Statistical Modeling of Small-Scale Fading Channels

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Electronic Theses, Treatises and Dissertations The Graduate School

12-14-2007

Statistical Modeling of Small-Scale FadingChannelsMahinga Leandra HekenoFlorida State University

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Recommended CitationHekeno, Mahinga Leandra, "Statistical Modeling of Small-Scale Fading Channels" (2007). Electronic Theses, Treatises and Dissertations.Paper 4134.

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THE FLORIDA STATE UNIVERSITY

FAMU – FSU COLLEGE OF ENGINEERING

STATISTICAL MODELING OF SMALL-SCALE FADING CHANNELS

By

MAHINGA HEKENO

A Thesis submitted to the Department of Electrical Engineering

in partial fulfillment of the requirements for the degree of

Master of Science

Degree Awarded: Spring Semester, 2008

Copyright © 2007 Mahinga Hekeno

All Rights Reserved

ii

The members of the Committee approve the thesis of Mahinga Hekeno defended on

December 14th, 2007

________________________

Bing W. Kwan

Professor Directing Thesis

________________________

Ming Yu

Committee Member

________________________

Krishna Arora

Committee Member

Approved:

______________________________________

Victor DeBrunner, Chair, Department of Electrical and Computer Engineering

______________________________________

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

The Office of Graduate Studies has verified and approved the above named committee

members.

iii

In Memory of Jennifer Marealle-Hekeno

iv

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my supervisor Dr. Bing W. Kwan for his

valuable guidance and continuous support throughout my graduate program. His tireless

efforts and willingness to share his in-depth knowledge and experience provided me with

essential ingredients for my academic development. Furthermore, I would like to thank

my committee members, Dr. Krishna Arora and Dr. Ming Yu, for their willingness to be

part of the thesis committee and their generous advice and interest.

I would like to say a special thanks to Dr. Primus Mtenga for his financial support that

facilitated the completion of my graduate studies.

I would also like to thank the academic and administrative staff in the Department of

Electrical and Computer Engineering as well as my colleagues at the Information

Processing and Transmission Engineering laboratory (IPTEL); Hung Khong, Hai Hoang,

Ma Xiaoguang, Turgay Koklu, Khue Ngo and Robert Hunter.

Furthermore, I would like to thank all the friends I have met over my years at Florida

State University; especially, Donation Mkubulo, Wolta Shiyo, Dr. Victor Mchuruza,

Doreen Kobelo, Saidi Siuhi, Judith Lwitiko, Cathbert Akaro and Geophrey Mbatta. I

have learned a lot from each of you.

Finally, I would like to thank my family for believing in me.

v

TABLE OF CONTENTS

LIST OF TABLES............................................................................................................ vii

LIST OF FIGURES ......................................................................................................... viii

ABSTRACT...................................................................................................................... xii

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

1.1. Overview............................................................................................................. 1

1.2. Background......................................................................................................... 1

1.3. Scope of Work .................................................................................................... 2

1.3.1. The Communication System.......................................................................2

2. DETERMINATION OF THE PHYSICS-BASED CHANNEL MODEL................. 4

2.1. Multipath Channel Characterization................................................................... 4

2.2. Linear Time-Varying Channel............................................................................ 4

2.2.1. Doppler Shift............................................................................................... 5

2.2.2. Multipath Delay .......................................................................................... 6

2.2.3. The Field Strength Amplitude .................................................................... 6

2.3. Development of the Physics-based channel Model............................................ 9

2.4. The Simulation Model ...................................................................................... 12

2.4.1. Channel Space........................................................................................... 13

2.5. Statistical Properties of the Physics-based channel Model............................... 16

3. CHANNEL TYPE ASSESSMENT.......................................................................... 18

3.1. Channel Parameters of Interest ......................................................................... 18

3.2. Types of Multipath Fading in Wireless Channel .............................................. 20

3.2.1. Effects of Fading as a Result of Time Dispersion of Multipath Channel. 21

3.2.2. Effects of Fading as a Result of Doppler Spread...................................... 21

4. AUTOREGRESSIVE MODEL................................................................................ 24

4.1. Levinson-Durbin Recursion.............................................................................. 25

4.1.1. Steps to Solve Normal Equations for Autocorrelation Method................ 28

5. COMPUTATION AND SIMULATION RESULTS ............................................... 29

5.1. Simulation of Flat Slow-Fading (FSF) Channel ............................................... 31

vi

5.2. Simulation of Flat Fast Fading (FFF) Channel ................................................. 40

5.3. Simulation of Frequency-Selective Slow Fading (FSSF) Channel................... 47

5.4. Simulation of Frequency-Selective Fast Fading (FSFF) Channel .................... 55

6. CONCLUSION AND RECOMMENDATIONS ..................................................... 60

APPENDIX A: Levinson-Durbin parameters................................................................... 61

APPENDIX B: Matlab Codes........................................................................................... 74

BIBLIOGRAPHY............................................................................................................. 83

BIOGRAPHICAL SKETCH ............................................................................................ 84

vii

LIST OF TABLES

Table 3.1: Summary of the properties of Small scale fading............................................22

Table 5.1: Input values – FSF simulation No.1 ................................................................ 31



Table 5.4: Input values – FFF simulation No.1 ................................................................ 40



Table 5.7: Input values – FSSF simulation No.1.............................................................. 47



Table 5.10: Input values – FSFF simulation No.1 ............................................................ 55

Table 5.11: Input values – FSFF simulation No.2 ............................................................ 57

Table A.1: Levinson-Durbin coefficients and ACF parameters of AR signal.................. 61

viii

LIST OF FIGURES

Figure 1.1: Communication system .................................................................................... 3

Figure 2.1: Receiver in the presence of reflecting objects.................................................. 4

Figure 2.2: Doppler shift due to motion of the receiver ..................................................... 5

Figure 2.3: The field strength as a function of distance...................................................... 7

Figure 2.4: Impulse response .............................................................................................. 9

Figure 2.5: Pulse signal with width Tp ............................................................................. 11

Figure 2.6: Problem definition.......................................................................................... 13

Figure 2.7: LOS path and Ground reflection .................................................................... 14

Figure 2.8: Fresnel zone scenario ..................................................................................... 16

Figure 3.1: Different kinds of fading, depending on the relation between the signal and

the channel main parameters [Bla07]. ............................................................ 23

Figure 4.1: Autoregressive Model .................................................................................... 24

Figure 5.1: Physics-based channel model........................................................................ 29

Figure 5.2: AR signal model............................................................................................. 30

Figure 5.3: Simulation Design .......................................................................................... 31

Figure 5.4: FSF received multipath signal (στ = 9.98×10-9s, Tc = 0.0054s, Ts = 4.00×10-

6s, ∆τ = 5 ns)................................................................................................... 32

Figure 5.5: FSF power delay profile (στ = 9.98×10-9s, Tc = 0.0054s, Ts = 4.00×10-6s ∆τ =

5 ns)................................................................................................................. 33

Figure 5.6: FSF – comparison of autocorrelation functions Rx and Rx (MSE = 5.06×10-5,

στ = 9.98×10-9s, Tc = 0.0054s, Ts = 4.00×10-6s, ∆τ = 5 ns) .......................... 34

Figure 5.7: FSF – Typical waveform realized by the AR model( στ = 9.98×10-9s, Tc =

0.0054s, Ts = 4.00×10-6s, ∆τ = 5 ns) .............................................................. 34

Figure 5.8: FSF power delay profile (στ = 1.12×10-7s, Tc = 0.0054s, Ts = 4.00×10-6s, ∆τ

= 5 ns) ............................................................................................................. 35

ix

Figure 5.9: FSF – comparison of autocorrelation functions Rx and Rx (MSE = 5.67×10-4,

στ = 1.12×10-7s, Tc = 0.0054s, Ts = 4.00×10-6s, ∆τ = 5 ns) .......................... 36

Figure 5.10: FSF – Typical waveform realized by the AR model (στ = 1.12×10-7s, Tc =

0.0054s, Ts = 4.00×10-6s, ∆τ = 5 ns) .............................................................. 37

Figure 5.11: FSF power delay profile (στ = 7.58×10-7s, Tc = 0.0054s, Ts = 7.14×10-6s, ∆τ

= 89.2857ns) ................................................................................................... 38

Figure 5.12: FSF – comparison of autocorrelation functions Rx and Rx (MSE = 5.91×10-

5, στ = 7.58×10-7s, Tc = 0.0054s, Ts = 7.14×10-6s, ........................................ 39

Figure 5.13: FSF – Typical waveform realized by the AR model (MSE = 5.91×10-5, στ =

7.58×10-7s, Tc = 0.0054s, Ts = 7.14×10-6s, ∆τ = 89.2857ns)........................ 39

Figure 5.14: FFF power delay profile (στ = 3.80×10-7s, Tc = 4.41×10-4s, Ts = 5.00×10-

4s, ∆τ= 312.5ns).............................................................................................. 40

Figure 5.15: FFF – comparison of autocorrelation functions Rx and Rx (MSE = 4.33×10-

5, στ = 3.80×10-7s, Tc = 4.41×10-4s, Ts = 5.00×10-4s, ∆τ= 312.5ns)............. 41

Figure 5.16: FFF – Typical waveform realized by the AR model (στ = 3.80×10-7s, Tc =

4.41×10-4s, Ts = 5.00×10-4s, ∆τ= 312.5ns).................................................... 42

Figure 5.17: FFF power delay profile (στ = 1.26×10-6, Tc = 4.41×10-4s, Ts = 5.00×10-4s,

∆τ = 312.5ns) .................................................................................................. 43


5, στ = 1.26×10-6s, Tc = 4.41×10-4s, Ts = 5.00×10-4s, ∆τ = 312.5ns)........... 44

Figure 5.19: FFF – Typical waveform realized by the AR model (MSE = 7.06×10-5, στ =

1.26×10-6s, Tc = 4.41×10-4s, Ts = 5.00×10-4s, ∆τ = 312.5ns)....................... 44

Figure 5.20: FFF power delay profile (στ = 4.78×10-6s, Tc = 4.41×10-4s, Ts = 5.00×10-4s,

∆τ = 312.5ns) .................................................................................................. 45


4, στ = 4.78×10-6s, Tc = 4.41×10-4s, Ts = 5.00×10-4s, ∆τ = 312.5ns) ........... 46

Figure 5.22: FFF – Typical waveform realized by the AR model (MSE = 1.50×10-4, στ =

4.78×10-6s, Tc = 4.41×10-4s, Ts = 5.00×10-4s, ∆τ = 312.5ns)....................... 47

Figure 5.23: FSSF power delay profile (στ = 5.05×10-6s, Tc = 0.0054s, Ts = 2.00×10-6s,

∆τ = 10ns) ....................................................................................................... 48

x

Figure 5.24: FSSF – comparison of autocorrelation functions Rx and Rx (MSE =

9.13×10-5, στ = 5.05×10-6s, Tc = 0.0054s, Ts = 2.00×10-6s, ∆τ = 10ns) ....... 49

Figure 5.25: FSSF – Typical waveform realized by the AR model (στ = 5.05×10-6s, Tc =

0.0054s, Ts = 2.00×10-6s, ∆τ = 10ns) ............................................................. 49

Figure 5.26: FSSF power delay profile (MSE = 9.11×10-5, στ = 7.23×10-6s, Tc = 0.0054s,

Ts = 2.00×10-6s, ∆τ = 10ns) ........................................................................... 50


9.11×10-5, στ = 7.23×10-6s, Tc = 0.0054s, Ts = 2.00×10-6s, ∆τ = 10ns)....... 51

Figure 5.28: FSSF – Typical waveform realized by the AR model (MSE = 9.11×10-5, στ

= 7.23×10-6s, Tc = 0.0054s, Ts = 2.00×10-6s, ∆τ = 10ns).............................. 52

Figure 5.29: FSSF power delay profile (MSE = 8.44×10-4, στ = 3.26×10-6s, Tc = 0.002s,

Ts = 2.00×10-6s, ∆τ = 10ns) ........................................................................... 53


8.44×10-4, στ = 3.26×10-6s, Tc = 0.002s, Ts = 2.00×10-6s, ∆τ = 10ns) ......... 54

