khonghthesis

8/6/2019 KhongHThesis

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FLORIDA STATE UNIVERSITY

FAMU FSU COLLEGE OF ENGINEERING

A COMPUTER SIMULATION MODEL FOR MICROWAVE LINK

PATH LOSS PREDICTION

By

HUNG HUY KHONG

A Thesis submitted to the

Department of Electrical and Computer Engineeringin partial fulfillment of the

requirements for the degree ofMaster of Science

Degree Awarded:

Summer Semester, 2009

Copyright 2009

Hung Huy KhongAll Rights Reserve


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The members of the Committee approve the Thesis of Hung Huy Khong defended on

April 27th, 2009

__________________________________________

Bing W. KwanProfessor Directing Thesis

_________________________________________

Leonard J. Tung

Committee Member

__________________________________________Simon Y. Foo

Committee Member

Approved:

Victor DeBrunner, Chair, Department of Electrical and Computer Engineering

Ching-Jen Cheng, Dean, FAMU-FSU College of Engineering

The Graduate School has verified and approved the above named committee members.


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TABLE OF CONTENTS

LIST OF TABLES .............................................................................................. vi

LIST OF FIGURES ............................................................................................. vii

LIST OF ABBREVIATIONS.............................................................................. ix

ABSTRACT ...................................................................................................... x

1. INTRODUCTION .......................................................................................... 1

1.1. Overview of Radio Wave Propagation ................................................ 1

1.2. Brief Review of Path-loss models........................................................ 3

1.3. Problem Statement ............................................................................... 5

1.4. Thesis Organization ............................................................................. 6

2. NARROWBAND PATH-LOSS MODELS ................................................... 7

2.1. Free Space Path-loss for LOS Environment ........................................ 7

2.2. Path-loss for NLOS Environment........................................................ 7

2.3. Semi-empirical Path-loss Models for Macro-cell Areas...................... 8

2.3.1. Okumura Model.......................................................................... 9

2.3.2. Hata Model.................................................................................. 9

2.3.3. COST-231 Model........................................................................ 11

2.3.4. Lee Model ................................................................................... 12

2.3.5. The Model in [3] ......................................................................... 13

2.3.6. Comparison of Semi-empirical Models...................................... 14

3. COMPUTER SIMULATION PATH-LOSS MODEL................................... 16

3.1. Ray Tracing Technique........................................................................ 16

3.1.1. Two-Ray Model.......................................................................... 16

3.2. Development of the Computer Simulation Path-loss model................ 18

3.3. Channel Space...................................................................................... 21

4. COMPUTER SIMULATION RESULTS....................................................... 23

4.1. Simulation Procedure........................................................................... 23

4.2. Statically Regularity of Received Signals............................................ 23


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LIST OF TABLES

Table 2.1: Urban environment correction factors for the Hata model................. 11

Table 2.2: Suburban and Rural environment correction factors for theHata model......................................................................................... 11

Table 2.3: Environment parameters for the Lee model ....................................... 13

Table 2.4: Environment parameters for the model in [3]..................................... 14

Table 4.1: Mean path loss exponents and intercept values for three differentgroups of objects................................................................................ 33

Table 4.2: Mean path loss exponents and intercept values for three differentground reflection coefficients ............................................................ 36


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LIST OF FIGURES

Figure 1.1: A communication system ................................................................. 1

Figure 2.1: Median loss relative to free space ),( dfAmu .................................. 10

Figure 2.2: Correction factor AREAG for different environments ..................... 10

Figure 2.3: Path-loss predicted by Hata, Lee, and the model in [3] for urban area 15

Figure 3.1: Two-ray model ................................................................................. 17

Figure 3.2: Two two-ray models characterizing a two-hop multipath signal ..... 19

Figure 3.3: Channel Space .................................................................................. 21

Figure 4.1: Typical multipath electric-field phasors received at short distance

(d= 1 km) withN= 100 scattering objects ....................................... 24

Figure 4.2: Typical multipath electric-field phasors received at long distance

(d= 20 km) withN= 100 scattering objects ..................................... 24

Figure 4.3: Path losses due to 1000 random distributions ofN= 10 scatteringobjects ............................................................................................... 25

Figure 4.4: The MSE curve fitted through the mean path losses versus logdistance forN= 10 objects yielding a path-loss exponent 0.8

and intercept 84 dB ........................................................................... 26

