# Connectivity Analysis of Wireless Ad Hoc Networks With Beamforming

**ABSTRACT** In this paper, we present an analytical model for evaluating the impact of shadowing and beamforming on the connectivity of wireless ad hoc networks accommodating nodes equipped with multiple antennas. We consider two simple beamforming schemes: random beamforming, where each node selects a main beam direction randomly with no coordination with other nodes, and center-directed beamforming, where each node points its main beam toward the geographical center of the network. Taking path loss, shadowing, and beamforming into account, we derive an expression for the effective coverage area of a node, which is used to analyze both the local network connectivity (probability of node isolation) and the overall network connectivity (1-connectivity and path probability). We verify the correctness of our analytical approach by comparing with simulations. Our results show that the presence of shadowing increases the probability of node isolation and reduces the 1-connectivity of the network, although moderate shadowing can improve the path probability between two nodes. Furthermore, we show that the impact of beamforming strongly depends on the level of the channel path loss. In particular, compared with omnidirectional antennas, beamforming improves both the local and the overall connectivity for a path loss exponent of alpha < 3. The analysis in this paper provides an efficient way for system designers to characterize and optimize the connectivity of wireless ad hoc networks with beamforming.

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Page 1

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 9, NOVEMBER 2009 5247

Connectivity Analysis of Wireless Ad Hoc

Networks With Beamforming

Xiangyun Zhou, Student Member, IEEE, Salman Durrani, Member, IEEE, and Haley M. Jones

Abstract—In this paper, we present an analytical model for eval-

uating the impact of shadowing and beamforming on the connec-

tivity of wireless ad hoc networks accommodating nodes equipped

with multiple antennas. We consider two simple beamforming

schemes: random beamforming, where each node selects a main

beam direction randomly with no coordination with other nodes,

andcenter-directedbeamforming,whereeachnodepointsitsmain

beam toward the geographical center of the network. Taking path

loss, shadowing, and beamforming into account, we derive an

expression for the effective coverage area of a node, which is used

to analyze both the local network connectivity (probability of node

isolation) and the overall network connectivity (1-connectivity and

path probability). We verify the correctness of our analytical ap-

proach by comparing with simulations. Our results show that the

presence of shadowing increases the probability of node isolation

and reduces the 1-connectivity of the network, although moderate

shadowing can improve the path probability between two nodes.

Furthermore, we show that the impact of beamforming strongly

depends on the level of the channel path loss. In particular, com-

paredwithomnidirectionalantennas,beamformingimprovesboth

the local and the overall connectivity for a path loss exponent of

α < 3. The analysis in this paper provides an efficient way for

system designers to characterize and optimize the connectivity of

wireless ad hoc networks with beamforming.

Index Terms—Beamforming, connectivity, effective coverage

area, shadowing, wireless ad hoc networks.

I. INTRODUCTION

A

no need for any pre-existing network infrastructure [1]. In such

networks, connectivity is a fundamental requirement, i.e., any

node pair should be connected either directly or via multiple

direct links between intermediate nodes. The study of con-

nectivity of wireless ad hoc networks can broadly be catego-

rized based on whether the individual nodes are equipped with

omnidirectional antennas or beamforming antennas. Tradition-

ally, ad hoc networks are assumed to employ omnidirectional

antennas, which transmit a signal in all directions with the

WIRELESS ad hoc network consists of self-organizing

mobile nodes that can dynamically form a network, with

Manuscript received November 8, 2008, revised April 15, 2009. First pub-

lished June 26, 2009; current version published November 11, 2009. This

paper was presented in part at the 2008 IEEE International Symposium on

Personal, Indoor, and Mobile Radio Communications Conference, Cannes,

France, September 2008, and in part at the 2007 International Conference

on Signal Processing and Communication Systems, Gold Coast, Australia,

December 2007. The review of this paper was coordinated by Dr. J. Li.

The authors are with the College of Engineering and Computer Science,

The Australian National University, Canberra, ACT 0200, Australia

(e-mail:xiangyun.zhou@anu.edu.au;

jones@anu.edu.au).

Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2009.2026049

salman.durrani@anu.edu.au; haley.

same power. Recently, there has been growing interest in using

beamforming to improve the connectivity of wireless ad hoc

networks [2]–[5]. Beamforming allows the formation of a nar-

rowantenna beam thatcan be steered tofocus mostof thesignal

energy toward a desired direction. In this paper, we provide an

analytical model for characterizing the impact of beamforming

on the connectivity of wireless ad hoc networks.

For omnidirectional antennas, previous work has analyzed

the connectivity of wireless ad hoc networks using different

methods and connectivity metrics [6]–[15]. A widely used

approach in this regard is the geometric disk model [6], [7]. It

is assumed that two nodes can communicate with each other if

their distance apart is smaller than a given transmission radius.

Using the geometric disk model, a semi-analytical procedure

for the determination of the critical node density for an almost

surely connected network for the case of simple path loss

channels was considered in [9]. The results were extended

to a shadowing environment in [10], and it was shown that

the channel randomness caused by shadowing can improve

network connectivity by reducing the number of isolated nodes.

In [11], a probability density function (pdf) of the distance

between two nodes in a rectangular or hexagonal region was

analytically derived using a space decomposition method and

wasusedtocalculatetheaveragenumberofneighborsofanode

(i.e., node degree) with a simple path loss model. The results

wereextendedforthecaseofmultihopnetworksinashadowing

environment in [12]. An alternative analytical method, which is

based on the concept of effective coverage area, was proposed

in [13] to analyze the effect of path loss and shadowing on

the connectivity of wireless ad hoc networks. It must be noted

that the shadowing channel model used in [10], [12], and

[13] increases the average channel gain, whereas in a practical

wireless channel, shadowing affects only the randomness and

not the average value of the channel gain.