Figure 5.31: FSSF – Typical waveform realized by the AR model (MSE = 8.44×10-4, στ

= 3.26×10-6s, Tc = 0.002s, Ts = 2.00×10-6s, ∆τ = 10ns) ................................ 54

Figure 5.32: FSFF power delay profile (στ = 2.62×10-6s, Tc = 7.93×10-7s, Ts = 2.00×10-

6s, ∆τ = 10ns).................................................................................................. 55

Figure 5.33: FSFF – comparison of autocorrelation functions Rx and Rx (MSE =

5.48×10-5, στ = 2.62×10-6s, Tc = 7.93×10-7s, Ts = 2.00×10-6s, ∆τ = 10ns) . 56

Figure 5.34: FSFF – Typical waveform realized by the AR model (στ = 2.62×10-6s, Tc =

7.93×10-7s, Ts = 2.00×10-6s, ∆τ = 10ns) ....................................................... 57

Figure 5.35: FSFF power delay profile (στ = 2.98×10-6s, Tc = 6.61×10-7s, Ts = 2.00×10-

6s, ∆τ = 10 ns)................................................................................................. 58

Figure 5.36: FSFF – comparison of autocorrelation functions Rx and Rx ((MSE =

1.43×10-4, στ = 2.98×10-6s, Tc = 6.61×10-7s, Ts = 2.00×10-6s, ∆τ = 10 ns) 59

Figure 5.37: FSFF – Typical waveform realized by the AR model ((MSE = 1.43×10-4, στ

= 2.98×10-6s, Tc = 6.61×10-7s, Ts = 2.00×10-6s, ∆τ = 10 ns) ....................... 59

xi

LIST OF ACRONYMS

AR Autoregressive

FFF Flat Fast Fading

FSF Flat Slow Fading

FSFF Frequency Selective Fast Fading

FSSF Frequency Selective Slow Fading

IEEE Institute of Electrical and Electronics Engineers

LOS Line of sight

MSE Mean Square Error

PCS Personal Communication System

PDP Power Delay Profile

QoS Quality of Service

RMS Root Mean Square

WLAN Wireless Local Area Network

WSSUS Wide Sense Stationarity with Uncorrelated Scattering

xii

ABSTRACT

With the increase of wireless networks, consumers are increasingly aware of the

importance and convenience of wireless technology. Wireless technologies such as

WLANs, mobile phones, blue tooth or PCS rely on a range of mechanisms to provide for

high Quality of Service (QoS), the core of which would be accurate modeling of the

wireless channels.

The radio channel emanates time-variant linear channel characteristics. In this research,

the analysis of the statistics of the underlying channel behavior is investigated using a

developed physics-based channel model that characterizes small-scale fading behavior

the wireless channels. Specifically, we investigate Flat Slow Fading, Flat Fast Fading,

Frequency-Selective Slow Fading and Frequency-Selective Fast Fading propagation

channels.

This thesis will provide for computer simulation of a physics-based channel model to

define the essential channel parameters, and subsequently reproduce the characterized

channel by appropriately utilizing the autoregressive process to remodel the attained

channel data. The principal method for this study is the use of Levinson-Durbin recursion

to build a signal model for channel analysis.

The motivation for this research is, given a set of channel parameters obtained from the

physics-based channel model, the proposed autoregressive signal model can reproduce

the physical channel parameters and accurately predict the nature of small scale fading

present in a channel whether it is Flat Slow Fading, Flat Fast Fading, Frequency-

Selective Slow Fading or Frequency-Selective Fast Fading.

Performance comparisons are then made from the generated physical properties of the

channel with the simulation results of the constructed autoregressive model built by the

use of statistical comparison analysis such as autocorrelation properties to demonstrate

the merits of the approach.

xiii

This manuscript is organized as follows; Chapter one provides an introduction and

background information of communication systems. Chapter two describes random time-

varying channels, different parameters affecting the propagation of signals in the

communication channel; phenomena such as Doppler shift and multipath delay are

discussed. The physics-based channel is developed in chapter two. Chapter three

discusses different parameters that can be used to categorize wireless channels and types

of multipath fading that can happen in a wireless channel. Autoregressive channel

modeling using Levinson-Durbin recursion is discussed in chapter four. Simulation

results of the developed model are provided and discussed in chapter five. Chapter six

gives a conclusion and discusses areas where further studies need to be carried out.

1

CHAPTER ONE

1. INTRODUCTION

1.1. Overview

The emergence of more wireless services requires a good understanding of the radio

channel behavior. To understand wireless communications channel, one needs to be

acquainted with the properties that governs radio propagation. The basic drawback on

wireless communication system is the determination of its channel characteristics on a

particular environment. As the channel is subjected to time varying distortions due to

noise, multi-path fading, interference, etc., these limitations poses a great challenge for

building a reliable communications system. It is important to correctly understand the

radio channel parameters so as to determine its capabilities in handling the transmission

of data in wireless systems.

1.2. Background

The wireless systems are expected to provide multimedia services with transmission

capabilities that are able to handle higher data rates and higher mobility. The reliability of

the channel depends on the understanding of its mechanism and how it behaves given a

specific environment. There have been a lot of publications on wireless channel

modeling. The principles behind them on channel modeling are centered on the use of

deterministic and stochastic approaches to describe the wireless channel parameters.

Deterministic and Empirical Models – Empirical models are channel models that depend

on observation and measurement data of a particular location.

Stochastic Models – Stochastic models use the first and second order statistical

properties of the channels impulse response to characterize the channel behavior.

2

They model the variability properties of channels environment as a series of random

variables.

Although deterministic and empirical channel models are more accurate, they are highly

unavailable and/or expensive to implement [Gol05]. This thesis makes use of the

stochastic approach to model wireless propagation channel.

1.3. Scope of Work

The radio channel which is defined as the area between the transmitter and receiver is

influenced by the properties that govern the electromagnetic waves. These phenomena

cause attenuation of the transmitted signals or what is popular known as fading effect of

the transmission channel. Doppler spread and delay spread are the key phenomena for

multipath fading. Multipath fading is the interference of signal at the receiving antenna

due to many reflected objects found in the propagation paths.

1.3.1. The Communication System

A communication system contains three main subsystems (Figure 1.1): the transmitter,

the channel and the receiver. The main concern of this work is the second subsystem, the

channel. During transmission the signal is affected by the channel’s environment. If the

channel characteristics are known, then it is possible to predict the effects on the

transmitted signal at the receiver.

3

Figure 1.1: Communication system

To determine a channel model, mathematical descriptions of the propagation effect

between the transmitter and receiver must be known. If the channel characteristics are

known, it helps to study and understand the performance of different communication

systems.

The following chapter shows mathematical modeling of the fading channel and its

characterization as a stochastic process.

4

CHAPTER TWO

2. DETERMINATION OF THE PHYSICS-BASED

CHANNEL MODEL

2.1. Multipath Channel Characterization

A signal propagating through a multipath radio channel experiences signal variations due

to changes in delay, amplitude and polarization [Vau03]. The statistical description of the

propagation parameters are derived from the study of the random linear time varying

channel.

2.2. Linear Time-Varying Channel

The properties of a radio propagation channel in a multipath fading environment can be

modeled from the analysis of the linear time varying radio channel as it gives an insight

of the physical properties that characterize the channel behavior such as scatterers and

reflectors that are found in the channel.

Receiver

Figure 2.1: Receiver in the presence of reflecting objects

5

Let the impulse response of the linear time varying multipath channel to be represented

by h(t, τ) and is described below as [Rap02]: -

( ) ( ) ( )( )( )

∑=

− −=tN

nn

tjn tetth n

0

),( ττδατ φ (2.1)

Where;

N(t) – the number of the multipath signal components present in the channel (at

time t)

τn(t) – the delay of the nth multipath signal component

αn(t) – the amplitude of the nth signal multipath component

φn(t) – the phase shift associated with the nth multipath component

The phase element, ( )tnφ contains the contributions from multipath delay, ( )tnτφ and

doppler shift, ( )tndφ of each multipath component and is denoted as:

( ) ( ) ( )tttnn dn φφφ τ += (2.2)

2.2.1. Doppler Shift

A signal experiences a Doppler shift if its frequency changes due to relative motion of the

receiver. A receiver moving with velocity, v and carrier frequency, fc experiences n

number of multipath signals that are reflected and each arrives at an angle θn relative to

the direction of motion (See Figure 2.2).

Figure 2.2: Doppler shift due to motion of the receiver

6

The doppler shift of the nth multipath component, ( )tfnd due to the receiver’s motion is

given by:

( ) ( )λθ tv

tfnd

cos= (2.3)

The receiver’s motion gives rise to phase shifts of each reflected multipath element. As a

result the signal amplitude varies randomly. The angle of arrival is assumed to be

uniformly distributed between [-π, π]. The phase shift associated with the Doppler

frequency is:

∫=t

nd dttfdn

)(2πφ (2.4)

2.2.2. Multipath Delay

Multipath delay, τn(t) is the measure of the time that the nth multipath signal takes to

arrive at the receiver. Delay for individual multipath component at the receiver is

calculated by using their corresponding path length, rn(t) and the speed of light, c.

c

trt n

n

)()( =τ (2.5)

The phase shift related to the delay is

( )tfjn

nce τπτφ 2−= (2.6)

2.2.3. The Field Strength Amplitude

Let the transmitting signal has a transmitting power Pt and gain Gt. The receiving signal

power Pr with gain Gr can be calculated using:

LGGPP rttr

1= (2.7)

The path loss, L is given by:

7

24

=λπd

L (2.8)

Where d is the free space distance in meters from transmitter to the receiver and λ is the

wavelength of the transmitted signal.

The power density, S (W/m2) at a distance d from the transmitting antenna is given by

24 d

GPS tt

π= (2.9)

The field strength of the receiving antenna, Er (V/m) is related to the power density by the

following formula:

π120

2

rES = (2.10)

Therefore from equations (2.8), (2.9) & (2.10); the field strength becomes:

d

GPE tt

r

30= (2.11)

LOStodued

GPE tt

r

30=

tE

Figure 2.3: The field strength as a function of distance

8

If a signal is traveling and encounter an nth scattering object as shown in the above figure,

the field strength, En would be:

'''

30

nn

ttn dd

GPE

+Γ= (2.12)

Where:

• Γ – the reflection coefficient

• '''nn dandd are the distances traveled by signal before and after its

reflection from the nth reflecting object

Reflection coefficient conditions:

1. Depending on the encountered reflecting object, the reflection coefficient of the

reflecting objects ranges from 0 to 1

10 <Γ<⇒ (2.13)

2. Reflection coefficient Γ is related to the absorption coefficient Τ by:

122 =Τ+Γ (2.14)

Lower bound calculations of reflection coefficients of reflecting objects:

Assuming there is 50% absorption during the encounter of the signal with the reflecting

object, from equation (2.14) it follows that the lower bound of the reflection coefficient is

given as:

5.02 =Γ

707.0=Γ⇒ (2.15)

Combining equations (2.13) and (2.15) to determine the reflection coefficient range as:

0.707 < Γ< 1

Consequently;

d

GP

dd

GP tt

nn

tt 3030'''

≤+

Γ (2.16)

9

This implies at the receiver, maximum field strength, Er-max;

d

GPE tt

r

30max =−

(2.17)

Equation (2.17) is equivalent to equation (2.11) and it describes the field strength due to

LOS line.

2.3. Development of the Physics-based channel Model

The received signal y(t) is the convolution of the transmitted signal x(t) and the time

varying impulse response, h(t, τ). By definition, this means: -

( ) ( ) ( )∫∞

∞−−= τττ dthtxty ,

( ) ( ) ( )ττ ,* thxty = (2.18)

( )τ,th

Figure 2.4: Impulse response The equation (2.1) describes the impulse response h(t, τ), hence the received signal from

equation (2.18) becomes:

( ) ( ) ( )( )( )

∑=

− −=tN

nn

tjn txetty n

0

)( ττα φ (2.19)

If the channel is time invariant or can be characterized as having wide sense stationarity

over a small scale distance or over a short-time interval, then the channel impulse

response has the complex form defined as:

10

( ) ( )∑−

=

− −=1

0

N

nn

jn

neh ττδατ φ (2.20)

This simplified version describes the time invariant nature of the channel’s impulse

response. It means that a response at time t to an impulse t- τ is the same as response at

time t+T to an impulse at time t+T- τ [Gol05]. Hence the time varying parameters in the

time varying impulse response becomes constant; therefore,

( ) ( ) ( )τττ hTthth =+= ,, (2.21)

This assumption makes use of the fact that the time delay resolution of the channel

impulse response model accurately resolves every multipath component over the local

area [Rap02].

