Page 1

Data Fusion Improves the Coverage of Wireless Sensor

Networks

Guoliang Xing1; Rui Tan2; Benyuan Liu3; Jianping Wang2; Xiaohua Jia2; Chih-Wei Yi4

1Department of Computer Science & Engineering, Michigan State University, USA

2Department of Computer Science, City University of Hong Kong, HKSAR

3Department of Computer Science, University of Massachusetts Lowell, USA

4Department of Computer Science, National Chiao Tung University, Taiwan

glxing@msu.edu; {tanrui2@student., jianwang@, csjia@}cityu.edu.hk;

bliu@cs.uml.edu; yi@cs.nctu.edu.tw

ABSTRACT

Wireless sensor networks (WSNs) have been increasingly

available for critical applications such as security surveil-

lance and environmental monitoring.

formance measure of such applications is sensing coverage

that characterizes how well a sensing field is monitored by

a network. Although advanced collaborative signal process-

ing algorithms have been adopted by many existing WSNs,

most previous analytical studies on sensing coverage are con-

ducted based on overly simplistic sensing models (e.g., the

disc model) that do not capture the stochastic nature of sens-

ing. In this paper, we attempt to bridge this gap by explor-

ing the fundamental limits of coverage based on stochastic

data fusion models that fuse noisy measurements of multi-

ple sensors. We derive the scaling laws between coverage,

network density, and signal-to-noise ratio (SNR). We show

that data fusion can significantly improve sensing coverage

by exploiting the collaboration among sensors. In particu-

lar, for signal path loss exponent of k (typically between 2.0

and 5.0), ρf = O(ρ1−1/k

ties of uniformly deployed sensors that achieve full coverage

under the fusion and disc models, respectively. Our results

help understand the limitations of the previous analytical re-

sults based on the disc model and provide key insights into

the design of WSNs that adopt data fusion algorithms. Our

analyses are verified through extensive simulations based on

both synthetic data sets and data traces collected in a real

deployment for vehicle detection.

An important per-

d

), where ρf and ρd are the densi-

Categories and Subject Descriptors

C.2.1 [Computer-Communication Networks]: Network

Architecture and Design—Network topology; G.3 [Probability

and Statistics]: Stochastic processes

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MobiCom’09, September 20–25, 2009, Beijing, China.

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General Terms

Performance, Theory

Keywords

Data fusion, target detection, coverage, performance limits,

wireless sensor network

1.INTRODUCTION

Recent years have witnessed the deployments of wireless

sensor networks (WSNs) for many critical applications such

as security surveillance [16], environmental monitoring [25],

and target detection/tracking [21]. Many of these applica-

tions involve a large number of sensors distributed in a vast

geographical area. As a result, the cost of deploying these

networks into the physical environment is high. A key chal-

lenge is thus to predict and understand the expected sensing

performance of these WSNs. A fundamental performance

measure of WSNs is sensing coverage that characterizes how

well a sensing field is monitored by a network. Many recent

studies are focused on analyzing the coverage performance

of large-scale WSNs [4,19,24,33,38,41,43].

Despite the significant progress, a key challenge faced by

the research on sensing coverage is the obvious discrepancy

between the advanced information processing schemes adopted

by existing sensor networks and the overly simplistic sens-

ing models widely assumed in the previous analytical stud-

ies. On the one hand, many WSN applications are designed

based on collaborative signal processing algorithms that im-

prove the sensing performance of a network by jointly pro-

cessing the noisy measurements of multiple sensors. In prac-

tice, various stochastic data fusion schemes have been em-

ployed by sensor network systems for event monitoring, de-

tection, localization, and classification [8, 10, 11, 16, 20, 21,

29, 34].On the other hand, collaborative signal process-

ing algorithms such as data fusion often have complex com-

plications to the network-level sensing performance such as

coverage. As a result, most analytical studies1on sensing

coverage are conducted based on overly simplistic sensing

models [3,4,14,18,19,23,24,33,38,39,43]. In particular, the

1Among the total six papers on the coverage problem of

WSNs that have been published at MobiCom since 2001,

five of them adopted the disc sensing model. Similarly, the

disc model is also assumed by seven out of nine relevant

papers published at MobiHoc since 2001.

Page 2

1

0.8

0.6

0.4

0.2

200150100 500

Detection probability PD

Distance from the vehicle (meters)

t = 0.01

t = 0.05

Figure

probability vs.

tance from the vehicle.

1: Detection

the dis-

0.10

0.08

False alarm rate PF

0.06

0.04

0.02

0

0.15 0.100.050.01

Detection threshold t

Figure 2:

rate vs. detection thresh-

old.

False alarm

sensing region of a sensor is often modeled as a disc with

radius r centered at the position of the sensor, where r is

referred to as the sensing range. A sensor deterministically

detects the targets (events) within its sensing range. Al-

though such a model allows a geometric treatment to the

coverage problem, it fails to capture the stochastic nature of

sensing.

To illustrate the inaccuracy of the disc sensing model, we

plot the sensing performance of an acoustic sensor in Fig. 1

and 2 using the data traces collected from a real vehicle de-

tection experiment [1]. In the experiment, the sensor detects

moving vehicles by comparing its signal energy measurement

against a threshold (denoted by t). Fig. 1 plots the probabil-

ity that the sensor detects a vehicle (denoted by PD) versus

the distance from the vehicle. No clear cut-off boundary be-

tween successful and unsuccessful sensing of the target can

be seen in Fig. 1. Similar result is observed for the rela-

tionship between the sensor’s false alarm rate (denoted by

PF) and the detection threshold shown in Fig. 2. Note that

PF is the probability of making a positive decision when no

vehicle is present.

In this work, we develop an analytical framework to ex-

plore the fundamental limits of coverage of large-scale WSNs

based on stochastic data fusion models. To characterize the

inherent stochastic nature of sensing, we propose a new cov-

erage measure called (α,β)-coverage where α and β are the

upper and lower bounds on the system false alarm rate and

detection probability, respectively. Compared with the clas-

sical definition of coverage, (α,β)-coverage explicitly cap-

tures the performance requirements imposed by sensing ap-

plications.For instance, the full (0.05,0.9)-coverage of a

region ensures that the probability of detecting any event

occurring in the region is no lower than 90% and no more

than 5% of the network reports are false alarms.

The main focus of this paper is to investigate the fun-

damental scaling laws between coverage, network density,

and signal-to-noise ratio (SNR). To the best of our knowl-

edge, this work is the first to study the coverage performance

of large-scale WSNs based on collaborative sensing models.

Our results not only help understand the limitations of the

existing analytical results based on the disc model but also

provide key insights into designing and analyzing the large-

scale WSNs that adopt stochastic fusion algorithms. The

main contributions of this paper are as follows.

• We derive the (α,β)-coverage of random networks un-

der both data fusion and probabilistic disc models.

Based on these results, we can compute the minimum

network density that is required to achieve a desired

level of sensing coverage. Moreover, the existing ana-

lytical results based on the disc model can be naturally

extended to the context of stochastic event detection.

• We study the fundamental scaling laws of (α,β)-coverage.

Let ρd and ρf denote the minimum network densities

for achieving full coverage under the disc and fusion

models, respectively. We prove that ρf = O(2r2

where r is the radius of sensing disc and R is the fu-

sion range within which the measurements of all sen-

sors are fused2. As fusion range can be much greater

than sensing range, ρf is much smaller than ρd. Fur-

thermore, when the optimal fusion range is adopted,

ρf = O(ρ1−1/k

ponent that typically ranges from 2.0 to 5.0. In par-

ticular, when k = 2 (which typically holds for acoustic

signals), ρf = O(√ρd). This result shows that data

fusion can effectively reduce the network density com-

pared with the disc model. Furthermore, the existing

analytical results based on the disc model significantly

overestimate the network density required for achiev-

ing coverage.

