Statistical Analysis and Reduction of Multiple Access...

Statistical Analysis and Reduction ofMultiple Access Interference in

MC-CDMA Systems

Xuan Li

Faculty of Built Environment and Engineering

Queensland University of Technology

A thesis submitted for the degree of

Doctor of Philosophy

2008

Abstract

Multicarrier code division multiple access (MC-CDMA) is a very promising

candidate for the multiple access scheme in fourth generation wireless communi-

cation systems. During asynchronous transmission, multiple access interference

(MAI) is a major challenge for MC-CDMA systems and significantly affects their

performance. The main objectives of this thesis are to analyze the MAI in asyn-

chronous MC-CDMA, and to develop robust techniques to reduce the MAI effect.

Focus is first on the statistical analysis of MAI in asynchronous MC-CDMA.

A new statistical model of MAI is developed. In the new model, the derivation

of MAI can be applied to different distributions of timing offset, and the MAI

power is modelled as a Gamma distributed random variable. By applying the new

statistical model of MAI, a new computer simulation model is proposed. This

model is based on the modelling of a multiuser system as a single user system

followed by an additive noise component representing the MAI, which enables

the new simulation model to significantly reduce the computation load during

computer simulations.

MAI reduction using slow frequency hopping (SFH) technique is the topic of

the second part of the thesis. Two subsystems are considered. The first sub-

system involves subcarrier frequency hopping as a group, which is referred to as

GSFH/MC-CDMA. In the second subsystem, the condition of group hopping is

dropped, resulting in a more general system, namely individual subcarrier fre-

quency hopping MC-CDMA (ISFH/MC-CDMA). This research found that with

the introduction of SFH, both of GSFH/MC-CDMA and ISFH/MC-CDMA sys-

tems generate less MAI power than the basic MC-CDMA system during asyn-

chronous transmission. Because of this, both SFH systems are shown to outper-

form MC-CDMA in terms of BER. This improvement, however, is at the expense

of spectral widening.

In the third part of this thesis, base station polarization diversity, as another

MAI reduction technique, is introduced to asynchronous MC-CDMA. The com-

bined system is referred to as Pol/MC-CDMA. In this part a new optimum com-

bining technique namely maximal signal-to-MAI ratio combining (MSMAIRC) is

proposed to combine the signals in two base station antennas. With the applica-

tion of MSMAIRC and in the absents of additive white Gaussian noise (AWGN),

the resulting signal-to-MAI ratio (SMAIR) is not only maximized but also in-

dependent of cross polarization discrimination (XPD) and antenna angle. In

the case when AWGN is present, the performance of MSMAIRC is still affected

by the XPD and antenna angle, but to a much lesser degree than the traditional

maximal ratio combining (MRC). Furthermore, this research found that the BER

performance for Pol/MC-CDMA can be further improved by changing the angle

between the two receiving antennas. Hence the optimum antenna angles for both

MSMAIRC and MRC are derived and their effects on the BER performance are

compared. With the derived optimum antenna angle, the Pol/MC-CDMA system

is able to obtain the lowest BER for a given XPD.

Contents

1 Introduction 1

1.1 Need for multicarrier code division multiple access . . . . . . . . . 2

1.1.1 Frequency division multiple access . . . . . . . . . . . . . 2

1.1.2 Time division multiple access . . . . . . . . . . . . . . . . 3

1.1.3 Code division multiple access . . . . . . . . . . . . . . . . 4

1.1.4 Orthogonal frequency division multiplexing . . . . . . . . . 6

1.1.5 Multicarrier code division multiple access . . . . . . . . . . 8

1.2 Multiple access interference problem in MC-CDMA . . . . . . . . 9

1.3 Thesis objectives and contributions . . . . . . . . . . . . . . . . . 13

1.4 Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . 15

2 Statistical analysis of MAI in asynchronous MC-CDMA systems 19

2.1 Problems with the existing statistical analysis of MAI in asyn-

chronous MC-CDMA systems . . . . . . . . . . . . . . . . . . . . 20

2.2 Asynchronous MC-CDMA model . . . . . . . . . . . . . . . . . . 22

2.2.1 Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.2 Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Derivation of test statistic . . . . . . . . . . . . . . . . . . . . . . 24

iii

CONTENTS

2.4 Statistical modelling of MAI and its power . . . . . . . . . . . . . 30

2.5 Mean and variance of MAI power with different distributions of

timing offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.6 Applications of the new developed MAI model: A tool for analyz-

ing the effect on BER . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.6.1 Theoretical derivation of BER . . . . . . . . . . . . . . . . 39

2.6.2 Simulation results of the effects on BER . . . . . . . . . . 41

2.7 Application of the new developed MAI model: computer simulations 43

3 Multiple access interference reduction techniques 49

3.1 Spreading sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Multiuser detection . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.1 Minimum mean-square error detector . . . . . . . . . . . . 51

3.2.2 Subtractive interference cancellation . . . . . . . . . . . . 53

3.3 Slow frequency hopping . . . . . . . . . . . . . . . . . . . . . . . . 54

4 Group subcarrier frequency hopping MC-CDMA 57

4.1 Asynchronous GSFH/MC-CDMA model . . . . . . . . . . . . . . 58

4.2 MAI analysis in different detection scenarios . . . . . . . . . . . . 61

4.2.1 Scenario A . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2.2 Scenario B . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2.3 Scenario C . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2.4 Scenario D . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3 Average overall MAI Power . . . . . . . . . . . . . . . . . . . . . 73

4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.5 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

iv

CONTENTS

5 Individual subcarrier frequency hopping MC-CDMA 79

5.1 Asynchronous ISFH/MC-CDMA model . . . . . . . . . . . . . . . 80

5.2 MAI analysis for ISFH/MC-CDMA . . . . . . . . . . . . . . . . . 82

5.3 Special case: GSFH/MC-CDMA . . . . . . . . . . . . . . . . . . . 86

5.4 MAI power comparison between asynchronous MC-CDMA and

asynchronous ISFH/MC-CDMA . . . . . . . . . . . . . . . . . . . 87

5.5 Bit error rate analysis . . . . . . . . . . . . . . . . . . . . . . . . 88

5.6 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.6.1 MAI power ratio . . . . . . . . . . . . . . . . . . . . . . . 90

5.6.2 Bit error rate . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6 Asynchronous MC-CDMA with base station polarization diver-

sity 95

6.1 Polarization diversity versus other diversity techniques . . . . . . 96

6.2 The mechanism of polarization diversity . . . . . . . . . . . . . . 97

6.3 Two factors affecting polarization diversity . . . . . . . . . . . . . 98

6.3.1 Correlation coefficient . . . . . . . . . . . . . . . . . . . . 99

6.3.2 Cross polarization discrimination (XPD) . . . . . . . . . . 100

6.4 Diversity combining technique . . . . . . . . . . . . . . . . . . . . 101

6.4.1 Selection combining . . . . . . . . . . . . . . . . . . . . . . 102

6.4.2 Maximal ratio combining . . . . . . . . . . . . . . . . . . . 102

6.4.3 Equal gain combining . . . . . . . . . . . . . . . . . . . . . 103

6.5 Base station polarization diversity reception model and its appli-

cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

v

CONTENTS

6.6 System model for Pol/MC-CDMA system . . . . . . . . . . . . . 106

6.7 Derivation of the test statistic for Pol/MC-CDMA . . . . . . . . . 109

6.8 Maximal signal-to-MAI ratio combining . . . . . . . . . . . . . . . 113

6.8.1 Performance analysis of Pol/MC-CDMA in the presence of

both AWGN and MAI . . . . . . . . . . . . . . . . . . . . 116

6.8.2 Results and Discussions for MSMAIRC . . . . . . . . . . . 120

6.9 Optimum antenna angle . . . . . . . . . . . . . . . . . . . . . . . 124

6.9.1 Optimum antenna angle for MRC . . . . . . . . . . . . . . 127

6.9.2 Optimum antenna angle for MSIRC . . . . . . . . . . . . . 127

6.9.2.1 Special Case 1 . . . . . . . . . . . . . . . . . . . 128

6.9.2.2 Special Case 2 . . . . . . . . . . . . . . . . . . . 128

6.9.3 Results and discussion for the antenna angle effect . . . . . 128

7 Conclusions and future works 131

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.2 Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Appendices 136

A Statistical modeling of sum of correlated Rayleigh random vari-

ables 139

B Proof of (5.15) 145

C Proof of (5.20) 147

D Proof of (6.27) and (6.28) 151

vi

List of Figures

1.1 Frequency division multiple access (FDMA) [1] . . . . . . . . . . . 2

1.2 Time division multiple access (TDMA) [1] . . . . . . . . . . . . . 3

1.3 Code division multiple access (CDMA) [1] . . . . . . . . . . . . . 5

1.4 MAI generation in asynchronous MC-CDMA systems . . . . . . . 10

1.5 Effect of MAI on the performance of asynchronous MC-CDMA

system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Asynchronous MC-CDMA transmitter and receiver model . . . . 22

2.2 Effective timing offset scenarios . . . . . . . . . . . . . . . . . . . 28

2.3 PDF of MAI in asynchronous MC-CDMA (with N=K=16, uni-

formly distributed τk) . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4 PDF of MAI in asynchronous MC-CDMA (with N=K=16, expo-

nentially distributed τk) . . . . . . . . . . . . . . . . . . . . . . . 31

2.5 PDF of MAI in asynchronous MC-CDMA (with N=K=16, Gaus-

sian distributed τk) . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.6 PDF of MAI power for asynchronous MC-CDMA (N=K=8, uni-

formly distributed τ e) . . . . . . . . . . . . . . . . . . . . . . . . . 34

vii

LIST OF FIGURES

2.7 PDF of MAI Power for asynchronous MC-CDMA (N=32 K=8,

Gaussian distributed τ e) . . . . . . . . . . . . . . . . . . . . . . . 34

2.8 BER comparison between uniformly distributed τk and exponen-

tially distributed τk with SNR = 10dB . . . . . . . . . . . . . . . 41

2.9 BER comparison between uniformly distributed τk and Gaussian

distributed τk with SNR = 10dB . . . . . . . . . . . . . . . . . . 42

2.10 Proposed asynchronous MC-CDMA simulation model . . . . . . . 43

2.11 MAI noise generator for asynchronous MC-CDMA . . . . . . . . . 44

4.1 Asynchronous GSFH/MC-CDMA transmitter and receiver model 58

4.2 Frequency spectrum of asynchronous GSFH/MC-CDMA signals . 59

4.3 Asynchronous GSFH/MC-CDMA detection interval with length of

2Nh symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4 Asynchronous GSFH/MC-CDMA detection scenario A . . . . . . 64

4.5 Asynchronous GSFH/MC-CDMA dectection scenario B . . . . . . 69

4.6 Asynchronous GSFH/MC-CDMA detection scenario C . . . . . . 72

4.7 Asynchronous GSFH/MC-CDMA detection scenario D . . . . . . 73

4.8 GSFH/MC-CDMA MAI power ratio for different number of sub-

carrier frequency groups (N=K=16) . . . . . . . . . . . . . . . . . 76

4.9 GSFH/MC-CDMA MAI power ratio for different spreading factors 76

5.1 Asynchronous ISFH/MC-CDMA transmitter and receiver Model . 80

5.2 MAI power ratios (as a percentage) for asynchronous ISFH/MC-

CDMA systems with different values of Q and N . . . . . . . . . 90

5.3 MAI power ratio for asynchronous GSFH/MC-CDMA and ISFH/MC-

CDMA systems for N=K=8 . . . . . . . . . . . . . . . . . . . . . 91

viii

LIST OF FIGURES

5.4 Bit error rate performance for asynchronous ISFH/MC-CDMA

with (N = K = 16 and the correlation coefficient for subcarrier

fading is 0.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.1 Two-branch receiver model for base station . . . . . . . . . . . . . 107

6.2 MC-CDMA receiver with polarization diversity . . . . . . . . . . 109

6.3 BER comparison between Pol/MC-CDMA with MRC and MC-

CDMA (with α = π/4; δr = 0 and N = K = 16) . . . . . . . . . 118

6.4 BER comparison between Pol/MC-CDMA with MSMAIRC and

MC-CDMA (with α = π/4; δr = 0 and N = K = 16) . . . . . . . 119

6.5 BER comparison between Pol/MC-CDMA with MSMAIRC and

MRC (with α = π/4; δr = 0 and N = K = 16) . . . . . . . . . . . 119

6.6 Threshold XPD for Pol/MC-CDMA with MSMAIRC and MRC

(with α = π/4; δr = 0 and N = K = 16) . . . . . . . . . . . . . . 120


given different number of users (α = π/4; δr = 0 and N = 16) . . 123


for 2 users (α = π/4; δr = 0 and N = 16) . . . . . . . . . . . . . . 124





6.11 Effect of antenna angle when Eb/N0 = 10dB . . . . . . . . . . . . 129

6.12 Effect of antenna angle when Eb/N0 = 30dB . . . . . . . . . . . . 129

ix

LIST OF FIGURES

A.1 Approximation of a sum of correlated Rayleigh random variables

(N=16 and correlation coefficient = 0.7) . . . . . . . . . . . . . . 142

A.2 Approximation of a sum of Rayleigh random variables (8 correlated

Rayleigh random variables + 8 independent Rayleigh random vari-

ables and correlation coefficient = 0.7) . . . . . . . . . . . . . . . 142

x

List of Tables

2.1 Distribution fitting for conditional MAI power in asynchronous

MC-CDMA: Kullback-Leibier divergence values . . . . . . . . . . 33

2.2 Pair T-test results for BER samples obtained in the mathematical

model and the proposed model (System 1) . . . . . . . . . . . . . 46

2.3 Pair T-test results for BER samples obtained in the mathematical

model and the proposed model (System 2) . . . . . . . . . . . . . 47

A.1 Table of Fitting Distributions . . . . . . . . . . . . . . . . . . . . 140

A.2 Fitting Result of Correlated B (Kullback-Leibier divergence values) 141

xi

Acronyms

4G The fourth generation of mobile

communication systems

AWGN Additive white Gaussian noise

BER Bit error rate

bps Bits per second

BPSK Binary phase shift keying

BS Base station

CDMA Code division multiple access

DS-CDMA Direct-sequence CDMA

EGC Equal gain combining

FDMA Frequency division multiple access

FFT Fast Fourier transform

GSFH/MC-CDMA Group subcarrier frequency hopping

MC-CDMA

Hpol Horizontal polarization

ICI Inter-channel interference

i.i.d. Independent and identically distributed

ISFH/MC-CDMA Individual subcarrier frequency hopping

MC-CDMA

ISI Inter-symbol interference

LOS Line of sight

MAI Multiple access interference

MC-CDMA Multicarrier CDMA

MC-DS-CDMA Multicarrier DS-CDMA

MMSE Minimum mean-square error

MRC Maximal ratio combining

MS Mobile subscriber

MSMAIRC Maximal signal-to-MAI ratio combining

MUD Multiuser detection

NLOS No line-of-sight

OFDM Orthogonal frequency division multiplexing

PDF Probability density function

SFH Slow frequency hopping

SMAIR Signal-to-MAI ratio

SMAINR Signal-to-MAI-plus-noise ratio

SNR Signal-to-noise ratio

TDMA Time division multiple access

Vpol Vertical polarization

Authorship

The work contained in this thesis has not been previously submit-

ted to meet requirements for an award at this or any other higher

education institution. To the best of my knowledge and belief, the

thesis contains no material previously published or written by another

person except where due reference is made.

Signed: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Date: . . . . . . . . . . . . . . . . . .

List of Publications

Conference papers

• X. Li and B. Senadji, “Statistical Analysis of Interference In

Asynchronous MC-CDMA Systems”, The 1st Australian Con-

ference on Wireless Broadband and Ultra Wideband Communi-

cations, Sydney, 2006.

• X. Li, D. Carey and B. Senadji, “Interference Reduction and

Analysis for Asynchronous MCCDMA Using a Dual Frequency

Switching Technique”, The 5th Workshop on the Internet, Telecom-

munications and Signal Processing, Hobart, 2006.

• X. Li and B. Senadji, “Multiple access interference analysis in

asynchronous GSFH/MC-CDMA systems”, Wireless Communi-

cation and Networking Conference, 2007. IEEE, pp.197-201, 11-

15 March 2007.

• X. Li, Y.C. Huang and B. Senadji, “MAI Analysis of an Asyn-

chronous MC-CDMA System With Polarization Diversity”, The

1st International Conference on Signal Processing and Commu-

nication Systems (ICSPCS 2007), Gold Coast, 2007.

Papers submitted to journals

• X. Li, D. Carey and B. Senadji, “Statistical Analysis of Multi-

ple Access Interference In Asynchronous MC-CDMA Systems”,

IEEE Transactions of Vehicular Technology, [Submitted].

• X. Li and B. Senadji, “Performance Analysis of Asynchronous

MC-CDMA with Subcarrier Frequency”, IEEE Transactions of

Vehicular Technology, [Submitted].

• X. Li, Y.C. Huang and B. Senadji, “Maximal Signal-to-MAI Ra-

tio Combining for MC-CDMA with Base Station Polarization

Diversity”, IEEE Transactions of Vehicular Technology, [Sub-

mitted].

Acknowledgements

I would like to thank my principal supervisor, Dr. Bouchra Senadji, for her

continuous support in my PhD program. Bouchra has guided me throughout the

entire PhD program. She is always there, giving advice to help me solve difficult

problems. Especially I would like to thank Bouchra for her patience in helping

me to improve my fluency in the English needed for this subject area.

I would like to thank my associate supervisor, Prof. Sridha Sridharan, for

giving me opportunities to work for him during the past four years. It is my

great honor to work for Prof. Sridharan, who has given me superb opportunities

to broaden my knowledge in the area of telecommunication. I would also like to

thank him for his support in my PhD program.

I would like to thank my fellow PhD student, Brian Huang, for his contribution

in the later part of my PhD candidature. Brian is smart and hardworking. It was

a great pleasure for me to work with him. Finally, I would like to thank Patrick

Lau, one of my best friends over the last seven years, for his advice on my thesis

writing.

I am so lucky to meet these wonderful people. Without them I could not have

come this far. Thank you all.

Chapter 1

Introduction

In telecommunication systems, communication resources refer to the time and

frequency bandwidth that is available in a given system. With multiple users

on a system, resources (time/bandwidth) need to be shared among the users

in order to establish communication links between the mobile subscriber (MS)

and the base station (BS). However, the available resources are often limited

for any given user, as the total bandwidth on a system is restricted. In order

to improve the efficiency of resource allocations, multiple access techniques have

been developed, with an ideal system illustrating the following qualities [2]: 1)

available resources are fully utilized; 2) all resources are shared equally among

users; 3) interference is not introduced between users i.e. no multiple access

interference (MAI), and 4) the capacity of the system is maximized.

1

1. INTRODUCTION

1.1 Need for multicarrier code division multiple

access

Three major multiple access schemes exist: frequency division multiple access

(FDMA), time division multiple access (TDMA) and code division multiple access

(CDMA) [1]. In the following sections, the advantages and disadvantages of these

techniques will be reviewed, and the need for multicarrier code division access

(MC-CDMA) will be discussed.

1.1.1 Frequency division multiple access

Figure 1.1: Frequency division multiple access (FDMA) [1]

Frequency division multiple access (FDMA) was the first multiple access tech-

nique, developed in the early 1900s [2]. With FDMA, the total frequency band-

width is divided into frequency channels that are assigned to each user perma-

2

1.1 Need for multicarrier code division multiple access

nently, resulting in multiple user signals that are both spectrally separated and

simultaneously transmitted and received. This has been graphically represented

in Fig. 1.1.

The FDMA systems requires a relatively simple algorithm and implementa-

tion compared to TDMA and CDMA [1], but there are several drawbacks. Firstly,

due to the permanent assignment of FDMA channels, unused channels cannot be

utilized by other users, resulting in wasted communication resources. Secondly,

nonlinearities in the power amplifier can cause signal spreading in the frequency

domain, causing inter-channel interference (ICI) in other FDMA channels. Fi-

nally, the capacity of an FDMA system is limited by the number of channels

available.

1.1.2 Time division multiple access

Figure 1.2: Time division multiple access (TDMA) [1]

3

1. INTRODUCTION

Time division multiple access (TDMA) has been developed with a similar

concept to FDMA, but with TDMA, multiple user signals are separated in the

time domain rather than in the frequency domain. Fig. 1.2 shows a TDMA

system with the transmission time divided into a number of cyclically repeating

time slots that can be assigned to individual users, allowing all users access to all

of the available bandwidth.

Compared to FDMA systems, TDMA systems offer more flexibility in the

assignment of time slots whereby different numbers of time slots can be allocated

to different users depending on the service demanded. Furthermore, because

TDMA users can transmit signals only in their own time slots, the transmission

of TDMA signal is noncontinuous and occurs in bursts, resulting in less battery

power consumption. However, the TDMA signal requires a large synchronization

overhead due to its non-continuous transmission. Inter-symbol interference (ISI),

caused by multipath propagation, is also a major problem for TDMA, especially

during high data rate transmissions.

1.1.3 Code division multiple access

Over the last decade, code division multiple access (CDMA) has been developed

to overcome the disadvantages of other multiple access techniques such as TDMA

and FDMA [3].

Fig. 1.3 demonstrates multiple CDMA users signals that are separated by

spreading sequences. In particular, each user signal is spread using a pseudo-

random sequence which is orthogonal to the sequence of other users. As a result,

only the intended user-receiver can despread and receive the information cor-

4


Figure 1.3: Code division multiple access (CDMA) [1]

rectly; other users on the system perceive the signal as noise, resulting in multiple

user signals that can be transmitted within the same bandwidth simultaneously.

The main advantage with CDMA is that the system capacity is limited only

by the amount of interference; with a lower level of interference the system can

support a higher number of users [1]. CDMA systems are also robust to narrow

band jamming as the receiver signal can spread the jamming signals’ energy over

the entire bandwidth making it insignificant in comparison to the signal itself

[2]. If the spreading sequence is perfectly orthogonal, it is possible to transmit

multiple CDMA signals without introducing multiple access interference (MAI)

during synchronous transmission [3]

Various types of CDMA such as direct-sequence CDMA (DS-CDMA) and

wideband CDMA (W-CDMA), have been developed and utilized in both 2G

and 3G systems similar to cdmaOne (IS-95), UMTS and CDMA2000 [4]. These

5

1. INTRODUCTION

techniques are considered to be single-carrier CDMA systems. Unfortunately

when moving into the fourth generation of wireless communication systems (4G),

in which data is transmitted at a rate as high as 1 Giga bits-per-second (bps) [5],

single-carrier CDMA systems are not suitable. This is because

1. With high data rates the symbol duration will become shortened, resulting

in the channel delay spread exceeding the symbol duration causing ISI [6].