Using equations (2.3), (2.4), and (2.6), equation (2.22) can equivalently be rewritten as:

( ) ( )n

tfjN

nn

nnc

eth ττδα λθπτπ

−=

+−−

=∑

cos221

0

(2.22)

Similarly, for the time invariant system, the received signal y(t) is the convolution of the

transmitted signal x(t) with the impulse response:

( ) ( ) ( )txetyN

nn

jn

n ∗−=∑−

=

−1

0

ττδα φ

( ) ( )∑−

=

− −=1

0

N

nn

jn xety n ττα φ (2.23)

The received signal corresponds to a transmitted signal x(t), which is a pulse p(t) with

unit amplitude, Tp width, and duty circle D, namely;

( ) ( ) ≤≤

==otherwise

Tttptx

p

0

01

s

p

T

TD =

Where sT denotes the symbol duration.

11

Figure 2.5: Pulse signal with width Tp

For M-ary system, it entails that the symbol rate Rs (symbols/s) is;

M

RR b

s =

Where;

Rb is the bit rate in bits/sec

Hence the symbol duration is denoted as:

ss R

T1=

The pulse is transmitted as a modulated waveform at carrier frequency cf , which may be

represented as;

( ) ( ) tfj cetptx π2=

The received signal has the complex representation;

( ) ( ) ( )twetpety tfjN

nn

jn

cn +−=∑−

=

− πφ τα 21

0

Where;

( ) noise white=tw

The equivalent baseband representation of y(t) is then given by

( ) ( )nj

n

N

nb tpety n τα φ −= −

−

=∑ .

1

0

(2.24)

1

Tp

t

Ts

12

The received signal can be further derived as follows: -

( ) ( )

−= ∑−

=

− tfjN

nn

jn

cn etpety πφ τα 21

0

.Re

( ) ( ) ( )

−⋅−⋅

−= ∑−

=tfjtftpjty ccn

N

nnnnn ππτφαφα 2cos2coscoscosRe

1

0

( )

( )n

N

n

N

ncnncnn

N

ncnn

N

ncnn

tptftfj

tfjtfty

τπφαπφα

πφαπφα

−⋅

+−

+=

∑ ∑

∑∑

−

=

−

=

−

=

−

=

1

0

1

0

1

0

1

0

2sinsin2cossin

2sincos2coscosRe

( )

( )n

N

ncnn

N

ncnn

N

ncnn

N

ncnn

tptftfj

tftfty

τπφαπφα

πφαπφα

−⋅

−+

+=

∑∑

∑∑

−

=

−

=

−

=

−

=

1

0

1

0

1

0

1

0

2cossin2sincos

2sinsin2coscosRe

( ) ( ) ( ) tftptftpty c

xComponentQuadrature

N

nnnnc

xcomponentinphase

N

nnnn

QI

πτφαπτφα 2sinsin2coscos

,

1

0

,

1

0

⋅−⋅+⋅−⋅= ∑∑−

=

−

= 444 3444 21444 3444 21

( ) ( )

( ) ( )∑

∑−

=

−

=

−⋅=

−⋅=⇒

1

0

1

0

sin component, quadrature theand

cos component, inphase The

N

nnnnQ

N

nnnnI

tpty

tpty

τφα

τφα

The envelope of the received signal is described as:

( ) ( ) ( )txtxta QIr22 += (2.25)

2.4. The Simulation Model

Let the transmitted signal, x(t) with bandwidth Bu and carrier frequency fc, be represented

by a pulse whose width is Tp as follows;

( ) ( ){ }tfj cetptx π2Re= (13)

Where;

13

( ) ≤≤

=otherwise

Tttp

p

0

01

p(t) is the equivalent lowpass signal of x(t).

2.4.1. Channel Space

The area of interest is within an ellipsoid defined by region S which surrounds the

transmitter and receiver and is within kth Fresnel zone. See the figure below: -

''nd''

3d

''1d

''2d

'nd

'1d

'2d

'3d

iE

rE

kh

S

0d

Figure 2.6: Problem definition Where;

hk – Fresnel zone defined by index k

d’n – The path distance before the nth reflective object

d’’ n – The path distance after the nth reflective object

14

d0– Distance between transmitter and receiver

Ei – Incident signal’s field strength

Er – Reflected signal’s field strength

The envelope of the received signal strength at the receiver is derived from the equation

(2.25). The received signal strength is the combination of field signal strength due to LOS

path, ground reflected and reflections due to scatterers.

scatterersectedgroundreflLOSreceived EEEE ++=

''1d

'1d

th

rh

0d

g

Figure 2.7: LOS path and Ground reflection

Where;

g – The point of ground reflection

Consequently, the received signal is categorized into three scenarios: -

1. Received Field strength due to the direct signal

( )

+≤≤=

otherwise

Tc

dt

c

dtp p

0

1 00

15

Factors of equation (2.24) are defined as;

Amplitude, α0: 0

0

30

d

GP tt=α

Delay, τ0 : c

d00 =τ

Doppler frequency, 0df :

λθπ 0cos2

0

vf d =

2. Received Field strength due to ground reflection with reflection coefficient, Γg

( )

+

+≤≤

+=

otherwise

Tc

ddt

c

ddA

tp pp

0

2''

1'1

''1

'1

Similarly, factors of equation (2.24) for ground reflection scenario are defined as;

Amplitude, α1: ''1

'1

1

30

dd

GP ttg +

Γ=α

Delay, τ1 : c

dd ''1

'1

1

+=τ

Doppler frequency, 1df :

λθπ 1cos2

1

vfd =

3. Received Field strength due to reflection of multipath components from scatterers

( )

++≤≤+

=otherwise

Tc

ddt

c

ddA

tp pnnnn

p

0

2''''''

Likewise, factors of equation (2.24) for the reflecting objects are defined as;

Amplitude, αn: '''

30

nn

ttnn dd

GP

+Γ=α

16

Delay, τn : c

dd nnn

''' +=τ

Doppler frequency, ndf :

λθπ n

dv

fn

cos2=

Fresnel Zone:

The obstacles are defined up to kth Fresnel zone. The excess path traveled by signal is

equal to 2λk compared to the direct line of sight distance. This means:

2''' λ

kddd nn +=+ ; k=1,2, 3, …

'nd

''nd

obstaclehkh

rh

Figure 2.8: Fresnel zone scenario

2.5. Statistical Properties of the Physics-based channel Model

By definition an autocorrelation function of the random process is given as [Cou01]

( ) 2121212121 ,)()()()(),( dxdxxxftxtxtxtxttR xx ∫ ∫∞

∞−

∞

∞−

=⋅=

17

A random process x(t) is said to be wide sense stationary if

a) Its mean is independent of time

tconstx tan)( =

b) Its autocorrelation function is a function of only time difference, τ.

( ) ( )( ) ( )τxx

xx

RttR

ttRttR

=−=

21

2121

,

,

So for our process, the autocorrelation of the time invariant impulse response is given as

follows as it is independent of time, t:

( ) ( ){ } ( )tRtththE h ∆=∆+⋅ ;,;; 212*

1 ττττ

It also follows that the effect of the channel associated with multipath delay of one

component i.e. τ1 does not have an effect on channel response associated with another

multipath component with delay τ2. As a result, the channel is said to exhibit WSSUS –

Wide Sense stationarity with uncorrelated scattering. The autocorrelation function

becomes:

( ) ( ){ } ( ) ( ) ( )tRtRtththE hh ∆=−∆=∆+⋅ ,;;; 21121* τττδτττ

18

CHAPTER THREE

3. CHANNEL TYPE ASSESSMENT

As we have seen in the developed channel model; the received signal that has been

subjected to time varying channel, is the sum of amplitudes and time delays of the

multipath components that have arrived at the receiver at a particular time instance.

From the signal’s power delay profile, channel parameters such as mean excess delay,

RMS delay spread and excess delay spread are determined. Power delay profile shows

the signal’s strength and arrival time of each multipath component at the receiver. This

profile helps in determining the main parameters of multipath radio channel.

3.1. Channel Parameters of Interest

Channel parameters that describe time dispersive nature of the channel are delay spread

and coherence bandwidth.

1) The mean excess delay τ is the first moment of the power delay profile of the

pulse signal and is defined as [Bla07]:

( )( )∑

∑

∑

∑==

nn

nnn

nn

nnn

P

P

τ

ττ

α

τατ

2

2

2) The RMS delay spread δτ is the square-root of the second moment of the power

delay profile. The RMS delay spread is a measure of time dispersion in the

channel.

( )22 ττσ τ −=

Where:

19

( )( )∑

∑

∑

∑==

nn

nnn

nn

nnn

P

P

τ

ττ

α

τατ

2

2

22

2

3) Coherence bandwidth Bc is the statistical measure of frequencies over which the

channel can be considered “flat” [Rap02]. It describes the point at which two

frequencies of a signal are likely to experience comparable amplitude fading. This

property is related to the rms delay spread as follows;

- For a bandwidth over which the frequency correlation function is above

90% or 0.9, the coherence bandwidth is given as [Bla07];

τσ50

1≈cB

- If a coherence bandwidth Bc describes a bandwidth over which the

frequency correlation function is above 50% or 0.5 then it is approximated

as follows;

τσ5

1≈cB

Parameters describing the varying nature of the channel are Doppler spread and

Coherence time.

1) Doppler spread, BD shows the spectral broadening due to effect of the doppler

shift, fd. It is the effect from the movements of the receiver/reflectors in the

channel. These movements cause the variation of signal’s frequency during

transmission through each multipath. This means signals traveling in different

multipath have different Doppler shift. The Doppler spread shows the spectral

spreading caused by the time rate of change of the wireless propagation channel

due to the relative motions of receiver/reflectors with respect to the base stations

[Bla07].

mD fB =

Where fm is the maximum Doppler shift and is denoted as;

20

λv

fm =

2) Coherence time Tc is inversely related to Doppler spread and is expressed as

follows;

mc f

T1≈

Coherence time is the statistical measure of the time duration over which the

channel impulse response is essentially invariant, and quantifies the similarity of

the channel response at different times [Rap02]. It describes the duration of time

at which two signals at the receiver experience possible amplitude correlation.

- If this coherence time is defined at the time over which the time

correlation function is over 50% or 0.5 then coherence time is denoted as:

mc f

Tπ16

9≈

As a rule of thumb, the coherence time is approximated as a geometric mean

of the two above equation.

mc f

T423.0≈

The analysis of multipath fading channel using these time and frequency dispersion

parameters help to evaluate type of wireless channel that is of concern.

3.2. Types of Multipath Fading in Wireless Channel

In assessing the type of fading that is being experienced in a channel during small scale

fading, the comparison between the nature of the transmitted signal and different channel

parameters need to be examined. The multipath wireless channels are categorized in four

different types; namely, Flat slow fading, Flat fast fading, Frequency-selective slow

fading and Frequency-selective fast fading. These categories stem from the two

independent propagation mechanisms; multipath delay spread and Doppler spread.

21

3.2.1. Effects of Fading as a Result of Time Dispersion of Multipath Channel

The time delay spread due to multipath results in flat or frequency selective channel.

Flat Fading:

A channel is said to convey a flat fading effect if it has a constant gain and linear phase

response over a bandwidth which is more than that of the transmitted signal [Gol02]. Flat

fading channel has its multipath time delay spread δτ much smaller than the transmitted

signal symbol duration Ts and the channels coherence bandwidth Bc is greater than the

transmitted signal’s bandwidth Bs. As a result, flat fading channel affects all frequency

components of a narrow band transmitted signal the same way and hence the signal will

experience the same magnitude of fading at the receiver.

