeh Fazel,Member, IEEE,†Milica Stojanovic,Fellow, IEEE, and↑Geert Leus,Fellow, IEEE

Abstract—This article considers the joint problem of packetscheduling and self-localization in an underwater acoustic sensornetwork where sensor nodes are distributed randomly in anoperating area. In terms of packet scheduling, our goal is tominimize the localization time, and to do so we consider twopacket transmission schemes, namely a collision-free scheme(CFS), and a collision-tolerant scheme (CTS). The requiredlocalization time is formulated for these schemes, and throughanalytical results and numerical examples their performancesare shown to be generally comparable. However, when the packetduration is short (as is the case for a localization packet),and theoperating area is large (above 3km in at least one dimension), thecollision-tolerant scheme requires a smaller localization time thanthe collision-free scheme. After gathering enough measurements,an iterative Gauss-Newton algorithm is employed by each sensornode for self-localization, and the Cramer Rao lower bound isevaluated as a benchmark. Although CTS consumes more energyfor packet transmission, it provides a better localizationaccuracy.Additionally, in this scheme the anchor nodes work independentlyof each other, and can operate asynchronously which leads toasimplified implementation.

I. I NTRODUCTION

After the emergence of autonomous underwater vehicles(AUVs) in the 70s, developments in computer systems andnetworking have been paving a way towards fully autonomousunderwater acoustic sensor networks (UASNs) [2], [3]. Mod-ern underwater networks are expected to handle many tasksautomatically. To enable applications such as tsunami moni-toring, oil field inspection or shoreline surveillance, thesensornodes measure various environmental parameters, encode theminto data packets, and exchange the packets with other sensornodes or send them to a fusion center. The data packets areusually meaningless if they are not labeled with the time andthe location of their origin. In this sense, localization isanindispensable task for the network.Due to the challenges of underwater acoustic communicationssuch as low data rates and long propagation delays with

↑The authors are with the Faculty of Electrical Engineering,Mathematicsand Computer Science, Delft University of Technology, 2826CD Delft, TheNetherlands. e-mails:{h.mashhadiramezani, g.j.t.leus}@tudelft.nl.

†The authors are with the Department of Electrical and Com-puter Engineering, Northeastern University, 02611, MA, USA. e-mails:{ffazel,millitsa}@ece.neu.edu.

* Corresponding author: Hamid Ramezani, phone: (+31)152786280, fax:(+31)152786190, e-mail: h.mashhadiramezani@tudelft.nl.

The research leading to these results has received funding in part from theEuropean Commission FP7-ICT Cognitive Systems, Interaction, and Roboticsunder the contract #270180 (NOPTILUS), NSF grant CNS-1212999, andONR grant N00014-09-1-0700. Part of this work is accepted inthe Proceedingof IEEE ICC 2014 Workshop on Advances in Network Localization andNavigation (ANLN), 10-14 June 2014, Sydney, Australia [1].

variable sound speed [4], a variety of localization algorithmshave been introduced and analyzed in the literature [5] [6].These algorithms are relatively different from the ones studiedfor terrestrial wireless sensor networks (WSNs). For instance,in the terrestrial WSNs, a sensor node can be equipped witha GPS module to determine its location. On the other hand,GPS signals (radio-frequency signals) are highly attenuatedunderwater, and cannot propagate more than a few meters.Therefore, acoustic signals are usually used for underwatercommunications. In addition, sensor nodes in WSNs experi-ence low propagation delays in packet exchanging becauseradio-frequency signals travel almost with light speed. Incontrast, acoustic signals propagate very slowly in comparisonwith light speed, and that introduces long propagation delaysbetween the underwater nodes.

For an underwater sensor node to determine its location,it can measure the time of flight (ToF) to several anchorswith known positions, estimate its distance to them, and thenperform multilateration. Other approaches may be employedfor self-localization, such as finger-printing [7] or angleofarrival estimation [8]. Nevertheless, packet transmissions fromanchors are required in all these approaches.

Many factors determine the accuracy of self-localization.Other than noise, the number of anchors, their constellation,relative position of the sensor node [9], propagation lossesand fading also affect the localization accuracy. Some ofthese parameters can be adjusted to improve the localizationaccuracy.

Although a great deal of research exists on underwaterlocalization algorithms [2], little work has been done todetermine how the anchors should transmit their packets tothe sensor nodes. In long base-line (LBL) systems wheretransponders are fixed on the sea floor, an underwater nodeinterrogates the transponders for round-trip delay estimation[10]. In the underwater positioning scheme of [11], a masteranchor sends a beacon signal periodically, and other anchorstransmit their packets in a given order after the reception of thebeacon from the previous anchor. The localization algorithmin [12] addresses the problem of joint node discovery andcollaborative localization without the aid of GPS. The algo-rithm starts with a few anchors as primary seed nodes, andas it progresses, suitable sensor nodes are converted to seednodes to help in discovering more sensor nodes. The algorithmworks by broadcasting command packets which the nodesuse for time-of-flight measurements. The authors evaluate theperformance of the algorithm in terms of the average networkset-up time and coverage. However, physical factors such as

2 (DRAFT)

packet loss due to fading or shadowing and collisions have notbeen reviewed, and it is not established whether this algorithmis optimal for localization. In reactive localization [13], anunderwater node initiates the process by transmitting a “hello”message to the anchors in its vicinity, and those anchorsthat receive the message correctly transmit their packets.Anexisting medium access control (MAC) protocol may be usedfor packet exchanging [14]; however, there is no guaranteethat it will perform satisfactorily for the localization task. Theperformance of localization under different MAC protocolsis evaluated in [15], where it is shown that a simple carriersense multiple access (CSMA) protocol performs better thanthe recently introduced underwater MAC protocols such asT-Lohi [16].

In our previous work, we considered optimal collision-freepacket scheduling in a UASN for the localization task insingle-channel [17] and multi-channel [18] scenarios. There,the position information of the anchors is used to minimize thelocalization time. In spite of the remarkable performance overother algorithms (or MAC protocols), they are highly demand-ing. Their main drawback is that they require a fusion centerwhich gathers all the position information of the anchors, anddecides on the time of packet transmission from each anchor.In addition, they need the anchors to be synchronized andequipped with radio modems in order to exchange informationfast. In contrast, in this paper we consider packet schedulingalgorithms that do not need a fusion center or synchronizedanchors.

We assume a single-hop UASN where anchors are equippedwith half-duplex acoustic modems, and can broadcast theirpackets based on two classes of scheduling: a collision-freescheme (CFS), where the transmitted packets never collidewith each other at the receiver, and a collision-tolerant scheme(CTS), where the collision probability is controlled by thepacket transmission rate in such a way that each sensor nodecan receive sufficient error-free packets for self-localization.Our contributions are listed as below.

