on combination and interference free window spreading sequences

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On Combination and Interference Free

Window Spreading Sequences

Gregory Cresp BSc(Hons), BE(Hons)

This thesis is presented for the degree of

Doctor of Philosophy

of

The University of Western Australia.

School of Electrical, Electronic, and Computer Engineering

Crawley, WA 6009

May 2008


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Abstract

Spread spectrum techniques have a number of different applications, including range

finding, synchronisation, anti-jamming systems and multiple access communication

systems. In each of these applications the properties of the resulting systems depend

heavily on the family of spreading sequences employed. As such, the design ofspreading sequences is an important area of research.

Two areas of spreading sequence design are of particular interest in this work,

combination techniques and Interference Free Window (IFW) sequences.

Combination techniques allow a new sequence family to be constructed by com-

bining two or more existing families. Such an approach allows some of the desirable

properties of the components to be maintained, whilst mitigating the components

disadvantages. In addition, it can facilitate the construction of large families at a

greatly reduced computational cost. Combination families are considered throughthe construction of two new classes of sequences, modified Unified Complex Hadam-

ard Transform (UCHT) sequences, and combination Oppermann sequences, respect-

ively based on UCHT sequences and periodic Oppermann sequences. Numerical

optimisation techniques are employed to demonstrate the favourable performance of

sequences from these classes compared to conventional families.

Second, IFW sequences are considered. In systems where approximate, but not

perfect, synchronisation between different users can be maintained, IFW sequences

can be employed to greatly reduce both interference between users and interferenceresulting from multipath spread of each users signal.

Large Area Synchronous (LAS) sequences are a class of sequences which both

result from combination techniques and exhibit an IFW. LAS sequences are pro-

duced by combining Large Area (LA) sequences and LS sequences. They have been

demonstrated to be applicable to multiple access communication systems, particu-

larly through their use in LAS2000, which was proposed for third generation mobile

telephony. Work to date has been restricted to only a very small range of examples

of these families.

In order to examine a wider range of LAS sequences, the construction and result-

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ABSTRACT

ing properties of LA and LS families are considered. The conditions an LA family

must satisfy are codified here, and algorithms which can be used to construct LA

families with given parameters are presented. The construction of LS sequences

is considered, and relationship between each of the parameters used in this con-struction and the properties of the final family is examined. Using this expanded

understanding of both these sequence families, a far wider range of LAS families,

potentially applicable to a wider range of applications, can be considered.

Initially, the merits of proposed sequences are considered primarily through their

correlation properties. Both maximum and mean squared correlation values are con-

sidered, depending on the context. In order to demonstrate their practical applicab-

ility, combination Oppermann, modified UCHT and LAS sequences are employed in

a simulated communications system, and the resulting bit error rates are examined.

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Acknowledgements

This thesis has led me on a journey covering four years, two continents, more pages

than I care to count, and the complete emotional spectrum. This journey wouldnt

have happened without a great number of people.

First, my principal supervisor Prof. Dr.-Ing. Hans-Jurgen Zepernick. The areahe suggested for research allowed me to find common ground between my inner

mathematician and my inner engineer, and his consistently high standards helped

to push me to higher quality output than I thought myself capable of.

My co-supervisor Dr Hai Huyen Dam was always a source of useful comments

and advice. She showed by quiet example that ones inner mathematician doesnt

have to stay so inner.

A key part of any research work is being part of a research community. My time

at WATRI would not have been the same without my fellow post-grads, particularly,

but not limited to, the 10 different people with whom Ive shared an office over the

years.

My studies were made possible through financial support from The University

of Western Australia, through the Hackett Scholarship, and the Commonwealth of

Australia, through the Australian Telecommunications CRC.

The inter-continental nature of my studies was made possible by the Department

of Signalbehandling at Blekinge Tekniska Hogskola. I am very grateful for the

hospitality shown to me by everybody there.

I owe a debt of thanks to my friends, in particular my proof-readers, David

Clifton, Wayne Griffiths and Andrew Sharp, who may not have known what they

were getting themselves into when they offered.

This journey ultimately started with my parents, Judy and John Cresp. Their

support throughout my education transcends the pages ability to express gratitude.