R2 · ρd)

d

) where k is the signal’s path loss ex-

• We study the impact of signal-to-noise ratio (SNR) on

the network density when full coverage is required. We

prove that ρf/ρd = O(SNR2/k). This result suggests

that data fusion is more effective in reducing the den-

sity of low-SNR network deployments, while the disc

model is suitable only when the SNR is sufficiently

high.

• To verify our analyses, we conduct extensive simula-

tions based on both synthetic data sets and real data

traces collected from 20 sensors. Our simulations show

that our analytical results can accurately predict the

stochastic coverage of WSNs under a variety of realis-

tic settings.

The rest of this paper is organized as follows. Section 2

reviews related work. Section 3 introduces the background

and problem definition. We study the (α,β)-coverage under

the disc and fusion models in Section 4 and 5, respectively.

In Section 6, we investigate the impact of data fusion on

asymptotic sensing coverage. Section 7 presents simulation

results and Section 8 concludes this paper.

2.RELATED WORK

Many sensor network systems have incorporated various

data fusion schemes to improve the system performance. In

the surveillance system based on MICA2 motes [16], the

system false alarm rate is reduced by fusing the detection

decisions made by multiple sensors. In the DARPA Sen-

sIT project [1], advanced data fusion techniques have been

employed in a number of algorithms and protocols designed

for target detection [8,21], localization [20,34], and classi-

fication [10,11]. Despite the wide adoption of data fusion

in practice, the performance analysis of large-scale fusion-

based WSNs has received little attention.

There is a vast of literature on stochastic signal detec-

tion based on multi-sensor data fusion. Early works [5,37]

2We adopt the following asymptotic notation: 1) f(x) =

O(g(x)) means that g(x) is the asymptotic upper bound of

f(x); 2) f(x) = Θ(g(x)) means that g(x) is the asymptotic

tight bound of f(x).

Page 3

focus on small-scale powerful sensor networks (e.g., several

radars). Recent studies on data fusion have considered the

specific properties of WSNs such as sensors’ spatial distri-

bution [10, 11, 29] and limited sensing/communication ca-

pability [8]. However, these studies focus on analyzing the

optimal fusion strategies that maximize the system perfor-

mance of a given network. In contrast, this paper explores

the fundamental limits of sensing coverage of WSNs that

are designed based on existing data fusion strategies. Re-

cently, irregular sampling theory has been applied for re-

constructing physical fields in WSNs [30,31]. Different from

these works that focus on developing sampling schemes to

improve the quality of signal reconstruction, we aim to ana-

lyze sensors’ spatial density for achieving the required level

of coverage.

As one of the most fundamental issues in WSNs, the cov-

erage problem has attracted significant research attention.

Previous works fall into two categories, namely, coverage

maintenance algorithms/protocols and theoretical analysis

of coverage performance. These two categories are reviewed

briefly as follows, respectively.

Early work [22,26,27] quantifies sensing coverage by the

length of target’s path where the accumulative observations

of sensors are maximum or minimum [22,26,27]. However,

these works focus on devising algorithms for finding the tar-

get’s paths with certain level of coverage. Several algorithms

and protocols [41,42] are designed to maintain sensing cov-

erage using the minimum number of sensors. However, the

effectiveness of these schemes largely relies on the assump-

tion that sensors have circular sensing regions and determin-

istic sensing capability. Several recent studies [2,17,32,40]

on the coverage problem have adopted probabilistic sensing

models. The numerical results in [40] show that the coverage

of a network can be expanded by the cooperation of sensors

through data fusion. However, these studies do not quantify

the improvement of coverage due to data fusion techniques.

Different from our focus on analyzing the fundamental lim-

its of coverage in WSNs, all of these studies aim to devise

algorithms and protocols for coverage maintenance.

Theoretical studies of the coverage of large-scale WSNs

have been conducted in [4,14,18,19,23,24,33,38,43]. Most

works [18,19,23,33,38,43] focus on deriving the asymptotic

coverage of WSNs. The critical conditions for full k-coverage

(i.e., any physical point is within the sensing range of at

least k sensors) over a bounded square area [19,33,38,43] or

barrier area [18,23] are derived for various sensor deployment

strategies. The coverage of randomly deployed networks is

studied in [24]. The existing theoretical results on coverage

for both static and mobile sensors/targets are surveyed in

[4]. However, all the above theoretical studies are based on

the deterministic disc model. In this paper, we compare

our results obtained under a data fusion model against the

results from [4,24].

3.BACKGROUND AND PROBLEM DEFI-

NITION

In this section, we first describe the preliminaries of our

work, which include sensor measurement, network, and data

fusion models. We then introduce the problem definition.

3.1Sensor Measurement and Network Model

We assume that sensors perform detection by measuring

Table 1: Summary of Notation∗

Definition Symbol

S original signal energy emitted by the target

mean and variance of noise energy

peak signal-to-noise ratio (PSNR), δ = S/σ

path loss exponent

signal decay function, w(x) = Θ(x−k)

distance from the target

attenuated signal energy, si= S · w(di)

noise energy, ni∼ N(µ, σ2)

signal energy measurement, yi= si+ ni

false alarm rate / detection probability

upper / lower bound of PF / PD

hypothesis that the target is absent / present

network density

the set of sensors within fusion range of point p

the number of sensors in F(p)

upper bound of target localization error

∗The symbols with subscript i refer to the notation of sensor i.

µ, σ2

δ

k

w(·)

di

si

ni

yi

PF / PD

α / β

H0 / H1

ρ

F(p)

N(p)

ǫ

the energy of signals emitted by the target3. The energy

of most physical signals (e.g., acoustic and electromagnetic

signals) attenuates with the distance from the signal source.

Suppose sensor i is dimeters away from the target that emits

a signal of energy S. The attenuated signal energy si at the

position of sensor i is given by

si = S · w(di), (1)

where w(·) is a decreasing function satisfying w(0) = 1,

w(∞) = 0, and w(x) = Θ(x−k). The w(·) is referred to as

the signal decay function. Depending on the environment,

e.g., atmosphere conditions, the signal’s path loss exponent

k typically ranges from 2.0 to 5.0 [15, 20]. We note that

the theoretical results derived in this paper do not depend

on the closed-form formula of w(·). We adopt the following

signal decay function in the simulations conducted in this

paper:

w(x) =

1

1 + xk. (2)

The sensor measurements are contaminated by additive

random noises from sensor hardware or environment. De-

pending on the hypothesis that the target is absent (H0) or

present (H1), the measurement of sensor i, denoted by yi, is

given by

H0 :

H1 :

yi = ni,

yi = si+ ni,

(3)

(4)

where ni is the energy of noise experienced by sensor i. We

assume that the noise ni at each sensor i follows the nor-

mal distribution, i.e., ni ∼ N(µ,σ2), where µ and σ2are

the mean and variance of ni, respectively. We assume that

the noises, {ni|∀i}, are spatially independent across sensors.

Therefore, the noises at sensors are independent and iden-

tically distributed (i.i.d.) Gaussian noises. In the presence

of target, the measurement of sensor i follows the normal

3Several types of sensors (e.g., acoustic sensor) only sample

signal intensity at a given sampling rate. The signal energy

can be obtained by preprocessing the time series of a given

interval, which has been commonly adopted to avoid the

transmission of raw data [8,10,11,20,34].