• Under the conditions of packet loss and collision, thelocalization time is formulated for each scheme, and itsminimum is obtained analytically for a predeterminedprobability of successful localization for each sensornode.

• An iterative Gauss-Newton self-localization algorithm isintroduced for a sensor node which experiences packetloss or collision. Furthermore, it is explained how thisalgorithm can be used for each packet scheduling scheme.

• The Cramer Rao lower bound (CRB) on localizationis derived for each scheme. Other than the distance-dependent signal to noise ratio on each measurement, theeffects of packet loss due to fading or shadowing, colli-sions, and the probability of successful self-localizationare included in this derivation.

The structure of the paper is as follows. Section II describesthe system model, and explains self-localization. The problemof minimizing the localization time in the collision-free andcollision-tolerant packet transmission schemes is formulatedand analyzed in Section III-A and Section III-B, respectively.

The self-localization algorithm is introduced in Section IV.The average energy consumption is analyzed in Section V,and Section VI compares the two classes of localization packetscheduling through several numerical examples. Finally, weconclude the paper in Section VII, and outline some futurework.

II. SYSTEM MODEL

We consider a UASN consisting ofM sensor nodes andN anchors as shown in Fig. 1. Each anchor in the networkencapsulates information about its ID, its location, time ofpacket transmission, and a predetermined training sequencefor the time of flight estimation. The so-obtained localizationpacket is broadcast to the network based on a given protocol,e.g., periodically, or upon the reception of a request from asensor node [19]. The system structure is specified as follows.• Anchors and sensor nodes are equipped with half-duplex

acoustic modems, i.e., they cannot transmit and receivesimultaneously.

• Sensor nodes are located randomly in an operating areaaccording to some probability density function (pdf). Weassume that the distance between a sensor node and ananchor is distributed according to a pdfgD(z). It is furtherassumed that the pdf of the distance between the anchorsis fD(z). The pdfs can be estimated from the empiricaldata gathered during past network operations.

• Although the concept of this article can be extendedto a multi-hop network, in this work we consider asingle-hop network where all the nodes are within thecommunication range of each other.

• It is assumed that in the absence of collision, the probabil-ity of packet loss due to fading or shadowing between ananchor and a sensor node ispl which is further assumedto be distance-independent.

The considered localization algorithms are assumed to bebased on ranging, whereby a sensor node determines itsdistance to several anchors via ToF or round-trip-time (RTT).Each sensor node can determine its location if it receivesat leastK different localization packets fromK differentanchors. The value ofK depends on the geometry (2D or3D), and other factors such as whether depth information ofthe sensor node is available, and whether the sound speedestimation is required or not. The value ofK is usually 3for a 2D operating environment with known sound speed and4 for a 3D one. In a situation where the underwater nodesare equipped with pressure sensors, three different successfulpackets would be enough for a 3D localization algorithm [20].

Successful reception

Packet lossAnchors

Unlocalized nodes

Fig. 1: Anchors and sensor nodes are uniformly distributed in a rectangulararea.

H 3

The localization procedure starts either periodically forapredetermined duration (in a synchronized network), or uponreceiving a request from a sensor node (in any kind of network,synchronous or asynchronous) as explained below.

M1) Periodic localization: If all the nodes in the networkincluding anchors and sensor nodes are synchronizedwith each other, a periodic localization approach may beemployed. In this approach, after the arrival of a packetfrom thej-th anchor, them-th sensor node can estimateits distance to that anchor asdm,j = c(tRm,j − tTj ) wherec is the sound speed,tTj is the time at which the anchortransmits its packet, andtRm,j is the estimated time atwhich the sensor node receives this packet. The departuretime tTj is obtained by decoding the received packet (theanchor inserts this information in the localization packet),and the arrival timetRm,j can be calculated by e.g.,correlating the received signal with the known trainingsequence. The estimated time of arrival is related to theactual arrival time throughtRm,j = tRm,j + nm,j, wherenm,j is zero-mean Gaussian noise with powerσ2

m,j whichvaries with distance and can be modeled as [21]

σ2m,j = kEdn0

m,j , (1)

with dm,j the distance between thej-th anchor and them-th sensor node,n0 the path-loss exponent (spreadingfactor), andkE a constant that depends on system pa-rameters (such as signal bandwidth, sampling frequency,channel characteristics, and noise level).

M2) On-demand localization: In this procedure (which canbe applied to a synchronous or an asynchronous network)a sensor node initiates the localization process. This ishandled by transmitting a high power frequency toneimmediately before the request packet. This tone wakesup the anchors from their idle mode, and puts them inthe listening mode. The request packet may be used fora more accurate estimation of the arrival time. In thispaper, we assume that all the anchors have been correctlynotified by this frequency tone. After the anchors receivethis frequency tone, they reply with their localizationpackets. In this case, the time when the request hasbeen received by an anchor and the time at which alocalization packet is transmitted have to be included inthe localization packet, and this information will be usedby the sensor node to estimate its round-trip-time (whichis proportional to twice the distance) to the anchor. Theround-trip-time can be modeled as

tRTTm,j = (tRm,j − tTm)− (tRj,m − tTj ) + nj,m + nm,j . (2)

Therefore, the estimated distance to anchorj would be

dm,j =1

2ctRTT

m,j . (3)

After the sensor node estimates its location, it broadcastsits position to other sensor nodes. This enables the sensornodes which have overheard the localization processto estimate their positions without initializing anotherlocalization task.

Frequency tone (it may be followed by a request packet)

Rx mode

! !!" ! and ! Distance between anchors

Localization packet from an anchor

sensor node

anchor 1

anchor 2

anchor

Localization time

time

!"# !#

$

!""!

$

!%

$

!&

$

Fig. 2: Packet transmission from anchors in the collision-free scheme. Here,each anchor transmits its packets according to its index value (ID number).All links between anchors are assumed to function properly in this figure(there are no missing links).

Frequency tone (it may be followed by a request packet)

Localization packet from an anchor

Active mode

Rx mode

!"#

! #

Localization time

time

! "

!#"

#anchor

anchor 2

anchor 1

sensor node

Fig. 3: Packet transmission from anchors in the collision-tolerant scheme.Here, each anchor transmit its packets at random according to a Poissondistribution.

The time it takes for an underwater node to gather at leastK different packets fromK different anchors is called thelocalization time. A shorter localization time allows for amoredynamic network, and leads to a better network efficiency interms of throughput. In the next section, we formally definethe localization time, and show how it can be minimized forthe collision-free and collision-tolerant packet transmissionschemes.

III. PACKET SCHEDULING

A. Collision-free packet scheduling

Collision-free localization packet transmission is analyzedin [17], where it is shown that in a fully-connected (single-hop)network, based on a given sequence of the anchors’ indices,each anchor has to transmit immediately after receiving theprevious anchor’s packet. Furthermore, it is shown that thereexists an optimal ordering sequence which minimizes thelocalization time. However, to obtain that sequence, a fusioncenter is required that knows the positions of all the anchors.In a situation where this information is not available, wemay assume that anchors simply transmit in order of theirID numbers as illustrated in Fig. 2.