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iv


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Contents

Abstract i

Acknowledgements iii

Contents v

List of Tables xi

List of Figures xiii

List of Abbreviations xvii

List of Common Symbols xix

1 Introduction 1

1.1 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Major Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Combination sequences . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 LAS sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 Bit error rate results . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Background 9

2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Spread Spectrum Systems . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Performance Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Energy efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.2 Auto and cross-correlation . . . . . . . . . . . . . . . . . . . . 15

2.3.3 Ratios of correlations . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.4 Mean squared correlation measures . . . . . . . . . . . . . . . 18

2.3.5 Maximum value measures . . . . . . . . . . . . . . . . . . . . 19

2.4 Introducing Sequence Families . . . . . . . . . . . . . . . . . . . . . . 21

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CONTENTS

2.4.1 Maximal length linear shift register sequences . . . . . . . . . 21

2.4.2 Gold codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.3 Orthogonal sequence families . . . . . . . . . . . . . . . . . . 22

2.4.4 Interference free window families . . . . . . . . . . . . . . . . 222.4.5 Walsh-Hadamard sequences . . . . . . . . . . . . . . . . . . . 23

2.4.6 Unified Complex Hadamard Transform sequences . . . . . . . 23

2.4.7 Frank-Zadoff-Chu sequences . . . . . . . . . . . . . . . . . . . 24

2.4.8 Oppermann sequences . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Performance Measure Bounds . . . . . . . . . . . . . . . . . . . . . . 26

2.5.1 Welch bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.2 Sidelnikov bound . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5.3 Levenshtein bound . . . . . . . . . . . . . . . . . . . . . . . . 272.5.4 Mean squared Welch bound . . . . . . . . . . . . . . . . . . . 28

2.6 Bit Error Rate Estimation . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7 Optimisation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7.1 Genetic algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7.2 Branch and bound . . . . . . . . . . . . . . . . . . . . . . . . 32

2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Combination Sequences 35

3.1 Modified Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1.1 Constructing modified UCHT sequences . . . . . . . . . . . . 39

3.1.2 Designing modified UCHT sequences . . . . . . . . . . . . . . 40

3.2 Periodic Oppermann Sequences . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 Constructing periodic Oppermann sequences . . . . . . . . . . 41

3.2.2 Properties of periodic Oppermann sequences . . . . . . . . . . 42

3.2.3 Designing periodic Oppermann sequences . . . . . . . . . . . . 45

3.3 Combination Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.1 Constructing combination Oppermann sequences . . . . . . . 473.3.2 Properties of combination Oppermann sequences . . . . . . . 48

3.3.3 Designing combination Oppermann sequences . . . . . . . . . 52

3.4 Kronecker Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5.1 Cost functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5.2 Designing families by genetic algorithm . . . . . . . . . . . . . 55

3.5.3 Modified UCHT results . . . . . . . . . . . . . . . . . . . . . . 56

3.5.4 Periodic Oppermann results . . . . . . . . . . . . . . . . . . . 59

3.5.5 Combination Oppermann results . . . . . . . . . . . . . . . . 60

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CONTENTS

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Interference Free Window Families 65

4.1 Interference Free Windows . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Windowed Correlation Measures . . . . . . . . . . . . . . . . . . . . . 68

4.2.1 Windowed mean squared cross-correlation . . . . . . . . . . . 69

4.2.2 Windowed mean squared autocorrelation . . . . . . . . . . . . 69

4.3 Contemporary Work on IFW Families . . . . . . . . . . . . . . . . . . 70

4.3.1 Overview of Existing IFW families . . . . . . . . . . . . . . . 70

4.3.2 Bounds on the IFW . . . . . . . . . . . . . . . . . . . . . . . . 72

4.4 LA Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4.1 Pulse positions and pulse spacings . . . . . . . . . . . . . . . . 74

4.4.2 Constructing LA sequences . . . . . . . . . . . . . . . . . . . . 75

4.4.3 Permutation LA families . . . . . . . . . . . . . . . . . . . . . 76

4.5 Golay Pairs and Mates . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.5.1 Golay pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.5.2 Mated Golay pairs . . . . . . . . . . . . . . . . . . . . . . . . 78

4.6 LS Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.6.1 Basic LS sequence structure . . . . . . . . . . . . . . . . . . . 80

4.6.2 Lis European patent . . . . . . . . . . . . . . . . . . . . . . . 81

4.6.3 Stanczaks construction . . . . . . . . . . . . . . . . . . . . . . 81

4.6.4 Weis variation . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.6.5 LS sequences in LAS2000 . . . . . . . . . . . . . . . . . . . . 82

4.6.6 LS sequences from complete complementary codes . . . . . . . 83

4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Large Area Sequences 85

5.1 Relationships Between the LA Criteria . . . . . . . . . . . . . . . . . 88

5.2 Conditions on Pulse Spacings . . . . . . . . . . . . . . . . . . . . . . 915.2.1 Lis original conditions . . . . . . . . . . . . . . . . . . . . . . 92

5.2.2 The weak LA condition . . . . . . . . . . . . . . . . . . . . . . 93

5.2.3 The periodic weak LA condition . . . . . . . . . . . . . . . . . 97

5.2.4 The strong LA condition . . . . . . . . . . . . . . . . . . . . . 99

5.2.5 Partial conditions . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2.6 Interaction between the LA conditions . . . . . . . . . . . . . 103

