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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

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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