2. When data rate goes beyond a hundred Mega bps, it becomes difficult to

synchronize, as the data is sequenced at high speeds [7].

3. Due to multipath propagation, signal energy is scattered in the time domain:

in single-carrier CDMA systems such as DS-CDMA, RAKE receivers are

often used to combine the multipath signals. However, not all paths of

signals can be successfully received. If the number of fingers in the RAKE

receiver is less than the number of resolvable paths, some of the received

signal energy can not be combined, thus a portion of the signal energy is

lost [8]. But if the number of fingers in the RAKE receiver is more than

the number of resolvable paths, noise will be enhanced.

Therefore a conventional single-carrier CDMA such as DS-CDMA is not practical

for 4G systems where a high data rate is required.

1.1.4 Orthogonal frequency division multiplexing

Orthogonal frequency division multiplexing (OFDM) proposed in [9] has the abil-

ity to support higher data rate transmission. When using OFDM, the channel

bandwidth is divided into a number of equal bandwidth subchannels, with each

6


subchannel utilizing a subcarrier to transmit a data symbol. The frequency sep-

aration of adjacent subcarriers is chosen to equal the inverse of the symbol du-

ration, resulting in all the subcarriers being orthogonal to one another over one

symbol interval. Hence, OFDM technique can transmit a large number of dif-

ferent data symbols over multiple subcarriers simultaneously, enabling this tech-

nique to support a higher data rate transmission. In addition the bandwidth of

each subchannel is designed to be so narrow that the frequency characteristics of

each subchannel are constant, making OFDM signals robust to frequency selec-

tive fading [10]. The other advantage of OFDM is that the signal can be easily

and efficiently modulated and demodulated using fast Fourier transform (FFT)

devices [11]. As FFT can be easily implemented, the receiver complexity does

not increase substantially while transmission rate can be largely increased.

Despite all these advantages, OFDM still have some drawbacks due to its im-

plementation of multicarrier modulation. OFDM suffers a high peak-to-average

power ratio that occurs when all the signals in the subcarriers are added con-

structively [12, 13]. This results in the saturation of the power amplification at

the transmitter, causing inter-modulation distortion. OFDM is very sensitive to

frequency offset, as the spectrums of the subcarriers are overlapping [14, 15]. Any

frequency offset can lead to ICI, which suggests that OFDM requires a high de-

gree of synchronization of subcarriers. Besides, the conventional OFDM systems

can support only a single user, raising the need for multicarrier code division

multiple access (MC-CDMA).

7

1. INTRODUCTION

1.1.5 Multicarrier code division multiple access

Based on the combination of OFDM and DS-CDMA, multicarrier code division

multiple access (MC-CDMA) is proposed [16]. Unlike DS-CDMA, which spreads

the original data stream into the time domain, MC-CDMA spreads the original

data stream into the frequency domain by initially converting the input data

stream from serial to parallel then multiplying this stream by the spreading chips

in different OFDM subcarriers, resulting in a MC-CDMA signal which takes on

the advantages of both DS-CDMA and OFDM. The advantages of MC-CDMA

are:

1. The capacity is interference limited [17] and any techniques that reduce

interference are capable of increasing the capacity of MC-CDMA.

2. The signal is robust to frequency selective fading and can support high data

rate transmission.

3. Bandwidth is used more efficiently as the spectra of subcarrier overlap [18].

4. Since the received signal is combined in the frequency domain, a MC-CDMA

receiver can employ all the received signal energy scattered in the frequency

domain [19]. This is a significant advantage over DS-CDMA, where part

of the signal energy can be lost due to insufficient number of fingers in the

RAKE receiver.

5. The transmitter and receiver signals can be implemented using FFT, which

does not increase the degree of complexity.

However, as MC-CDMA is still a multi-carrier modulation technique, it in-

evitably has the same drawback as OFDM. Problems such as inter-modulation

8

1.2 Multiple access interference problem in MC-CDMA

distortion and ICI can still be found in MC-CDMA systems. Furthermore as with

other CDMA techniques, MC-CDMA suffers multiple access interference (MAI)

during asynchronous transmission, which significantly degrades the performance

of MC-CDMA systems. This research will focus on the analysis and reduction of

MAI in MC-CDMA systems.

1.2 Multiple access interference problem in MC-

CDMA

In MC-CDMA systems, as with DS-CDMA systems, each user is assigned a

unique orthogonal pseudo random spreading sequence which allows the receiver

to distinguish individual user signals from one another. During asynchronous

transmissions, however, where multiple user signals arrive at the receiver with

different timing offsets, the orthogonality between users is lost [1]. This creates

the first type of multiple access interference (MAI) in MC-CDMA. This type of

MAI, denoted as Is, is the interference created by non-reference users transmit-

ting information over the same subcarrier frequencies and it is commonly found

in both asynchronous DS-CDMA and MC-CDMA systems. The second type of

MAI, denoted as Id, is exclusive to MC-CDMA systems and it is due to the nature

of the multicarrier transmission. In MC-CDMA systems the frequency separa-

tion between subcarriers is chosen as an integer multiple of 1/Ts (Ts denoting

the symbol duration), such that subcarriers are orthogonal to each other. During

asynchronous transmission, however, this orthogonality between subcarriers is

also lost, which gives rise to the production of the second type of MAI. This MAI

9

1. INTRODUCTION

Figure 1.4: MAI generation in asynchronous MC-CDMA systems

term represents the interference generated by transmission over different subcar-

rier frequencies by the non-reference users. Fig. 1.4 illustrates the generation of

the two types of MAI for a two-user system. In this figure, both reference user r

and interferer k are transmitting signals through N subcarriers. It can be seen

from Fig. 1.4 that in the reference user r the signal with subcarrier frequency

fr,1 is suffering two types of MAI i.e. Is(1) and Id(1, j) (j = 2 . . . N). The first

type of MAI Is(1) (represented by a solid line) is generated due to the interferer

signal with subcarrier frequencies fk,1, with fk,1 = fr,1. Meanwhile the signals in

other interferer subcarriers fk,j (for j = 2 . . . N) contribute to the second type of

MAI Id(1, j) (represented by dash lines). The total MAI in the reference user,

denoted as I, is the sum of both types of MAI in all N subcarriers, which means

I =N∑

i=1

Is(i) +N∑

i=1

N∑

j=1;j 6=i

Id(i, j) . (1.1)

The effect of MAI on the performance of asynchronous MC-CDMA was first

10

1.2 Multiple access interference problem in MC-CDMA

Figure 1.5: Effect of MAI on the performance of asynchronous MC-CDMA system

(Note: The full capacity of the system is 16 users. The level of noise isindicated by Eb/No, where Eb is the bit energy and No is the single-sided noisespectral density)

11

1. INTRODUCTION

studied in [20] in a frequency selective fading channel. Then later in [21] and [22],

similar analyses were given in the case of the Nakagami fading channel. All this

research showed that during asynchronous transmission, the bit error rate (BER)

performance of MC-CDMA is significantly degraded by the presence of MAI.

BER is a performance measurement for telecommunication systems. It rep-

resents the percentage of bits that have errors relative to the total number of

bits received in a transmission [2]. A small BER indicates that the system has a

small probability of receiving bits in error, hence, the performance of the system

is good. On the contrary, a large BER indicates a large probability of receiving

error bits, hence, the system has a poor performance.

Fig. 1.5 illustrates the effect of MAI on the BER of asynchronous MC-CDMA.

This figure compares a single user system (i.e. no MAI) to a multiple user system

with MC-CDMA. As seen in Fig. 1.5, MC-CDMA is capable of offering a low

BER for the single user system, and this BER can be further reduced by decreas-

ing the level of noise (indicating by increasing Eb/No). However, when two or

more users exist in the system, MAI is introduced, and the BER is increased, in-

dicating a performance degradation. In the case when the system is fully loaded,

the MAI becomes the dominant factor of the interference. As a result, further

reduction of noise cannot decrease the BER, and the system has an irreducible

error floor. Therefore MAI is one of the major factors that limits the performance

of asynchronous MC-CDMA.

12

1.3 Thesis objectives and contributions

1.3 Thesis objectives and contributions

The main objectives of this thesis are to analyse the multiple access

interference in asynchronous MC-CDMA and to develop robust tech-

niques to reduce the MAI effect.

In order to achieve these objectives, this thesis has been separated into three

parts. The first part primarily considers the statistical behavior of MAI in asyn-

chronous MC-CDMA, and provides statistical MAI analysis and a statistical

model for asynchronous MC-CDMA. The principal contributions in this part

of the research are listed below

1. Define effective timing offset and derive its statistics.

2. Derive the exact expression of MAI for MC-CDMA as a function of timing

offset.

3. Establish a statistical model for the PDF of the time varying MAI power

using Gamma distribution.

4. Propose a computer simulation model for asynchronous MC-CDMA sys-

tems.

All the above contributions led to the publications of [23] and [24].

In the second part, slow frequency hopping (SFH) as a MAI reduction tech-

nique will be introduced to MC-CDMA (SFH/MC-CDMA). SFH/MC-CDMA is

further divided into two subsystems, group subcarrier frequency hopping MC-

CDMA (GSFH/MC-CDMA) [25, 26] and individual subcarrier frequency hop-

ping MC-CDMA (ISFH/MC-CDMA). Listed below are the lists of contributions

presented in this part.

13

1. INTRODUCTION

1. Provide a MAI analysis for GSFH/MC-CDMA with asynchronous trans-

missions.

2. Show that the MAI power in GSFH/MC-CDMA is not affected by the

length of hopping period.

3. Derive a more general SFH/MC-CDMA system based on the application of

individual subcarrier frequency hopping (ISFH/MC-CDMA).

4. Derive the analytical expression of the MAI for ISFH/MC-CDMA under

asynchronous conditions and compare it to that of MC-CDMA as well as

GSFH/MC-CDMA.

5. Identify the relationship between the MAI power generated in ISFH/MC-

CDMA and MC-CDMA.

6. Demonstrate that GSFH/MC-CDMA is a special case of ISFH/MC-CDMA.

The publications that are associated with this part are [27, 28, 29].

The third part is based on developing a system which combines base sta-

tion polarization diversity and MC-CDMA, referred to as Pol/MC-CDMA. The

original contributions presented in this part of the thesis are as follows:

1. Introduce base station polarization diversity to MC-CDMA and derive the

expression of MAI and its power for Pol/MC-CDMA.

2. Propose an optimum combining method to combine the signal in two base

station antennas, called maximal signal-to-MAI ratio combining (MSMAIRC)

for Pol/MC-CDMA system.

14

1.4 Organization of this thesis

3. Derive the optimum antenna angle for Pol/MC-CDMA with MSMAIRC as

well as the traditional maximal ratio combining (MRC). With this optimum

antenna angle the signal-to-interference-plus-noise ratio is maximized and

the BER performance of Pol/MC-CDMA is the lowest.

4. Compare the BER performance of Pol/MC-CDMA with MRC and MS-

MAIRC.

These contributions are documented in the publications of [30] and [31].


After the introductory chapter, the thesis develops through seven chapters.

Chapter 2 focuses on the statistical analysis of MAI for asynchronous MC-

CDMA systems and a new statistical model of MAI is derived. Unlike existing

statistical models where uniform distributed timing offsets and constant MAI

power are assumed, in the new statistical model of MAI, different distributions

of timing offset can be applied and the MAI power is found to be Gamma dis-

tributed. Utilizing the new statistical model of MAI, a computer simulation

model for asynchronous MC-CDMA systems is proposed. Current models re-

quire a heavy computational load because multiple user signals must be simu-

lated simultaneously. In the new simulation model, the computer simulation of

the multiuser system is replaced by a single user system, followed by an additive

noise component representing the MAI. As a result, the proposed model requires

significantly less computing. The proposed model is validated by using statistical

measurements such as paired t-test.

15

1. INTRODUCTION

Chapter 3 reviews the three main techniques for MAI reduction: optimal

spreading sequence design, multiuser detection (MUD) and slow frequency hop-

ping (SFH). It provides the literature reviews and background information for

Chapter 4 and Chapter 5.

Chapter 4 provides a solution for reducing the MAI effect by introducing

a group subcarrier frequency hopping technique to MC-CDMA (GSFH/MC-

CDMA). In this chapter, the MAI performance of GSFH/MC-CDMA under

asynchronous transmission is analysed. Three different detection scenarios are

considered. The expression of MAI power is derived and verified using Monte-

Carlo simulation results.

In Chapter 5, the condition of group hopping in GSFH/MC-CDMA is dropped

and a new system referred to as the individual subcarrier frequency hopping

MC-CDMA (ISFH/MC-CDMA) is proposed. In this chapter, a thorough MAI

analysis for asynchronous ISFH/MC-CDMA is provided. The MAI power in

ISFH/MC-CDMA is then compared with MC-CDMA as well as with the previ-

ously proposed GSFH/MC-CDMA. ISFH/MC-CDMA is found to generate less

MAI power than the basic MC-CDMA system during asynchronous transmis-

sion. Because of this, ISFH/MC-CDMA is shown to outperform MC-CDMA in

terms of the bit error rate (BER). Moreover, in this chapter it is shown that

GSFH/MC-CDMA is a special case of ISFH/MC-CDMA; in particular, when the

number of available subcarrier frequencies are equal, both systems generate the

same amount of MAI power. The expression of BER for ISFH/MC-CDMA is

derived and all theoretical results are confirmed using Monte-Carlo simulations.

Chapter 6 firstly reviews the literature for the existing research on polar-

ization diversity. Then base station polarization is introduced as another MAI

16


reduction technique to asynchronous MC-CDMA. In this chapter, a new diversity-

combining technique is proposed for the asynchronous MC-CDMA, with two

branch base station polarization diversity (Pol/MC-CDMA). The new combining

technique, called the maximal signal-to-MAI ratio combing (MSMAIRC), aims

to maximize the received signal-to-MAI ratio (SMAIR) at the diversity combiner.

Bit error rate (BER) performance of Pol/MC-CDMA using MSMAIRC is anal-

ysed and compared with maximal ratio combining (MRC). It is found that when

the level of additive white Gaussian noise (AWGN) is small in terms of the Eb/N0,

MSMAIRC generally outperforms MRC for most values of cross polarization dis-

crimination (XPD). Moreover, MSMAIRC is also found to be less sensitive to the

change in XPD. This indicates that MSMRC is able to guarantee a more stable

performance if the transmission environment is changing. Finally, in this chapter,

the optimum antenna angles for Pol/MC-CDMA with both MSMAIRC and max-

imum ratio combining (MRC) are also derived. By setting the antenna angles to

the derived optimal values, Pol/MC-CDMA systems can obtain the lowest BER.

Chapter 7 summarizes the research results and discuss the future research

directions.

17

1. INTRODUCTION

18

Chapter 2

Statistical analysis of MAI in

asynchronous MC-CDMA

systems

In this chapter, two problems with the existing statistical analysis of MAI in

asynchronous MC-CDMA systems have been identified. By solving these two

problems, a new statistical model of MAI is developed. This statistical model

can be applied in two applications. Firstly, it can be used as a tool to analyze

the performance of asynchronous MC-CDMA systems with different assumptions

of MAI and its power, including different distributions of timing offsets and dif-

ferent statistical model of MAI power. The second application of this statistical

model is computer simulations, and an efficient computer simulation model for

asynchronous MC-CDMA is proposed.

19

2. STATISTICAL ANALYSIS OF MAI IN ASYNCHRONOUSMC-CDMA SYSTEMS

2.1 Problems with the existing statistical analy-

sis of MAI in asynchronous MC-CDMA sys-

tems

During the analysis of the performance in asynchronous MC-CDMA, MAI is

generally approximated as a zero mean Gaussian random variable [20, 21, 22, 32].

This is mainly because the random phase of each subcarrier is assumed to be

independent and identically distributed (i.i.d.) for each subcarrier and for each

user. As a result, the MAI from each subcarrier of interferers is uncorrelated.

According to the central limit theorem, the total MAI can be approximated as

Gaussian distributed and its power has been derived as a constant in [20]. This is

the generally accepted statistical property of MAI in asynchronous MC-CDMA

[20, 21, 32, 33, 22].

The analysis, however, is not complete. In previous analyses of MAI [20, 21,

32, 33, 22], the timing offset, which is defined as the arrival time difference between

the reference user and the interferer, is assumed to be uniformly distributed

within one symbol duration. But this assumption has been contested by many

other research studies such as [34, 35, 36, 37]. The main objection is that the

distribution of the timing offsets between users is strongly dependent on the user

distribution within a cell, and is not necessarily uniform. For example, when the

user distribution is uniform, the distribution of the timing offsets is shown to

be linear [34]. Hence the MAI analysis should not be restricted to the uniform

distributed timing offset; a more general derivation which can be applied to any

possible distribution of the timing offsets is required.

20

2.2 Asynchronous MC-CDMA model

Another problem with the existing statistical analysis of MAI in asynchronous

MC-CDMA systems relates to the assumption of a constant MAI power. Under

this assumption the MAI is considered stationary in a channel that is known to

be non-stationary [38]. A more realistic assumption in a non-stationary channel

is that the MAI is a non-stationary process, in which case the MAI power must

be time varying, indicating that the MAI power is also a random variable.

The statistical properties of the MAI power are not widely documented. [39]

and [40] are among the few to study the statistical behavior of the MAI power

for asynchronous MC-CDMA. In [39], the probability density function (PDF)

of the MAI power is derived. Then in [40] the MAI powers of MC-CDMA and

DS-CDMA are compared. It is found that the MAI power in asynchronous MC-

CDMA is lower than in DS-CDMA. Unfortunately in both studies only one type

of MAI is considered, and the presence of the other type of MAI is ignored, which

suggests that the developed results are incomplete.

The aim of this chapter is to complete the analysis for MAI and its power in

asynchronous MC-CDMA. By using this result, an accurate statistical model for

MAI can be established. This statistical model is flexible which can be adjusted

to different distributions of timing offsets according to different transmission en-

vironments, and it does not limit to a constant MAI power which is more realistic.

Finally, this statistical model will lead to the proposal of an efficient model for

computer simulations in asynchronous MC-CDMA systems.

21


Figure 2.1: Asynchronous MC-CDMA transmitter and receiver model


Fig. 2.1 shows the mathematical model for the asynchronous MC-CDMA trans-

mitter and receiver considered in this chapter. In this model, originally proposed

in [20], the transmission of binary data, bk(t) (k = 1 . . . K) for K users, is con-

sidered. It is assumed that bk(t) are i.i.d. random variables. As shown in Fig.

2.1, the data from the kth user are first BPSK modulated, before the spread-

ing chip sequence ck,i (i = 1 . . . N) is applied over N orthogonal subcarriers fi

(i = 1 . . . N). The carrier frequency for the ith subcarrier, fi, is defined by

fi = f1 + i−1Ts

for i = 1...N , (2.1)

where Ts represents symbol duration. The transmitted signal for the kth user is

given by

sk (t) =N∑

i=1

√2Pbk (t)ck,i cos (2πfit + φk,i) , (2.2)

22


where P is the signal power of each subcarrier, and φk,i is the random phase

introduced by the BPSK modulator. For each subcarrier, the random phase φk,i

is assumed to be uniformly and i.i.d. over the interval [0, 2π).

2.2.1 Channel

Following the approach developed in [20], it is assumed that each subcarrier signal

passes through a frequency non-selective Rayleigh fading channel, and that the

fading remains constant for at least two symbol durations. For a given user, the

Rayleigh fading channels for transmitting subcarriers are assumed correlated. For

different users, however, the Rayleigh fading channels are assumed independent.

Finally, the statistics of the Rayleigh fading for all channels are assumed to be

identical. The complex impulse response of the Rayleigh fading channel of the

ith subcarrier in the kth user can then be written as

νk,i (t) = βk,iejθk,iδ (t − τk) (2.3)

where τk represents the timing offset of the kth user with respect to the reference

user, δ (·) is the Dirac’s delta function. The random phase introduced by the

channel, θk,i, is modelled as uniformly distributed over the interval of [0, 2π),

and assumed i.i.d. for each subcarrier and each user. Finally, βk,i is a Rayleigh

random variable with second-order moment E[β2k,i] = σ2.

In previous research [20, 21, 32, 33, 39, 22], the timing offsets τk were generally

assumed to be uniformly distributed between 0 and Ts. In communication system

analysis, however, the assumption of uniform distribution does not always hold,

and τk can take different distributions depending on the assumptions made on

23


the user distribution within a cell [34, 35, 36, 37]. Due to the uncertainty of the

distribution of the timing offset, in this chapter, timing offsets are allowed to

follow any distribution. Although the results for the uniform, exponential, and

Gaussian distributions will be demonstrated as examples, readers can extent the

same concept to almost any possible distributions found in practical environment.

2.2.2 Receiver

The signal received at the base station is written as

r (t) =K∑

k=1

N∑

i=1

√2Pβk,ibk (t − τk) ck,i cos (2πfit + ζk,i) + n (t) . (2.4)

In (2.4), ζk,i = θk,i + φk,i − 2πfiτk represents the total phase effect for the ith

subcarrier allocated to the kth user and n (t) is additive white Gaussian noise

(AWGN) with double-sided power spectral density N0/2.

At the receiver site, the signal is demodulated and despread. Equal gain

combining (EGC) is then applied to the despread signals. In EGC, signals re-

ceived from each subcarrier i (i = 1 . . . N) are weighted equally by parameters µr,i

(i = 1 . . . N) and summed. In this thesis, µr,i = 1 (i = 1 . . . N). The combined

signal is then passed through a matched filter followed by a maximum likelihood

detector.

2.3 Derivation of test statistic

The test statistic at the output of the matched filter is denoted as [Z]mc, where

[ · ]mc represents symbols that are exclusive to MC-CDMA systems. With l de-

24


fined to be an arbitrary integer, [Z]mc can be obtained as

[Z]mc =N∑

i=1

∫ (l+1)Ts

lTs

r (t) µkcr,i cos (2πfk,it) dt

= [D]mc + [η]mc + [I]mc . (2.5)

[Z]mc can be written as a sum of three components, as indicated above by [D]mc,

[η]mc and [I]mc. [D]mc is the component which represents the desired signal, and

is given by [20]

[D]mc =

√P

2TsNbr (l) Br , (2.6)

where

Br =N∑

i=1

βr,i . (2.7)

It represents the sum of Rayleigh fading coefficients of the N subcarrier in the

reference user. The data bit of the reference user in the current detection interval

[lTs, (l + 1) Ts] is denoted by br (l). The interference due to the AWGN is repre-

sented by [η]mc. It is a Gaussian random variable with zero mean and variance

given as [20]

V ar [η]mc =N0NTs

4. (2.8)

The [I]mc term represents the undesired multiple access interference. For MC-

CDMA, the MAI term was shown to be composed of two independent terms

referred to [Is]mc and [Id]mc [20]. The expressions for [I]mc, [Is]mc and [Id]mc can

25


be written respectively as

[I]mc = [Is]mc + [Id]mc , (2.9)

where

[Is]mc =K∑

k=1;k 6=r

N∑

i=1

Is,k (i) , (2.10)

[Id]mc =K∑

k=1;k 6=r

N∑

i=1

N∑

j=1;j 6=i

Id,k (i, j) . (2.11)

In (2.10), Is,k(i) represents the MAI generated by user k using subcarrier fre-

quency fk,i, which is also used by the reference user r, that is fk,i = fr,i. Similarly,

in (2.11), Id,k(i, j) represents the MAI generated by user k using subcarrier fre-

quency fk,j, which is different from that used by reference user r, that is fk,j 6= fr,i.