5.3 Testing LA Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.3.1 The weak LA condition . . . . . . . . . . . . . . . . . . . . . . 106

5.3.2 The periodic weak LA condition . . . . . . . . . . . . . . . . . 108

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5.3.3 The strong LA condition . . . . . . . . . . . . . . . . . . . . . 108

5.4 The LA Conditions and Permutation Families . . . . . . . . . . . . . 109

5.5 Bounds on Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.5.1 Bounds for the LA conditions . . . . . . . . . . . . . . . . . . 1145.5.2 Bounds for the k-partial weak LA condition . . . . . . . . . . 118

5.6 Generating LA Families . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.6.1 Ad hoc techniques . . . . . . . . . . . . . . . . . . . . . . . . 120

5.6.2 Choi and Hanzo . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.6.3 Periodic Choi and Hanzo . . . . . . . . . . . . . . . . . . . . . 125

5.6.4 Branch and bound algorithms . . . . . . . . . . . . . . . . . . 127

5.7 Numerical Results for LA Families . . . . . . . . . . . . . . . . . . . 131

5.7.1 Lower bounds on LA sequence length . . . . . . . . . . . . . . 1315.7.2 Achievable LA sequence lengths . . . . . . . . . . . . . . . . . 132

5.7.3 LA sequences with a novel number of pulses . . . . . . . . . . 136

5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6 LS and Large Area Synchronous Sequences 139

6.1 Constructing LS Families . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.1.1 N-quasi complete complementary codes . . . . . . . . . . . . . 141

6.1.2 Producing an N-quasi cross-complementary code . . . . . . . 142

6.1.3 Expanding an N-quasi cross-complementary code to an N-quasi complete complementary code . . . . . . . . . . . . . . . 144

6.1.4 Producing LS sequences . . . . . . . . . . . . . . . . . . . . . 146

6.1.5 Variations in LS construction . . . . . . . . . . . . . . . . . . 147

6.2 LS Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.3 The IFW of LS sequences . . . . . . . . . . . . . . . . . . . . . . . . 150

6.3.1 IFW radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.3.2 LS family size compared to theoretical bounds . . . . . . . . . 1 5 3

6.3.3 Correlation close to the IFW . . . . . . . . . . . . . . . . . . . 1546.4 Selection of LS Parameters . . . . . . . . . . . . . . . . . . . . . . . . 155

6.4.1 Selecting the orthogonal matrix . . . . . . . . . . . . . . . . . 155

6.4.2 Selecting Golay pair length and zero gap width . . . . . . . . 1 5 8

6.4.3 The choice of the selection value . . . . . . . . . . . . . . . . . 161

6.4.4 Variation of cross-correlations with zero gap width . . . . . . . 162

6.4.5 Variation of correlations with Golay pair length . . . . . . . . 1 6 4

6.5 LAS Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.5.1 Absolute encoding of LA sequences . . . . . . . . . . . . . . . 165

6.5.2 Cross-correlations of encoded LA families . . . . . . . . . . . . 166

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6.5.3 Encoding LA families with non IFW families . . . . . . . . . . 1 6 8

6.5.4 Encoding LA sequences with an LS family . . . . . . . . . . . 171

6.6 Constructing an LAS family . . . . . . . . . . . . . . . . . . . . . . . 172

6.6.1 Permutation LAS families . . . . . . . . . . . . . . . . . . . . 174

6.6.2 Construction of LAS2000 . . . . . . . . . . . . . . . . . . . . . 175

6.7 Comparing Full LAS and Base LAS Families . . . . . . . . . . . . . . 178

6.8 Assigning LAS Sequences to Users . . . . . . . . . . . . . . . . . . . . 179

6.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7 Bit Error Rate Performance 183

7.1 Bit Error Rate Simulation . . . . . . . . . . . . . . . . . . . . . . . . 185

7.2 Combination Families . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7.2.1 Sequence families . . . . . . . . . . . . . . . . . . . . . . . . . 187

7.2.2 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . 188


7.3 Comparing Base and Full LAS Families . . . . . . . . . . . . . . . . . 192

7.3.1 The LAS family . . . . . . . . . . . . . . . . . . . . . . . . . . 192



7.4 Permutation LAS Families . . . . . . . . . . . . . . . . . . . . . . . . 198



7.4.3 Comparison of cellular LAS bit error rate results . . . . . . . . 2 0 6

7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

8 Conclusions 209

8.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . 210

8.2 Suggestions for Future Work . . . . . . . . . . . . . . . . . . . . . . . 213

Appendix A Proofs of Combination Sequence Results 215

A.1 Periodic Oppermann Sequence Results . . . . . . . . . . . . . . . . . 215

A.2 Combination Oppermann Sequence Results . . . . . . . . . . . . . . . 219

Appendix B Golay Pairs 221

B.1 Primitive Golay Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

B.2 Constructing Golay Pairs . . . . . . . . . . . . . . . . . . . . . . . . . 221