Figure 4.5: Histograms of path loss exponents and intercept values forN= 10

scattering objects ............................................................................... 27

Figure 4.6: Histograms of path loss exponents and intercept values forN= 50scattering objects ............................................................................... 27

Figure 4.7: Histograms of path loss exponents and intercept values forN= 150scattering objects ............................................................................... 28

Figure 4.8: Path loss based on computer simulations forN= 5 objects.= 0.08and exponent = 0.677 ...................................................................... 30

Figure 4.9: Path loss based on computer simulations forN= 5 objects.= 0.8and exponent = 2.606 (different object distribution from Figure 4.8) 30


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CHAPTER 1

INTRODUCTION

1.1. Overview of Radio Wave Propagation

Figure 1.1: A communication system [17]

A typical communication system consists of three main systems: the transmitter (TX), the

receiver (RX) and the transmission channel as depicted in Figure 1.1.

It is desirable to understand the channels statistical characteristics in order to predict the channel

behavior. The signal processing techniques then will be developed and used properly to ensure

that the transmission channel between the TX and the RX is as reliable as possible. Therefore,

understanding the channel characteristics plays an important role in designing and optimizing a

communication system.

The channel is subject to noise, distortion and other interference sources leading to the variation

of the received signal power. The models that characterize the signal fluctuations over small

distances in the order of the wavelength or short-time duration are called small-scale models.

While the propagation models that predict the mean signal strength over the large distances

between the TX and the RX or long-time duration are called large-scale or path-loss models.

Destination

Input

Signal

Received

Signal

Source

Transmitted

Signal Output

Signal

Noise, Interference,

Distortion

Transmitter ReceiverTransmission

Channel


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km to 20 km has been proposed by Hata [1], Lee [2], and several other research groups [3, 4].

Hata proposed the path-loss models for different environments such as urban areas, suburban

areas, and rural areas based on measurement data. All of these models are similar with respect to

the signal attenuation rate (path loss exponents). Lee also performed measurement works in

different cities around the world and proposed alternate path-loss models. Several other research

groups also investigated the path-loss models for narrowband communication systems based on

measurement [3]. Those models are applicable not only for different environments, but also for

varying carrier frequency, antenna heights, the TX-RX separation, etc. For small distances

between the TX and the RX, several other research groups have also proposed the path-loss

models for micro-cell areas [4]. It is noticed that all the semi-empirical models predict the

median path losses.

There are some techniques used to develop the path-loss model using computer simulation.The

two-ray models are most commonly used to predict the path loss and calculate the signal

strength. These models use the ray-tracing technique that is based on geometrical optics to

account for the three mechanisms discussed above [5, 6]. This technique assumes a finite number

of multipath signals from the TX to the RX. The knowledge about the physical elements of the

channel is essential, including the geometry of the scattering objects in the channel. To simplify

the solution and procedure, however, the ray tracing technique often makes approximations to

obtain an excess path length. Typically, it is assumed that the nearest scattering objects are in the

far-field regions of the TX and the RX. In addition, the scattering objects are electrically large [7

10]. Several research efforts reported in [11 14] have employed the ray-tracing technique. In

[13], it shows a simple deterministic-plus-stochastic path-loss model for small outdoor areas

(such as parking lots and intersections). It is found that for these areas with well defined

geometries, a model (such as 2-ray, 4-ray or multipath reflection model) may be postulated

initially and the corresponding measurement will be performed to verify the accuracy of the

assumed models. Simulation results of path loss and delay spread are also presented in [14]. This

research effort finds that the propagation loss depends on shaped objects and is affected by the

physical elements of the media, such as the dielectric constants and the roughness factors of the

objects.


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1.3. Problem Statement

One may notice that the research works described in section 1.2 have several limitations as

described in the following:

The collecting measurement data are generally very tedious and costly because largevolume of data needs to be acquired and specialized expensive equipments are involved.

The measurement data are inherently site-specific and vary from site to site. The ray tracing technique often makes approximations to obtain excess path lengths. Some existing models have been proposed and verified by measurement for very small

outdoor areas characterized by the TX-RX separation less than 50 meters.