The expressions of Is,k(i) and Id,k(i, j) have been developed by Gui and Ng

in [20] under the assumption of uniformly distributed timing offsets. They are

given respectively as

Is,k (i) =

√P

2µr,ick,icr,i cos ζk,i

· [βk,i (l − 1) bk (l − 1) τk + βk,i (l) bk (l) (Ts − τk)] , (2.12)

Id,k (i, j) =√

2Pµr,ick,jcr,iTs

4π∆i,j

· [βk,j (l − 1) bk (l − 1) − βk,j (l) bk (l)]

·[sin

(2π

∆i,jτk

Ts

+ ζk,j

)− sin ζk,j

], (2.13)

where bk (l) and βk,i (l) are respectively as the binary data and the channel fading

in the current detection interval, that is [lTs, (l + 1) Ts]. Similarly bk (l − 1) and

26


βk,i (l − 1) represent respectively the binary data and the channel fading associ-

ated with the previous detection interval, that is [(l − 1) Ts, lTs] respectively. In

(2.13), ∆i,j denotes as the spectral distance between fr,i and fk,j and is defined

as

∆i,j = i − j . (2.14)

The derivations of (2.12) and (2.13) are, however, based on the assumption

that the timing offsets lie only within 0 and Ts. When the timing offsets are not

assumed to be uniformly distributed between 0 and Ts, they can take any real

value between −∞ and +∞. As a result, the expressions derived in (2.12) and

(2.13) become invalid.

Fig. 2.2 shows the three possible scenarios involving timing offsets. In sce-

nario 1, the timing offset between the reference user and the interferer fall within

[0, Ts). This is the scenario that was originally considered in [20]. In scenario 2,

the arrival time difference, τk, between the reference user and interferer is larger

than Ts, while in scenario 3, the interfering signal arrives earlier than the reference

signal, resulting in a negative timing offset.

What is important to understand, is that the amount of MAI affecting the

reference user is not determined by the timing offset, τk, but by the time mis-

alignment between the reference user symbol br(l) and the interferer symbol bk(l).

This time misalignment is referred to as the effective timing offset, and denoted

it as τ ek .

In scenario 1 of Fig. 2.2, the time misalignment between the reference symbol

br(l) and interfering symbol bk(l) is equal to the timing offset, that is τ ek = τk.

27


Figure 2.2: Effective timing offset scenarios

28


In scenario 2, as illustrated in Fig. 2.2, the timing offset τk is greater than Ts.

The timing offset in this case is calculated as τk mod Ts, where mod denotes the

modulo operator. Finally, in scenario 3, the signal of the interfering user arrives

before the reference user, and generalizes the case for τ ek < 0. In this case, the

effective timing offset is calculated as τ ek = Ts − (τk mod Ts).

In summary, given a timing offset τk between −∞ and +∞, the effective tim-

ing offset τ ek responsible for MAI always lies within [0, Ts), and can be expressed

as

τ ek =

τk mod Ts for τk > 0

Ts − (τk mod Ts) for τk < 0

(2.15)

Further, the probability density function (PDF) of the effective timing offset

τ ek can be expressed in terms of the PDF of the timing offset τk as:

For 0 ≤ τ ek < Ts prob (τ e

k) =∞∑

l=−∞prob (sgn (l) · τk + lTs) for 0 ≤ τk < Ts

For |τ ek | ≥ Ts prob (τ e

k) = 0,

(2.16)

where l can take any integer values from −∞ to +∞.

As the effective timing offset is always within one symbol duration, the two

MAI terms, (2.17) and (2.18) previously calculated in [20] for timing offsets, τk,

in the range [0, Ts), can be generalized to any distribution of the timing offset

29


as

Is,k (i) =

√P

2µr,ick,icr,i cos ζk,i

· [βk,i (l − 1) bk (l − 1) τ ek + βk,i (l) bk (l) (Ts − τ e

k)] , (2.17)

Id,k (i, j) =√

2Pµr,ick,jcr,iTs

4π∆i,j

· [βk,j (l − 1) bk (l − 1) − βk,j (l) bk (l)]

·[sin

(2π

∆i,jτek

Ts

+ ζk,j

)− sin ζk,j

]. (2.18)

These results will be used in the next section for modelling the statistics of the

MAI power.

2.4 Statistical modelling of MAI and its power

In asynchronous MC-CDMA, due to the assumption that φk,i and θk,i are i.i.d. for

different k and i, both types of MAI, that is Is,k (i) and Id,k (i, j) defined respec-

tively in (2.17) and (2.18), are independent for different subcarriers and different

users. Hence their corresponding summation over i and k, that is [Is]mc and [Id]mc

defined in (2.10) and (2.11), are also uncorrelated. As a result, according to the

central limit theorem, the total MAI defined in (2.9) can be modelled as Gaussian

distributed [20]. Figure 2.3 to 2.5 show three examples of the PDF for the MAI

in asynchronous MC-CDMA, with timing offset being uniformly, exponentially

and Gaussian distributed respectively. All three examples show close fits between

the PDF of the MAI and that of a zero-mean Gaussian PDF.

Problems remain in the statistical modelling of MAI power. Previous research

assumes that the MAI power is a constant in time. In a non-stationary environ-

ment, however, the MAI power becomes randomly time-varying. Although the

30


Figure 2.3: PDF of MAI in asynchronous MC-CDMA (with N=K=16, uniformlydistributed τk)

Figure 2.4: PDF of MAI in asynchronous MC-CDMA (with N=K=16, exponen-tially distributed τk)

31


Figure 2.5: PDF of MAI in asynchronous MC-CDMA (with N=K=16, Gaussiandistributed τk)

exact PDF of the MAI power is unknown, a good statistical model can be ob-

tained through the fitting of its PDF to existing well-known PDFs.

Because the MAI power is a positive random variable, the fitting process will

be limited to positive distributions, namely Gamma, Nakagami, Rice and Weibull

distributions. The statistical modelling is also performed for various distributions

of the effective timing offset, and for various numbers of users and subcarriers.

The Kullback-Leibier divergence (KLD) is used as a measure of closeness be-

tween the PDF of the MAI power and that of the positive distributions considered

in this chapter. The expression of the KLD is given as [41]

KLD =∑

i∈Ψ

prob (V ar [I]mc = i) · log

prob (V ar [I]mc = i)

prob(

V ar [I]mc = i)

(2.19)

where (V ar [I]mc = i) represents the ith realization of the MAI power, Ψ is the

total number of realizations, prob (V ar [I]mc = i) is the true PDF of V ar [I]mc

32


Table 2.1: Distribution fitting for conditional MAI power in asynchronous MC-CDMA: Kullback-Leibier divergence values

τk Spreading factor DistributionDistribution & user numbers Gamma Nakagami Rice Weibull

Uniform N=8; K=8 0.0068 0.0082 0.0107 0.0916N=16; K=10 0.0090 0.0100 0.0119 0.0917N=32;K=8 0.0080 0.0094 0.0118 0.0940N=64;K=6 0.0151 0.0161 0.0197 0.0926

Gaussian N=8; K=8 0.0078 0.0088 0.0113 0.0898with N=16; K=10 0.0081 0.0093 0.0114 0.0964

mean = 0.8Ts N=32;K=8 0.0081 0.0095 0.0119 0.0965variance = 0.6Ts N=64;K=6 0.0135 0.0155 0.0192 0.1018

Exponential N=8; K=8 0.0059 0.0063 0.0079 0.0764with N=16; K=10 0.0058 0.0062 0.0077 0.0856

mean = 0.5Ts N=32;K=8 0.0068 0.0075 0.0100 0.0863variance = 0.5Ts N=64;K=6 0.0086 0.0102 0.0129 0.0874

and prob(

V ar [I]mc = i)

is the approximated PDF.

The results of the statistical modelling over 10000 realizations are summarized

in Table 2.1. Three different distributions of the effective timing offset have been

considered, namely the uniform, exponential and Gaussian distributions. The

numbers of subcarriers N and active users K have also been varied. Table 2.1

shows that the Gamma distribution consistently displays the best closeness of

fit with respect to the various distributions tested, and the multiple scenarios

considered. Figures 2.6 and 2.7 are two examples that confirm the choice of the

Gamma distribution as a model for the PDF of the MAI power.

The PDF of a Gamma distributed random variable, x, is given by

prob (x) =1

baΓ (a)xa−1e−x/b , (2.20)

where a and b are the parameters of the distribution that is aimed to be estimated.

33


Figure 2.6: PDF of MAI power for asynchronous MC-CDMA (N=K=8, uniformlydistributed τ e)

Figure 2.7: PDF of MAI Power for asynchronous MC-CDMA (N=32 K=8, Gaus-sian distributed τ e)

34

2.5 Mean and variance of MAI power with different distributions oftiming offset

a and b are completely determined by the knowledge of the mean and variance

of x as shown below [42]

a =E2 [x] /V ar [x] (2.21)

b =V ar [x] /E [x] . (2.22)

It follows that the PDF of the MAI power for asynchronous MC-CDMA can be

completely determine by the mean power and its variance. Therefore, the next

section illustrates the derivation of the mean and variance of MAI power.

2.5 Mean and variance of MAI power with dif-

ferent distributions of timing offset

In this section, an expression for MAI power conditional to τ ek is derived. The

derived expression can be used to obtain the mean and variance of MAI power

for the various distribution of the effective timing offset. The expressions of the

conditional MAI power are

E[I2s |τ e

k

]

mc=Eb(t),β,ζ

[I2s

]

=NPσ2

4

K∑

k=1;k 6=r

[(τ e

k)2 + (Ts − τ ek)2]

(2.23)

E[I2d |τ e

k

]

mc=Eb(t),β,ζ

[I2d

]

= (K − 1)PT 2

s σ2

4π2

N∑

i=1

N∑

j=1j 6=i

1

∆2i,j

=E[I2d

]. (2.24)

35


In (2.23) and (2.24), E[I2s |τ e

k

]

mcand E

[I2d |τ e

k

]

mcrepresent the MAI power con-

ditional to τ ek . Eb(t),β,ζ [ · |τ e

k ] represent the expectation operation respect to b (t),

β, ζ and it is conditional to τ ek . It is interesting to show that in (2.24) the power

of Id is independent of τ ek . As a result, different distributions of the timing offset

will affect the power of Is but not Id.

The sum of E[I2s |τ e

k

]

mcand E

[I2d |τ e

k

]

mcgives total conditional MAI power for

asynchronous MC-CDMA, which is written as

E[I2|τ e

k

]

mc=

NPσ2

4·[X + (K − 1)

1

π2C]

, (2.25)

where

X =K∑

k=1;k 6=r

(τ ek)2 + (Ts − τ e

k)2 , (2.26)

and the following relationship is applied [22]

C =1

N

N∑

i=1

N∑

j=1;j 6=i

1

∆i,j

. (2.27)

After obtaining the conditional MAI power expression, the mean and variance

of the MAI power can be obtained by further taking the expectation to include

τ ek . The results are shown respectively as

E[I2]

mc=

NPσ2

4·[Eτe

k[X] + (K − 1)

C

π2

], (2.28)

V ar[I2]

mc=

(NPσ2

4

)2 [Eτe

k

[X2]−(Eτe

k[X]

)2]

, (2.29)

36

2.5 Mean and variance of MAI power with different distributions oftiming offset

where Eτek[X] and Eτe

k[X2] can be derived as

Eτek[X] =

K∑

k=1;k 6=r

(2 · Eτe

[(τ e

k)2]− 2Ts · Eτe

k[τ e

k ] + T 2s

), (2.30)

Eτek

[X2]

= (K − 1)

4Eτek

[(τ e

k)4]+ T 4

s − 4T 3s E [τ e

k ]

+8T 2s Eτe

k

[(τ e

k)2]− 8TsEτe

k

[(τ e

k)3]

+ (K − 1) (K − 2)(2Eτe

k

[(τ e

k)2]+ T 2

s − 2TsE [τ ek ])2

, (2.31)

In (2.28)-(2.31), Eτek[ · ] represents the expectation operation in respect to τ e

k . It

is found that the key to obtaining the mean and variance of MAI power lies in the

evaluation of the first four order moments of the effective timing offset. Hence

the knowledge of the distribution of the effective timing offset, τ ek is required.

As a result, the jth moment of the effective timing offset can be easily calcu-

lated as

Eτek

[(τ e

k)j]

=∫ Ts

0(τ e

k)j · prob (τ ek)dt . (2.32)

The average MAI power for asynchronous MC-CDMA for various distributions of

timing offset can finally be obtained by substituting (2.32) into (2.30) and (2.31).

It follows that the mean and variance of the Gamma distribution representing

the MAI power, and therefore the parameters of the Gamma distribution, are

completely determined by the knowledge of the number of active users K, the

symbol duration Ts and the first four moments of the effective timing offset, τ ek .

For example, when the timing offsets are uniformly distributed between [0, Ts],

the effective timing offsets are also uniformly distributed between [0, Ts] and the

37


fist four moments of the effective timing offset are given as

E [τ ek ] =

Ts

2, (2.33)

E[(τ e

k)2]

=T 2

s

3, (2.34)

E[(τ e

k)3]

=T 3

s

4, (2.35)

E[(τ e

k)4]

=T 4

s

5. (2.36)

Hence

E [X] = (K − 1)2

3T 2

s , (2.37)

E[X2]

= (K − 1) T 4s

[7

15+ (K − 2)

4

9

]. (2.38)

Therefore, by substituting (2.37) and (2.38) into (2.28) and (2.29), the mean and

variance of the MAI power are given by

E[I2]

= (K − 1)NPσ2

4

(2

3T 2

s +C

π2

), (2.39)

V ar[I2]

= (K − 1)

(NPT 2

s σ2

4

)21

45. (2.40)

Applying (2.21) and (2.22), the parameters for the Gamma distributed MAI

power can be obtained as

a =45 (K − 1)

(2

3+

C

T 2s π2

)2

, (2.41)

b =NPσ2

180(

23

+ CT 2

s π2

) . (2.42)

38

2.6 Applications of the new developed MAI model: A tool foranalyzing the effect on BER

Up to this point, the problems with the existing research on statistical anal-

ysis of MAI have been corrected. The general expressions of the MAI and its

power, which can accommodate different distributions of timing offset, have been

developed and the MAI power has been modelled as a timing varying random

variable with PDF that follows Gamma distributions. In the next section the

bit error rate (BER) performance of asynchronous MC-CDMA is derived and the

effect of the new developed MAI model on the BER is shown.

2.6 Applications of the new developed MAI model:

A tool for analyzing the effect on BER

2.6.1 Theoretical derivation of BER

Based on the analysis for MAI, in this section the bit error rate (BER) for asyn-

chronous MC-CDMA is derived. The mean of the test statistic is the desired

signal component [D]mc, which is given in (2.6). The corresponding variance of

the test statistic is the sum of the AWGN power in (2.8) and the total MAI power,

which is derived in (2.28). Hence the BER for the system is given by

pe|Br = Q

√√√√ E [Z]2mc

V ar [Z]mc

= Q

√√√√√[D]2mc

V ar [η]mc + E[I2]

mc

, (2.43)

where Q (·) is the Q function. Notice that pe|Br is conditional to Br which is the

sum of Rayleigh fading random variables in the subcarriers of the reference user.

To obtained the average BER, expectation should be taken with respect to Br

39


which gives

pe =

+∞∫

0

pe|Br · prob(Br)dBr. (2.44)

To evaluate the above integral, numeric methods such as Monte-Carlo integration

[43] can be applied. The PDF of Br can be approximated as Nakagami-m PDF

(see Appendix A) for details. Hence

prob (Br) =2mmB2m−1

r

Γ (m) Ωmexp

(−m

ΩB2

r

). (2.45)

with parameters

Ω =E[B2

r

], (2.46)

m =E [B2

r ]

E [B2r ] − (E [Br])

2 . (2.47)

The associated first and second order moments for Br can be derived respectively

as

E [Br] =Nσ√

π/4 , (2.48)

E[B2

r

]=σ2

N + n (n − 1)(ρ − π

4ρ + π

4

)

+ (N − n) (N + n − 1) π4

, (2.49)

where n is the number of correlated subcarriers, n ≤ N and ρ is the correlation

coefficient between correlated subcarriers.

40

2.6 Applications of the new developed MAI model: A tool foranalyzing the effect on BER

Figure 2.8: BER comparison between uniformly distributed τk and exponentiallydistributed τk with SNR = 10dB

2.6.2 Simulation results of the effects on BER

In this section the effect on BER for different statistical assumptions of MAI is

shown through Monte Carlo simulations. An asynchronous MC-CDMA system

described in Section 2.2 was simulated. The second-order moment of the Rayleigh

fading channel is assumed equal to unity and the number of realizations used

for the Monte-Carlo simulations is 100, 000. Further, it is assumed that all the

fadings in the subcarriers are correlated with correlation coefficient ρ = 0.6, and

the signal-to-noise ratio (SNR) is set to 10dB.

Fig. 2.8 shows the BER comparison between uniformly distributed τk and

exponentially distributed τk. It is found that the BER of exponentially distributed

τk decreased when the parameter λ increased. In exponential distribution, λ

represents the mean as well as the variance. Fig. 2.8 also shows that when

λ = 0.5Ts, the resulting BER is close to the one with uniformly distributed τk.

41


Figure 2.9: BER comparison between uniformly distributed τk and Gaussiandistributed τk with SNR = 10dB

In fact, if λ is further increased, the BER obtained will be close to the BER of

uniformly distributed τk.

Similar results can be observed in Fig. 2.9 for the comparison between uni-

formly distributed τk and Gaussian distributed τk. Different sets of the parameters

of the Gaussian distributed timing offset will lead to different BER. For example,

when the mean and the standard deviation are 0.1Ts, the PDF of the effective

timing offset is significantly different from that of the uniform distribution, result-

ing in a BER that is different from that of the uniformly distributed τk. However,

when the mean is 0.6Ts and the standard deviation is 0.7Ts, the resulting BER

is close to the one obtained from uniformly distribution τk.

42

2.7 Application of the new developed MAI model: computersimulations

Figure 2.10: Proposed asynchronous MC-CDMA simulation model

2.7 Application of the new developed MAI model:

computer simulations

So far in this chapter, it has been shown that the Gamma distribution provides

the best fitting results for the PDF of MAI power. Furthermore the Gamma dis-

tribution can be characterized by using the expressions of the mean and variance

of the MAI power, derived respectively in (2.28) and (2.29). Therefore the statis-

tical behavior of the MAI power can now be completely described. Utilizing this

result, this section proposes a new computer simulation model for asynchronous

MC-CDMA systems, referred to as the meta-model.

In section 2.2, the mathematical model of asynchronous MC-CDMA is de-

scribed. If this mathematical model is used for computer simulations, the com-

putation load is increased with the number of active users K. This is because in

the mathematical model, MAI is generated through the simulation of K−1 inter-

ferer signals with different randomly generated timing offsets. When the number

of users K is large, substantial computing resources need to be allocated to the

generation of MAI. For the performance analysis of asynchronous MC-CDMA,

however, only the signal from the reference user is interested and all the other

43


Figure 2.11: MAI noise generator for asynchronous MC-CDMA

K − 1 interferer signals are acting as interference only.

With this concept in mind, a more efficient way to simulate asynchronous

MC-CDMA systems has been found and a new model is proposed for computer

simulations referred to as the meta-model. In the meta-model, shown in Fig. 2.10,

only the transmission and reception of the reference user signal are simulated.

The simulation of the K − 1 interferer signals is replaced by a noise component

representing MAI, which is added to the reference signal before passing into the

channel. Therefore, the proposed meta-model is essentially a single-user simula-

tion model which required significantly less computation load than a multiuser

simulation.

The additive MAI noise component is a MAI generator, shown in Fig. 2.11.

It is a Gaussian random variable generator with mean and power as inputs. The

mean input is set to zero and its power input is the output from a Gamma random

variable generator. The Gamma random variable generator also has the mean

and variance as inputs, which can be determined using the derived equation in

(2.28) and (2.29). The output random variable of the MAI noise generator has

44


the identical statistics of the MAI for asynchronous MC-CDMA.

The performance of the proposed meta-model is validated by comparing,

through Monte-Carlo simulations, the BER between the mathematical model

described in Section 2.2 and the simplified model in Fig. 2.10.

The models for two systems with randomly chosen parameters are built. In

system 1, N = K = 16 is assumed and all fadings in subcarriers are correlated

with the correlation coefficient ρ = 0.6. Three different distributions of timing

offset are considered here. They are uniform distributed timing offset, Gaussian

distributed timing offset with mean equal to 0.5Ts and standard deviation equal to

0.2Ts, and exponential distributed timing offset with mean and standard deviation

set to 0.2Ts.

In system 2, N = K = 32 is assumed and the correlation coefficients for all

subcarriers fading are set to 0.8. System 2 also applied three different distribu-

tions of timing offset, namely uniform distribution, Gaussian distribution with

mean 0.7Ts and standard deviation 0.3Ts, and exponential distributed with mean

and standard deviation set to 0.6Ts.

For both systems, the number of realizations used for the Monte Carlo sim-

ulations is 100, 000. Further, it is assumed that the second-order moment of the

Rayleigh fading channel is set to unity.

For each system, a paired t-test [44] is performed for the average BER samples

obtained in both models. The paired t-test is a statistical test that compares the

means of two groups of observations. In this case, the paired t-test determines

whether the BER samples from two models differ from each other in a signifi-

cant way under the assumptions that the paired differences are independent and

identically normally distributed.

45


Table 2.2: Pair T-test results for BER samples obtained in the mathematicalmodel and the proposed model (System 1)

t-test for system with N = K = 16 and ρ = 0.695% confident interrval

Timing Offset Eb/N0 p for the differenceDistributions in dB Value Lower UpperUniform 0 0.5794 -0.0144 0.0081

10 0.3040 -0.0108 0.003420 0.4870 -0.0036 0.0076

Gaussian 0 0.2612 -0.0046 0.0168mean = 0.5Ts 10 0.1788 -0.0050 0.0077√

variance = 0.2Ts 20 0.6352 -0.0070 0.0043Exponential 0 0.4121 -0.0069 0.0168mean = 0.2Ts 10 0.6309 -0.0066 0.0108√

variance = 0.2Ts 20 0.3013 -0.0104 0.0033

During the test, 100 samples are taken for the average BER from both models

and the significant level is set to 0.05. The null hypnosis of the test is that the

difference between the average BER samples is a zero mean Gaussian process.