B.3 Correlation Values of Golay Pairs . . . . . . . . . . . . . . . . . . . . 223

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Appendix C Additional Results for Large Area Sequences 225

C.1 Mean Squared Correlation Properties of LA Families . . . . . . . . . 225

C.2 Results for the Periodic Weak LA Condition . . . . . . . . . . . . . . 227

C.3 Intermediate Results for Optimality of the Choi and Hanzo Algorithm 230

Appendix D Additional Results for LS and Large Area Synchronous

Families 233

D.1 Cross-correlation Results for LS families . . . . . . . . . . . . . . . . 233

D.1.1 Regions 1 and 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 234

D.1.2 Region 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

D.2 An Intermediate Result for the LS IFW . . . . . . . . . . . . . . . . . 238

D.3 Calculating the WMSCC Value of an LAS Family . . . . . . . . . . . 238

Appendix E Bit Error Rate Results for a Rayleigh Channel 241

E.1 Simulation Environment . . . . . . . . . . . . . . . . . . . . . . . . . 241

E.2 Combination Sequence Results . . . . . . . . . . . . . . . . . . . . . . 242

E.3 Comparing Full and Base LAS Families . . . . . . . . . . . . . . . . . 243

E.4 Permutation LAS Families . . . . . . . . . . . . . . . . . . . . . . . . 245

Bibliography 249

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List of Tables

3.1 Sequence designs for minimum cmax(U) using the genetic algorithm. . 57

3.2 Modification vectors for the optimised modified families. . . . . . . . 57

3.3 Optimal periodic Oppermann sequences with period 62. . . . . . . . . 59

3.4 Optimal Oppermann sequences of length 62. . . . . . . . . . . . . . . 59

3.5 Optimal combination Oppermann families of length 711 and 13 11. 61

3.6 Optimal Oppermann families of length 77 and 143. . . . . . . . . . . 61

4.1 Some examples of IFW families. . . . . . . . . . . . . . . . . . . . . . 70

5.1 Relationships between the LA criteria and the different LA conditions. 92

5.2 Pulse spacings achieving the minimum sequence length Lmin for the

periodic weak LA condition with K = 16 pulses, compared to the ad

hoc results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.3 Example LA pulse spacings for a range of minimum spacings M and

number of pulses K. . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.1 The parameters of LS sequences. . . . . . . . . . . . . . . . . . . . . 148

6.2 MSAC and MSCC values for the 4 4 Sylvester and Huang form

Hadamard matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.3 Correlation results for (26, 4, 5) LS families from different Hadamard

matrices and selection values. . . . . . . . . . . . . . . . . . . . . . . 157

6.4 MSAC and MSCC values for three orthogonal families. . . . . . . . . 1 5 8

6.5 A comparison of(26, p, 5) LS families constructed from different or-

thogonal families. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.6 WMSCC values for the full (204, 22, 8) (4, 2, 3) LAS family under

different sequence assignments. . . . . . . . . . . . . . . . . . . . . . 181

7.1 Sequence families considered for BER simulations. . . . . . . . . . . . 188

B.1 Some primitive binary Golay pairs. . . . . . . . . . . . . . . . . . . . 222

B.2 Some primitive quaternary Golay pairs. . . . . . . . . . . . . . . . . . 222

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List of Figures

2.1 Block diagram of a DS-CDMA communications system. . . . . . . . . 12

3.1 Example correlation functions from the modified UCHT and UCHT

families with lowest maximum correlation. . . . . . . . . . . . . . . . 58

3.2 Correlation functions for the periodic Oppermann and Oppermann

families of length 62. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3 Correlation values for the 711 combination Oppermann and N = 77

Oppermann families. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4 MSAC and MSCC values versus the mean squared Welch bound for

13 11 combination Oppermann families. . . . . . . . . . . . . . . . 63

4.1 A cross-correlation from a family with an IFW of radius 9. . . . . . . 68

4.2 Sequences from a (75, 5, 8) LA family. . . . . . . . . . . . . . . . . . . 764.3 The base sequences of a (38, 8, 4) LA family and a permutation family. 77

4.4 The general structure of an LS sequence. . . . . . . . . . . . . . . . . 80

5.1 An aperiodic correlation which violates the constraint of LA Cri-

terion 1a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2 A periodic correlation which violates the constraint of LA Criterion 1b). 97