More recently, the connectivity of wireless ad hoc networks

with different beamforming schemes has been studied in [16]–

[22]. Although the size, cost, and power consumption issues

limit the applicability of large antenna arrays for wireless

mobile devices, the advent of low-cost digital signal processor

chips have made beamforming systems practical for commer-

cial use [23], [24] and beamforming is being widely considered

for wireless network standards such as IEEE 802.11, IEEE

802.16, and IEEE 802.15.3c [4]. A survey of different beam-

forming strategies for ad hoc networks was provided in [16]. It

is well known that the use of beamforming can improve the

network connectivity if each node has knowledge about the

locations of all the neighboring nodes. However, the discov-

ery of neighboring nodes in a decentralized network requires

0018-9545/$26.00 © 2009 IEEE

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5248IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 9, NOVEMBER 2009

significant signal processing for direction-of-arrival estimation

in the implementation of beamforming algorithms [17]. A very

simple solution that does not require neighbor node positioning

information is the random beamforming strategy [17]. A core

idea in random beamforming is that each node in the network

randomly selects a main beam direction without any coordi-

nation with other nodes. Therefore, the random beamforming

scheme has minimal communication overhead and hardware

complexity.Usingsimulations,itwasshownthat,whilerandom

beamforming may decrease the number of neighbors of a node,

it leads to an overall improvement in the network connectivity

[17]. A similar conclusion, based on simulation studies, was

drawn in [19], where the performance of random beamforming

is compared to center-directed beamforming, in which each

node in the network points its main beam toward the geographic

center of the network. A major limitation of the work in

[16]–[19] is that it only relies on simulations and provides no

analytical solutions. In [20], an analytical study was used to

show that random beamforming can both increase and decrease

the number of isolated nodes and network connectivity, but no

insight was given into when (i.e., under what channel condi-

tions)thisoccurs.Theaforementionedlimitationsareaddressed

in this paper.

A. Approach and Contribution

We present an analytical method to characterize the perfor-

mance of beamforming in the presence of both path loss and

shadowing. We extend the work in [13] for ad hoc networks

with omnidirectional antennas to include the effects of beam-

forming. We investigate both the local and the overall connec-

tivity with beamforming: 1) connectivity from the viewpoint of

a single node (probability of node isolation) and 2) connectivity

from the viewpoint of the entire network (1-connectivity and

path probability).

Our initial work in [21] considered the local connectivity

performance with random beamforming in simple path loss

channels, which was extended to 1-connectivity performance in

[22]. Our contribution and innovation in this paper differ from

[13] and our previous research in [21] and [22] in three major

respects. First, we propose a realistic unbiased shadowing

channel model that removes the bias in the average received

power due to the lognormal spread of the shadowing. Using the

unbiased shadowing channel model, the effect of shadowing

on the network connectivity is fundamentally different from

the existing results in the literature. Second, we consider both

the exact and a simplified beam pattern model in the analysis

of random beamforming and extend our analytical approach

to include the case of center-directed beamforming. Third, in

addition to new analytical results, we include simulation results

for path probability, which is a relatively moderate metric for

the overall network connectivity compared with 1-connectivity.

We show that the impact of shadowing or beamforming on

path probability can be different from that on 1-connectivity.

The analytical and simulation results in this paper provide

fundamental insights into how the channel and beamforming

conditions affect the connectivity properties of wireless ad hoc

networks.

The following is a summary of the main results in this paper.

1) We present a simple intuitive method to calculate the

effective coverage area of a node, taking beamforming

and a channel model incorporating path loss and unbiased

shadowing into account.

2) For a fixed path loss exponent α, we prove that the pres-

ence of shadowing always reduces the effective coverage

area of a node, thereby increasing the probability of node

isolation and reducing the network 1-connectivity. This

interesting result, which is the opposite of the conclusions

in [10], [13], and [22], is because of the realistic unbiased

shadowing channel model employed in this paper. In ad-

dition, we prove that the detrimental effect of shadowing

reaches its maximum at α = 4.

3) We show that moderate shadowing can improve the path

probability between two nodes, while it always reduces

the network 1-connectivity. This improvement in the path

probability is due to the randomness introduced in the

communication range of a node.

4) We show that the impact of beamforming strongly de-

pends on the level of the channel path loss. In particular,

beamforming improves the connectivity, compared with

the use of omnidirectional antennas, for a path loss expo-

nent of α < 3.

5) Comparing random beamforming and center-directed

beamforming, we find that both schemes give similar

performance for the local and the overall network con-

nectivity, with center-directed beamforming slightly out-

performing random beamforming.

The rest of this paper is organized as follows. In Section II,

we present the antenna and channel model. In Section III, we

derive the effective coverage area of a node and analytically

study the impact of shadowing. Section IV studies the impact of

bothrandombeamformingandcenter-directedbeamformingon

the effective coverage area. In Section V, we use the effective

coverage area results to characterize the connectivity of wire-

less ad hoc networks. In Section VI, we validate the proposed

model by comparing with simulation results and investigate the

local and the overall network connectivity. Finally, conclusions

are drawn in Section VII.

II. SYSTEM MODEL DESCRIPTION

Consider a wireless ad hoc network, as shown in Fig. 1.

The nodes with beamforming antennas are assumed to be ran-

domly distributed in a 2-D space according to a Poisson point

process. A homogeneous Poisson process provides an accurate

model for a uniform distribution of nodes as the network area

approaches infinity [25]. Let ρ denote the node density in

nodes per square meter. The probability mass function of the

number of nodes X in an area A is given by P(X = x) =

(μx/x!)e−μ, where the Poisson distribution parameter μ = ρA

is the expected number of nodes in the area A.