Table 2.2 and 2.3 show the test results for systems 1 and system 2 respectively.

It can be seen that, in all situations, the p values of the tests are larger than 0.05

and the 95% significant intervals are small and contain zero value, indicating that

the tests fail to reject the null hypnosis. These test results imply that there is no

significant difference between the average BER obtained from the mathemathical

model and from the proposed model, therefore validating the proposed model.

46


Table 2.3: Pair T-test results for BER samples obtained in the mathematicalmodel and the proposed model (System 2)

t-test for system with N = K = 32 and ρ = 0.8Timing Offset 95% confident interrvalDistributions Eb/N0 p for the difference

in dB Value Lower UpperUnifrom 0 0.3769 -0.0173 0.0066

10 0.3238 -0.0040 0.011920 0.6491 -0.0064 0.0102

Gaussian 0 0.1926 -0.0037 0.0182mean = 0.7Ts 10 0.4152 -0.0046 0.0111√

variance = 0.3Ts 20 0.4808 -0.0093 0.0044Exponential 0 0.5550 -0.0081 0.0150mean = 0.6Ts 10 0.7796 -0.0059 0.0078√

variance = 0.6Ts 20 0.5296 -0.0088 0.0045

47


48

Chapter 3

Multiple access interference

reduction techniques

There are a number of proposed techniques that are capable of reducing multi-

ple access interference (MAI) for DS-CDMA systems. These techniques involve

mainly the application of the optimal spreading sequences, multiuser detection

(MUD) and slow frequency hopping (SFH). However, not all these techniques can

be applied to MC-CDMA systems. In the following sections, a review of these

three MAI reduction techniques are shown and their applications on MC-CDMA

systems are also discussed.

3.1 Spreading sequence

As early as 1977, Pursley in [45] was among the first to derive the exact expression

of MAI for asynchronous DS-CDMA systems. He argued that the amount of MAI

in an asynchronous DS-CDMA system is determined by the discrete aperiodic

49

3. MULTIPLE ACCESS INTERFERENCE REDUCTIONTECHNIQUES

cross-correlation function of the spreading sequence. As a result, in order to

minimize the MAI in DS-CDMA, the cross correlation of the spreading sequence

is required to be low. Various spreading sequences have been developed to satisfy

this requirement, for example the m-sequence [46], the Gold sequence [47] and

the Kasami sequence [48].

For asynchronous MC-CDMA, the effect of spreading sequences has been stud-

ied in [49], where a selection criterion for the spreading sequence has been iden-

tified. The performances of asynchronous MC-CDMA are compared for four dif-

ferent spreading sequences namely, Walsh-Hadamard sequence, Gold sequence,

orthogonal Gold sequence [50] and Zadoff-Chu sequence [51]. It was determined

that Zadoff-Chu sequence was the optimal spreading sequence for asynchronous

MC-CDMA. Based on this result, Yip and Ng derived the upper and lower bound

for the bit error rate (BER) of asynchronous MC-CDMA [52] . Then in [53], the

analysis was completed by considering both types of MAI.

However, all the analysis discussed in [49, 52, 53] was limited to the case of

additive white Gaussian noise (AWGN) channels. When a frequency selective

fading channel was considered, the results in [32] show that the choice of the

spreading sequence has little effect on the power of the MAI. In fact, in [20],

the expression of the MAI power is shown to be independent of the spreading

sequence.

As frequency selective fading channels appear in most wireless communication

systems, the effect of MAI for asynchronous MC-CDMA cannot be reduced by

using the optimal spreading sequence.

50

3.2 Multiuser detection


Verdu in [54] proposed the first optimal multiuser detector, referred to as the

maximum-likelihood sequence (MLS) detector. He proved that, with his ap-

proach, systems with asynchronous multiuser signals can achieve a minimum

BER, equivalent to that of a single-user system when the level of AWGN is low.

In other words, all interferences including MAI can be eliminated. Despite this

great achievement, the MLS detector has not been widely adopted in practical

CDMA systems. This is mainly due to its high complexity cost. According to [55],

the complexity of the MLS detector is increased exponentially with the number

of active users.

Since then multiuser detector (MUD) research has been focused on reducing

complexity, and various types of suboptimal multiuser detectors have been devel-

oped for asynchronous CDMA systems. The two most common types of MUD

are minimum mean-square error (MMSE) detector and subtractive interference

cancellation.

3.2.1 Minimum mean-square error detector

MMSE detectors can be implemented linearly or adaptively. Linear MMSE de-

tectors are relying on the linear transformation of the soft outputs from the

conventional single-user detector [56]. The criterion of the linear transformation

is based on MMSE, which minimizes the mean-squared error between the actual

data and the soft output of the conventional detector. It has been shown that

linear MMSE detectors can successfully reduce MAI and other forms of interfer-

ence and thus give a significant improvement for the performance of asynchronous

51


DS-CDMA [56]. Furthermore, unlike other linear detectors, the MMSE detectors

do not enhance the background noise. However, the complexity of the system is

still a problem. In order to suppress interference, linear MMSE detectors require

the estimation of the phase, frequency, delays and amplitude of all users [55]. In

addition, it also involves the inversion of a large-size matrix which requires high

computation capacity [55].

Adaptive MMSE detector [57, 58, 59], the adaptive extension of the linear

MMSE detector, can solve both the assumed knowledge and complexity problems.

By making the filter parameters adaptive, adaptive MMSE detector does not

require any matrix inversion; only the rough timing of the desired user is needed

[57, 55]. Castoldi in [60] studied the performance of adaptive MMSE detectors

for asynchronous DS-CDMA. The trained adaptive MMSE detector, a subclass

of the adaptive MMSE detectors, is able to give a BER performance that is

similar to the nonadaptive linear MMSE detector. Moreover the trained adaptive

MMSE detector is robust to the imperfect timing recovery as well as the phase

and frequency offset.

The disadvantage of adaptive MMSE detector is discovered when it is ap-

plied in a frequency selective channel. In the research for the performance of

DS-CDMA, the authors in [61, 62, 63] argued that it is crucial for the receiver

to accurately track the channel fading parameters for all of the users’ signals;

otherwise substantial degradation in performance [61] and capacity [62] can re-

sult. The same problem happens in MC-CDMA, where the tracking of all the

users’ fading process is also essential for MC-CDMA [64]. Although a tracking

algorithm had been developed in [64], it inevitably increased the complexity of

the resulting multiuser receiver [62].

52


3.2.2 Subtractive interference cancellation

Subtractive interference cancellation is another common type of multiuser detec-

tor. The basic principle behind these detectors relies on the regeneration of the

MAI contributed from each user and then subtracting some or all of the MAI

from the desired user signal [65]. The subtraction can be performed in either a

serial or a parallel approach.

When the serial approach is taken, the multiuser detector is referred to as

the successive interference cancellation (SIC) detector [66]. Multiple stages are

required for SIC detectors. In each stage, the MAI from the strongest signal

is regenerated and then canceled out from the received signal, so that in the

next stage a smaller amount of MAI is presented. Holtzman in [67] studied the

performance of SIC detectors in DS-CDMA systems. He found that the SIC

detector can offer system performance that is significantly better than that of

the single-user detector. A similar conclusion was drawn in [68] for MC-CDMA

systems.

Parallel interference cancellation (PIC) detectors [69] take the parallel ap-

proach of interference cancellation. Similarly to SIC, PIC detectors also require

multiples stages. In each stage, an estimated data bit is produced for each user

and is then fed into the next stage. When all data estimations are correct, no

additional stage is needed and complete MAI elimination is achieved. Numerous

studies have investigated the performance of PIC detectors in DS-CDMA systems

e.g. [70, 71, 72]. PIC detectors are applied to MC-CDMA systems in [73]. All this

research shows that PIC detectors can improve system performance substantially.

However, both SIC and PIC detectors have similar drawbacks. Some of them

53


are listed below [60, 65, 74]:

• SIC detectors require the signal to be reordered continuously if the power

profile changes.

• Accurate amplitude estimations of all user signals are crucial for both de-

tectors. Any estimation error can cause additional noise.

• Extra delay is introduced; it increases linearly with the number of stages.

During high data rate transmission (e.g. 1G bps in 4G), where large delay is

prohibited, the number of stages in SIC and PIC detectors must be reduced.

As a result, the MAI can not be completely eliminated and the system

performance is degraded.

3.3 Slow frequency hopping

Due to the limitations of spreading sequence and multiuser detectors, it is neces-

sary to investigate other possible techniques that are capable of MAI reduction

for MC-CDMA. In this thesis, slow frequency hopping (SFH) technique is the

focus.

Frequency hopping technique involves changing or hopping carrier frequency

in every predetermined time interval [2]. For slow frequency hopping, the prede-

termined time interval is restricted to any values that are much larger than one

symbol duration.

SFH has been successfully implemented in DS-CDMA systems as a way to

reduce MAI during asynchronous transmission [75, 76]. SFH has also been im-

plemented in multicarrier DS-CDMA (MC-DS-CDMA) systems and has similarly

54

3.3 Slow frequency hopping

achieved a reduction in MAI [77, 78]. To the best of the author’s knowledge, the

introduction of SFH in MC-CDMA systems, as a possible solution for improving

the performance of MC-CDMA systems, has been attempted in only[25] and [26].

In both [25] and [26], the total number of available subcarrier frequencies

Q is divided into H groups. Each group has N subcarriers (where N is the

spreading factor) and the relative position for all subcarrier frequencies in the

spectrum is fixed for a given group. As a result, when hopping occurs, the whole

group of N subcarriers has to hop together. In [25] and [26], this technique

is referred to as frequency hopping MC-CDMA (FH/MC-CDMA). However, to

clearly represents the nature of hopping and to avoid confusion, in this research,

this technique is referred to as group subcarrier frequency hopping MC-CDMA

(GSFH/MC-CDMA). In [25] and [26], GSFH/MC-CDMA has been implemented

under synchronous conditions. It was found that GSFH/MC-CDMA can increase

the capacity of the system and outperform MC-CDMA in a tap delay line corre-

lated channel. However, in both papers, no MAI analysis has been undertaken.

This research gap will be filled in Chapter 4, where a thorough MAI analysis for

GSFH/MC-CDMA with asynchronous transmissions will be provided.

The problem for GSFH/MC-CDMA, however, is the condition of group hop-

ping, which is found to be unnecessary and which puts a constraint on the total

bandwidth to be multiples of N subcarrier frequencies. Therefore, in Chapter 5,

a new MC-CDMA system based on the use of SFH is proposed; this new system

is referred to as individual subcarrier frequency hopping MC-CDMA (ISFH/MC-

CDMA). In ISFH/MC-CDMA systems, each individual subcarrier can randomly

hop within the available bandwidth. As a result, at every hop, N subcarrier

frequencies are randomly chosen from the total available frequencies Q. The

55


randomness, in this case, exists for each individual subcarrier. In Chapter 5,

the analytical expression of the MAI for ISFH/MC-CDMA under asynchronous

conditions is derived, and comparison is made to that of MC-CDMA. It will be

shown that ISFH/MC-CDMA is capable of reducing MAI power and therefore

allows for better performance in terms of BER. Further, GSFH/MC-CDMA is

found to be a special case of ISFH/MC-CDMA when Q = N · H.

56

Chapter 4

Group subcarrier frequency

hopping MC-CDMA

Group subcarrier frequency hopping MC-CDMA (GSFH/MC-CDMA) has been

previously proposed in [25] and [26], and analysis has been given under syn-

chronous transmission. This chapter provides a MAI analysis for GSFH/MC-

CDMA under asynchronous transmission by considering four different detection

scenarios. These four detection scenarios represent all possible outcomes that

the MAI can generate from the interferer on the reference user. The expres-

sion of MAI and its power for GSFH/MC-CDMA in all four detection scenarios

will be derived and the total MAI power will be compared with that of basic

asynchronous MC-CDMA systems.

57

4. GROUP SUBCARRIER FREQUENCY HOPPING MC-CDMA

Figure 4.1: Asynchronous GSFH/MC-CDMA transmitter and receiver model

4.1 Asynchronous GSFH/MC-CDMA model

As shown in Fig. 4.1, the model for asynchronous GSFH/MC-CDMA is similar

to the model of asynchronous MC-CDMA described in Section 2.2. The main

difference lies in the choice of subcarrier frequencies. In basic MC-CDMA, the

subcarrier frequencies are fixed during transmission, while in GSFH/MC-CDMA

the subcarrier frequencies randomly change every Nh symbols. In the spectrum of

GSFH/MC-CDMA signals, the overall bandwidth is divided into H groups of N

orthogonal subcarrier frequencies. The configuration of the subcarrier frequency

groups is shown in Fig. 4.2.

At a given time, a user k is allocated group hk, where hk can take any integer

value between 1 and H. The group of subcarrier frequencies allocated to user k

58

4.1 Asynchronous GSFH/MC-CDMA model

Figure 4.2: Frequency spectrum of asynchronous GSFH/MC-CDMA signals

at that time is referred to as fi(hk), i = 1 . . . N and k = 1 . . . K and it is given as

fi(hk) = fi +(i − 1) + N(hk − 1)

Ts

, (4.1)

where hk is an integer random variable uniformly distributed between 1 and H.

The transmitted signal for the kth user in GSFH/MC-CDMA systems is given

as

[sk(t)]mg =N∑

i=1

√2Pbk(t)ck,i cos [2πfi(hk)t + φk,i] , (4.2)

where [ · ]mg is used to denote symbols that are exclusive to GSFH/MC-CDMA

systems. P is the signal power over each subcarrier. φk,i is the random phase

introduced by the BPSK modulator. It is assumed independent and identically

distributed (i.i.d.) for each subcarrier and each user, and is assumed uniformly

distributed over the interval [0, 2π).

The channel considered here is identical to the one considered for asynchronous

59


MC-CDMA, and to focus on the effect of GSFH, only uniform distributed timing

offset is considered. After the channel, the received signal for GSFH/MC-CDMA

can be written as

[r(t)]mg =K∑

k=1

N∑

i=1

√2Pβk,ibk(t − τk)ck,i · cos [2πfi(hk)t + ζk,i] + n(t) , (4.3)

where ζk,i = θk,i + φk,i − 2πfiτk represents the total phase.

At the receiver site, the signal is demodulated and despread. Equal gain

combining (EGC) is then applied to the despread signals: the signals received

from each subcarrier i (i = 1 . . . N) are multiplied by the gain parameter µr,i

(i = 1 . . . N) then added together. In this research, µr,i = 1 (i = 1 . . . N) is

set. The resulting signal is next passed through a matched filter, then through a

maximum likelihood detector.

The test statistic at the output of the matched filter is then composed of the

desired signal, an MAI term due to the asynchronous transmission and a noise

term due to the AWGN channel. The MAI term can be further decomposed into

two independent terms [Is]mg and [Id]mg. They represent, respectively, the MAI

term due to interference within the same subcarrier frequency, and the MAI term

due to interference within different subcarrier frequencies. In the next section,

the expressions of [Is]mg and [Id]mg and their corresponding MAI power will be

derived.

60

4.2 MAI analysis in different detection scenarios

Figure 4.3: Asynchronous GSFH/MC-CDMA detection interval with length of2Nh symbols

4.2 MAI analysis in different detection scenar-

ios

Fig. 4.3 represents two hopping intervals. Each interval has length Nh symbols.

Interferer k is delayed by a timing offset τk with respect to reference user r. In

this figure, the symbols for the reference user are denoted as br and the symbols

for the interferer are denoted as bk.

Let hr,−1 represents the subcarrier group index for the first hopping interval

and hr,0 represents the subcarrier group index for the second hopping interval.

The reference user is allocated subcarrier frequencies fi(hr,−1), i = 1 . . . N for

symbols br (−Nh + 1) to br (0), then, after the hop, subcarrier frequencies fi(hr,0),

i = 1 . . . N for symbols br (1) to br (Nh). Similarly, the interferer uses frequencies

fi(hk,−1), i = 1 . . . N for symbols bk (−Nh + 1) to bk (0), then frequencies fi(hk,0),

i = 1 . . . N for symbols bk (1) to bk (Nh).

The current detection interval is assumed to be between symbols br (1) and

61


br (Nh), and the MAI affecting these symbols will be analyzed. If br (l) is referred

to as the current symbol being detected, br (l) is affected by the corresponding

symbols in the interferer, referred to as bk (l), as well as the previous symbol,

referred to as bk (l − 1). More precisely, the MAI affecting each symbol in the

reference user can be written as the sum of four terms, the [Is]mg from bk (l) and

bk (l − 1) respectively, as well as the [Id]mg from bk (l) and bk (l − 1) respectively,

where both [Is]mg and [Id]mg are functions of the subcarrier frequencies used.

The overall MAI affecting each symbol depends on four different scenarios.

These four scenarios represent all the possible resulting MAI generating on the

reference user. By analyzing the MAI and its power generating in each scenario

and obtaining the probability of the occurrence of each scenario, the average MAI

power for GSFH/MC-CDMA can be derived. In Scenario A, the reference user

chooses a subcarrier frequency group that is identical to the one used by the

interferer in the current detection interval, but is different from the subcarrier

frequency group used by the interferer in the previous detection interval. Hence

fi(hr,0) =fi(hk,0) , (4.4)

fi(hr,0) 6=fi(hk,−1) . (4.5)

Scenario B is the opposite case of Scenario A. In Scenario B, the reference user

chooses a subcarrier frequency group that is different from the one used by the

interferer in the current detection interval, but is identical to the subcarrier fre-

62


quency group used by the interferer in the previous detection interval. Hence

fi(hr,0) 6=fi(hk,0) , (4.6)

fi(hr,0) =fi(hk,−1) . (4.7)

When the system is in Scenario C, the reference user chooses a subcarrier fre-

quency group that is different from the ones used by the interferer in both hopping

intervals which means

fi(hr,0) 6=fi(hk,0) , (4.8)

fi(hr,0) 6=fi(hk,−1) . (4.9)

The opposite of Scenario C is Scenario D, where the reference user chooses a

subcarrier frequency group that is identical to the ones used by the interferer in

both hopping intervals which gives

fi(hr,0) =fi(hk,0) , (4.10)

fi(hr,0) =fi(hk,−1) . (4.11)

In the following, the MAI power associated with each of the detection scenarios

will be analysed.

4.2.1 Scenario A

Scenario A is shown in Fig. 4.4. In this scenario, symbols br (2) to br (Nh) in

the reference user share the same subcarrier frequencies group with their cor-

63


Figure 4.4: Asynchronous GSFH/MC-CDMA detection scenario A

responding bk (l) and bk (l − 1). For example, for br (2), its corresponding bk (l)

and bk (l − 1) are symbols bk (2) and bk (1) in the interferer, which use the same

subcarrier frequency group as br (2). The situation here is identical to the ba-

sic MC-CDMA, where symbols in the reference user and the interferer share the

same frequencies. Hence the calculation of the MAI for br (2) is the same as the

one shown in basic MC-CDMA, and two types of MAI can be found.

Let[IAs

]

mgand

[IAd

]

mgrepresents the two types of MAI found in br (2). The

64


MAI affecting symbol br (2) can be derived as

[IAs

]

mg=

K∑

k=1;k 6=r

N∑

i=1

∫ (l+1)Ts

lTs

√2Pβk,ibk (l)ck,icr,i · cos [2πfi (hk,l) t + ζk,i]

· cos [2πfi (hr,l) t] dt

=K∑

k=1;k 6=r

N∑

i=1

√P

2ck,icr,i cos ζk,iβk,i [bk (l − 1) τk + bk (l) (Ts − τk)] , (4.12)

[IAd

]

mg=

K∑

k=1;k 6=r

N∑

i=1

N∑

j=1;j 6=r

∫ (l+1)Ts

lTs

√2Pβk,jbk (l)ck,jcr,i · cos [2πfj (hk,l) t + ζk,j]

· cos [2πfi (hr,l) t] dt

=K∑

k=1;k 6=r

N∑

i=1

N∑

j=1;j 6=i

√2PTsβk,jck,jcr,i

4π∆i,j

[bk (l − 1) − bk (l)]

·[sin

(2π∆i,j

Ts

τk

)− sin (ζk,j)

], (4.13)

where superscript A refers to scenario A, l can take any integer value and ∆i,j =

i− j. The MAI powers associated with these two terms are given respectively as

V ar[IAs

]

mg= (K − 1)

NPT 2s σ2

6, (4.14)

V ar[IAd

]

mg= (K − 1)

PT 2s σ2

4π2

N∑

i=1

N∑

j=1;j 6=i

1

∆2i,j

. (4.15)

Because symbols br (3) to br (Nh) in the reference user are in the same situation

as symbols br (2), their MAI powers are identical to (4.14) and (4.15).

For symbol br (1), however, the MAI analysis is different. Due to subcarrier

frequency hopping, the corresponding bk (l) and bk (l − 1) for symbol br (1), i.e.

symbol bk (1) and bk (0) in the interferer, are on different subcarrier frequencies.

As a result, the expressions of MAI generated by bk (1) and bk (0) are different

65


and they need to be derived separately.

