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Accepted in IEEE Wireless Communications Letters. This is the author’s version, for the published version please see IEEE Explore.
On the Probability of Line-of-Sight
in Urban Environments
Akram Al-Hourani, Senior Member, IEEE
Abstract—Line-of-Sight communication will play a major role
in the next generation of wireless technologies; from UAV-to-UAV
networks, to land mobile satellite services, and mmWave cellular
coverage. In this paper we present a model for predicting the ge-
ometric line-of-sight probability in different urban environments
formulated based on stochastic geometry. The model combines
simplicity of formulation with high degree of accuracy when
compared to real-life geographic data. For two points in the 3D
space, it takes the height of each point and the distance between
them as input, also it takes 4 other simple parameters that
describe the geometric properties of the underlying environment.
Based on these inputs, the proposed model produces consistent
analytic results aligned with ray-tracing of the geographic data.
Index Terms—Line-of-sight probability, mmWave communica-
tion, UAV-to-UAV channel modeling, land mobile satellite.
I. INTRODUCTION
THE availability of line-of-sight (LoS) has a large effect
on the performance of wireless channels, this influence
becomes even more important for applications like mmWave,
land mobile satellite, and UAV communications that are envi-
sioned to take a prominent role in the next wave of hybrid
communication systems. In these applications, the wireless
signal is either too week to be delivered using non-line-of-
sight modes (the case of UAV and satellites) or it does suffer
from an extreme penetration loss due to absorbent obstacles
(the case of mmWave).
Throughout the years, there have been many good models
to capture the LoS probability developed by researchers and
engineers, where the majority are specific to certain applica-
tions (or scenarios). The most known of which are the models
recommended by the International Telecommunication Union
(ITU-R M.2135-1) [1] focusing on common cellular scenar-
ios such as Macro and Micro cellular sites with restriction
on the acceptable range of nodes heights. With the advent
of mmWave in 5G, extensive empirical studies have been
conducted by researchers and industry to determine the radio
LoS probability as part of the overall efforts in studying the
propagation properties of mmWave, example papers cited here
[2]–[4] among many other good works. 3GPP is also actively
working on expanding its traditional 2D-based channel models
[5] into 3D incorporating the LoS probability, this mainly
stems from the intrinsic limitation of planner models in
capturing mmWave and high-rise indoor scenarios. Most of the
mentioned studies are based on empirical measurements with-
out the need to rely on developing complicated geometrical
models. On the other hand, the study in [6] attempts to develop
a theoretical basis for the LoS probability parameterized with
A. Al-Hourani is with the School of Engineering, RMIT University,
Australia. Email: akram.hourani@rmit.edu.au.
Fig. 1. The geometric representation for some buildings in the Central
Business District in Melbourne as part of the data used for the ray-tracing
simulations. Geodata source from the City of Melbourne Open Data [7].
three main geometric descriptors of an urban environment,
although the framework is interesting, the study does not try
to apply the model into realistic/empirical footprint data from
cities.
In this paper we present a framework for modeling the LoS
probability in different urban environments, the framework
is derived using tools from stochastic geometry representing
buildings with scattered cylinders having random heights. The
model in this paper builds upon the simplicity of the Pois-
son point process in representing randomly scattered points,
where, at the same time, it yields very reliable results when
compared with ray-tracing obtained from urban footprint data
(an example of such data is shown in Fig. 1). The model
accepts four geometric parameters, one of which is based on
the ITU recommendation [8] and the remaining are extension
to this recommendation. Based on these parameters the model
then predicts the LoS probability between two points in space
for any arbitrary distance and heights. For convenience we list
the mathematical notations in Table I.
TABLE I
NOTATIO NS AN D SYMBOLS
Symbol Units Explanation
αo- Ratio of the built-up area to the total area
λobuildings/m2Density of buildings (structures)
G(h)- CCDF of buildings heights
mo, vo[m]Mean and std. dev. of building heights
µo, σo- Log-normal distribution parameters
ro[m]Mean building radius
ho[m]Common height of the two points
h1, h2[m]Height of the first and the second point
d[m]Distance between the two points
II. GE OM ET RI C MOD EL
In order to allow a tractable formulation of the LoS prob-
ability, we model the buildings (or obstacles) as random
Accepted in IEEE Wireless Communications Letters. This is the author’s version, for the published version please see IEEE Explore.