Page 4

distribution, i.e., yi|H1 ∼ N(si+ µ,σ2). Due to the inde-

pendence of noises, the sensors’ measurements, {yi|∀i,H1},

are spatially independent but not identically distributed as

sensors receive different signal energies from the target. We

define the peak signal-to-noise ratio (PSNR) as δ = S/σ

which quantifies the noise level. Table 1 summarizes the

notation used in this paper.

The above signal decay and additive i.i.d. Gaussian noise

models have been widely adopted in the literature of multi-

sensor signal detection [2, 5, 8, 20, 24, 27, 29, 34, 37, 40] and

also have been empirically verified [15,20]. In practice, the

parameters of these models (i.e., S, w(·), µ, and σ2) can

be estimated using training data. The normal distribution

might be an approximation to the real noise distribution in

practice. As discussed in Section 5.1, the assumption of i.i.d.

Gaussian noises can be relaxed to any i.i.d. noises.

We consider a network deployed in a vast two-dimensional

geographical region. The positions of sensors are uniformly

and independently distributed in the region. Such a deploy-

ment scenario can be modeled as a stationary two-dimensional

Poisson point process. Let ρ denote the density of the under-

lying Poisson point process. The number of sensors located

in a region A, N(A), follows the Poisson distribution with

mean of ρ||A||, i.e., N(A) ∼ Poi(ρ||A||), where ||A|| repre-

sents the area of the region A. We note that the uniform

sensor distribution has been widely adopted in the perfor-

mance analysis of large-scale WSNs [4,19,24,33,38]. There-

fore, this assumption allows us to compare our results with

previous analytical results.

3.2Data Fusion Model

Data fusion can improve the performance of detection sys-

tems by jointly considering the noisy measurements of mul-

tiple sensors. There exist two basic data fusion schemes,

namely, decision fusion and value fusion. In decision fusion,

each sensor makes a local decision based on its measurements

and sends its decision to the cluster head, which makes a sys-

tem decision according to the local decisions. The optimal

decision fusion rule has been obtained in [5]. In value fu-

sion, each sensor sends its measurements to the cluster head,

which makes the detection decision based on the received

measurements. In this paper, we focus on value fusion, as

it usually has better detection performance than decision

fusion [37].Under the assumptions made in Section 3.1,

the optimal value fusion rule is to compare the following

weighted sum of sensors’ measurements to a threshold (the

derivation can be found in Appendix A):

Yopt =

X

i

si

σ· yi.

However, as sensor measurements contain both noise and

signal energy (see (4)), the weight

ceived by sensor i, is unknown.

to adopt equal constant weights for all sensors’ measure-

ments [8,29,40]. Since the measurements from different sen-

sors are treated equally, the sensors far away from the target

should be excluded from data fusion as their measurements

suffer low SNRs. Therefore, we adopt a fusion scheme as

follows.

For any physical point p, the sensors within a distance of

R meters from p form a cluster and fuse their measurements

to detect whether a target is present at p. R is referred

to as the fusion range and F(p) denotes the set of sensors

si

σ, i.e., the SNR re-

A practical solution is

within the fusion range of p. The number of sensors in F(p)

is represented by N(p). A cluster head is elected to make

the detection decision by comparing the sum of measure-

ments reported by member sensors in F(p) against a detec-

tion threshold T. Let Y denote the fusion statistics, i.e.,

Y =P

We assume that the cluster head makes a detection based

on snapshot measurements from member sensors without us-

ing temporal samples to refine the detection decision. Note

that such a snapshot scheme is widely adopted in previous

works on target surveillance [8,20,29,34,40]. Fusion range R

is an important design parameter of our data fusion model.

As SNR received by sensor decays with distance from the

target, fusion range lower-bounds the quality of information

that is fused at the cluster head. In Section 5.2, we will dis-

cuss how to choose the optimal fusion range. The above data

fusion model is consistent with the fusion schemes adopted

in [8,29,40]. If more efficient fusion models are employed,

the scaling laws proved in this paper still hold as discussed

in Section 6.5.

We assume that the target keeps stationary after appear-

ance and the position of a possible target can be obtained

through a localization algorithm. For instance, the target

position can be estimated as the geometric center of a num-

ber of sensors with the largest measurements. Such a simple

localization algorithm is employed in the simulations con-

ducted in this paper. The localized position may not be

the exact target position and the distance between them is

referred to as localization error.

calization error is upper-bounded by a constant ǫ. The lo-

calization error is accounted for in the following analyses.

However, we show that it has no impact on the asymptotic

results derived in this paper.

The above data fusion model can be used for target detec-

tion as follows. The detection can be executed periodically

or triggered by user queries. In a detection process, each sen-

sor makes a snapshot measurement and a cluster is formed

by the sensors within the fusion range from the possible tar-

get to make a detection decision. The cluster formation may

be initiated by the sensor that has the largest measurement.

Such a scheme can be implemented by several dynamic clus-

tering algorithms [6]. The fusion range R can be used as an

input parameter of the clustering algorithm. The communi-

cation topology of the cluster can be a multi-hop tree rooted

at the cluster head. As the fusion statistics Y is an aggrega-

tion of sensors’ measurements, it can be computed efficiently

along the routing path to the cluster head. In this work, we

are interested in the fundamental performance limits of cov-

erage under the fusion model and the design of clustering

and data aggregation algorithms is beyond the scope of this

paper.

i∈F(p)yi. If Y ≥ T, the cluster head decides H1;

otherwise, it decides H0.

We assume that the lo-

3.3Problem Definition

The detection of a target is inherently stochastic due to

the noise in sensor measurements.

mance is usually characterized by two metrics, namely, the

false alarm rate (denoted by PF) and detection probability

(denoted by PD). PF is the probability of making a positive

decision when no target is present, and PD is the probabil-

ity that a present target is correctly detected. In stochastic

detection, positive detection decisions may be false alarms

caused by the noise in sensor measurements. In particular,

The detection perfor-

Page 5

Figure 3: Coverage un-

der the disc

Sensing range r = 17m,

which is computed by

(7).

model.

Figure

under the fusion mo-

del.Grayscale repre-

sents the value of PD.

4:Coverage

although the detection probability can be improved by set-

ting lower detection thresholds, the fidelity of detection re-

sults may be unacceptable because of high false alarm rates.

Therefore, PF together with PD characterize the sensing

quality provided by the network. For a physical point p,

we denote the probability of successfully detecting a tar-

get located at p as PD(p). Note that PF is the probability

of making positive decision when no target is present, and

hence is location independent.

Our focus is to study the coverage of large-scale WSNs.

We introduce a concept called (α,β)-coverage that quantifies

the fraction of the surveillance region where PF and PD are

bounded by α and β, respectively.

Definition 1

α ∈ (0,0.5) and β ∈ (0.5,1), a physical point p is (α,β)-

covered if the false alarm rate PF and detection probability

PD(p) satisfy

((α,β)-coverage). Given two constants

PF ≤ α,PD(p) ≥ β.

The (α,β)-coverage of a region is defined as the fraction of

points in the region that are (α,β)-covered.

The full coverage of a region refers to the case where

the (α,β)-coverage of the region approaches one, i.e., the

false alarm rate is below α and the probability of detect-

ing a target present at any location is above β. In practice,

mission-critical surveillance applications [11–13,16] require

a low false alarm rate (α < 5%) and a high detection prob-

ability (β ≫ 50%).