Frequency-Selective Fading:

A wireless channel will reflect frequency-selective fading on the received signal if it has

a constant gain and linear phase response over a bandwidth which is less than the

transmitted signal’s bandwidth. This channel has multipath delay spread that is larger

than the transmitted signal’s symbol duration and its channel coherence bandwidth is

much smaller than the transmitted signal’s bandwidth. As a result, the Different

frequency components of the signal therefore experience decorrelated fading.

3.2.2. Effects of Fading as a Result of Doppler Spread

A transmitted signal changes as compared to the rate of change of the wireless channel. If

the rate of change of the channel’s impulse response is faster or slower compared to the

symbol duration; then the channel may be classified as to be a fast fading or slow fading

channel respectively. The Doppler spread shows this frequency dispersion characteristic

of the channel due to motion.

The motion of the receiver or surrounding objects compared to the properties of the

baseband signal determines how fast or slow the multipath channel will fade.

22

Slow fading

Slow fading occurs when the Doppler spread of the wireless channel is much less

compared to the baseband bandwidth of the transmitted signal and channel’s coherence

time is greater than the transmitted signal symbol duration.

Fast fading

When channel’s Doppler spread is more than baseband bandwidth of the transmitted

signal then the channel undergoes fast fading. This time channel’s coherence time is

smaller than the transmitted signal’s symbol duration.

Table 3.1: Summary of the properties of Small scale fading Signal’s bandwidth is less

than channel’s bandwidth

Flat Fading

Delay spread is less than

symbol period

Signal’s bandwidth is

greater than channel’s

bandwidth

Mul

tipat

h D

elay

Spr

ead

Frequency Selective Fading

Delay spread is greater

than Symbol period

Low Doppler spread

Coherence time is greater

than symbol duration

Slow Fading

Slower channel variations

compared to baseband

signal variations

High Doppler spread

Coherence time is less

than the symbol duration

Dop

pler

Spr

ead

Fast fading

Faster channel variations

compared to baseband

signal variations

23

In order to categorize further the wireless channel Table 3.2 is used to show the summary

of channel assessment by using transmitted baseband signal and channel parameters.

Figure 3.1: Different kinds of fading, depending on the relation between the signal and the channel main parameters [Bla07].

24

CHAPTER FOUR

4. AUTOREGRESSIVE MODEL

Autoregressive model is of the form;

Figure 4.1: Autoregressive Model If p is the model order, an AR process is obtained by filtering white noise w(n) with an

all pole filter of the form [Hay96];

( ) ( ) [ ]nh

zazazazA

zHp

k

kk

pp

↔+

=+++

==

∑=

−∗−−

1

*1*1 1

1

...1

11

Where;

k = 1, 2,…, p

The above expressions are Yule-Walker equations and can be reproduced in matrix form

as follows;

][nx

( )zA

1

][nw

22εσσ =w

White Noise

Output Process

25

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

( )( )

( )

=

−−

−−

0

0

0

1

2

1

1

021

2012

1101

210

**

**

***

MM

L

MOMMM

L

L

L

p

p

p

p

xxxx

xxxx

xxxx

xxxx

pa

a

a

RpRpRpR

pRRRR

pRRRR

pRRRR

ε (4.1)

Equivalently it can be written as;

Rpap =εpu1

The above equations are described as normal equation for autocorrelation method of pth

order and they are a combination of equation (4.2) and (4.3) ;

[ ] ( ) [ ]∑=

=−+p

mxpx mkRmakR

1

0 (4.2)

With the modeling error εp;

[ ] ( ) [ ]∑=

=−+=p

mxpxp mkRmakR

1

0ε (4.3)

Levinson-Durbin recursion can be applied to solve the above linear equations.

4.1. Levinson-Durbin Recursion

Levinson-Durbin recursion solves linear equations such as autocorrelation normal

equations of the form as described in equation (4.1). Levinson-Durbin recursions help in

obtaining the coefficients of the linear equations; ap(1), ap(2),…, ap(p) . The Levinson-

Durbin recursion shows how the solution for jth-order normal equations helps to derive

the (j+1)st-order equations. If aj(i) is the solution to the jth order normal equations then:

26

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

( )( )

( )

=

−−

−−

0

0

0

2

1

1

021

2012

1101

210

**

**

***

MM

L

MOMMM

L

L

L j

j

j

j

xxxx

xxxx

xxxx

xxxx

ja

a

a

RjRjRjR

jRRRR

jRRRR

jRRRR ε

Which is equivalent to:

Rjaj =εju1

Knowing aj enables the derivation of the solution for (j+1)st-order normal equations as

shown below:

Rj+1aj+1 =εj+1u1

The vector aj is appended a zero and the results are multiplied by Rj+1. It follows:

[ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ]

[ ][ ]

[ ][ ]

[ ][ ]

[ ][ ]

[ ][ ]

( )( )

( )

=

+−−

+

−−−

+

j

j

j

j

j

x

x

x

x

x

x

x

x

x

x

xxxxx

xxxxx

xxxxx

ja

a

a

R

R

R

R

jR

jR

jR

jR

jR

jR

jRjRRRR

jRjRRRR

jRjRRRR

γ

ε

0

0

0

0

2

1

1

0

1

1

0

1

21

1

12012

1101

1210

*

**

***

****

MM

L

LMMOMMM

L

L

L

(4.4)

Where:

[ ] ( ) [ ]∑=

−+++=j

ixjxj ijRiajR

1

11γ (4.5)

Equation (4.4) can be rewritten as:

[ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ]

[ ][ ]

[ ][ ]

[ ][ ]

[ ][ ]

[ ][ ]

( )( )

( )

=

−

+−−

+

−−−

+

j

j

j

j

j

x

x

x

x

x

x

x

x

x

x

xxxxx

xxxxx

xxxxx

a

ja

ja

R

R

R

R

jR

jR

jR

jR

jR

jR

jRjRRRR

jRjRRRR

jRjRRRR

ε

γ

0

0

0

0

1

1

1

0

1

1

0

1

21

1

12012

1101

1210

*

**

***

****

MM

L

LMMOMMM

L

L

L

(4.6)

27

By taking the complex conjugate of equation (4.6) and combine the results with equation

(4.4), then the outcome is given by equation (4.7) for any complex constant χj+1.

Rj+1

( )( )

( )

( )( )

( )

+

=

−+

−++

*

*

1

*

*

*

1

0

0

0

0

0

0

1

1

1

1

0

1

1

1

j

j

j

j

j

j

j

j

j

j

j

j

a

ja

ja

a

ja

ja

ε

γ

χ

γ

ε

χMMMM

(4.7)

Setting,

*1

j

jj ε

γχ −=+ (4.8)

Then equation (4.7) becomes:

Rj+1aj+1 =εj+1u1

Where:

aj+1 =

( )( )

( )

( )( )

( )

−+

−+

1

1

1

1

0

1

1

1

*

*

*

1

j

j

j

j

j

j

j

a

ja

ja

a

ja

ja

MMχ (4.7)

This is the (j+1)st order normal equations. In addition εj+1 is the (j+1)st order modeling

error given as:

*11 jjjj γχεε ++ +=

[ ]2

11 1 ++ −=⇒ jjj χεε (4.9)

If we set;

( ) 10 =ja and

28

( ) 011 =++ ja j

Then;

( ) ( ) ( )1*11 +−+= ++ ijajaia jjjj χ

For i=0,1,…,j+1

4.1.1. Steps to Solve Normal Equations for Autocorrelation Method

In order to solve normal equations using the Levinson-Durbin recursion the following

steps need to be considered when using the Levinson-Durbin recursion [Hay96]:

1. The conditions to initiate the Levinson-Durbin recursion are defined using the

solution for the model of order j=0. Hence:

( ) 100 =a

( )00 xR=ε

2. For j=0, 1, …, p -1; the (j+1)st order model is determined using jth order model.

Using the equations below to determine the complex constant χj+1.

[ ] ( ) [ ]∑=

−+++=j

ixjxj ijRiajR

1

11γ

*1j

jj ε

γχ −=+

Then the recursion is updated to determine a j+1(i) from a j(i).

( ) ( ) ( )1*11 +−Γ+= ++ ijajaia jjjj

For i=1,2,..,j

( ) 11 1 ++ Γ=+ jj ja

3. The error εj+1 is then updated using:

[ ]2

11 1 ++ Γ−= jjj εε

The above equation can also be written as:

( ) ( ) ( )∑+

=++ +=

1

111 0

j

ixjxj iRiaRε

29

CHAPTER FIVE

5. COMPUTATION AND SIMULATION RESULTS

The physics-based channel model was simulated to generate data that depict the four

environment categories for wireless channel; Flat Fast Fading, Flat Slow Fading,

Frequency-Selective Fast Fading and Frequency-Selective Slow Fading. The parameters

for each category were examined to confirm the appropriate characteristics of each

environment.

For each scenario, autoregressive model was used to approximate the random channel

parameters generated by the channel models. Comparison was made of the two

autocorrelation functions generated from the data obtained from the physics-based

channel model, and the data obtained from the autoregressive process of each individual

wireless channel category. The two autocorrelation functions for each case describe: -

1. The output of the physics-based channel where p(t) is the input. The

autocorrelation function of the output, y(t) is Ry.

( )tp( )ty( )τ,th

Figure 5.1: Physics-based channel model

30

2. The output of the AR model after being driven by white noise, w[n]. The

autocorrelation function of the output x[n] is Rx.

( )zA

1[ ]nw [ ]nx

Figure 5.2: AR signal model Mean Square Error (MSE) was used to compare the two autocorrelation functions. MSE

is the cumulative squared error between autocorrelation obtained from the channel model

Ry to the one obtained form the AR process Rx. Mathematically, it is described as:

M

mRmR

MSE

M

m

yx

2

1

)()(

−

=∑

=

(5.1)

The following figure shows the simulation design that was used to analyze different

scenarios of small scale fading channels. The results are grouped according to the

considered nature of wireless channel environment.

31

Figure 5.3: Simulation Design

5.1. Simulation of Flat Slow-Fading (FSF) Channel

Simulation 1:

a) Input Values:

Table 5.1: Input values – FSF simulation No.1 Parameter Value

Range, d(m) 1000

Carrier frequency, fc (Hz) 900M

32

Parameter Value

Transmitter height, ht (m) 30

Receiver height, hr (m) 2

Vmax (m/s) 26

Number of obstacles, N 5

Data rate, Rb (bits/s) 1M

Fresnel zone, k=1e2

b) Received signal amplitude at a delay τn:

Figure 5.4: FSF received multipath signal (στ = 9.98×10-9s, Tc = 0.0054s, Ts = 4.00×10-6s, ∆τ = 5 ns)

33

c) Power delay profile:

Power delay profile was obtained from calculations of the power of each

multipath signal, yn received at the receiver by using the following formula.

)(log*10 210 nn yP =

Figure 5.5: FSF power delay profile (στ = 9.98×10-9s, Tc = 0.0054s, Ts = 4.00×10-6s

∆τ = 5 ns)

d) Comparison of autocorrelation functions Rx and Ry:

The Levinson-Durbin solution solved for coefficients from the linear equations

described by the autocorrelation function of the physics based channel. The

coefficients were used to determine the autoregressive process that described an

autocorrelation function similar to the one obtained from the physics based

channel (See Appendix A).

