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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 9, NOVEMBER 20095247
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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5248 IEEE 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 BEAMFORMING5249
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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5250 IEEE 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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5252 IEEE 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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Page 8
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
Probability ofnode isolation
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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ZHOU et al.: CONNECTIVITY ANALYSIS OF WIRELESS AD HOC NETWORKS WITH BEAMFORMING 5255
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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5256 IEEE 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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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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