The two types of MAI generated by bk (0) in the interferer can be derived as

[IA,1s,l−1

]

mg=

K∑

k=1;k 6=r

N∑

i=1

∫ lTs+τk

lTs

√2Pβk,ibk (l − 1) ck,icr,i

· cos [2πfi (hk,l−1) t + ζk,i] · cos [2πfi (hr,l) t] dt

=K∑

k=1;k 6=r

N∑

i=1

√2PTsβk,ibk (l − 1) ck,icr,i

4πδ

·[sin

(2πδ

Ts

τk + ζk,i

)− sin ζk,i

], (4.16)

[IA,1d,l−1

]

mg=

K∑

k=1;k 6=r

N∑

i=1

N∑

j=1;j 6=i

∫ lTs+τk

lTs

√2Pβk,jbk (l − 1) ck,jcr,i

· cos [2πfj (hk,l−1) t + ζk,j] · cos [2πfi (hr,l) t] dt

=K∑

k=1;k 6=r

N∑

i=1

N∑

j=1;j 6=i

√2PTsβk,jbk (l − 1) ck,jcr,i

4π (∆i,j + δ)

·[sin

(2π (∆i,j + δ)

Ts

τk + ζk,j

)− sin ζk,j

], (4.17)

where δ = N · (hk,−1 − hr,0). Meanwhile another two types of MAI are generated

66


by bk (1) in the interferer, given respectively as

[IA,1s,l

]

mg=

K∑

k=1;k 6=r

N∑

i=1

∫ (l+1)Ts

lTs+τk

√2Pβk,ibk (l) ck,icr,i

· cos [2πfi (hk,l) t + ζk,i] · cos [2πfi (hr,l) t] dt

=K∑

k=1;k 6=r

N∑

i=1

√P

2Tsβk,ibk (l) ck,icr,i cos ζk,i , (4.18)

[IA,1d,l

]

mg=

K∑

k=1;k 6=r

N∑

i=1

N∑

j=1;j 6=i

∫ lTs+τk

lTs

√2Pβk,jbk (l) ck,jcr,i

· cos [2πfj (hk,l) t + ζk,j] · cos [2πfi (hr,l) t] dt

=K∑

k=1;k 6=r

N∑

i=1

N∑

j=1;j 6=i

√2PTsβk,jbk (l) ck,icr,i

4π∆i,j

·[sin ζk,j − sin

(2π (∆i,j + δ)

Ts

τk + ζk,j

)]. (4.19)

Taking the summation of the MAI contributed from bk (0) and bk (1) in the in-

terferer then

[IA,1s

]

mg=[IA,1s,l−1

]

mg+[IA,1s,l

]

mg, (4.20)

[IA,1d

]

mg=[IA,1d,l−1

]

mg+[IA,1d,l

]

mg. (4.21)

As a result the corresponding MAI powers affecting symbol br (0) in the reference

67


user are given as

V ar[IA,1s

]

mg= (K − 1)

NPT 2s σ2

2

(1

4π2δ2+

1

6

)

=γ1V ar[IAs

]

mg, (4.22)

V ar[IA,1d

]

mg= (K − 1)

PT 2s σ2

8π2

N∑

i=1

N∑

j=1;j 6=i

1

(∆i,j + δ)2 +1

∆2i,j

=γ2V ar[IAd

]

mg, (4.23)

where

γ1 =3

4π2δ2+

1

2, (4.24)

γ2 =1

2

N∑i=1

N∑j=1;j 6=i

1(∆i,j+δ)2

N∑i=1

N∑j=1;j 6=i

1∆2

i,j

+ 1

. (4.25)

Therefore the average MAI power over Nh symbols in Scenario A is given by

V ar[IA]

mg=

1

Nh

(γ1V ar

[IAs

]

mg+ γ2V ar

[IAd

]

mg

)

+Nh − 1

Nh

(V ar

[IAs

]

mg+ V ar

[IAd

]

mg

). (4.26)

4.2.2 Scenario B

Fig. 4.5 demonstrates the detection Scenario B for GSFH/MC-CDMA. In this

scenario, because for all symbols from br (2) to br (Nh) the subcarrier frequencies

used by the reference user are different from the ones used by interferers, there is

no [Is]mg and only one type of MAI, [Id]mg, presents.

68


Figure 4.5: Asynchronous GSFH/MC-CDMA dectection scenario B

In Scenario B, the MAI affecting each symbols from br (2) to br (Nh) in the

reference user, is calculated as

[IBd

]

mg=

K∑

k=1;k 6=r

N∑

i=1

N∑

j=1

∫ (l+1)Ts

lTs

√2Pβk,jbk (l)ck,jcr,i

· cos [2πfj (hk,l) t + ζk,j] · cos [2πfi (hr,l) t] dt (4.27)

For ease of presentation, the MAI obtained in (4.27) is decomposed into two

independent terms referred to as[IBd,1

]

mgand

[IBd,2

]

mg.[IBd,1

]

mgis the result of

(4.27) when j = i in the double summation term. In contrast,[IBd,2

]

mgis the

69


result of (4.27) when j 6= i. They can be obtained respectively as

[IBd,1

]=

K∑

k=1;k 6=r

N∑

i=1

√2Pβk,iTsck,jcr,i

4πδ[bk (l − 1) − bk (l)]

·[sin

(2πδ

Ts

τk

)− sin ζk,i

], (4.28)

[IBd,2

]=

K∑

k=1;k 6=r

N∑

i=1

N∑

j=1;j 6=i

√2Pβk,jTsck,jcr,i

4π (∆i,j + δ)· [bk (l − 1) − bk (l)]

·[sin

(2πδ (∆i,j + δ)

Ts

τk

)− sin ζk,j

]. (4.29)

and their respective power is obtained as

V ar[IBd,1

]

mg= (K − 1)

NPT 2s σ2

4π2δ2

=γ3V ar[IAs

], (4.30)

V ar[IBd,2

]

mg= (K − 1)

PT 2s σ2

8π2

N∑

i=1

N∑

j=1;j 6=i

1

(∆i,j + δ)2

=γ4V ar[IAd

]. (4.31)

where

γ3 =3

2π2δ2, (4.32)

γ4 =

N∑i=1

N∑j=1;j 6=i

1(∆i,j+δ)2

N∑i=1

N∑j=1;j 6=i

1∆2

i,j

. (4.33)

For symbol br (1), its corresponding bk (l) and bk (l − 1), i.e. bk (1) and bk (0),

are on different subcarrier frequencies. In addition, the subcarrier frequencies in

bk (0) are the same as the subcarrier frequencies used by the reference user in

70


the current detection interval. Hence, unlike other symbols, symbol br (1) suffers

both types of MAI and they should be derived in the same way as in Scenario A.

As a result,

V ar[IB,1s

]

mg=V ar

[IA,1s

]

mg= γ1V ar

[IAs

]

mg, (4.34)

V ar[IB,1d

]

mg=V ar

[IA,1d

]

mg= γ2V ar

[IAd

]

mg. (4.35)

Finally the average MAI power over Nh symbols in Scenario B can be obtained

as

V ar[IB]

mg=

1

Nh

(γ1V ar

[IAs

]

mg+ γ2V ar

[IAd

]

mg

)

+Nh − 1

Nh

(γ3V ar

[IAs

]

mg+ γ4V ar

[IAd

]

mg

). (4.36)

4.2.3 Scenario C

As shown in Fig. 4.6, in Scenario C, because all symbols in the reference user

and interferer use different subcarrier frequencies, only the second type of MAI,

[Id]mg can be found. The MAI power affecting all symbols including symbol br (1)

in the reference user is derived in the same way as was done in Scenario B for

br (2). Again the MAI in this scenario is divided into two individual terms. The

power associated with each of these two MAI terms is given below

V ar[ICd,1

]

mg= V ar

[IBd,1

]

mg= γ3V ar

[IAs

]

mg, (4.37)

V ar[ICd,2

]

mg= V ar

[IBd,2

]

mg= γ4V ar

[IAd

]

mg. (4.38)

71


Figure 4.6: Asynchronous GSFH/MC-CDMA detection scenario C

As a result the average MAI power for Scenario C can be obtained as

V ar[IC]

mg=V ar

[ICd,1

]

mg+ V ar

[ICd,2

]

mg

=γ3V ar[IAs

]

mg+ γ4V ar

[IAd

]

mg. (4.39)

4.2.4 Scenario D

The last detection scenario, Scenario D, for GSFH/MC-CDMA is shown in Fig.

4.7. In this scenario, all symbols in the reference user and interferer share the

same subcarrier frequencies. Hence, for all symbols in the reference user, their

affecting MAI are identical to the ones calculated for symbol br (2) in Scenario

A. Therefore, the average MAI power in Scenario D is given as

V ar[ID]

mg= V ar

[IAs

]

mg+ V ar

[IAd

]

mg(4.40)

72

4.3 Average overall MAI Power

Figure 4.7: Asynchronous GSFH/MC-CDMA detection scenario D

4.3 Average overall MAI Power

From the above analysis, γ1, γ2, γ3 and γ4 are functions of δ = N · (hk,−1 − hr,0),

determined by the distance in the spectrum between two hopping patterns in

adjacent hopping intervals. Since the subcarrier frequencies group is assumed to

hop following a uniform pattern, the probability density function (PDF) of δ is

derived as

prob (|δ|) =

2NH

(NH−|δ|

H−1

)for 0 ≤ |δ| ≤ N (H − 1)

0 elsewhere, (4.41)

73


where | · | represents the absolute value operator. Because γ1 and γ2 can be

expressed as

γ1 =1

2γ3 +

1

2, (4.42)

γ2 =1

2γ4 +

1

2. (4.43)

Only the average values of γ3 and γ4 are required. The results are

γ3 =N(H−1)∑

δ=0

prob (|δ|) ·(

3

2π2δ2

), (4.44)

γ4 =N(H−1)∑

δ=0

prob (|δ|) ·

N∑i=1

N∑j=1;j 6=i

1(∆i,j+δ)2

N∑i=1

N∑j=1;j 6=i

1∆2

i,j

. (4.45)

After considering the four scenarios in Section 4.2, the average overall MAI

power expression can now be obtained by multiplying the MAI power by its

corresponding probability and then taking the summation. The resulting average

MAI power is

V ar [I]mg =Eγ

[V ar

[IA]

mg· prob (A)

]+ Eγ

[V ar

[IB]

mg· prob (B)

]

+ Eγ

[V ar

[IC]

mg· prob (C)

]+ V ar

[ID]

mg· prob (D) , (4.46)

where Eγ[·] represents the expectation operation respect to γ3 and γ4. prob(A)

denotes the probability of Scenario A which is given as

prob(A) =H − 1

H2. (4.47)

74

4.4 Simulation Results

It represents the probability of choosing fi(hr,0) = fi(hk,0) and fi(hr,0) 6= fi(hk,−1).

Similarly, prob(B) represent the probability of Scenario B given as

prob(B) =H − 1

H2, (4.48)

and it represents the probability of choosing fi(hr,0) = fi(hk,−1) and fi(hr,0) 6=

fi(hk,0). The probability of Scenario C is given as

prob(C) =(H − 1)2

H2, (4.49)

and it represents the probability of both fi(hr,0) 6= fi(hk,−1) and fi(hr,0) 6=

fi(hk,0). Finally, the probability of Scenario D is given as

prob(D) =1

H2, (4.50)

Substituting (4.44), (4.45), (4.47), (4.48), (4.49) and (4.50) into (4.46), the

overall MAI power can be obtained as

V ar [I]mg =1

H

(V ar

[IAs

]

mg+ V ar

[IAd

]

mg

)

+H − 1

H

(γ3V ar

[IAs

]

mg+ γdV ar

[IAs

]

mg

). (4.51)

It is interesting to note from (4.51) that the MAI power in asynchronous

GSFH/MC-CDMA systems is independent of the hopping interval Nh.

75


Figure 4.8: GSFH/MC-CDMA MAI power ratio for different number of subcarrierfrequency groups (N=K=16)

Figure 4.9: GSFH/MC-CDMA MAI power ratio for different spreading factors

76



In this section the results of Monte-Carlo simulation based on the asynchronous

GSFH/MC-CDMA model described in Section 4.1 are shown. A full user system

i.e. K = N was considered. The second-order moment of the Rayleigh parameter

is set to unity. Walsh-Hadamard spreading sequences are applied and the number

of realizations is set to be 100,000. The simulation results are shown in Fig. 4.8

and 4.9, with the theoretical results (shown as continuous lines) matching very

closely to the Monte-Carlo simulation results (shown as markers).

The performance of the conventional asynchronous MC-CDMA with a uni-

formly distributed timing offset is used as a benchmark. In order to describe

the MAI power reduction effect of GSFH/MC-CDMA systems, a new term, MAI

power ratio [Θ]mg is introduced. This term is defined as

[Θ]mg =V ar [I]mg

V ar [I]mc

, (4.52)

which is the ratio between the MAI power obtained from asynchronous GSFH/MC-

CDMA and the MAI power obtained from conventional asynchronous MC-CDMA.

To achieve better system performance, it is desirable to have the MAI power ratio

as low as possible. (Note: From this chapter, for simplicity reason, V ar [I]mc is

used to represent the average MAI power for MC-CDMA instead of E[I2]

mc)

In Fig. 4.8, showing the MAI power ratio for asynchronous GSFH/MC-CDMA

with different values of H, the MAI power ratio decreases with the increasing

number of subcarrier frequency groups H. This result indicates that, by allocat-

ing extra frequencies and allowing the subcarrier to hop, GSFH can successfully

reduce the amount of MAI power for asynchronous MC-CDMA systems.

77


Moreover, the amount of MAI power reduction is not affected by the spreading

factors. As shown in Fig. 4.9, where the MAI power ratio is plotted against

different spreading factors, N , the MAI power ratio for GSFH/MC-CDMA is

nearly unchanged for systems with different spreading factors.

4.5 Discussions

In previous sections, it was shown that by allowing the subcarriers to hop as

a group, MAI power reduction is achieved. Further, it was found that with the

number of groups increased, the MAI power for GSFH/MC-CDMA reduced which

suggesting better performance than the conventional MC-CDMA. However, this

MAI power reduction comes with several costs. Firstly, as shown in Fig. 4.1, both

of the transmitter and receiver of GSFH/MC-CDMA require the installment of

frequency synthesizers to determined the group hopping pattern for subcarriers.

Secondly, hopping pattern information is required to be known on both transmit-

ter and receiver. Both of these two requirements increase the complexity of the

system.

Finally, in GSFH/MC-CDMA, the whole group of subcarriers has to hop to-

gether every hopping interval. As a result, the frequency spectrum for GSFH/MC-

CDMA system has to be equal to the multiple of the spreading factor N and num-

ber of groups available H. In situations where this frequency spectrum require-

ment is not satisfied, GSFH/MC-CDMA cannot be applied. To overcome this

disadvantage, a new developed system namely, individual subcarrier frequency

hopping MC-CDMA (ISFH/MC-CDMA, is proposed in the next chapter.

78

Chapter 5

Individual subcarrier frequency

hopping MC-CDMA

In Chapter 4, group subcarrier frequency hopping MC-CDMA (GSFH/MC-CDMA)

is introduced. Although MAI power reduction is achieved, GSFH/MC-CDMA has

a limitation of inflexible frequency spectrum requirement, restricting the widely

application of GSFH/MC-CDMA. In this chapter, the condition of group hop-

ping for group subcarrier frequency hopping MC-CDMA (GSFH/MC-CDMA) is

dropped and a system with a more general form of subcarrier frequency hop-

ping is proposed. This chapter then continues with the performance analysis for

the proposed system, which is referred to as the individual subcarrier frequency

hopping MC-CDMA (ISFH/MC-CDMA).

79

5. INDIVIDUAL SUBCARRIER FREQUENCY HOPPINGMC-CDMA

Figure 5.1: Asynchronous ISFH/MC-CDMA transmitter and receiver Model

5.1 Asynchronous ISFH/MC-CDMA model

Fig. 5.1 shows the model for the transmitter and receiver of the proposed asyn-

chronous ISFH/MC-CDMA systems. This model is similar to the asynchronous

MC-CDMA model described in Section 2.2, as well as to the GSFH/MC-CDMA

model described in Section 4.1. The main difference between them lies in the

transmission subcarrier frequencies. With basic MC-CDMA the transmission

subcarrier frequencies remain unchanged during transmission. With GSFH/MC-

CDMA, the total bandwidth is divided into H groups of subcarrier frequencies,

and at every hop the transmitter randomly chooses one of the H groups as the

transmission frequency group. The subcarrier frequencies are still hopping in

ISFH/MC-CDMA but the frequency choice process is different. In ISFH/MC-

CDMA, the total bandwidth is divided into Q orthogonal frequencies. At every

80

5.1 Asynchronous ISFH/MC-CDMA model

hop, the ISFH/MC-CDMA transmitter randomly chooses N subcarrier frequen-

cies from all the available Q frequencies, Q ≥ N , where N represents the spreading

factor and the number of transmitting subcarriers.

The expression of the randomly hopped subcarrier frequencies for the kth user

is

[fk,i]mi = fi +pk,i

Ts

for i = 1 . . . N , (5.1)

where [ · ]mi is used to indicate symbols for ISFH/MC-CDMA systems, and pk,i

is a random integer with values ranging between 0 and Q − 1. The sequence

pk = [pk,1, pk,2, . . . , pk,N ] is referred to as the frequency hopping pattern for the

kth user. pk is the output of the frequency synthesizer and it changes every

Nh × Ts where Nh is the number of symbols transmitted per hop. For slow

frequency hopping, Nh is restricted to taking any positive integer values that are

larger than 1. For a given user, all N subcarrier frequencies must be different

during the same hopping interval.

The transmitted signal for the kth user is then given by

[sk(t)]mi =N∑

i=1

√2Pbk(t − τk)ck,i(t − τk) cos [2πfk,i(t − τk) + φk,i] , (5.2)

When comparing (5.2) to (2.2), the expressions of the transmitted signal in

ISFH/MC-CDMA and MC-CDMA are similar. The only difference between them

is the expression of the subcarrier frequency.

The channel considered in this case is the same as the one considered in the

case of asynchronous MC-CDMA. The received signal for asynchronous ISFH/MC-

81


CDMA is

[r(t)]mi =K∑

k=1

[rk(t)]mi , (5.3)

where [rk(t)]mi is the received signal for the kth user, and is given as

[rk(t)]mi =N∑

i=1

√2Pβk,ibk(t − τk)ck,i(t − τk) cos(2π [fk,i]mi t + ζk,i) + n(t). (5.4)

At the receiver site, the signal is first dehopped according to the reference

user’s hopping pattern, pr, and then passed through a MC-CDMA receiver. The

signal at the output of the matched filter also has three components: the desired

signal, a MAI term and an AWGN noise term. In the next section, the expressions

of the MAI term for the asynchronous ISFH/MC-CDMA are derived.

5.2 MAI analysis for ISFH/MC-CDMA

Unlike MC-CDMA, where all users share the same N subcarrier frequencies, in

ISFH/MC-CDMA the reference user and a given interferer might only share q

subcarrier frequencies with q ≤ N , where q is random and is referred to as the

number of hits. Because the MAI received in the q subcarriers have different

characteristics from the MAI received in the remaining N − q subcarriers, for the

purpose of MAI analysis, the subcarriers in the reference user are divided into

two groups.

The first group consists of the q subcarriers that have frequencies shared with

the interferer. The MAI calculations for this group of subcarriers are similar

to those of the MAI in MC-CDMA. The expressions of the two types of MAI

affecting this groups are similar to those of (2.10) and (2.11), except that the

82


number of subcarrier is q instead of N . Hence

[Is]mi =K∑

k=1;k 6=r

q∑

i=1

Is,k (i) (5.5)

[Id,1]mi =K∑

k=1;k 6=r

q∑

i=1

N∑

j=1;j 6=i

Ik,d (i, j) , (5.6)

where [Is]mi represents the MAI generated by the subcarriers with the same fre-

quencies in the interferer, and [Id,1]mi represents the MAI generated by the sub-

carriers with different frequencies in the interferer. In (5.5) and (5.6), Is,k(i) and

Id,k(i, j) are given in (2.12) and (2.13) respectively.

In the second group of subcarriers, no common frequencies are shared between

the reference user and interferers. In the case of q hits, there will be N − q

subcarriers that can be found in this group. Unlike the first group, only one type

of MAI can be found here and it is referred to [Id,2]mi. The source of [Id,2]mi in

ISFH/MC-CDMA is the same as [Id]mc in MC-CDMA, i.e. it is due to the existing

of subcarriers with different frequencies in interferers. Hence the calculation of

[Id,2]mi can be based on the calculation of [Id]mc stated in (2.11), which leads to

[Id,2]mi =K∑

k=1;k 6=r

N∑

i=q+1

N∑

j=1

Ik,d (i, j) (5.7)

where Id,k(i, j) is given in (2.13).

As stated in Section 2.2, Is,k(i) and Id,k(i, j) are uncorrelated for different i

and k. Hence [Is,k]mi, [Id,k,1]mi and [Id,k,2]mi are also uncorrelated. As a result,

[Is,k]mi, [Id,k,1]mi and [Id,k,2]mi can be approximated as Gaussian random variables.

By taking the expectation respect to βk,i, bk and ζk,i, the mean of [Is,k]mi, [Id,k,1]mi

83


and [Id,k,2]mi are found to be zero and their powers are derived respectively as

V ar [Is]mi = (K − 1)qPT 2

s σ2

6(5.8)

V ar [Id,1]mi = (K − 1)PT 2

s σ2

4π2

q∑

i=1

N∑

j=1;j 6=i

1

∆2i,j

(5.9)

V ar [Id,2]mi = (K − 1)PT 2

s σ2

4π2

N∑

i=q+1

N∑

j=1

1

∆2i,j

. (5.10)

Therefore the total power of the MAI for asynchronous ISFH/MC-CDMA is

the sum of (5.8), (5.9) and (5.10), which is given as

V ar [I|q, ∆i,j]mi = V ar [Is]mi + V ar [Id,1]mi + V ar [Id,2]mi . (5.11)

In (5.11) above, V ar [I|q, ∆i,j]mi is conditional to the number of hits q and the

spectral separation between subcarriers ∆. In order to obtain the average MAI

power the probability density functions (PDF) of q and ∆ are required.

The probability of choosing N out of Q available subcarrier frequencies and

having q hits follows the hypergeometric distribution [79] and is given by

prob(q) =

N

q

Q − N

N − q

Q

N

. (5.12)

84


The average value of q is given by [79]

E[q] =N∑

q=qmin

q · prob(q) =N2

Q(5.13)

where qmin = 2N − Q is the minimum number of hits for a given Q and N .

In MC-CDMA, the spectral distance ∆i,j is deterministic, whereas in ISFH/MC-

CDMA, ∆i,j is a random variable with its absolute value ranging between 1 and

(Q − 1). The general form of the probability density function (PDF) for |∆i,j|

can be found as

prob (∆i,j) =

2(Q−|∆i,j |)

Q2−Qfor (1) ≤ |∆| ≤ (Q − 1)

0 elsewhere(5.14)

In (5.14) | · | represents the absolute value.

After obtaining the PDF of q and ∆i,j,the average power of the total MAI can

be calculated by taking the expectation of (5.11) further with respect to q and

∆i,j. The result is derived as (Proof can be found in Appendix B)

V ar [I]mi = (K − 1)Pσ2T 2

s N2

6Q

1 +3

π2Q

Q−1∑

i=1

Q

i2−

Q−1∑

i=1

1

i

. (5.15)

Meanwhile, the average MAI power for MC-CDMA, derived in (2.28), can be

rewritten as

V ar [I]mc = (K − 1)Pσ2T 2

s N

6

[1 +

3

π2N

(N−1∑

i=1

N

i2−

N−1∑

i=1

1

i

)], (5.16)

85


where the following relation is applied

N∑

i=1

N∑

j=1;j 6=i

1

∆2= 2

(N−1∑

i=1

N

i2−

N−1∑

i=1

1

i

). (5.17)

5.3 Special case: GSFH/MC-CDMA

In the GSFH/MC-CDMA system stated in Chapter 4, one out of H groups of

subcarriers is chosen for each user. Because each group has N subcarriers, the

total number of subcarriers used in GSFH/MC-CDMA is N · H. In ISFH/MC-

CDMA, N subcarriers are randomly chosen from the pool of Q available sub-

carriers for each user, provided Q > N . Since there is no restriction on the

selection of subcarrier frequencies in ISFH/MC-CDMA, when the pool of sub-

carriers in ISFH/MC-CDMA is equal to the total number of subcarriers required

for GSFH/MC-CDMA , i.e. Q = N · H, the spectrum of the GSFH/MC-CDMA

signal is equivalent to one of the possible outcomes resulting from the random

selection of subcarriers for ISFH/MC-CDMA. Hence GSFH/MC-CDMA can be

viewed as a special case of ISFH/MC-CDMA. As a result, when Q = N · H, the

MAI power for asynchronous GSFH/MC-CDMA is the same as the MAI power

for asynchronous ISFH/MC-CDMA, i.e.