34

Figure 5.6: FSF – comparison of autocorrelation functions Rx and Rx (MSE =

5.06×10-5, στ = 9.98×10-9s, Tc = 0.0054s, Ts = 4.00×10-6s, ∆τ = 5 ns)

e) Typical waveform realized by the AR model:

Figure 5.7: FSF – Typical waveform realized by the AR model( στ = 9.98×10-9s, Tc = 0.0054s, Ts = 4.00×10-6s, ∆τ = 5 ns)

35

Simulation 2:

a) Input Values:


Range, d(m) 1000




Vmax (m/s) 26



Fresnel zone, k=1e3

b) Power delay profile:

Figure 5.8: FSF power delay profile (στ = 1.12×10-7s, Tc = 0.0054s, Ts = 4.00×10-6s, ∆τ = 5 ns)

36

c) Comparison of autocorrelation functions Rx and Ry:






Figure 5.9: FSF – comparison of autocorrelation functions Rx and Rx (MSE =

5.67×10-4, στ = 1.12×10-7s, Tc = 0.0054s, Ts = 4.00×10-6s, ∆τ = 5 ns)

d) Typical waveform realized by the AR model:

37

Figure 5.10: FSF – Typical waveform realized by the AR model (στ = 1.12×10-7s, Tc = 0.0054s, Ts = 4.00×10-6s, ∆τ = 5 ns)

Simulation 3:

a) Input Values:


Range, d(m) 1000




Vmax (m/s) 26



Fresnel zone, k=1e4

38


Figure 5.11: FSF power delay profile (στ = 7.58×10-7s, Tc = 0.0054s, Ts = 7.14×10-6s, ∆τ = 89.2857ns)







39

Figure 5.12: FSF – comparison of autocorrelation functions Rx and Rx (MSE = 5.91×10-5, στ = 7.58×10-7s, Tc = 0.0054s, Ts = 7.14×10-6s, ∆τ = 89.2857ns)


Figure 5.13: FSF – Typical waveform realized by the AR model (MSE = 5.91×10-5, στ = 7.58×10-7s, Tc = 0.0054s, Ts = 7.14×10-6s, ∆τ = 89.2857ns)

40

5.2. Simulation of Flat Fast Fading (FFF) Channel

Simulation 1:

a) Input Values:

Table 5.4: Input values – FFF simulation No.1 Parameter Value

Range, d(m) 1000

Carrier frequency, fc (Hz) 2.4G



Vmax (m/s) 120


Data rate, Rb (bits/s) 8K

Fresnel zone, k=1e4


Figure 5.14: FFF power delay profile (στ = 3.80×10-7s, Tc = 4.41×10-4s, Ts =

5.00×10-4s, ∆τ= 312.5ns)

41







Figure 5.15: FFF – comparison of autocorrelation functions Rx and Rx (MSE =

4.33×10-5, στ = 3.80×10-7s, Tc = 4.41×10-4s, Ts = 5.00×10-4s, ∆τ= 312.5ns)

42


Figure 5.16: FFF – Typical waveform realized by the AR model (στ = 3.80×10-7s, Tc = 4.41×10-4s, Ts = 5.00×10-4s, ∆τ= 312.5ns)

Simulation 2:

a) Input Values:


Range, d(m) 1000




Vmax (m/s) 120


43

Parameter Value


Fresnel zone, k=1e5


Figure 5.17: FFF power delay profile (στ = 1.26×10-6, Tc = 4.41×10-4s, Ts = 5.00×10-4s, ∆τ = 312.5ns)







44

Figure 5.18: FFF – comparison of autocorrelation functions Rx and Rx (MSE =

7.06×10-5, στ = 1.26×10-6s, Tc = 4.41×10-4s, Ts = 5.00×10-4s, ∆τ = 312.5ns)


Figure 5.19: FFF – Typical waveform realized by the AR model (MSE = 7.06×10-5, στ = 1.26×10-6s, Tc = 4.41×10-4s, Ts = 5.00×10-4s, ∆τ = 312.5ns)

45

Simulation 3:

a) Input Values:


Range, d(m) 1000




Vmax (m/s) 120



Fresnel zone, k=1e6


Figure 5.20: FFF power delay profile (στ = 4.78×10-6s, Tc = 4.41×10-4s, Ts = 5.00×10-4s, ∆τ = 312.5ns)

46







Figure 5.21: FFF – comparison of autocorrelation functions Rx and Rx (MSE = 1.50×10-4, στ = 4.78×10-6s, Tc = 4.41×10-4s, Ts = 5.00×10-4s, ∆τ = 312.5ns)

47


Figure 5.22: FFF – Typical waveform realized by the AR model (MSE = 1.50×10-4, στ = 4.78×10-6s, Tc = 4.41×10-4s, Ts = 5.00×10-4s, ∆τ = 312.5ns)

5.3. Simulation of Frequency-Selective Slow Fading (FSSF) Channel

Simulation 1:

a) Input Values:

Table 5.7: Input values – FSSF simulation No.1 Parameter Value

Range, d(m) 1000




48

Parameter Value

Vmax (m/s) 26



Fresnel zone, k=1e4


Figure 5.23: FSSF power delay profile (στ = 5.05×10-6s, Tc = 0.0054s, Ts =

2.00×10-6s, ∆τ = 10ns)





49




9.13×10-5, στ = 5.05×10-6s Tc = 0.0054s, Ts = 2.00×10-6s, ∆τ = 10ns)


Figure 5.25: FSSF – Typical waveform realized by the AR model (στ = 5.05×10-6s, Tc = 0.0054s, Ts = 2.00×10-6s, ∆τ = 10ns)

50

Simulation 2:

a) Input Values:


Range, d(m) 1000




Vmax (m/s) 26



Fresnel zone, k=1e4


Figure 5.26: FSSF power delay profile (MSE = 9.11×10-5, στ = 7.23×10-6s, Tc = 0.0054s, Ts = 2.00×10-6s, ∆τ = 10ns)

51








9.11×10-5, στ = 7.23×10-6s, Tc = 0.0054s, Ts = 2.00×10-6s, ∆τ = 10ns)

52


Figure 5.28: FSSF – Typical waveform realized by the AR model (MSE = 9.11×10-5, στ = 7.23×10-6s, Tc = 0.0054s, Ts = 2.00×10-6s, ∆τ = 10ns)

Simulation 3:

a) Input Values:


Range, d(m) 1000




Vmax (m/s) 26


53

Parameter Value


Fresnel zone, k=1e4


Figure 5.29: FSSF power delay profile (MSE = 8.44×10-4, στ = 3.26×10-6s, Tc = 0.002s, Ts = 2.00×10-6s, ∆τ = 10ns)







54

Figure 5.30: FSSF – comparison of autocorrelation functions Rx and Rx (MSE = 8.44×10-4, στ = 3.26×10-6s, Tc = 0.002s, Ts = 2.00×10-6s, ∆τ = 10ns)


Figure 5.31: FSSF – Typical waveform realized by the AR model (MSE = 8.44×10-4, στ = 3.26×10-6s, Tc = 0.002s, Ts = 2.00×10-6s, ∆τ = 10ns)

55

5.4. Simulation of Frequency-Selective Fast Fading (FSFF) Channel

Simulation 1:

a) Input Values:

Table 5.10: Input values – FSFF simulation No.1 Parameter Value

Range, d(m) 1000

Carrier frequency, fc (Hz) 2400G



Vmax (m/s) 80



Fresnel zone, k=1e6


Figure 5.32: FSFF power delay profile (στ = 2.62×10-6s, Tc = 7.93×10-7s, Ts =

2.00×10-6s, ∆τ = 10ns)

56







Figure 5.33: FSFF – comparison of autocorrelation functions Rx and Rx (MSE =

5.48×10-5, στ = 2.62×10-6s, Tc = 7.93×10-7s, Ts = 2.00×10-6s, ∆τ = 10ns)

57


Figure 5.34: FSFF – Typical waveform realized by the AR model (στ = 2.62×10-6s, Tc = 7.93×10-7s, Ts = 2.00×10-6s, ∆τ = 10ns)

Simulation 2:

a) Input Values:

Table 5.11: Input values – FSFF simulation No.2 Parameter Value

Range, d(m) 1000

Carrier frequency, fc (Hz) 2400G



Vmax (m/s) 80

58

Parameter Value



Fresnel zone, k=1e6


Figure 5.35: FSFF power delay profile (στ = 2.98×10-6s, Tc = 6.61×10-7s, Ts = 2.00×10-6s, ∆τ = 10 ns)







59

Figure 5.36: FSFF – comparison of autocorrelation functions Rx and Rx ((MSE =

1.43×10-4, στ = 2.98×10-6s, Tc = 6.61×10-7s, Ts = 2.00×10-6s, ∆τ = 10 ns)


Figure 5.37: FSFF – Typical waveform realized by the AR model ((MSE = 1.43×10-4, στ = 2.98×10-6s, Tc = 6.61×10-7s, Ts = 2.00×10-6s, ∆τ = 10 ns)

60

CHAPTER SIX

6. CONCLUSION AND RECOMMENDATIONS

This research proposes a systematic procedure to simulate channel data in different

scenarios in small-scale fading environment. It was shown that the simulated channel

data from the physics-based model was then successfully estimated by an AR signal

model for different channel types. This procedure is an efficient way to generate inputs

for other channel estimation or tracking studies.

Typically, the channel data comes from the experimental measurements. However, these

practical measurements require some expensive devices such as transceivers and signal

processing boards. In addition, the information provided not only is it limited by

hardware but also it is bounded by geographic area. This proposed channel model

provides for cost-effective way of generating channel data. The modeling technique used

for this work can facilitates the design of optimal receivers such as that of MIMO

systems.

Recommendations and Future work:

1. Different channel type categories were realized (Flat Slow Fading, Flat Fast

Fading, Frequency-selective Slow Fading and Frequency-selective Fast Fading

channels); however the inputs used to categorize FSFF according to the definition

described in chapter four were not practical. For instance the carrier frequency

was in the order of 10-12. Additional work needs to be done to take into account

the practical realization of FSFF category of channel classification.

2. More work needs to be done to understand how the spatial distribution of

scattering objects and their velocities affect the channel characteristics, namely

the coherence time and RMS delay spread.

3. Finding ways to minimize the order p of the AR model.

61

APPENDIX A: Levinson-Durbin parameters

Table A.1: Levinson-Durbin coefficients and ACF parameters of AR signal

Type of Channel

Simulation No.

FFF 1 0 1 2 3 4 5

LD coefficients 1.00 -

0.62 -

0.04 0.21 -

0.11 -

0.08

ACF parameters 1.00 0.60 0.33 0.12 0.12 0.15 0 1 2 3 4 5 6 7 8

2 LD coefficients 1.00 -

0.57 0.07 -

0.41 0.10 -

0.08 0.20 -

0.20 0.08

ACF parameters 1.00 0.67 0.52 0.56 0.44 0.35 0.27 0.28 0.22 9 10 11 12 13 14 15 16 17

LD coefficients 0.01 0.10 0.04 -

0.02 -

0.19 0.10 -

0.18 0.14 -

0.05

ACF parameters 0.13 0.12 0.09 0.12 0.16 0.13 0.18 0.13 0.12 18 19 20 21 22

LD coefficients 0.00 0.04 0.11 -

0.07 -

0.02

ACF parameters 0.13 0.05 0.02 0.05 0.04 0 1 2 3 4 5 6 7 8

3 LD coefficients 1.00 0.00 -

0.36 -

0.01 -

0.22 -

0.29 0.17 -

0.15 -

0.12

ACF parameters 1.00 0.35 0.55 0.34 0.46 0.50 0.35 0.50 0.34 9 10 11 12 13 14 15 16 17

LD coefficients -

0.07 -

0.10 0.02 0.27 0.14 -

0.03 -

0.17 -

0.06 0.17

ACF parameters 0.47 0.37 0.38 0.25 0.29 0.35 0.37 0.37 0.20 18 19 20 21 22 23 24 25 26

LD coefficients 0.14 0.11 -

0.09 -

0.27 -

0.17 -

0.09 0.07 -

0.09 0.13

ACF parameters 0.28 0.19 0.32 0.30 0.26 0.30 0.20 0.33 0.14 27 28 29 30 31 32 33 34 35