5.3 Relationships between the different LA conditions. . . . . . . . . . . . 104

5.4 Relationships between the LA conditions and the LA criteria. . . . . 105

5.5 The cross-correlation between the base sequences of a (124, 12, 8) LA

family and a permutation family. . . . . . . . . . . . . . . . . . . . . 112

5.6 A comparison of the bounds on LA sequence length L. . . . . . . . . 1 3 2

5.7 Bounds on LA sequence length L under the strong LA condition. . . 132

5.8 The minimal LA sequence length L achievable using the different

algorithms for K = 16 pulses. . . . . . . . . . . . . . . . . . . . . . . 133

5.9 Achievable lengths for the strong LA condition with K = 8 pulses

compared to the theoretical bound. . . . . . . . . . . . . . . . . . . . 136

6.1 The first sequence of a (8, 2, 7) LS family with selection value n = 1. . 147

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6.2 Autocorrelation functions of the first sequences from two LS families

of the same length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.3 Comparing correlation functions for two (16, 2, 10) LS families. . . . . 162

6.4 Correlation measures for (26, 8, W0) LS families. . . . . . . . . . . . . 164

6.5 Correlation measures for (N, 8, 5) LS families. . . . . . . . . . . . . . 165

6.6 The absolute encoding of an LA sequence . . . . . . . . . . . . . . . . 166

6.7 An example cross-correlation between encoded LA sequences at small

shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.8 Cross-correlation values for the absolute encoding of a (284, 32, 8) LA

family by a modified UCHT family. . . . . . . . . . . . . . . . . . . . 169

6.9 Cross-correlation values for absolute encoding of a (284, 32, 8) LA

family by an Oppermann family. . . . . . . . . . . . . . . . . . . . . . 1706.10 A comparison of data transmission using full and base LAS. . . . . . 172

6.11 Cross-correlation values for a full (204, 22, 8) (4, 2, 3) LAS family. . 174

6.12 The structure of an LS sequence within an LAS sequence. . . . . . . 177

7.1 A block diagram of the DS-CDMA system used to calculate BERs. . 186

7.2 BER results for the original and modified UCHT families comparing

branch and bound and removal algorithms. . . . . . . . . . . . . . . . 189

7.3 BER results for the original and modified Walsh-Hadamard families. . 189

7.4 BER results for the Oppermann and combination Oppermann famil-

i es of l ength 77. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

7.5 BER versus maximum delay for the four (728, 38, 16) (4, 4, 3) LAS

systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

7.6 BER results for the four (728, 38, 16) (4, 4, 3) LAS systems at fixed

maximum delays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

7.7 The simplified cellular system. . . . . . . . . . . . . . . . . . . . . . . 198

7.8 BER results for a base LAS cellular system using permutation families

for varying secondary cell attenuation. . . . . . . . . . . . . . . . . . 201

7.9 BER results for the two cell LAS systems when ACI,min = 0. . . . . . 202

7.10 BER results for the two cell LAS systems with sequence reuse when

ACI,min = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

D.1 The three cross-correlation regions between LS sequences. . . . . . . . 234

E.1 A typical Rayleigh fading envelope. . . . . . . . . . . . . . . . . . . . 242

E.2 BER results for the UCHT and modified UCHT families comparing

branch and bound and removal algorithms on a Rayleigh channel. . . 243

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LIST OF FIGURES

E.3 BER results for the Oppermann and combination Oppermann famil-

ies of length 77 on a Rayleigh channel. . . . . . . . . . . . . . . . . . 243

E.4 BER versus maximum delay for the four (728, 38, 16) (4, 4, 3) LAS

systems in Rayleigh fading. . . . . . . . . . . . . . . . . . . . . . . . 244E.5 BER results for the four (728, 38, 16) (4, 4, 3) LAS systems at fixed