A. Antenna Model

We assume that all nodes are equipped with identical beam-

forming array antennas for transmission and reception. The

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ZHOU et al.: CONNECTIVITY ANALYSIS OF WIRELESS AD HOC NETWORKS WITH BEAMFORMING 5249

Fig. 1.

antennas, where the beams are pointed in random directions. The two indicated

nodes are a distance d apart.

Wireless ad hoc network with nodes equipped with beamforming

antennas are lossless and devoid of any mutual coupling. In

general, the antenna gain is given by [26]

G =

|E(θ,φ)|2

0|E(θ,φ)|2sin(θ)dθdφ

1

4π

?2π

0

?π

(1)

where φ ∈ [0,2π) is the angle from the x-axis in the xy plane,

θ ∈ [0,π) is the angle from the z-axis, and E(θ,φ) is the

electric field strength of the antenna array.

We consider that a uniform-circular-array (UCA) config-

uration is employed at each node. A UCA configuration is

chosen because a circular array has a single main lobe, and

the beamwidth of the main lobe is almost independent of the

main beam direction [27]. By comparison, a uniform linear

array has two main lobes due to symmetry, and the beamwidth

significantly varies with the main beam direction [27]. This can

lead to an increase in the interference level and an ambiguity

in the direction of the incoming signal at a receiver [28]. In

addition, UCAs are known to outperform uniform rectangular

arrays [29]. It should be noted that our analytical results on

beamforming in Section IV and connectivity in Section V are

general and apply to any antenna array configuration.

For a UCA of M identical antenna elements, the electric field

strength is given by [27]

E(θ,φ) =

M

?

m=1

E0γmexp[jkasin(θ)cos(φ − φm)]

(2)

where E0is the electric field pattern of the constituent omnidi-

rectional antennas, which is set to 1 without loss of generality,

a is the radius of the circular array, k = 2π/λ is the wave

number, λ is the wavelength of the propagating signal, φm=

2πm/M is the angular position of the mth element, and γm

is the complex excitation for each antenna element. Since

the nodes are located on the 2-D xy plane, we consider all

beamforming directions to be on the xy plane as well. For

classical 2-D beamforming, γmis given by [27]

γm= exp[−jkasin(θ0)cos(φ0− φm)]

(3)

where θ0= π/2 (i.e., the xy plane), and φ0is the azimuth angle

of the desired main beam.

Substituting (3) into (2) and (2) into (1), we can calculate the

antenna gain for any azimuthal angle φ. Note that the resulting

antenna gain G from (1) is a function of φ, φ0, and M. Thus,

we denote it as G(φ,φ0,M). In the remainder of this paper, we

will use the antenna gain G(φ,φ0,M) to evaluate the impact of

beamforming on the network connectivity.

B. Channel Model

We assume that the channel gain between a transmitting and

receiving node pair is affected by path loss attenuation and

shadowing effects. The severity of the path loss is characterized

by the path loss exponent α, which usually ranges from 2 to 5

[30]. The shadowing S is modeled as a random variable drawn

from a log-normal distribution given by

S = 10w/10

(4)

where w is a Gaussian random variable with zero mean and

standard deviation σ (hence, S is normal in decibels) [31].

A typical value of σ ranges from 4 to 13 dB [32]. Note that

both path loss and shadowing are multiplicative factors of the

received signal power.

Let PT denote the transmit power of each node. With path

loss and shadowing, the received signal power PRis given by

PR= ζ

1

dαC GTGRPT

(5)

where d is the distance between the transmitting and receiving

nodes, C = (λ/(4π))2is a constant, GT and GR are the

antenna gains of the transmitting and receiving nodes, respec-

tively, ζ = S/E[S] is the normalized shadowing variable, and

E[·] denotes statistical expectation.

It is important to note that, unlike the signal models in [10],

[12], [13], and [22], we normalize the shadowing term S by

its mean value E[S] in the shadowing variable ζ. In practi-

cal scenarios, shadowing is caused by the variation of local

propagation conditions at different locations. The presence of

shadowingintroducesavariationinthereceivedsignalstrength,

but it does not change the average value determined by the path

loss model [30]. The proposed normalization in ζ removes the

bias in the received signal power so that the average received

power does not artificially increase with the lognormal spread

of the shadowing. The consequences of this normalization will

be discussed in the next section.

III. EFFECTIVE COVERAGE AREA ANALYSIS

In this section, we derive the effective coverage area of a

node, taking into account beamforming, path loss, and shad-

owing. Without loss of generality, we can normalize (5) with

respect to the constant C so that the power attenuation is

expressed as

β(d) =PT

PR

=1

ζ

dα

GTGR.

(6)

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5250IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 9, NOVEMBER 2009

Assuming identical node hardware and negligible internode

interference, two nodes separated by a distance d are connected

if β(d) < βth, where βthis the threshold signal power attenu-

ation. From (6), the probability of having no direct connection

between two nodes separated by a distance d is given by

?1

=P

(βthζGTGR)

P(β ≥ βth) =P

ζ

dα

GTGR

≥ βth

?

?

1

α≤ d

?

.

(7)

For simplicity, we define the random variable R as

R = (βthζGTGR)

1

α.

(8)

Substituting (8) into (7), we get P(β ≥ βth) = P(R ≤ d).

Hence, the random variable R can be referred to as the commu-

nication range. That is, the node is able to communicate with

all nodes lying within a distance R. The coverage area of a node

can thus be considered as a disk with radius R centered at the

node. Since R is a random variable, the effective coverage area

is defined as the expected value of the coverage area given by

E[πR2] = πE[R2], where E[R2] is the second moment of the

communication range. As shown in [13], the effective coverage

area is strongly related to the connectivity properties of the

network. Hence, we now study the impact of shadowing and

beamforming on the effective coverage area E[πR2].