We now illustrate the (α,β)-coverage by an example, where

δ = 1000 (i.e., 30dB), α = 5%, β = 95%, and R = 50m.

Fig. 3 and 4 illustrate the coverage under the disc and fusion

models, respectively. In Fig. 4, when a target (represented

by the triangle) is present, the sensors within the fusion

range from it fuse their measurements to make a detection.

The gray area is (α,β)-covered, where grayscale represents

the value of PD at each point. As shown in Fig. 3, the cov-

ered region under the disc model is simply the union of all

sensing discs. As a result, when a high level of coverage is

required, a large number of extra sensors must be deployed

to eliminate small uncovered areas surrounded by sensing

discs. In contrast, data fusion can effectively expand the

covered region by exploiting the collaboration among neigh-

boring sensors.

In the rest of this paper, we consider the following prob-

lems:

1. Although a number of analytical results on coverage

[4,19,24,33,38,41–43] have been obtained under the

classical disc model, are they still applicable under the

definition of (α,β)-coverage which explicitly captures

the stochastic nature of sensing? To answer this ques-

tion, we propose a probabilistic disc model such that

the existing results can be naturally extended to the

context of stochastic detection (Section 4).

2. How to quantify the (α,β)-coverage when sensors can

collaborate through data fusion? Answering this ques-

tion enables us to evaluate the coverage performance of

a network and to deploy the fewest sensors for achiev-

ing a given level of coverage (Section 5).

3. What are the scaling laws between coverage, network

density, and signal-to-noise ratio (SNR) under both

the disc and fusion models? The results will provide

important insights into understanding the limitation

of analytical results based on the disc model and the

impact of data fusion on the coverage of large-scale

WSNs (Section 6).

4.COVERAGE UNDER PROBABILISTIC

DISC MODEL

As the classical disc model deterministically treats the de-

tection performance of sensors, existing results based on this

model [4,19,24,33,38,41–43] cannot be readily applied to

analyze the performance or guide the design of real-world

WSNs. In this section, we extend the classical disc mo-

del based on the stochastic detection theory [37] to capture

several realistic sensing characteristics and study the (α,β)-

coverage under the extended model.

In the probabilistic disc model, we choose the sensing range

r such that 1) the probability of detecting any target within

the sensing range is no lower than β, and 2) the false alarm

rate is no greater than α. As we ignore the detection prob-

ability outside the sensing range of a sensor, the detection

capability of sensor under this model is lower than in reality.

However, this model preserves the boundary of sensing region

defined in the classical disc model. Hence, the existing re-

sults based on the classical disc model [4,19,24,33,38,41–43]

can be naturally extended to the context of stochastic de-

tection.

We now discuss how to choose the sensing range r under

the probabilistic disc model. The optimal Bayesian detec-

tion rule for a single sensor i is to compare its measurement

yi to a detection threshold t [37]. If yi exceeds t, sensor i

decides H1; otherwise, it decides H0. Therefore, the PF and

PD of sensor i are given by

PF = P(yi ≥ t|H0) = Q

„t − µ

„t − µ − si

σ

«

, (5)

PD = P(yi ≥ t|H1) = Q

σ

«

, (6)

where P(·) is the probability notation and Q(·) is the comple-

mentary cumulative distribution function (CDF) of the stan-

dard normal distribution, i.e., Q(x) =

As PD is non-decreasing function of PF [37], it is maxi-

mized when PF is set to be the upper bound α.

the optimal detection threshold can be solved from (5) as

topt = µ+ σQ−1(α), where Q−1(·) is the inverse function of

1

√2π

R∞

xe−t2/2dt.

Hence

Page 6

Q(·). By replacing t = topt and si = S ·w(r) in (6), we have

„Q−1(α) − Q−1(β)

where w−1(·) is the inverse function of w(·). If the target

is more than r meters from the sensor, the detection per-

formance requirements, i.e., α and β, cannot be satisfied

by setting any detection threshold. Note that a similar def-

inition of sensing range is proposed in [40] for stochastic

detection. From (7), the sensing range of a sensor varies

with the user requirements (i.e., α and β) and PSNR δ. For

instance, the sensing range r is 3.8m if α = 5%, β = 95%,

δ = 50 (i.e., 17dB)4and w(·) is given by (2) with k = 2.

As w(·) is a decreasing function, w−1(·) is also a decreasing

function. Therefore, r increases with the PSNR δ according

to (7). This conforms to the intuition that a sensor can de-

tect a farther target if the noise level is lower (i.e., a greater

δ).

We now extend the coverage of random networks [4,24]

derived under the classical disc model to (α,β)-coverage.

Under both the classical and probabilistic disc models, a lo-

cation is regarded as being covered if it is within at least one

sensor’s sensing range. Accordingly, the area of the union

of all sensors’ sensing ranges is regarded as being covered

by the network. The coverage of random networks under

the classical disc model has been extensively studied based

on the stochastic geometry theory [4,24]. Specifically, the

coverage of a network deployed according to a Poisson point

process of density ρ is given by

r = w−1

δ

«

, (7)

c = 1 − e−ρπr2. (8)

If the sensing range r is chosen by (7), Eq. (8) computes the

(α,β)-coverage of a random network under the probabilistic

disc model. This result will be used as the basis for studying

the impact of data fusion on network coverage in Section 6.

5. COVERAGEUNDERDATAFUSIONMO-

DEL

Although the probabilistic disc model discussed in Sec-

tion 4 captures the stochastic nature of sensing, it does not

exploit the collaboration among sensors. In this section, we

first derive the (α,β)-coverage under the fusion model, then

illustrate the analytical results using numerical examples.

5.1 DerivingCoverageunder DataFusionMo-

del

We have the following lemma regarding the (α,β)-coverage

of random networks.

Lemma 1. The (α,β)-coverage of a uniformly deployed

network under the data fusion model, denoted by c, is given

by

c = P

P

i∈F(p)si

pN(p)

≥ σ`Q−1(α) − Q−1(β)´

!

, (9)

where p is an arbitrary physical point in the network.

4The PSNR is set according to the measurements from the

vehicle detection experiments based on MICA2 [12] and

ExScal [13] motes.

Proof. We first discuss the necessary and sufficient con-

dition that p is (α,β)-covered. When no target is present, all

sensors measure i.i.d.noises and hence Y |H0 =P

P(Y ≥ T|H0) = Q

N(p)

threshold. As PD is a non-decreasing function of PF [37], it

is maximized when PF is set to be the upper bound α. Such

a scheme is referred to as the Constant False Alarm Rate

detector [37]. Let PF = α, the optimal detection threshold

can be derived as Topt = µN(p) + σQ−1(α)pN(p).

N(µN(p) +P

T −µN(p)−P

i∈F(p)ni ∼

N(µN(p),σ2N(p)). Therefore, the false alarm rate is PF =

„

T−µN(p)

σ√

«

, where T is the detection

When the target is present, Y |H1 =P

probability at p is given by

i∈F(p)si + ni ∼

i∈F(p)si,σ2N(p)). Therefore, the detection

PD(p)=P(Y ≥T|H1)=Q

i∈F(p)si

σpN(p)

!

By replacing T with Toptand solving PD(p) ≥ β, we have the

necessary and sufficient condition that p is (α,β)-covered:

P

i∈F(p)si

pN(p)

≥ σ`Q−1(α) − Q−1(β)´.(10)

As the random network is stationary, the fraction of covered

area equals the probability that an arbitrary point is covered

by the network [24]. Therefore, the (α,β)-coverage of the

network is given by (9).