V ar [I]mg = V ar [I]mi when Q = N · H (5.18)

where V ar [I]mg refers to the average MAI power for asynchronous GSFH/MC-

CDMA. The proof of this equation can be easily obtained by substituting Q =

N ·H into (5.15) and comparing the result with the expression of V ar [I]mg given

86

5.4 MAI power comparison between asynchronous MC-CDMA andasynchronous ISFH/MC-CDMA

in (4.51).

Based on this result, it can be further concluded that the configuration of

subcarrier groups in GSFH/MC-CDMA does not affect the amount of MAI power

as long as full user systems (i.e. K = N) are considered. This is because any

configuration of subcarrier groups for GSFH/MC-CDMA is just another example

of a possible outcome from the random subcarrier selection in ISFH/MC-CDMA

and its MAI power is again equivalent to the MAI power in ISFH/MC-CDMA

for Q = N · H.

5.4 MAI power comparison between asynchronous

MC-CDMA and asynchronous ISFH/MC-

CDMA

In order to describe the MAI power reduction effect of ISFH/MC-CDMA sys-

tem, the MAI power of the basic asynchronous MC-CDMA system is used as a

benchmark and MAI power ratio, [Θ]mi, is introduced, which is defined as

[Θ]mi =V ar [I]mi

V ar [I]mc

=N

Q

[1 + 3

π2Q

(Q−1∑i=1

Qi2−

Q−1∑i=1

1i

)]

[1 + 3

π2N

(N−1∑i=1

Ni2−

N−1∑i=1

1i

)] (5.19)

where V ar [I]mc is defined in (5.16) and V ar [I]mi is defined in (5.15). In (5.19),

when Q = N , ISFH/MC-CDMA generates the same amount of MAI power as

MC-CDMA systems. For a given N , the MAI power ratio decreases when Q

87


increases. When N is large, [Θ]mi approaches to a constant value

[Θ]mi ≈N

Q. (5.20)

The proof of (5.20) can be found in Appendix C.

5.5 Bit error rate analysis

In this section, the MAI power derived in the previous sections will be used to

obtain the bit error rate (BER) for ISFH/MC-CDMA systems. To derive the

BER expression, the mean and variance of the test statistic are required. The

mean of the test statistic E [Z] is the desired signal component D. The expression

of D in ISFH/MC-CDMA are identical to the one in M-CDMA and it is given in

(2.6). Hence for both MC-CDMA and ISFH/MC-CDMA

E [Z] = D =

√P

2Tsbr(t)Br , (5.21)

where

Br =N∑

i=1

βr,i . (5.22)

The corresponding variance of Z is the summation of AWGN variance and MAI

power. The AWGN variance is given in (2.8) and it has equal value for both

MC-CDMA and ISFH/MC-CDMA. Hence

V ar [η]mc = V ar [η]mi . (5.23)

88

5.6 Simulation results

The MAI power for ISFH/MC-CDMA is given in (5.15). Hence the variance of

Z can be written as

V ar [Z]mi = V ar [η]mi + V ar [I]mi . (5.24)

The general form of bit error rate (BER) conditional to Br is given by

pe|Br =Q

√√√√ E2 [Z]

V ar [Z]

=B2

r

(K − 1) σ2N2

3Q

[1 + 3

π2Q

(Q−1∑i=1

Qi2−

Q−1∑i=1

1i

)] , (5.25)

where Q(·) represents the Q function. This BER, however, is conditional to Br

which represents the sum of N Rayleigh fading coefficients in the reference user.

To find the average BER, expectation is taken in respect to Br and it is calculated

as

E [pe] =∫ +∞

−∞pe|Br · prob(Br)dBr , (5.26)

where prob(Br) represents the PDF for Br and is given in (2.45). It has been

derived in Appendix A. Numerical methods such as Monte Carlo integration [43]

can be applied to evaluate the integral in (5.26).


Monte-Carlo simulations based on the asynchronous ISFH/MC-CDMA model

described in Section 5.1 were used to verify the theory developed in this chapter.

89


Figure 5.2: MAI power ratios (as a percentage) for asynchronous ISFH/MC-CDMA systems with different values of Q and N

Full user systems are considered i.e. N = K. The second order moments of the

Rayleigh fading coefficients for all channels are set to unity. 100, 000 realizations

were used for the Monte-Carlo simulations. The simulation results of the MAI

power ratio are shown in Fig. 5.2 and Fig. 5.3. In all figures, the theoretical

values are shown as continuous lines and the Monte-Carlo simulation results are

shown as markers.

5.6.1 MAI power ratio

The MAI power ratio for asynchronous ISFH/MC-CDMA is shown in Fig. 5.2. In

this figure, a fixed number of extra subcarrier frequencies (i.e. Q−N) is allocated

(e.g. 2, 5, 10 and 20). As it shows, when no extra subcarrier frequencies is allo-

cated, i.e. Q = N , the MAI power ratios (shown as a percentage) are remaining

as in 100% for all values of N and K, indicating that no MAI power reduction

occurs, and introducing slow frequency hopping in this case does not reduce the

MAI power. When two extra subcarrier frequencies, i.e. Q = N +2, are allocated

90


Figure 5.3: MAI power ratio for asynchronous GSFH/MC-CDMA and ISFH/MC-CDMA systems for N=K=8

to enable hopping, the MAI power ratios start to drop, which suggests that the

MAI power in ISFH/MC-CDMA is now lower than the basic MC-CDMA. Then

when more extra subcarrier frequencies are allocated for hopping, the resulting

MAI power ratios are lower. For example, when N = K = 32 and an extra 20

subcarrier frequecies are allocated, i.e. Q = N +20, the MAI power in ISFH/MC-

CDMA is only 60% of the MAI power in the basic MC-CDMA. Therefore, for

a given N and K, the MAI power ratio decreases as the number of available

subcarriers Q increases; however, this is at the expense of bandwidth. It can

also be seen that the MAI power reduction effect of using subcarrier frequency

hopping is more significant for small spreading factors (a small number of users).

On the other hand, where large spreading factors and a large number of users are

applied, the MAI power ratio tends to approach its limit, defined in (5.20).

The MAI power ratios for asynchronous GSFH/MC-CDMA and ISFH/MC-

CDMA are shown in Fig. 5.3. It can be seen that at the left end, where the

number of available subcarrier Q equals the spreading factor N , both GSFH/MC-

91


CDMA and ISFH/MC-CDMA are equivalent to the basic MC-CDMA, hence

the MAI power ratio is 100%. When Q increases, the MAI power ratios for

GSFH/MC-CDMA and ISFH/MC-CDMA decrease. It should be noted that at

the point where Q = N · H, the amounts of MAI power generated from asyn-

chronous GSFH/MC-CDMA and ISFH/MC-CDMA are identical, which verifies

the validity of (5.18).

5.6.2 Bit error rate

Fig. 5.4 shows the bit error rate performance for the asynchronous ISFH/MC-

CDMA system with N = K = 16 and the correlation coefficient for subcarrier

fading is set to 0.7. The BER for basic asynchronous MC-CDMA is also shown

in Fig. 5.4 for comparison purposes. It can be seen that the introduction of

the SFH technique to MC-CDMA can improve the BER performance due to

the reduction of MAI power. Furthermore, with an increase for Q, the BER of

ISFH/MC-CDMA can be further reduced.

5.7 Discussion

In Sections 5.2 to 5.6, it has been shown that the introduction of SFH to MC-

CDMA can improve the performance of the system by generating less MAI power.

This improvement, however, is at the expense of spectral widening because extra

subcarrier frequencies must be allocated to the system to enable hopping. In the

situation when a fixed bandwidth is given and when the MAI is so large that the

system performance (e.g. BER) cannot meet the minimum quality requirement,

the capacity of the system must be reduced, i.e. the maximum number of users

92

5.7 Discussion

Figure 5.4: Bit error rate performance for asynchronous ISFH/MC-CDMA with(N = K = 16 and the correlation coefficient for subcarrier fading is 0.7)

must be less than the number of available subcarriers. In this case, the use

of ISFH/MC-CDMA will allow for a higher capacity than that achieved by the

standard MC-CDMA system.

93


94

Chapter 6

Asynchronous MC-CDMA with

base station polarization diversity

This chapter firstly provides the literature review for polarization diversity. Com-

parisons between polarization diversity and other diversity techniques are given.

Then the two factors affecting polarization diversity are discussed, followed by the

review of the existing diversity combining technique. After the literature review,

this chapter proposes a new system which combines the MC-CDMA system with

a base station polarization diversity scheme. The new system will be referred to

as Pol/MC-CDMA. The performance of Pol/MC-CDMA in asynchronous trans-

mission is analyzed and, in particular, an expression for the multiple access inter-

ference (MAI) is derived. Furthermore, this chapter also proposes a new diver-

sity combining technique, namely maximal signal-to-MAI ratio combining (MS-

MAIRC), to combine the signals in two diversity antennas for Pol/MC-CDMA.

Finally, in this chapter the effects of antenna angles are studied and the optimal

antenna angles for both MRC and MSMAIRC are derived.

95

6. ASYNCHRONOUS MC-CDMA WITH BASE STATIONPOLARIZATION DIVERSITY

6.1 Polarization diversity versus other diversity

techniques

During wireless communication transmission, the envelope amplitudes of the re-

ceived signals are subjected to fading due to multipath propagation. To reduce

the fading effect, various diversity techniques have been developed. The most

commonly used techniques are 1) time diversity, 2) frequency diversity, 3) space

diversity and 4) polarization diversity.

The simplest form of diversity is time diversity, which involves transmitting

information repeatedly at a time spacing that exceeds the coherence time of the

channel [1]. The time-spacing set-up is to make sure that the signals carrying

repeated information are subjected to independent fading in different transmis-

sion time. Despite its simplicity, this form of diversity is wasting important time

communication resources [80].

Frequency diversity involves transmitting the same information on two or

more carrier frequencies. If these frequencies are separated by more than the co-

herence bandwidth of the channel, the signals on different transmitted frequencies

are approximately uncorrelated [81]. However, frequency diversity is expensive in

terms of transmitters and required bandwidth. It has hardly been used because

frequency bandwidth is one of the most scarce resources in wireless communica-

tions [1].

Space diversity is shown to be more economical in terms of communication

resources, than both time and frequency. With space diversity, the same infor-

mation is transmitted in two antennas that are separated in space [80]. However,

space diversity has spacing issues in the base station (BS) because BS is nor-

96

6.2 The mechanism of polarization diversity

mally elevated above most local reflectors, signals arrive through narrow angles

and hence need wide separation in space to keep the correlation coefficient be-

tween signals below 0.7 [82], the threshold for an effective diversity scheme [80].

Experimental results show that space diversity at a BS requires antenna spacing

of up to about 20 to 30 wavelengths for the broadside case and more for the

in-line case [83, 82]. Giving the limited size of BS, space diversity is impractical.

Polarization diversity technique has overcome all the disadvantages of the

previous three techniques. It involves transmitting and receiving the same infor-

mation with two antennas that are in vertical and horizontal polarization states

[82]. Unlike time and frequency diversity, polarization diversity does not require

either extra time or extra frequency bandwidths. The two polarized antennas

can be co-located, which solves the spacing problem for space diversity. Since the

signals in both polarizations are utilized, polarization diversity has the benefit of

recovering the energy residing in the polarization orthogonal to the polarization

of the receive antenna [84]. Therefore, due to the above advantages over other

forms of diversity, polarization diversity has been chosen to be the focus of this

research. Its capability of reducing the effect of MAI will be explored.

6.2 The mechanism of polarization diversity

During the propagation between mobile and base stations, signals in both ver-

tical polarization (Vpol) and horizontal polarization (Hpol) undertake multiple

reflections due to the presence of obstacles (such as buildings and vehicles) in the

propagation path. These multiple reflections can cause two effects which make

polarization diversity possible.

97


First, because signals in different polarization states generally have different

reflection coefficients; when reflection occurs, both Vpol and Hpol signals undergo

different phase shifts [85]. Such phase shifts can decorrelate the signals in Vpol

and Hpol, resulting in independent signals in two polarizations [86]. Secondly,

because reflections occur in three dimensions, the signal power of one polarization

can be cross coupled into its orthogonal polarization [85]. For example, assume

that the signals are transmitted in Vpol. After a sufficient number of random

reflections, some of the transmitted signal power in Vpol can be decoupled into

Hpol. As a result, the receiver can have independent signals carrying identical

information available in both Vpol and Hpol, indicating that a receiving polar-

ization diversity system can be established.

6.3 Two factors affecting polarization diversity

Diversity systems are built to counter the fading effect, but not all diversity

systems can achieve this goal. For a diversity system to be effective, signals in

all branches of the diversity must be independent (or weakly correlated) to each

other and they must have comparable mean signal levels [87].

Consequently two factors affect polarization diversity: 1) the correlation co-

efficient, which is the measure of the degree of independency between signals in

two polarizations, and 2) the cross polarization discrimination (XPD), which is

the measure for the mean signal levels in Vpol and Hpol.

98

6.3 Two factors affecting polarization diversity

6.3.1 Correlation coefficient

For a diversity system to be effective, the correlation coefficients between branches

are required to be below 0.7 [80]. Many experiments have been conducted to mea-

sure the correlation coefficients for polarization diversity in different transmission

environments and in different transmission frequencies. For example, as far back

as 1953, through experimenting on ionospherically propagated radio signal, and

by showing the joint distribution of the signal amplitudes received by vertical

and horizontal antennas, Glaser and Faber [88] declared that the fading of sig-

nals received on vertically and horizontally polarized antennas is approximately

independent, but no specific value of correlation coefficient was claimed at that

time. In 1972, Lee et al. [89] conducted their experiment in a suburban environ-

ment at 836MHz and formally proposed the polarization diversity. They showed

that the correlation between two polarized waves is low, with value mostly below

0.2. Then in [85], an experiment, conducted in 463MHz for both suburban and

urban areas, showed that the correlation coefficient between the signal in Vpol

and Hpol is 0.019 for the suburban area and −0.003 for the urban area. In [87],

Turkmani et al. conducted the experiments in the 1800MHz frequency band, with

five different environments chosen for measurements, including urban and subur-

ban areas. They reported at least 95% of the collected data for the correlation

coefficient to be less than 0.7 in all environments. Lempiainen and Laiho-Steffens

[90] conducted another experiment in the 1800MHz frequency band, focusing on

small cell system only. All experiment results in [90] were collected in semiurban

area and the analyses were separated by two situations: 1) with a line of sight

(LOS) path and 2) without a line of sight (NLOS) path. They found that in LOS

99


and NLOS there are no considerable differences between correlation coefficients.

Moreover, the average correlation coefficient is less than 0.2. A more detailed

experiment was conducted by Dietrich et al. in [91]. The frequency used by

the experiment was 2.05GHz. Eight different locations were considered including

urban, suburban, rural, indoor and outdoor-to-indoor areas. The correlation co-

efficients were reported to be between 0.02 for the indoor area and 0.32 for the

outdoor area.

Therefore in almost all experiments conducted so far, the signals in Vpol and

Hpol have a correlation coefficient much less than 0.7. In some cases, such as the

experiment conducted in [85], the two signals in Vpol and Hpol can be considered

as independent.

6.3.2 Cross polarization discrimination (XPD)

Another important factor for polarization diversity is the cross polarization dis-

crimination (XPD) factor. XPD is defined as the ratio between the signal power

in Vpol and the signal power in Hpol. In polarization diversity the mean signal

difference between the signals in Vpol and Hpol is measured by XPD. For systems

to achieve effective polarization diversity, XPD is desired to have values that are

close to 0dB.

Empirical studies [85, 87, 90, 91] found that XPD is largely dependent on the

transmission environment. For example, in the experiment conducted in [85], it

was reported that XPD is 7dB in urban area and 12dB in suburban area. In [87],

XPD was found to be in the range between 9.5dB for urban area and 12.6dB for

Rural area. Lempiainen et al. in [90] tried to separate the analyses of the cases

100

6.4 Diversity combining technique

LOS and NLOS. They reported that for a pure LOS connection the XPD can

be as high as 14dB and for a completely NLOS environment XPD can still be

7.5dB. The more comprehensive study in [91] showed that XPD is in the range

between 6 and 7dB for an outdoor area but that for an indoor area XPD can be

as low as 2.3dB.

In general XPD is relatively low in urban and indoor areas. This is because

in urban and especially in indoor area, the large number of obstacles in the

propagation path causes sufficient reflections and refractions; these in term lead

to effective polarization cross coupling [91]. On the other hand, in suburban

and rural area, the number of obstacles is relatively small. Hence reflection and

refraction are insufficient. As a result a larger XPD is obtained.

XPD is the major factor that limits the wide implementation of polarization

diversity. Having a large XPD means the receiver can received the signals in one

branch only and thus the advantage of introducing polarization diversity scheme

is lost.


Three types of diversity combining techniques can be used to combine the outputs

in polarization diversity, namely selection combining, equal gain combing (EGC)

and maximum ratio combing (MRC). Discussion of each of these techniques fol-

lows.

101


6.4.1 Selection combining

With selection combining, the system simply selects the diversity branch output

with the largest signal-to-noise ratio (SNR) and discards the output into other

branches [80]. Such branch selection can increase the average SNR of the system

and thus offers a better performance [1].

The advantage of selection combining is its simple implementation, as it re-

quires only a side monitoring station and an antenna switch at the receiver [1].

However, this technique is not optimal because it does not fully utilize the signals

available in all possible diversity branches [1, 3].

6.4.2 Maximal ratio combining

Maximal ratio combining (MRC) overcomes the limitations of selection combin-

ing: it combines the input signals in all diversity branches. MRC has been

considered as the optimal combining technique in the presence of additive white

Gaussian noise (AWGN) due to its ability to maximize the instantaneous output

SNR [1]. This is demonstrated as below.

Assume a system with Nd diversity branches, the instantaneous output SNR

is given by [3]

SNR =(

Eb

N0

)

∣∣∣∣∣Nd∑i=1

µiβiejθi

∣∣∣∣∣

2

Nd∑i=1

|µi|2(6.1)

where Eb is bit energy; N0 is noise spectral density, µi is the combining weight

and βi and θi are the magnitude and phase of the received signal respectively.

102


To obtain the maximum instantaneous output SNR, Cauchy-Schwarz inequal-

ity is applied, giving the maximum value as [3]

SNR ≤(

Eb

N0

)Nd∑i=1

|µi|2∣∣∣∣∣Nd∑i=1

βiejθi

∣∣∣∣∣

2

Nd∑i=1

|µi|2=(

Eb

N0

) Nd∑

i=1

β2i =

Nd∑

i=1

SNRi (6.2)

The only condition to reach this maximum value is to set [3]

µi = cβie−jθi for i = 1...Nd (6.3)

where c is some arbitrary complex constant. Therefore, according to (6.3), in

MRC, the magnitude of the combining weight is proportional to the magnitude

of the received signal, and the phase of the combining weight is the negative value

of the phase of the received signal.

The maximum SNR in (6.2) also suggests that MRC can produce an output

SNR equal to the sum of the individual SNRs in each diversity branch. It follows

that MRC can offer the advantage of producing an acceptable output SNR even

when none of the SNR in individual branches is acceptable [1].

6.4.3 Equal gain combining

Although MRC has the ability to maximize the instantaneous output SNR, in

certain cases it is difficult to track the magnitude and phase of the received signal

in order to produce a time varying combining weight [3, 1]. Equal gain combining

(EGC) provides a more convenient solution whereby the combining weights in all

branches are equal to a constant [80].

103


With constant gain, EGC can not maximize the instantaneous output SNR.

Hence it is not optimal and its performance is inferior to MRC. However, EGC

still has the ability to exploit the signals in all diversity branches, allowing it to

give better performance than selection combining [1].

6.5 Base station polarization diversity reception

model and its applications

To overcome the XPD limitations and to make polarization diversity more fea-

sible, the Kozono model was proposed in [83], as a base station reception model

using polarization diversity. In the Kozono model, the MS transmits signals in

Vpol. The receiver in the BS is composed of two antennas elements which have

±α inclination angles from the vertical axises. This BS antenna configuration is

aimed to equalize the received mean signal level in two antenna elements. How-

ever, this equalization is at the expense of raised correlation coefficients [85]. As

reported in [83], by letting α = 45 degree, the value of the mean signal level differ-

ences between two antenna elements can be lower than 2.5dB but the correlation

coefficient will increase to a value not larger than 0.6. The increased correlation

coefficient, however, is still lower than the 0.7 threshold for effective diversity

scheme. Hence the Kozono model is able to unlock the potential advantages of

polarization diversity, providing a solution for countering multipath fading in the

BS.

Despite its significance and simplicity, very few studies have been done regard-

ing the application of polarization diversity using the Kozono configuration with

104

6.5 Base station polarization diversity reception model and itsapplications

CDMA systems. To the best of the author’s knowledge, the analysis of DS-CDMA

system using a polarization diversity scheme at the BS has been attempted only

in [92] and [93]. In both papers, a new BS receiver architecture which combines

multistage interference cancellation and polarization diversity using the Kozono

configuration is proposed. The performance of uplink DS-CDMA using this re-

ceiver was analyzed and it was claimed that the proposed receiver can achieve

significant performance gains over the conventional DS-CDMA receiver.

However, the application of the Kozono model to MC-CDMA has never been

studied before. One of the aims of this research is to fill this gap. In the following

section, a new system which combines the Kozono model with asynchronous MC-

CDMA is proposed. The new system is referred to as Pol/MC-CDMA and its

performance during asynchronous transmission is analyzed.

Further, in this chapter an optimum combining method referred to as maximal

signal-to-MAI ratio combining (MSMAIRC) is proposed for the Pol/MC-CDMA

system. Unlike MRC, which is aimed to maximize the instantaneous SNR and to

provide optimal bit error rate (BER) in the presence of additive white Gaussian

noise (AWGN), MSIRC is aimed to maximize the instantaneous signal-to-MAI

ratio (SMAIR) and provide optimal BER in the presence of MAI. Hence in asyn-

chronous MC-CDMA systems, where performance is typically limited by the MAI,

applying MSMAIRC is more appropriate than applying MRC.