In the event of a packet loss, a subsequent anchor will notknow when to transmit. If an anchor does not receive a packet

4 (DRAFT)

from a previous anchor, it waits for a predefined time (countingfrom the starting time of the localization process), and thentransmits its packet, similarly as introduced in [22]. Withaslight modification of the result from [22], the waiting timeforthe j-th anchor could be as short astj,k +(j−k)

(

Tp +Daac

)

,wherek is the index of the anchor whose packet is the lastone which has been received by thej-th anchor,tj,k is thetime at which this packet was received by thej-th anchor(counting from the starting time of the localization process),c is the sound speed,Daa

cis the maximum propagation delay

between two anchors, andTp is the packet length. The packetlength is related to the system bandwidthB (or symbol timeTs ≈ 1

B), number of bits in each symbolbs, number of bits

in each packetbp, and guard timeTg as formulated in

Tp = Tg +bp

bsTs. (4)

Let us denote byu a (N − 1)× 1 vector whose elementuj iseither 1 or 0, whereuj = 1 indicates that there is a packet loss(with probabilitypl) between anchorsj andj+1, anduj = 0represents no packet loss between these anchors. Let us alsodefined as a(N − 1)× 1 vector whosej-th element,dj,j+1,is the distance between the anchorsj and j + 1. With thesedefinitions, the localization time for the collision-free schemecan be formulated as

TCF =1

c(1 − u)Td+

Daa

cuT1 +NTp +

ds

c+

de

c, (5)

whereds is the distance between the requesting sensor nodeand anchor node 1 which initiates the localization process,deis the distance between the last anchor and the farthest sensornode in the network, and1 is a (N − 1) × 1 vector whoseentries are 1. Since the anchors are unaware of the positionof the farthest sensor node, we setde = Dsa, whereDsa isthe distance corresponding to the maximally separated sensor-anchor pair. The addition of this maximum propagation delayensures that the last transmitted packet will reach the farthestnode. Note that in the case of periodic localizationds = 0because no request from a sensor node is required in order tostart the localization process. Givenu, the conditional pdf ofTCF depends only on the pdfs of the distances between anchor-anchor and anchor-sensor pairs, and can be obtained as

fTCF|u(t) =

c fD(ct) ∗ fD(ct) ∗ . . . ∗ fD(ct)︸ ︷︷ ︸

N−1−nu times

∗ gD(ct) ∗ δ

(

t −NTp − nu

Daa

c− Dsa

c

)

,

(6)

wherenu is the number of lost packets, andδ(t) is the Diracfunction. Using (6), the pdf ofTCF can be obtained as

fTCF(t) =

N−1∑

n=0

fTCF|u(t)P (nu = n) (7)

whereP (nu = k) =

(N − 1

k

)

pkl (1− pl)N−1−k . (8)

Given the pdf of the collision-free localization time,fTCF(z),

the minimum collision-free localization timeTminCF

for the N

anchors to finish transmitting their packets with probability

Ptt can be obtained by solving

Ptt =

∫ TminCF

0

fTCF(t)dt. (9)

Sincedj,j+1 for j = 1, ..., N − 1, ds, andu are independentof each other, the average localization time can be obtainedas

T avgCF

=NTp + (N − 1)(1 − pl)davg

c+

(N − 1)plDaa

c+

Dsa

c+

ds

c,

(10)

wheredavg is the average distance between two anchors, andcan be formulated as

davg =

∫ Daa

0

zfD(z)dz, (11)

and ds is the average ofds which can be either 0 ords,avg

(periodic or on-demand localization, respectively), withds,avg

obtained similarly as (11).Under the condition of no packet loss (pl = 0), the anchors

do not need to wait for the maximum propagation delay, andthey can transmit their packets immediately after the completereception of the previously transmitted packet. As a result, thelocalization time is the shortest and given by

TCF|u=0

= NTp +1

c

N−1∑

j=1

dj,j+1 +ds

c+

Dsa

c. (12)

For a givenPtt the lower bound on the localization time,TminCF

,denoted asT low

CF, can now be obtained by solving

Ptt =

∫ T lowCF

0

fTCF|u=0

(z)dz. (13)

In contrast, in the worst case, all the packets betweenanchors are lost, and the requesting sensor node is located atits farthest distance to the initiating anchor. This case yieldsthe longest localization time given by

T uppCF

= NTp + (N − 1)Daa

c+

Dsa

c+

Dsa

c, (14)

which is equivalent to packet transmission based on time divi-sion multiple access (TDMA) with time-slot durationTp +

Dc

(assumingD = Dsa= Daa).Another figure of merit is the probability with which a

node can localize itself. If this probability is required tobeabove a design valuePss, the necessary number of anchors isdetermined as the smallestN for which

P locCF

=N∑

k=K

(Nk

)

pkCF(1− pCF)

N−k ≥ Pss (15)

wherepCF is the probability that a transmitted packet reachesa sensor node correctly, and it can be calculated as

pCF = (1 − pl)

∫ ∞

γ0N0B

fX0(x)dx, (16)

where N0B is the noise power,γ0 is the minimum SNRat which a received packet can be detected at the receiver,and fX0(x) is the pdf of the received signal power whichwill be derived in the next subsection. Note that in one-hopcommunications, the transmission power will be set in sucha way that for any distance, in the collision-free scheme, theSNR is greater thanγ0, i.e., pCF = (1− pl).

H 5

It is worth mentioning that instead of increasing the num-ber of anchors, in a mobile scenario one can repeat packettransmissions fromK anchors multiple times. That wouldchange (5) and the pdf of the localization time (7) to someextent; however, this approach is not considered in the presentanalysis.

