Radio Channel Modeling and Simulation for a Stratospheric Platform Link

POLITECNICO DI TORINO

Facolt�a di IngegneriaCorso di Laurea in Ingegneria delle Telecomunicazioni

Tesi di Laurea

Radio channel modeling andsimulation for a stratospheric

platform link

Relatori:

Prof. Patrizia Savi

Prof. Marina Mondin

Ing. Fabio Dovis

Candidato:

Roberto Fantini

Marzo 2001

Summary

A High Altitude Long Endurance (HALE) vehicle is a low cost stratospheric

payload platform, ying at altitudes between 15 and 30 km, with a potential

endurance of weeks to months. HALE platforms could play a role similar to

arti�cial satellites, with the major advantage of being less expensive, more

adaptable and closer to ground.

Thanks to its high position, a stratospheric platform can provide telecom-

munication services over a wide area. The radius of the cell covered by the

platform can be evaluated geometrically, by supposing rectilinear propaga-

tion. In fact, the non-rectilinear propagation of the electromagnetic �eld

through the atmosphere can be taken into account by substituting the Earth

radius RE with a modi�ed radius R0E = �RE, whose aim is to reform the local

geometry of the Earth in order to transform curvilinear lines into rectilinear

lines. Typically, in temperate zone � = 4=3, and in this case the radius of

the cell covered by the platform is about 100 km, so that the lower elevation

angle at which the platform can be seen is 10�.

In this thesis we have studied the propagation channel that characterizes

the transmission from a stratospheric platform, and we have developed a

simulation model of the channel. We have focused on describing both the

average path loss at a given distance from the transmitter (large scale fading),

as well as the rapid uctuation that a�ects the received signal strength while

moving in a small area (small scale fading).

Free space attenuation, interference due to ground re ection, di�raction

by obstructions and rain attenuation are the basic mechanisms that a�ect

large scale propagation. The e�ects of those mechanism when transmitting

from a Ground Station (GS) or from a Platform Station (PS) to a receiver at

a given ground distance has been compared. The ground distance has been

I

de�ned as the distance from the receiver to the GS, or, in the case of the

platform, from the point vertically below the platform to the receiver.

The free space model developed by Friis enlightens the major disadvan-

tage of the platform. Being placed at an height of 17 km, no transmission

can be done from the platform with less attenuation than Afs = 118:6 dB;

in general, for a given ground distance, a signal transmitted from a GS has

to cover a shorter path in order to be received, thus undergoing to a smaller

free-space path loss.

Interference due to ground presence can be estimated in a GS system

using the two ray model. In this model the total received E-�eld is the

result of a direct line-of-sight component (ELOS) and of a ground re ected

component (Eg). When the separation distance between the transmitter and

the receiver is largely greater than their heights, ground tends to re ect like

a perfect conductor, and the component Eg is equal in magnitude to ELOS,

but near to be 180� out of phase, thus causing destructive interference.

In the platform case ground can no longer be considered a perfect con-

ductor. Thus, we have modi�ed the two ray model, and the component Eg

has been evaluated accordingly to the laws of re ection, by means of the

Fresnel re ection coe�cient. In this case interference due to Eg is strongly

reduced and the received power falls o� more slowly with distance. As a

consequence for the �rst eight kilometers of coverage a GS system undergoes

to less attenuation because the transmitter is closer than in the PS case, but

for greater ground distances the propagation conditions in the PS case are

less severe and the received power is higher.

Thanks to its high position, the platform should grant a direct link be-

tween the transmitter and the receiver. In that sense, a land mobile satellite

(LMS) system experiment carried out in the framework of ESA's PROSAT

progress showed that in rural environment, for an elevation angle between

39� and 13�, the platform should be in LOS for a percentage of time between

90% and 60%. However, in urban environment and for low elevation angles,

the direct ray can be shadowed. In that case, di�raction allows the radio

signal to propagate up to the receiver. In a GS system path loss in urban

environment can be expressed as the sum of free-space attenuation (Lfs),

multiple di�raction past rows of buildings (Lmd), and di�raction from one

II

building rooftop down to the receiver at street-level (Lrts). We have proved

that in a PS system only one building can interfere with the signal trans-

mission, because only one building can be inside the First Fresnel Zone, thus

the total attenuation is given only by Lfs and Lrts. As a consequence the

attenuation due to shadowing, whenever is present, is weaker than in a GS

based system.

Rain attenuation is proportional to the length of the path covered by the

E-�eld under the rain. Rain is generated in the �rst 3 km of the atmosphere;

it follows that in a PS system, for ground distances greater than 3.048 km,

the E-�eld interacts with rain drops for a shorter path than in the GS case,

thus undergoing to a lower level of attenuation.

Small scale fading occurs when the received signal is the sum of di�erent

replicas of the same transmitted signal, that arrive to the receiver following

di�erent paths. Small scale fading is characterized by two functions, the

power delay pro�le, that describes dispersion in time of the received signal,

and the Doppler spectrum, that is related to the movement of the transmitter

and of the receiver and characterizes dispersion in frequency. Power delay

pro�les for terrestrial systems are well known, and usually have been obtained

by measurements. In this work, the power delay pro�le for the stratospheric

channel has been estimated analytically, by modifying two models for ground

based system, that were two-dimensional in nature. The models have been

extended in order to be three-dimensional, so that they can take into account

also the height of the platform and the fact that scatterers can be found only

in a layer concentrated over the ground. Both the models yield to the same

power delay pro�le, and show that dispersion in time is largely lower in the

PS case than in the GS case. In fact, being the elements of scattering only in

a layer at ground level, the number of echoes in a PS based system is lower

than in a GS based system. Moreover, the presence of a direct ray from

the transmitter to the receiver makes the ratio between the power of the

transmitted signal and the power of the interfering rays very high. It follows

that the time support and rms delay spread of the power delay pro�le are

smaller in the case of the stratospheric channel than in traditional terrestrial

systems. Doppler spectrum for ground system, where the transmitter is

still, is given by the typical Jakes' model, while for the platform must be

III

evaluated taking into account that both the receiver and the platform move:

as a consequence, dispersion in frequency is greater in the platform case.

However, the reduced dispersion in time, and the presence of a direct ray

that is more powerful than the interfering echoes, widely reduce the small

scale fading e�ects in a PS system, as pointed out by the simulation results

obtained with the simulation model developed for TOPSIM.

IV

Table of contents

Summary I

1 Introduction 1

1.1 The HALE platforms - an historical overview . . . . . . . . . . 1

1.2 HeliNet and HeliPlat . . . . . . . . . . . . . . . . . . . . . . . 4

2 Large Scale Fading 10

2.1 Free space and 2-ray model . . . . . . . . . . . . . . . . . . . 10

2.2 Visibility and Line Of Sight . . . . . . . . . . . . . . . . . . . 22

2.3 Di�raction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Absorption and Rain Attenuation . . . . . . . . . . . . . . . . 33

3 Small Scale Fading and Multipath - A Theoretical Introduc-

tion 36

3.1 Multipath Channel Baseband Impulse Response . . . . . . . . 37

3.2 Power Delay Pro�le and Coherence Bandwidth . . . . . . . . . 41

3.3 Doppler Spectrum and Coherence Time . . . . . . . . . . . . . 42

4 Small Scale Fading in a Platform Based System 49

4.1 Extension of the Rappaport's geometrical model to the plat-

form case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Extension of Burr's channel model to the platform case . . . . 55

4.3 Comparison between the results obtained with Rappaport's

and Burr's models . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4 Doppler Spectrum for a Platform Based System . . . . . . . . 71

V

TABLE OF CONTENTS

4.5 Coherence Bandwidth and Coherence Time of the Platform

Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5 The Simulation Program 77

5.1 Descriptions of Simulation Models for Di�erent Mobile Radio

Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2 Description of the TOPSIM Simulation Block MUPATP . . . 85

5.3 MUPATP Block Validation . . . . . . . . . . . . . . . . . . . . 90

5.4 Simulation Results Using the MUPATP Block . . . . . . . . . 103

Conclusions 107

A The Simulation Block MUPATP 109

B The Fortran Subroutine POWDEL 115

Bibliography 119

VI

Chapter 1

Introduction

1.1 The HALE platforms - an historical overview

In the recent years a great interest has arisen for the development of Un-

manned Aerial Vehicles High Altitude Long Endurance (UAV-HALE) plat-

forms. A HALE vehicle can be described as a low cost stratospheric payload

platform, tailored for a wide range of applications within telecommunica-

tions and remote sensing, ying at altitudes between 15 and 30 km, with a

potential endurance of weeks to months.

Historically the HALE evolution can be seen as the convergence of three

technology development paths for high-altitude surveillance aircraft, long

endurance experimental aircraft and Unmanned Aerial Vehicles.

� Historical piloted aircraft designed for high altitude surveillance opera-

tions, such as the U-2 and its modern derivatives, have been developed

with a capacity for ights at altitudes above 20 km but with an en-

durance of the order of hours.

� An experimental aircraft optimized for long endurance, the Voyager,

has circumnavigated the Earth with a crew of two people. With a

composite materials design, using conventional piston engines, it proved

an endurance in excess of nine days at low altitude.

� Unmanned drones or RPV's (Remotely Piloted Vehicles) are receiving

1

1 { Introduction

increasing attention around the world for military applications. The re-

search is primarily focused on the development of operational vehicles

with improved performance in terms of autonomy; this step is essen-

tial to make feasible the transition from RPV's to more autonomous

UAV's. More recently, however, military high altitude designs are being

developed for operational use.

In the last decades several development programs have aimed to combine

these three �elds into an operational HALE system [13] [52] [43] [50]. In the

late 1980's this culminated in an unmanned military technology demonstra-

tion aircraft, the Boeing Condor, that proved capacity to maintain a payload

of several hundred kilograms above 20 km altitude for more than 48 hours

[1]. With a limited energy supply, fuel-burning platforms are, however, by

de�nition limited in endurance.

Several approaches to reach a longer presence in the Stratosphere have

therefore been considered. Dominating in this category are solar powered

solutions and supply system based on remote microwave transmission of en-

ergy from the ground. Solar powered vehicles have roots in the development

of extremely low-speed and light-weight human powered aircraft, such as the

Gossamer Albatross that crossed the English channel. Solar ight during

day-time at low altitude and over considerably distances has been demon-

strated by vehicles such as the Solar Challenger that crossed the English

channel.

The Icare, a German university e�ort, proved that a conventional glider

aircraft can be converted into a relatively robust solar powered aircraft ca-

pable of ight in less than perfect conditions at low altitudes. Solar pow-

ered long endurance aerial platform has proved to be feasible by the NASA

Path�nder, an experimental solar powered vehicle that has exceed 15 km

altitude in a 12 hours ight. In the end, recent study produced within the

NASA ERAST Program [3] has proven the viability of a solar powered air-

craft for high altitude and long endurance ight using a new stratospheric

platform called Centurion, that was de�ned the �rst \near eternal plane",

and its last evolution, the Helios.

Even if in the beginning HALE platforms were developed as a military

2

1 { Introduction

project, they seem suitable for a wide class of civilian applications, includ-

ing: pollution monitoring, meteorologic measurement, real time monitoring

of seismal or coastal regions and terrestrial structures, agriculture support,

telecommunication services such as cellular/PCS, video-surveillance [18]. As

a matter of fact they could play a role similar to arti�cial satellites, with

the major advantage of being less expensive, more adaptable and closer to

ground; moreover, HALE platform could be moved on demand in order to

cover di�erent regions and, in case of breakdown, they could land, be repaired

and take o� again.

At present, in the international community, many projects of airborne

platforms for telecommunication and video-surveillance services are under

development. ESA/ESTEC has performed a feasibility study of HALEs, y-

ing at height of 15-30 km, powered by means of solar cells or microwave

transmission, for scienti�c, video-surveillance or telecommunication applica-

tions [21]. NASA in 1993 started the Environmental Research Aircraft and

Sensor Technology (ERAST) program [3] for the realization of unmanned air-

crafts ying at subsonic speed, at a height of 30 km and with a 96 hour long

endurance. In the �eld of speci�c telecommunication applications, such as

digital broad-band communications, the Stratospheric Telecommunications

Service from SkyStation Inc. [5] and the HALO Network (High Altitude

Long Operation) project from Angel Technologies Inc. [2] are worth to be

mentioned: the former plans to set up a 250 airships network, whereas the

latter employs conventional manned aircrafts operating 24 hours a day over

the supported metropolitan areas. In the military �eld HALE UAV are expe-

riencing a strong development (Global Hawk project by Darpa [1]), and it is

expected that the development of military projects will provide the required

technology, infrastructure and maturity for future civil HALE applications.

In this work we will refer to the HeliPlat project, currently being devel-

oped at Politecnico di Torino and supported by the European Commission

[4] [44].

3

1 { Introduction

1.2 HeliNet and HeliPlat

HeliPlat is an unmanned, solar powered aircraft, especially tailored for long

endurance operations at altitude of 17 km, which enables a payload of 100 kg

and o�ers an available power for telecommunications applications of 500-800

W.

The project being developed by Politecnico di Torino was funded by the

European Commission within the Fifth Framework programme, and involves

a network of HeliPlats named HeliNet (i.e HELIplat NETwork) that will

provide several services, such as: tra�c surveillance, environmental monitor-

ing, wireless telephone and broadband communication services, integrated

and interactive remote sensing services and positioning services [32] [31]. In

that sense, HeliNet should be intended as a network infrastructure that can

provide services integrated with other systems, such as GPS or the European

EGNOS and Galileo, or replace terrestrial networks whenever they are not

cost-e�ective, such as in rural or impervious regions, or when they became

unavailable due to natural disaster or emergency.

Platforms are expected to y in the stratosphere, an environment which

signi�cantly di�ers from the lower part of the atmosphere. In Figure 1.1 the

variations with respect to the altitude of few parameters (air density,pressure,

temperature) are reported [51].

It can be noted that temperature has a non-linear characteristic: in the

troposphere, from ground level to about 11 km, temperature decreases with

altitude, whereas in the tropopause temperature ceases to decrease. This

point is usually taken as the border between troposphere and stratosphere,

and is quite distinct, being in the order of a hundred of meter in altitude.

Above this level, in the stratosphere, temperature slowly begin to increase.

Practically all weather related phenomena that may be detrimental to

ight, such as clouds and thunder storms, are limited to the troposphere,

thus wind speed is the only speci�c factor that could interest the platform's

operations. Wind speed generally increases with altitude from ground level to

about 7 to 11 km, then steadily decreases in the lower stratosphere, reaching

a minimum around 16 to 21 km. This is the best height to place the platform,

being the wind speed relatively modest and comparable to ground level winds.

4

1 { Introduction

0

5

10

15

20

25

30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Altitude [km]

Relative to Ground Level

Atmospheric Conditions

TemperatureAir DensityPressure

Figure 1.1. Atmospheric conditions as a function of increasing altitude

Above this level wind speed resumes to increase again.

Following the previous consideration an altitude of 17 km is foreseen as

operating altitude for HeliPlat. From such an altitude it can theoretically

cover a very large area. In fact it can be easily proved [20] that the radius

of the cell covered by the platform, taking into account only geometrical

considerations and supposing rectilinear propagation (see Figure 1.2), can be

written as:

R = RE

��

2� �� arcsin

�RE

RE + hcos�

��(1.1)

where RE = 6370 km is the Earth radius, h=17 km the platform altitude,

and � the elevation angle of the platform with respect of the user position.

The non-rectilinear propagation of the electromagnetic �eld through the

atmosphere can be taken into account by using the modi�ed radius approach,

thus substituting the Earth radius RE with a modi�ed radius R0E = �RE

5

1 { Introduction

α

R

R

h

r

Edge of the cell

Platform

Figure 1.2. Geometrical model for the estimation of cell radius.

whose aim is to reform the local geometry of the Earth in order to transform

curvilinear lines into rectilinear lines. The parameter � depends upon the

variation of the refraction index with the altitude and upon the meteorologi-

cal conditions. For temperate zones it can be taken as � = 4=3. In Figure 1.3

is reported the cell radius as a function of the platform altitude for di�erent

elevation angles.

Usually, when dealing with satellite based system, 10� is the lower ele-

vation angle taken into account in order to de�ne the coverage area of the

system (see [30] or [48]); in the platform case, for an altitude of 17 km, this

value of the elevation angle corresponds to a coverage radius of about 100

km, that in this thesis will be used as the maximum radius of the coverage

area.

In the following Chapters we will present a study on the propagation

conditions from a stratospheric platform: as an example, we will refer to

transmissions at a center frequency of 1.2 GHz.

Propagation models have traditionally focused on describing both the

6

1 { Introduction

1

10

100

1000

0 5 10 15 20 25 30

Log(R)

Altitude [km]

Cell radius R as a function of altitude and elevation angle

α = 0α = 10α = 20α = 30α = 40α = 50

Figure 1.3. Cell radius versus platform's altitude, for various elevationangles

average path loss at a given distance from the transmitter, as well as the

rapid uctuation that a�ects the received signal strength while moving in a

small area.

A propagation model that predicts the mean signal strength for an ar-

bitrary transmitter-receiver separation is useful in estimating the coverage

area of a transmitter and is usually referred to as a large-scale propagation

model.

On the other hand small-scale or fading models characterize the rapid

variations of the received power over very small travel distances (of a few

wavelengths) or short time durations (on the order of seconds). Small scale

fading occurs when the received signal is the sum of di�erent replicas of the

same transmitted signal, that arrive to the receiver following di�erent paths.

7

1 { Introduction

As a result, each replica has a random phase and arrives with a di�erent delay:

thus their sum may vary widely as the receiver is moved of a small fraction of

a wavelength. Moreover it is well known that a mobile that receives a signal

while moving at a certain velocity v experiences a shift in the center frequency

of the received signal. This change in frequency, known as the Doppler shift,

is related to the mobile velocity v and the angle of arrival of the received

wave. When the received signal is the sum of di�erent contributions that

followed di�erent paths, each contribution comes to the receiver with an

arbitrary angle of arrival, thus experiencing a di�erent change in frequency.

This causes the received signal level to vary widely in time.

In Chapter 2 the e�ects of large scale fading are taken into account: the

visibility of the platform is investigated, and the e�ect of re ection, di�rac-

tion and scattering from rain drops are evaluated in order to compare the

performance of a transmitter placed at ground level with the platform solu-

tion.

In Chapter 3 the theoretical aspect of small scale fading are presented:

the concepts of dispersion in time and frequency are illustrated, as well as the

de�nition of power delay pro�le and Doppler spectrum of a fading channel.

A taxonomy of the radio mobile channel is presented and the de�nitions of

fast/slow and at/selective fading channel are given.

