Graphical Interface for propagation models in urban

environments

João Pedro Apolinário

Instituto de telecomunicações, Instituto Superior Técnico

Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Abstract – This work consists on the development

of a graphical application that allows simulating

and demonstrating some of the aspects related to

the propagation of electromagnetic waves in

presence of the Earth within mobile communication

systems. This application was developed using

MATLAB® and includes the following features:

Representation of the field interference pattern due

to ground reflection, ray tracing in an atmosphere

with a given refraction index profile horizontally

stratified with or without obstacles, visualization of

the effects caused by the inversion of the refraction

index (mirages), representation of the electrical

field close to obstacles and evolution of the

electrical field in the presence of multiple buildings

in an urban environment (mobile communication

scenario).

I. INTRODUCTION

Radio propagation is the behaviour of radio waves

when they are transmitted, or propagated from one

point on the Earth to another, or into various parts of the

atmosphere. As a form of electromagnetic radiation,

radio waves are affected by the phenomena of

reflection, refraction, diffraction, absorption, polarization

and scattering [1].

The studies of all this phenomena are of great

relevance for the development of wireless and mobile

communications systems. Applications for education

purpose, allowing the visualization of the different

aspects of the propagation of electromagnetic-waves in

complex environments, find great application, not only

in education, but also in the design of mobile

communications systems and other radio wave

communication systems.

In recent years there has been a grown in the data

transferred through mobile terminals due to the

increased capacity and functionalities of these devices.

Some of the functionalities as video streaming, video

call, TV broadcast, high quality video games through

internet and others need a good binary rate and

efficiency from the systems. Therefor the mobile

communication and radio wave communication systems

need to be capable of satisfy all the requests and data

growth. This way, the radio propagation and mobile

communication models continue to have a very strong

interest in order to develop new and more sophisticated

systems.

In this paper is present a visualization tool developed

in MATLAB®

designed to allow the real-time

visualization of several phenomena related to the

propagation, reflection, refraction, and diffraction of

electromagnetic waves in an urban macro-cell

environment as well in long distance point-to-point

communication services.

This application is constituted by four modules. The

first module allows the representation of the field

strength in the presence of the ground, through a colour

graph reproducing different intensities, along a certain

distance and height for the reception antenna. The

influence of the frequency in the field is demonstrated

as well as the influence of an array of antennas and the

contribution of the polarization in the field maximums

and minimums. The second module corresponds to the

refraction module where the influence of the refraction

index of an atmosphere is demonstrated in the ray

tracing. Also is demonstrated the duct effect and the

distortion of images in super-refraction situations with

special attention to the mirage affect. The next module

is about diffraction, with the knife-edge model being

used to simulate an obstacle and the filed attenuation

due to obstacle. The influence of the frequency in de

model is demonstrated as well. A final module

simulates the field distribution in certain urban

environments, such as multi-path macro-cell based on

the Walfisch-Bertoni model as presented in [2]. In this

module is given special attention to the effect of the

reflection coefficient of the buildings and to the

frequency.

II. REFLECTION

This chapter shows the effect of the reflected ray in

the electrical field. The interference between the direct

ray and the reflected ray produces maximums and

minimums in the electrical field around a mean value

(free-space field). The polarization is another factor

analysed in this section as well as the frequency and

the use of an array of antennas as transmitter.

The equation that represents the electrical field with

the reflected ray included is shown next

[ | | ( )]

where √

represents the free-space field, the

reflection coefficient of the ground and the phase

difference. The coefficient reflection depends of the

polarization. The polarization can be vertical or

horizontal as represented in Fig. 1.

Fig. 1 - Vertical and horizontal polarizations [3]

The reflection coefficients are given by

√

√

for horizontal polarization and

√

√

for vertical polarization. The phase difference is related

with the trajectory difference between the direct ray and

the reflected ray (Fig. 2). The phase difference is

represented by

{ }

with .

Fig. 2 - Representation of direct and reflected rays [3]

From the equation of the electrical field the maximums

and minimums are given by

(

) | |

(

) | |

and they occur when ( ) or ( ) ,

which results in

{ }

The first demonstration in this section is the variation

of the electrical field in a given distance. Using vertical

polarization is possible to get the evolution and the

visualization of the maximums and minimums of the

field.

Fig. 3 - Electrical field strength with distance using vertical

polarization

As seen in Fig. 3 is possible to distinguish the

maximums and minimums of the electrical field (blue

line) around the free-space field (red line) due to the

interference of the reflected ray with the direct ray. In

this figure the distance between antennas varies.

The electrical field presents maximums and minimums

with the variation of the distance between antennas.