However, in reality the performance of asynchronous MC-CDMA is not sub-

jected to only AWGN or MAI but both of them. Hence the overall BER perfor-

mance for asynchronous MC-CDMA is determined by signal-to-MAI-plus-noise

ratio (SMAINR) instead of SMAIR and SNR only. In this chapter, it will be

shown that when MSIRC and MRC are applied to Pol/MC-CDMA, the SMAINR

105


of the system can be affected by the antenna angle α, and then the optimum an-

tenna angles for both combining methods are derived.

6.6 System model for Pol/MC-CDMA system

The transmitter structure for the Pol/MC-CDMA system model is identical to

the one for asynchronous MC-CDMA systems, described in Section 2.2, but since

this chapter is dealing with base station polarization diversity, it needs to be

emphasized that each user signal in the Pol/MC-CDMA system model is trans-

mitted through a single vertically polarized antenna. The channel considered in

Pol/MC-CDMA is the same as the one considered in the case for asynchronous

MC-CDMA. However, to focus on the effect of polarization diversity, only uniform

distributed timing offsets are considered in this analysis.

The main differences between the asynchronous Pol/MC-CDMA and MC-

CDMA models lie in the structures of the base station receiver. As illustrated

in Fig. 6.1, the base station receiver architecture is based on the two-branch

polarization diversity configuration proposed by Kozono et al. The two base

station antennas, V1 and V2, are co-located and are inclined at angles ±α relative

to the vertical axis. Azimuthal dependence of user k is introduced with δk; it is

assumed as a random variable uniformly distributed between −π/2 and π/2.

Polarization diversity is achieved through the reception of signals that have

undergone independent fading in both polarizations. Even though the mobile sub-

scriber (MS) is transmitting in a principally vertical polarization, the presence of

obstacles acting as reflection, scattering sources in the mobile environment causes

signal power cross coupling between two polarizations. As a consequence, some

106

6.6 System model for Pol/MC-CDMA system

Figure 6.1: Two-branch receiver model for base station

of the power in the vertical polarization (Vpol) is decoupled into the horizontal

polarization (Hpol) [85].

At the receiver the resulting signal is modelled as a signal that contains both a

vertically polarized component rk,v and a horizontally polarized component rk,h.

The expressions of rk,v and rk,h are given respectively as

rk,v =N∑

i=1

√2Pbkβk,v,ick,i cos [2πfi (t − τk) + φk,i + θk,v,i] (6.4)

rk,h =N∑

i=1

√2Pbkβk,h,ick,i cos [2πfi (t − τk) + φk,i + θk,h,i] . (6.5)

The random phases introduced by the vertically and horizontally polarized chan-

nels are modelled with θv,k,i and θh,k,i, respectively. It is assumed that θv,k,i and

θh,k,i are independent and are uniformly distributed over the interval [0, 2π). The

107


Rayleigh fadings associated with the Vpol and Hpol are introduced with the βk,v,i

and βk,h,i parameters respectively. From the experimental results published in

[85, 87, 89], it is reasonable to assume that βk,v,i and βk,h,i are uncorrelated. The

power of βk,v,i and βk,h,i is related by cross polarization discrimination (XPD),

which is defined as the ratio between the available power in the Vpol and the

available power in the Hpol.

χ =E[β2

k,v,i

]

E[β2

k,h,i

] =σ2

v

σ2h

. (6.6)

The signal received in the diversity antennas is the summation of the resultant

projections of rk,v and rk,h onto V1 and V2 [83]. V1 and V2 are calculated as

V1 =K∑

k=1

rk,h(t) sin α cos δk + rk,v(t) cos α , (6.7)

V2 =K∑

k=1

−rk,h(t) sin α cos δk + rk,v(t) cos α . (6.8)

The newly proposed receiver and combiner are an extension to a single user MC-

CDMA receiver. Each of the separate diversity branches as realized by V1 and V2

is fed into a separate instance of the single user MC-CDMA receiver, as shown in

Fig. 6.2. The signals received in the two branches are detected independently and

combined together before being input through the decision device to determine

the transmitted bit. Notice that diversity combining of the V1 and V2 branches

occurs after the matched filter stage. This type of combining is called post-

detection combining, and signal combination is achieved when the output test

statistics Z1 and Z2 from the correlation receivers are first weighted with a branch

gain εq (q = 1, 2) and then summed to give the final test statistic [Z]mp where

108

6.7 Derivation of the test statistic for Pol/MC-CDMA

Figure 6.2: MC-CDMA receiver with polarization diversity

[ · ]mp represents symbols that are exclusive for Pol/MC-CDMA systems.

6.7 Derivation of the test statistic for Pol/MC-

CDMA

The final test statistic [Z]mp is a weighted sum of the individual test statistics

output from each of the diversity branches and it is calculated as

[Z]mp =2∑

q=1

Zqεq =∫ (l+1)Ts

lTs

2∑

q=1

N∑

i=1

Vqcr,i cos (2πfit) µq,iεq dt

= [D]mp + [η]mp + [I]mp . (6.9)

The l parameter can be any arbitrary integer, and is defined to select the symbol

of interest. µq,i are the combining gain parameters used to weight each of the

109


individual subcarriers prior to the matched filter. At this stage, equal gain com-

bining (EGC) is assumed, and, for the sake of simplicity, these parameters have

all been set to unity, such that µq,i = µi = 1 for q = 1, 2 and i = 1 . . . N .

The resulting calculation for the final test statistic [Z]mp in (6.9) is most easily

represented as a sum of three components, the desired signal [D]mp, the additive

white Gaussian noise [η]mp, and the multiple access interference component [I]mp.

The desired signal component [D]mp is calculated as

[D]mp =

√P

2Tsbr (l) [sin α cos δr (ε1 − ε2) Bh + cos α (ε1 + ε2) Bv] , (6.10)

where

Bv =N∑

i=1

βr,v,i (6.11)

Bh =N∑

i=1

βr,h,i . (6.12)

The data bit of the reference used in the current detection interval [lTs, (l + 1) Ts]

is denoted by br (l). The interference due to noise introduced by the AWGN

channel is represented by [η]mp. It is a Gaussian random variable with zero mean

and variance given as

V ar [η]mp =N0TsN

4

(ε21 + ε2

2

). (6.13)

The MAI component, as a result of the loss in orthogonality between user

codes for the other K − 1 active users, is represented by [I]mp. In asynchronous

MC-CDMA, the MAI is analyzed by decomposing [I]mp into two independent

110

6.7 Derivation of the test statistic for Pol/MC-CDMA

terms, [Is]mp and [Id]mp [20]. [Is]mp represents the MAI generated by other users

using the same subcarrier frequencies, while [Id]mp represents the MAI generated

by other users using different subcarrier frequencies. The expressions for [Is]mp

and [Id]mp can be written respectively as

[Is]mp =

√P

2Ts

K∑

k=1;k 6=r

[bk (l − 1) τk + bk (l) (Ts − τk)]

·N∑

i=1

ck,icr,iµi

βk,h,i cos ζk,h,i sin α cos δk (ε1 − ε2)

+βk,v,i cos ζk,v,i cos α (ε1 + ε2)

, (6.14)

[Id]mp =√

2PK∑

k=1;k 6=r

[bk (l − 1) − bk (l)]

·N∑

i=1

µi

N∑

j=1;j 6=i

ck,jcr,iTs

2∆i,j

βk,h,j sin α cos δk (ε1 − ε2)

· [sin (∆i,jτk + ζk,h,j) − sin ζk,h,j]

+βk,v,j cos α (ε1 + ε2)

· [sin (∆i,jτk + ζk,v,j) − sin ζk,v,j]

. (6.15)

where bk (l) is denoted as the binary data for the kth user in the current de-

tection interval i.e. [lTs, (l + 1) Ts]. Similarly, bk (l − 1) represents the binary

data associated with the previous detection interval i.e. [(l − 1) Ts, lTs]. The

ζ parameters has been introduced to represent the total combined phase, where

ζk,h,i = θk,h,i + φk,i + 2πfiτk and ζk,v,i = θk,v,i + φk,i + 2πfiτk. In (6.15), ∆ denotes

the spectral distance between subcarrier i of the reference user and subcarrier j

of the interfering user k. The spectral distance is defined as ∆ = i − j.

Due to the assumption that φk,i, θk,v,i and θk,h,i are i.i.d. for different users

and different subcarriers, [Is]mp and [Id]mp are also independent for different sub-

111


carriers and different users. Because of this, the central limit theorem allows the

approximation of [Is]mp and [Id]mp as Gaussian random variables [20]. When the

expectation is taken with respect to βk,i, bk, ζk,v,i and ζk,h,i, the average values of

both [Is]mp and [Id]mp are found to be zero and their variance given respectively

as

V ar [Is]mp =(K − 1) PT 2

s σ2vN

6

[sin2 α

2χ(ε1 − ε2)

2 + cos2 α (ε1 + ε2)2

](6.16)

V ar [Id]mp =(K − 1) PT 2

s σ2vNC

4π2

[sin2 α

2χ(ε1 − ε2)

2 + cos2 α (ε1 + ε2)2

](6.17)

where

C =1

N

N∑

i=1

N∑

j=1;j 6=i

1

∆2i,j

. (6.18)

The total MAI power V ar [I]mp is calculated as the summation of (6.16) and

(6.17), which gives

V ar [I]mp =V ar [Is]mp + V ar [Id]mp

= (K − 1) PT 2s σ2

vN[1

6+

C

4π2

]·[sin2 α

2χ(ε1 − ε2)

2 + cos2 α (ε1 + ε2)2

].

(6.19)

The three main combining techniques employed in practice are selection combin-

ing, equal gain combining and maximal ratio combining. When selection combin-

ing is used, the diversity branch with the highest instantaneous baseband SNR

is chosen and fed into the receiver. In equal gain combining, ε1 is set equal to

112

6.8 Maximal signal-to-MAI ratio combining

ε2, where they are typically chosen to equal one. If MRC is applied, each of

the diversity branches is weighted by their respective SNRs, cophased and then

summed. In the absence of MAI, the MRC gives the best statistical reduction

to fading, and the resultant SNR at the baseband is maximum [1]. Nevertheless,

where Pol/MC-CDMA is considered, when the contribution of interference from

MAI is significant, MRC can not guarantee the best combining performance be-

cause the combining gains do not account for the interference due to MAI. The

objective of this paper is to derive the optimal combining gain parameters, ε1

and ε2, for Pol/MC-CDMA, in the presence of MAI.


During asynchronous MC-CDMA transmission, MAI is the major limiting factor

of system performance. Better performance is expected when the combining gains

are designed to maximize the signal-to-MAI ratio (SMAIR). In this section, the

combining gains that will maximize the instantaneous SMAIR is derived. This

new combining scheme will be referred to as the maximal signal-to-MAI ratio

combining (MSMAIRC).

The instantaneous SMAIR of an asynchronous MC-CDMA signal is derived

by taking the ratio between the power of the desired signal and the power of the

MAI

SMAIR =[sin α cos δr (ε1 − ε2) Bh + cos α (ε1 + ε2) Bv]

2

2 (K − 1) σ2vN

(16

+ C4π2

) [sin2 α

2χ(ε1 − ε2)

2 + cos2 α (ε1 + ε2)2] .

(6.20)

113


To further simplify the instantaneous SMAIR expression, let Υs = (ε1 + ε2) , and

Υd = (ε1 − ε2). The numerator and the denominator are then divided by Υ2d.

SMAIR =

(sin α cos δrBh + cos αBv

Υs

Υd

)2

2 (K − 1) σ2vN

(16

+ C4π2

) [sin2 α

2χ+ cos2 α

(Υs

Υd

)2] . (6.21)

The new combining gains to yield the maximum instantaneous SMAIR are

derived by taking the gradient of (6.21) relative to Υs/Υd, and setting the result

to zero. This equation is then solved for Υs/Υd.

Υs

Υd

=sin αBv

2χ cos α cos δrBh

. (6.22)

The interpretation of this result is that, as long as the combining gains ε1 and ε2

satisfy the relationship stated in (6.22), the maximum SMAIR is achieved. One

such example of this is to set the combining gains as

ε1 = sin αBv + 2χ cos α cos δrBh , (6.23)

ε2 = sin αBv − 2χ cos α cos δrBh . (6.24)

Using these combining gains, the maximum obtainable instantaneous SMAIR can

be calculated by substituting (6.22) into (6.21).

SMAIRmax =2χ cos2 δrB

2h + B2

v

(K − 1) σ2vN

(16

+ C4π2

) . (6.25)

From this analysis, one of the important details to note when using MSMAIRC in

the absence of AWGN is that the result of the maximum obtainable instantaneous

114


SMAIR in (6.25) is independent of the antenna angle α. It should be also note

that if the expectation of the SMAIRmax is taken with respect to B2v and B2

h, the

average SMAIR becomes independent of the XPD. The process of calculation is

shown below.

E [SMAIRmax] =2χ cos2 δrE [B2

h] + E [B2v ]

2 (K − 1) σ2vN

(16

+ C4π2

) . (6.26)

The second-order moments E [B2v ] and E [B2

h] are required to evaluate the expres-

sion for the average SMR. The details of the calculations are found in Appendix

D.

E[B2

v

]=Nσ2

v

[1 + (N − 1)

(ρv −

π

4ρv +

π

4

)](6.27)

E[B2

h

]=Nσ2

h

[1 + (N − 1)

(ρh −

π

4ρh +

π

4

)]

=Nσ2

v

χ

[1 + (N − 1)

(ρh −

π

4ρh +

π

4

)], (6.28)

where ρv and ρh are the correlation coefficients for the fading between two sub-

carriers in Vpol and Hpol, respectively. By substituting (6.27) and (6.28) into

(6.26), this research shows that the average of SMR is independent of the XPD.

E [SMAIRmax] =

2 cos2 δr

[1 + (N − 1)

(ρh − π

4ρh + π

4

)]

+[1 + (N − 1)

(ρv − π

4ρv + π

4

)]

2 (K − 1)(

16

+ C4π2

) . (6.29)

115


The notable characteristics of using the new MSMAIRC for Pol/MC-CDMA

in the absence of AWGN are: 1)the instantaneous SMAIR is maximized;

2)the instantaneous SMAIR is invariant of the antenna angle; 3)the

average SMAIR is independent of XPD.

6.8.1 Performance analysis of Pol/MC-CDMA in the pres-

ence of both AWGN and MAI

Even though the contribution to the source of interference for MC-CDMA is dom-

inated by the MAI, in practice the interference as a result of the AWGN cannot be

ignored. In this section, both the MAI and the AWGN are considered as sources

of interference, and the performance analysis of Pol/MC-CDMA is presented by

evaluating the instantaneous signal-to-MAI-plus-noise ratio (SMAINR) and the

bit error rate (BER).

The general formula for the instantaneous SMAINR is given as

SMAINR =

[P2T 2

s b2r (l)

]· (sin α cos δrΥdBh + cos αΥsBv)

2

[(K − 1) PT 2

s σ2vN

(16

+ C4π2

)]·(

sin2 α2χ

Υ2d + cos2 αΥ2

s

)

+(

N0TsN8

)· (Υ2

d + Υ2s)

. (6.30)

The instantaneous BER can then be calculated with the application of the Q

function to the obtained SMAINR [33], which gives,

pe|Bv,Bh = Q(√

SMAINR)

, (6.31)

where Q ( · ) is the Q function. The average BER is obtained by taking the

116


expectation with respect to Bv and Bh, which is presented here by

E [pe] =∫ +∞

−∞

∫ +∞

−∞pe|Bv, Bh · prob (Bv, Bh) dBvdBh . (6.32)

Previously, in section 6.6, it was assumed that the fadings of the Vpol and

Hpol are independent, hence the joint distribution of Bv and Bh, prob (Bv, Bh)

can be written as the product of their respective distributions, prob (Bv) and

prob (Bh). As a result, the average BER can be written as

E [pe] =∫ +∞

−∞

∫ +∞

−∞pe|Bv, Bh · prob (Bv) · prob (Bh) dBvdBh . (6.33)

The evaluation of the average BER requires knowledge of the PDF for Bh

and Bv i.e. prob (Bv) and prob (Bh). Because the Rayleigh random variable is

a special case of the Nakagami-m random variable (with m = 1), the PDF for

the sum of N independent Rayleigh random variables can be approximated by a

Nakagami-m distribution [94]. In the case where the Rayleigh random variables

are correlated, the exact expression of the PDF for B has been derived in [95].

However, the evaluation of this PDF requires the summation of an infinite series,

which is impractical especially when the number of random variable N is large.

Through statistical (Shown in Appendix A), this research found that the PDF

for the sum of N correlated Rayleigh random variables can also be approximated

by the Nakagami-m distribution. prob (Bv) and prob (Bh) are approximated by

prob (Bv) =2mv

mvB2mv−1v

Γ (mv) Ωmvv

exp(−mv

Ωv

B2v

)(6.34)

prob (Bh) =2mh

mhB2mh−1h

Γ (mh) Ωmh

h

exp(−mh

Ωh

B2h

), (6.35)

117


Figure 6.3: BER comparison between Pol/MC-CDMA with MRC and MC-CDMA (with α = π/4; δr = 0 and N = K = 16)

with parameters of the Nakagami-m distribution given as

Ωv = E[B2

v

](6.36)

Ωh = E[B2

h

](6.37)

mv ≈ E [B2v ]

5[E [B2

v ] − (E [Bv])2] (6.38)

mh ≈ E [B2h]

5[E [B2

h] − (E [Bh])2] . (6.39)

The associated first order moments for Bv and Bh are E [Bv] = Nσv

√π/4 and

E [Bh] = Nσh

√π/4 respectively. The expressions of E [B2

v ] and E [B2h] have been

given earlier in (6.27) and (6.28) respectively. Finally, the double integral for the

average BER is evaluated using the Monte Carlo integration technique [43].

118


Figure 6.4: BER comparison between Pol/MC-CDMA with MSMAIRC and MC-CDMA (with α = π/4; δr = 0 and N = K = 16)

Figure 6.5: BER comparison between Pol/MC-CDMA with MSMAIRC and MRC(with α = π/4; δr = 0 and N = K = 16)

119


Figure 6.6: Threshold XPD for Pol/MC-CDMA with MSMAIRC and MRC (withα = π/4; δr = 0 and N = K = 16)

6.8.2 Results and Discussions for MSMAIRC

This section compares of the BER performance against the XPD for three sys-

tems, namely, the basic MC-CDMA, Pol/MC-CDMA with MRC, and Pol/MC-

CDMA with MSMAIRC. For fair comparison, both Pol/MC-CDMA systems have

their antenna angles set to π/4.

In Fig. 6.3 a comparison between the basic MC-CDMA against Pol/MC-

CDMA using MRC is presented. For all levels of Eb/N0, it is shown that Pol/MC-

CDMA with MRC generates lower BER than the basic MC-CDMA, indicating

the advantages of introducing polarization diversity. Nevertheless, the advantage

disappears as the XPD is increased. When the XPD approaches infinity, Pol/MC-

CDMA with MRC yields the same BER as the basic MC-CDMA. This results

indicated that the performance of MRC is largely depends on the XPD.

However, ideally it is desired to have a superior system performance that is

independent of XPD. With the application of the newly proposed MSMAIRC and

120


with a high level of Eb/N0, this goal can be achieved. In Fig. 6.4, the BER of

the Pol/MC-CDMA using MSMAIRC is compared against the basic MC-CDMA.

Notice that when Eb/N0 = 30dB, Pol/MC-CDMA with MSMAIRC significantly

outperforms the basic MC-CDMA for all values of XPD between 0 and 20dB,

and the resulting BER is almost invariant for XPD. However, when the Eb/N0

becomes lower, the advantage of using MSMAIRC is fading out. In particular,

Fig. 6.4 shows that for Eb/N0 = 0dB, which indicates a case where there is a

significant level of AWGN, the Pol/MC-CDMA with MSMAIRC can only out-

perform MC-CDMA when the XPD is less than 5dB. This is to be expected, as

the MSMAIRC is only designed to optimize against the MAI to yield maximum

SMAIR. This value of the performance threshold can be increased by increasing

the Eb/N0. For example, when Eb/N0 is increased to 10dB, Pol/MC-CDMA with

MSMAIRC outperforms the basic MC-CDMA for values of the XPD less than

12dB. Therefore, the performance of MSMAIRC depends on the level of AWGN,

rather than XPD, as in the case of MRC.

From Fig. 6.3 and Fig. 6.4 it can be seen that introducing polarization diver-

sity to MC-CDMA can improve the BER. However, depending on the combining

method that is chosen, the BER improvement is subjected to different conditions.

For MRC, the BER improvement is most sensitive to the values of XPD, whereas

for MSMAIRC, even though still a function of the XPD, is far more sensitive to the

changes in Eb/No. It is of interest to determine which combining method is better

under specific conditions. Firstly the BER of Pol/MC-CDMA for both MRC and

MSMAIRC are compared and presented in Fig. 6.5. This indicates that, gen-

erally, for low values of Eb/N0, performance of MRC is found to be better than

MSMAIRC. On the other hand, with larger values of Eb/N0, MSMAIRC yields

121


better results because the contribution of interference from the MAI is dominant

over the contribution of AWGN. It can be seen that when Eb/N0 = 0dB, MRC

outperforms MSMAIRC for all values of the XPD. When Eb/N0 is increased to

5dB, MSMAIRC starts to outperform MRC up to the XPD values of 5dB. Here

the value of the XPD under which MSMAIRC can outperform MRC is defined

as the XPD threshold. As shown in Fig. 6.5, the threshold XPD value can be

increased by increasing Eb/N0. For example, when Eb/N0 = 10dB, MSMAIRC

can generate lower BER than MRC up to the XPD values of 11dB. Finally, when

Eb/N0 is increased to more than 20dB, MSMAIRC gives lower BER performance

in almost all values of the XPD.

In the two previous figures, it has been shown that both the Eb/N0 and the

XPD affect the performance of MSMAIRC and MRC. In Fig. 6.6, the XPD

threshold values against the levels of Eb/N0 is presented. The line shown in this

figure demonstrates the conditions (as a combination of Eb/N0 and XPD) under

which MSMAIRC is better than MRC. For example, when Eb/N0 = 12dB, the

threshold XPD value is 13.5dB, which indicates that, given this level of AWGN,

MSMAIRC can outperform MRC for all values of the XPD up to 13.5dB. In

essence, the region below this threshold XPD line indicates the conditions in

which MSMAIRC has more favorable performance over the MRC. In mobile com-

munications, the value of Eb/N0 is typically greater than 10dB. Combining this

with the experimental results in the literature showing that the values of the XPD

in a dense urban environment is typically below 10dB [91, 87, 85] shows that the

conditions are therefore suitable for the application of MSMAIRC in favor over

the MRC.