In Chapter 4 the small scale fading characteristic of the platform channel

are described. The power delay pro�le is obtained on the basis of two di�erent

theoretical models, presented respectively by Rappaport with Liberti [40] [41]

and Burr [11], that have been modi�ed in order to suit the platform based

system. The Doppler spectrum has been evaluated using the theoretical

results obtained by M.Pent in [35] following the model for Low Earth Orbit

(LEO) satellites presented in [6].

In Chapter 5 the results obtained in the previous Chapters are used in

order to develop a simulation model of the stratospheric channel; the model

has been integrated as a block for TOPSIM, a simulation tool for the analysis

and design of communication systems developed by the Telecommunication

8

1 { Introduction

Group of Electronic Department of Politecnico di Torino. The block is pre-

sented, together with some simulation results that allow to highlight the

behaviour of the block and the characteristics of the stratospheric channel.

9

Chapter 2

Large Scale Fading

In this Chapter we will focus our attention on the description of the three

basic mechanisms that a�ect large scale propagation: ground re ection,

di�raction and rain attenuation. We aim at comparing the e�ects of those

mechanisms in the performance of a ground-level station with respect to a

transceiver on-board of a stratospheric platform. In particular, the total path

loss experienced when transmitting from a Ground Station (GS) or from a

Platform Station (PS) to a receiver at a given ground distance d will be com-

pared. The ground distance can be de�ned as the distance from the receiver

to the GS or, in the case of a platform, from the platform sub-point (i.e. the

point vertically below the intended location of the platform) to the receiver,

as shown in Figure 2.1. To achieve our purpose we will use some traditional

large-scale propagation models and develop their extension to the platform

case.

2.1 Free Space Propagation and 2-ray Prop-

agation Model

The �rst model used in this section is the classical free space model [7].

Although this model is very simple, it enlightens the major disadvantage of

a platform based transmission, i.e. the increased distance from the receiver.

A second more realistic model, the two-ray model [56] [38], will be used to

investigate the e�ects of ground re ection on the path loss. It will be shown

10

2 { Large Scale Fading

RXGS

PS

α

- h-height

Platform

Ground Distance -d-

Figure 2.1. Side view of a stratospheric platform operating scenario

that in the platform case, the interference caused by the presence of the

ground is reduced, so that the disadvantage due to the increased distance

can be overcome.

Free space propagation accounts for the signal attenuation due to the

spherical spreading of the wavefront excited by a point source. The free

space attenuation in the signal strength received at a distance r0 from the

transmitter, is given by the Friis free space Equation [7]:

Afs(r0) =

�4�r0�

�2

(2.1)

where � = c=f is the wavelength, c = 3 108 m/s the speed of light and

f = 1:2 GHz the carrier frequency.

As already mentioned, we consider a stratospheric platform operating at

an altitude of 17 km. As a consequence, in the platform case the distance

between the transmitter and the receiver is always at least r0 min = 17 km.

11


In this situation no transmission can be done with less attenuation than

Afs = 118:6 dB (2.2)

In general, for a given ground distance, a signal transmitted from a GS has

to cover a shorter path in order to be received, thus undergoing to a smaller

free-space path loss. In fact, for a ground distance d, the T-R separation r0

in the GS case is just r0 = d, so that

Afs GS(d) =

�4�d

�

�2

(2.3)

while in the PS case is r0 =pd2 + h2, being h = 17 km the platform height,

so that free space attenuation is:

Afs PS(d) =

4�pd2 + h2

�

!2

(2.4)

Figure 2.2 shows a comparison of the free-space path loss experienced

when transmitting from a GS or from a PS to a receiver with ground distance

d. Using this model leads to conclude that a stronger path loss is experienced

in a platform based system. However the free space propagation model should

be used when interference from surrounding environment can be neglected,

for example in satellite, or in microwave line-of-sight transmission. Both in

the GS and in the PS case, for a carrier frequency of about 1.2 GHz, the

direct path is seldom the only physical means for propagation, and hence the

free-space propagation model is inaccurate.

In order to achieve a reasonable prediction of the large scale signal strength,

the 2-ray ground re ection model may be used. This model is based on ge-

ometric optics and considers both the direct path and a ground re ected

propagation path between the transmitter and the receiver.

Ground Link

In the two-ray model (see Figure 2.3), the total received E-�eld (ETOT ) is the

result of the direct line-of-sight component (ELOS) and the ground re ected

12


50

60

70

80

90

100

110

120

130

0 5 10 15 20 25 30 35 40

Path Loss [dB]

d, ground distance [km]

Free Space Attenuation

Platform StationGround Station

Figure 2.2. Free-space attenuation at a given ground distance d for a PSand for a GS.

TX

RX

h

d

t

Ei

ELOS

E = Egrr

θi θr

h

d"

d’

Figure 2.3. Two-ray ground re ection model

13


component (Eg). Note that in most mobile communication systems the max-

imum T-R separation distance is at most only a few tens of kilometers, so

that Earth surface may be assumed to be at.

In the free space model, if E0 is the free space E-�eld (in units of V/m)

at a reference distance d0 from the transmitter, then for d > d0 the E-�eld

is given by

E(d;t) =E0d0d

cos

�!c

�t� d

c

��(2.5)

where jE(d;t)j = E0d0=d represents the envelope of the E-�eld at d meters

from the transmitter.

In the two-ray model, two propagating waves arrive at the receiver: the

direct wave that travels for a distance d0, and the re ected wave that travels

for a distance d00. Suppose that the E-�eld is linearly polarized and perpen-

dicular to the plane of incidence, so that the model is easier to handle. The

E-�eld due to the line-of-sight component at the receiver can be expressed as

ELOS(d0;t) =

E0d0d0

cos

�!c

�t� d0

c

��(2.6)

whereas the E-�eld for the ground re ected wave, which has a propagation

distance of d00, can be expressed as

Eg(d00;t) = �?

E0d0d00

cos

�!c

�t� d00

c

��(2.7)

Equation (2.7) can be easily found according to the laws of re ection [36].

In general, re ection occurs when a radio wave propagating in one medium

impinges upon another medium of di�erent electrical properties. If the plane

wave is incident on a perfect dielectric, part of the energy is transmitted in

the second medium and part of the energy is re ected back into the �rst one,

with no loss of energy. If the second medium is a perfect conductor, then all

incident energy is re ected back into the �rst medium without loss of energy.

The electric �eld intensity of the re ected (Eg) and transmitted (Et) waves

can be related to the incident wave (Ei) in the medium of origin through the

Fresnel re ection coe�cient �. It can be shown that (see [38]):

Eg = �Ei (2.8)

Et = (1 + �)Ei (2.9)

14


and that

�i = �r (2.10)

being �i the angle of incidence of Ei, and �r the re ection angle of Eg. The

re ection coe�cient � is a function of the material properties and it generally

depends on the wave polarization, angle of incidence, and the frequency of

the propagating wave. In particular, as reported in [38], for the perpendicular

polarization �? can be expressed as:

�? =sin �i �

p�r � cos2 �i

sin �i +p�r � cos2 �i

(2.11)

From Equation 2.11 it can be seen that for small values of �i (grazing inci-

dence) the re ected wave is equal in magnitude and 180� out of phase with

the incident wave, i.e. ground tends to re ect like a perfect conductor, having

�? = �1 and Et = 0.

It follows that the electric �eld ETOT (d;t), sum of the line of sight com-

ponent (2.6) and of the ground re ected component (2.7), can be expressed

as:

ETOT (d;t) =E0d0d0

cos

�!c

�t� d0

c

��+ (�1)E0d0

d00cos

�!c

�t� d00

c

��(2.12)

Using the method of images [36], as shown in Figure 2.4, the path dif-

ference � between the line of sight and the ground re ected paths can be

expressed as

� = d00 � d0 =p(ht + hr)2 + d2 �

p(ht � hr)2 + d2 (2.13)

where ht is the height of the transmitter and hr is the height of the receiver.

When the T-R separation d is very large if compared to ht + hr, Equation

(2.13) can be simpli�ed using a Taylor series approximation

� = d00 � d0 � 2hthrd

(2.14)

As a consequence, the phase di�erence �� between the two E-�eld compo-

nents and the time delay �d between the arrivals of the two components are:

�� =2��

�=

2�fc�

c(2.15)

15


TX

RX

E

ELOS

h th

g

d

h -h rt

h +ht r

r

d’

d"

Figure 2.4. The method of images used to �nd the path di�erence betweendirect path and the re ected one

�d =�

c=

��2�fc

(2.16)

It should be noted that as d becomes large, the di�erence between the

distances d0 and d00 becomes very small, and the amplitudes of ELOS and

Eg are virtually identical, so that the two E-�eld components di�er only in

phase. That is ��E0d0d

�� E0d0d0

�� E0d0d00

�� (2.17)

Thus, Equation (2.12) can be expressed as

ETOT (d;t) =E0d0d0

cos

�!c

�t� d00

c

�+ !c

d00 � d0

c

�� E0d0

d00cos

�!c

�t� d00

c

��

� E0d0d

�cos

�!c

�t� d00

c

�+ ��

�� cos

�!c

�t� d00

c

��(2.18)

Using Equation (2.18) and referring to the phasor diagram of Figure 2.5,

the amplitude of the received total E-�eld at a distance d from the transmitter

16


E

θ

E d /d"

E d /d’

-E d /d" 0 0

0 0TOT

∆

0 0

Figure 2.5. Phasor diagram showing the E-�eld components of the LOS,ground-re ected, and total received E-�elds derived from Equation 2.18

can be written as

jETOT (d)j =s�

E0d0d

�2

(cos �� 1)2 +

�E0d0d

�2

sin2 �� (2.19)

or

jETOT (d)j = E0d0d

p2� 2 cos �� (2.20)

Using trigonometric identities, Equation (2.20) can be expressed as

jETOT (d)j = 2E0d0d

sin

��2

�(2.21)

Note that when ��=2 is less than 0.3 radian sin(��=2) � ��=2, so that

Equation (2.21) can be simpli�ed. Using Equations (2.14) and (2.15)

��2� 2�hthr

�d< 0:3 rad (2.22)

which implies that

d >20�hthr

3�� 20hthr

�(2.23)

It follows that, as long as d satis�es (2.23), the received E-�eld can be ap-

proximated as

ETOT (d) � 4E0d0�hthr�d2

(2.24)

17


The total received power is related to the electric �eld amplitude by

PRX =jETOT j2Z0

Aeq (2.25)

where Z0 = 120� ' 377 is the intrinsic impedance of free space and Aeq is

the e�ective aperture of the receiving antenna, which can be expressed as:

Aeq =�2

4�GRX (2.26)

being GRX the gain of the receiving antenna in the direction of the radio

wave arrival.

Substituting (2.24) in (2.25), remembering the (2.26) and that at a dis-

tance d0 the power ux density can be expressed as

Pd =PTXGTX

4�d20=jE0j2Z0

(2.27)

so that

(E0d0)2 =

Z0

4�PTXGTX (2.28)

the total received power at a distance d from the transmitter can be expressed

as

PRX(d) = PTXGTXGRXh2th

2r

d4(2.29)

At large distances the received power falls o� with distance raised to the

fourth power, that is a much more rapid path loss than is experienced in free

space. This is due to the interference of the ground re ected ray with the

direct one: at grazing incidence (� � �1), and being d0 � d00, the re ected

E-�eld component and the line-of-sight E-�eld component have the same

magnitude. However, the former is near to be 180� out of phase with the

latter, so that interference is near to be destructive, thus causing a very rapid

power fall-o�.

Stratospheric Platform Link

The approximations previously made no longer hold on if the transmitter is

payload of a stratospheric platform. In the following it will be shown that

18


in this case the interference due to the re ected wave is less severe, so that

power falls o� more slowly with distance.

For a platform based communication system, the total received E-�eld,

supposing that the transmitted E-�eld is linearly polarized and perpendicular

to the plane of incidence, can still be expressed as the sum of a direct E-�eld,

ELOS, and a re ected one, Eg, thus obtaining:

ETOT (d;t) =E0d0d0

cos

�!c

�t� d0

c

��+ �?

E0d0d00

cos

�!c

�t� d00

c

��(2.30)

As already mentioned, the re ection coe�cient �? is a function of the

angle of incidence �i; when �i approaches zero the ground tends to act as

a perfect conductor, so that �? ' �1. However, as it can be seen from

Figure 2.4

tan �i =ht + hr

d(2.31)

thus �i � 0 if the ground distance d is very large compared to ht + hr. In

the platform case, being ht = 17 km, this condition cannot be satis�ed. In

order to evaluate �?, it is required the knowledge of the dielectric relative

permittivity �r of the re ecting ground. Typical values are �r = 4 for poor

ground, �r = 15 for typical ground and �r = 25 for good ground [38].

E d /d" EΓ 00 TOT

E d /d’0 0

θ∆

Figure 2.6. Phasor diagram showing the E-�eld components of the LOS,ground-re ected, and total received E-�elds derived from Equation 2.30

19


Equation 2.9 is no longer valid and the magnitude of ETOT is in this case:

jETOT (d)j =s�

E0d0d0

+ �?E0d0d00

cos

�!cd00 � d0

c

��2

+

��?

E0d0d00

sin

�!cd00 � d0

c

��2

(2.32)

as can be easily found from the phasor diagram of Figure 2.6. Note that

Equation (2.32) can be written in the form

jETOT (d)j = E0d0pf(d) (2.33)

where

f(d) =

�1

d0+�?d00

cos

�!cd00 � d0

c

��2

+

��?d00

sin

�!cd00 � d0

c

��2

(2.34)

is a function of the ground distance d through �?, d0 and d00. With reference

to Equations (2.25), (2.26) and (2.28) the total received power is then:

PRX(d) =jETOT j2Z0

�2

4�GRX = PTXGTXGRX

��

4�

�2

f(d) (2.35)

Once that the received power as a function of the ground distance between

the transmitter and the receiver has been obtained, a comparison between

the path loss in a PS based and in a GS based communication system can be

done. Path loss in a GS based system can be found from Equation (2.29):

AGS =d4

h2th2r

(2.36)

while path loss in a PS based system can be derived from Equation (2.35):

APS =

�4�

�

�21

f(d)(2.37)

Equation (2.37) requires to evaluate d0, d00 and �? for each ground distances

d; d0 and d00 can be obtained as already done by using the method of images

(see Figure 2.4 and Equation (2.13)):

d0 =p(ht � hr)2 + d2 (2.38)

d00 =p(ht + hr)2 + d2 (2.39)

20


0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30 35 40

Path Loss [dB]


Attenuation with the 2-ray model

Platform StationGround Station

Figure 2.7. Comparison between attenuation in a GS or PS based com-munication system

-140

-120

-100

-80

-60

-40

-20

0

20

40

0 5 10 15 20 25 30 35 40

Gain [dB]


Gain of a PS with respect to a GS

Figure 2.8. Gain of a PS based system with respect to a GS based system

21


In Figure 2.7 path losses as a function of ground distance d for a GS and

a PS based communication system are shown, while in Figure 2.8 the gain

of a PS based system with respect to a GS based one is reported. For the

�rst eight kilometers of coverage, a GS system undergoes to less attenuation.

This is due to the fact that the transmitter is closer than in the PS case.

However, for greater ground distances, the propagation conditions in the

case of a stratospheric platform are less severe, and the received power from

a PS is higher.

We can conclude that a stratospheric platform is more suitable than a

ground station to cover a wide area where no obstruction are present, like

in rural environment, because the received �eld undergoes to a lower atten-

uation. Moreover, it should be noted that in the case of the platform the

received power level varies more slowly than in the case of a ground-level

transceiver and this can be a major advantage for certain services that re-

quire a constant received signal strength.

2.2 Visibility and Line Of Sight

In the previous section we have supposed that a direct path between the

transmitter and the receiver can be found. In general, however, this is sel-

dom a valid assumption for a transmission system. Shadowing from hills,

mountains or from high buildings generally occurs, so that no direct path

exists and the performance of the communication system is decreased. Being

placed to an elevated height, the PS is less vulnerable to similar obstructions,

so that Line of Sight (LOS) transmission can be generally taken into account.

In this section we want to investigate the probability that a direct path exists

in a PS based communication system.

With reference to Figure 2.1, it is intuitive that the higher is the elevation

angle �, the lower is the probability of �nding an obstruction along the direct

path. A measure of the percentage of time in which no shadowing occurs for

di�erent values of the elevation angle has been obtained in 1987 by means of

a land mobile satellite system propagation experiment developed in Europe

22


in the framework of ESA's PROSAT (phase II) progress [8]. The experiment

was carried out using a 1.5 GHz signal transmitted from the MARECS satel-

lite, and measuring the signal received from a van traveling through Spain,

France and Sweden. Data were registered in urban, suburban and rural ar-

eas. Measurements taken in Spain correspond to an elevation angle of 39�, in

North France to 26�, whereas in Sweden to 13�. In the platform case those

elevation angles correspond to a ground distance of respectively 21 km, 35

km and 74 km.

Environment Elevation Time in LOS Time with Time withangle faded signal no signal

urban 13� 1 % 39 % 60 %26� 50 % 30 % 20 %39� 70 % 20 % 10 %

rural 13� 60 % 35 % 5 %26� 80 % 18 % 2 %39� 90 % 9 % 1 %

Table 2.1. Percentage of time with a certain signal level in di�erentconditions.

In Table 2.1 the results obtained from the measure campaign are summa-

rized.

The cumulative distributions of Figure 2.9 have been obtained from mea-

surements in urban environments, and give the percentage of time in which

the received signal level, normalized to the line of sight received power, was

above a threshold level. In Figure 2.10 the same cumulative distributions are

reported for rural environment.

As it was expected, the probability of being in LOS increases with the

increasing of the elevation angle and it was obviously lower in urban environ-

ment than in rural environment. However, it should be noted that for high

value of the elevation angle even in urban environment it is likely to have a

direct path to the transmitter. This is a major advantage with respect to a

ground level transceiver, that in urban environment usually cannot provide

a LOS transmission, because high buildings shadow the transmitted signal.

Thanks to its higher position, a stratospheric platform can be used to

cover a large urban area, having a radius of tens of kilometers, granting

23


Figure 2.9. Cumulative distribution in time of received power level inurban environments

Figure 2.10. Cumulative distribution in time of received power level inrural environments

reliable transmission even in street canyons, where users cannot be reached

by traditional ground-level stations.

As already mentioned in Chapter 1, the coverage radius of a platform

24


based system should not be greater than 100 km, corresponding to an ele-

vation angle of about 10�, that is also the lower elevation angle taken into

account in satellite based system (see [30] or [48]). As a consequence, mea-

surements made in Sweden can be considered a worst case for the platform

based system.