LD coefficients 0.21 -

0.06 -

0.02 0.20 -

0.07 0.02 0.03 -

0.02 -

0.03

ACF parameters 0.17 0.15 0.22 0.19 0.22 0.15 0.14 0.11 0.14

62

3 36 37 38 39 40 41 42 43 44

LD coefficients 0.09 -0.09 -0.08 0.03 -0.02 -0.21

-0.03 0.06 0.04

ACF parameters 0.14 0.13 0.16 0.12 0.14 0.15 0.11 0.14 0.11

45 46 47 48 49 50 51 52 53

LD coefficients 0.11 0.16 -0.12 0.01 0.02 -0.08 0.05 -0.02 -0.05

ACF parameters 0.16 0.11 0.15 0.08 0.09 0.11 0.08 0.13 0.07

54 55 56 57 58 59 60 61 62

LD coefficients 0.03 -0.07 -0.02 0.06 -0.05 0.00 0.05 0.02 0.12

ACF parameters 0.12 0.06 0.12 0.04 0.08 0.05 0.08 0.07 0.05

63 64 65 66 67 68 69 70 71

LD coefficients 0.04 -0.06 0.05 0.03 -0.10 -0.05

-0.07 0.01 -0.04

ACF parameters 0.08 0.05 0.06 0.05 0.08 0.05 0.07 0.05 0.06

72 73 74 75 76 77 78 79 80

LD coefficients 0.08 0.07 -0.02 -0.02 0.00 -0.01 0.09 0.01 -0.06

ACF parameters 0.04 0.04 0.05 0.05 0.05 0.04 0.05 0.05 0.05

81 82 83 84 85 86 87 88 89

LD coefficients -0.04 -0.02 0.00 -0.01 0.01 -0.04

-0.04 0.03 0.08

ACF parameters 0.05 0.04 0.05 0.05 0.05 0.03 0.05 0.04 0.04

90 91 92 93 94 95 96 97 98

LD coefficients 0.01 0.02 0.03 -0.02 -0.02 -0.01 0.02 -0.01 -0.06

ACF parameters 0.04 0.04 0.03 0.02 0.04 0.03 0.03 0.02 0.04

99 100 101 102 103 104 105

LD coefficients 0.02 0.03 0.00 0.03 -0.04 -0.01 0.03

ACF parameters 0.01 0.02 0.02 0.02 0.02 0.01 0.03

FSF 1 0 1 2 3 4 5 6 7 8

LD coefficients 1.00 -0.53 0.28 -0.38 0.33 -0.27 0.27 -0.27 0.00

ACF parameters 1.00 0.38 0.09 0.17 0.00 0.05 0.00 0.13 0.22

63

FSF 2 0 1 2 3 4 5 6 7 8

LD coefficients 1.00 -0.39 0.16 -0.25 -0.10 -0.10 0.03 -0.09 -0.08

ACF parameters 1.00 0.42 0.23 0.38 0.38 0.32 0.27 0.30 0.32

9 10 11 12 13 14 15 16 17

LD coefficients 0.03 -0.01 -0.20 0.18 -0.04 0.09 0.03 0.04 -0.15

ACF parameters 0.23 0.21 0.35 0.21 0.17 0.18 0.17 0.16 0.24

18 19 20 21 22 23 24 25 26

LD coefficients 0.08 0.08 -0.16 0.23 -0.14 -0.09 0.03 -0.13 -0.03

ACF parameters 0.15 0.11 0.19 0.06 0.19 0.23 0.16 0.19 0.20

27 28 29 30 31 32 33 34 35

LD coefficients 0.05 0.03 0.01 -0.04 0.12 -0.02 0.02 0.01 -0.07

ACF parameters 0.14 0.15 0.14 0.16 0.15 0.07 0.15 0.19 0.19

36 37 38 39 40 41 42 43 44

LD coefficients 0.06 -0.03 -0.02 -0.05 -0.09 0.13

-0.18 0.12 -0.03

ACF parameters 0.12 0.15 0.13 0.18 0.21 0.09 0.17 0.14 0.07

45 46 47 48 49 50 51 52 53

LD coefficients 0.13 -0.11 0.08 -0.10 0.08 -0.01 0.01 -0.03 0.10

ACF parameters 0.06 0.15 0.12 0.12 0.04 0.10 0.12 0.07 0.05

54 55 56 57 58

LD coefficients -0.02 0.05 -0.06 0.04 -0.04

ACF parameters 0.01 0.03 0.05 0.07 0.02

3 0 1 2 3 4 5 6 7 8

LD coefficients 1.00 -0.02 0.08 0.06 -0.03 -0.07 0.00 -0.32 -0.01

ACF parameters 1.00 0.20 0.12 0.06 0.17 0.21 0.16 0.24 0.06

9 10 11 12 13 14 15 16 17

LD coefficients -0.41 -0.33 -0.21 -0.02 -0.10 0.10

-0.17 0.09 0.19

ACF parameters 0.35 0.32 0.24 0.04 0.07 0.10 0.22 0.19 0.00

18 19 20 21 22 23 24 25 26

LD coefficients 0.19 0.23 0.18 -0.02 0.19 0.00 0.06 -0.08 0.01

ACF parameters 0.08 0.04 0.14 0.13 0.00 0.02 0.03 0.19 0.01

64

FSF 3 27 28 29 30 31 32 33 34

LD coefficients -0.12 -0.08 -0.09 0.05 -0.02 0.01 -0.08 -0.04

ACF parameters 0.04 0.00 0.00 0.01 0.01 0.01 -0.01 0.10

FSSF 1 0 1 2 3 4 5 6 7 8

LD coefficients 1.00 -0.38 0.14 -0.05 0.02 -0.01 0.00 0.00 0.00

ACF parameters 1.00 0.29 0.00 -0.01 -0.02 0.00 0.00 0.01 0.01

9 10 11 12 13 14 15 16 17

LD coefficients 0.00 0.00 0.00 -0.01 0.02 -0.05 0.11 -0.33 0.05

ACF parameters 0.01 0.01 0.00 -0.01 0.00 -0.01 -0.01 0.25 0.13

18 19 20 21 22 23 24 25 26

LD coefficients 0.01 -0.01 0.01 -0.02 0.03 -0.02 0.01 0.00 0.00

ACF parameters 0.01 0.00 -0.02 -0.02 -0.01 -0.01 -0.02 0.00 0.01

27 28 29 30 31 32 33 34 35

LD coefficients -0.01 0.02 -0.02 0.02 -0.01 0.04 0.05 -0.03 0.01

ACF parameters -0.01 -0.01 0.00 0.00 -0.01 -0.01 0.00 0.01 0.00

36 37 38 39 40 41 42 43 44

LD coefficients -0.02 0.04 -0.05 0.03 -0.02 0.01 -0.01 0.01 0.00

ACF parameters 0.00 -0.01 -0.01 -0.02 -0.01 0.00 0.01 -0.01 0.00

45 46 47 48 49 50 51 52 53

LD coefficients -0.01 0.04 -0.09 0.07 -0.06 0.03 -0.02 0.04 -0.07

ACF parameters 0.00 0.00 0.08 -0.01 0.00 0.00 0.00 0.01 0.06

54 55 56 57 58 59 60 61 62

LD coefficients 0.06 -0.04 0.02 -0.01 0.01 0.00 0.00 -0.01 0.05

ACF parameters 0.00 0.00 0.00 -0.01 0.00 -0.01 -0.01 -0.01 -0.02

63 64 65 66 67 68 69 70 71

LD coefficients -0.22 0.02 0.02 -0.01 0.02 -0.04 0.06 -0.04 0.02

ACF parameters 0.17 0.09 0.01 0.01 -0.01 -0.01 0.00 0.01 0.01

72 73 74 75 76 77 78 79 80

LD coefficients -0.01 0.00 0.00 0.00 -0.01 0.04 -0.07 0.12 0.03

ACF parameters 0.00 0.00 0.01 0.01 0.00 -0.02

-0.01 -0.01 -0.01

65

81 82 83 84 85 86 87 88 89

LD coefficients -0.03 0.02 -0.02 0.06 -0.06 0.02 0.00 0.00 0.00

ACF parameters 0.01 0.02 0.00 -0.01 -0.02 0.00 0.00 -0.01 0.00

90 91 92 93 94 95 96 97 98

LD coefficients 0.00 0.00 0.00 0.00 -0.03 0.02

-0.06 0.03 -0.03

ACF parameters 0.00 0.01 0.00 -0.01 0.03 -0.01

-0.01 0.00 0.02

99 100 101 102 103 104 105 106 107

LD coefficients 0.06 -0.10 0.08 -0.03 0.01 0.00 0.00 0.01 -0.02

ACF parameters 0.01 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00

108 109 110 111 112 113 114 115 116

LD coefficients 0.04 -0.06 0.09 -0.05 0.03 -0.01 0.00 0.02 -0.15

ACF parameters 0.01 0.01 0.01 0.01 0.00 -0.01

-0.01 -0.01 0.14

117 118 119 120 121 122 123 124 125

LD coefficients 0.00 0.02 -0.01 0.00 0.02 -0.02 0.03 -0.02 0.01

ACF parameters 0.07 0.00 0.00 -0.01 0.00 0.00 0.00 0.00 -0.01

126 127 128 129 130 131 132 133 134

LD coefficients 0.02 0.01 0.00 -0.02 0.03 -0.04 0.08 0.03 -0.02

ACF parameters 0.00 0.01 0.00 0.00 -0.01 -0.02

-0.02 -0.01 0.00

135 136 137 138 139 140 141 142 143

LD coefficients 0.00 0.01 -0.02 0.02 -0.01 -0.01 0.03 -0.04 0.00

ACF parameters -0.01 -0.02 -0.02 0.00 0.01 0.01 0.00 0.00 0.00

144 145 146 147 148 149 150 151 152

LD coefficients -0.01 0.01 0.01 -0.04 0.02 -0.03 0.00 0.01 -0.02

ACF parameters 0.00 0.00 -0.01 0.04 -0.01 -0.01 0.01 0.00 0.00

FSSF 2 0 1 2 3 4 5 6 7 8

LD coefficients 1.00

-0.27 0.12

-0.06 0.03

-0.12

-0.05

-0.17 0.02

ACF parameters 1.00 0.30 0.04 0.09 0.08 0.16 0.19 0.24 0.12

9 10 11 12 13 14 15 16 17

LD coefficients 0.02 0.01

-0.03 0.05

-0.13

-0.03

-0.05 0.05 0.02

ACF parameters 0.06 0.10 0.11 0.11 0.22 0.17 0.11 0.02 0.08

66

FSSF 2 18 19 20 21 22 23 24 25 26

LD coefficients

-0.06 0.11

-0.21 0.04

-0.01

-0.12 0.03

-0.07

-0.02

ACF parameters 0.15 0.09 0.26 0.13 0.06 0.14 0.09 0.13 0.13

27 28 29 30 31 32 33 34 35

LD coefficients 0.07 0.08 0.02

-0.02 0.00 0.00 0.04

-0.04 0.01

ACF parameters 0.08 0.04 0.04 0.11 0.13 0.07 0.06 0.08 0.08

36 37 38 39 40 41 42 43 44


-0.03

-0.03

-0.09

-0.06

-0.02

-0.04

-0.05 0.05

ACF parameters 0.07 0.12 0.16 0.12 0.14 0.10 0.05 0.14 0.11

45 46 47 48 49 50 51 52 53


-0.03 0.05

-0.07

-0.12

-0.03 0.04

ACF parameters 0.07 0.03 0.03 0.08 0.05 0.11 0.14 0.07 0.00

54 55 56 57 58 59 60 61 62

LD coefficients

-0.02 0.04 0.03 0.09 0.02 0.02

-0.02

-0.07

-0.03

ACF parameters 0.07 0.07 0.03 0.03 0.05 0.05 0.07 0.15 0.08

63 64 65 66 67 68 69 70 71

LD coefficients 0.07 0.01 0.08 0.00

-0.04 0.01 0.00

-0.11 0.04

ACF parameters 0.02 0.03 0.04 0.04 0.06 0.07 0.05 0.08 0.05

72 73 74 75 76 77 78 79 80

LD coefficients

-0.05 0.05

-0.03

-0.01 0.03 0.03

-0.06

-0.04 0.01

ACF parameters 0.06 0.03 0.10 0.09 0.04 0.03 0.07 0.09 0.06

81 82 83 84 85 86 87 88 89


-0.14 0.05

-0.04 0.02

-0.05 0.00 0.02

ACF parameters 0.01 0.05 0.10 0.07 0.08 0.02 0.06 0.04 0.03

90 91 92 93 94 95 96 97 98

LD coefficients 0.03 0.04 0.05 0.05 0.00

-0.02

-0.01

-0.07

-0.07

ACF parameters 0.05 0.03 0.05 0.02 0.02 0.04 0.02 0.08 0.13

99 100 101 102 103 104 105 106 107

LD coefficients

-0.02 0.01 0.07

-0.08 0.11 0.05

-0.12

-0.02

-0.05

ACF parameters 0.08 0.03 0.01 0.08 0.03 0.03 0.15 0.07 0.02

67

FSSF 2 108 109 110 111 112 113 114 115 116


-0.05

-0.02

-0.07 0.06 0.04

-0.02

-0.05

-0.05