B. Collision-tolerant packet scheduling

To avoid the need for coordination among anchor nodes,in a collision-tolerant packet scheduling, anchors work in-dependently of each other. During a localization period orupon receiving a request from a sensor node, they transmitrandomly, e.g. according to a Poisson distribution with anaverage transmission rate ofλ packets per second. Packetstransmitted from different anchors may now collide at a sensornode, and the question arises as to what is the probability ofsuccessful reception. This problem is a mirror image of theone investigated in [23] where sensor nodes transmit theirpackets to a common fusion center. Unlike [23] however,where the sensors know their location, and power control fullycompensates for the known path-loss, path-loss is not knownin the present scenario, and there is no power control. Theaverage received signal strength is thus different for differentlinks (this signal strength, along with a given fading model,determines the probability of packet loss). In this regard,thesignal received at them-th sensor node from thej-th anchoris

vm,j (t) = cm,jvj(t) + im(t) +wm(t), (17)

wherevj(t) is the signal transmitted from thej-th anchor,cm,j

is the channel gain,wm(t) is the additive white Gaussian noisewith powerN0B, andim(t) is the interference caused by otheranchors whose packets overlap with the desired packet,

im(t) =∑

k 6=j

cm,kvk(t − τk), (18)

whereτk is the difference in the arrival times of the interferingsignals w.r.t. the desired signal, and it is modeled as anexponentially distributed random variable. The SNR at thereceiver depends on the interference level, and is given by

γ =X0

I0 +N0B, (19)

whereX0 = |cm,j |2P0 is the power of the signal of interest

with P0 the anchor’s transmit power, and whereI0 is the totalinterference power which can be expressed as

I0 =

q∑

i=1

|cm,ki|2P0 (20)

with q the number of interferers, andki the index of thei-thinterferer. Using a simple path-loss model we can formulatethe attenuation of the signal power as

|cm,j |2 = α0 (d0/dm,j)n0 , (21)

whereα0 is a constant andd0 is the reference distance. Using(21), the pdf of the received signal power of the desired signalis

fX0(x) =

d0

n0

(P0α0)1

n0

(1

x

) 1n0

+1

gd

(

d0(P0α0

x)

1n0

)

, (22)

and the pdf of the interference can be obtained as

fI0 (x) = fX0(x) ∗ fX0

(x) ∗ . . . ∗ fX0(x)

︸ ︷︷ ︸

q times

. (23)

The probability that a packet is received correctly by a sensornode is then [23]

ps = (1− pl)

N−1∑

q=0

P (q)ps|q, (24)

where P (q) =(2NλTp)

q

q! e−2NλTp is the probability thatqpackets interfere with the desired packet, andps|q is the prob-ability that the desired packet “survives” under this condition,

ps|q ={∫∞

γ0N0BfX0

(x)dx q = 0∫∞γ0

∫∞N0B

fX0(γw)fI (w −N0B)wdwdγ q ≥ 1

(25)

wherew = I0 +N0B.In addition, it should be noted that redundant successfully

received packets from an anchor are not useful for localization.However, they may be used to reduce the effects of noise onthe range estimation (see Section IV), or in mobile scenarioswhere the anchors are moving they can be used for rangetracking [24]. However, we will not consider the mobile casein this paper.

The probability of receiving a useful packet from an anchorduring transmission timeTT can now be approximated by [23]

pCT = 1− e−psλTT , (26)

and the probability that a sensor node accomplishes self-localization duringTT seconds usingN anchors can be ob-tained as

P locCT

=N∑

k=K

(Nk

)

pkCT(1− pCT)

N−k , (27)

which is equivalent to the probability that a node receives atleastK different localization packets duringTT seconds.

It can be shown thatP locCT

is an increasing function ofTT

(see Appendix A), and as a result for any value ofpsλ 6= 0,there is aTT that leads to the probability of self-localizationequal or greater thanPss. The minimum value for the requiredTT can be obtained at a point wherepsλ is maximum (λopt).It can be proven that the lower bound ofλopt is λlow

opt =1

2NTp,

and its upper bound isN+12NTp

(see Appendix B).Given the number of anchorsN , and a desired probability

of successful self-localizationPss, one can determinepCT from(27), andλ and the minimum localization time jointly from(24) and (26). Similarly to the collision-free scheme, we thenadd the time of requestds

c, and the maximum propagation

delay between anchor-sensor pairDsac

to the (minimum)TT thatis obtained from (24) and (26). This value is then consideredasthe (minimum) localization time (Tmin

CT) TCT, for the collision-

tolerant scheme.

IV. SELF-LOCALIZATION PROCESS

As explained before, a sensor node requires at leastK

distinct packets (or time-of-flight measurements) in ordertodetermine its location. However, it may receive more thanK

different packets as well as some replicas, i.e,qj packets from

6 (DRAFT)

anchorj, wherej = 1, ..., N . In this case, it uses all of thisinformation for self-localization. Note that in the collision-freescheme,qj is either zero or one; however, in the collision-tolerant schemeqj can be more than 1. Packets received fromthe j-th anchor can be used to estimate the sensor node’sdistance to that anchor, and the redundant packets would adddiversity (or reduce measurement noise) for this estimate.Inthe next two subsections, we show how all of the correctlyreceived packets can be used in a localization algorithm, andhow the CRB of the location estimate can be obtained for theproposed scheduling schemes.

A. Localization algorithm

After the anchors transmit their localization packets, eachsensor node hasQ measurements. Each measurement is con-taminated by noise whose power is related to the distancebetween the sensor and the anchor from which the measure-ment has been obtained. Thel-th measurement obtained fromthej-th anchor is related to the sensor’s position,x, as (sensornode index is omitted for simplicity)

tl = f(x) + nl, (28)

wherenl is the noise andf(x) is

f(x) =1

c‖x− xj‖2 (29)

where xj is the j-th anchor’s position. Stacking all themeasurements gives us aQ × 1 vector t. The number ofmeasurements can be formulated as

Q =N∑

j=1

qj , (30)

whereqj is the number of measurements which are obtainedcorrectly from thej-th anchor. In CFS,qj is a Bernoullirandom variable with success probabilitypCF, in CTS qj isa Poisson random variable with distribution

Pnj = P (qj = n) =

(psλTT )n

n!e−psλTT . (31)

Since the measurement errors are independent of each other,the maximum likelihood solution forx is given by

x = argminx

∥∥t− f(x)

∥∥2, (32)

which can be calculated using a method such as the Gauss-Newton algorithm specified in Algorithm 1. In this al-

Algorithm 1 Gauss-Newton Algorithm

Start with an initial location guess.Set i = 1 andE = ∞.while i ≤ I andE ≥ ǫ do

Next state:x(i+1) = x

(i)−η(

∇f(x(i))T∇f(x(i)))−1

∇f(x(i))T(

f(x(i))− t)

E = ||x(i+1) − x(i)||

i = i+ 1end whilex = x

(i)

gorithm, η controls the convergence speed,∇f(x(i)) =

[

∂f1∂x

, ∂f2∂x

, . . . , ∂fQ∂x

]T

x=x(i)

represents the gradient of the vec-

tor f w.r.t. the variablex at x(i), x(i) is the estimate in the

i-th iteration, and∂fl∂x

=[

∂fl∂x

, ∂fl∂y

, ∂fl∂z

]T

wherel = 1, . . . ,Q.Here,I and ǫ are the user-defined limits on the stopping cri-terion that determines when the algorithm exits the loop. Theinitial guess is also an important factor for the algorithm.Onemay obtain the initial guess through geometrical properties ofa triangulation, similarly as explained in [25].

B. Cramer-Rao bound

The Cramer-Rao bound is a lower bound on the varianceof any unbiased estimator of a deterministic parameter. In thissubsection, we derive the CRB for the location estimate of asensor node.