We assume that the nodes choose the beamforming strat-

egy independently of the shadowing effects at their locations.

From (8), we have

?

=π(βth)2/αE[ζ2/α]E

E[πR2] =πE

(βthζGTGR)

2

α

?

?

(GTGR)2/α?

.

(9)

From (9), we can see that the effects of the shadowing and

beamforming on the effective coverage area are characterized

by the shadowing factor E[ζ2/α] and the beamforming factor

E[(GTGR)2/α], respectively. In the following, we present the

results of the shadowing factor. The analysis of the beamform-

ing factor will be performed in the next section.

Theorem 1: The effect of shadowing on the effective cover-

age area of a randomly chosen node is given by

??ln10

Proof: See the Appendix.

From (10), we see that the effect of shadowing depends

on both the lognormal standard deviation σ and the path loss

exponent α. Since α > 2 for any practical wireless channel,

the exponent in (10) is always negative. Therefore, Theorem 1

implies that shadowing always results in a reduction of the ef-

fective coverage area. This result contradicts previous results in

[10], [13], and [22], which do not normalize the shadowing fac-

tor. It must be noted that if the normalization is not used, then

the impact of shadowing on the effective coverage area is given

by E[S2/α] = exp{(σ ln10/5α)2/2} (see the Appendix),

where the exponent is always positive, and consequently, shad-

E[ζ2/α] = exp

10

σ

?2?2 − α

α2

??

.

(10)

Fig. 2.

values of the shadowing lognormal standard deviation σ (in decibels).

Shadowing factor in (10) versus path loss exponent α for different

owing always increases the coverage area, as concluded in [10],

[13], and [22].

Furthermore, by examining the first and second derivatives

of the shadowing factor E[ζ2/α] in (10) w.r.t. α, the following

corollary can be obtained.

Corollary 1: For any fixed value of σ, the shadowing factor

reduces as α increases from 2 to 4, reaching a minimum at

α = 4, and increases as α increases beyond 4.

Corollary 1 implies that shadowing results in a maximum

reduction of the effective coverage area at α = 4 for any fixed

σ. Similarly, one can fix α and investigate the effect of σ

on the shadowing factor. For a fixed α, the first term in the

exponent in (10) implies that the shadowing factor decreases

as σ increases. Fig. 2 shows the shadowing factor versus α for

different values of σ, which confirms the results in Corollary 1.

Furthermore, we see from Fig. 2 that the shadowing factor does

not change much with α for α > 3. On the other hand, the

shadowing factor significantly varies with σ for any fixed α.

IV. BEAMFORMING ANALYSIS

In this section, we study the effects of random beamform-

ing and center-directed beamforming on the effective cover-

age area, which is characterized by the beamforming factor

E[(GTGR)2/α].

A. Random Beamforming

Random beamforming is a simple scheme that requires

no knowledge of the positions of individual nodes or any

geographical information about the network. It allows each

node in the network to randomly select a main beam di-

rection. Fig. 3(a) shows a pair of transmitting (TX) and re-

ceiving (RX) nodes in a random beamforming scenario. The

parameters shown in the figure are defined as follows:

d = distance between the TX and RX nodes, φ = direction of

the RX node from the TX node, with respect to the

x-axis, φT= main beam direction of the TX node, and φR=

main beam direction of the RX node.

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ZHOU et al.: CONNECTIVITY ANALYSIS OF WIRELESS AD HOC NETWORKS WITH BEAMFORMING5251

Fig. 3.Relative positions of a transmitting and receiving node pair in (a) random beamforming and (b) center-directed beamforming.

TABLE I

BEAMFORMING FACTOR FOR RANDOM BEAMFORMING

For a network utilizing random beamforming, the pdf of the

main beam angle has a uniform distribution. Since the nodes

are deployed according to a uniform distribution, the direction

of any other node from a chosen node has a uniform distrib-

ution as well. Therefore, angles φ, φT, and φRhave identical

but mutually independent pdf’s, being uniformly distributed

over [0,2π). Using this argument, the beamforming factor is

given by

?

2π

?

000

E

(GTGR)

2

α

?

=

1

(2π)3

·

2π

?

2π

?

(G(φ,φT,M)G(π+φ,φR,M))

2

αdφRdφTdφ

(11)

where G(φ,φT,M) and G(π + φ,φR,M) are the transmit and

receive antenna gains, which can be determined from (1).

In general, a closed-form expression for the beamforming

factor in (11) with an exact expression for the antenna gains

cannot be obtained. However, it can be seen from (11) that

the beamforming factor depends on the path loss exponent α

which usually ranges from 2 to 5, and the number of antenna

elementsM,whichisusuallylessthan10inpracticalscenarios.

Hence, the form of the aforementioned integral is such that it

can quickly and accurately be numerically computed for the

values of α and M in the range of interest. Therefore, the

computational complexity of studying the beamforming factor

using the exact beam pattern is still low.

Using the antenna gain of a UCA given in Section II-A,

we numerically evaluate (11), and the results are summarized

in Table I for different values of α and M. Note that, for

omnidirectional antennas, the beamforming factor is unity. We

can see from Table I that the beamforming factor decreases as

α increases. However, for a fixed α, the beamforming factor

stays relatively constant over a practical range of M (e.g.,

M < 10). For α < 3, the beamforming factor is greater than

unity, and beamforming increases the effective coverage area.

On the other hand, it decreases the effective coverage area

when α > 3.