2.3 Di�raction

In the previous section it was shown that for high elevation angles a direct

path between the PS and the receiver can usually be found. In this section

we will show that, even when a direct path does not exist, the platform high

position can weaken the e�ect of shadowing from obstructions, making the

path loss from a PS weaker than the GS one.

It is well known that even when a direct path does not exist di�raction

allows the radio signals to propagate up to the receiver. In urban environ-

ment, for example, high buildings usually shadow the transmitted signal, but

the transmission is possible thanks to the di�racted �eld. As a consequence,

the received �eld will be attenuated, and a deterministic prediction of the

received signal strength cannot be practically done.

Traditionally, in order to facilitate the propagation prediction for mobile

radio systems in urban and suburban environments, radio signal variation

over distance is described by an inverse power law, in association with log-

normal shadowing. That is, the received signal from a mobile station placed

at a distance r0 from a base station is proportional to

Lurb = 10�=10=r 0 (2.40)

where � is a random shadowing component and is the deterministic slope

index of the inverse power law. In typical land mobile radio environments,

the logarithm of the shadowing component � is found to be a zero mean

Gaussian random variable, with a standard deviation of 8 dB. The path loss

component of the received signal, on the other hand, can be expressed in

25


logarithmic scale as a linear function of r0

L(dB) = C + 10 log r0 (2.41)

where is typically ranging from 2 to 5 and C is an intercept point gen-

erally taken as the path loss at the distance of 1 kilometer. Experimental

measurements show that, beyond 1 kilometer, radio signal generally decays

continuously with distance. Hence, the slope index and the intercept point

C, provide a simple meaningful measure of path loss, and are used to de-

velop empirical path loss models. The slope index and the intercept point

are generally obtained by �eld measurements made in typical propagation

environments.

However, since they are developed on the basis of in site speci�c mea-

surements, empirical models are not suitable for a di�erent propagation en-

vironment like the scenario of a platform based communication system.

In order to investigate the propagation characteristics in urban and subur-

ban environments when transmitting from a PS, we will refer to an analytical

model developed by H.H.Xia and H.L.Bertoni et al. (see [53], [55], [57] and

[29]). This model has been veri�ed by extensive measurements made in the

United States [58] and Europe [28], and has been partially incorporated in

the COST231 model developed under the European Cooperation in the Field

of Scienti�c and Technical Research (COST) program for GSM system de-

sign [12]. Since the model in its original format involves multiple dimension

integration, in this study we will use the simpli�ed version presented in [54].

Unlike empirical models, the analytical model derives the path loss in

urban and suburban environments as a result of signal reduction due to free

space wave front spreading, multiple di�raction past rows of buildings, as

well as building shadowing. The structure of the model makes it suitable

to investigate both the transmission from a GS than from a PS system. In

fact, as explained in the following, in PS transmissions only the free space

attenuation and the building shadowing have to be taken into account, while

usually in GS transmissions also the di�raction from rows of buildings must

be considered.

It is well known that the e�ect of an obstructing object on the received

power is negligible only if the object is out of the �rst Fresnel zone [36].

26


Fresnel zones represent successive regions where secondary waves, due to the

di�racted �eld, have a path length from the transmitter to the receiver which

are n�=2 greater than the total path length of a line-of-sight path.

��

��

z

TX

RX

z 0

h

xx 0 x

r0

l

Figure 2.11. First Fresnel zone in the platform case

In the case of the platform the border of the �rst Fresnel zone is an ellipse

having the platform and the receiver as foci. Referring to Figure 2.11, the

ellipse equation can be derived from

p(x� x0)2 + z2 +

px2 + (z � z0)2 =

�

2+qx20 + z20 (2.42)

which leads to

4(x20 � k2)x2 + 4(z20 � k2)z2 � 8x0z0xz +

+ [8x0z20 � 4x0(r

20 � k2)]x +

+ [8z0x20 � 4z0(r

20 � k2)]z +

+ (r20 � k2)2 � 4x20z20 = 0 (2.43)

being r0 =px20 + z20 the line-of-sight path length and k = �=2 + r0.

Solving Equation (2.43) for z = h allows to �nd out the position xl where

an object of height h enters the �rst Fresnel zone.

27


x0, receiver �x = jxl � x0jposition (in km) (in m)

0 61 7.15 11.510 17.520 29.530 42.340 55.350 69.4

Table 2.2. Minimum distance �x to consider an object of height 18 metersout of the �rst Fresnel zone.

Table 2.2 reports the minimum distance needed to consider a building of

height 18 meters out of the �rst Fresnel zone. It can be seen that although

di�raction from one building may occur, in uence from a second row of

buildings is negligible. In fact, a second row of building will be separated

from the receiver by the �rst building and by one street width, thus being at

an average distance of 50 meters.

As a result, accordingly to [54], the path loss in the platform case can

be expressed as the sum of two contributions: one due to the free space

propagation (Lfs) and the other to di�raction from a building rooftop down

to the street level (Lrts). Total path loss is thus:

Lp = Lfs + Lrts (2.44)

Both mechanisms are well understood and can be represented with the

simple formulas reported below.

Free space loss accounts for the signal attenuation due to spherical spread-

ing of the wavefront excited by a point source, and is given by:

Lfs = �10 log�

�

4�r0

�2

(2.45)

where � is the wavelength and r0 is the T-R separation.

The di�raction process of the radio wave traveling from the rooftop down

to the street level is represented by the di�racted rays generated at the

building edges. These di�racted rays can be calculated by means of the

28


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

x

z

-HITX

TX -EQ

TX -LO∆h b-LO

∆h b-HI

d b ∆x

θ

rrts ∆h mRX

Figure 2.12. Geometry used for the evaluation of di�raction terms in theattenuation of the received signal.

Geometrical Theory of Di�raction (GTD), thus obtaining the excess path

loss Lrts due to a building presence. Being �hm the height di�erence between

the average rooftop level and the receiver, and �x the horizontal distance

between the receiver and the di�racting edge (see Figure 2.12), Lrts is given

by:

Lrts = �10 log"

�

2�2rrts

�1

�� 1

2� + �

�2#

(2.46)

where � = arctan(�hm=�x) and rrts =p(�hm)2 + (�x)2.

Usually, in a ground based system multiple screen di�raction (md) due

to propagation past row of building also occurs. Based on the studies by Xia

and Bertoni [57], this third factor is given by

Lmd = �10 log(Q2M) (2.47)

The factor QM can be expressed in terms of Boersma functions IM�1;q [57],

QM =pM

1Xq=0

1

q!(2g

p|�)qIM�1;q (2.48)

for M di�racting screens. The dimension-less parameter g is given by

g =�hbp�db

(2.49)

29


where �hb is the transceiver antenna height with respect to the average

rooftop level, and db is the average separation distance between the rows of

buildings (see Figure 2.12). The base station antenna can be above, at, or

below the average rooftop level. The Boersma functions satisfy the recursion

relation,

IM�1;q =(M � 1)(q � 1)

2MIM�1;q +

1

2p�M

M�2Xn=1

In;q�1pM � 1� n

(2.50)

with initial terms,

IM�1;0 =1

M2=3

IM�1;1 =1

4p�

M�1Xn=0

1

n2=3(M � n)2=3(2.51)

The slope index of Equation (2.40), appearing in the distance depen-

dence 1=r 0 of the received signal in the GS case, can be calculated [55] as

= 2(1 + s) (2.52)

where

s = � log(QM+1=QM)

log[(M + 1)=(M)](2.53)

This slope index starts at a value greater than 4 and decreases to a value

less than 4 as the base station antenna height varies from below to above the

rooftops.

As demonstrated in [54], a simpli�ed version of the factor QM (2.48) can

be found in three di�erent conditions:

� Base station antenna near to average roof top level

QM =dbr0

(2.54)

� Base station antenna above the average roof top level

QM � 2:35

�hbr0

rdb�

!0:9

(2.55)

30


� Base station antenna below the average rooftop level

QM =

�db

2�(r0 � db)

�2�p

(�hb)2 + d2b

�1

�� 1

2� + �

�2

(2.56)

being � = arctan(�hb=db)

140

160

180

200

220

240

260

280

0 5 10 15 20 25 30

Path Loss [dB]


Path Loss from GSs of Different Heights

Platform Station Ground Station - LOGround Station - EQGround Station - HI

Figure 2.13. Comparison between path loss due to di�raction using a PSor a GS with di�erent antenna's height

In Figure 2.13 path loss in a PS system is compared with path loss in a

GS based system having di�erent antenna's height:

� �hb = 0 m, GS antenna at roof level (EQ case)

� �hb = 15 m, GS antenna above roof level (HI case)

� �hb = �5 m, GS antenna below roof level (LO case)

while Figure 2.14 reports gains of a PS based system with respect to GS

systems.

31


-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30

Gain [dB]


Gain of a PS with respect to a GS

LOEQHI

Figure 2.14. Gain of PS system with respect to GS systems having di�er-ent antenna's height

As already seen, while di�raction from one building is equally strong in

platform case or in GS case, free space path loss is generally greater when

transmitting from a platform.

However, when GS antenna is placed at or below rooftop level, the third

factor Lmd, that accounts for propagation past row of building and that is

not present in the transmission from a stratospheric platform, is so strong to

overcome free space path loss, thus obtaining a greater total path loss in the

GS based system.

On the other hand, when GS antenna is placed above rooftop level, Lmd is

less severe, so that for small ground distances GS based system still undergoes

to a weaker total path-loss. However, for ground distances greater than 6

km, Lmd increases so much that total path loss becomes greater in the GS

based system.

We can conclude that a stratospheric platform is suitable to be used in

32


urban environment not only because it is more visible, as shown in previous

section, but also because shadowing, whenever it is present, is weaker than

in a GS based system.

2.4 Absorption and Rain Attenuation

Rain is one of the most important atmospheric factor that can decrease the

performance of a radio communication system. In fact, an electro-magnetic

�eld that travels through a rain layer interacts with rain drops, so that part of

the transmitted energy is scattered throughout the region of interaction. As

a result the received �eld is attenuated. The speci�c attenuation r (dB/km)

is obtained from the rain rate r (mm/h) using the power law relationship [23]:

r = kr� (2.57)

The values of k and � can be obtained from ITU-R PN 838-1 and depend

on climatic zone and transmission frequency. We can observe that for high

frequency rain can cause a signi�cant amount of attenuation, thus preventing

great ground link lengths. This e�ect is reduced when transmitting from a

stratospheric platform, because rain is generated in the �rst 2.5 - 3 km of the

atmosphere. In a GS based system rain attenuation for a T-R separation d

is simply:

Arg = rd (2.58)

whereas in a PS based system is

Arp = rlr (2.59)

being lr the length of the path in which radio waves interact with rain drops.

With reference to Figure 2.15, for a ground distance d, lr can be expressed

as:

lr = hrl

s1 +

�d

h

�2

(2.60)

where hrl is the height of the rain layer and h the height of the PS.

33


PS

RX

Ground Distance - d -

Platform

- h -

RainLayer

height

- h -rl

Figure 2.15. Side view of a PS operating scenario with a hrl high rainlayer

Figure 2.16 reports the gain of a PS based system with respect to a GS

system for a 3 km high rain layer, which can be easily found as the ratio

of Equation (2.59) and (2.58). Note that this gain does not depend on the

speci�c attenuation r, and so it is not a function of frequency. As it was

expected, a ground distance dc from which rain attenuation is less severe in

a PS system exists. The cross-over distance dc can be easily found, being

the point in which lr(dc) = dc. Remembering Equation (2.60), dc can be

expressed as:

dc =hrlq

1� �hrld

�2 (2.61)

Cross-over distances for typical rain layer height are reported in Table 2.3.

hrl, rain layer dc, cross-overheight (in km) distance (in km)

2 2.0143 3.048

Table 2.3. Cross-over distance from which rain attenuation is less severefor a PS than for a GS

34


-15

-10

-5

0

5

10

0 5 10 15 20 25 30

Gain [dB]


Gain in Rain Attenuation of a PS w.r.t. a GS

Figure 2.16. Gain in rain attenuation of a PS with respect to a GS for a3 km high rain layer

35

Chapter 3

Small Scale Fading and

Multipath - A Theoretical

Introduction

Small scale fading is used to describe the rapid uctuation of the received

signal strength when traveling through short distances or when considering

short time interval, so that large scale path loss e�ects can be neglected.

Small scale fading, or simply fading, is caused by the interference of several

versions of the same transmitted radio signal, that arrive to the receiver

following di�erent paths. These radio waves, calledmultipath waves, combine

at the receiver giving a resultant signal which can vary widely in amplitude

and phase. These variations depend on the distribution of the delays and

relative power of the multipath waves, on the speed of the receiver, and on

the time-frequency characteristics of the transmitted signal.

The three most important small scale fading e�ects are:

� Time dispersions (echoes) caused by multipath propagation delays

� Rapid uctuation on the received signal strength when traveling through

short distances

� Random frequency modulation due to varying Doppler shifts on di�er-

ent multipath waves

36

3 { Small Scale Fading and Multipath - A Theoretical Introduction

As a result of time dispersion and frequency modulation the received signal

fades. In the following it will be shown that two signal su�ciently separated

in frequency (time) can experience di�erent attenuation levels because of

time dispersion (frequency modulation).

In this Chapter we will focus on the description of small scale fading as

it is experienced in a typical GS based transmission. It will be shown that

the properties of a fading channel can be described by two functions, the

power delay pro�le and the Doppler Spectrum; we will show how to derive

the coherence bandwidth and the coherence time from the power delay pro�le

and from the Doppler spectrum, and we will discuss how those parameters

interfere in de�ning whether a channel is at or selective in frequency, and

fast or slow in time.

3.1 Multipath Channel Baseband Impulse Re-

sponse

The small scale variations of a mobile radio signal can be directly related to

the impulse response of the mobile radio channel. A mobile radio channel

can be modeled as a linear �lter with a time varying impulse response: the

�ltering nature of the channel is caused by the sum of amplitudes and delays

of the multiple arriving waves at any instant of time, while the time variation

is due to the motion of the receiver, as well as to the motion of the transmitter

or of other elements in the propagation environment. As an example consider

the case where time variation is due strictly to the receiver motion in space

(see Figure 3.1): for a �xed position d, the channel between the transmitter

and the receiver can be modeled as a linear time invariant system. Note

that the di�erent multipath waves have propagation delays which vary over

di�erent spatial locations of the receiver. Thus, the impulse response of the

linear time invariant channel is a function of the position of the receiver and

of time: h(d;t). Let x(t) represent the transmitted signal, then the received

signal y(d;t) at a certain position d can be expressed as a convolution of x(t)

37


d

Figure 3.1. A receiver experiencing di�erent transmission channels whilemoving in space

with h(d;t).

y(d;t) = x(t) h(d;t) =

Z1

�1

x(�)h(d;t� �) d� (3.1)

For a causal system, h(d;t) = 0 for t < 0, and Equation (3.1) reduces to

y(d;t) =

Z t

�1

x(�)h(d;t� �) d� (3.2)

If the receiver moves on the ground at a constant velocity v, the position of

the receiver can be expressed as

d = vt (3.3)

Substituting (3.3) in (3.2), we obtain

y(vt;t) =

Z t

�1

x(�)h(vt;t � �) d� (3.4)

Since v is a constant, y(vt;t) is just a function of t. As a consequence,

Equation (3.4) can be expressed as

y(t) =

Z t

�1

x(�)h(vt;t� �) d� = x(t) h(vt;t) (3.5)

Therefore x(t) represents the transmitted bandpass waveform, y(t) the

received waveform, and h(t;�) the time variant impulse response of the radio

38


channel. The variable t represents the time variations due to motion, whereas

� represents the channel multipath delay for a �xed value of t (see Figure 3.2).

One may think of � as being a vernier adjustment of time.

If the multipath channel is assumed to be a band-limited bandpass chan-

nel, which is reasonable, then it may equivalently be expressed by a complex

baseband impulse response hb(t;�) [37], with the input and output being the

complex envelope representations of the transmitted and received signals,

respectively:

1

2r(t) =

1

2c(t) 1

2hb(t;�) (3.6)

where c(t) and r(t) are related to x(t) and y(t) through:

x(t) = Refc(t) exp(j2�fct)g (3.7)

y(t) = Refr(t) exp(j2�fct)g (3.8)

The lowpass characterization removes the high frequency variations caused

by the carrier, making the signal analytically easier to handle.

In order to �nd out an expression for the channel baseband response

hb(t;�), it is useful to discretize the multipath delay axis � into equal time

delay segments usually called excess delay bins [38], where each bin has a time

width of �� = �i� �i�1: any signal received within the ith bin is represented

as a single multipath component having excess delay �i. For convention,

�0 = 0 is the excess time delay of the �rst arriving multipath component,

and neglects the propagation delay between the transmitter and the receiver,

while �i = i�� is the relative delay of the ith multipath component as com-

pared to the �rst arriving component. This technique of quantizing the delay

bins determines the time delay resolution of the channel model, and the use-

ful frequency span of the model can be shown to be 1=(2��). Therefore, the

model may be used to analyze signals having a bandwidth not greater than

1=(2��).

Since the received signal in a multipath channel is the sum of di�erent

echoes, each having a di�erent amplitude, delay and phase, the baseband

impulse response of the channel can be expressed as:

39


hb(t;�) =N�1Xi=0

ai(�;t) exp(j2�fc�i(t) + �(�;t))�(� � �i(t)) (3.9)

where ai(�;t) and �i(t) are respectively the real amplitude and excess delay

of the ith multipath component. Note that some excess delay bins may have

no multipath components at certain time t, since ai(�;t) may be zero. The

phase term 2�fc�i(t) + �(�;t) in (3.9) represents the phase shift due to free

space propagation of the ith multipath component, plus any phase shift that

the transmitted signal experiences along the transmission path. In general

the phase term is simply represented by a single variable �i(�;t) which lumps

together all the mechanism for phase shifts of a single multipath component

within the ith path.

t

h(t,

τ

τ

τ

τ

∆τ

)

Figure 3.2. An example of the time varying impulse response model for amultipath radio channel

In the following section it will be shown that the channel baseband re-

sponse can be used to obtain the power delay pro�le of the channel, which is

a useful mean to describe the time dispersive properties of the channel.