In order to find the CRB, the Fisher information matrix(FIM) has to be calculated. The Fisher information is ameasure of information that an observable random variablet carries about an unknown parameterx upon which the pdfof t depends. The elements of the FIM are defined as

I(x)i,j = −E

[∂2

∂xi∂xj

log h(t;x)|x]

(33)

wherex is the location of the sensor node,h(t;x) is the pdfof the measurements parametrized by the value ofx, and theexpected value is over the cases where the sensor is localizable.

In a situation where the measurements (ToFs or RTTsbetween a sensor node and the anchors) are contaminatedwith Gaussian noise (whose power is related to the SNR orequivalently to the mutual distance between a sensor-anchorpair), the elements of the FIM can be formulated as

I(x)i,j =1

P loc

QN∑

qN=0

. . .

Q2∑

q2=0

Q1∑

q1=0

s.t.{q1,...,qN} enable self-localization{

∂f

∂xi

T

R−1w

∂f

∂xj

+1

2tr

[

R−1w

∂Rw

∂xi

R−1w

∂Rw

∂xj

]}

ΠNj=1P

qjj

(34)

whereP loc is the localization probability,Qi = 1 for CFS,and∞ for CTS,Rw is theQ×Q noise covariance matrix

∂Rw

∂xi

= diag

(∂[Rw]11

∂xi

,∂[Rw]22

∂xi

, ...,∂[Rw]QQ

∂xi

)

, (35)

and∂f

∂xi

=

[∂f1

∂xi

,∂f2

∂xi

, ...,∂fQ

∂xi

]T

, (36)

with fi a ToF (or RTT) measurement.Once the FIM has been computed, the lower bound on the

variance of the estimation error can be expressed as CRB=∑3

i=1 CRBxiwhere CRBxi

is the variance of the estimationerror in thei-th variable and it is defined as

CRBxi=

[I−1(x)

]

ii. (37)

Note that the CRB is meaningful if the node is localizable( 1P loc in (34)), meaning that a sensor node has at leastK dif-

ferent measurements. Hence, only∑N

k=K

(

Nk

)

possible stateshave to be considered in order to calculate (34) for collision-free scheduling, while the number of states is countless for

H 7

collision-tolerant scheduling. Nonetheless, it can be shown thatthe number of possible states in CTS can be dropped to thatof CFS (see Appendix C).

V. ENERGY CONSUMPTION

In this section, we consider the average energy consumedduring localization. In CFS, the receiver of anchorj is on fortj seconds. If the power consumption in listening mode isPL ,and in transmitting modeP0, the total energy consumption inCFS can be formulated as

ECF = NTpP0 +

N∑

j=1

tjPL , (38)

where the processing energy consumption has been ignored.The receiving time is a random variable and can be formulatedas

tj =1

c(1j − uj)

Tdj +

Daa

cuj1j , for j = 2, ...,N (39)

whereuj (1j) is a (j − 1) × 1 vector whose elements arethe firstj − 1 elements ofu (1). Note that in our localizationprocedurest1 = 0, because the first anchor does not listento the channel in periodic localization (M1), and the timethat it receives a request in on-demand localization (M2) isnegligible. The average time that the receiver of each anchoris on can be calculated as

tavgj =

j − 1

c[(1− pl)davg+ plDaa], (40)

which results inEavg

CF=NTpP0+

PL

[

(1 − pl)davg

c+ pl

Daa

c

]N(N − 1)

2,

(41)

whereEavgCF

is the average energy consumption by CFS duringeach localization procedure. As is clear from (40), an anchorwith a higher index value consumes more energy in com-parison with the one that has a lower index. To overcomethis problem, anchors can swap indices in each localizationprocedure.

In CTS, the anchors do not need to listen to the channel andthey only transmit at an average rate ofλ packets per second.The average energy consumption is thus

EavgCT

= λTTNTpP0. (42)

For small ratiosPLP0

, the average energy consumption of CTS isalways greater than that of CFS. However, asλ gets smaller(or equivalentlyTCT get larger), the energy consumption byCTS reduces.

VI. N UMERICAL RESULTS

To illustrate the results, a two-dimensional rectangular-shape operating area with lengthDx and widthDy is con-sidered with uniformly distributed anchors and sensors. Thereis no difference in how the anchors and sensor nodes aredistributed, and therefore we havefD(d) = gD(d) which can

be obtained as (see Appendix D)

fD(d) = (43)2d

D2xD

2y

[

d2(sin2 θe − sin2 θs) + 2DxDy(θe − θs)

+ 2Dxd(cos θe − cos θs)− 2Dyd(sin θe − sin θs)]

whereθs andθe are related tod as given in Table I.

TABLE I: Values ofθs andθe based on distanced.

distance θs θe0 ≤ d ≤ Dy 0 π

2

Dy ≤ d ≤ Dx 0 sin−1 Dy

d

Dy ≤ d ≤√

D2x +D2

y cos−1 Dx

dsin−1 Dy

d

The parameter values for the numerical results are listed inTable II, and for all numerical results, we use these valuesunless otherwise stated.

The number of bits in each packet is set tobp = 200which is sufficient for the position information of each anchor,time of transmission, (arrival time of the request packet),andthe training sequence. Assuming QPSK modulation (bs = 2),guard timeTg = 50ms, and a bandwidth ofB = 2kHz thelocalization packet length isTp = 100ms (see (4)). In addition,kE is set to10−8 for the sake of simulation. In theory it canacquire much smaller values.

Fig. 4 shows the probability of successful self-localizationin the collision-tolerant scheme as a function ofλ and theindicated value forTCT. It can be observed that there is anoptimal value ofλ (denoted byλopt) which corresponds to theminimal value ofTCT (Tmin

CT) which satisfiesP loc

CT≥ Pss. The

highlighted area in Fig. 4 shows the predicted region (obtainedin Appendix B) whereλopt is. As it can be seen,λopt is closeto λlow

opt , and it gets closer to this value asPs|q>0 gets smaller.In addition, for the values ofTCT greater thanTmin

CT, a range of

values forλ ∈ [λlow, λupp] can attain the desired probability ofself-localization. In this case, the lowest value forλ should beselected to minimize the transmission energy consumption.

Fig. 5 shows the probability of correct packet receptionversus the number of interferers (the effect of packet loss dueto fading is not included in the figure, and the desiredPss is

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

λlowopt

TCT

= 5.6s

TCT

= 6.1s

TCT

= 6.9s

TminCT

= 8.1sTCT

= 18s

TCT

= 12s

TCT

= 10s

Pss

λlow

λupp

λopt

Probability of localization vs. λ and Tloc

, Pss

= 0.90

Rate of packet transmission per second, λ

Plo

c

Fig. 4: Probability of successful localization for different values ofλ andTCT.