0 10 20 30 40 50 60
Building height [m]
0
0.2
0.4
0.6
0.8
1
CDF
Dense urban
Urban
Suburban
Log-normal fit
Fig. 2. The height distribution of the buildings in three different urban
environments, comparing the log-normal model and the empirical data. The
fitting parameters [µo, σo]for this example are [2.70,0.75],[1.78,0.43],
[1.42,0.23] for the dense-urban, urban, suburban respectively.
points following the Poisson point process (PPP) with density
λo[building/m2]. Buildings are represented as cylinders of a
fixed radius roand a random height Hfollowing an arbitrary
distribution of f(H), where the choice of the distribution
does not impact the derivation leading to the main results of
this paper. However, in this study we adopt the log-normal
distribution as an example based on the empirical analysis for a
large set of buildings footprint data (around 45,000 buildings)
in three different urban environments in Victoria, Australia;
(i) suburban, (ii) urban, and (iii) high-rise urban. Fig. 2 shows
the empirical cumulative distribution function (CDF) of these
areas with the suggested log-normal model given by
F(H) = 1
2+1
2erf ln H−µo
√2σo,(1)
where µoand σoare the mean and standard deviation of the
height’s logarithm calculated as
µo= ln
mo
q1 + vo
m2
o
, σ2
o= ln 1 + vo
m2
o,(2)
where parameter moand vorepresent the mean and variance of
the heights. Thus a certain region with homogeneous buildings
is represented by the vector M= [λo, ro, µo, σo]. We later
show in Section V how these parameters can be estimated
from a given geographic representation of the buildings in the
study region. Such representation is usually stored as a list of
polygons tagged with a particular height, where an example
is shown in Fig. 1.
III. MODELING LINE-OF -SIG HT PRO BABILITY
A. The Special-Case (Scenario 1)
We start with the simplified scenario when the two points A
and Bare at the same height hofrom the ground. In this case
the the geometrical line-of-sight condition will be achieved
when no building is obstructing the straight line connecting
the two points Aand B. This condition is equivalent to having
the connecting rectangle with the two inward semicircle ends
void of any building centers (seeds). The exact shape of this
region (also dubbed as the critical region), is illustrated in Fig.
3 and denoted as Ssuch that
S(A, B, ro)∆
= Rect(A, B, 2ro)\b(A, ro)\b(B , ro),(3)
Fig. 3. The upper figure shows the critical region Sthat has to be void of
any building seeds (point). The lower figure is the elevation view (profile) of
the line-of-sight in the generalized case.
where b(A, ro)is the circle centered at point Awith radius ro
and Rect(A, B, 2r0)is the rectangle with width 2r0having
points Aand Bon its length axis. The reason for removing
the two semi circles from the critical region is that we are
conditioning the existence of transmitters in outdoor, i.e., both
points Aand Bare conditioned to be free from obstructing
structures within radius ro. We can see here the reason for
choosing the circle for modeling the shape of the buildings
is that the orientation of the circle does not impact the
formulation of the critical region, any other shape will require
the orientation to be defined and would be orientation-specific.
Using geometric reasoning, the area of such shape can be
shown to be
|S(A, B, ro)|= 2dro−πr2
o+l(d, ro),(4)
where l(d, ro)is the area of the symmetric lens formed by
the intersection of two circles when Aand Bare too close to
each others, i.e., 0< d ≤2ro, and it is given by
l(d, ro) = (2r2
ocos−1d
2ro−1
2dp4r2
o−d2: 0 < d ≤2ro
0 : d > 2ro.