ACF parameters 0.04 0.09 0.09 0.10 0.08 0.04 0.02 0.06 0.07

117 118 119 120 121 122 123 124 125


-0.12 0.01 0.06

-0.02 0.05 0.01 0.02

ACF parameters 0.04 0.07 0.12 0.09 0.01 0.08 0.05 0.01 0.06

126 127 128 129 130 131 132 133 134


-0.01 0.00 0.08

-0.01 0.05

-0.03 0.07

-0.05

ACF parameters 0.06 0.03 0.04 0.00 0.01 0.02 0.01 0.02 0.05

135 136 137 138 139 140 141 142 143


-0.02 0.00 0.04

-0.05

-0.01

-0.04

-0.03

ACF parameters 0.03 0.00 0.04 0.03 0.04 0.06 0.04 0.03 0.02

145 146 147 148 149 150 151 152 153

LD coefficients

-0.06 0.15

-0.08 0.07

-0.04 0.05 0.04

-0.05

-0.03

ACF parameters 0.03 0.02 0.04 0.01 0.07 0.03 0.00 0.01 0.01

154 155 156 157 158 159 160 161 162

LD coefficients

-0.02

-0.09 0.06

-0.03 0.00 0.03

-0.04 0.04

-0.01

ACF parameters 0.06 0.06 0.00 0.02 0.08 0.05 0.06 0.03 0.02

163 164 165 166 167 168 169 170 171


-0.04 0.02

-0.04 0.08 0.00

-0.04 0.00 0.00

ACF parameters 0.01 0.03 0.03

-0.01 0.01 0.02 0.02 0.01 0.01

FSSF 3 0 1 2 3 4 5 6 7 8


-0.30

-0.05

-0.10 0.00 0.02

-0.04

-0.12

-0.28

ACF parameters 1.00 0.39 0.20 0.20 0.17 0.18 0.22 0.30 0.38

9 10 11 12 13 14 15 16 17

LD coefficients 0.07 0.00 0.11 0.00

-0.12

-0.05 0.04

-0.04 0.08

ACF parameters 0.18 0.15 0.10 0.12 0.25 0.23 0.16 0.19 0.11

18 19 20 21 22 23 24 25 26

LD coefficients

-0.04

-0.01

-0.03 0.03 0.07 0.06

-0.02 0.01

-0.08

ACF parameters 0.14 0.15 0.14 0.14 0.10 0.08 0.11 0.08 0.22

68

FSSF 3 27 28 29 30 31 32 33 34 35

LD coefficients

-0.07 0.13

-0.05

-0.11

-0.07 0.02

-0.03

-0.06

-0.01

ACF parameters 0.22 0.06 0.10 0.16 0.15 0.12 0.15 0.22 0.13

36 37 38 39 40 41 42 43 44


-0.02 0.01 0.08 0.05

-0.05

-0.07

ACF parameters 0.03 0.03 0.03 0.12 0.13 0.04 0.05 0.08 0.11

45 46 47 48 49 50 51 52 53


-0.06 0.06

-0.03

-0.02 0.04 0.03 0.00

-0.13

ACF parameters 0.09 0.10 0.10 0.08 0.05 0.06 0.05 0.13 0.20

54 55 56 57 58 59 60 61 62


-0.04 0.07 0.02

-0.12 0.04

-0.21 0.13 0.11

ACF parameters 0.11 0.07 0.07 0.07 0.13 0.10 0.25 0.16 0.05

63 64 65 66 67 68 69 70 71

LD coefficients

-0.06

-0.03 0.03 0.05 0.03 0.06

-0.02

-0.11

-0.08

ACF parameters 0.09 0.06 0.04 0.11 0.09 0.10 0.07 0.10 0.10

72 73 74 75 76 77 78 79 80


-0.11 0.01 0.04 0.05 0.01 0.01

ACF parameters 0.02 0.06 0.10 0.09 0.07 0.04 0.05 0.10 0.05

81 82 83 84 85 86 87 88 89


-0.06 0.00 0.01 0.00 0.09 0.02

-0.10 0.03

ACF parameters 0.01 0.06 0.09 0.08 0.03 0.00 0.08 0.06 0.03

90 91 92 93 94 95 96 97 98


-0.01 0.03

-0.08 0.02

-0.01 0.04

-0.01 0.03

ACF parameters 0.04 0.06 0.05 0.04 0.00 0.03 0.00 0.03 0.00

69

FSSF 3 99 100 101 102 103 104 105 106 107

LD coefficients

-0.05 0.00 0.04

-0.03 0.02 0.03

-0.02 0.01 0.00

ACF parameters 0.01 0.02 0.01 0.02 0.02 0.01 0.04 0.04 0.00

108 109 110 111 112 113 114 115 116


-0.06

-0.03

-0.04 0.04

-0.02 0.03

-0.10

ACF parameters 0.00 0.01 0.03 0.01 0.01 0.06 0.07 0.01 0.03

117 118 119 120 121 122 123 124 125


-0.01 0.02

-0.04

-0.03 0.06 0.07

-0.08

ACF parameters 0.00 0.02 0.03 0.01 0.04 0.06 0.02 0.00 0.01

126 127 128 129 130 131 132 133 134

LD coefficients

-0.04 0.02 0.00

-0.02 0.00 0.01

-0.03 0.00 0.00

ACF parameters 0.04 0.02 0.05 0.06 0.03 0.01 0.03 0.02 0.03

135 136 137 138 139 140 141 142 143


-0.01

-0.03 0.00

-0.01 0.02 0.05 0.03

ACF parameters 0.03 0.05 0.04 0.03 0.02 0.05 0.04 0.01 0.00

144 145 146 147 148 149

LD coefficients

-0.04 0.02 0.02

-0.03 0.01

-0.05

ACF parameters 0.01 0.02 0.02 0.03 0.05 0.03

FSFF 1 0 1 2 3 4 5 6 7 8


-0.37 0.15

-0.06 0.03

-0.03 0.07

-0.07 0.04

ACF parameters 1.00 0.28 0.00 0.01 0.00 0.01 0.01

-0.01

-0.01

9 10 11 12 13 14 15 16 17

LD coefficients

-0.03 0.05

-0.09 0.06

-0.04 0.03

-0.06 0.11

-0.26

ACF parameters

-0.01

-0.01 0.08 0.01 0.01 0.00 0.00 0.00 0.18

70

FSFF 1 18 19 20 21 22 23 24 25 26


-0.01 0.01 0.00 0.00

-0.02 0.00 0.03

-0.09

ACF parameters 0.09 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.06

27 28 29 30 31 32 33 34 35


-0.34 0.08

-0.02 0.00 0.02 0.01

-0.06 0.09

ACF parameters

-0.01 0.24 0.13

-0.01 0.00 0.01 0.00 0.06 0.01

36 37 38 39 40 41 42 43 44

LD coefficients

-0.04 0.00

-0.02 0.05

-0.05 0.06

-0.08 0.14

-0.17

ACF parameters 0.00 0.05

-0.01 0.00 0.01 0.01

-0.01

-0.01

-0.01

45 46 47 48 49 50 51 52 53


-0.05 0.04

-0.03 0.02

-0.05 0.08

-0.05 0.05

ACF parameters

-0.01

-0.01

-0.01 0.00 0.01

-0.01

-0.01 0.00 0.02

54 55 56 57 58 59 60 61 62

LD coefficients

-0.16

-0.02 0.06 0.03

-0.06 0.10

-0.15 0.09

-0.01

ACF parameters 0.17 0.08

-0.01 0.00 0.00 0.00 0.10 0.01

-0.01

63 64 65 66 67 68 69 70 71


-0.04 0.06

-0.02 0.00 0.02

-0.04 0.05

-0.06

ACF parameters 0.01 0.00 0.00

-0.01

-0.01

-0.01 0.01 0.00 0.07

72 73 74 75 76 77 78 79 80


-0.09 0.01 0.00

-0.02 0.03 0.04

-0.07 0.06

ACF parameters 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.00 0.00

81 82 83 84 85 86 87 88 89

LD coefficients

-0.05 0.09 0.00 0.02

-0.06 0.07

-0.05

-0.08

-0.05

ACF parameters 0.00 0.00 0.01 0.00 0.00 0.00 0.01 0.20 0.12

71

FSFF 2 0 1 2 3 4 5 6 7 8


-0.50

-0.01

-0.12 0.15

-0.18

-0.06 0.04

-0.05

ACF parameters 1.00 0.52 0.34 0.29 0.19 0.25 0.27 0.17 0.13

9 10 11 12 13 14 15 16 17


-0.02 0.00

-0.01

-0.01

-0.02 0.02

ACF parameters 0.08 0.04 0.05 0.10 0.11 0.10 0.09 0.11 0.11

18 19 20 21 22 23 24 25 26

LD coefficients

-0.07

-0.06

-0.05 0.15

-0.04

-0.04 0.01

-0.02

-0.03

ACF parameters 0.15 0.18 0.18 0.09 0.15 0.16 0.14 0.17 0.15

27 28 29 30 31 32 33 34 35


-0.02

-0.07

-0.08

-0.01

-0.01 0.19

-0.01

-0.15

ACF parameters 0.07 0.11 0.16 0.17 0.15 0.12 0.02 0.07 0.18

36 37 38 39 40 41 42 43 44


-0.03

-0.17 0.14

-0.18 0.12

-0.14 0.08

ACF parameters 0.10 0.10 0.15 0.17 0.12 0.17 0.12 0.15 0.14

45 46 47 48 49 50 51 52 53

LD coefficients

-0.05 0.08 0.02

-0.02 0.10 0.01

-0.04

-0.05

-0.01

ACF parameters 0.10 0.07 0.04 0.03 0.01 0.01 0.05 0.07 0.05

54 55 56 57 58 59 60 61 62


-0.07 0.15

-0.15 0.10

-0.08 0.06

-0.07 0.12

ACF parameters 0.07 0.08 0.00 0.11 0.11 0.12 0.14 0.13 0.06

63 64 65 66 67 68 69 70 71

LD coefficients

-0.10

-0.02

-0.09 0.05 0.04 0.00

-0.02 0.08 0.06

ACF parameters 0.12 0.13 0.14 0.11 0.06 0.01 0.04 0.01 0.01

72

FSFF 2 72 73 74 75 76 77 78 79 80

LD coefficients

-0.11 0.13

-0.11 0.05

-0.09 0.03

-0.04 0.00 0.06

ACF parameters 0.03 0.02 0.06 0.04 0.05 0.07 0.05 0.07 0.07

81 82 83 84 85 86 87 88 89

LD coefficients

-0.09 0.05 0.06 0.10

-0.11 0.05

-0.13 0.06 0.01

ACF parameters 0.09 0.09 0.06 0.03 0.05 0.04 0.10 0.05 0.06

90 91 92 93 94 95 96 97 98


-0.05

-0.02 0.02

-0.03 0.03 0.07

-0.11 0.07

ACF parameters 0.05 0.04 0.07 0.07 0.08 0.07 0.05 0.05 0.04

99 100 101 102 103 104 105 106 107

LD coefficients

-0.07 0.13

-0.08 0.03

-0.05 0.06 0.01

-0.03 0.00

ACF parameters 0.05 0.02 0.02 0.03 0.03 0.05 0.03 0.04 0.04

108 109 110 111 112 113 114 115 116


-0.02 0.01 0.03 0.00

-0.08 0.02 0.01

ACF parameters 0.01 0.01 0.01 0.01 0.01 0.01 0.00 0.01 0.02

117 118 119 120 121 122 123 124 125


-0.01

-0.04 0.06

-0.05 0.07

-0.02 0.01

-0.04

ACF parameters 0.02 0.02 0.03 0.02 0.03 0.03 0.01 0.03 0.02

126 127 128 129 130 131 132 133 134


-0.06

-0.04 0.03 0.01 0.00

-0.02 0.07

-0.03

ACF parameters 0.02 0.03 0.03 0.02

-0.01

-0.01 0.00 0.00 0.00

135 136 137 138 139 140 141 142 143


-0.02 0.00 0.03

-0.07 0.04

-0.02 0.03

-0.06

ACF parameters

-0.01

-0.01 0.00 0.01 0.03 0.03 0.02 0.03 0.01

73

FSFF 2 144 145 146 147 148


-0.05 0.02

ACF parameters 0.02 0.03 0.05 0.05 0.03

74

APPENDIX B: Matlab Codes

Main Program:

close all

clear all

clc

%************************************************** ********************

%Input parameters

%************************************************** ********************

c=3e8;%speed of light

fc = input('Enter carrier frequency: ');

d = input('Enter the distance between Tx and Rx: ');

ht = input('Enter height of transmitter: ');

hr = input('Enter height of receiver: ');