In the previous part, the performance of MSMAIRC whilst changing the con-

122


Figure 6.7: Threshold XPD for Pol/MC-CDMA with MSMAIRC and MRC givendifferent number of users (α = π/4; δr = 0 and N = 16)

ditions for the Eb/N0 is examined, in order to observe the effects of varied levels

of AWGN power. In this part, the effects of the threshold XPD as a function of

the number of users is examined. By changing the number of users, effectively

the effect of the threshold XPD is examined as a result of different levels of the

MAI. Since the MAI is linearly proportional to the number of users, as shown in

(6.19), a decrease in the number of users indicates lower MAI. Fig. 6.7 presents

the threshold XPD values against the levels of Eb/N0 for different number of

users. It can be seen here that as the number of users increase, the area which

indicates the conditions more favorable to MSMAIRC increases. An increase in

the number of users in Pol/MC-CDMA translates to an increase in the levels of

MAI and it follows that when the same levels of Eb/N0 are compared, the XPD

threshold is higher for a larger number of users. When the system is operating at

full capacity (in this case, 16 users), the XPD threshold takes its highest value.

This reflects the characteristics of the MSMAIRC, since MSMAIRC is optimised

123


Figure 6.8: Threshold XPD for Pol/MC-CDMA with MSMAIRC and MRC for2 users (α = π/4; δr = 0 and N = 16)

for the signal to MAI power ratio.

On closer inspection of the results, Fig. 6.8, Fig. 6.9 and Fig. 6.10 are

presented, which shows the threshold XPD for Pol/MC-CDMA with MSMAIRC

given two, four and eight users respectively. The Eb/N0 and threshold XPD

axis have been deliberately selected to reflect the typical operating parameters

in mobile communications, such that 0dB < XPD < 10dB and 10dB < Eb/N0 <

30dB. The figures show that, despite a reduction in the levels of MAI power,

the regions indicating better performance covered by MSMAIRC are still in a

majority over the MRC. For cases where the number of users is greater than 8,

MSMAIRC always outperforms MRC in the given window.

6.9 Optimum antenna angle

In this section, both the MAI and the AWGN are considered as sources of in-

terference, and the optimization of the average signal-to-MAI-plus-noise ratio

124




125


(SMAINR) is presented. Optimization is done using the variation of the diversity

antenna angles.

Previously, in Section 6.8, the expression for the instantaneous SMAIR has

been derived. It has been shown that for any given antenna angle, MSMAIRC

always guarantees the optimal SMAIR. Further optimization can be carried out

with respect to the AWGN by using the angles of the diversity antennas to pro-

duce an optimal SMAINR. Since SMAINR is used as the measure for the overall

performance of the system, the application of the optimal antenna angle to max-

imize the SMAINR leads to the best overall performance for Pol/MC-CDMA. In

the proceeding work, the optimal antenna angles are derived for both MRC and

MSIRC techniques.

The general formulas for instantaneous and average SINR are given respec-

tively as

SMAINR =

[P2T 2

s b2r (l)

]· (sin α cos δrΥdBh + cos αΥsBv)

2

[(K − 1) PT 2

s σ2vN

(16

+ C4π2

)]·(

sin2 α2χ

Υ2d + cos2 αΥ2

s

)

+(

N0TsN8

)· (Υ2

d + Υ2s)

,

(6.40)

E [SMAINR] =∫ ∞

−∞

∫ ∞

−∞SINR · prob (Bh, Bv)dBhdBv

=∫ ∞

−∞

∫ ∞

−∞SINR · prob (Bh) prob (Bv) dBhdBv· . (6.41)

The general approach to deriving the optimum antenna angles is to firstly specify

the combining gains ε1, ε2 and then to calculate Υs and Υd. Following this,

substitutions are made into (6.40), where the average SINR can then be derived.

The antenna angle to achieve maximum average SINR is calculated by taking the

126


derivative of the average SINR with respect to α, setting the expression to equal

zero, and solving for α.

6.9.1 Optimum antenna angle for MRC

For MRC, Υd = 2 sin α cos δrBh and Υs = 2 cos αBv are defined, and the opti-

mized antenna angle is calculated as

αmrc = arcsin

√√√√√√√√√√√√√√√√√

4[(K − 1) PT 2

s σ2vN

(16

+ C4π2

)]· cos2 δr · χ

+(

N0TsN4

)· (cos2 δr − χ) · χ

2[(K − 1) PT 2

s σ2vN

(16

+ C4π2

)]· cos2 δr · (1 + 2χ)

−(

N0TsN4

)· (cos2 δr − χ)

2

. (6.42)

6.9.2 Optimum antenna angle for MSIRC

In the calculation for the optimum antenna angles for MSIRC, the set of combin-

ing gains as derived in (6.23) and (6.24) are used. The optimized antenna angle

is calculated as:

αMSIRC = arcsin

√√√√ 2√

χ cos δr

2√

χ cos δr + 1. (6.43)

The result shows that the optimum antenna angle for MSIRC depends on neither

the level of the MAI nor the noise power from the AWGN. The only parameters

which affect α are the XPD, and the location of the reference user in the azimuth

plane.

127


6.9.2.1 Special Case 1

It is interesting to note that when χ = 0dB and when the azimuth angle for the

reference user δr = 0, the optimum antenna angle for both the MRC and MSIRC

are equal. Moreover, the values of the SINR are also equal.

6.9.2.2 Special Case 2

Another special case arises when the contribution of the AWGN is comparatively

small relative to the MAI, such that the AWGN noise power component is ap-

proximately zero. In this case, the treatment of the SINR can be approximated

with the SIR. The resulting expression of the optimum antenna angle for the

MRC can be reduced to

αmrc = arcsin

(√2χ

2χ + 1

). (6.44)

With this antenna angle for MRC, the resulting SIR is maximized, and its value

is identical to the average SIR obtained with the application of MSIRC. In other

words, given negligible contribution from AWGN, the application of MRC with

an optimized antenna angle gives the same average SIR as MSIRC.

6.9.3 Results and discussion for the antenna angle effect

In this section, the results of the BER as a function of the antenna angles for

Pol/MC-CDMA with MSMAIRC and Pol/MC-CDMA with MRC are presented.

Fig. 6.11 and Fig. 6.12 demonstrate the performance comparison between the

two systems for Eb/N0 = 10dB and Eb/N0 = 30dB respectively.

In the case for Eb/N0 = 10dB, it can be seen that for 0.25π < α < 0.45π,

128


Figure 6.11: Effect of antenna angle when Eb/N0 = 10dB

Figure 6.12: Effect of antenna angle when Eb/N0 = 30dB

129


MSMAIRC can outperform MRC in two regards: 1) the BER of MSMAIRC is

less sensitives to the change of α than the BER of MRC;and, 2) For most values

of α, MSMAIRC yields lower BER than MRC.

The advantages of applying MSMAIRC become more apparent as Eb/N0 is

increased. In the case when Eb/N0 = 30dB, the improvement gained by applying

MSMAIRC is significant. As shown in Fig. 6.12, the BER of MSMAIRC outper-

forms MRC for almost all antenna angles and for all values of the XPD. As well

as this, the BER variation of MSMAIRC as a function of α is considerably small

for angles 0.1π < α < 0.45π. It can be seen that within this operating range of

α, the BER of MRC remains almost constant.

130

Chapter 7

Conclusions and future works

7.1 Conclusions

Multicarrier code division multiple access (MC-CDMA) is a promising multiplex-

ing technique supporting the high data rate requirement in the fourth genera-

tion of wireless communication systems. During synchronous transmission, MC-

CDMA has the capability to offer excellent bit error rate (BER) performance.

During asynchronous transmission, however, multiple access interference (MAI)

still remains a major challenge for MC-CDMA systems and significantly affects

their performance. The primary objectives of this thesis were to analyze the

MAI in asynchronous MC-CDMA systems and to develop robust techniques for

reducing the MAI effect.

In achieving this objective, this thesis firstly focused on the statistical analysis

of MAI. Previous statistical analysis of MAI was challenged here because the

uniform assumption of timing offsets and the constant assumption of MAI power

do not always hold. To rectify this problem, the concept of effective timing offset

131

7. CONCLUSIONS AND FUTURE WORKS

was introduced, alleviate the need for uniformly distributed timing offset. The

MAI power was also modelled as a time-varying process, and its time variation

was found to follow the Gamma distribution. In order to characterize the Gamma

distributed MAI power, the expressions of the mean and variance for the MAI

power were derived. Unlike other MAI analysis, the derivations presented here

were not restricted to the uniform distribution, and could be applied to any

possible distribution of the timing offsets.

Based on the new statistical model of MAI, a meta-model of asynchronous

MC-CDMA was proposed: this research showed that the meta-model was equiv-

alent to the mathematical model in terms of average BER. However, during sim-

ulations the meta-model required a significantly less computational load, because

in the meta-model only the reference user needs to be simulated and the simula-

tion of other user signals was replaced by an additive noise generator representing

the MAI.

Next in this thesis, slow frequency hopping as a MAI reduction technique was

applied to MC-CDMA. The MAI analysis for asynchronous GSFH/MC-CDMA

has been provided. Four different detection scenarios were considered and the

MAI power occurring in each scenario was derived. After applying the probabil-

ity of each scenario, the total MAI power for asynchronous GSFH/MC-CDMA

was finally obtained. Both theoretical and simulation results showed that group

subcarrier frequency hopping can successfully reduce the MAI power in asyn-

chronous MC-CDMA. However, the GSFH/MC-CDMA system has two disad-

vantages: 1)extra number of subcarriers (i.e. bandwidth) must be allocated for

the purpose of hopping; 2)the number of subcarriers allocated is fixed at integer

multiples of N .

132

7.1 Conclusions

The second disadvantage of GSFH/MC-CDMA is overcome by introducing

an individual subcarrier frequency hopping technique. This research found that

if the condition of group subcarrier hopping is dropped, the combination of slow

frequency hopping and MC-CDMA can result in a more generalized system, the

ISFH/MC-CDMA. Analysis showed that ISFH/MC-CDMA generates less MAI

power than MC-CDMA. Moreover, there is an interesting relationship between

the MAI power of asynchronous MC-CDMA, GSFH/MC-CDMA and ISFH/MC-

CDMA. Asynchronous GSFH/MC-CDMA was proved to be a special case of

ISFH/MC-CDMA in terms of generating MAI power. As a result, the spec-

trum allocation for asynchronous GSFH/MC-CDMA has no effect on its gen-

erating MAI power. Finally Monte-Carlo simulations showed that the BER of

ISFH/MC-CDMA increased with the increasing number of subcarriers subjected

to correlated fading.

MAI reduction using base station polarization diversity was another topic of

this thesis. This research found that although base station polarization diversity

can successfully improve the signal to interference ratio (SMAIR), the improve-

ment is subjected to the amount of cross polarization discrimination (XPD) as

well as to the value of antenna angle in the base station. This problem has

been solved by proposing the new optimum combining technique called maximal

signal-to-MAI ratio combining (MSMAIRC). With the application of MSMAIRC,

the average SMAIR of Pol/MC-CDMA is maximized and is independent of XPD

and antenna angles.

Further, the BER for Pol/MC-CDMA is found to be affected by the value of

angle between the two receiving antennas. With the derivation of the optimal

antenna angle for both MSMAIRC and MRC, this research showed that the BER

133


of Pol/MC-CDMA can be improved by applying the optimal angle. However,

in the comparison between MRC and MSMAIRC, MSMAIRC is less sensitive

to the variations of antenna angle than MRC, especially when the amount of

additive white Gaussian noise is small. Therefore the proposed new MSMAIRC

allows Pol/MC-CDMA to give the best and, more importantly, the most reliable

performance during asynchronous transmission.

7.2 Future works

In Chapter 4 and 5, slow frequency hopping technique was introduced to MC-

CDMA. Research results indicated that slow frequency hopping can successfully

reduce the MAI power. These results are based on the assumption that each

subcarrier frequencies has an equal chance of being selected during a hopping

interval. Although this subcarrier frequencies selection scheme is simple and

practical, future research based on the results of this research can develop more

sophisticated subcarrier frequencies selection schemes which can further reduce

the MAI power.

As another MAI reduction technique, in Chapter 6, base station polarization

diversity was introduced to MC-CDMA, and an antenna configuration, known as

the Kozono model, was applied. In Kozono model, the transmitted signal of the

mobile user is assumed to be vertically polarized. This assumption holds only

when the antenna of the mobile user is aligned with vertical axis.

However, in reality, the antenna of the mobile user generally has a inclination

angle with respect to vertical axis. Hence the transmitted signals of the mobile

user is no longer vertical polarized. Future works can extent the results of this

134

7.2 Future works

research to address this issue of antenna orientation.

Furthermore, in terms of MAI power reduction and signal quality improve-

ment, other forms of diversity such as time, frequency and space diversity can be

introduced to MC-CDMA systems. Although each of these forms of diversity has

its own limitation, it would be interesting for future research to evaluate to cost

benefit relations for each of these combinations.

135


136

Appendices

137

Appendix A

Statistical modeling of sum of

correlated Rayleigh random

variables

This appendix provides a statistical model for the sum of correlated Rayleigh

random variables i.e.

Br =N∑

i=1

βi (A.1)

where βi is a Rayleigh random variable with its second-order moment defined as

σ2.

Because the Rayleigh random variable is a special case of the Nakagami-

m random variable, the PDF for the sum of N independent Rayleigh random

variables can be approximated by Nakagami-m distribution [94]. In the case

when Rayleigh random variables are correlated, the exact expression of the PDF

139

A. STATISTICAL MODELING OF SUM OF CORRELATEDRAYLEIGH RANDOM VARIABLES

Table A.1: Table of Fitting DistributionsDistributions Probability Density Function Notes

Nakagami-m [94] p(x) = mmxm−1

xmΓ(m)e−

mxx m = x2

var(x)

Rayleigh p(x) = xb2

e

(−x2

2b2

)

b = x√π/2

Gaussian p(x) = 1√var(x)2π

e−(x−x)2

2var(x)

Exponetial p(x) = 1xe

xx

Note: x = mean of x; var(x) = variance of x; Γ(x) =∞∫

0e−ttx−1dt

for Br has been derived in [95]. However, the evaluation of this PDF requires the

summation of infinite series, which is impractical especially when the number of

random variable N is large.

A good approximation can be obtained by fitting the PDF of correlated Br

into the well-known PDFs listed in Table A.1.

The differences between the actual PDF of Br and the fitting PDFs are mea-

sured by Kullback-Leibier divergence (KLD). The expression of KLD is given as

[41]

KLD =∑

i∈Ψ

prob (Br = i) · log

prob (Br = i)

prob(Br = i

)

(A.2)

where i is a realization of correlated Br. Ψ is the total number of realizations.

prob (Br = i) is the true PDF of correlated Br and prob(Br = i

)is the approx-

imated PDF. In (A.2), a KLD value closer to 0 indicates that the difference

between the true PDF of Br and its fitting PDF is small; hence the fitting is

accurate. On the other hand, a large KLD value indicates that the true PDF of

B is very different from the fitting PDF; hence, the fitting is poor.

The PDF fitting result is shown in Table A.2. In this table both fully cor-

140

Table A.2: Fitting Result of Correlated B (Kullback-Leibier divergence values)Total No. of No. of Correlated DistributionsSubcarriers Subcarriers Na Ra Ga Ex8 8 0.0216 0.0304 0.0354 0.4222

4 0.0071 0.3551 0.0211 0.929816 16 0.0196 0.0376 0.0343 0.4414

12 0.0069 0.1749 0.0348 0.68748 0.0060 0.4226 0.0200 1.0114

32 32 0.0190 0.0413 0.0354 0.449224 0.0079 0.1883 0.0298 0.707716 0.0072 0.4684 0.0300 1.0641

Note: Na=Nakagami-m; Ra=Rayleigh; Ga=Gaussian; Ex=Exponential

related cases and partial correlated cases are considered. In the fully correlated

cases, all N Rayleigh random variables are correlated, while in the partial cor-

related cases only part of the total N Rayleigh random variables are correlated,

with the rest remaining independent of each other. As shown in the table, in all

cases Nakagami-m distributions offer the smallest KLD values among the four

possible PDFs and all these KLD values are smaller than 0.03. Therefore, it can

be concluded that Nakagami-m distribution provides the best fit for the PDF of

correlated Br.

Fig. A.1 shows an example PDF of Br which equals the sum of 16 correlated

Rayleigh random variables. Fig. A.2 shows another example PDF of Br which

is also equal to the sum of 16 Rayleigh fading random variables. In Fig. A.2,

however, only 8 out of the 16 Rayleigh random variables are correlated; the rest

are independent of each other. The approximated Nakagami-m distributions for

both examples are also shown in Fig. A.1 and Fig. A.2 respectively. Both exam-

ples show that the Nakagami-m distributions provide an excellent approximation

for the PDF of Br.

141


Figure A.1: Approximation of a sum of correlated Rayleigh random variables(N=16 and correlation coefficient = 0.7)

Figure A.2: Approximation of a sum of Rayleigh random variables (8 correlatedRayleigh random variables + 8 independent Rayleigh random variables and cor-relation coefficient = 0.7)

142

Therefore the PDF of B, prob (Br), is approximated by a Nakagami-m PDF

as

prob (Br) =2mmB2m−1

r

Γ (m) Ωmexp

(−m

ΩB2

r

). (A.3)

with parameters

Ω =E[B2

r

], (A.4)

m =E [B2

r ]

E [B2r ] − (E [Br])

2 . (A.5)

The associated first- and second- order moments for Br can be derived respectively

as

E [Br] =Nσ√

π/4 , (A.6)

E[B2

r

]=σ2

N + n (n − 1)(ρ − π

4ρ + π

4

)

+ (N − n) (N + n − 1) π4

, (A.7)

where n is the number of correlated subcarriers, n ≤ N and ρ is the correlation

coefficient between correlated subcarriers.

143


144

Appendix B

Proof of (5.15)

By taking the expectation of q and ∆i,j, (5.11) can written as

V ar [I|q, ∆i,j]mi = (K − 1) Pσ2

[E [q]

T 2s

6+

N2 − E [q]

4π2· E

[1

∆2i,j

]]. (B.1)

By substituting

E [q] =N2

Q(B.2)

E

[1

∆2i,j

]=

Q−1∑

∆i,j=1

1

∆2i,j

· prob(

1

∆2i,j

)

=Q−1∑

∆i,j=1

1

∆2i,j

· 2 (Q − |∆i,j|)Q2 − Q

, (B.3)

the average MAI power for ISFH/MC-CDMA can be obtained as

V ar [I]mi = (K − 1)Pσ2T 2

s N2

6Q

1 +3

π2Q

Q−1∑

∆i,j=1

(Q − |∆i,j|)∆2

i,j

. (B.4)

145

B. PROOF OF (5.15)

Since

Q−1∑

∆i,j=1

(Q − |∆i,j|)∆2

i,j

=

Q−1∑

i=1

Q

i2−

Q−1∑

i=1

1

i

, (B.5)

The average MAI power for ISFH/MC-CDMA can be rewritten as

V ar [I]mi = (K − 1)Pσ2T 2

s N2

6Q

1 +3

π2Q

Q−1∑

i=1

Q

i2−

Q−1∑

i=1

1

i

(B.6)

146

Appendix C

Proof of (5.20)

The average MAI power for ISFH/MC-CDMA can be written as

V ar [I]mi = (K − 1)N

Q

Pσ2T 2s N

6

1 +3

π2Q

Q−1∑

i=1

Q

i2−

Q−1∑

i=1

1

i

(C.1)

For any positive integer number L,L∑

i=1(1/i2) and

L∑i=1

(1/i) have the following

bounds [96, 97]

1

2 (L + 1)+ ln L + γ <

L∑

i=1

1

i<

1

2L+ ln L + γ (C.2)

π2

6

(2L

2L + 1

)(2L − 1

2L + 1

)<

L∑

i=1

1

i2<

π2

6

(2L

2L + 1

)(2L + 2

2L + 1

)(C.3)

147

C. PROOF OF (5.20)

In (C.2) γ represents the Euler-Mascheroni Constant. The bounds in (C.2) and

(C.3) are tight when L is large. Hence it can be approximated that

L∑

i=1

1

i≈ 1

2L+ ln L + γ (C.4)

L∑

i=1

1

i2≈ π2

6(C.5)

Applying these approximation methods, when Q is a large integer number, the

total MAI power can be rewritten as

V ar [I]mi ≈(K − 1) Pσ2T 2

s

6

N2

Q

[1 +

3

Qπ2

(π2Q

6− 1

2Q− ln Q − γ

)](C.6)

Similarly the total MAI power for basic asynchronous MC-CDMA, shown in

(5.16), can be simplified to (when N is large)

V ar [I]mc ≈(K − 1) Pσ2T 2

s N

6

[1 +

3

Nπ2

(π2N

6− 1

2N− ln N − γ

)](C.7)

Therefore the MAI power ratio for ISFH/MC-CDMA can be obtained as

[Θ]mi =N

Q·

1 + 3Qπ2

(π2Q

6− 1

2Q− ln Q − γ

)

1 + 3Nπ2

(π2N

6− 1

2N− ln N − γ

) (C.8)

For large values of Q and N , the second multiplication term shown in (C.8) is

approaching to 1, giving the MAI power ratio as

[Θ]mi =N

Q(C.9)

148

C. PROOF OF (5.20)

150

Appendix D

Proof of (6.27) and (6.28)

E [B2v ] and E [B2

h] represents the power of the sum of N Rayleigh random vari-

ables. This means that for B2v , its mean can be expanded as

E[B2

v

]= E

(

N∑

i=1

βr,v,i

)2

=N∑

i=1

E[β2

r,v,i

]+

N∑

i=1

N∑

j=1;j 6=i

E [βr,v,i · βr,v,j] . (D.1)

The correlation coefficient between the fading in any two subcarriers, βr,v,i and

βr,v,j, is defined by

ρ =E [βr,v,i · βr,v,j] − E [βr,v,i] E [βr,v,j]√

V ar [βr,v,i] ·√

V ar [βr,v,j]=

E [βr,v,i · βr,v,j] − σ2v

π4

σ2v

(1 − π

4

) (D.2)

where the following relationship is applied

E [βr,v,i] =E [βr,v,j] = σv

√π

4(D.3)

V ar [βr,v,i] =V ar [βr,v,j] = σ2v

(1 − π

4

)(D.4)

151

D. PROOF OF (6.27) AND (6.28)

Hence

E [βr,v,i · βr,v,j] = σ2v

(ρ − π

4ρ +

π

4

)(D.5)

Substitute (D.5) into (D.1) and assume all subcarriers have the same statistics,

E [B2v ] can be obtained as

E[B2

v

]=NE

[β2

r,v,i

]+ N (N − 1) E [βr,v,i · βr,v,j]

=Nσ2v

[1 + (N − 1)

(ρv −

π

4ρv +

π

4

)](D.6)

The same approach can be applied to obtained E [B2h].

152

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