The same behaviour also happens with the variation of

the height of the receiver antenna. In the next figure

(Fig. 4), that evolution is shown and from a certain point

it’s possible to see once again the maximums and

minimums of the field but this time both appear in

distance and height. From the Fig. 4 it is possible to see

that are combinations of the distance and height of the

receiver antenna that can have a better signal than

others.

10-1

100

101

102

103

104

105

106

100

102

104

106

108

1010

distance [m]

E [

V/m

]

Field Strenght variation with distance

Electrical Field

Free-Space Field

surrounding

Fig. 4 - Evolution of the electrical field with ground

reflection

The next analysis consists on the influence of an array

of antennas in the electrical field. The antennas used

are half-wave dipoles and depending on the separation

between antennas and on the phase currents, the array

exhibits different behaviours as shown in Fig. 5.

Fig. 5 - Behaviour of the array with different configurations

The image in the upper left corner is the behaviour of

the array using only one antenna. In the upper right

corner the antennas are separated by with the

currents in phase. In the bottom, the antennas are

separated by but in the first the currents are in phase

and in the second they are lagged 45º. As we can see

changing the separation between the antennas change

the main lobe and that gets narrower and begins to

appear secondary lobes. The lagged in the current

change the inclination of the major lobes. Relatively to

the field, the array change the electrical field intensity

as represented in Fig. 6.

Fig. 6 - Electrical field with reflection using an array of 2 antennas separated by and with currents in phase

III. REFRACTION

The refraction index influences the ray trajectory.

Using the modified refraction index to describe different

atmospheres was possible to simulate the trajectory of

the rays and demonstrate the influence of the index in

them.

The modified refraction index is given by

where represents the refraction index, the height

and the Earth radius. The refractivity, , is given by

( )

with .

Using the last two equations is possible to obtain the

ray tracing. There are two cases to simulate. The case

where the standard atmosphere is used and the case

where special conditions occur, called duct.

To represent the trajectory an analytical model was

used, where

(√ √ )

√ and represents the height of the

transmitter.

The standard atmosphere is when the modified index

refraction is linear and doesn’t change. The next figures

represent the behaviour of the rays in that specific

atmosphere.

200 400 600 800 1000 1200 1400 1600 1800 2000

10

20

30

40

50

60

70

80

90

100

distance [m]

heig

ht

[m]

Electrical field with ground reflection [dB V/m]

-30

-20

-10

0

10

20

200 400 600 800 1000 1200 1400 1600 1800 2000

10

20

30

40

50

60

70

80

90

100

distance [m]

heig

ht

[m]

Electrical field with ground reflection [dB V/m]

-40

-30

-20

-10

0

10

20

Fig. 7 – Ray trajectory and , with ground

reflection

Fig. 8 – Ray trajectory for and

Fig. 9 – Ray trajectory for and

Fig. 10 – Ray trajectory for and

From the analysis of the figures it is possible to

conclude the influence of the modified refraction index

in the ray tracing. When the index is positive the rays

tend to rise and when the index is negative the rays

tend to sink.

When a special condition occurs, ducts may be

formed, where the rays can travel longer distances and

cause interferences in other systems. The duct takes

place when the atmosphere is constituted by two or

three layers. Each layer presents a different modified

refraction index, and in two consecutive layers the

signals of the indexes are opposite. The duct with two

layers is called surface duct and the duct with three

layers is called raised duct. The Fig. 11 and Fig. 12

demonstrate the effect of ducts on ray propagation.

Fig. 11 – Surface duct

Fig. 12 – Raised duct

From the observation of the figures we can conclude

that the rays are contained in the layer where the

modified refraction index is negative. Because of that,

the rays can travel longer distances contrary to normal

conditions. If the rays during propagation find an

obstacle, they can change trajectory backwards if they

collide sideways with the obstacle, or continue the

trajectory if they collide with the top of the obstacle. The

next figures represent the situation where a ray collides

sideway or in the top of the obstacle.