Rb = input('Enter data rate: ');

l= input('Enter Fresnel zone index: ');

Vmax= input('Enter maximum velocity: ');

K= input('Enter the number of reflecting objects: ');

kk=2; %M-ary signaling kk=log2M

S=51;

A_p=1; %Pulse amplitude

E0=1.55; %sqrt(60*EIRP)- transmitted field strength

g1=0; %no reflection coeffient on direct path

g2=-1; %ground reflection coefficient

75

%************************************************** ********************

%Derived parameters

%************************************************** ********************

lda=c/fc;%wavelength

fm=Vmax/lda;%maximum Doppler frequency

Rs=Rb/kk;%Symbol rate

Ts=inv(Rs);%Symbol duration

M=2^kk;% M-ary signaling

Tp=Ts/M;%pulse width

delta_s=Tp/(S-1);%sampling interval

Ns=(Ts/delta_s)+1;%total samples per symbol duration

d1=unifrnd(0.5*lda,d-(lda/2),K+2,1);%max x-position of the reflecting object

h=sqrt((l*lda.*d1.*(d-d1))/d); % max y-position of the reflecting object

vobs=unifrnd(0,Vmax,1,K+2);

gK=unifrnd(0.7, 1, [K,1]); %reflection coefficient of reflecting objects

gamma=[g1; g2; gK];

D=Tp/Ts; %Duty cycle

%************************************************** ********************

%Received field strength at a LOS path

%************************************************** ********************

tau_n(1)=d/c; %delay due to LOS

E(1)=E0/sqrt(d^2+(ht-hr)^2);%received field strength due to LOS

%************************************************** ********************

%Received field strength due to ground reflection

%************************************************** ********************

Rg=d*ht/(ht+hr); %Rg is the point of reflection at the ground

76

dd1(2)=sqrt(ht^2+Rg^2); %distance travelled by incident signal

dd2(2)=sqrt(hr^2+(d-Rg)^2); %distance travelled by reflected signal

tau_n(2)=(dd1(2)+dd2(2))/c; %delay due to ground reflection

E(2)=gamma(2)*E0/(dd1(2)+dd2(2));%received field strength due to ground

%reflection

% ************************************************* ********************

%Received field strength due to reflecting objects in the space

% ************************************************* ********************

for n=3:K+2

hobs(n) = unifrnd(0,h(n),1,1);

dd1(n)=sqrt((d1(n))^2+(hobs(n)-ht)^2); %transmitter-obstacle distance

dd2(n)=sqrt((d-d1(n))^2+(hobs(n)-hr)^2); %obstacle - receiver distance

E(n)=E0*gamma(n)/(dd1(n)+dd2(n)); %field strength amplitude

tau_n(n)=(dd1(n)+dd2(n))/c; %delay due to reflectin objects

end

tau=tau_n; %delay vector

EdB=10*log10(E); % field strength amplitude in dB

% ************************************************* ********************

% Received pulses at each multipath delay

% ************************************************* ********************

for j=1:20

Tdist=[d dd2(2:end)]; %array of distance after reflection

v=[Vmax vobs(2:end)];

theta=rand(1,K+2)*2*pi-pi; %angle of arrival at the terminal

alpha=E; %each path field strength amplitude

77

delay=2*pi*fc*tau; %delay of each multipath

doppler=(2*pi*(v.*cos(theta)*fc)/c).*Tdist;

tm=1:Ns*40;

for k=1:K+2

for m=1:Ns*40

tlower(k)=tau(k); %lower bound of time scale after delay

t(k,m)=tlower(k)+(m-1)*Ts/(Ns-1); %time steps associated with each delay

if m < round(tlower(k)/delta_s)

rt_in(k,m)=0;

rt_q(k,m)=0;

rt_env(k,m)=0;

elseif m==round(tlower(k)/delta_s)% & m<=round(D*Ns)

rt_in(k,m)=alpha(k)*cos(delay(k)+doppler(k));

rt_q(k,m)=alpha(k)*sin(delay(k)+doppler(k));

rt_env(k,m)=sqrt(rt_in(k,m)^2+rt_q(k,m)^2);

p(k,m)=A_p; %first part of input pulse

else

rt_in(k,m)=0;

rt_q(k,m)=0;

rt_env(k,m)=0;

p(k,m)=0*A_p; %second part of input pulse

end

end

end

rt_sum=sum(rt_env);

RP(j,:)=rt_sum;

end

t_scale=round(tlower/delta_s);

RP_mean=mean(RP);

78

ReceivedPulse=RP_mean((min(t_scale)):(max(t_scale)));

time=tm(min(t_scale):max(t_scale));

RPpwr=ReceivedPulse.^2;

%************************************************** ********************

%Autocorrelation of the generic data

%************************************************** ********************

[nom_A_C,lags]=xcorr(ReceivedPulse,'coeff'); %normalized ACF of the received

signal

%************************************************** ********************

%Building an autoregressive model

%************************************************** ********************

length=size(lags);

mid_pt=round(length(2)/2);

for i=1:20

AC_GENERIC=nom_A_C(mid_pt:end); %Half side of the ACF from generic

model

[AR_Coeff, var]=rtoa(AC_GENERIC); %AR coefficients from the ACF_gen +

variance

noise=sqrt(var)*randn(1000,1); %noise as input to AR model

wgnfil=filter(1,AR_Coeff,noise); %AR model driven by noise with variance, var

[filcoeff,error_LD]=acm(wgnfil,mid_pt-1); %filter coefficient of signal reproduced

by AR model

AC_AR=ator(filcoeff); %Half side of the ACF of AR reproduced signal

flip_AC_AR=fliplr(AC_AR);

AC_AR_fct_Full_length=[flip_AC_AR AC_AR(2:end)];

AC_AR_fct_matrix(i,:)=AC_AR_fct_Full_length;

end

AC_AR_fct=sum(AC_AR_fct_matrix)/i; %ACF of AR reproduced signal

79

t_lag=sort(tau/delta_s);

%************************************************** ********************

%Mean Square Error Calculations

%************************************************** ********************

vect_error=(nom_A_C-AC_AR_fct).^2; %square error of the two ACFs

mse=mean(vect_error) %Mean square error

%************************************************** ********************

% RMS delay spread and coherence time

%************************************************** ********************

mean_excessdelay=sum(RPpwr.*time*delta_s)/sum(RPpwr); %mean excess delay

MEDsquare=sum((delta_s*time).^2.*RPpwr)/sum(RPpwr); %mean square excess

delay

rms_delay_spread=sqrt(MEDsquare-mean_excessdelay^2) %rms delay spread

T_c=0.423/fm %channel's coherence time

Ts %symbol duration

delta_tau=delta_s*1e9

AR_Coeff

ACF=AC_AR_fct'

%************************************************** ********************

%Different Plots

%************************************************** ********************

figure(1);

stem(time,ReceivedPulse*1e3)

xlabel('Time in units of \Delta\it\tau')

80

ylabel('Received signal amplitude (mV)')

figure(2);

stem(time,10*log10(RPpwr))

xlabel('Time in units of \Delta\it\tau')

ylabel('Received signal power (dB)')

figure(3);

stem(lags,nom_A_C,'b*')

hold on

stem(lags,AC_AR_fct,'ro')

xlabel('Time lag in units of \rm\Delta\it\tau ')

ylabel('Autocorrelation function')

legend('Channel Model','AR Model')

figure(4);

stem(wgnfil(950:end))

xlabel('Sample time in units of \rm\Delta\it\tau ')

ylabel('Received signal(mV)')

Levinson-Durbin recursion: From autocorrelation function to AR coefficients [Hay96]

function [a,epsilon]=rtoa(r)

%RTOA Levinson-Durbin recursion.

%----

%USAGE: [a,epsilon]=rtoa(r)

%

% Solves the Toeplitz normal equations

% R a = epsilon [1 0 ... 0]'

% where R=toeplitz(r) is a Toeplitz matrix that contains

81

% the autocorrelation sequence r(k). The ouputs are the

% coefficients of the all-pole model a(k) and the constant

% epsilon.

%---------------------------------------------------------------

r=r(:);

p=length(r)-1;

a=1;

epsilon=r(1);

for j=2:p+1;

gamma=-r(2:j)'*flipud(a)/epsilon;

a=[a;0] + gamma*[0;conj(flipud(a))];

epsilon=epsilon*(1 - abs(gamma)^2);

end

Inverse Levinson-Durbin recursion: From AR coefficients to autocorrelation function

[Hay96]

function r=ator(a,b)

%ATOR Inverse Levinson-Durbin recursion.

%----

%USAGE: r=ator(a,b)

% Finds the autocorrelation sequence r(k) of an

% autoregressive process that is generated by

% filtering unit variance white noise with the filter

% H(z)=b(0)/A(z)

% With r = ator(a) the autocorrelation sequence is

% normalized so that r(0)=1.

%

% see also ATOG, GTOA, GTOR, RTOA, RTOG

82

%---------------------------------------------------------------

% copyright 1996, by M.H. Hayes. For use with the book

% "Statistical Digital Signal Processing and Modeling"

% (John Wiley & Sons, 1996).

%---------------------------------------------------------------

p=length(a)-1;

gamma=atog(a);

r=gtor(gamma);

if nargin == 2,

r = r*b^2/prod(1-abs(gamma).^2);

end;

83

BIBLIOGRAPHY

[You00] David J. Young, ‘The Generation of Correlated Rayleigh Random Varietes by Inverse Discrete Fourier Transform’, IEEE Transactions on Communications, Vol 48, no. 7, July 2000

[Bad05] Kareem E. Baddour, Norman C. Beaulieu, ‘Autoregressive Modeling for

Fading Channel Simulations’, IEEE Trans on Wireless Communications, Vol. 4, no. 4, July 2005

[Mat05] Suhas Mathur, ‘Small Scale Fading’, Rutgar University, 2005 [Hay96] Monson H, Hayes, Statistical Digital Signal Processing and Modeling,

John wiley & Sons, Inc., New Jersey, 1996 [Pap91] A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd

Edition, McGraw Hill New York, 1991 [Vau03] Vaughan, Rodney and Anderson, Jørgen Bach, Channels, Propagation

and Antennas for Mobile Communications, Institute of Electical Engineers, 2003

[Gol05] Andrea Goldsmith, Wireless Communications. Cambridge University

Press, 2005 [Rap02] T. S. Rappaport, Wireless Communications, Principles and Practice,

Prentice Hall, New Jersey, 1996 [Ath 00] G. E. Athanasiadou, A. R. Nix, and J. P. McGreehan, “A microcellular

ray- tracing propagation model and evaluation of its narrowband and wideband predictions,” IEEE Journal on Selected Areas in Communications, Wireless Communications series, vol. 18, pp. 322-335, March 2000

[Cou01] Leon W. Couch, II, Digital and Analog Communication systems, Prentice

Hall, New Jersey, 2001 [Bla07] Nathan Blaunstein, Christos G. Christodoulou, Radio Propagation and

Adaptive Antennas for Wireless Communications Links, John Wiley & Sons, Inc., New Jersey, 2007

84

BIOGRAPHICAL SKETCH

Mahinga Hekeno was born and raised in Dar-es-Salaam, Tanzania. She is currently a

Master’s student of Electrical and Computer Engineering at Florida State University. Her

Master’s thesis research is on statistical modeling of small-scale fading channels, and is

under the direction of Professor Bing W. Kwan.

She has been a master’s student in Electrical and Computer Engineering at Florida State

University since January 2006. She has also held a Bachelors of Science degree in

Electrical and Electronic Engineering from Department of Electrical and Computer

Engineering of California State University, Sacramento.

Documents

Statistical Modeling of Small-Scale Fading Channels