Analysis With a Simplified Beam Pattern Model: A simple

beam pattern model was proposed in [33], where the beam

pattern consists of a flat main lobe and a flat sidelobe, which

is also referred to as the keyhole beam pattern. The param-

eters associated with the model are the main-lobe width ω

(normalized w.r.t. 2π) and the sidelobe attenuation factor ν,

with the main-lobe gain being M and the sidelobe gain being

νM. Parameters ω and ν are determined based on a given beam

pattern by preserving the first- and second-order moments of

the beam pattern, i.e., ωM + (1 − ω)νM = E[G(φ,φ0,M)],

and ωM2+ (1 − ω)(νM)2= E[G2(φ,φ0,M)]. This model

was found to be accurate for the capacity and outage probability

analysis of beamforming in wireless cellular systems [33], [34].

Using this simplified model, it can be shown that (11),

after some manipulations, reduces to a closed-form expression

given by

?

where

E?G2(φ,φ0,M)?− (E [G(φ,φ0,M)])2

ν =1

ME [G(φ,φ0,M)] − M

We compute the beamforming factors using the simplified

model given in (12) (the results are not shown for brevity).

Similar to Table I, we observe that the beamforming factor

decreases as α increases. However, the actual values calculated

using (12) are usually different from those in Table I. The

reason for this difference is that the simplified model only

matches the first and second moments of G(φ,φ0,M), while

E

(GTGR)

2

α

?

=

?

ωM2/α+ (1 − ω)(νM)2/α?2

(12)

ω =

E [G2(φ,φ0,M)] − 2E [G(φ,φ0,M)]M + M2

E?G2(φ,φ0,M)?− E [G(φ,φ0,M)]M

(13)

.

(14)

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5252IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 9, NOVEMBER 2009

TABLE II

BEAMFORMING FACTOR FOR CENTER-DIRECTED BEAMFORMING

the computation of the beamforming factor in (11) requires an

accurate match of G2/α(φ,φ0,M), with 2 ≤ α ≤ 5. Thus, we

can conclude that the simplified beam pattern model proposed

in [33] for the capacity and outage analysis of beamforming in

cellular networks is not suitable for the connectivity analysis of

beamforming in ad hoc networks.

B. Center-Directed Beamforming

The major drawback of random beamforming is the border

effect. That is, the nodes located at the border of the network

may happen to steer their main beams in a direction outside the

network area, causing loss of connectivity [17]. If the area of

the network is relatively fixed, it is possible for the nodes to

be given or learn the geographical information of the network.

The cost and complexity of obtaining the information on the

center of the network is much lower than that of obtaining the

positions of individual nodes. In this scenario, center-directed

beamforming can be adopted, where each node points its main

beam toward the center of the network.

Fig. 3(b) shows a scenario with center-directed beamform-

ing, where each node directs its main beam toward the ge-

ographical center of the network. In Fig. 3(b), dC denotes

the distance of the TX node from the geographical center of

the network, and the remaining parameters are defined as in

Fig. 3(a).

To simplify the analysis, we make the assumption that the

distances between the directly (i.e., one-hop) connected node

pairs are much smaller than their distances from the geograph-

ical center of the network, such that d ? dC. This assumption

is reasonably accurate in many practical situations where the

majority of one-hop connected node pairs are a relatively large

distance from the center of the network, compared with the

distance between the node pairs themselves. Using this assump-

tion, it can be shown that the one-hop-connected node pairs

have approximately the same main beam direction, as they all

point their main beams toward the center, such that φR≈ φT.

By letting φR= φT for all node pairs, we can establish a

simplified model to approximate the beamforming factor for the

center-directed beamforming scheme.

As previously discussed, for a uniformly distributed network,

φ and φT have uniform distributions over [0,2π). Therefore,

an approximation of the beamforming factor for center-directed

beamforming can be expressed as

E

?

(GTGR)

2

α

?

(G(φ,φT,M)G(π+φ,φT,M))

≈

1

(2π)2

2π

?

0

2π

?

0

2

αdφTdφ.

(15)

We can see that (15) depends on the path loss exponent α

and the number of antenna elements M, and it can quickly and

accurately be numerically computed for the values of α and

M in the range of interest. Therefore, we numerically evaluate

(15) and summarize the results in Table II. Similar to random

beamforming, we also compute the beamforming factor using

the simplified beam pattern model (the results are not shown

for brevity)1and find that the values are usually different from

the ones in Table II, which again indicates that the simplified

beam pattern model is not suitable for the connectivity analysis

of beamforming in wireless ad hoc networks.

Comparing the results in Tables I and II, we see that the

beamforming factors for random beamforming and center-

directed beamforming behave very similarly, with the beam-

forming factors for center-directed beamforming being slightly

higher for given values of α and M. Therefore, the effective

coverage area of a node is slightly larger in a center-directed

beamforming network than that in a random beamforming

network.

V. CONNECTIVITY ANALYSIS

In this section, we characterize the connectivity of wire-

less ad hoc networks with beamforming. We consider both

the local network connectivity (probability of node isolation)

and the overall network connectivity (1-connectivity and path

probability).

A. Local Network Connectivity

The probability of node isolation, which is denoted by

P(iso), is defined as the probability that a randomly selected

node in an ad hoc network has no connections to any other

nodes. It is a measure of the local network connectivity, and

its general expression is given by [9]

P(iso) = exp{−E[D]}

(16)

where D denotes the node degree, which is defined as the

number of direct links that any given node has to other nodes.

For a node deployment following a homogeneous Poisson point

process with density ρ, the node degree has a Poisson distribu-

tion with parameter ρE[πR2] [13]. Therefore, the average node

degree E[D] is given by

E[D] =ρE[πR2]

=ρπ(βth)2/αE[ζ2/α]E

?

(GTGR)2/α?

(17)

1With the simplified beam pattern model given in [33], it can be

shown that the beamforming factor in (15) reduces to E[(GTGR)2/α] ≈

2ων2/αM4/α+ (1 − 2ω)(νM)4/α.