40


3.2 Power Delay Pro�le and Coherence Band-

width

For small scale channel modeling, the power delay pro�le P (�) of the channel

is found by taking the spatial average of jhb(t;�)j2 over a local area. The

power delay pro�le is used to determine some parameters that quantify the

time dispersive properties of a multipath channel. The most commonly used

are the mean excess delay (�) and the rms delay spread(��). The mean excess

delay is the �rst moment of the power delay pro�le and is de�ned to be

� =

Pk a

2k�kP

k a2k

=

Pk P (�k)�kPk P (�k)

(3.10)

The rms delay spread is the square root of the second central moment of

the power delay pro�le and is de�ned to be

�� =

q� 2 � (�)2 (3.11)

where

� 2 =

Pk a

2k�

2kP

k a2k

=

Pk P (�k)�

2kP

k P (�k)(3.12)

It should be noted that the power delay pro�le and the magnitude fre-

quency response (the spectral response) of a mobile radio channel are related

through the Fourier transform. Thus, an equivalent description of the chan-

nel in the frequency domain can be obtained using its frequency response

characteristic. Analogous to the delay spread parameter in the time domain,

the coherence bandwidth Bc is used to characterize the channel in the fre-

quency domain. Coherence bandwidth is a statistical measure of the range

of frequencies over which the channel can be assumed to be at, i.e. a channel

which passes all frequencies with approximately equal gain and linear phase.

On the other hand, when a transmitted signal has a bandwidth greater than

Bc the channel must be considered frequency selective. Coherence bandwidth

and rms delay spread are related to each other: it can be shown [26] that if the

coherence bandwidth is de�ned as the bandwidth over which the frequency

correlation is above 0.9, then Bc is approximately

Bc =1

50��(3.13)

41


Power delay pro�les are usually obtained through measurements taken in

di�erent environments. In Table 3.1 are reported some results obtained from

di�erent measurements, while Table 3.2 reports the continuous power delay

pro�le de�ned in the recommendation CEPT/GSM 05.05.

Environment Frequency RMS Delay Notes Ref.(MHz) Spread (�� )

urban 910 1300 ns Avg. New York City [14]600 ns Std. Dev.3500 ns Max.

urban 892 10-25 us Worst case San Francisco [39]Suburban 910 200-310 ns Averaged typical case [15]Suburban 910 1960-2110 ns Averaged extreme case [15]Indoor 850 270 ns Max. O�ce building [17]Indoor 1900 70-94 ns Avg. Three San Francisco [46]

1470 ns Max. buildings

Table 3.1. Typical measured value of RMS Delay Spread

In general, urban environment is subject to a stronger time dispersion,

because there are more scatterers surrounding the receiver, while in rural

environment time dispersion is less severe. Note that in hilly environments

it is possible to �nd a two-peaks power delay pro�le, the second peak being

caused by a cluster of strong scatterers that are far away from the receiver,

such as mountains, skyscrapers or buildings.

3.3 Doppler Spectrum and Coherence Time

Delay spread in time domain and coherence bandwidth in frequency domain

are parameters which describe the time dispersive nature of the channel in

a local area. However, they do not give any informations about the time

varying nature of the channel caused by either relative motion between the

receiver and the transmitter, or by movement of objects in the channel. In

the following it will be shown that these e�ects are taken into account by the

Doppler Spectrum of the channel.

42


Urban non-hilly Area - Typical case P (�) =

�e�� 0 < � < 7 us0 elsewhere

Urban hilly Area - Bad case P (�) =

8<:

e�� 0 < � < 5 us0:5e(5��) 5 < � < 10 us0 elsewhere

Rural non-hilly Area - Typical case P (�) =

�e�9:2� 0 < � < 0:7 us0 elsewhere

Rural hilly Area - Typical case P (�) =

8<:

e�3:5� 0 < � < 2 us0:1e(15��) 15 < � < 20 us0 elsewhere

Table 3.2. Examples of power delay pro�les de�ned in RecommendationCEPT/GSM 05.05

It is well known that a mobile, moving at a certain velocity v, which

receives a radio wave of frequency fc, experiences a Doppler shift in the

received frequency, given by

�f =v cos�

cfc (3.14)

where � is the angle of arrival of the received wave with reference to the

direction of motion of the mobile, and c is the speed of light.

In a multipath channel echoes experience di�erent shifts in frequency,

since they have di�erent angles of arrival. Therefore the multipath channel

is dispersive in frequency. A simple model that describes the frequency dis-

persive nature of a multipath channel was introduced by Jakes [24]. The

model considers plane waves from stationary scatterers that are incident on

a mobile which is traveling in the x-direction with velocity v, as shown in

Figure 3.3. The x-y plane is assumed to be horizontal. The vehicle motion

43


x

y

n-th echo

αn

v

RX

Figure 3.3. The geometry considered in Jakes' model for the evaluationof the Doppler spectrum.

introduces a Doppler shift in every wave

!n = 2�fn = 2�v

cfc cos�n (3.15)

being fc the frequency of the transmitted carrier frequency. If the transmitted

signal is vertically polarized, the �eld component seen at the receiver are:

Ez = E0

NXn=1

Cn cos(!ct+ �n) (3.16)

Hx = �E0

Z0

NXn=1

Cn sin�n cos(!ct+ �n) (3.17)

Hy =E0

Z0

NXn=1

Cn cos�n cos(!ct+ �n) (3.18)

(3.19)

where

�n = !nt+ �n (3.20)

and Z0 is the free-space wave impedance, E0Cn is the real amplitude of

the nth wave in the Ez �eld. The �n are random phase angles uniformly

44


distributed from 0 to 2�. Furthermore, the Cn are normalized so that the

ensemble average is <PN

n=1C2n >= 1. It can be shown that for a number of

multipath waves N su�ciently high, the envelope r of the received Ez �eld

is a random variable, having a Rayleigh probability density:

p(r) =

(rbe�r

2=2b; r � 0

0; r < 0(3.21)

where b = E20=2 is the mean power of the received signal. Furthermore, if

the distribution of power with the echoes' Direction Of Arrival (DOA) is a

uniform distribution, the power spectrum of Ez is:

SEz(f) =b

�fm

"1�

�f � fcfm

�2#�1=2

(3.22)

where fm = v=c fc is the maximum Doppler shift.

In some environments the random multipath components are superim-

posed on a stationary dominant signal. In this case, the envelope of the

received signal is a random variable that follows a Ricean distribution [38]:

p(r) =

(rbe�

(r2+A2)2b I0

�Arb

�; (A � 0;r � 0)

0; r < 0(3.23)

The parameter A denotes the peak amplitude of the dominant signal and I0

is the modi�ed Bessel function of the �rst kind and zero order. The power

spectrum of the received signal is:

S 0Ez(f) = SEz(f) + A2�(f � fc � fa) (3.24)

where fa = fm cos�0 is the Doppler shift experienced by the direct wave,

having an angle of arrival �0.

The model shows that when a pure sinusoidal tone of frequency fc is

transmitted, the received signal spectrum, also called the Doppler spectrum,

will have components in the range fc � fm to fc + fm. It can be shown

that if the baseband signal bandwidth is much greater than fm the e�ects of

Doppler spread are negligible at the receiver [24].

Coherence time TC is used to characterize the time varying nature of the

frequency dispersiveness of the channel in the time domain. The maximum

45


Doppler spread and coherence time are inversely proportional to one other.

That is,

TC � 1

fm(3.25)

Coherence time is actually a statistical measure of the time duration over

which the channel impulse response is essentially invariant, and quanti�es

the similarity of the channel response at di�erent times. In other words,

coherence time is the time duration over which two received signal have

a strong potential for amplitude correlation [24]. If the coherence time is

de�ned as the time over which the time correlation function is above 0.5,

then the coherence time is approximately [47]

TC � 9

16�fm(3.26)

If the channel impulse response changes rapidly within the symbol period

of the transmitted signal, the channel is said to be a fast fading channel,

otherwise is said to be a slow fading channel.

The time variant properties of the Rayleigh fading signal can be described

with the aim of two statistics that were computed by Rice [42] for a math-

ematical problem which is similar to Jakes' fading model. Those statistics

provide simple expressions for computing the average number of level cross-

ing and the duration of fades, and are known as the level crossing rate (LCR)

and the average fade duration [42].

The level crossing rate is de�ned as the expected rate at which the

Rayleigh fading envelope, normalized to the local rms signal level, crosses

a speci�ed level in a positive-going direction. The number of level crossing

per second is given by

NR =

Z1

0

�rp(R;

�r) d

�r =

p2�fm�e

��2 (3.27)

where�r is the time derivative of r(t) (i.e., the slope ), p(R;

�r) is the joint

density function of r and�r at r = R, fm is the maximum Doppler frequency

and � = R=Rrms is the value of the speci�ed level R, normalized to the local

rms amplitude of the fading envelope. Equation (3.27) gives the value of

46


NR, the average number of level crossing per second at speci�ed R. The level

crossing is a function of the mobile speed as is apparent from the presence of

fm in Equation (3.27). From this Equation it can be derived that the signal

envelop experiences very deep fade only occasionally, whereas shallow fades

are frequent.

The average fade duration is de�ned as the average period of time for

which the received signal is below a speci�ed level R. For a Rayleigh fading

signal, this is given by

�R =1

NRPfr � Rg (3.28)

where Pfr � Rg is the probability that the received signal is less than R and

is given by

Pfr � Rg = 1

T

Xi

�f;i (3.29)

where �f;i is the duration of the fade and T is the observation interval of

the fading signal. The probability that the received signal is less than the

threshold R is found from the Rayleigh distribution as

Pfr � Rg =Z R

0

p(r) dr = 1� e��2

(3.30)

where p(r) is the pdf of a Rayleigh distribution. Thus, using Equations

(3.27), (3.28) and (3.30), the average fade duration as a function of � and fm

can be expressed as

�R =e��

2 � 1

�fmp2�

(3.31)

Average fade duration primarily depends upon the speed of the mobile,

and decreases as the maximum Doppler frequency fm becomes large. This

can be understood remembering that a channel with a high value of fm is a

channel with short coherence time, i.e. a channel that varies rapidly.

As pointed out by E.Damosso and L.Stola in [16], theoretically only the

bandwidth fm of the Doppler spectrum a�ects the values of NR and �R, so

that the general trend of the Doppler spectrum is not really determinant in

47


evaluating the performance of a communication channel. As a consequence

Jakes' Doppler spectrum, that is the correct Doppler spectrum only in the

case of omni-directional antennas that receive echoes with uniformly dis-

tributed DOAs, is used also in the modeling of radio channels that do not

respect those conditions.

48

Chapter 4

Small Scale Fading in a

Platform Based System

As shown in Chapter 3, a mobile radio channel is characterized by the Power

delay pro�le and the Doppler spectrum. With the growth of cellular and

mobile communications, several theoretical and experimental models for the

mobile radio channel have been presented. However, usually those models

and measurements deal with ground communication systems, and with a still

transmitter, being usually a �xed antenna. In this Chapter we will study the

small-scale characteristics of a channel in a scenario in which the transmitter

is payload of a stratospheric platform.

The �rst part of the Chapter is devoted to the study of the Power de-

lay pro�le for a platform based system. Power delay pro�les derived from

measures at ground level are in site speci�c and cannot be adapted to the

platform case. In order to �nd out a realistic power delay pro�le for the plat-

form case, we have considered two analytical models proposed respectively

by Rappaport with Liberti [40] [41] and Burr [11], and we have extended

them to the platform case.

In the second part of this Chapter we will focus on the Doppler spec-

trum of the platform based system, and we will present an analytical model

developed by M.Pent et al. [35].

49

4 { Small Scale Fading in a Platform Based System

4.1 Extension of the Rappaport's geometri-

cal model to the platform case

The geometrical model presented by Rappaport considers a transmitter and

a receiver having separation d0 (Figure 4.1). Scatterers are assumed to lie

in the horizontal plane which includes the transmitter and a receiver. If

a LOS path exists between the transmitter and the receiver, then the �rst

component of the total received signal will arrive with a path delay t0 = d0=c

where c is light speed. It is assumed that scatterers are uniformly distributed

in space and have equal scattering cross sections. As noted in [34] all the

scatterers giving rise to single bounce components arriving between time t0

and time ti = t0+ �i (i.e all scatterers giving rise to echoes with excess delay

lower than �i), lie in the region bounded by an ellipse with parameters

a =cti2

(4.1)

b =pa2 � f 2 (4.2)

where f = d0=2, as shown in Figure 4.1.

As shown in [27], it is possible to �nd out an expression for the cumulative

distribution function (cdf) of the excess delay �i, by introducing a maximum

path delay, �m, which de�nes a windows such that 0 � �i < �m for all �i. The

ratio of the area of the ellipse corresponding to delay �i, AGS(�), to the area

of an ellipse corresponding to �m, AGS(�m), leads to an expression for the cdf

FGS(�):

FGS(�) =AGS(�)

AGS(�m)(4.3)

By di�erentiating the cdf, we obtain the probability density function (pdf).

The same steps can be followed in the platform case. The geometry of

the system is more complex, because the receiver and the transmitter are no

longer on the same plane, in fact also the height of the transmitter must be

considered. Introducing a Cartesian reference system, consider the receiver

placed at a point R of coordinates (x0,0,0), while the transmitter is placed in

T (0,0,z0). The direct transmission path is then r0 =px20 + z20 , which leads

to a propagation delay �0 = r0=c, being c the light speed. A re ected wave

50


TX RX

a/2

b/2

y

x

f

Figure 4.1. The ellipse containing scatterers that cause echoes with anexcess delay lower than �i

impinging at the receiver with an excess delay � must have covered a path

length

k = r0 + c� (4.4)

The set of points for which the distance from the transmitter T plus the

distance from the receiver R is k is an ellipsoid � having T and R as foci.

A re ected ray with an excess delay between 0 and � must be originated

by a scatterer S inside the ellipsoid �: that is, a wave traveling from the

transmitter T to the scatterer S, impinges on the scatterer and is re ected

toward the receiver, covering a total path length shorter than k (see Fig-

ure 4.2). However, it is reasonable that scatterers can be found only in a

thin layer close to the ground: in fact scattered waves are mainly due to high

buildings, trees, poles, or hills, that cannot be found above a certain height

h. As a result, the volume V (�) containing scatterers that cause a re ected

wave with an excess delay between 0 and � is the intersection between the

ellipsoid � and the layer of height h. The volume V (�) is thus de�ned as the

51


RX

TX (0,0,z

(x ,0,0)

Sz < h

x

y

z

Σ )

s

0

0

Figure 4.2. The ellipsoid � containing scatterers that cause excess delayslower than �i, and a scatterer S of height zs < h contained in the ellipsoid

(Extension of the Rappaport model).

set of points (x;y;z) that satisfy:( p(x� x0)2 + y2 + z2 +

px2 + y2 + (z � z0)2 < k(�)

0 < z < h(4.5)

Starting from Equation (4.5), it is possible to obtain the expression of the

volume V as a function of � . In fact V (�) can be calculated by evaluating

the area A(z;�) of the ellipse due to the intersection of the ellipsoid with a

plane z = const (see Figure 4.3), and then by integrating this area with z

varying from 0 to h.

A(z;�) can be expressed as:

A(z;�) = �a(z;�)b(z;�) (4.6)

52


��

��

A(z)

RX

TX

x

y

z

Σ

Figure 4.3. The area A(z) intersection of the ellipsoid � with a planeparallel to x-y

where a e b are the semi-axes of the ellipse, and can be evaluated observing

that, for a given height z, the abscissa x of one point of the ellipse is:

x(y;z;�) = xc(z;�)� xd(y;z;�) (4.7)

where

xc(z;�) =�x30 + x0k

2 + 2x0z0z � x0z20

2k2 � 2x20(4.8)

and

xd(y;z;�) =k

2k2 � 2x20

pp(y;z;�) (4.9)

where

p(y;z;�) = 4z20z2 + 2x20z

20 + 4x20y

2 � 4k2z2 � 4z30z � 4k2y2 � 2x20k2

�2z20k2 + 4k2z0z + 4x20z2 + k4x40z

40 � 4x20z0z (4.10)

Thus, as we can observe in Figure 4.4:

a = xd(0;z;�) (4.11)

53


A(z)

x

y

a(z)

b(z)

x

xc

d

xc + xd

xc xd-

Figure 4.4. The ellipse of area A(z) and the semi-axes a(z) and b(z)

and b is the ordinate y for which

xd(b;z;�) = 0 (4.12)

Substituting a and b in (4.6) leads to

A(z;�) =�k(k2 � x20 � z20)

4p(k2 � x20)

3

�4z(z0 � z) + k2 � x20 � z20

�(4.13)

where A(z;�) is a function of � through k, which is given in (4.4). Equation

(4.13) can be easily integrated:

V (�) =

Z h

0

A(z;�) dz (4.14)

and yields to:

V (�) =��(c2� 2 + 2r0c�)(r0 + c�)

4p(z20 + c2� 2 + 2r0c�)3

�4

3h3 � 2z0h

2 � (c2� 2 + 2r0c�)h

�(4.15)

54


As previously done in the GS case, the cdf of the excess delay can be expressed

as the ratio between the volume V (�) and the volume that corresponds to

the maximum excess delay of the system, V (�m):

FPS(�) =V (�)

V (�m)(4.16)

while the pdf can be found by di�erentiating (4.16):

fPS(�) =1

V (�m)

dV

d�(4.17)

Equations (4.16)and (4.17) require the knowledge of the maximum path

delay �m. Even if values of �m for the GS based case are well known, there

are no measurements available for the platform case. The development of

the second model presented in this Chapter leads to value of �m that are in

the order of hundreds of nanoseconds, which is considerably less than the GS

based case.

In Figure 4.5 and 4.6 the distribution of delay and the pdf of delay, cal-

culated with �m = 0:150 us, x0 = 40 km and h = 21 m are reported. Values

given to x0 and h a�ect the shape of the distribution, as shown in Fig-

ure 4.7, 4.8, 4.9 and 4.10. In Figures 4.7 and 4.8 are shown the distributions

and pdfs of the delay for h = 21 m and x0 from 0 to 100 km with a step of

20 km, while Figures 4.9 and 4.10 report the distributions and pdfs of the

delay for x0 = 40 km and h varying from 11 m to 51 m with a step of 10 m.

Note that even if the curves obtained are di�erent, the general shape of the

distribution is maintained; in a �rst order of approximation it is reasonable,

in the development of a channel model, to use one speci�c value for x0 and h

and consider the related curve as a general distribution of the delay for the

platform case.

4.2 Extension of Burr's channel model to the

platform case

As shown by A. G. Burr in [11], it's possible to obtain the power-delay pro�le

of a radio channel assuming a Poisson-distributed statistic for the echo delays.

55


0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

F(

τ)

Excess Delay τ [µs]

Distribution of the Delay

x0= 40 km

Figure 4.5. Delay's distribution obtained using the extension of Rappa-port's geometrical model, with �m = 0:150 �s, x0 = 40 km and h = 21

m.

This assumption was already suggested by A. S. Bawja in [9], and by Saleh

and Venezuela in [45], and was used to get an analytical expression for the

experimental distribution of delays measured in some environments.