8 (DRAFT)

TABLE II: Simulation parameters. Note that, in this table some parameterssuch asN , Daa, Tg , etc. are related to other parameters, e.g.,N depends onthe values of thepl, andPss.

Description Parameter Value Unit

Number of anchor nodes N 5 -Number of sensor nodes M 100 -Sound speed c 1500 m/sNumber of required different packets K 3 -Area size inx-axis Dx 3c mArea size iny-axis Dy 3c mmaximum anchor-anchor distance Daa 3c

√2 m

maximum anchor-sensor distance Dsa 3c√2 m

Guard time for localization packet Tg 50 msNumber of bits per sample bs 2 -Number of bits per packet bp 200 -System bandwidth B 2 kHzLocalization packet length Tp 100 msPacket loss probability pl 0.1 -Noise power N0B −47.5 dBToF noise power coefficient kE 10−8

Transmit power P0 15 wReference distance d0 1 mPower coefficient α0 1 mPath-loss exponent n0 1.4 -Required SNR for packet detection γ0 6 dBRequest packet arrival delay ds/c 0 sRequired probability of successful

localization Pss 0.99 -Required probability that all packet are

transmitted beforeTminCF

in CFS Ptt 0.90 -

set to0.90 in this example) for different values of the path-loss exponentn0. As it was mentioned before, when there isno interference, the probability of packet reception is 1. Yet,when there is an interferer, the chance of correct receptionofa packet becomes very small (0.06 for n0 = 1.4), and as thenumber of interferers grows, it gets smaller.

The probability that two or more packets overlap witheach other is also depicted in part (b) of this figure for thethree values ofλ shown in Fig. 4. It can be seen that asthe value ofλ is reduced fromλopt (which is equivalent toa largerTCT), the probability of collision gets smaller. Thisincreases the chance of correct packet reception, and reducesthe energy consumption as explained in Section V. In addition,it can be observed that although usingλupp results in thesame performance asλlow, it relies on the packets that havesurvived collisions, which is not energy-efficient in practicalsituations neither for anchors (required energy for multiplepacket transmissions) nor for sensor nodes (processing energyneeded for packet detection).

Fig. 6 shows the minimum required time for localizationversus the probability of packet loss. Packet loss is a phe-nomenon that is common in underwater acoustic systemsbecause of many reasons such as location-dependent fading,shadowing, noise, and so on. Aspl increases, more anchors arerequired for collision-free localization. In Fig. 6, for a givenpl, the number of anchorsN is calculated using (15), whichis then used to calculate the minimum required time for thecollision-free and collision-tolerant localization. Each increasein T upp

CFin CFS indicates that the number of anchors has been

increased by one. We also note that for a given number ofanchors, the lower and upper bounds of the collision-freealgorithm are constant over a range ofpl values because

0 1 2 3 40.00

0.05

0.10

1.00

a) Number of interferes

ps|

q, n

o pa

cket

loss

is a

ssum

ed

ps|q

for different values of path−loss exponent

← 6

.4%

← 4

.5%

← 3

.4%

← 2

.8%

100%

n0 = 2.0

n0= 1.4

n0 = 1.0

0 1 2 3 40

0.2

0.4

0.6

0.8

1

b) Number of interfereing packets, q

P(q

) =

q, P

orba

bilit

y th

at q

pac

kets

c

ollid

e w

ith th

e de

sire

d pa

cket

← 1

1.2%

← 3

6.4%

← 3

.4%

← 0

.7%

← 2

0.9%

← 8

.6%

← 0

% ← 8

%←

14.

2%

← 0

%←

2.3

% ← 1

7.6%

88%

← 3

1.7%

← 0

.7%

λ

low

λopt

λupp

Fig. 5: a) Probability of successful packet reception versus different numberof interferers. b) Probability thatq interferers collide with the desired packet.For this figure,λlow, λopt andλupp are chosen from Fig. 4.

5

10

15

20

25

30

10−3↔ 10−2.41

Loca

lizat

ion

time

in s

econ

dN = 3

TuppCF

TminCF

TavgCF

TlowCF

TminCT

10−2.41↔ 10−1.38

N = 4

10−1.37↔ 10−0.98

N = 5

10−0.98↔ 10−0.76

N = 6

Probablity of packet loss, pl

Fig. 6: Effect of link-loss probability on the minimum required time forlocalization. The greater the value ofpl is, the more anchors are required inthe collision-free protocols.

they are not affected by that; however, the actual performanceof both schemes becomes worse aspl gets larger. Still, thecollision-tolerant approach performs better for a wide rangeof pl, and as the number of anchors increases its performanceslightly changes for high values ofN . Therefore, it can be usedin a system with limited anchors, and can be implemented inpractice with low computational complexity since the anchorswork independently of each other.

Many factors such as noise power or packet length aredirectly dependent on the operating frequency and systembandwidth. Assuming single-hop communication among thesensor nodes, an optimum frequency band exists for a givenoperating area. As the size of the operating area increases,a lower operating frequency (with less bandwidth) is used tocompensate for the increased attenuation. Furthermore, asthedistance increases the amount of available bandwidth for theoptimum operating frequency also gets smaller [26]. As it was

H 9

0.05 0.1 0.15 0.2 0.25 0.3 0.358

10

12

14

16

18

20

22

24

Packet length in second, including guard−band

Req

uire

d lo

caliz

atin

tim

e to

mee

t Plo

c>P

ss

Required localization time vs. packet length

TuppCF

TminCF

TavgCF

TlowCF

TminCT

Fig. 7:Effect of packet length on the minimum required time for localization.

0.5 1 1.5 2 2.5 3 3.50

5

10

15

20

Dimention of the operating area (normalized by c), Dx/c=D

y/c

Req

uire

d lo

caliz

atin

tim

e to

mee

t Plo

c>P

ss

Required localization time vs. packet length

TuppCF

TminCF

TavgCF

TlowCF

TminCT

Fig. 8: Effect of the operating area size on the required localization time.

mentioned before, the localization packet is usually shortinterms of the number of bits, but its duration (in seconds) stilldepends on the system bandwidth. In this part, we investigatethe effect of packet length (or equivalently system bandwidth)on the localization time.

As it is shown in Fig. 7, the length of the localization packetplays a significant role in the collision-tolerant algorithm. Theminimum localization time grows almost linearly w.r.t.Tp inall cases; however, the rate of growth is much higher for thecollision-tolerant system than for the collision-free one. Atthe same time, as shown in Fig. 8, the size of the operatingarea has a major influence on the performance of the CFS,while that of the CTS does not change very much. It canbe deduced that in a network where the ratio of the packetlength to the maximum propagation delay is low, the collision-tolerant algorithm outperforms the collision-free one.