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ZHOU et al.: CONNECTIVITY ANALYSIS OF WIRELESS AD HOC NETWORKS WITH BEAMFORMING5253

where (9) is used to obtain (17). Substituting (17) into (16), we

can determine the probability of isolation with beamforming

to be

?

where E[ζ2/α] is given by (10), and E[(GTGR)2/α] is given

by (11) or (15).

Remark 1: From Theorem 1 in Section III, we know that

the shadowing factor E[ζ2/α] is always negative for wireless

channels with α > 2. Therefore, the presence of shadowing

always increases the probability of node isolation given in

(18). Furthermore, from the values of the beamforming factor

E[(GTGR)2/α] given in Tables I and II, we conclude that the

use of beamforming, compared with omnidirectional antennas,

reduces the probability of node isolation when the path loss

exponent is lower than 3, and it increases node isolation when

the path loss exponent is higher than 3. It must be noted that the

reduction in the local connectivity is not always detrimental.

For example, the reduction in the node degree may result in a

reduction in the interference level if the internode interference

needs to be considered.

P(iso)=exp

−ρπ(βth)2/αE[ζ2/α]E

?

(GTGR)2/α??

(18)

B. Overall Network Connectivity

1-connectivity, which is denoted by P(1-con), is defined as

the probability that every node pair in the network has at least

one path connecting them. It is a relatively strong measure

of the overall network connectivity. An upper bound for the

network 1-connectivity is given by [9]

P(1 − con) < exp{−ρAP(iso)}

where A is the area of the network, P(iso) is given in (16),

and exp{−ρAP(iso)} is the probability of no isolated nodes in

the network. It can be shown from the theorems of geometric

random graphs [7] that the bound for 1-connectivity in (19) is

tight as the probability approaches unity for a path-loss channel

[9]. The tightness of the bound at high connectivity has also

been found for shadowing channels [10].

For1-connectivityanalysis,wefocusonthenodedensitythat

yields an almost surely connected network, that is, the density

atwhichP(1-con) = 0.99[10].Thisisreferredtoasthecritical

node density, which is denoted by ρc. Since the bound in (19)

is very tight at P(1-con) ≈ 1, we can use it to calculate ρc. The

critical node density can be solved from (19) using Lambert’s

W function as

?E[πR2]ln0.99

= −

?ln0.99

where W−1 denotes the real-valued nonprincipal branch of

Lambert’s W function.

(19)

ρc= −

1

E[πR2]W−1

A

?

1

π(βth)2/αE[ζ2/α]E?(GTGR)2/α?

W−1

A

π(βth)2/αE[ζ2/α]E

?

(GTGR)2/α??

(20)

Remark 2: With the upper bound on 1-connectivity given in

(19), we know that 1-connectivity reduces as the probability of

node isolation increases. Since the presence of shadowing and

the use of beamforming at a high path loss exponent (α>3)

increase node isolation, a higher critical node density ρc is

required to establish an almost surely connected network in

these scenarios. On the other hand, ρccan be reduced by using

beamforming when the path loss exponent is low (α < 3).

Another measure of the overall network connectivity is path

probability, which is denoted by P(path) and is defined as the

probability that two randomly chosen nodes are connected via

either a direct link or a multihop path. It is a relatively moderate

metric compared with 1-connectivity. An analytical expression

for the path probability is still an open research problem [17].

Therefore, we will carry out simulations in the next section to

illustrate the effects of beamforming on path probability. Our

analytical approach, however, allows us to make the following

intuitive observations regarding the effects of shadowing and

beamforming on path probability.

Remark3: Theeffectsofshadowing andtheeffectsofbeam-

forming, particularly random beamforming, are very similar, as

both create randomness in the node communication range. The

presence of shadowing introduces randomness in the received

signal power for nodes at different locations. On the other hand,

a beamforming node loses links to closely located neighbors in

some directions, while it creates links to nodes that are farther

away in other directions. Furthermore, it has been shown in [17]

and [19] that a reduction in the local network connectivity may

result in an improvement in the path probability by the use of

beamforming, particularly in sparse networks. A beamforming

nodecreateslinkstonodesthatarefartherawayincertaindirec-

tions, and it is the long links that improve the path probability.

As the effect of shadowing is very similar to that of beamform-

ing, we can expect that the presence of shadowing may improve

the path probability, particularly in sparse networks, although it

reduces the local network connectivity.

VI. RESULTS

In this section, we present the numerical results of the effect

of shadowing and beamforming on the connectivity of wireless

ad hoc networks. We verify the analytical models and insights

given in Sections III, IV, and V by comparing them with

simulations. In the simulations, nodes are randomly distributed

accordingtoauniformdistributiononasquareofareaB m2.To

eliminate border effects, we use the sub-area simulation method

[9], such that we only compute the connectivity measures for

nodes located on an inner square of area A m2, where A is

sufficiently smaller than B. The simulation results are then cal-

culated by averaging over 5000 Monte Carlo simulation trials.

A. Effect of Shadowing

To validate the analytical model for studying the effect of

shadowing in Section III, we compare the analytical and simu-

lation results on the probability of node isolation. Fig. 4 shows

the probability of node isolation versus node density for βth=

50 dB, M = 1 (omnidirectional antennas), α = 2.5,3,and 4,

and σ = 0,4,and8dB. The analytical results shown using lines

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5254IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 9, NOVEMBER 2009

Fig.

with

eters

(lines = analytical results from (18) and markers = simulation results).

4.