In view of the limitations described above, it is desirable to have a computer simulation model to

predict the path loss for different environments and to validate the existing semi-empirical

models. To achieve this objective, a computer simulation model is proposed that will require

minimal information about the environment. More specifically, only the number of scattering

objects and their locations are used to predict the path loss. Such an approach essentially makes

the proposed model widely applicable to different environments. The added benefit is that it does

not require the high costs associated with the measurement efforts. Moreover, the computer

simulation model may lead to criteria for identifying the proper conditions that are best suited for

a specific existing semi-empirical model.

To develop a flexible computer model that possesses the attractive features described previously,the following solution approach is taken. Multipath signals resulted from the scattering by any

number of randomly located objects are considered. Each multipath signal is a two-hop signal

that involves a single scattering object. The wave (signal) interactions between the TX and a

scattering object are based on the two-ray model without making restrictive assumptions. The

same approach is used to account for the wave interactions between a scattering object and the

RX. The two-ray model considers the effects of wave propagation along the direct path as well

as the indirect path due to ground reflection. The phase and magnitude of the received signals are

taken into account in order to properly understand the received signal characteristics. In addition,

the proposed model allows the flexibility in studying the dependence of path loss on the density

of scattering objects and the distance between the TX and the RX.

The key step in the development of the model is the proper representation of the electric field

due to the direct path and ground-reflected path. The power transmitted from the TX to the


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CHAPTER 2

NARROWBAND PATH-LOSS MODELS

2.1 Free Space Path-loss for LOS Environment

Free space path loss provides a means to predict the received signal power when there is no

object obstructing the LOS path between the TX and the RX.

The model for path loss in a LOS environment is straightforward. The received power rP is

related to the transmitted power tP via the Friis transmission formula [15]:

2

2

2 )4(4 d

GGPA

d

EIRPP rtter

==(W) (2.1)

In (2.1), ttGPEIRP = . The transmitting antenna has gain tG while the receiving antenna has

gain rG . Distance d is the separation between the TX and the RX. The RX has an effective

aperture given by 4/2re GA = , where is the signal wavelength. The path loss is defined by

the term

2

2 44

===

ddG

A

GGP

PL

r

e

rtt

r

(2.2)

It is shown that for the LOS environment, the power received will fall off with the square of the

distance between the TX and the RX.

2.2. Path-loss for NLOS Environment

Since there are scattering objects in the channel, it is quite likely that the transmitted signal

cannot reach the RX directly since the LOS path is blocked by these objects. These objects can

greatly impact the average signal strength at the RX.

Unlike the LOS case, the modeling of path loss in the NLOS environment is more complex and

involves more environmental parameters in the model. A popular NLOS path-loss model is semi-

empirical and assumes the form [10], [15]:


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[dB])(log10)()(0

100 dSd

dndPdL L +

+= (2.3)

In (2.3), )( 0dPL is the path loss at a reference distance 0d from the TX, S(d) accounts for the

lognormal shadow fading effects, and n is the path-loss exponent.Many experiments and results have been reported for both LOS and NLOS environments,

covering the frequency range from 100 MHz to 2 GHz, which is the frequency band for most

narrowband transmission systems. According to (2.3), path-loss is assumed to be the linear

function of log distance (to the base 10) with loss exponent n and intercept )( 0dPL . These loss

exponents and intercept values are subject to change for different environments and generally do

not equal to the values for LOS environment. Normally the loss exponents and intercept values

are typically determined by taking several measurements at various distances and performing a

linear regression to obtain a least square fit to the measured data.

2.3. Semi-empirical Path-loss Models for Macro-cell Areas

There are numerous semi-empirical models in the literature aiming to predict the path loss over

different environments. The models demonstrated below are models derived from measured data

corresponding to macro cellular environments with path distances ranging from 1 km to over 10

km. The Hata and Lee models are the most commonly used models for signal prediction in urban

areas nowadays although they were developed long ago. Many telecommunication providers are

using these models in designing and optimizing their radio networks. The model in [3] was

recently developed and will also be compared to the computer simulation model proposed in this

thesis. The common characteristic of all the models is the linear dependence of path-loss on the

distance from the TX to the RX, as depicted in (2.3). Various path loss exponents and intercept

values are reported in these models with respect to different environments. Three typical models:

the Hata model, the Lee model, and the model in [3], which are best suitable for urban areas, willbe served as the reference for comparison with the proposed computer simulation model reported

later in Chapter 4.