If we use a Poisson distribution, then the probability that at some time

� the number N of echo arrivals is equal to i is:

PfN = ig = �(�)ie��(�)

i!(4.18)

The Poisson parameter �(�) gives the probability of an echo's arrival

within an in�nitesimal time interval, divided by that time interval:

PfN�2[md�;(m+1)d� ] = 1g = �(�)d� (4.19)

As shown by Burr, if we consider a transmitter and a receiver both lying

over the ground at a distance d, surrounded by a uniform distribution of

scatterers, Equation (4.19) can also be expressed as the probability that a

56


0

2

4

6

8

10

12

14

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

f(

τ)


Pdf of the Delay

x0= 40 km

Figure 4.6. Delay's pdf obtained using the extension of Rappaport's geo-metrical model, with �m = 0:150 �s, x0 = 40 km and h = 21 m.

scatterer will be found within an elliptical annulus of in�nitesimal thickness

corresponding to that time interval. If we assume that scatterers are uni-

formly distributed over the ground with a density per unit area �, then the

probability of �nding a scatterer in the elliptical annulus of area dA is:

PfN�2[md�;(m+1)d� ] = 1g = �dA = ��A

��d� (4.20)

which yields to:

�(�) d� =�c�(2c2� 2 � d2)

4pc2� 2 � d2

d� (4.21)

However we are interested in studying a system in which the transmitter

is on a platform ying at a certain height z0. In this case the model developed

by Burr can no longer be used, because the scatterers are concentrated on a

layer over the ground, so that they surround only the receiver.

57


0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

F(

τ)



h= 21 m

x0= 0 kmx0= 100 km

Figure 4.7. Delay's distribution with �m = 0:150 �s, h = 21 m and x0varying from 0 to 100 km with step 20 km.

Using this particular geometry, Equation (4.19) must be expressed as the

probability that a scatterer can be found in a volume dV corresponding to

echoes with a delay in the range [md�;(m+1) d� ], i.e. in which the distance

of the scatterer from the transmitter added to the distance from the receiver

ranges between kmin = r0 + c m d� and kmax = r0 + c (m + 1) d� , being r0

the distance between transmitter and receiver, and where the height of the

scatterer is between 0 and a certain maximum height h.

If we assume that the scatterers are uniformly distributed over the ground

with a density �, we obtain:

PfN�2[md�;(m+1)d� ] = 1g = �dV (4.22)

which yields to

�(�)d� = �dV = ��V (�)

��d� (4.23)

An expression for the volume V (�) has been already given in Equation (4.15).

By deriving Equation (4.15) it is possible to �nd an analytical expression for

58


0

2

4

6

8

10

12

14

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

f(

τ)


Pdf of the Delay

h= 21 m

x0= 0 km

x0= 100 km

Figure 4.8. Delay's pdf with �m = 0:150 �s, h = 21 m and x0 varyingfrom 0 to 100 km with step 20 km

the Poisson parameter �:

�(�) = ��V (�)

��(4.24)

As mentioned before, � represents the density of scatterers in the area of

interest. In general, surfaces of buildings, as well as trees, poles or vehicles

are the most important elements of scattering. In this work we will assume

that a scatterer can be found once every 10 m, which is reasonable, thus

obtaining � = 106 km�3.

From the Poisson assumption previously made it follows that the inter-

arrival time between two echoes is exponentially distributed with mean delay

1=�(�). As a consequence, the Poisson parameter given in Equation (4.24)

can be interpreted as the frequency of echo arrivals and the average num-

ber n(�) of echoes received in an interval of time �T around the instant

corresponding to a delay � can be approximately expressed as:

n = �(�) �T = ��V (�)

��T (4.25)

59


0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

F(

τ)



x0= 40 km

h= 51 mh= 1 m

Figure 4.9. Delay's distribution with �m = 0:150 �s, x0 = 40 km and hvarying from 11 to 51 m with step 10 m

Using Equations (4.21) and (4.24) it is possible to compare the frequency

of arrivals in the case of a GS based system and in the case of a platform

based system: Figure 4.11 shows the comparison in the case in which the

transmitter and the receiver are at a ground distance of 40 km. In the

GS case a scatters'density � = 104 km�2 has been taken, so that even in the

ground case a scatterer can be found once every 10 m. It can be seen that for

small values of excess delay the frequency of arrivals is lower in the platform

case than in the GS case. It will be shown that echoes with small delay are

less attenuated than those with a great delay, so causing a major interference

with the direct ray. Having a smaller number of powerful echoes, platform

based system undergoes to a less severe fading.

Given the average number of echoes received in an interval around � as

a function of � (4.25), it is possible to �nd out an expression for the average

power delay pro�le PPS(�) of the PS based system. In fact, the average total

power received in � can be expressed as the product between the average

60


0

2

4

6

8

10

12

14

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

f(

τ)


Pdf of the Delay

x0= 40 km

h= 51 m

h= 1 m

Figure 4.10. Delay's pdf with �m = 0:150 �s, x0 = 40 km and h varyingfrom 11 to 51 m with step 10 m

power of a single echo having delay � , Ps(�), with the average number of

echoes having delay � , n(�):

PPS(�) = Ps(�)n(�) (4.26)

Equation (4.26) requires the knowledge of Ps(�). In the following we derive

an expression for Ps(�) assuming all scatterers of the same height zs.

As already stated, an echo received with an excess delay � must have

covered a path length k(�) = r0 + c� , being r0 the distance between the

stratospheric platform and the receiver. In the assumption of scatterers of the

same height, the scatterer S that produced the echo can be found somewhere

in the ellipse shown in Figure 4.12. If the scatterers are uniformly distributed

in space, we can assume that the coordinate ys of the scatterer S is uniformly

distributed between �b(zs) and b(zs) (see Figure 4.12), where b can be found

accordingly to Equation (4.12) . Following Equation (4.7), the abscissa xs is:

xs� = xc � xd(ys) (4.27)

61


35

40

45

50

55

60

65

70

0 1 2 3 4 5 6 7

10 Log10(

λ)


Poisson Parameter λ for d= 40 km

PS with ρ=106 km-3

GS with σ=104 km-2

Figure 4.11. Poisson parameter in the GS case (dotted line) and in thePS case (solid line)

Assuming a uniform distribution of the scatterers, the abscissa is, with equal

probability, xs+ = xc + xd(ys) and xs� = xc � xd(ys), that is:

Pfxs = xs+ = xc + xd(ys)g = Pfxs = xs� = xc � xd(ys)g = 0:5 (4.28)

As a consequence, the distance between the scatterer S and the receiver can

be written as:

rs�(ys) =qx2s� + y2s (4.29)

Thus, as it can be seen from Figure 4.13, an echo with excess delay � covers

a path length k � rs� from the transmitter to the scatterer S (path TX-S)

and a path length rs� from the scatterer S to the receiver (path S-RX).

As shown in Chapter 2 usually a stratospheric platform and a point at

ground level are in line of sight, and the height of the platform is su�cient

to neglect e�ects due to ground re ection, so that free space propagation can

be taken into account for path TX-S. Power received from the scatterer S

62


x

y

xc

b(z )

A(z )

a(z )

s

s

s

S

Figure 4.12. The ellipse of area A(zs) and the semi-axes a(zs) and b(zs)

is scattered all around according to the radar cross section (RCS) �r of the

scatterer [51]. The total power re ected in the direction of the receiver is

then:

Pr s (rs�(ys)) = PTXGTX�r

4�(k � rs�(ys))2(4.30)

If we assume free space propagation also for the path S-RX, then the received

power due to a scatterer of ordinate ys is:

Ps (rs�(ys)) = PTXGTX�r

4�(k � rs�(ys))2�2

(4�rs�(ys))2GRX (4.31)

Note that Ps is a function of the random variable ys, whose distribution is

ys � U(�b(zs);b(zs)), and of the values taken by rs that can be either rs+

or rs� according to the abscissa xs�. The average value of Ps can thus be

calculated as:

Ps = E fPs(rs+)gPfxs� = xs+g+ E fPs(rs�)gPfxs� = xs�g (4.32)

63


-+

-+k - r s

rs

k

r0

x

y

z

TX

RX

S

Figure 4.13. The path covered by an echo having delay � , impinging ona scatterer S.

that is

Ps =

�Z b

�b

Ps(rs+(ys))

2bdys

�0:5 +

�Z b

�b

Ps(rs�(ys))

2bdys

�0:5 (4.33)

Using Equations (4.33), (4.25) and (4.26) we have numerically evaluated

the power delay pro�le for the platform case.

In Figure 4.14 is shown the power delay pro�le obtained assuming the

receiver at x0 = 20 km, a density of scattering point of � = 106 km�3, and

using a radar cross section �r of 20 m2. This value corresponds to the radar

cross section of a at metal plate having an edge long 0.5 m, in the case of

normal incidence. In fact, as shown in [25], the RCS of a metal plate in this

case can be approximated to

�r =4�A2

�2(4.34)

where A is the physical area of the plate. For a wavelength of 0.25 m, i.e a

carrier frequency of 1.2 GHz, �r = 20 m2 is obtained for A = 0:32 m2

64


-35

-30

-25

-20

-15

-10

-5

0

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

P/PLOS [dB]


Normalized Power Delay Profile

Figure 4.14. Power delay pro�le for a platform based system

Note that the model holds as long as the free space propagation law used

in (4.31) is valid, i.e. as long as the receiver is in the far �eld region with

respect to the scatterers. This condition is satis�ed when:

rs(ys)� �

4�(4.35)

It can be seen from Figure 4.12 that, for a given value of � , rs�(ys) takes

its smallest value for ys = 0, with abscissa xs+. In this case (see Figure 4.15)

rs becomes:

(x0 + rs)2 + z20 = (k � rs)

2 (4.36)

The far �eld condition is therefore:

rs =k2 � x20 � z202(k + x0)

� �

4�(4.37)

that leads to

k � �

4�+

rz20 + (x0 +

�

4�)2 (4.38)

65


SRX

TX

x

z

z0

k- rs

x0

sr

Figure 4.15. Evaluation of the smallest rs

By substituting k = r0 + c� we obtain:

� ��4�

+qz20 + (x0 +

�4�)2 � r0

c(4.39)

The power delay pro�le of Figure 4.14 obtained with a carrier frequency of

1.2 GHz and receiver position x0 = 20 km, is valid for � � 0:12 ns.

Note that the power delay pro�le has been evaluated by assuming free

space propagation also for the path S-RX, even if it is covered at ground

level. As mentioned in Chapter 2, for ground level transmission a higher

order attenuation law is most commonly used instead of the second order

free space attenuation law, so that interference due to ground re ection is

taken into account. However, the ground re ection model can be used only

if the distance between the transmitter and the receiver satis�es Equation

2.23. For the path S-RX Equation 2.23 is satis�ed when

rs >20hshRX

�(4.40)

where hs is the height of the scatterer and hRX is the height of the receiver.

Assuming hs = hRX = 1:5 m, and following the same steps as from Equation

66


(4.36) to (4.39), we obtain that the two-ray ground re ection model is valid

for echoes with an excess delay � > 1:08 us. However, it can be seen from

Figure 4.14 that echoes with so a great delay are already too much attenuated

to be taken into account.

Table 4.1 reports a comparison between some parameters derived from

the power delay pro�le PPS(�) and the same parameters obtained by mea-

surements made for Land Mobile Satellite (LMS) communications. The rms

delay spread �� obtained from PPS(�) is �� = 21 ns, while measurements

reported in [33] present a rms delay spread varying between 8 ns to 102 ns,

as shown in Figure 4.16. In the case presented in [33], the transmission is at

L-Band (1540-1560 MHz) and S-Band (2315-2335 MHz), and it was radiated

from an helicopter acting as an alternative platform, with the aim of simu-

lating a low orbit satellite. The helicopter ew at altitudes of 450 m to 730

m above sea level, and it allowed measurements for elevation angles in the

range 15� � 80�. In Figure 4.17 are presented the measurements of the ratio

between the carrier power and the multipath content of the received signal

(C/M ratio) obtained in [33]. The measurements are in the range between

14.6 dB and 19.7 dB, that is in good agreement with the calculated value of

18.4 dB.

We de�ne the maximum excess delay �max as the excess delay for which

the average received power is attenuated more than 30 dB with respect to the

direct signal. From the power delay pro�le PPS(�) we �nd out a maximum

excess delay �max = 128 ns. In [19] plots of power delay pro�les obtained

from a channel simulator for land mobile satellite systems are shown. In this

simulator the mobile's surrounding are characterized by topographical and

land use data, as well as by random obstacles generated by algorithms based

on the Landscape Generation Model (LGM). Using ray-tracing techniques,

the simulator provided the power delay pro�les of Figure 4.18. It can be

seen that a two peaks power delay pro�le can be experienced, thus causing

a maximum excess delay in the order of 500 ns. But when a single peak is

present, the maximum excess delay is between 150 ns and 200 ns, in good

agreement with the value obtained from PPS(�).

67


0

10

20

30

40

50

60

70

80

90

100

110

30 35 40 45 50 55 60 65 70 75 80

Delay Spread

σ τ [ns]

Elevation Angle α [Deg.]

Delay Spread vs. Elevation Angle

L-BandS-Band

Figure 4.16. Results of Delay Spread at L-Band (continuous line) andS-Band (dotted line) presented by Parks in [33]

Value obtained Value obtainedby PPS(�) by measurements

Mean excess delay (�) 12 nsrms delay spread (�� ) 21 ns 8-102 ns [33]C/M ratio 18.4 dB 14.6 - 19.7 dB [33]�max at -30 dB 128 ns � 200 ns [19]

Table 4.1. A comparison between parameters evaluated from PPS(�), andvalues obtained by measurements for LMS system

4.3 Comparison between the results obtained

with Rappaport's and Burr's models

Starting from the Burr's model we have �nd out an expression for the power

delay pro�le that characterizes the transmission channel of a stratospheric

platform. The power delay pro�le is expressed as the product of the average

power of an echo having delay � , Ps(�), and the average number of echo

68


14

15

16

17

18

19

20

30 35 40 45 50 55 60 65 70 75 80

C/M ratio [dB]

Elevation Angle α [Deg.]

C/M Ratio vs. Elevation Angle

L-BandS-Band

Figure 4.17. Results of C/M ratio at L-Band (continuous line) and S-Band(dotted line) presented by Parks in [33]

having delay � , n(�), where n(�) is found assuming a Poisson distribution

for the number of echoes arrivals.

On the other side, using Rappaport's geometrical model an expression for

the distribution of the excess delay can be found. In this section it will be

shown that starting from this distribution it is possible to �nd out the average

number of echo arrivals n(�) in a form equivalent to the Burr's model.

Let's assume that N echoes are received with an excess delay in the

range f0;�mg, each echo being independent and having the distribution for

the excess delay given in (4.16). The probability that an echo has an excess

delay in an interval �T around � can be approximated to:

p(�) = fPS(�)�T (4.41)

69


Figure 4.18. Plots of the power delay pro�le obtained with the channelsimulator developed in [19].

where fPS(�) is the pdf of the echo delay given in (4.17). Thus, the probabil-

ity of having n(�) echoes in the interval �T can be expressed as a binomial:

Pfn(�)g =

N

n

!pn(�)(1� p(�))N�n (4.42)

whose mean value is

n(�) = Efn(�)g = Np(�) =N

V (�m)�T

dV

d�(4.43)

Comparing Equation (4.43) with the average number of echoes obtained

by Burr's model (4.25) we note that they both depend on the di�erential of

the volume V (�), and they di�ers only for a constant multiplicative term.

In the Burr's model, this term is the density of the scatterers �, while in

the Rappaport's model is N=V (�m), that is strongly correlated with �, being

the ratio between the total number of signi�cant echoes received, N , and the

volume from which they come, V (�m).

We can conclude that both models lead to the same average power delay

pro�le, the only di�erence being a multiplicative constant that in both cases

represents the density of scatterers.

70


4.4 Doppler Spectrum for a Platform Based

System

The Doppler spectrum for a platform based system was already evaluated

by M.Pent et al. in [35]. In the following the developed channel model is

summarized.

The relative position of the platform and the mobile is de�ned through

the use of the azimuthal angles �p, �m (that identify the platform and the

mobile movement directions with respect to their line of conjunction on the

horizontal plane), '0 (the azimuth angle under which the mobile see the plat-

form), and the distance d0 between the mobile and the platform projection

on the ground, as shown in Figure 4.19.

mφ

d 0

0

p

h

x-z plane

p

m

d 0

θ θ

x-y plane

mp

Figure 4.19. Relative positions and velocity vectors of the platform p andthe mobile m

With the help of this de�nitions three di�erent zones in the area below

the platform can be identi�ed:

� the inside-orbit zone (IOZ): 0 < d0 < 10 km, '0 2 [60�;90�]

� the out-orbit zone (OOZ): 10 < d0 < 30 km, '0 2 [30�;60�]

� the far-orbit zone (FOZ): 30 < d0 < 100 km, '0 2 [10�;30�]

The following three di�erent positions of the mobile are considered:

� '0 = 75� in the IO zone

� '0 = 45� in the OO zone

71


IOZ �180� < � < +180�

60� < � < 90�

OOZ �45� < � < +45�

30� < � < 60�

FOZ �30� < � < +30�

10� < � < 30�

Table 4.2. DOA angle distribution for the signals arriving to the platformbase station

� '0 = 15� in the FO zone

In this conditions the mean signi�cant scatterers are (see Table 4.2):

� uniformly distributed within the IO zone. The platform catches echoes

from azimuthal angles � uniformly distributed in [�180�; + 180�] and

elevation angles � 2 [60�;90�]:

� uniformly distributed inside an annular sector around the mobile in

the OO zone. The platform receives signal components from angles

� 2 [�45�;+ 45�] and � 2 [30�;60�];

� uniformly distributed inside an annular sector aroun the mobile in the

FO zone. The signals impinging the platform antenna come from uni-

formly distributed angles � 2 [�30�;30�] and � 2 [10�;30�].

The study of a system with both moving antennas can be found in [6],

where it is shown that the global Doppler spectrum S(f) can be obtained as

the convolution of the two Doppler spectra obtained assuming that only the

�rst (the second) is moving, while the second (the �rst) is still:

S(f) =

Z +1

�1

Sm(�)Sp(f � �) d� (4.44)

For the previous hypothesis, the Doppler spectrum Sm(f) in the case that

only the receiver is moving is the well known Jakes' model Doppler spectrum

given in (3.22), where the maximum Doppler shift is fd;m = vmf0=c, being

vm the mobile's speed.