maximum delays in Rayleigh fading. . . . . . . . . . . . . . . . . . . 244

E.6 BER results for a base LAS cellular system using permutation families

for varying secondary cell attenuation over a Rayleigh channel. . . . . 246

E.7 BER results for the two cell Rayleigh faded LAS systems with ACI,min =

0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

E.8 BER results for the two cell Rayleigh faded LAS systems with se-

quence reuse when ACI,min = 10. . . . . . . . . . . . . . . . . . . . . . 247

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List of Abbreviations

ACI Adjacent Cell Interference

AWGN Additive White Gaussian Noise

BER Bit Error Rate

BPSK Binary Phase Shift KeyCDMA Code Division Multiple Access

DS-CDMA Direct Sequence CDMA

FDMA Frequency Division Multiple Access

FZC Frank-Zadoff-Chu

IFW Interference Free Window

ISI Inter-symbol Interference

JPL Jet Propulsion Laboratories

LA Large Area

LAS Large Area Synchronous

LCZ Low Correlation Zone

MAI Multiple Access Interference

MC-DS-CDMA Multicarrier DS-CDMA

MSAC Mean Squared aperiodic Autocorrelation

MSCC Mean Squared aperiodic Cross-Correlation

PAPR Peak to Average Power Ratio

QPSK Quadrature Phase Shift Key

SINR Signal to Interference plus Noise Ratio

SNIR Signal to Noise and Interference Ratio

SNR Signal to Noise Ratio

TDMA Time Division Multiple Access

UCHT Unified Complex Hadamard Transform

WMSAC Windowed Mean Squared aperiodic Autocorrelation

WMSCC Windowed Mean Squared aperiodic Cross-Correlation

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List of Common Symbols

General

a b (mod N) a and b give the same remainder after division by N

a = b mod N a is equal to the remainder of b after division by N

br,c Entry in row r, column c of the matrix B

BT Transpose of matrix B

BH Hermitian transpose of matrix B

dec(b) Decimal representation of the unipolar binary vector b

diag(b) Diagonal matrix whose main diagonal is the vector b

ab Dirac-delta function

exp() Natural exponent, e, raised to the power of ()

gcd(a, b) Greatest common divisor of integers a and b

j Square root of1n

k

Number of ways to choose k elements from a set of size n

(N) Eulers totient function evaluated at integer N

R[] Real part of ()

Ceiling of ()

Floor of ()

() Complex conjugate of ()

Sets

[a, b] Interval a to b, inclusive

(a, b) Interval a to b, exclusive

A \ B Difference set ofA and B

A Cardinality of the set A

C Set of complex numbers

M({di}ki=0, a , b) Modulo summation of the set {di}ki=0 from index a to index b

Z Set of integers

Z+ Set of positive integers

Zm Group of integers modulo m

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LIST OF COMMON SYMBOLS

Performance Measures

E(uk) Energy of sequence uk

E(U) Average energy of sequence family U

(U) Energy efficiency of sequence family UCuk (l) Aperiodic autocorrelation of uk at shift l

Cuk,r(l) Aperiodic cross-correlation between uk and ur at shift l

Cuk,vr(l) Aperiodic cross-correlation between uk and vr at shift l

uk(l) Periodic autocorrelation of uk at shift l

uk,r(l) Periodic cross-correlation between uk and ur at shift l

uk ,vr(l) Periodic cross-correlation between uk and vr at shift l

cam(U) Maximum aperiodic out-of-phase autocorrelation ofU

ccm(U) Maximum aperiodic cross-correlation ofUcmax(U) Maximum non-trivial aperiodic correlation ofU

am(U) Maximum periodic out-of-phase autocorrelation ofU

cm(U) Maximum periodic cross-correlation ofU

max(U) Maximum non-trivial periodic correlation ofU

Rac(U) Mean squared aperiodic out-of-phase autocorrelation ofU

Rcc(U) Mean squared aperiodic cross-correlation ofU

Wac(U, lmax) Windowed mean squared aperiodic out-of-phase autocorrela-

tion ofU with maximum shift lmax

Wcc(U, lmax) Windowed mean squared aperiodic cross-correlation ofUwith

maximum shift lmax

Sequence Families

F Family of Frank-Zadoff-Chu (FZC) sequences

G Family of Gold sequences

Hn Family of Walsh-Hadamard sequences of size 2n

Hmn Family of modified Walsh-Hadamard sequences of size 2n

La

Family of Large Area (LA) sequences(L,M,K) Parameter set for an LA family containing sequences of length

L, each with K pulses and minimum pulse spacing M

Ls Family of LS sequences

(N,p,W0) Parameter set for an LS family containing 2p sequences using

Golay pairs of length N and zero gaps of width W0

Las Family of Large Area Synchronous (LAS) sequences

O Family of Oppermann sequences

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LIST OF COMMON SYMBOLS

(N,m,n,p) Parameter set for an Oppermann family of length N with

exponents m, n and, optionally, p

Op Family of periodic Oppermann sequences

Oc Family of combination Oppermann sequencesTn Family of Unified Complex Hadamard Transform (UCHT) se-

quences of size 2n

(1, 2, 3) Parameter set for the UCHT family constructed using the

values 1, 2 and 3

Tmn Family of modified UCHT sequences of size 2n

T Cyclic left shift operator

uk kth sequence from the family U

uk(i) Index i of the sequence uku Sequence formed by reversing the terms of sequence u

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Chapter 1

Introduction

Spread spectrum is a paradigm arising from advances in anti-jamming techniques

before and during the Second World War. As their name suggests, spread spectrum

techniques take signals with a narrow bandwidth and spread them over a wider

frequency range. Such techniques can be employed to achieve resistance to jamming,

precise synchronisation, range finding, or to allow multiple users to co-exist on a

shared channel [1].

In contrast to previous multiple access techniques such as Time Division Mul-

tiple Access (TDMA) and Frequency Division Multiple Access (FDMA), a spread

spectrum communications system does not have hard limits on the number of usersor the data rate it can support. Instead the system has soft limits relating to the

total interference [2]. Whilst TDMA and FDMA differentiate between different users

by using different time and frequency slots, respectively, the most common variant

of spread spectrum multiple access, Code Division Multiple Access (CDMA), dif-

ferentiates between users by assigning unique spreading sequences. Interference is

primarily dependant on the family of spreading sequences employed by the system,

and hence the design of these sequences is a key concern for any such system.