(5)
In the PPP process, the probability of a region Rto be
void of points is exp(−λ|R|)where λis the density of the
points and |.|is the area measure. Given the assumption of
this scenario where both points are located at height ho,
the ratio of the buildings that are above this height is the
complimentary cumulative distribution function of the heights,
i.e., G(ho)=1−F(ho). Thus the density of buildings that
are above hois λoG(ho). Accordingly, the probability of
LoS in this special-case scenario is given by PLoS1(d, ho) =
P[Φb∩ S =∅] = exp (−λoG(ho)|S|), thus
PLoS1(d, ho) = exp−λoG(ho)2rod−πr2
o+l(d, ro)(6)
where Φbis the buildings’ point process.
Accepted in IEEE Wireless Communications Letters. This is the author’s version, for the published version please see IEEE Explore.
B. The Generalized Case (Scenario 2)
In the generalized scenario we relax the common height
condition such that points Aand Bcould now have arbitrary
heights h1,h2respectively. In order to solve this case we
divide the critical region Sinto infinitesimal partitions with
a shape of meniscus lenses as shown in Fig. 4 where these
partitions S1,S2, . . . Sn, . . . SNare formed by walking along
the LoS axis with increments of ∆x. In order to satisfy the
LoS condition, all these partitions should be void of building
seeds, this condition is expressed as follows
PLoS2(d, h1, h2) = P[Φb∩ S =∅] = P"Φb
N
\
n=1 Sn=∅#,
(a)
= lim
N→∞
N
Y
n=1
exp (−λoG(h) [2ro∆x−∆l]) ,
= lim
N→∞ exp −
N
X
n=1
λoG(h) [2ro∆x−∆l]!,(7)
step (a) follows from the void probability of the PPP where
the term 2ro∆xrepresents the area of the meniscus lens Sn,
and the term ∆lis the area of the lightly shaded red region
in Fig. 4, this term is required to account for the incomplete
partitions when reaching the circle b(B, ro). The infinitesimal
area of this region is calculated as
∆l=l(d−x−∆x, ro)−l(d−x, ro),
∆l|∆x→0=−d
dxl(d−x, ro) dx=p4r2
o−(d−x)2dx.
(8)
By substituting in (7) and using the theorem of Riemann
integration, it can be shown that the summation converges to
a Riemann integral as the number of partitions N→ ∞, thus
PLoS2(d, h1, h2) = (9)
exp −Zd
0
λoG(h)h2ro−pmax (4r2
o−(d−x)2,0)idx!,
where the operation max() is added such that the effect of the
incomplete partition will only be accounted when x>d−2ro.
The height at the integration partition, as indicated in Fig. 4
can be obtained using geometrical reasoning as follows
h=x
d(h2−h1) + h1,(10)
which is the height of the line-of-sight at distance xfrom the
first point, thus we applied the thinning G(h)to reflect the
reduced density of buildings at this height. The integral in (9)
can be further simplified by adjusting the integration limit as
follows
PLoS2(d, h1, h2) = exp −2roλoZd−π
2ro
0
G(h)dx!(11)
where it can be proven, using geometric reasoning, that the
integration regions in (9) and (11) are equal for d > π
2ro1.
1which is usually the case in many practical scenarios, however, one can still
choose to use (9) instead of (11) for cases of very short distances d < π
2ro
Fig. 4. The integration region in the generalized case (scenario 2).
Fig. 5. Illustration of Monte-Carlo ray-tracing simulation setup indicating an
example of the tracing process.
As discussed earlier this is the generalized scenario when
h1and h2are two arbitrary values, where it can be shown that
the relation in (9) reduces to (6) when h1=h2=ho, i.e.,
PLoS2(d, ho, ho) = PLoS1(d, ho).
IV. MOD EL VERIFICATION
In order to check the theoretical validity of the proposed an-
alytic formulas in (6) and (11) we run extensive Monte-Carlo
simulations for different heights and distances. To achieve
this, large PPP realizations are generated based on a given
building density λowhere each point constitutes the center of
a cylinder (or a circle in case of scenario 1) with radius ro.
Then pairs of test points are placed randomly, with uniform
distribution, across the simulation region. For each of these
pairs, the connecting line is traced to find if any intersection
with the cylindrical buildings exists. If an intersection is found,
then the pair are registered as NLoS, see Fig. 5 for an example
of the line-tracing (ray-tracing) method. After running all the
generated point pairs, the estimate of the LoS probability is
found by counting the LoS pairs divided by the total number
of simulated pairs (within the same height and distance group).