The derivation of the Doppler spectrum Sp(f) is more complicated be-

cause both the elevation angle and the radiation pattern of the antenna on

72


board have to be considered. A study on the e�ect of a moving antenna in a

three dimension space can be found in [24]. The essential result is that the

Doppler power spectrum can be written as

Sp(f) = kp

IG()P�() �

�f � f0 � f0

cvp � ar()

�d (4.45)

where:

� is the solid angle of the signal impinging the platform antenna

d = cos(�)d�d� (4.46)

� G() = G(�;�), is the power gain of the platform antenna;

� vp is the platform speed vector, constant for short time intervals

vp � jvpj (4.47)

� ar() is the unit propagation vector for a wave impinging with angle

,

vp � ar() = vp cos(� � �p) cos� (4.48)

� kp is a multiplicative constant function of the average received power;

� P () is the probability that a ray is received at the platform, coming

from direction ar(). On the basis of previous assumptions

P (�;�) =

(P0 �1 < � < �2;�1 < � < �2

0 elsewhere(4.49)

where the intervals (�1;�2) and (�1;�2) are the limit angles de�ned in

Table 4.2. Moreover the following Equation must hold:IP () d = 1 (4.50)

so that

P0 =1

(�2 � �1)(sin�2 � sin�1)(4.51)

73


Being the direction of arrival (DOA) of echoes bounded in the limit of

Table 4.2, and because of the non-uniform antenna radiation pattern, the

value of �p in uences the Doppler spectrum. The support of the Doppler

power spectrum Sp(f) is [fm;fM ], where

fm =vpf0c

min�fcos(�p � �)g

fM =vpf0c

max�fcos(�p � �)g (4.52)

as can be found from Equations (4.45) and (4.48). These values are ap-

proximate, since they do not re ect the presence of a non-omnidirectional

antenna.

�p Zone of the fm=fd;p fM=fd;preceiver

0� -1 145� IOZ -1 190� -1 10� 0.703 145� OOZ 0 190� -0.703 0.7030� 0.866 145� FOZ 0.260 0.96690� -0.5 0.5

Table 4.3. Normalized support of Sp(f), as a function of �p and the zoneof interest; fd;p = vpf0=c is the maximum Doppler shift.

In Table 4.3 are reported values of fm and fM normalized to the maximum

Doppler shift due to the platform movement fd;p = vpf0=c. The values are

presented for the three zone previously introduced (IOZ, OOZ, FOZ), and for

di�erent values of �p, corresponding to the platform approaching the receiver

(�p = 0), in an oblique movement with respect to the receiver (�p = 45�),

and moving perpendicular to the receiver (�p = 90�).

From Equation (4.44) it follows that the support of the global Doppler

power spectrum is the sum of the support of Sm(f) and of Sp(f). This leads

to a global support of [fm � fd;m;fM + fd;m].

74


Even if theoretically from Equations (4.44), (4.45) and (3.22) it is possible

to evaluate the correct Doppler spectrum for the platform system, it was

already stated that the trend of the spectrum is not relevant in evaluating

the performance of the channel. For this reason, it is simpler to model the

channel with a Jakes' Doppler Spectrum having a bandwidth equal to the

support of the correct platform Doppler spectrum.

4.5 Coherence Bandwidth and Coherence Time

of the Platform Channel

Once that the Power delay pro�le and the Doppler Spectrum of a platform

based system are obtained, it is possible to evaluate the coherence bandwidth

and the coherence time of the channel. As it is shown in Chapter 3, those

parameters allow to predict if the fading that will a�ect the transmission

over the channel will be respectively selective or at and fast or slow.

From the power delay pro�le an rms delay spread �� = 21 ns has been

obtained. Using Equation (3.13) it yields to a coherence bandwidth Bc equal

to:

Bc =1

50��= 952:38 KHz (4.53)

Assuming the speed of the platform vp = 150 km/h and the speed of the

receiver vm = 50 km/h, the maximum Doppler Spread fm (see Table 4.3) is:

fm = fd;p + fd;m = [vp + vm]f0c= 222:16 Hz (4.54)

where f0 = 1:2 GHz is the carrier frequency and c = 3108 m/s is light speed.

Using Equation (3.26) this value of fm leads to a coherence time Tc of:

Tc =9

16�fm= 0:806 ms (4.55)

In Figure 4.20 the condition of fading to which undergoes a transmitted

signal having a bandwidth Bs and a symbol duration Ts obtained with (4.53)

and (4.55) are shown. Note that the platform channel, having a low delay

spread and thus a negligible dispersion in time, is selective only in the case

75


Slow Fading

Slow Fading

Flat Fading

Selective Fading Selective Fading

Flat Fading

Fast Fading

Fast Fading

Ts (ms)

Bs (KHz)

Tc = 0.806

952=cB

Figure 4.20. A diagram representing the channel fading properties forvarious symbol duration Ts and signal bandwidth Bs.

of wide band signals. On the other hand, the dispersion in frequency is more

severe than in usual transmission channel, because both the movement of the

platform and of the mobile contribute to the Doppler spread.

76

Chapter 5

The Simulation Program

In Chapter 4 we have found an expression for the power delay pro�le in

a platform based system and evaluated the Doppler spread introduced by

the movements of the platform and of the receiver. Using the Power delay

pro�le and the Doppler spectrum that characterize the mobile radio chan-

nel, it is possible to develop simulation programs that allow to evaluate the

performance of di�erent transmission systems operating over the channel.

In this Chapter it is described the simulation model that we have de-

veloped using the theoretical functions previously evaluated. The model

has been realized as a BLOCK for TOPSIM. TOPSIM [49] is a simulation

tool for the analysis and design of communication systems developed by the

Telecommunication Group of Electronic Department of Politecnico di Torino

with the support of the European Space Agency (ESA/ESTEC). The usual

description of a communication system is based on block diagrams, and in

TOPSIM the system under study can be modeled as a set of blocks con-

nected by input and output signals. TOPSIM has been written in standard

FORTRAN 77 and C language. The TOPSIM library, composed by more

than 400 models called BLOCKs, provides the user with most of the models

needed to design communication systems, however a block to simulate the

stratospheric channel was not available yet.

In the �rst part of the Chapter we will present di�erent simulation models

that are used to simulate di�erent type of mobile radio channels. The second

77

5 { The Simulation Program

part will be focused on the model used for the development of the TOPSIM

simulation block. Finally, we will report some simulation results that describe

the behaviour of the block.

5.1 Descriptions of Simulation Models for Dif-

ferent Mobile Radio Channels

The simplest model for a mobile radio channel is the one used when the

transmitted signal is a�ected by at and slow fading. In this case the band-

width of the received signal is not su�ciently wide to resolve the multipath

component and, as a consequence, the dispersion in time can be neglected.

Moreover the channel varies slowly if compared to the transmission dura-

tion and can be considered constant for the whole transmission. If there is

no direct path between the transmitter and the receiver, the received signal

will be composed of di�erent replicas of the transmitted signal, each replica

having a di�erent random attenuation level, phase shift and Doppler shift.

As shown in Chapter 3, in this condition the received signal envelope is de-

scribed by a random Rayleigh distributed variable. The simulation model

consists simply of a multiplier for a constant r that is randomly generated

following a Rayleigh distribution, as shown in Figure 5.1. The variance �2rof the random variable is given by the total received power. White gaussian

noise ~n(t) can then be added through an adder.

r

x(t)~

n(t)~

y(t)~

Figure 5.1. Diagram of the simulation model used for a slow and atfading channel, in the absence of the direct signal. ~x(t) is the complexenvelope of the input signal, r is a Rayleigh distributed random variable

and ~n(t) represents white gaussian noise.

If a direct path can be found between the transmitter and the receiver, so

that line of sight transmission is possible, the received signal will be the sum

78


of the Rayleigh attenuated signal (generated by the replicas of the direct

signal) and the direct signal itself. The simulation model is depicted in

Figure 5.2.

y(t)~x(t)~

n(t)~k

r

Figure 5.2. Diagram of the simulation model used for a slow and atfading channel, in the presence of the direct signal. ~x(t) is the complexenvelope of the input signal, r is a Rayleigh distributed random variable, kis a multiplicative constant that allows to vary the ratio between the line

of sight and the di�used signal, ~n(t) is white gaussian noise.

The direct ray is split in two branches. The �rst one simply leads to a

gain block, controlled by a parameter k, and it is used to simulate the direct

component of the received signal. The second branch lead to a multiplier

for a Rayleigh distributed variable r. Usually, the variance of the Rayleigh

variable r is set to 1, so that the parameter k in the direct ray branch allow

to regulate the importance of the direct ray with respect to the multipath

component. These two signals are added and generate the received signal

that follows a Ricean distribution as shown in Chapter 3. White gaussian

noise can then be added to the received signal.

Let us suppose that the channel is no longer slow fading, that is, the

channel varies during the transmission. However, suppose that the e�ect

of dispersion in time can still be neglected. The channel is subject to at

but fast fading. If no direct path exists the simulation model is the one of

Figure 5.3.

The time variant nature of the model is simulated by the fading process

79


x(t)

r(t)~

y(t)~~

n(t)~

Figure 5.3. Diagram of the simulation model used for a fast and at fadingchannel, in the absence of the direct signal. ~x(t) is the complex envelopeof the input signal, ~r(t) is the power fading process, ~n(t) represents noise

~r(t), whose power spectrum is given by the Doppler spectrum of the channel.

As already stated (see Chapter 3), the correct shape of the Doppler spectrum

is not relevant, being the attributes of Level Crossing Rate and Average Fade

Duration of the channel dependent on the support width of the Doppler

Spectrum only. As a consequence, the spectrum of the process ~r(t) is usually

assumed to be the Jakes' Doppler Spectrum given in Equation (3.22).

If a direct ray can be received, the correct model for the channel is shown

in Figure 5.4. Once again the transmitted signal is split into two branches.

y(t)~x(t)~

n(t)~

r(t)~

k

Figure 5.4. Diagram of the simulation model used for a fast and at fadingchannel, in the presence of the direct signal. ~x(t) is the complex envelopeof the input signal, k is a multiplicative constant that allows to vary theratio between the line of sight and di�used signal, ~r(t) is the unitary power

fading process, ~n(t) represents noise

The �rst refers to the direct component of the received signal, and allows

to control its power through a gain block of parameter k, while the second

80


produces the multipath component through a multiplier for the process ~r(t).

As for the case with no direct ray, ~r(t) has a power spectrum given by the

Doppler spectrum of the channel and its power is usually unitary. By means

of k it is possible to control the power level of the direct ray with respect to

the multipath component.

The simulation models previously presented do not take into account the

time dispersive nature of a multipath channel. As shown in Chapter 3, when

the bandwidth of the signal is greater than the coherence bandwidth of the

channel, time dispersion cannot be neglected and the channel is selective in

frequency.

In order to simulate a channel that is both dispersive in frequency (i.e.

subject to fast fading) and in time (i.e. subject to selective fading), two

di�erent approaches can be followed.

The �rst method try to reproduce the physics of the channel, and holds

on the mathematical model of a multipath channel presented by Bello [10]

and known as the Gaussian Wide Sense Stationary Uncorrelated Scattering

Channel (GWSSUS). The basic formula used for the computation of the

output signal y(t) is:

y(t) = Re�v(t)ej2�f0t

= Re

(NXi=1

�iu(t� �i) exp[j2�t(f0 � fi)� j2�f0�i � j�i]

)(5.1)

where v(t) represents the complex envelope of y(t), u(t) the complex envelope

of the input signal s(t):

s(t) = Re�u(t)ej2�f0t

(5.2)

In the previous relationships f0 is the central frequency of the pass band

signal and N the number of discrete scattering points that should generate

the received signal. Each point is characterized by an amplitude �i, an excess

delay �i, a Doppler shift fi and phase �i.

In order to have a GWSSUS channel, the following assumptions are made:

� �i and �i are statistically independent;

81


� �i is a random variable uniformly distributed in the range 0-2�;

� �i and �j are statistically independent;

� �i is a random variable Rayleigh distributed;

By means of Equation (5.1) it is possible to compute the system function,

obtained as the response of the channel to an input complex envelope

u(t) = ej2�ft (5.3)

corresponding to the real function cos[2�(f0 + f)t]. By omitting the factor

ej2�f0t we obtain the system function for the complex envelopes in the form:

T (t;f) =NXi=1

�ie�j2�[f�i+fit]e�j2�f0�i+�i (5.4)

By means of the independence between �i and �j, we obtain:

E[T (t;f)T �(t0;f 0)] =NXi=1

E[�i��

i ]e�j2�[(f�f 0)�i+fi(t�t0)] (5.5)

where E[�] stands for the ensamble average operator.Equation (5.5) can be written as:

E[T (t;f)T �(t� �;f + )] =DXi=1

FiXj=1

e�j2��i+j2��fjE[j�ijj2] (5.6)

where � = t � t0, = f 0 � f and the scattering points are combined in D

groups each having the same delay �i and Fi di�erent values of the Doppler

shift.

Considering a continuous distribution of delays and Doppler shifts, Equa-

tion (5.6) can be transformed in the following way:

E[T (t;f)T �(t� �;f + )] =

Z ZDF

�(fd;�r)e�j2��rej2��fd d�r dfd (5.7)

where �r is the continuous variable representing the delays, fd is the con-

tinuous variable representing the Doppler shifts, and DF is the domain in

the (fd;�r) plane where the function �(fd;�r) is not vanishing. Note that

82


E[T (t;f)T �(t� �;f + )] = R� (;�) represents the time frequency correla-

tion function of the channel, and it is related to the function �(fd;�r) by a

two dimensional Fourier transform. The function �(fd;�r) can thus be inter-

preted as the power density of the scattering points, and it is strictly related

to the power delay pro�le and the Doppler Spectrum of the channel. In fact

it can be shown that:

P (�r) =

Z�(fd;�r) dfd (5.8)

is the power delay pro�le of the channel, while

S(fd) =

Z�(fd;�r) d�r (5.9)

is the Doppler spectrum of the channel. As a result �(fd;�r)dfdd�r represents

the incremental value of average power due to echoes having Doppler shift

between fd and fd + dfd and delay between �r and �r + d�r.

As proposed by Falciasecca, Frullone and Riva in [22], the simulation of

the channel can be done by implementing Equation (5.1). When generating

the random variable for Equation (5.1), it is possible to consider the �(fd;�r)

function, suitable normalized, as the probability density function for a dis-

tribution of echoes having equal amplitude. In such a way �(fd;�r) dfd d�r

represents the probability of an echo occurring with Doppler shift between

fd and fd+ dfd and delay between �r + d�r. With this assumption the values

of fd and �r carrying out a great amount of power will be represented with

a major number of echoes. The generation of the random variable fi and �i

for Equation (5.1) can be done with the following assumption:

�(fd;�r) = P (�r)S(fd) (5.10)

P (�r) and S(fd) can be separately normalized, and P (�r) can be taken as the

distribution of the �i, while S(fd) can be taken as the distribution of fd.

In order to approximate in a suitable way the continuous function �(fd;�r),

it is necessary to use a great number of pointsN to implement Equation (5.1).

As a consequence, even if the model correctly reproduce the behaviour of a

mobile radio channel, its computational cost is too large.

83


A second approach to reproduce a fast and selective fading channel is the

simulation model presented by Rappaport in [38].

The transmitted signal is split into several branches, each branch repre-

senting a cluster of echoes arriving at the receiver in an interval around an

excess delay �i. Each branch carries an amount of power given by the power

delay pro�le. If the echoes in the cluster are separated by a delay that is

much lower than the duration of a transmitted symbol, each branch will un-

dergo to a at (but potentially fast) fading. Dispersion in time is therefore

introduced only by the presence of several branches. The simulation model

is shown in Figure 5.5.

y(t)~

n(t)~

τ0

τ i

0

x(t)~

τ N

ri(t)~

r (t)~

r~N (t)

iP

0P

PN

Figure 5.5. Diagram of the simulation model used for a fast and at fadingchannel, in the presence of the direct signal. ~x(t) is the complex envelopeof the input signal, �i and Pi are the excess delays and power attenuationsgiven by the power delay pro�le, ~ri(t) are unitary power fading processes

and ~n(t) represents noise.

A �nite number N of replicas of the input signal ~x(t) is produced, each

84


replica is delayed of �i by a delay block, then the fast fading e�ect is intro-

duced by multiplying for a unitary power fading process ~ri(t). Finally, the

signal is attenuated ofpPi accordingly to the power delay pro�le. The repli-

cas are added to the direct component through an adder. White gaussian

noise ~n(t) can be added to the received signal through another adder.

This model requires to discretize the power delay pro�le, so that a dis-

crete number N of delays �i and gain factors Pi are produced. As already

mentioned in Chapter 3, a discrete version of the power delay pro�le can

be used as long as the bandwidth of the transmitted signal is not greater

than 1=(2��), being �� the time step between �i and �i+1. Values of Pi,

as suggested in the recommendation CEPT/GSM 05.05, should be chosen in

order to maintain the time dispersive properties of the channel, in particular

the mean excess delay � introduced and the rms delay spread �� .

5.2 Description of the TOPSIM Simulation

Block MUPATP

In Chapter 4 we have shown that the coherence bandwidth of the platform

channel is Bc = 952 KHz, so that in most transmission systems the channel is

expected to be at. In this case, a simulation model that takes into account

at fading is enough for the platform channel. In particular, as explained

in Chapter 2, line-of-sight transmission can be usually achieved from the

platform, so that reliable simulation results can be obtained by using the

model for fast and at fading with direct ray of Figure 5.4. However, in order

to simulate the platform radio channel, we have used the model presented

in Figure 5.5. As already stated this model is used to simulate channels

that undergo to fast and selective fading. When dealing with stratospheric

platform transmissions, this model is usually more accurate than what is

really necessary, however it allows to fully comprehend the behaviour of the

channel and to obtain a simulation program that is valid for every kind of

transmission.

The routine code is presented in appendix A. In a TOPSIM simulation

program this block is called as follows:

85


Y < MUPATP(F0,D0,CVEL,MVEL,PVEL,G0,NTAP) < X

where X is the input analytic signal, while Y is the output analytic sig-

nal produced by the block. F0 is the frequency of the carrier, D0 is the

distance between the receiver and the projection over the ground of the plat-

form, CVEL, MVEL, PVEL are respectively light speed, the mobile receiver

speed, and the platform speed. G0 is the ratio in dB between the power car-

ried by the signal and the power carried by multipath waves, and NTAP is

the number of tap used to simulate the power delay pro�le of the channel (i.e,

equals N in the diagram of Figure 5.5). Note that D0 must be expressed in

kilometers, while CVEL, MVEL and PVEL must be expressed as kilometers

over a time dimension that is taken accordingly to the normalization used

to express simulation time (that is, if simulation time is given in microsec-

onds, speed parameters must be expressed as km/us, if simulation time is in

milliseconds, speeds parameters must be in km/ms, and so on).