0 1000 2000 3000 4000 5000 60000

5

10Modified refraction index

M

h[km

]

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

Ray trajectory

z[km]

h[km

]

0 1000 2000 3000 4000 5000 60000

5

10Modified refraction index

M

h[km

]

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

Ray trajectory

z[km]

h[km

]

-5000 -4000 -3000 -2000 -1000 0 10000

5

10Modified refraction index

M

h[km

]

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

Ray trajectory

z[km]

h[km

]

-5000 -4000 -3000 -2000 -1000 0 10000

5

10Modified refraction index

M

h[km

]

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

Ray trajectory

z[km]

h[km

]

0 5 10 15 20 250

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Ray trajectory

z[km]

h[k

m]

100 200 300 4000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Modified refraction index

M

h[k

m]

0 5 10 15 20 250

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Ray trajectory

z[km]

h[k

m]

300 400 500 6000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Modified refraction index

M

h[k

m]

Fig. 13 - Sideway reflection on an obstacle in a surface duct

Fig. 14 - Top reflection on an obstacle in a surface duct

Although is only represented the reflection in an

obstacle for the surface duct, in the others situations

(raised duct, and normal conditions) the rays exhibit the

same behaviour as shown in Fig. 13 and Fig. 14.

A consequence of the atmosphere with super-

refraction situations is the distortion of images captured

by the human eye, usually called mirages. This mirage

effect happens when the atmosphere presents different

modified refraction index in height. If the atmosphere is

stratified in two layers an upper mirage occur, if the

atmosphere is stratified in three layers a lower mirage

occur. The normal and distortion situation are

represented in Fig. 15 and Fig. 16.

Fig. 15 - Normal image captured by the human eye

Fig. 16 - Distortion of the image captured by human eye

due to super-refraction conditions

As we can see in Fig. 16 the rays have different

trajectories from the normal case and an image

captured by the human eye can be deformed due to

that fact.

To simulate the mirage effect, an image is divided in

equal vertical intervals, depending on the number of

lines in the image. Each line corresponds to a ray, and

after the ray tracing they arrive to a determined interval

in the image. The rays are numbered in an increasing

order of the departure angle. The arrived point is stored

in a matrix. The original image is then divided in two

halves. The superior half corresponds to the vertical

plan and the inferior half corresponds to the horizontal

plan. The Fig. 17 represents the image splitting.

Fig. 17 - Image splitting in two orthogonal planes

Now using the stored information in the previous

matrix, if a ray reaches the maximum distance he is in

the vertical plan, otherwise he is in the horizontal plan.

After defining where each ray belongs, the ray lengths

and heights are converted to a scale according with the

number of rays. These values are stored in a position

matrix and are converted in a new image.

An upper mirage is when the objects in the ground

level are copied to the sky. The Fig. 18 represents the

simulation of a upper mirage. In the image it is

represented the modified refraction index, with the

resulted ray tracing. The bottom left image is the

original image, and the bottom right image is the result

of the ray tracing due to the super-refraction situation.

0 20 40 60 80 1000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Ray trajectory

z[km]

h[k

m]

100 200 300 4000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Modified refraction index

M

h[k

m]

0 20 40 60 80 1000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Ray trajectory

z[km]

h[k

m]

100 200 300 4000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Modified refraction index

M

h[k

m]

Fig. 18 – Upper mirage

For the lower mirage, the sky is copied to the ground

and is usually confused with a puddle. As represented

in Fig. 18, the lower mirage is shown in Fig. 19.

Fig. 19 - Lower mirage

IV. DIFRACTION

Usually in a long distance communication system the

path between the antennas isn’t flat, contrariwise. There

are mountains and other obstacles that create

additional attenuations on the signal. Several models

were created to study this effect. The one used in this

article is the knife-edge model.

The knife-edge model considers the obstacle as a

semi-infinite plan. The Fig. 20 represents the knife-edge

model geometry, where

̅

√

̅

Fig. 20 - Knife-edge geometry [4]

The attenuation is strongly dependent of the gap ( ̅)

between the ray and the top of the obstacle. The

attenuation is given by

( )

[

( )]

[

( )]

where ( ) and ( ) are the Fresnel integrals. The

Fig. 21 gives the attenuation as function of .

Fig. 21 - Knife-edge attenuation as function of the

penetration

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2Trajectória do raio

z[km]

h[km

]

-1500 -1000 -500 0 5000

0.05

0.1

0.15

0.2Variação da refractividade modificada com a altura

M

h[km

]

Imagem Original Atmosfera com M variável

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1Trajectória do raio

z[km]

h[km

]

-3000 -2000 -1000 0 10000

0.2

0.4

0.6

0.8

1Variação da refractividade modificada com a altura

M

h[km

]

Imagem Original Atmosfera com M variável

-3 -2 -1 0 1 2 3 4 5

-5

0

5

10

15

20

25

30

Knife-edge model attenuation, A [dB]

v

Knife-e

dge a

ttenuation,

A in d

B,

bellow

fre

e-s

pace

As said before, the attenuation is strongly dependent of

the gap between the ray and the top of the obstacle.