The localization accuracy is related to the noise level atwhich a ToF measurement is taken, and to the anchors’constellation. If a sensor node in a 2D operating systemreceives packets from the anchors which are (approximately)located on a line, the sensor node is unable to localize itself(or it experiences a large error). To evaluate the localizationaccuracy of each algorithm, we consideredM = 50 sensornodes, and run a Monte Carlo simulation (103 runs) to extractthe results. The number of iterations in Algorithm 1 is set toI = 50, and the convergence rate isη = 1

5 . The TCF was setequal to the average localization time of CFS. In this specialcase whereTmin

CFis lower thanT avg

CT, the successful localization

probability (P loc) of CTS would be better than that of CFS.

The probability distribution of the localization error‖x− x‖is illustrated in Fig. 9 for both schemes. In this figure, theroot mean square error (RMSE), and root CRB (R-CRB) arealso shown with the dashed and dash-dotted lines, respectively.It can be observed that in CTS the pdf is concentrated atlower values of the localization error in comparison withCFS, because each sensor in CTS has a chance of receivingmultiple copies of the same packet, and that reduces the rangeestimation error.

0 20 40 60 80 1000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Localizaion error ‖x− x‖ in meter

Pdf

of l

ocal

izat

ion

erro

r

TlocCT

= TavgCF

= 12.07s, σavgd

= 34.3m

CF coverage = 0.9914

CT coverage = 0.9996

CF Err PdfCT Err PdfCF RMSECT RMSECF R−CRBCT R−CRB

Fig. 9:Probability distribution of the localization error, and its correspondingCRB for CTS and CFS.

Measurement noise plays a major role in the localizationaccuracy. For a fixed signal bandwidth, the accuracy of rangeestimation is only a function of the SNR. Since the distancebetween the nodes is a random variable, the SNR is alsorandom. In Fig. 10, we change the ToF measurement noisepower (or equivalently the transmit power) to adjust thelevel of the ranging error at the average distance defined as

σavgd = c

(

kEdn0avg

)12 . As it can be anticipated from the theory

of CRB, for low ToF noise power, the RMSE approaches itsCRB, while for high noise power, it deviates from the CRB.

10−1

100

101

102

0

5

10

15

20

25

Standard deviation of the ranging error at average distance, σavgd

RM

SE

and

CR

B o

f loc

aliz

atio

n

Localization error and its CRB for CFS and CTS

CF RMSECT RMSECF R−CRBCT R−CRB

Fig. 10:CRB of the localization estimate for each packet schedulingschemeversusσavg

d.

In order to compare the total average energy consumptionof the two schemes, the transmit and listening power valuesare selected from an actual underwater acoustic modem, theEvologics S2CR 12/24 [27] as shown in Table II. Usingequations (41) and (42), the average energy consumed by CFSand CTS is 30.14w and 12.72w, respectively. This indicatesthe higher energy consumption by CTS.

10 (DRAFT)

VII. C ONCLUSION

We have considered two classes of packet scheduling forself-localization in an underwater acoustic sensor network,one based on a collision-free design and another based on acollision-tolerant design. In collision-free packet scheduling,the time of the packet transmission from each anchor is set insuch a way that none of the sensor nodes experiences a colli-sion. In contrast, collision-tolerant algorithms are designed soas to control the probability of collision to ensure successfullocalization with a pre-specified reliability. We have alsopro-posed a simple Gauss-Newton based localization algorithm forthese schemes, and derived their Cramer-Rao lower bounds.The performance of the two classes of algorithms was shownto be comparable in terms of the time required for localization.When the ratio of the packet length to the maximum propaga-tion delay is very low, the collision-tolerant protocol requiresless time for localization in comparison with the collision-freeone for the same probability of successful localization. Otherthan average energy consumption by anchors, the collision-free scheme has other advantages with the major advantagebeing its simplicity of implementation with no requirementson synchronization. For a practical (non-zero) packet lossrate, collision-tolerant scheduling takes less time to localizea sensor node. In addition, the anchors work independently ofeach other, and as a result the scheme is spatially scalable,withno need for a fusion center. Finally, its localization accuracyis always better than that of the collision-tolerant schemedue to the multiple receptions of the desired packets fromanchors. These features make the collision-tolerant localizationscheme appealing for a practical implementation. In the future,we will extend this work to a multi-hop network where thecommunication range of the acoustic modems is much shorterthan the size of the operating area.

APPENDIX AP LOC

CTIS AN INCREASING FUNCTION OFTCT

In this appendix, we show that the probability of successfullocalization is an increasing function of the localizationtime.According to (26), and the fact thatpsλ is independent ofTT,it is clear thatpCT is an increasing function ofTT. Therefore,P loc

CTis an increasing function ofTT, if P loc

CTis an increasing

function of pCT. The derivative ofP locCT

w.r.t. thepCT is

∂P locCT

∂pCT

=N∑

k=K

(Nk

)

(k −NpCT)pk−1

CT(1− pCT)

N−k−1. (44)

With a simple modification we have

∂P locCT

∂pCT

=1

pCT(1 − pCT)

{

[N∑

k=0

(Nk

)

kpkCT(1− pCT)

N−k −K−1∑

k=0

(Nk

)

kpkCT(1− pCT)

N−k

]

−

NpCT

[N∑

k=0

(Nk

)

pkCT(1 − pCT)

N−k −K−1∑

k=0

(Nk

)

pkCT(1− pCT)

N−k

]}

.

(45)

Using the properties of binomial random variables we havethat

N∑

k=0

(Nk

)

kpkCT(1 − pCT)

N−k = NpCT, (46)

andN∑

k=0

(Nk

)

pkCT(1− pCT)

N−k = 1. (47)

Now, equation (45) (or equivalently (44)) is equal to

∂P locCT

∂pCT

=

K−1∑

k=0

(Nk

)

(NpCT − k)pk−1

CT(1 − pCT)

N−k−1. (48)

It can be observed that (44) is always positive forpCT <KN

, and (48) is always positive forpCT > KN

, and as a result∂P loc

CT∂pCT

is positive for any value ofpCT; consequently,P locCT

is anincreasing function ofpCT, and consequentlyTT.

APPENDIX BMAXIMUM VALUE OF psλ

The first and second derivatives ofpsλ w.r.t. λ can beobtained as

∂psλ

∂λ=

N∑

q=0

ps|qxqe−x

q!(q − x+ 1), (49)

(∂psλ)2

∂2λ=

N∑

q=0

ps|qxq−1e−x

q![(q − x)(q − x+ 1) − x], (50)

wherex = 2NλTp. It can be observed that forx < 1 thederivative in (49) is positive, and forx > N +1 it is negative.Therefore,psλ has at least one maximum withinx ∈ [1, N +1]. In practical scenarios the value ofps|q for k > 0 is usuallysmall, so that it can be approximated by zero. For a specialcase whereps|q>0 = 0, (49) is zero ifx = 1, and (50) isnegative, and as a resultλlow

opt =1

2NTpmaximizesP loc

CT. This

corresponds to a lower bound on the optimal point in a generalproblem (i.e.,ps|q>0 6= 0).