For the special-case, i.e., scenario 1, we depict the results of
both the analytic and Monte-Carlo simulations in Fig. 6, where
both degrees of freedom (distance dand common height ho)
are varied. For the generalized case, i.e., scenario 2, we have
three degrees of freedom (distance d, height of the first point
h1, and height of the second point h2), accordingly we choose
to fix one variable h1=1.5 m, without the loss of generality,
and vary the remaining two variables. Both scenarios show a
high degree of matching between Monte-Carlo simulation and
the analytic results.
V. TUNING TH E GEO ME TR IC PARAMET ER S
Recall from Section II that the statistics of the geome-
try of a certain region is captured in the proposed vector
Accepted in IEEE Wireless Communications Letters. This is the author’s version, for the published version please see IEEE Explore.
0 50 100 150 200 250
Horizontal distance (d) [m]
0
0.2
0.4
0.6
0.8
1
LoS probability
h = 1.5m, 15m, 30m, 60m, 120m
Monte Carlo simulation
Analytic
Fig. 6. Scenario 1: comparing the analytic results, as per (6), with Monte-
Carlo simulations.
0 50 100 150 200 250
Horizontal distance (d) [m]
0
0.2
0.4
0.6
0.8
1
LoS probability
h2 = 1.5m, 15m, 30m, 60m, 120m
h1 = 1.5
Monte Carlo simulation
Analytic
Fig. 7. Scenario 2: comparing the analytic results, as per (11), with Monte-
Carlo simulations.
M= [λo, ro, µo, σo], these parameters can be estimated from
the geodata as follows;
•λois calculated by dividing the number of polygons to
the total area,
•rois obtained as ro=Eq|Poly|
π, where |Poly|is the
polygon area of the buildings,
•µoand σoare straight forward to get by fitting the
buildings height to the log-normal distribution.
Although the above method is a reasonable starting point, we
found that the accuracy of the model can be further enhanced
by optimizing against the empirical ray-tracing simulation (or
actual measurements if available). The optimization problem
can be formulated by defining the loss function as the Eu-
clidean distance between the analytic solution PLoS|Mand
the ray-tracing. Thus optimizing these parameters against the
ray tracing is formulated as follows
M∗= arg min
MkPLoS|M−PLoSRayTracing k2,(12)
The comparison between the analytic results and the geodata
ray-tracing simulations are depicted in Fig. 8 and Fig. 9 for
scenario 1 and scenario 2 respectively, where in these examples
we have utilized the well-established optimization function
fmincon() in Matlab with the interior-point method. The
optimized parameter vectors are M1=1
1430 ,8.07,3.04,0.9
and M2=1
50 ,1.04,1.12,1.7for scenario 1 and 2 respec-
tively in dense-urban environment. The average root mean
square error in this example is less than 2.85%.
0 50 100 150 200 250
Horizontal distance (d) [m]
0
0.2
0.4
0.6
0.8
1
LoS probability
h = 1.5m, 15m, 30m, 60m, 120m
Real data simulation
Analytic
Fig. 8. Scenario 1: comparing the analytic results, as per (6), with the ray-
tracing based on the geographic data of Melbourne.
0 50 100 150 200 250
Horizontal distance (d) [m]
0
0.2
0.4
0.6
0.8
1
LoS probability
h2 = 1.5m, 15m, 30m, 60m, 120m
h1 = 30
Real data simulation
Analytic
Fig. 9. Scenario 2: comparing the analytic results, as per (11), with the
ray-tracing based on the geographic data of Melbourne.
VI. CONCLUSION
We presented a modeling approach for the geometric line-
of-sight probability based on stochastic geometry. The model
is captured in simple-to-compute analytic formulas that adapt
based on the underlying geometric parameters of the urban en-
vironment, and it shows high-degree of consistency with real-
life geographic data. The model has wide range of applications
in radio channel predictions in next generation communication
technologies.
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