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2.3.1. Okumura Model

The Okumura model [4] is applied when frequencies range from 150 KHz to 1920 MHz and

distances from the TX to the RX, range from 1 km up to 100 km. The model is based on

measured data taken in Japan. This is among the simplest models and becomes the standard for

path-loss modeling. The standard deviation in predicting the path loss is around 10 dB.

The model is described by

AREArtmuF GhGhGdfALdBL += )()(),()( (2.4)

where L(dB) is the median path loss, FL is the free space path loss calculated by (2.2),

),( dfAmu , the median loss relative to free space, is a set of curves developed from measured

data using omni-directional antennas, )( thG is the base station (BS) gain factor with height th ,

)( rhG is the MS gain factor with height rh , and AREAG is the environment gain factor. These

gain factors are given by

mhmh

hG tt

t 301000,200

log20)( 10 >>

= (2.5)

>>

=

mhmh

mhh

hG

rr

rr

r

310,3

log20

3,3

log10

)(

10

10

(2.6)

The median loss ),( dfAmu and the environment gain factor AREAG are available as Okumura

curves shown in Figures 2.1 and 2.2.

2.3.2. Hata Model

Hata further developed the Okumura model from the semi-empirical path loss data provided by

Okumura. The Hata model [1] is mostly applicable for urban areas that have not been addressed

in the Okumura model. The model is best accurate within a frequency range from 150 KHz to

1500 MHz. The basic formula for the median path lossL(dB) given by Hata is

KdhhahfdBL kmtrtMHz ++= log)log55.69.44()(log82.13log16.2655.69)( (2.7)

Where ht and hr are base station and mobile station antenna heights in meters, respectively, dkm

is the link distance or cell radius in kilometers,fMHz is the centre frequency in megahertz, and the


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terms a(hr)and Kare vehicular station antenna height-gain correction factor and environment

correction factor, respectively, which depend upon the environment. These correction factors for

different environments are tabulated in Tables 2.1 and 2.2 below.

Figure 2.1: Median loss relative to free space ),( dfAmu [15]

Figure 2.2: Correction factor AREAG for different environments [15]


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Table 2.1: Urban environment correction factors for the Hata model

Environment Type K a(hr)

Urban indoor large city 0MHzfh MHzr 200,0.151.1)54.1log(29.8

2

Urban large city 0MHzfh MHzr 200,1.1)54.1log(29.8

2

Urban small city 0 )8.0log56.1()7.0log11.1( MHzrMHz fhf

Table 2.2: Suburban and Rural environment correction factors for the Hata model

Environment Type K A(hr)

Suburban 4.528

log2

2

+

MHzf 0

Rural indoor (quasi-open) 1094.35log331.18)(log78.4 2 + MHzMHz ff 0

Rural (quasi-open) countryside

94.35log331.18)(log78.4 2 + MHzMHz ff 0

Rural (open) desert 94.40log331.18)(log78.4 2 + MHzMHz ff 0

2.3.3. COST-231 Model

COST-231 model [15] is the extended version of the Hata model developed by EURO-COST.

This model extends the Hata models frequency range from 150 KHz to2000 MHz.The model has a slight difference in the intercept value but the same exponent compared with the

Hata model. The models path loss is given by

MkmtrtMHz CdhhahfdBL +++= log)log55.69.44()(log82.13log9.333.46)( (2.8)


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=

(W)10

(W)PowerTransmit3 (2.13)

=

4

dipolehwavelengthalfrespect togain withantennaBS4 (2.14)

MSatfactorcorrectiongainantennadifferent5 =

The variable v in 2 equals 2 when the MS antenna height is greater than 10 m and equals 1

when MS antenna height is smaller than 3 m.