APPENDIX CCRAMER RAO LOWER BOUND FORCTS

As stated before, the upper bound on the sum operationin (34) for CTS is∞, and this makes the CRB calculationvery difficult even if it is implemented numerically. In orderto reduce the complexity of the problem, the observationof a sensor node from thej-th anchor is divided into twoparts: Either a sensor node does not receive any packet fromthis anchor (no information is obtained), or it receives oneor more packets. Since the anchor and the sensor node donot move very much during the localization procedure, theirdistance can be assumed almost constant, and therefore thenoise power is the same for all measurements obtained froman anchor. When a sensor node gathers multiple measurementscontaminated with independent noise with the same power(diagonal covariance matrix), the calculation of the CRB canbe done with less complexity. We will explain this complexityreduction for the first anchor, and then we generalize this ideafor the other anchors too. Considering the first anchor, eachelement of the FIM can be calculated in two parts; no correctpacket reception from the first anchor, and one or more thanone correct packet reception from this anchor which can beformulated as

I(x)i,j = P 0I(x|q1 = 0)i,j + P>0

I(x|q1 > 0)i,j , (51)

H 11

whereP 0 = P 01 is the probability that no packet is received

from an anchor which is same for all anchors, andP>0 =P>01 =

∑∞q1=1 P

k1 is the probability that one or more than

one packets are received from an anchor which is also samefor all anchors. The second term in (51) can be expanded as

I(x|q1 > 0)i,j =1

P loc

QN∑

qN=0

. . .

Q2∑

q2=0

s.t.{q1,...,qN} enable self-localization{

1σ−21

∂f1

∂xi

∂f1

∂xj

+ c1 + 1σ−41

∂σ21

∂xi

∂σ21

∂xj

+ c2

}

P 11 /P

>01 ΠN

j=2Pqjj +

{

2σ−21

∂f1

∂xi

∂f1

∂xj

+ c1 + 2σ−41

∂σ21

∂xi

∂σ21

∂xj

+ c2

}

P 21 /P

>01 ΠN

j=2Pqjj +

...{

kσ−21

∂f1

∂xi

∂f1

∂xj

+ c1 + kσ−41

∂σ21

∂xi

∂σ21

∂xj

+ c2

}

P k1 /P

>01 ΠN

j=2Pqjj +

...(52)

wherec1 andc2 are only affected by measurements from theother anchors. Using a simple factorization we have

I(x|q1 > 0)i,j =1

P loc

QN∑

qN=0

. . .

Q2∑

q2=0

s.t.{q1,...,qN} enable self-localization{

gCT

[

σ−21

∂f1

∂xi

∂f1

∂xj

+ σ−41

∂σ21

∂xi

∂σ21

∂xj

]

+ c1 + c2

}

ΠNj=2P

qjj

(53)

where

gCT =

∑∞q1=1 kP

k1

∑∞q1=1 P

k1

=psλTT1− P 0

(54)

can be calculated analytically and be used in (53). That enablesus to calculate only two possible states for the sum overq1.Now, we defineaN×1 where itsk-th elementak is either zero(if qk = 0) or 1 (if qk > 0). We also definebN×1 with its k-thelementbk =

[

σ−2k

∂fk∂xi

∂fk∂xj

+ σ−4k

∂σ2k

∂xi

∂σ2k

∂xj

]

. Then, we have

I(x|a)i,j = gCT

1

P locaTb(P 0

)N−na(1− P 0

)na . (55)

where na is the number of non-zero elements ina. Thismeans that to evaluateI(x)i,j for the localizable scenarios

only(

NK

)

possible states (different realizations ofa whichlead to localizable scenarios) have to be considered which isthe same as that of CFS.

APPENDIX DDISTRIBUTION OF THE MUTUAL DISTANCE

In this appendix, we derive the pdf of the distance betweentwo nodes located uniformly at random in a rectangular regionas shown in Fig. 11. Under this condition the pdfs of thexandy projections of the distance are

f∆X(∆x) =

2

D2x

(Dx −∆x), 0 ≤ ∆x ≤ Dx (56a)

f∆Y(∆y) =

2

D2y

(Dy −∆y), 0 ≤ ∆y ≤ Dy , (56b)

and since they are independent, the joint pdf in polar coordi-nates (see Fig. 12) is

fD,Θ(d, θ) =4d

D2xD

2y

(Dx − d cos θ)(Dy − d sin θ). (57)

Fig. 11: Two randomly located nodes in a rectangular operating area.

Fig. 12: Illustration of the parameters and their relations to each other incalculating the pdf of the distance between two nodes located uniformly atrandom.

By taking an integral overθ, the pdf of the distance follows(43). This pdf is shown in Fig. 13.

Dy Dxdm

R1 R2 R3

dm =2(Dx+Dy)

3 −

√

4(Dx+Dy)2−3DxDyπ

3

approximatelylinear region

Fig. 13:Probability density function of the distance between two uniformlyrandomly located nodes.dm is the point at which the maximum of the pdfoccurs.

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[2] L. Paull, S. Saeedi, M. Seto, and H. Li, “AUV navigation and localiza-tion: A review,” IEEE Journal of Oceanic Engineering, vol. 39, pp. 131– 149, 2013.

[3] S. Chatzicristofis, A. Kapoutsis, E. Kosmatopoulos, L. Doitsidis,D. Rovas, and J. Sousa, “The NOPTILUS project: Autonomous multi-AUV navigation for exploration of unknown environments,” in IFACWorkshop on Navigation, Guidance and Control of UnderwaterVehicles(NGCUV2012), vol. 3, 2012, pp. 373–380.

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[7] H. Jamali-Rad, H. Ramezani, and G. Leus, “Sparsity-aware multi-sourceRSS localization,”Elsevier Signal Processing, vol. 101, pp. 174–191,2014.

[8] P. Kuakowski, J. Vales-Alonso, E. Egea-Lpez, W. Ludwin,and J. Garca-Haro, “Angle-of-arrival localization based on antenna arrays for wirelesssensor networks,”Computers and Electrical Engineering, vol. 36, no. 6,pp. 1181 – 1186, 2010.

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[18] ——, “DMC-MAC: Dynamic multi-channel MAC in underwateracous-tic networks,” accepted in the European Signal Processing Conference(EUSIPCO’13), Marrakech, Morocco,, 2013.

[19] P. Carroll, S. Zhou, K. Mahmood, H. Zhou, X. Xu, and J.-H.Cui, “On-demand asynchronous localization for underwater sensor networks,” inOceans Conference, 2012.

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[24] H. Ramezani, H. Jamali-Rad, and G. Leus, “Target localization andtracking for an isogradient sound speed profile,”IEEE TRANSACTIONSON SIGNAL PROCESSING, vol. 61, no. 6, pp. 1434–1446, March 2013.

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