The previous figure shows exactly that. As we can see

if the gap goes to negative the penetration ( ) goes

positive and the attenuation gets bigger. In the other

hand, if the gap is positive the penetration is negative

because the ray doesn’t pass throw the obstacle and

the attenuation is low or zero if the first Fresnel ellipsoid

is unobstructed.

After understanding the influence of the penetration

on the attenuation, it’s possible to describe how the

electrical field is calculated. Using the same expression

of the reflection section and adding the attenuation, we

get the follow equation

[ | | ( )

where and are the attenuation for the direct ray

and for the reflected ray. The Fig. 22 represents the

evolution of the electrical field as function of the height

and distance of the receiver antenna, using the

previous equation.

Fig. 22 - Knife-edge model at 300MHz

From the analysis of the figure, it is possible to see

the attenuation caused by the obstacle. This creates a

shadowing zone where the field presents very low

values.

Another very important factor that influences the knife-

edge model is the frequency. The higher the value of

the frequency the higher the attenuation caused by the

obstacle as shown in Fig. 23.

Fig. 23 - Knife-edge model at 1800MHz

As we can see, the field shows lower values after the

obstacle and the shadowing zone increases.

V. URBAN PROPAGATION MODELS

The urban environments are divided in cells. There

are three types of cells:

Macro-cell

Micro-cell

Pico-cell

The macro-cells are the coverage zones with

dimensions in the order of 2-3 km, where the base

stations are usually in the top of the buildings and the

mobile terminal are in the shadowing zone of the

obstacles.

One model to study the electrical field strength along

the cell is the Walfisch-Bertoni model. In this model, the

biggest contribution to the field is due to diffraction in

the top of the buildings. Two types of attenuation are

considered in this model:

The attenuation due to multiple obstacles that

interfere from the transmitter to the receiver

The attenuation associated to the diffraction

from the top of the building to the street.

The second attenuation is associated with the multi-

path effect caused by two buildings. The mobile

terminal is reached by several rays but only two have

preponderant contributions. Those are the direct ray

and the ray that reflects once in the front building. The

Fig. 24 represents the geometry of the multi-path.

200 400 600 800 1000 1200 1400 1600 1800 2000

10

20

30

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50

60

70

80

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100

distance [m]

heig

ht

[m]

Difraction caused by an obstacle - PV [dB V/m]

-60

-40

-20

0

20

40

200 400 600 800 1000 1200 1400 1600 1800 2000

10

20

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50

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distance [m]

heig

ht

[m]

Difraction caused by an obstacle - PV [dB V/m]

-80

-60

-40

-20

0

20

40

Fig. 24 - Multi-path geometry

Considering the buildings as semi-infinite plans and

the inclination ( ) of the wave relative to horizontal, the

attenuation ( ) is given by

| ( )| {

[

( )]

[

( )]

}

where ( ) and ( ) are the Fresnel integrals, and

are the equivalent height given by

√

[( )

]

√

( )[( ) ( )

]

The angle street ( ) is considered 90º because the

wave is in line with the buildings and is equal to

(

)

Lastly the electrical field is given by

| ( )|

| ( )|

| |

where is the direct ray, is the reflected ray and

| | is the reflection coefficient of the building.

The attenuation due to multiple buildings is only

applies when the buildings have all the same height and

the same spacing between them. In this situation, the

buildings are replaced by semi-infinite plans. The Fig.

25 shows the geometry used to calculate the

attenuation.

Fig. 25 Attenuation model for multi obstacles [1]

The attenuation ( ) induct by the multiple

obstacles is determined by the following expression

with given by

√

This expression used to calculate the attenuation

( ) can only be applied when and

where is the number of buildings and

{

}.

The total electrical field, after calculated all the

attenuations is given by

√

The last expression is the one used to calculate the

electrical field between the buildings.

To simulate the field at any distance and height, we

need to divide the figure in three zones. The first zone

is the one over the building, the second is the zone

between the buildings and the third is the zone after the

last building. Each zone has a different way to calculate

the electrical field. In the first zone is used the

expression of the free space field because there is no

interference from the reflected ray. The second zone

was calculated using the Walfisch-Bertoni model as

explain above. The third zone is calculated using the

knife-edge model without the reflected ray. The Fig. 26

represents the simulation of the Walfisch-Bertoni model

with the three zones represented.

Fig. 26 - Walfisch-Bertoni Model

The previous figure was simulated using the following

specifications:

From the analysis of the Fig. 26 it is possible to see

that the electrical field decreases very quickly between

the buildings and is very low after the last building. The

blue line and red line represents the space where the

Walfisch-Bertoni model is valid.