As p is an arbitrary point in the network, N(p) is a Pois-

son random variable, i.e., N(p) ∼ Poi(ρπR2). Moreover,

{si|i ∈ F(p)} are also random variables. However, we have

no closed-form formula for computing (9) due to the diffi-

culty of deriving the CDF of

P

i∈F(p)si

√

N(p)

. We now give an

approximation to (9) in the following lemma. The proof is

given in Appendix B.

Lemma 2. Let µs and σ2

of si|i ∈ F(p) for arbitrary point p, respectively. The (α,β)-

coverage of a uniformly deployed network under the data fu-

sion model can be approximated by

sdenote the mean and variance

c ≃ Q

γ(R) − ρπR2

pρπR2

!

,(11)

where γ(R) =

„

Q−1(α)σ−Q−1(β)√

σ2

s+σ2

µs

«2

.

We note that the formulas of µs and σ2

and (18), respectively. As Central Limit Theorem (CLT) is

applied in the derivation of (11), this approximation is ac-

curate when N(p) ≥ 20 [28]. This condition can be easily

met in many applications. For example, it is shown in [12]

that the detection probability is only about 40% when four

MICA2 motes are deployed in a 10×10m2region. Suppose

R = 20m and the network density is the same as in [12],

N(p) will be about 50. With the approximate formula, we

can evaluate the coverage performance of an existing net-

work or compute the minimum network density to achieve

the desired level of coverage under the fusion model. Our

simulation results in Section 7 show that (11) can provide

accurate prediction of coverage under the fusion model. We

note that the localization error has little impact on the accu-

racy of the approximate formula when R ≫ ǫ. Recent sensor

sare given by (17)

Page 7

network localization protocols can achieve a precision within

0.5m in large-scale outdoor deployments [36].

We now derive the lower bound of (α,β)-coverage under

the fusion model, which will be used in the derivations of

scaling laws in Section 6. We denote FPoi(·|λ) as the CDF

of the Poisson distribution Poi(λ), which is formally given

by FPoi(x|λ) =P⌊x⌋

Lemma 3. The lower bound of (α,β)-coverage of a uni-

formly deployed network under the data fusion model, de-

noted by cL, is given by

k=0

e−λλk

k!

.

cL = 1 − FPoi(Γ(R)|ρπR2),

Q−1(α)−Q−1(β)

δ

(12)

where Γ(R) =

large enough,

“”2

·

1

w2(R+ǫ). When ρπR2is

cL = Q

Γ(R) − ρπR2

pρπR2

P

!

. (13)

Proof. For any point p,

N(p), as si ≥ S · w(R + ǫ) for any sensor i in F(p). If

S·w(R+ǫ)·N(p)

√

N(p)

Therefore, by solving N(p), the sufficient condition that p

is (α,β)-covered is N(p) ≥ Γ(R).

Poi(ρπR2), we have

i∈F(p)si ≥ S · w(R + ǫ) ·

≥ σ`Q−1(α) − Q−1(β)´, Eq. (10) must hold.

Moreover, as N(p) ∼

c=

≥

=

P(point p is (α,β)-covered)

P(N ≥ Γ(R))

1 − FPoi(Γ(R)|ρπR2).

Therefore, the lower bound of c is given by (12). When ρπR2

is large enough, the normal distribution N(ρπR2,ρπR2) ex-

cellently approximates the Poisson distribution Poi(ρπR2).

Therefore, Eq. (12) can be approximated by (13).

In the proofs of above lemmas, the fusion statistics Y

has a componentP

{ni} are i.i.d.. Therefore, the assumption of i.i.d. Gaussian

noises made in Section 3.1 can be relaxed to i.i.d. noises that

follow any distribution, when the number of sensors taking

part in data fusion is large enough. In practice, the accuracy

of this approximation is satisfactory when N(p) ≥ 20 [28].

In particular, the distribution of noise will not affect the

asymptotic scaling laws in Section 6, as N(p) is large in the

asymptotic scenarios where c → 1.

5.2 Numerical Examples

In this section, we provide several numerical results to

help understand the coverage performance under the data

fusion model. We adopt the signal decay function given by

(2) with k = 2. Fig. 5 plots the approximate coverage com-

puted by (11). We can see from Fig. 5 that the coverage

initially increases with fusion range R, but decreases to zero

eventually. Intuitively, as the fusion range increases, more

sensors contribute to the data fusion resulting in better sens-

ing quality. However, as R becomes very large, the aggre-

gate noise starts to cancel out the benefit because the target

signal decreases quickly with the distance from the target.

In other words, the measurements of sensors far away from

the target contain low quality information and hence fusing

them leads to lower detection performance. An important

i∈F(p)ni. According to the CLT, this

component approximately follows the normal distribution if

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

02

Fusion range R (m)

46810

Coverage c

ρ = 1.0

ρ = 0.7

Figure 5:

fusion range (δ = 4, α =

5%, β = 95%).

Coverage vs.

0

20

40

60

80

100

0.02 0.04 0.06 0.08

Network density ρ

0.1

Ropt(m)

Figure 6: Optimal fusion

range vs. density(δ = 100,

α = 5%, β = 95%).

question is thus how to choose the optimal fusion range (de-

noted by Ropt) that maximizes the coverage. First, the Ropt

can be obtained through numerical experiments. Fig. 6 plots

the optimal fusion ranges under different network densities,

which are obtained by numerically maximizing the coverage.

Second, it is possible to obtain the analytical Roptby solving

dc

dR= 0. For instance, when the signal decay function w(·)

is given by (2) with k = 2, Ropt satisfies

and hence Ropt increases with network density ρ.

Ropt

ln Ropt= Θ(√ρ)

6. IMPACT OF DATA FUSION ON COVER-

AGE

Many mission-critical applications require a high level of

coverage over the surveillance region. As an asymptotic case,

full coverage is required, i.e., any target/event present in

the region can be detected with a probability of at least β

while the false alarm rate is below α. As a higher level of

coverage always requires more sensors, the network density

for achieving full coverage is an important cost metric for

mission-critical applications.

Under the disc model, the sensing regions of randomly

deployed sensors inevitably overlap with each other when a

high level coverage is required. According to (8), we have

dρ =

sensors (i.e., dρ) are required to eliminate a small uncovered

area (i.e., dc). Moreover, the situation gets worse when c

increases. In this section, we are interested in how much

network density can be reduced by adopting data fusion.

Specifically, we study the asymptotic relationships between

the network densities for achieving full coverage under the

probabilistic disc and data fusion models. The results pro-

vide important insights into understanding the limitation of

the disc model and the impact of data fusion on coverage.

1

πr2·

1

1−c·dc. If c is close to 1, a large number of extra

6.1Full Coverage using Fixed Fusion Range

We first study the relationship between the network den-

sities for achieving full coverage under the disc and fusion

models when fusion range R is a constant. We have the

following theorem.

Theorem 1. Let ρdand ρf denote the minimum network

densities required to achieve the (α,β)-coverage of c under

the disc and fusion models, respectively. If the fusion range

R is fixed, we have

ρf = O

„2r2

R2· ρd

«

,c → 1. (14)

Page 8

Proof. As ρf is large to provide a high level of cov-

erage under the fusion model, the lower bound of (α,β)-

coverage, cL, is given by (13) according to Lemma 3. We

define h1(ρf) =

cL = Q(h1(ρf)−h2(ρf)). When ρf → ∞, h2(ρf) dominates

h1(ρf) aslim

ρf→∞

Q(−√πR ·√ρf) when ρf → ∞. Define x = Q−1(c). We

have ρf ≤

Under the disc model, by replacing c = Q(x) = 1−Φ(x) in

(8) and solving ρd, we have ρd = −

is the CDF of the standard normal distribution. Hence, we

have

Γ(R)

√πR·

1

√ρf, h2(ρf) =√πR ·√ρf and hence

h1(ρf)

h2(ρf)= 0. Hence, c ≥ cL = Q(−h2(ρf)) =

1

πR2x2when c → 1.