For convenience, the path-loss models for different environments proposed by Lee are

summarized as follows:

[dB]log10)(log)( 01010

++=

of

fndYXdBL (2.15)

Table 2.3: Environment parameters for Lee model

Type of Environment X Y

Free space 96.92 20

Open area 86.12 43.5

US suburban 99.86 38.4

US urban 108.49 36.8

Newark, NJ 101.2 43.11

Tokyo, Japan 123.77 30.5

2.3.5. The Model in [3]

The model developed in [3] is best suitable for systems with a carrier frequency of 1.9 GHz and

base station antenna height of 10 m to 80 m; however, it can be applied to other frequencies with

a small error. The model is divided into three types according to the environment, namely A, B

and C. Type A is associated with the maximum path loss and is appropriate for hilly

environments with moderate to heavy foliage densities. Type C is associated with the minimum

path loss and applies to flat environments with light tree densities. Type B is characterized with


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either mostly flat environments with moderate to heavy tree densities or hilly environments with

light tree densities. The model may be described as follows:

00

10 ,log10)( ddsXXd

dndBL hf >+++

+= (2.16)

In (2.16), is the free space path loss given by (2.2), fX is the correction factor for the carrier

frequency, hX is the correction factor for the mobile station antenna height, s is the shadow

fading component, which is a zero-mean Gaussian variable, the

parameter nt

t xh

cbhan ++= )( is called the path loss exponent and depends on the base

station antenna height th and environment type, x is a zero-mean Gaussian variable of unit

standard deviation ]1,0[ . n normally ranges from 2 to 4. The values of a, b, c, x, and n fordifferent environments are tabulated in Table 2.4 below.

Table 2.4: Environment parameters for the model in [3]

Model

parameter

Environment

type A

Environment

type B

Environment

type C

a 4.6 4.0 3.6

b 0.0075 0.0065 0.005

c 12.6 17.1 20

n 0.57 0.75 0.59

2.3.6. Comparison of Semi-empirical Models

For comparison, the path losses for urban areas predicted by the Hata model, Lee model, and the

model in [3] for different environments are plotted in Figure 2.3. The specific parameter values

used for this plot are: antenna heights m2m,70 == rt hh , antenna gains 1== rt GG , and a

carrier frequency of 900 MHz. The Hata model and the model in [3] for environment A are


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applied. The Lee models for different cities including Philadelphia, Newark, and Tokyo are also

plotted.

One may observe the path-loss exponent predicted by these models varies from 2 to 4 and the

intercept value ranges from 90 to 120 dB. These values will be used for comparison later with

the model proposed in this thesis.

0 0.2 0.4 0.6 0.8 1 1.2 1.490

100

110

120

130

140

150

160

170

Distance log10(d)

Pathloss(dB

)

Free Space

Hata Urban

Lee Philadenphia

Lee Newark

Lee Tokyo

Model in [3]

n = 3.26

n = 2

n = 4.31

n = 4.2n = 3.68

n = 3.05

Figure 2.3: Path-loss predicted by Hata model, Lee model, and the model in [3] for urban areas


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CHAPTER 3

COMPUTER SIMULATION PATH-LOSS MODEL

3.1. Ray Tracing Technique

3.1.1. Two-Ray Model

In this section, the two-ray model based on ray tracing technique, which is the most reasonably

accurate to predict the path loss and calculate the signal strength, will be presented. This model is

especially suitable for the urban environments and radio systems that use high tower base station.

Many research efforts have used the ray-tracing technique in predicting the path loss [7]-[10],

[11]-[14].

To account for the rich multipath characteristics of the NLOS environment, it is necessary to

consider numerous scattering objects in the channel, namely the intervening medium between the

TX and the RX. For the sake of simplicity, it is assumed that the dominant received signal power

comes primarily from the one-hop direct signal along the LOS path and the two-hop multipath

signals due to single reflections.

The basis of the two-ray model is the proper representation of the electric field incident upon a

scattering object. The signal at the distance dfrom the transmitting antenna can be expressed in

terms of the electric field strength of the form:

)(cos),( 0 dc ttd

EdtE = (3.1)

In (3.1), 2/0 EIRPE = with 377= for free space, c is the speed of light, and dt is the

time delay.

The two-ray model shown in Figure 3.1 is essential to the development of the multipath model

that includes an arbitrarily number of reflecting objects. According to (3.1), the total electric

field at the RX at the distance dfrom the TX is calculated as the sum of the electric field due to

the LOS path and the ground reflected path.


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)/''(cos)'',(

)/'(cos)',(

),(),(),(

0

0

cdt

d

EdtE

cdtd

EdtE

dtEdtEdtE

cgg

cLOS

gLOSTOTAL

=

=

+=

(3.2)

In (3.2), g denotes the reflection coefficient of the ground.