The design of sequence families has attracted a great deal of attention for a num-ber of years [36]. It has been argued that the sequences should appear random, that

is without prior knowledge their values should be unpredictable [1]. This is particu-

larly important for resistance to intentional jamming. Randomly selected sequences

have been shown to perform well in some cases [7]. The use of sequences where each

term is generated according to some rule has many advantages, particularly the ease

in describing and storing such sequences.

Whilst the first stages of sequence design are undertaken at a theoretical level,

it must still be noted that the resulting family will ultimately be implemented in a

real system. Spread spectrum techniques are commonly employed in mobile devices,

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INTRODUCTION

thus the issue of the computational complexity of producing the sequences is an

important consideration. One possible approach to reduce complexity is to produce

the final spreading sequences by combining two or more intermediate sequences,

each of which is simpler to generate [8]. This can be employed to produce a largernumber of sequences, longer sequences, sequences with improved properties, or some

combination of the three.

Previously, a number of different combination techniques have been applied to

binary families. By way of introduction to combination sequences, three such tech-

niques are considered here: the modification method [9], the combination method [8]

and the Kronecker product [10]. In cases where combination sequences are not con-

sidered, it has been demonstrated that non-binary sequences can achieve improved

performance and increased design flexibility over their binary counterparts [11]. Inlight of this, this thesis considers the combination of non-binary sequences. In par-

ticular, the modification method and the combination method are used to construct

new classes of polyphase sequence families, respectively modified Unified Complex

Hadamard Transform (UCHT) and combination Oppermann sequences.

The ultimate achievement in sequence design, whether using combination tech-

niques or not, would be to completely remove both interference caused by multiple

paths of the same users signal and Multiple Access Interference (MAI) between

different users sharing the common channel. If the delays between different usersare completely random it is impossible to achieve this using a single carrier [12].

However, in the special class of systems where there is some known upper bound

on delays, a sequence family can be designed to be free of interference within the

range of delays experienced [13]. This class of sequences is known as Interference

Free Window (IFW) sequences.

Two IFW families of particular interest are Large Area (LA) sequences [14] and

LS1 sequences [18]. LA sequences are of interest because of their very large IFW and

very low interference outside the IFW [19]. However, for their length they support

a small number of users [16, 18], and their energy efficiency is very low [20]. LS

sequences exhibit a smaller IFW, but have both high energy efficiency [18] and, for

their length and IFW radius, support a large number of users.

The class of Large Area Synchronous (LAS) sequences [14] combines the con-

cepts of combination families and IFW families. An LAS sequence is produced

by combining LA and LS families, and exhibits an IFW equal to that of the LS

family. Work on LAS sequences to date has shown that they are capable of sup-

1

As noted by [15], LS is not an abbreviation. Some sources use it as an abbreviation of LooselySynchronous [16,17], referring to the same class of sequences.

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1.1. Thesis Structure

porting a large number of users at high data rates with low interference [21]. As

such, they were proposed as part of a third generation mobile telephony standard,

LAS2000 [22]. However, previous work has been greatly restricted by the lack of a

theoretical framework for LA families, and has thus concentrated on the LAS familyemployed by the LAS2000 standard [20, 2325]. This lack of a framework restricts

the number of LA families which can be constructed, and hence the applicability of

LAS families to a wider range of systems.

A rigorous treatment of LA families is thus required in order to expand the

understanding and application of LAS families. This thesis formally defines an LA

family and determines the minimum conditions required to satisfy this definition.

From this, alternatives to the previous ad hoc techniques for producing these families

are developed, which allow for a wider range of LA families to be produced.Whilst a number of different variations on the construction of LS sequences

have been proposed [18, 21, 26], the significance of each component used in these

constructions has not been considered. The effect of varying these parameters, such

as the family size and the Golay pair used to construct the LS family, is considered

here.

Armed with an expanded understanding of LA and LS families, the properties of

LAS families can be further examined. Two modes of operation of an LAS system

exist. These trade off between the number of supported users and the data rate

per user. The relative advantages of these systems are compared, and demonstrated

through numerical Bit Error Rate (BER) results.

1.1 Thesis Structure

The first part of the thesis presents background material. This material considers in

turn the general theory of spreading sequences, methods for producing combination

sequences, and finally IFW sequences. The second part of this work considers LA,LS and LAS sequences in detail. In particular, new methods to construct a wide

range of these families, beyond the small number of examples from the literature,

are considered. Finally, the BERs of the sequence families considered in this work

are presented via a simulation environment.