1

πr2lnΦ(x), where Φ(x)

lim

c→1

ρf

ρd

≤

lim

x→−∞

1

πR2x2

1

−

πr2lnΦ(x)= −r2

R2

lim

x→−∞

x2

lnΦ(x).

As lim

x→−∞

ρf

ρd≤2r2

is given by (14).

x2

lnΦ(x)= −2 (derived in Appendix C), we have

R2. Therefore, the asymptotic upper bound of ρf

lim

c→1

Theorem 1 shows that in order to achieve full coverage,

ρf is smaller than ρd if R >

ing range r is a constant independent of network density.

On the other hand, fusion range R is a design parameter of

the fusion model, which is mainly constrained by the com-

munication overhead. In practice, the condition R >

can be easily satisfied. For instance, the acoustic sensor on

MICA2 motes has a sensing range of 3m to 5m if a high

performance (e.g., α = 5% and β = 95%) is required [12].

On the other hand, the fusion range can be set to be much

larger. For example, Fig. 6 shows that Ropt ranges from 5m

to 100m when network density increases from 1.5 × 10−3

to 0.1. Therefore, according to Theorem 1, the fusion mo-

del with the optimal fusion range can significantly reduce

network density for achieving a high level of coverage.

√2r. According to (7), sens-

√2r

6.2 FullCoverageusingOptimalFusionRange

As discussed in Section 5.2, we can obtain the optimal

fusion range via numerical experiment or analysis.

fusion with the optimal fusion range allows the maximum

number of informative sensors to contribute to the detection.

The scaling law obtained with optimal fusion range will help

us understand the maximum performance gain by adopting

the data fusion model. The following theorem shows that

ρf further reduces to O(ρ1−1/k

is optimal. The proof is given in Appendix D.

Data

d

) as long as the fusion range

Theorem 2. Let ρdand ρf denote the minimum network

densities required to achieve the (α,β)-coverage of c under

the disc and fusion models, respectively. If the optimal fusion

range Ropt is adopted, we have

ρf = O

“

ρ1−1/k

d

”

,c → 1.(15)

Theorem 2 shows that if the optimal fusion range is adopted,

the fusion model can significantly reduce the network den-

sity for achieving high coverage. In particular, from Theo-

rem 2, the density ratio

which means ρf is insignificant compared with ρdfor achiev-

ing high coverage. Theorem 2 is applicable to the scenarios

where the physical signal follows the power law decay with

ρf

ρd= O(ρ−1/k

d

) = 0 when c → 1,

path loss exponent k, which are widely assumed and verified

in practice. We note that the path loss exponent k typically

ranges from 2.0 to 5.0 [15,20]. In particular, the propagation

of acoustic signals in free space follows the inverse-square

law, i.e., k = 2, and therefore ρf = O(√ρd).

6.3Impact of Signal-to-Noise Ratio

In this section, we study the impact of PSNR on the re-

sults derived in the previous sections. PSNR is an impor-

tant system parameter which is determined by the property

of target, noise level, and sensitivity of sensors. We have the

following corollary.

Corollary 1. For fixed fusion range R, we have

ρf

ρd

= O(δ2/k),c → 1.(16)

Proof. As w(x) = Θ(x−k), w−1(x) = Θ(x−1/k). Ac-

cording to (7), the sensing range r = Θ(δ1/k). As lim

c→1

ρf

ρd≤

2r2

R2 = Θ(δ2/k), we have (16).

Corollary 1 suggests that for a fixed R, the relative cost

between the fusion and disc models is affected by the PSNR

δ. Specifically, the fusion model requires fewer sensors to

achieve full coverage than the disc model if the PSNR is

low. On the other hand, the disc model suffices only if the

PSNR is sufficiently high. Intuitively, sensor collaboration

is more advantageous when the PSNR is low to moderate.

However, when the PSNR is sufficiently high, the detection

performance of a single sensor is satisfactory and the collab-

oration among multiple sensors may be unnecessary.

6.4 Implications of Results

We now summarize the implications of theoretical results

derived in this section.

6.4.1

According to Theorem 2, when the required coverage ap-

proaches one, ρdincreases significantly faster than ρf, espe-

cially for a small decay exponent. For instance, when k = 2

(which typically holds for acoustic signals), ρf = O(√ρd).

This result implies that the existing analytical results based

on the disc model (e.g., [4, 19, 24, 33, 38, 43]) significantly

overestimate the network density required for achieving full

coverage. On the other hand, Corollary 1 shows that the

disc model may lead to similar or even lower network density

than the fusion model if PSNR is sufficiently high. The noise

experienced by a sensor in real systems comes from various

sources, e.g., the random disturbances in the environment

and the electronic noise in sensor circuit. In practice, the

PSNRs in the applications based on low-cost sensors are usu-

ally low. For instance, the PSNRs in the vehicle detection

experiments based on MICA2 [12] and ExScal [13] motes are

about 50 (i.e., 17dB). In such a case, ρd≥ 2ρf for achieving

a high level of coverage if R is set to be greater than 8m.

The limitations of disc model

6.4.2

Our results provide several important guidelines on the de-

sign of data fusion algorithms for large-scale WSNs. First,

data fusion is very effective in improving sensing coverage

and reducing network density.

suggests that the performance gain of data fusion increases

when the PSNR is lower. Therefore, data fusion should be

Design of data fusion algorithms

In particular, Theorem 2

Page 9

50

100

150

200

250

300

350

400

450

500

0.50.55 0.60.650.70.750.80.850.90.95

The number of deployed sensors

(α,β)-coverage

Probabilistic disc model

Fusion model (R = 100 m)

Fusion model (R = 200 m)

Figure 7:

achieved (α,β)-coverage.

The number of deployed sensors vs.

employed for low-SNR deployments when a high level of cov-

erage is required. Second, Theorems 1 and 2 suggest that

fusion range plays an important role in the achievable per-

formance of data fusion. As discussed in Section 5.2, the op-

timal fusion range that maximizes coverage increases with

network density and can be numerically computed. How-

ever, a larger fusion range requires more sensors to fuse their

measurements resulting in higher communication overhead.

Investigating the optimal fusion range under both coverage

and communication constraints is left for future work.

6.5 Discussion

We now discuss several issues that have not been ad-

dressed in this paper.

The main objective of this paper is to explore the fun-

damental limits of coverage based on data fusion model in

target surveillance applications, in which sensors measure

the signals emitted by the target. The proofs of Lemma 1-3

and Theorem 1 are not dependent on the form of the sig-

nal decay function w(·). Therefore, these results hold under

arbitrary bounded decreasing function w(·). However, The-

orem 2 and Corollary 1 are only applicable for the applica-

tions where the target signal follows the power law decay,

i.e., w(x) = Θ(x−k). We acknowledge that most mechan-

ical and electromagnetic waves follow the power law decay

in propagation. In particular, in open space, inverse-square

law (i.e., k = 2) [9] applies to various physical signals such

as sound, light and radiation. In our future work, we will

extend our analyses to address other decay laws such as ex-

ponential decay in diffusion processes [35].