Figure 3.1: Two-ray model

The path difference between the LOS path and the ground reflected path can be well

approximated by

d

hhdd rt

2''' = (3.3)

In (3.3), ht and hr are the transmitted and received antenna heights, respectively. The

approximation is valid as long as the sum of the transmitted and received antenna heights is

small compared to d. This two-ray model is intended to include the significant effect of the

ground reflection.

The phase difference is then defined as

=

2(3.4)


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Consequently, the total electric field at the RX at the distance d from the TX is computed by

=

2sin2),( 0

d

EdtETOTAL (3.5)

As long as the distance d satisfies the condition

rthh

d

20

> , the total electric field can be

approximated by

2),(

d

kdtETOTAL = (3.6)

In (3.6), k is a constant related to 0E , the antenna heights, and the signal wavelength. The

received power for the two-ray model can be computed as the function of ),( dtETOTAL and the

path loss is then obtained as

4

22

d

hh

GGP

PL rt

rtt

r == (3.7)

It is seen that the path loss for the two-ray model falls off with the fourth power of the distance

between the TX and the RX. In our proposed model described subsequently, an arbitrary number

of two ray models will be applied.

3.2. Development of the Computer Simulation Path-loss Model

In this study, an arbitrary number of two-ray models will be applied for developing the path-loss

model without making any restrictive approximation, which is used by the two-ray model for one

object presented above.

In the intervening medium between the TX and the RX, it is assumed that there are numerous

scattering objects to characterize the rich NLOS multipath environment. Each scattering object

absorbs the direct signal power from the TX along the LOS path coupled with the indirect signal

power reflected off the ground. Consider scattering object i as shown in Figure 3.2. It is at

distance )(1 id from the TX along the LOS path and distance )(1 id along the ground-reflected

path. The total power absorbed is then re-radiated by object i to RX, which is at distance )(2 id

along the direct path and distance )(2 id along the ground-reflected path. The power absorbed by

object i depends on its radar cross section, which is taken to be a circular area with radius )(iR .


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Figure 3.2: Two two-ray models characterizing a two-hop multipath signal

The electric field incident upon scattering object i can be expressed in terms of the electric fieldstrength:

+=

)('2

1

)(2

101

11

)(')(

1)(

idjgidje

ide

idEiE

(3.8)

In (3.8), 2/0 EIRPE = with intrinsic impedance 377= for free space, and g denotes

the ground reflection coefficient. The time-average incident signal power density at object i is

calculated as:

)(W/m)(2

1 221)( iES iobject

= (3.9)

The EIRP of object i (accounting for the total incident signal power absorbed) is then given by:

2

)()()()(

2

1)()(

22122

1)(iRiE

iRiEiSiEIRP siobject

=== (3.10)


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It is noted in (3.10) that )(is is called the radar cross section of object i . Radar cross section of

a scattering object is defined as the ratio of the power density of the signal scattered in the

direction of the RX to the power density of the radio signal incident upon the object [15]. The

radar cross section can be approximated by the objects surface area. Without loss of generality,

each object i used in this model is assumed to have a circular shape represented by radius )(iR .

Therefore, the radar cross section or area surface of object i is given by 2)()( iRis = .

The corresponding power density due to object i at the RX is then given by:

22

221

22

1)(8

)()(

)(4

)()(

id

iRiE

id

iEIRPiSRx

== (LOS path) (3.11)

22

221

222 )('8

)()(

)('4

)()(

id

iRiE

id

iEIRPiS

Rx

== (Ground-reflected path) (3.12)

The electric field at the RX due to object i can be expressed by

)('2

22

1)(

2

22

1

)('2

2

)(2

1

22

22

)('2

)()(

)(2

)()()(

2)(2)()(

idj

g

idj

r

idj

Rxg

idj

Rxr

eid

iRiEe

id

iRiEiE

eiSeiSiE

+=

+= (3.13)

In an NLOS environment consisting ofNscattering objects, the total electric field at the RX is

given by the sum

=

=N

irtotal iEE

1

)( (3.14)

The time-average incident signal power density at the RX is given by

)(W/m2

1 2*totaltotal EES =

(3.15)

Accordingly the total average power received by the RX may be viewed as the average power

intercepted by the receiving antenna with effective aperture eA and is calculated by

(W)2

2total

eer E

AASP

== (3.16)

Finally, the path loss is then obtained as

rtt

r

GGP

PL = (3.17)


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CHAPTER 4

COMPUTER SIMULATION RESULTS

4.1. Simulation Procedure

In this section, the simulation result for the path loss presented in the model above will be

shown. For the computer simulation runs, set km20MHZ,900 0 == sfc . For more

convenience in calculation, EIRP is set at 1 kW since path loss does not depend on the EIRP.