M = 1

βth= 50

Probabilityof nodeisolation

antennas)

versus

and

and

node

system

density

param-(omnidirectional

dB,

α = 2.5,3,and

4,

σ = 0,4,and8

dB

TABLE III

CRITICAL NODE DENSITY: EFFECT OF SHADOWING

are calculated from (18), and the simulation results are indi-

cated by markers. We see that the simulation results are in

excellent agreement with the analytical results in all cases. For

a fixed α, we see that P(iso) increases as σ increases. That

is, shadowing results in more isolated nodes in the network. In

addition, we see that the negative effect of shadowing becomes

morenoticeable asα increasesfrom2.5to4.Thesetrendsagree

with our earlier observations in Sections III and V-A.2

For 1-connectivity, we focus on the critical node density ρc,

which ensures an almost connected network, i.e., P(1-con) =

0.99. Table III summarizes the values of ρcfor system param-

eters βth= 50 dB, A = 105m2, and M = 1. It also includes

the corresponding ρcfrom [10] for comparison. Our results in

Table III show that, for any given α, the presence of shadowing

increases the critical node density, while the results in [10]

show the opposite trend due to the absence of normalization

in the shadowing factor. For example, at α = 3, we see that

the network requires an additional 11% of the total number

of nodes to be almost surely connected from a non-shadowing

channel to a shadowing channel with σ = 4 dB. This increases

to an additional 52% of the total number nodes required if

σ = 8 dB. We also see that the percentage increase in the

critical node density due to shadowing becomes larger when

α increases from 3 to 4, which agrees with Corollary 1 in

Section III. These results show that the presence of shadowing

is detrimental for 1-connectivity.

2We have also verified our analytical results for other values of M and have

observed the same trends. The results are not shown for brevity.

Fig. 5.

tional antennas) and system parameters βth= 50 dB, α = 3, and σ =

0,4,8,and 12 dB.

Path probability versus node density with M = 1 (omnidirec-

Fig. 6.

4 antennas and system parameters βth= 50 dB, α = 2.5,3, and 4, and

σ = 4 dB (lines = analytical results from (18), markers = simulation results,

RB = random beamforming, and CDB = center-directed beamforming).

Probability of node isolation versus node density with M =

As mentioned in Section V-B, an analytical expression for

the path probability is still an open research problem; hence, we

present simulation results and focus on the overall trends. Fig. 5

shows the path probability versus node density for M = 1, α =

3, βth= 50 dB, A = 2.5 × 105m2, and different shadowing

environments with σ ranging from 0 (no shadowing) to 12 dB.

Unlike 1-connectivity, we see that the presence of moderate

shadowing, e.g.,σ = 4or8dB,significantlyimproves P(path).

For example, P(path) at ρ = 6 × 10−4m−2for σ = 0 dB is

0.35, and that for σ = 8 dB is 0.92. This agrees with our

expectation in Section V-B that the presence of shadowing

may increase the path probability, although it reduces the local

network connectivity. However, P(path) does not always in-

crease with σ. As shadowing becomes severe, e.g., σ = 12 dB,

P(path) is lower than that achieved in a moderate shadowing

environment. We confirmed that these trends are also observed

at other values of α.

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TABLE IV

CRITICAL NODE DENSITY: EFFECT OF BEAMFORMING

B. Effect of Beamforming

Fig. 6 shows the probability of node isolation versus node

density for βth= 50 dB, α = 2.5,3,and4, σ = 4 dB, and

M = 4 antennas. The analytical results for random and center-

directed beamforming are shown in solid lines and dashed lines,

respectively.Theanalyticalresultsforomnidirectionalantennas

are shown in dash-dotted lines for reference. The simulation

results for random beamforming are again in excellent agree-

ment with the analytical results in all cases, which validates

the analytical model for random beamforming presented in

Section IV-A. For center-directed beamforming, the simulation

results are accurate for α = 4. Although the accuracy slightly

reduces as α decreases from 4 to 3 and 2.5,3the analytical

model presented in Section IV-B provides a reasonably good

approximation. From Fig. 6, we see that the use of beamform-

ing results in a lower P(iso) when α < 3 and a higher P(iso)

when α > 3. This agrees with our analytical results for the

beamforming factor in Sections IV and V-A.4

Table IV illustrates the effect of beamforming on the critical

node density ρc for system parameters βth= 50 dB, A =

106m2, and σ = 8. We see that the use of beamforming de-

creases ρcat α = 2.5, while it increases ρcat α = 3 and 4. This

implies that beamforming improves 1-connectivity at small

path loss exponents and reduces 1-connectivity at large path

loss exponents. We also see that center-directed beamform-

ing slightly outperforms random beamforming in all cases.

These trends agree with the analytical results in Sections IV-A

and IV-B.

Next, we present the simulation results for path probability.

We focus on the general trends of the effect of random and

center-directed beamforming. Fig. 7 shows the path probability

versus node density with M = 4 in a moderate shadowing

environment, where σ = 8 dB for networks utilizing random

beamforming, center-directed beamforming, and omnidirec-

tional antennas. As the impact of beamforming strongly de-

pends on the path loss exponent α, we include P(path) for both

α = 3 and α = 4. Note that the threshold power attenuation

βth is chosen to be different in the two scenarios so that

all results can clearly be shown in one figure. By doing so,

3As α reduces, the distances between the directly connected node pairs in-

creases. Therefore, the assumption of d ? dCin the model for center-directed

beamforming becomes less accurate, which results in the slight mismatch

between the analytical and simulation results.

4We have also verified our analytical results for other values of σ and have

observed the same trends. The results are not shown for brevity.

Fig. 7.

system parameters βth= 50,67 dB, α = 3 and 4, and σ = 8 dB (RB =

random beamforming and CDB = center-directed beamforming).