The frequency influence the model in the same way

that influences the knife-edge model, which means that

the higher the frequency the higher the attenuation.

Furthermore the frequency influences the parameters

that validate the multi-obstacle attenuation. As we can

see from the expression of and , they depend on

the wavelength. The Fig. 27 shows the influence of the

frequency on the model.

Fig. 27 - Walfisch-Bertoni model at 500MHz

Other parameter of great influence is the reflection

coefficient of the buildings. If this parameter is zero, the

reflected ray does not contribute lowering the electrical

field. In this situation a shadowing zone appears next to

the previous building. This effect is represented in the

Fig. 28.

Fig. 28 - Walfisch-Bertoni model with

In the previous figure, the frequency used is 500MHz.

VI. CONCLUSIONS

This work aimed to create a didactic and project tool.

With this application is possible to demonstrate

graphically the theoretical models studied during the

course. In an academic context, permits to the students

analyse and understand how a parameter can influence

a model. The aspects addressed are the reflection,

diffraction, refraction and propagation models in

complex environments.

In the reflection topic, was represented graphically the

evolution of the electrical field with vertical polarization

considering the reflected ray in the ground. The

interference of the reflected ray with the direct ray was

demonstrated through the existence of maximums and

minimums of the electrical field. The influence of an

array of antennas was shown too. As conclusion, the

electrical field with ground reflection is influenced by the

frequency, by the distance and height of the receiver

antennas and by the polarization, all causing variations

of the field around a mean value.

In the refraction section the ray tracing was

demonstrated with a standard atmosphere and with

special conditions. When the modified refraction index

is negative, the ray goes down; otherwise, the ray goes

up. In special conditions, in a stratified atmosphere with

different refraction indexes a duct is formed and the

rays travel longer distances and are contained in the

layers where the index is negative. A special effect like

the mirages was demonstrated too. Due to the different

indexes in the atmosphere an image gets deformed in

200 400 600 800 1000 1200 1400 1600 1800 2000

10

20

30

40

50

60

70

80

90

100

distance [m]

heig

ht

[m]

Electrical field [dB V/m]

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-50

0

50

1

2

3

200 400 600 800 1000 1200 1400 1600 1800 2000

10

20

30

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distance [m]

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ht

[m]

Electrical field [dB V/m]

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0

20

40

200 400 600 800 1000 1200 1400 1600 1800 2000

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distance [m]

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ht

[m]

Electrical field [dB V/m]

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-20

0

20

40

the human eye. If the atmosphere has two layers, a

superior mirage happens, if the atmosphere has three

layers an inferior mirage occurs.

The knife-edge model was the one chosen to

demonstrate the diffraction on obstacles. In this model,

the obstacle is substituted with a semi-finite plan and

the attenuation is calculated as function of the

penetration. The penetration is the difference between

the ray and the top of the building. If the penetration is

positive, which means the ray passes through the

obstacle, the attenuation is big. Otherwise, the

attenuation is low. The obstacle creates a shadowing

zone, where the electrical field is very low. The knife-

edge is influenced by the frequency. The higher the

frequency, higher the attenuation is.

The last topic is about the propagation models in

urban environments. In a macro-cell type scenario, the

Walfisch-Bertoni model was demonstrated. In this

model, two types of attenuations are considered. The

first attenuation is due to multiple obstacles and the

second is due to multi-path from the antenna to the

mobile terminal. In the multi-path attenuation, two rays

were considered to the calculations, the direct ray and

the reflected ray in the front building. The electrical field

gets lower and lower along the buildings and after the

last building the knife-edge model was used. The

frequency influences the Walfisch-Bertoni model the

same way as influences the knife-edge mode, in other

words, the higher the frequency higher the attenuation

gets. Another parameter analysed was the reflection

coefficient of the building. If the reflection is zero, the

field gets low and a shadowing zone appears.

VII. REFERENCES

[1] Wikipedia, Radio_propagation,

(http://en.wikipedia.org/wiki/Radio_propagation)

[2] J.Walfsich and H.Bertoni, ‘A theoretical mode of

UHF propagation in urban environments’ IEEE trans.

Antennas Propagat. Vol 36, Nº 12, pp. 1788.1796,

Dec.1988

[3] Figanier, J. Fernandes, C.A, ‘Aspectos de

Propagação na Atmosfera’, Secção de Propagação e

Radiação, IST-DEEC, 2002

[4] Fernandes, C.A, ‘Radiopropagação Mobile Radio

Communications’, Slides disciplina Radiopropagação,

IST-DEEC

http://en.wikipedia.org/wiki/Radio_propagation