In the �rst part of the program the arrays that will be used are dimen-

sioned: ATAU and APOT are vectors of dimension NTAP that will contain

the values of delay and power attenuation used to discretize the power delay

pro�le (i.e. they contain respectively the values of �i and Pi of Figure 5.5),

XRITIN, XRIT, X1 are used in order to produce the delayed versions of the

input signal, Z1 is the unitary power fading process ~ri(t), and W1 is the

delayed echo that must be added to the input signal to obtain the output.

The array dimensioning is followed by the allocation of the HISTORY.

The HISTORY is a memory area used in TOPSIM to contain the value of

those variables that are characteristic of one instance of the block, so that,

at every simulation step, it is possible to keep memory of the value taken

in the previous step by one speci�c variable in one speci�c call of the block.

The HISTORY is composed of two arrays, the �rst, HIST, is used for real

variables, while the second, IST, is used for integer variables; the function

LOCH and LOCI allocate respectively a portion of the array HIST and IST,

and return the index that must be used in order to access the memory area

thus allocated.

In the MUPATP block the HISTORY is organized in the following way:

� HIST(L)...HIST(L+NTAP-1), array of amplitude attenuations APOT1=2;

86


� IST(LTIN)...IST(LTIN+NTAP-1), array of delays expressed as number

of samples;

� HIST(LTPAR)...HIST(LTPAR+NTAP-1), array of the remaining part

of the delays;

� HIST(LXR1)...HIST(LXR1+NTAP-1), previous value of the in-phase

part of the input signal X delayed of HIST(LTIN)..(LTIN+NTAP-1);

� HIST(LXR2)...HIST(LXR2+NTAP-1), previous value of the in-quadrature

part of the input signal X delayed of HIST(LTIN)..(LTIN+NTAP-1);

� HIST(LF), doppler spread FD;

� HIST(LAND), normalization coe�cient for the direct ray;

� HIST(LANM), normalization coe�cient for the multipath waves;

� IST(LNSIN), parameter for FADJAK

After the HISTORY is allocated, the block verify if the simulation is

already started, i.e. if the simulation time is greater than 0; if the simulation

is not started, then a procedure of initialization is executed: the block calls

the Fortran Subroutine POWDEL (see appendix B) with the statement

CALL POWDEL(F0,CVEL,DO,APOT,ATAU,NTAP)

that is used to calculate the power delay pro�le that characterizes the geom-

etry de�ned by D0, and returns in APOT and ATAU the NTAP values of Pi

and �i used to discretize the power delay pro�le. The arrays Pi and �i are

computed by POWDEL on the basis of Equations (4.26), (4.33) and (4.25).

POWDEL evaluates the power delay pro�le in one thousand instant of time

� , and then discretize it, by subdividing the time axis in NTAP intervals

and by calculating the average power of each intervals.

After the call to the subroutine POWDEL, a loop is executed. The

loop memorizes in the HISTORY the amplitude attenuations APOT1=2, and

expresses the excess delays as number of simulation time steps, given by

b�i=DELT c (where DELT is the simulation time step), plus the remaining

part of excess delay, that is a fraction of DELT, given by �i � b�i=DELT c.

87


The excess delay must be divided in this way, because TOPSIM with the

block DELAY allows to delay a signal only by a quantity that is a multiple

of the simulation time step DELT. In order to introduce the correct excess

delay �i, the block DELAY will be used �rst, in order to delay the signal of

b�i=DELT c time steps, and then the resulting signal will be interpolated in

order to introduce the remaining delay fraction of DELT.

Eventually, the initialization procedure evaluates the maximum Doppler

spread using Equation (4.54) and stores it in the HISTORY, and then eval-

uates and stores the normalization coe�cients for the direct and re ected

rays.

The normalization factors for the direct ray and the re ected waves are

evaluated by calculating the total power carried by the multipath waves. Sup-

posing that the echoes are not correlated, this is given by PROV=PNTAP

i=1 Pi.

In order to have the ratio between the power carried by the direct signal

and the power carried by the multipath waves equal to G0, the multipath

power is multiplied by GMULT = 10�G0=10=PROV , and, in order to have

an output signal with unitary power, both the direct ray and the multipath

waves are attenuated by 1 + 10�G0=10. In this way the the normalization

coe�cient for the direct ray is

HIST (LAND) =

r1

1 + 10�G0=10

while the normalization coe�cient for the multipath waves is

HIST (LANM) =

rGMULT

1 + 10�G0=10

The initialization procedure is followed by the core of the block code; �rst

of all, the LOS component in the output signal Y is calculated by multiply-

ing the input signal for the normalization factor ANORD = HIST (LAND).

Then, the multipath components are introduced with a loop that calculates

each NTAP taps representing the multipath contribute. In particular, in the

loop the input signal X is delayed of a multiple of the simulation time step

DELT with the block DELAY, thus obtaining XRITIN, and then XRITIN is

delayed of the remaining part of the delay �i, that is a fraction of the simu-

lation time step. Note that the last part of the delay has been introduced by

88


delaying the in-phase and in-quadrature parts of XRITIN and then by intro-

ducing a phase shift. In fact, starting from the in-phase and in-quadrature

representation of a signal

x(t) = xp(t) cos(2�f0t)� xq(t) sin(2�f0t) (5.11)

the delayed signal x(t� �i) can be expressed as:

x(t� �i) = xp(t� �i) cos[2�f0(t� �i)]� xq(t� �i) sin[2�f0(t� �i)] (5.12)

that leads to

x(t� �i) = fxp(t� �i) cos(2�f0�i) + xq(t� �i) sin(2�f0�i)g cos(2�f0t)�fxq(t� �i) cos(2�f0�i)� xp(t� �i) sin(2�f0�i)g sin(2�f0t)

(5.13)

DELT

x(t)

x(t)

t

x(t n )

x d

tn

iτ

t -n τ itn-1

x(t n-1 )

x(t - τ i)n

Figure 5.6. Method of interpolation used to introduce a delay that is afraction of the simulation time step DELT.

The in-phase and in-quadrature delayed part of XRITIN are obtained by

means of an interpolation. In fact, as shown in Figure 5.6, if the values of a

89


function x(t) are known in tn = n DELT and in tn�1 = (n� 1)DELT , it is

possible to approximate the delayed function x(tn � �i) with

x(tn � �i) ' xd = x(tn�1) +x(tn)� x(tn�1)

DELT(DELT� �i) (5.14)

that can be written as

x(tn � �i) '�1� �i

DELT

�x(tn) +

�iDELT

x(tn�1) (5.15)

The global delayed signal X1 is then multiplied for the unitary power

fading process Z1 generated by the FADJAK block, and afterwards it is

normalized by multiplying for the factor ANORM. Finally, this echo is added

to the output signal with the block BPSUM.

5.3 MUPATP Block Validation

In order to fully understand and verify the behaviour of the block, in the

following the results of some simulations executed with the block MUPATP

and with a modi�ed version of this block, called MUPNOF are presented.

MUPNOF omits the generation of the unitary power fading process Z1, that

is used to introduce phenomena related with dispersion in frequency. As a

consequence, the block MUPNOF can be used to highlight solely the e�ects

of dispersion in time in the behaviour of the stratospheric channel.

Let us take into account a discretized power delay pro�le having six

echoes. The value of Pi and �i generated in this case by the subroutine

POWDEL are reported in Table 5.1, and are presented in Figure 5.7.

The theoretical impulse response of the channel should consists of six

echoes, having amplitude given bypPi, delayed and shifted in phase accord-

ingly to �i. The in-phase and in-quadrature parts of the theoretical baseband

impulse response are reported in Figure 5.8 and 5.9.

Figures from 5.10 to 5.13 shown the in-phase part of the baseband im-

pulse responses obtained with the MUPNOF block for di�erent values of the

simulation time step DELT. In particular, DELT is taken equal to 250 ns,

62.5 ns, 15.625 ns and 3.90625 ns, that correspond to the time steps used in

the case of a transmission with a modulation that carries 2 bit per symbol,

90


Delay �i [us] Pi=PLOS Pi=PLOS [dB] Ai=ALOS

0.012905 0.06949 -11.5 0.263620.035654 0.00554 -22.5 0.074420.058403 0.00280 -25.5 0.052930.081152 0.00186 -27.3 0.043080.103901 0.00138 -28.5 0.037170.126650 0.00110 -29.5 0.03314

Table 5.1. Values of Pi and �i for the discretized power delay pro�le with6 taps, produced by the POWDEL block.

-30

-28

-26

-24

-22

-20

-18

-16

-14

-12

-10

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Pi/PLOS [dB]


Discretized Power Delay Profile of the Stratospheric Channel

Figure 5.7. Discretized Power Delay Pro�le with 6 echoes, normalized tothe line of sight (LOS) received power.

with 32 samples for each symbol, and with a bit rate of 0.25, 1, 4 and 16

Mb/s.

Note that TOPSIM approximates an impulse with the triangular shape

of Figure 5.14. When the impulse is delayed by a quantity �i that is not

an integer multiple of DELT, the interpolation used in order to produce the

91


-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

In-phase Part


Theoretical Impulse Response of the Stratospheric Channel

Figure 5.8. In phase part of the theoretical baseband impulse response ofthe stratospheric channel.

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

In-quadrature Part


Theoretical Impulse Response of the Stratospheric Channel

Figure 5.9. In quadrature part of the theoretical baseband impulse re-sponse of the stratospheric channel.

92


-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16

In-phase Part


Impulse Response generated by TOPSIM

Rb= 0.25 Mb/s

Figure 5.10. In phase part of the simulated baseband impulse response ofthe stratospheric channel, for a simulation time step DELT=250 ns.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1 1.5 2 2.5 3 3.5 4

In-phase Part



Rb= 1 Mb/s

Figure 5.11. In phase part of the simulated baseband impulse response ofthe stratospheric channel, for a simulation time step DELT=62.5 ns.

93


-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

In-phase Part



Rb= 4 Mb/s


-0.2

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25

In-phase Part



Rb= 16 Mb/s


94


delay yields to the trapezoidal shape of Figure 5.15, whose time support is

3 DELT. The simulated impulse response of the channel is thus dependent

on the simulation time step. In particular, if 3 DELT is greater than the

time interval �� = �i� �i�1 between two consecutive echoes, the trapezoidal

impulses overlap, as in Figures 5.10, 5.11 and 5.12, while if 3 DELT is not

greater than �� the echoes can be resolved and each echoes is visible in

the impulse response (see Figure 5.13). As a consequence, the e�ects of

dispersion in time are not negligible only in the case of elevated bit rates.

DELT

TIME

0

1

Figure 5.14. Shape of an impulse simulated by TOPSIM.

TIME(n-1)DELT n DELT

τ i

Figure 5.15. Shape of a delayed impulse generated by MUPATP or MUP-NOF.

95


The scattering diagrams obtained by simulating a system with a 4-DPSK

modulation scheme and a bit rate of 0.5, 2, 6 and 10 Mb/s are reported in

Figures from 5.16 to 5.19. In all this cases the channel is simulated by the

MUPNOF block.

Note that for low values of the bit rate, the e�ects of time dispersion

are largely negligible and the points of the 4-DPSK constellation are well

marked. However, when the bit rate increases, i.e the duration of a trans-

mitted symbol decreases, the e�ect of dispersion in time can be observed,

and intersymbolic interference (ISI) arises. In Figure 5.20 the area around a

point of the constellation of Figure 5.19 is zoomed. A replica of the 4-DPSK

constellation is clearly overlapped to the transmitted constellation, as it is

expected when dealing with ISI phenomena.

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

-1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25

QU

AD

RA

TU

RE

-

RB

=5.

0000

00E

-01

IN-PHASE

SCATTERING DIAGRAM OF XFRX - SAMPLED AT 7

Figure 5.16. Scattering diagram for a 4DPSK transmission at a bit rate of0.5 Mb/s. The transmission channel is simulated by the MUPNOF block.

Note that when the direct component is attenuated, the e�ects of ISI

become more evident. In Figure from 5.21 to 5.24 the scattering diagrams

in the case that the Carrier to Multipath (C/M) ratio is reduced to 6 dB,

instead of the 18 dB predicted for the system in Chapter 4 (see Table 4.1),

are reported.

96


-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

-1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25

QU

AD

RA

TU

RE

-

RB

=2.

0000

00E

+00

IN-PHASE


Figure 5.17. Scattering diagram for a 4DPSK transmission at a bit rateof 2 Mb/s. The transmission channel is simulated by the MUPNOF block.

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

-1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25

QU

AD

RA

TU

RE

-

RB

=6.

0000

00E

+00

IN-PHASE


Figure 5.18. Scattering diagram for a 4DPSK transmission at a bit rateof 6 Mb/s. The transmission channel is simulated by the MUPNOF block.

97


-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

-1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25

QU

AD

RA

TU

RE

-

RB

=1.

0000

00E

+01

IN-PHASE


Figure 5.19. Scattering diagram for a 4DPSK transmission at a bit rate of10 Mb/s. The transmission channel is simulated by the MUPNOF block.

Figures 5.25 and 5.27 show the curve of BER (Bit Error Rate) for a 4-

DPSK modulation scheme operating over the MUPNOF simulated channel,

for a C/M ratio of respectively 18 and 6 dB.

Note that for a C/M ratio of 18 dB the e�ect of dispersion in time is

highly negligible (see Figure 5.26), while it becomes visible for a C/M ratio

of 6 dB, even if only for high bit rates. However, it should be noted that in

these plots the phase of interfering echoes is �xed and depend on the delays

�i. In a realistic multipath channel, the phase of the echoes vary widely in

time and the phenomena of destructive interference may arise, thus causing

a great worsening in the performance of the system. In this respect, the

curves of Figures 5.25 and 5.27 represents the e�ect of one of the possible

interfering echoes con�guration, and cannot be interpreted as a measure of

the worsening caused by multipath. In order to have a realistic estimation

of the multipath e�ect, the time varying nature of the multipath channel by

using the MUPATP block must be reproduced.

98


0.675

0.7

0.725

0.75

0.775

0.8

0.825

0.85

0.875

-0.15 -0.1 -0.05 0 0.05

QU

AD

RA

TU

RE

-

RB

=1.

0000

00E

+01

IN-PHASE


Figure 5.20. Particular of the scattering diagram for a 4DPSK transmis-sion at a bit rate of 10 Mb/s. The transmission channel is simulated by

the MUPNOF block.

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

QU

AD

RA

TU

RE

-

RB

=5.

0000

00E

-01

IN-PHASE


Figure 5.21. Scattering diagram for a 4DPSK transmission at a bit rate of0.5 Mb/s. The transmission channel is simulated by the MUPNOF block

and the C/M ratio is set to 6 dB.

99


-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

QU

AD

RA

TU

RE

-

RB

=2.

0000

00E

+00

IN-PHASE


Figure 5.22. Scattering diagram for a 4DPSK transmission at a bit rateof 2 Mb/s. The transmission channel is simulated by the MUPNOF block


-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

QU

AD

RA

TU

RE

-

RB

=6.

0000

00E

+00

IN-PHASE




100


-0.6

-0.4

-0.2

0

0.2

0.4

0.6

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

QU

AD

RA

TU

RE

-

RB

=1.

0000

00E

+01

IN-PHASE




-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

8 10 12 14 16 18

Log(BER)

Eb/N0 [dB]

BER for a 4-DPSK Modulation, with C/M=18 dB

AWGN ChannelMupnof, 0.25 Mb/s

Mupnof, 1 Mb/sMupnof, 4 Mb/sMupnof, 8 Mb/s

Figure 5.25. BER for a 4-DPSK transmission scheme with a bit rate from0.25 to 8 Mb/s, using the MUPNOF block with C/M ratio of 18 dB.

101


-9.4

-9.39

-9.38

-9.37

-9.36

-9.35

-9.34

-9.33

-9.32

-9.31

-9.3

15.5 15.51 15.52 15.53 15.54 15.55 15.56

Log(BER)

Eb/N0 [dB]




Figure 5.26. A particular of the curve of BER for a 4-DPSK transmissionscheme with a bit rate from 0.25 to 8 Mb/s, using the MUPNOF block

with C/M ratio of 18 dB.

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

8 10 12 14 16 18

Log(BER)

Eb/N0 [dB]




Figure 5.27. BER for a 4-DPSK transmission scheme with a bit rate from0.25 to 8 Mb/s, using the MUPNOF block with C/M ratio of 6 dB.

102


5.4 Simulation Results Using the MUPATP

Block

In Figures 5.28 and 5.29 the scattering diagram of the received signal ob-

tained using respectively the Ricean model and the MUPATP block, for a

4-DPSK transmission scheme at a bit rate of 0.25 Mb/s are shown. The

time varying process are generated by the block FADJAK that introduces

the e�ects of dispersion in frequency. This process can produce deep fad-

ing, corresponding to the case in which the interfering rays are destructive

and in phase with each others. Both the diagrams show that the points of

the received constellation move in a wide area, thus strongly worsening the

performance of the transmission system.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

QU

AD

RA

TU

RE

IN-PHASE

SCATTERING DIAGRAM OF XFRXR - SAMPLED AT 7

Figure 5.28. Scattering diagram of the received signal, using a Riceanchannel with a C/M ratio of 18 dB.

In Figure 5.30, the BER estimated for the 4-DPSK scheme using the

Ricean model and the MUPATP block for a C/M ratio of 18 dB at a bit

rate of 0.25, 1 and 4 Mb/s is reported and compared with the BER obtained

over a AWGN channel. It is shown that for a bit rate of 0.25 and 1 Mb/s

the curves obtained with the MUPATP block substantially overlap with the

curves obtained with the Ricean model, while for a bit rate of 4 Mb/s the

103


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

QU

AD

RA

TU

RE

IN-PHASE

SCATTERING DIAGRAM OF XFRXM - SAMPLED AT 7

Figure 5.29. Scattering diagram of the received signal, using the MUPATPchannel with a C/M ratio of 18 dB with bit rate 0.25 Mb/s .

MUPATP curve shifts to the right. This value of bit rate corresponds to a

bandwidth of about 2MHz, that is greater than the coherence bandwidth of

the platform Bc = 952:38 KHz, found in Equation 4.53. In this case the

channel becomes selective in frequency and the performance of the system

are worse. However, being the C/M ratio very high, the worsening is quite

negligible. Obviously, the use of the simple white gaussian noisy channel

leads to estimate a better performance than the fading channel.