Theorem 1-2 and Corollary 1 give the upper bounds of

network density under the fusion model presented in Sec-

tion 3.2. If more efficient fusion models are employed, the

coverage performance will be further improved. In other

words, more efficient fusion model can reduce the network

density for achieving a certain level of coverage. As a re-

sult, the upper bounds of network density derived in this

paper still hold. Exploring the impact of efficiency of fusion

models on network density is left for future work.

7.SIMULATIONS

In this section, we conduct extensive simulations based

on real data traces as well as synthetic data to evaluate

the coverage performance in non-asymptotic and asymptotic

cases, respectively.

7.1 Trace-driven Simulations

We first conduct simulations using the data traces col-

lected in a real vehicle detection experiment [1].

experiments, 75 WINS NG 2.0 nodes are deployed to de-

tect military vehicles driving through the surveillance region.

We refer to [11] for detailed setup of the experiments. The

dataset used in our simulations includes the ground truth

data and the acoustic time series recorded by 20 nodes when

a vehicle drives through. The ground truth data include the

positions of sensors and the trajectory of the vehicle.

Sensors’ sensing ranges under the probabilistic disc mo-

del are determined individually to meet the detection per-

formance requirements (α = 5% and β = 95%). The re-

sulted sensing ranges are from 22.5m to 59.2m with the

average of 43.2m. Such a significant variation is due to sev-

eral issues including poor calibration and complex terrain.

In our simulation, we deploy random networks with size of

1000 × 1000m2. Each sensor in the simulation is associ-

ated with a real sensor chosen at random. For each deploy-

ment, we evaluate the (α,β)-coverage under both the disc

and fusion models. We divide the region into 1000 × 1000

grids. Under the disc model, the coverage is estimated as

the ratio of grid points that are covered by discs. Under the

fusion model, the coverage is estimated as the ratio of (α,β)-

covered grid points. Specifically, for a target that appears

at a grid point, each sensor makes a measurement which is

set to be the energy gathered by the associated real sensor

at a similar distance to vehicle in the data trace. A cluster

is formed around the sensor with the highest reading, which

fuses sensor measurements for detection.

Fig. 7 plots the the number of deployed sensors versus the

achieved (α,β)-coverage under various settings. We can see

that the disc model suffices if a moderate level of coverage

is required. However, the fusion model is more effective for

achieving high coverage.In particular, the fusion model

with a fusion range of 200m saves more than 50% sensors

when the coverage is greater than 0.75. Moreover, the trend

of density ratio also follows ρf = O(2r2

Section 6.1. We note that the average number of sensors

taking part in data fusion is within 30 and hence will not

introduce high communication overhead.

In the

R2 · ρd) derived in

7.2Simulations based on Synthetic Data

7.2.1

In addition to trace-driven simulations, we also conduct

extensive simulations based on synthetic data. These simu-

lations allow us to evaluate the theoretical results in a wide

range of settings. We adopt the signal decay function in

(2) with k = 2. Both the mean and variance of the Gaus-

sian noise generator, µ and σ2, are set to be 1. We set the

orginal energy of target, S, to be 4, 50, and 5000, so that

the SNRs in the simulations are consistent with several real

experiments [7,11–13].

As proved in Lemma 1, it suffices to measure the probabil-

ity that a point is covered for evaluating the coverage of the

whole network. Hence, we let the target appear at a fixed

point p and deploy random networks with size of 4R × 4R

centered at p. For each deployment, PD(p) is estimated as

the fraction of succesful detections. The (α,β)-coverage is

estimated as the fraction of deployments whose PD(p) is

greater than β.

We also evaluate the impact of localization error by inte-

grating a simple localization algorithm. Specifically, for each

Numerical Settings

Page 10

0

0.2

0.4

0.6

0.8

1

0.3 0.4 0.5

Network density ρ

0.6 0.70.80.9

Coverage c

analytical

SIM

SIM-LOC

Figure 8: Coverage vs. network

density (δ = 4, R = 5m).

-2

0

2

4

6

8

10

1 − 10−6

1 − 10−4

1 − 10−2

0

Density ratio

ρd

ρf

Coverage c

δ=4, R=5

δ=50, R=25

δ=5000, R=100

Figure 9:

coverage in log10scale with vari-

ous PSNRs.

Density ratio

ρd

ρf

vs.

0

0.5

1

1.5

2

2.5

3

0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85

ρf

6

4

2

0

√ρd

Ropt(m)

√ρd

Ropt

Figure 10:√ρd vs. ρf with opti-

mal fusion range Ropt (δ = 4).

detection, if a sensor’s reading exceeds S · w(R) + µ, it will

take part in the target localization. The target is localized

as the geometric center of the sensors participating in the

localization.

7.2.2

We first evaluate the accuracy of the approximate formula

given in Lemma 2. Fig. 8 plots the analytical and measured

coverage versus network density. The curves labelled with

SIM-LOC and SIM represent the measured results with and

without accounting for localization error, respectively. We

can see that the simulation result matches well the analyti-

cal result given by (11). A network density of 0.8 is enough

to provide high coverage under the fusion model, where the

SNR is very low (δ = 4). When there is localization error, a

maximum deviation of about 0.2 from the analytical result

can be seen from Fig. 8. The coverage decreases in the pres-

ence of localization error as sensors received weaker signals

when the target cannot be accurately localized. However,

the impact of localization error diminishes when c → 1.

The second set of simulations evaluate the impact of SNR

on the asymptotic network densities. Fig. 9 plots the net-

work density ratio

ρfversus the achieved coverage under

various PSNRs, where ρd is computed by (8) and ρf is ob-

tained in simulations, respectively. The x-axis is plotted in

log10scale. We can see that the density ratio increases with

the coverage, i.e., the fusion model becomes more effective

for achieving higher coverage. Moreover, the density ratio

decreases with the PSNR, which conforms to the result of

Corollary 1. For instance, to achieve a high coverage of 0.99,

the density ratio

ρfis about 8 when δ = 4. The density ra-

tio decreases to about 2 when δ = 50. This result shows

that data fusion is effective in the scenarios with low SNRs.

When δ = 5000, the disc model suffices. These results are

consistent with the analysis in Section 6.3.

The third set of simulations evaluate the asymptotic re-

lationship between ρd and ρf when the fusion range is op-

timized. In Fig. 10, the X- and Y -axis of each data point

represent the required network densities for achieving the

same coverage that approaches to one under the disc and

fusion models, respectively. Note that the Y -axis is plotted

in square root scale. The optimal fusion range Ropt plotted

in Fig. 10 is computed for each given ρf by numerically max-

imizing (11). We can see from Fig. 10 that the relationship

between√ρd and ρf is convex and therefore conforms to

the theoretical result ρf = O(√ρd) according to Theorem 2.

Simulation Results

ρd

ρd

Moreover, Ropt increases with ρf, which is also consistent

with the analysis in Section 5.2.

8.CONCLUSION

Sensing coverage is an important performance require-

ment of many critical sensor network applications. In this

paper, we explore the fundamental limits of coverage based

on stochastic data fusion models that jointly process noisy

measurements of sensors. The scaling laws between cover-

age, network density, and signal-to-noise ratio (SNR) are de-

rived. Data fusion is shown to significantly improve sensing

coverage by exploiting the collaboration among sensors. Our

results help understand the limitations of the existing ana-

lytical results based on the disc model and provide key in-

sights into the design and analysis of WSNs that adopt data

fusion algorithms. Our analyses are verified through simu-

lations based on both synthetic data sets and data traces

collected in a real deployment for vehicle detection.