The antenna heights m2m,70 == rt hh are used to match the parameters used in measurement

studies mentioned previously. Their antenna gains are 1== rt GG . The radii of the scattering

objects vary from 1 m to 5 m. The ground reflection coefficient g is set at 1 (assuming perfect

reflection). In order to reach statistically meaningful conclusions, 1000 simulation runs are

performed for a given number of scattering objects N. A typical simulation run involves three

key steps:

Step 1: Set the separation distance d between the TX and the RX. Select the number of

reflecting objects N, then randomly generate their locations ),,( iii zyx and radar cross section

radii )(iR , Ni ,2,1

= , such that

)(,)()(,0 0 iRr|z|iRrxiRsy ii


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In the case of small separations, there are fewer dominant received signals scattered by the

nearby objects as shown in Figure 4.1. These dominant signals vary considerably from

simulation to simulation, thereby resulting in large variance. In the case of large separations as in

Figure 4.2, most objects behave more or less equally as distant scatters, thus producing stationary

signals with smaller variance at the RX.

0.5 0 0.5 1 1.5 20.6

0.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

Real part of the electric field

Imaginarypartoftheelectricfield

Figure 4.1: Typical multipath electric-field phasors received at short distance (d= 1 km) withN= 100 scattering objects

0.025 0.02 0.015 0.01 0.005 0 0.005 0.01 0.0150.025

0.02

0.015

0.01

0.005

0

0.005

0.01

0.015

Real part of the electric field

Im

aginarypartoftheelectricfield

Figure 4.2: Typical multipath electric-field phasors received at long distance (d= 20 km) with

N= 100 scattering objects


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4.3. Path-loss vs. Distance

In this computer simulation study, the dependence of path loss on the separation distance d

between the TX and the RX is of interest. The observation that the variance of the total electric

field at the TX is smaller at large distances than at small distances as reported in the previous

section is indeed confirmed by the simulation results depicted in Figure 4.3. Shown in Figure 4.3

are the path losses due to 1000 random distributions of 100 scattering objects against log

distance. It clearly indicates that the path loss varies more widely at smaller distances than at

larger distances. This appears to suggest that the sum of multipath signals tends to reach

statistical regularity with increasing distance d. In measurement studies, there is no information

regarding this observation.

After a typical simulation run corresponding to N scattering objects with a given spatial

distribution, a straight line is constructed to fit the path losses versus log distance according to

the mean squared errors (MSE) or least-squares criterion. The slope of this best-fit line gives the

path-loss exponent, whereas its value at log10(1 km) is the intercept.

3 3.5 4 4.520

40

60

80

100

120

140

160

Distance log10(d)

Pathloss(dB)

Figure 4.3: Path losses due to 1000 random distributions ofN= 100 scattering objects


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3 3.5 4 4.5

70

75

80

85

90

95

100

105

Distance log10(d)

Pathloss(dB)

Figure 4.4: The MSE curve fitted through the mean path losses versus log distance for N= 100

objects yielding a path-loss exponent 1.69 and intercept 75 dB.

In Figure 4.4, the mean path loss is plotted against log distance for N= 100 scattering objects.

The mean path loss at a given distance d is obtained by computing the average of 1000 path loss

values resulted from simulation. In addition, a minimum MSE curve is fitted through the mean

path-loss values versus log distance to yield a path-loss exponent 1.69 and an intercept 75 dB.Figures 4.5, 4.6, and 4.7 are the histograms of the path loss exponents and intercepts generated

from 1000 simulation runs corresponding to the number of scattering objects N= 10, 50, and

150, respectively. It is evident that the path loss exponents cover the range from 2 to 4, while the

intercept value varies from 80 dB to 120 dB. These values are in very good agreement with those

predicted by the Hata model, the Lee model, and the model in [3] as shown in Figure 2.3.

Another observation is the path loss exponents tend to increase while the path loss intercept

values tend to decrease with the increase of the number of scattering objects.