The e�ects of multipath become more evident when the C/M ratio is set

to 6 dB, as in the case of Figure 5.31. Also in such a case, the curve of BER

obtained using a white gaussian noisy channel are compared with the Ricean

results and with those obtained with the MUPATP block at a bit rate of

0.25, 1 and 4 Mb/s. Being the contribute of multiple echoes higher than in

the case of Figure 5.30, the performance estimated by the MUPATP block

are highly worse than that provided by the simple Ricean channel. Also in

such a case a signi�cant worsening can be seen when the bandwidth of the

transmitted signal exceeds the coherence bandwidth, i.e. for a bit rate of 4

Mb/s. However, the e�ect of the frequency selective channel begins to be

visible even at a bit rate of 1 Mb/s.

104


-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

8 10 12 14 16 18 20

Log(BER)

Eb/N0 [dB]


AWGN ChannelRicean Channel

Mupatp, 0.25 Mb/sMupatp, 1 Mb/sMupatp, 4 Mb/s

Figure 5.30. BER for a 4-DPSK transmission scheme using a white gaus-sian noisy channel, a Ricean channel, and the MUPATP block with a C/M

ratio of 18 dB, at a bit rate of 0.25, 1, 4 Mb/s.

Eb=N0 for a BER=10�4

Gaussian Ricean MUPATP block at:channel channel 0.25 Mb/s 1 Mb/s 4 Mb/s

C/M=18 dB 10.78 dB 11.14 dB 11.09 dB 11.28 dB 11.49 dBC/M=6 dB 10.78 dB 14.62 dB 14.85 dB 15.14 dB 20.21 dB

Table 5.2. Estimated values of the signal to noise ratio necessary toachieve a bit error rate of 10�4, using the simple White gaussian noisychannel, the Ricean channel, and the MUPATP block at a bit rate of 0.25,

1 and 4Mb/s

In Table 5.2 the values of signal to noise ratio (Eb=N0) necessary to achieve

a Bit Error Rate of 10�4 are reported. Note that, in this case, the Ricean

model is not dependent on the bit rate because the coherence time Tc of the

channel is always largely greater than the symbol duration (Tc = 0:806 ms),

while for a bit rate of 0.25, 1 and 4 Mb/s, the symbol duration with a 4-DPSK

modulation is respectively 8 �s, 2 �s and 0.5 �s. The study has been limited

105


-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

10 15 20 25 30

Log(BER)

Eb/N0 [dB]


AWGN ChannelRicean Channel

Mupatp, 0.25 Mb/sMupatp, 1 Mb/sMupatp, 4 Mb/s

Figure 5.31. BER for a 4-DPSK transmission scheme using a white gaus-sian noisy channel, a Ricean channel, and the MUPATP block with a C/M

ratio of 6 dB, at a bit rate of 0.25, 1, 4 Mb/s.

to these values of bit rate because the duration of a simulation increases

linearly with the bit rate. As a consequence, the simulation of higher bit

rates is too long. In fact, in order to observe the complete behaviour of the

fading process, the behaviour of the system in a period of time greater than

the coherence time Tc of the fading process must be observed. Otherwise,

the results would be obtained observing symbols that undergoes to similar

level of attenuation, being Tc de�ned as the time duration over which two

received signal have a strong potential for amplitude correlation. Consider as

an example, a simulated time of nTc, then the number of symbols simulated

is nTc=Ts, where Ts is the symbol duration. However, in a 4-DPSK scheme,

Ts is related to the bit rate Rb by Ts = 2=Rb, thus the number of symbol

simulated is RbnTc=2. The greater Rb, the greater the number of symbols

that must be simulated (i.e. the computational duration of the simulation).

In the simulation n = 50 has been taken into account: the entire trend of

the fading process is thus observed, as can be seen by the scattering diagram

corresponding to a bit rate of 0.25 Mb/s (see Figure 5.29).

106

Conclusions

Flying at an altitude of 17 km a stratospheric platform can cover a wide

area, having a radius on the order of one hundred kilometers. In the previous

chapters we have studied how the propagation is a�ected by the high position

of the platform, and a simulation model for the stratospheric channel has

been presented, together with the simulation results obtained in the case of

a transmission system that uses a 4-DPSK simulation scheme.

As it was pointed out, the distance between a transmitter TX placed as

payload to the platform and a receiver RX at ground level is always greater

than the distance TX-RX in the case that TX is at ground level. As a

consequence, the transmission from a stratospheric platform undergoes to a

greater free space attenuation. However, the high position of the platform

grants some major advantages. In this work it has been proved that some

of the most important e�ects that are detriment to propagation (ground

re ection, di�raction, and scattering from rain) can be greatly decreased

thanks to the high altitude of the platform. As a consequence, when the

receiver is at a ground distance greater than a few kilometers, the total

attenuation experienced in a PS based system is lower than in a GS based

system.

In Chapters 3 and 4 we have studied the e�ects due to small scale fading,

and in particular we have evaluated the Doppler Spectrum and the Power

Delay Pro�le for the stratospheric channel. The analysis of these functions

allows to predict the e�ects of dispersion in frequency and time. In the PS

case the dispersion in frequency is greater than in the GS case, because also

the movement of the platform must be taken into account. On the other

hand, dispersion in time is greatly reduced. In fact in a platform based

107


system the elements of scattering that generate the interfering echoes can be

found only in a layer at ground level and thus only around the receiver. As a

consequence the number of echoes in a PS based system is lower than in a GS

based system. Moreover, the presence of a direct ray from the transmitter

to the receiver makes the ratio between the power of the transmitted signal

and the power of the interfering rays very high. It follows that the time

support and rms delay spread of the power delay pro�le are smaller in the

case of the stratospheric channel than in traditional terrestrial systems. As a

consequence, in the stratospheric case fading can be considered at even at

large bandwidth, on the order of 1 MHz, and the stratospheric channel can be

represented by a Ricean model, having a Carrier to Multipath (C/M) ratio

of 18 dB. Those conditions grant to the system better performance than the

traditional terrestrial mobile radio channel, where time dispersion cannot be

neglected for a signal bandwidth greater than 20 kHz1, and that is usually

represented by a Rayleigh channel model, because frequently the direct ray

cannot be distinguished by the interfering echoes.

In conclusion, a system based on a stratospheric platform is capable of

a wide coverage, in most part of its coverage area undergoes to a weaker

attenuation level than a terrestrial system, and is a�ected by less severe

fading e�ects, so that it can be used also for wideband applications.

1This is the coherence bandwidth for a channel having an rms delay spread �� of 1 us,as can be found using Equation (3.13).

108

Appendix A

The Simulation Block

MUPATP

SUBROUTINE MUPATP(F0,D0,CVEL,MVEL,PVEL,G0,NTAP,X,Y)

SAVE

C

C*******************************************************************

C PROGRAMMER ROBERTO FANTINI

C-------------------------------------------------------------------

C LATEST REVISION 30-01-2001

C-------------------------------------------------------------------

C SUBROUTINE CODE ????

C-------------------------------------------------------------------

C PURPOSES (Q)Diffuse Multipath scattering channel

C

C Simulate a diffuse multipath scattering channel from a

C stratospheric platform

C

C A tapped delay-line model is used

C

C-------------------------------------------------------------------

C PARAMETERS

C

109

A { The Simulation Block MUPATP

C F0 = center frequency - real variable

C D0 = receiver ground distance in KM - real variable

C CVEL = speed of light in KM/?? - real variable

C MVEL = receiver speed in KM/?? - real variable

C PVEL = platform speed in KM/?? - real variable

C G0 = C/M carrier to multipath power

C ratio in dB - real variable

C NTAP = number of reflected rays - integer variable

C-------------------------------------------------------------------

C INPUT SIGNALS

C

C X = input analytic signal

C-------------------------------------------------------------------

C OUTPUT SIGNALS

C

C Y = output analytic signal

C-------------------------------------------------------------------

C SIGNAL TYPE

C

C X = A

C Y = A

C-------------------------------------------------------------------

C TOPSIM CALLING MODE

C

C Y < MUPATP(F0,D0,CVEL,MVEL,PVEL,G0,NTAP) < X

C*******************************************************************

C COMMON BLOCKS

C (if applicable)

C

C (*) ZZCTRL KEEP,NCOUNT,TIME,DELT,FINTIM

C (*) ZZHIST HIST(20000)

C (*) ZZIST IST(20000)

C ...... ......

C

110


C (*) ARE STANDARD TOPSIM COMMON BLOCKS

C-------------------------------------------------------------------

C SUBPROGRAMS USED

C TSEDET SUBROUTINE (TOPSIM SERVICE LIBRARY)

C HIMULT BLOCK (TOPSIM LIBRARY)

C BPSUM BLOCK (TOPSIM LIBRARY)

C FADJAK BLOCK (TOPSIM LIBRARY)

C POWDEL SUBROUTINE (FORTRAN SUBROUTINE)

C*******************************************************************

C COMMON BLOCKS

C

COMMON/ZZCTRL/KEEP,NCOUNT,TIME,DELT,FINTIM

COMMON/ZZHIST/HIST(20000)

COMMON/ZZIST/IST(20000)

C-------------------------------------------------------------------

C DIMENSIONING ARRAYS

C

DIMENSION ATAU(NTAP)

DIMENSION APOT(NTAP)

DIMENSION X(3)

DIMENSION Y(3)

DIMENSION XRITIN(3)

DIMENSION Z1(3)

DIMENSION XRIT(2)

DIMENSION X1(3)

DIMENSION W1(3)

C------------------------------------------------------------------

C HISTORY REQUIREMENTS

C

L=LOCH(NTAP)

LTIN=LOCI(NTAP)

LTPAR=LOCH(NTAP)

LXR1=LOCH(NTAP)

LXR2=LOCH(NTAP)

111


LF=LOCH(1)

LAND=LOCH(1)

LANM=LOCH(1)

LNSIN= LOCI(1)

IF(TIME.GT.0.)GOTO 22222

C

C HISTORY ORGANIZATION

C

C HIST(L)...HIST(L+NTAP-1) array of amplitude

C attenuations APOT^(1/2)

C IST(LTIN)...IST(LTIN+NTAP-1) array of delays expressed as

C number of samples

C HIST(LTPAR)...HIST(LTPAR+NTAP-1) array of the remaining part

C of the delays

C HIST(LXR1)...HIST(LXR1+NTAP-1) previous value of the in-phase

C part of the input signal X delayed

C of HIST(LTIN)..(LTIN+NTAP-1)

C HIST(LXR2)...HIST(LXR2+NTAP-1) previous value of the in-quadrature

C part of the input signal X delayed

C of HIST(LTIN)..(LTIN+NTAP-1)

C HIST(LF) doppler spread FD

C HIST(LAND) normalization coefficient for the direct ray

C HIST(LANM) normalization coefficient for the multipath waves

C IST(LNSIN) parameter for fadjak

C-------------------------------------------------------------------

C INITIALIZATION PROCEDURE

C

CALL POWDEL(F0,CVEL,D0,APOT,ATAU,NTAP)

PROV=0.

DO 10 I=1,NTAP

J=L+I-1

HIST(J)=SQRT(APOT(I))

K=LTIN+I-1

IST(K)=INT(ATAU(I)/DELT)

112


KK=LTPAR+I-1

HIST(KK)=ATAU(I)-DELT*INT(ATAU(I)/DELT)

JJ=LXR1+I-1

HIST(JJ)=0

JK=LXR2+I-1

HIST(JK)=0

PROV=PROV+APOT(I)

10 CONTINUE

FD=(PVEL+MVEL)*F0/CVEL

HIST(LF)=FD

GMULT=10.**(-G0/10)/PROV

HIST(LAND)=SQRT( 1./(1.+10.**(-G0/10)) )

HIST(LANM)=SQRT( GMULT/(1.+10.**(-G0/10)) )

IST(LNSIN)=77

PI=4.*ATAN(1.)

C-------------------------------------------------------------------

22222 CONTINUE

C-------------------------------------------------------------------

C CHECK ON INPUT SIGNALS

C

CALL ANSCK(29023,X)

C-------------------------------------------------------------------

C PROGRAM STATEMENTS

C

FDTOT=HIST(LF)

ANORD=HIST(LAND)

ANORM=HIST(LANM)

NSIN=IST(LNSIN)

C-------------------------------------------------------------------

C LINE OF SIGHT TAP

C

Y(1)=ANORD*X(1)

Y(2)=ANORD*X(2)

Y(3)=F0

113


C-------------------------------------------------------------------

C REFLECTED PATHS

C

DO 30 I=1,NTAP

CALL DELAY(IST(LTIN+I-1),X,XRITIN)

XRIT(1)=(1-HIST(LTPAR+I-1)/DELT)*XRITIN(1) +

1 HIST(LTPAR+I-1)/DELT*HIST(LXR1+I-1)

XRIT(2)=(1-HIST(LTPAR+I-1)/DELT)*XRITIN(2) +

2 HIST(LTPAR+I-1)/DELT*HIST(LXR2+I-1)

X1(1)= XRIT(1)*COS(2*PI*F0*HIST(LTPAR+I-1)) +

3 XRIT(2)*SIN(2*PI*F0*HIST(LTPAR+I-1))

X1(2)= XRIT(2)*COS(2*PI*F0*HIST(LTPAR+I-1)) -

4 XRIT(1)*SIN(2*PI*F0*HIST(LTPAR+I-1))

X1(3)= F0

HIST(LXR1+I-1)=XRITIN(1)

HIST(LXR2+I-1)=XRITIN(2)

CALL FADJAK(NSIN,F0,FDTOT,Z1)

NSIN=NSIN+2

CALL HIMULT(X1,Z1,W1)

W1(1)=ANORM*HIST(L+I-1)*W1(1)

W1(2)=ANORM*HIST(L+I-1)*W1(2)

W1(3)=F0

CALL BPSUM(F0,Y,W1,Y)

30 CONTINUE

C-------------------------------------------------------------------

C RETURN TO THE CALLING PROGRAM

C

RETURN

C-------------------------------------------------------------------

C ERROR MESSAGES

C

C 29023 MUPATP Input signal not analytic

C-------------------------------------------------------------------

END

114

Appendix B

The Fortran Subroutine

POWDEL

SUBROUTINE POWDEL(F0,C,X0,POTI,TI,NTAP)

SAVE

C

C******************************************

C

C F0 = center frequency - real variable

C C = light speed

C X0 = user position in km

C NTAP = number of taps returned

C POTI = power associated to i-th tap - array

C TI = delay associated to i-th tap - array

C

C***************************************************************

C SUBPROGRAMS USED

C

C ATAN =FORTRAN intrinsic function

C INT =FORTRAN intrinsic function

C ALOG10 =FORTRAN intrinsic function

C*************************************************************

C DIMENSIONING ARRAYS

115

B { The Fortran Subroutine POWDEL

C

DIMENSION T(1000)

DIMENSION V(1000)

DIMENSION PRX(1000)

DIMENSION PS(1000)

DIMENSION TS(1000)

DIMENSION POTI(NTAP)

DIMENSION TI(NTAP)

N=1000

M=50

C

C*************************************************************

C

PI = 4*ATAN(1.)

Z0 = 17

H=0.018

RHO=10**6

SIGMA=20E-6

RDIR=SQRT(X0**2 + Z0**2)

T0=RDIR/C

TMAX=T0/400.

C

C*************************************************************

C

T(1) = 0

V(1) = 0

PRX(1) = 1

DT = TMAX/N

DO 101 I=2,N

T(I) = TMAX*(I-1)/N

V(I) = -PI*(RDIR+C*T(I))*(C**2*T(I)**2+2*RDIR*C*T(I))*

1 (4/3*H**3-2*Z0*H**2-(C**2*T(I)**2+2*RDIR*C*T(I))*H)/

2 (4*SQRT((Z0**2+C**2*T(I)**2 + 2*RDIR*C*T(I))**3))

DV=V(I)-V(I-1)

116


ECHI=RHO*DV

RK=RDIR+C*T(I)

B=(RK**2-RDIR**2)/(2*SQRT(RK**2-X0**2))

Y=0

DY=B/M

RTFIX=B**2*(RK**2+X0**2)/(RK**2-X0**2)

1 - X0**2/(RK**2-X0**2)*Y**2

RTVAR=2*B**2*X0*RK/(RK**2-X0**2)*

1 SQRT(1-Y**2/B**2)

RTPOS=SQRT(RTFIX+RTVAR)

RTNEG=SQRT(RTFIX-RTVAR)

OLDPRP=(RDIR**2*SIGMA/(4*PI))/

1 ((RK-RTPOS)**2*RTPOS**2)

OLDPRN=(RDIR**2*SIGMA/(4*PI))/

1 ((RK-RTNEG)**2*RTNEG**2)

PROVP=0

PROVN=0

DO 100 J=2,M

Y=Y+DY

RTFIX=B**2*(RK**2+X0**2)/(RK**2-X0**2)

1 - X0**2/(RK**2-X0**2)*Y**2

RTVAR=2*B**2*X0*RK/(RK**2-X0**2)*

1 SQRT(1-Y**2/B**2)

RTPOS=SQRT(RTFIX+RTVAR)

RTNEG=SQRT(RTFIX-RTVAR)

PRNPOS=(RDIR**2/(4*PI))/

1 ((RK-RTPOS)**2*RTPOS**2)*SIGMA

PRNNEG=(RDIR**2/(4*PI))/

1 ((RK-RTNEG)**2*RTNEG**2)*SIGMA

PROVP=PROVP + (PRNPOS+OLDPRP)/(2*M)

PROVN=PROVN + (PRNNEG+OLDPRN)/(2*M)

OLDPRP=PRNPOS

OLDPRN=PRNNEG

100 CONTINUE

117


PRXECO=0.5*PROVP+0.5*PROVN

PRX(I)=PRXECO*ECHI

101 CONTINUE

FLAGC=1

DO 110 I=1,N

IF ((PRX(I).LT.1).AND.(FLAGC.NE.0)) THEN

IMIN=I

FLAGC=0

END IF

110 CONTINUE

FLAGC=1

DO 120 I=1,N

IF ((10*ALOG10(PRX(I)).LE.(-30)).AND.(FLAGC.NE.0))

1 THEN

IMAX=I

FLAGC=0

END IF

120 CONTINUE

NUM=IMAX-IMIN+1

DO 130 I=1, NUM

PS(I)=PRX(IMIN+I-1)

TS(I)=T(IMIN+I-1)

130 CONTINUE

NSTEP= INT(NUM/NTAP)

DO 141 I=1,NTAP

TI(I)=TS(INT(NSTEP/2)+(I-1)*NSTEP)

PROV=0

DO 140 J=1, (NSTEP-1)

PROV=PROV + (PS((I-1)*NSTEP+J) +

1 PS((I-1)*NSTEP+J+1))/(2*NSTEP)

140 CONTINUE

POTI(I)=PROV

141 CONTINUE

END

118

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Radio Channel Modeling and Simulation for a Stratospheric Platform Link

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