Path probability versus node density with M = 4 antennas and

one can easily compare the connectivity improvement from

beamforming for different values of α. However, one cannot

directly compare the value of P(path) for α = 3 with that for

α = 4.WeseefromFig.7thattheuseofbeamformingprovides

a certain improvement in P(path) at α = 3 for a relatively

low node density, i.e., sparse networks. Moreover, we see that

center-directed beamforming generally outperforms random

beamforming. For example, P(path) at ρ = 4 × 10−4m−2is

0.62 for random beamforming and 0.73 for center-directed

beamforming. These trends are different from those observed

in the non-shadowing environment in [19], where it was found

that random beamforming usually outperforms center-directed

beamforming. In addition, we see that the improvement in

P(path) by either beamforming technique is negligible at α =

4. Since the beamforming factor decreases with α, we can

expect that there is little improvement or even a reduction in

path probability by using beamforming when α > 4.

C. Effect of Non-Uniform Distribution

In this paper, we have considered a uniform distribution of

nodes in the network. It can be argued that this assumption is

more accurate in the initial stage after the nodes have uniformly

been deployed at random. Due to node mobility, the spatial

distributionmaygraduallybecomenon-uniform,withthenodes

tending to form clusters in the network [35]. The analysis for

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5256IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 9, NOVEMBER 2009

TABLE V

SUMMARY OF RESULTS

the case of non-uniform distribution is outside the scope of

this paper. However, using the insights provided in this paper,

we can make the following observation regarding the effect

of non-uniform distribution. Since beamforming increases the

maximum communication range, it can assist nodes in different

clusters to communicate with each other, i.e., build bridges

between the clusters. Therefore, we expect that beamforming

performs better in a non-uniformly distributed network than in

a uniformly distributed network. The results of the effect of

beamforming presented in this paper can then be seen as worst-

case scenario results.

VII. CONCLUSION

In this paper, we have proposed an analytical model to

investigate the effects of shadowing and beamforming on the

connectivity of wireless ad hoc networks. Both random and

center-directed beamforming schemes for nodes equipped with

multiple antennas have been considered. We have derived the

effective coverage area of a node, taking into account path loss,

shadowing, and beamforming. A shadowing factor and a beam-

forming factor have been defined to characterize the effects

of shadowing and beamforming on the local and the overall

connectivity of the network. The accuracy of our analytical

model has been verified by comparison with simulation results.

Table V summarizes the important findings in this paper.

APPENDIX

PROOF OF THEOREM 1

To prove Theorem 1, we need to use the following result

given in [25].

Lemma 1: If a random variable Z = lnY has a normal

distribution with mean and standard deviation given by μZand

σZ, the mean of Y is given by E[Y ] = exp{μZ+ (σ2

Taking the natural logarithm of S in (4), we get

Z/2)}.

lnS = ln(10w/10) =w

10ln10 =

?ln10

10

?

w

(21)

where w is a Gaussian random variable with zero mean and

standard deviation σ. Therefore, using Lemma 1, the expected

value of S is given by

E[S] = exp

??ln10

10σ?2

2

?

.

(22)

Similarly, the expected value of S2/αis given by

E[S2/α] = E[10w/5α] = exp

??σ ln10

5α

2

?2

?

.

(23)

From (22) and (23), the impact of shadowing on the effective

coverage area is given by

E[ζ2/α] =

E[S2/α]

(E[S])2/α=

exp

?(σ ln 10

?(ln 10

?2?2

?2?2 − α

5α )

2

2

?

?

??

exp

10σ)

α

2

= exp

??ln10

??ln10

10

σ

α2−1

α

??

= exp

10

σ

α2

.

(24)

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers for

their valuable comments.

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pp. 1–5.

Xiangyun Zhou (S’08) received the B.E.(Hons.)

degree in electronics and telecommunications en-

gineering in 2007 from The Australian National

University, Canberra, ACT, Australia, where he is

currently working toward the Ph.D. degree in en-

gineering and information technology with the

Research School of Information Sciences and En-

gineering, College of Engineering and Computer

Science.

His research interests are in signal processing

for wireless communications, including multiple-

input–multiple-output systems, ad hoc networks, and relay and cooperative

networks.

Salman Durrani (S’00–M’05) received the B.Sc.

(first-class honors) degree in electrical engineering

from the University of Engineering and Technol-

ogy, Lahore, Pakistan, in 2000 and the Ph.D. de-

gree in electrical engineering from the University of

Queensland, Brisbane, Australia, in December 2004.

Since March 2005, he has been a Lecturer with

the College of Engineering and Computer Science,

The Australian National University, Canberra, ACT,

Australia. He has 32 publications to date in ref-

ereed international journals and conferences. His

current research interests include wireless and mobile networks, connectivity

of sensor/ad-hoc networks and vehicular networks, channel estimation, and

multiple-input–multiple-output and smart antenna systems.

Dr. Durrani is a Member of the Institution of Engineers, Australia. He was

the recipient of a University Gold Medal during his undergraduate studies and

an International Postgraduate Research Scholarship, which was funded by the

Australian government, for the duration of his Ph.D. studies.

Haley M. Jones received the B.E.(Hons.) degree in

electrical and electronic engineering and the B.Sc.

degree from the University of Adelaide, Adelaide,

SA, Australia, in 1992 and 1995, respectively, and

the Ph.D. degree in telecommunications engineering

from The Australian National University, Canberra,

ACT, Australia, in October 2002.

She has been with the College of Engineering and

Computer Science, The Australian National Univer-

sity, since January 2002. Her previous experience

includes time in industry and working on speech

codingwiththeCooperativeResearchCentreforRobustandAdaptiveSystems,

Canberra, Australia, from 1993 to 1999. Her research interests have included

wireless channel modeling, beamforming, and channel and topology issues in

mobile ad hoc networks. She has recently branched out into sustainable systems

with a particular emphasis on the cradle-to-cradle paradigm.

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