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5 0 50

10

20

30

40

50

60

70

Pathloss exponents

Frequencycounts

60 80 100 120 1400

5

10

15

20

25

30

35

40

Pathloss intercept values

Frequencycounts

Figure 4.5: Histograms of path loss exponents and intercept values forN= 10 scattering objects

2 0 2 40

10

20

30

40

50

60

70

80

90

Pathloss exponents

Frequencycounts

40 60 80 100 1200

5

10

15

20

25

30

35


Frequencycounts



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2 0 2 4 60

10

20

30

40

50

60

Pathloss exponents

Frequencycounts

40 60 80 1000

5

10

15

20

25

30

35


Frequencycounts


4.4. Simulations Key Findings

4.4.1. Negative Path Loss Exponents

It may occur as surprising that the histograms of the path loss exponent corresponding to N= 10

and 50 scattering objects exhibit negative values.

Specifically in the case when the number of objects N= 10, 16% of the exponents falls below 0,

81% of the exponents falls between 0 & 2, and 3% falls between 2 and 5.

A negative path loss exponent means the received signal is stronger than the transmitted signal,

indicating a physically unrealizable phenomenon. An explanation for such a physically

impossible result is the assumption that the path loss increases linearly with log distance may not

be always valid. To gain an understanding of such an abnormal result, the linear correlation of

the path loss and log distance is examined for two specific cases. The sample correlation

coefficient of the path loss and log distance is computed according to the formula [16]:


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= =

=

=M

m

M

mmm

M

mmm

LLdd

LLdd

1 1

22

1

)()(

)()(

(4.2)

In (4.2), Mis the number of data points resulted from varying the distances ,,,1, Mmdm =

between the TX and the RX. The path loss associated with md is denoted by mL . Further,

==

==M

mm

M

mm L

MLd

Md

11

1,

1(4.3)

The sample correlation coefficient close to 0 indicates a very weak linear relationship between

the path loss and log distance, whereas close to 1 signifies a strong positive linear relationship.

In this study, the simulations are conducted for M= 20 TXRX separations, and the distance md

ranges from 1 km to 20 km. Figure 4.8 depicts the results of simulation runs involving N= 5

scattering objects. This results in a sample correlation coefficient = 0.08. The subsequent

MSE-fit curve leads to a path loss exponent equal to 0.677. On the contrary, Figure 4.9

represents the simulation runs involving N= 5 scattering objects with a different distribution of

locations but instead have a sample correlation coefficient = 0.8. The resulted MSE-fit curve

has a path loss exponent equal to 2.606. This strongly suggests that, as decreases toward 0,

the linear model described by (2.3) may not be always applicable. Otherwise, the linear model

would yield a negative path loss exponent after applying the MSE curve fitting, especially when

the number of scattering objects is small. Since the semi-empirical models all have positive path

loss exponents, it is reasonable to suggest that the number of scattering objects is typically quite

large in a real environment. Hence, the linear model assumption is generally valid.


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3 3.5 4 4.580

90

100

110

120

130

140

Distance log10(d)

Pathloss(dB)

Figure 4.8: Path loss based on computer simulations forN= 5 objects.

= 0.08 and exponent = 0.677.

3 3.5 4 4.580

90

100

110

120

130

140

Distance log10(d)

Pathloss(dB

)

Figure 4.9: Path loss based on computer simulations forN= 5 objects.

= 0.8 and exponent = 2.606 (different object distribution from Figure 4.8)

4.4.2. Unnatural Path-loss Exponents

From Figure 4.5, 4.6 and 4.7 one may notice the unusual result that the path-loss exponents are

less than 2, which is the natural free-space path loss exponent. It is in fact only natural to expect


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the path loss exponent is no less than 2. To seek an explanation for such an unnatural

phenomenon, the computer simulation is adjusted to make the signals scattered from the objects

arrive at the RX coherently, namely the signals will add in phase. To ensure phase coherence of

the multipath signals, the location of scattering object i is so chosen that its distance

parameters )(and)( 22 idid as depicted in Figure 4.10 satisfy the condition in (19) subject to the

constraint in (20).

Ypidid += 2

2])()('[ 22 (4.4)

1)(4

)()2()())((cos1

2

22

222

khonghthesis

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