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APPENDIX

A.OPTIMAL VALUE FUSION RULE

Suppose there are N sensors taking part in the data fusion.

The optimal decision rule that minimizes the average cost

(i.e., Bayesian decision) is given by the likelihood ratio test:

p(y1,...,yN|H1)

p(y1,...,yN|H0)

H1

≷

H0

P0(C10− C00)

P1(C01− C11).

where P0 = P(H0), P1 = P(H1), and Cij is the cost that

we decide Hi when the ground truth is Hj. The left-hand

side is the likelihood ratio and the right-hand side is the

optimal Bayes threshold. As the sensors’ measurements are

independent Gaussians assumed in Section 3.1, we have

p(y1,...,yN|H1)

p(y1,...,yN|H0)=

N

Y

i=1

p(yi|H1)

p(yi|H0)= e

PN

i=1

2siyi−2µsi−s2

σ2

i

.

Page 12

Accordingly, the likelihood ratio test becomes

N

X

i=1

si

σ· yi

H1

≷

H0

1

2

N

X

i=1

2µsi+ s2

σ

i

+σ

2lnP0(C10− C00)

P1(C01− C11).

Therefore, the optimal fusion statistics for Bayesian decision

isPN

B. PROOF OF LEMMA 2

i=1

si

σ· yi where

si

σis the received SNR of sensor i.

Proof. We first prove that the {si|i ∈ F(p)} are i.i.d. for

given p and derive the formulas for µs and σ2

are deployed uniformly and independently, {di|i ∈ F(p)} are

i.i.d. for given p, where di is the distance between sensor i

and point p. To simplify our discussion, we now temporarily

assume that there is no localization error, i.e., ǫ = 0. There-

fore, {si|i ∈ F(p)} are i.i.d. for given p, as si is a function of

di (defined by (1)). Suppose the coordinates of point p and

sensor i are (xp,yp) and (xi,yi), respectively. The posterior

probability density function of (xi,yi) is f(xi,yi) =

where (xi − xp)2+ (yi − yp)2≤ R2. Hence, the posterior

CDF of di is given by F(di) =R2π

ZR

σ2

0

R2

s. As sensors

1

πR2

0

dθRdi

0

1

πR2· xdx =

d2

R2

i

where di ∈ [0,R]. Therefore, we have

sidF(di) =2S

µs =

0

R2·

ZR

0

xw(x)dx, (17)

s=

ZR

s2

idF(di) − µ2

s=2S2

ZR

0

xw2(x)dx − µ2

s.(18)

A straightforward approximation is to replaceP

the distribution ofP

P

P

bution, i.e., Y |H1 ∼ N(µsN(p)+µN(p),σ2

Therefore, the detection probability at point p is given by

i∈F(p)si

in (9) with its mean µsN(p). However, doing so ignores

i∈F(p)si. We approximateP

i∈F(p)si ∼ N(µsN(p),σ2

N(p) as a constant. When the target is present, Y |H1 =

i∈F(p)si +P

i∈F(p)si

as a Gaussian random variable according to the CLT, i.e.,

sN(p)). Note that here we treat

i∈F(p)ni. As the sum of two independent

Gaussians is also Gaussian, Y |H1 follows the normal distri-

sN(p)+σ2N(p)).

PD(p) = P(Y ≥ T|H1) ≃ Q

T − µsN(p) − µN(p)

√σ2

s+ σ2·pN(p)

!

.

By replacing T with the optimal detection threshold Topt

(derived in the proof of Lemma 1) and solving PD(p) ≥ β,

the condition that p is (α,β)-covered is given by N(p) ≥

γ(R). The approximate formula of (α,β)-coverage is then

given by

c ≃ P(N(p) ≥ γ(R)) = 1 − FPoi(γ(R)|ρπR2),

where FPoi(·|λ) is the CDF of the Poisson distribution Poi(λ).

When ρπR2is large enough, the Poisson distribution Poi(ρπR2)

can be excellently approximated by the normal distribution

N(ρπR2,ρπR2). Therefore, Eq. (19) can be further approx-

imated by (11).

(19)

C.TWO LIMITS USED IN THE PROOFS

OF THEOREMS 1 AND 2

Denote φ(x) as the probability density function of the

standard normal distribution, i.e., φ(x) =

1

√2πe−x2/2. Note

that Φ′(x) = φ(x) and φ′(x) = −xφ(x). For constant η < 0,

we have

z2

lnΦ(ηz)

z→∞

lim

z→∞

(*)

= lim

2z

1

Φ(ηz)φ(ηz)η=2

η

lim

z→∞

„

Φ(ηz)z

φ(ηz)

(*)

=2

η

lim

z→∞

φ(ηz)ηz+Φ(ηz)

−η2zφ(ηz)

η + lim

z→∞

= −2

φ(ηz)η

η3

η+ lim

z→∞

«

Φ(ηz)

zφ(ηz)

«

(*)

= −2

= −2

η3

„

„

φ(ηz) − η2z2φ(ηz)

η

1 − η2z2

η3

η + lim

z→∞

«

= −2

η2,

where the steps marked by (*) follow from the l’Hˆ opital’s

rule. Note that for η<0, lim

0. By replacing z=−x and η=−1, we have

x2

lnΦ(x)= −2.

z→∞Φ(ηz)z=0 and lim

z→∞zφ(ηz)=

lim

x→−∞

D.PROOF OF THEOREM 2

Proof. We choose R by

ξ

π·Γ(R)

R2

= ρf,(20)

where ξ is a constant and ξ > 1. It is easy to verify that the

chosen R is order-optimal for the lower bound of coverage

(i.e., cL). Moreover, it is easy to verify that both the chosen

R and Γ(R) increase with ρf. By replacing ρf in (13) with

(20), cL is given by

cL = Q

„„

1

√ξ−

p

ξ

«

·

p

Γ(R)

«

= 1 − Φ(ηz),

where η =

we have c ≥ cL = 1 − Φ(ηz). According to (8), the network

density under the disc model satisfies ρd= −

−

constant satisfies

1

√ξ−√ξ is a constant and z =

pΓ(R). Hence

1

πr2ln(1−c) ≥

1

πr2lnΦ(ηz). Hence, the ratio ρb

f/ρd where b is a positive

lim

c→1

ρb

ρd

f

≤ lim

R→∞

= −ξbr2

2ξbr2

πb−1η2· lim

`ξ

−

π

πr2lnΦ(ηz)

´b·Γb(R)

R2b

1

πb−1· lim

z→∞

z2

lnΦ(ηz)· lim

Γb−1(R)

R2b

R→∞

Γb−1(R)

R2b

=

R→∞

.

Note that lim

z→∞

z2

ln Φ(ηz)= −2

η2 (derived in Appendix C) in

the above derivation. As w(x) = Θ(x−k) and ǫ is constant,

Γ(R) = Θ(1/w2(R + ǫ)) = Θ((R + ǫ)2k) = Θ(R2k) and

hence Γb−1(R) = Θ(R2kb−2k). Therefore, lim

R→∞

Γb−1(R)

R2b

=

lim

R→∞R2kb−2k−2b. If b ≤

and hence lim

c→1

we have (15). We note that although the chosen R is not

optimal for c, the upper bound given by (15) still holds if R

is optimal for c.

k

k−1,lim

R→∞

Γb−1(R)

R2b

is a constant

ρb

ρdis upper-bounded by a constant. Hence,

f