The content of each chapter is as follows.

Chapter 2 provides background material on spreading sequences. The main ap-

plication, spread spectrum communications, is introduced. The performance meas-

ures which are of particular interest in this work are defined, and their relationships

are examined through theoretical bounds. Some examples of the most common

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INTRODUCTION

spreading sequences from the literature are presented. Finally, the genetic algorithm

and the branch and bound algorithm, which are used to optimise sequence families,

are introduced.

Chapter 3 examines the use of combination methods to construct sequence

families. The advantages of using combination techniques are shown by example.

Three combination methods are introduced, and the first two are used to create

new classes of sequence families. First the modification method is used to construct

the class of modified UCHT sequences. Second, a method of periodic repetition and

chip-wise multiplication is used to construct the class of combination Oppermann se-

quences. As an intermediate step, the class of periodic Oppermann sequences, which

are naturally periodic, is defined. It is shown that both periodic Oppermann and

combination Oppermann sequences retain many of the desirable properties of theOppermann sequences upon which they are based. The Kronecker product, whose

operation is similar to the chip-wise combination technique, is discussed. Given that

the Kronecker product does not as easily preserve the desirable properties of the ori-

ginal families, the chip-wise multiplication technique is used in preference. Using

the genetic algorithm introduced in Chapter 2, numerical results are produced for

each of the new classes of spreading sequence. It is shown that their performance

is generally equal to or superior to that of existing sequence families. The combina-

tion nature of these families results in additional benefits, such as a computationallysimpler construction.

Chapter 4 gives a review of IFW sequences. This both provides context and

motivation for the consideration of LA, LS and LAS families, and introduces the

theory necessary for the work of Chapters 5 and 6. The concepts behind and the

advantages of IFW sequences are considered, and a number of examples of IFW

families are presented. Detailed background on LA and LS sequences is presented

and new formal definition of an LA family is given.

Chapter 5 constructs a theoretical framework to describe LA sequences, build-ing on the definition given in Chapter 4. The minimum conditions a family must

satisfy in order to be an LA family are specified. Using these conditions, LA families

are classified into a number of categories. It is demonstrated that the very limited

number of examples presented thus far in the literature cover only a small propor-

tion of the possible LA families. In addition, simple tests to determine whether a

family satisfies these minimum conditions are presented. Based on these tests, a

number of algorithms, which are able to produce LA sequences with given paramet-

ers, are developed. The results from families produced by these algorithms are then

considered.

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1.2. Major Contributions

Chapter 6 covers LS sequences and LAS sequences, the latter of which are

produced by combining LA and LS sequences. The construction of LS sequences

is described in detail, and shown to be a special case of complete complementary

codes, a class of multicarrier spreading sequences. The relationship between theparameters used to construct an LS family and the correlation properties of the

resulting family is considered. The construction of LAS sequences is described, and

two different ways they may be used in a transmission system, base and full LAS, are

considered. Ways of minimising the interference observed when a system employing

LAS sequences loses quasi-synchronisation by a small amount are discussed.

Chapter 7 presents BER simulations of the sequence families presented in this

work. Examples of the combination families presented in Chapter 3 are simulated in

a multiple access environment and compared to traditional sequence families. Thedifferent modes of operation of LAS sequences are tested and compared, and it is

shown that a full LAS system in which each user is assigned multiple sequences

results in significantly lower BERs than the corresponding base LAS system. The

performance of LAS sequences in a cellular environment is also considered. It is

shown that the use of permutation LAS families can result in significant improvement

in BER compared to different cells reusing the same LAS sequences.

Finally, Chapter 8 summarises the findings of this work, and presents possibil-

ities for future research on these topics.

1.2 Major Contributions

The contributions of this thesis relate to two main areas, combination sequences

in general, and LAS sequences, including their component LA and LS sequences.

The theoretical claims of these contributions are then supported by numerical BER

results.

1.2.1 Combination sequences

Modified UCHT sequences

The new class of modified UCHT sequences is produced by applying the modifica-

tion technique to UCHT families. This is done analogously to the construction of

modified Walsh-Hadamard sequences.

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INTRODUCTION

Combination Oppermann sequences

The class of periodic Oppermann sequences, which are a special case of Oppermann

sequences that exhibit natural periodicity, is introduced. These sequences are thencombined to produce the new class of combination Oppermann sequences. Nu-

merical results are presented to show that combination Oppermann sequences can

achieve similar performance to the original Oppermann sequences, with the addi-

tional benefits of simplified construction and implementation, and orthogonality.

1.2.2 LAS sequences

LA sequences

The definition and construction of LA sequences is treated rigourously. A formal

definition is given, and three conditions which can be used to test different degrees

of adherence to this definition are developed. Using these conditions, LA families

are divided into a number of overlapping classes. From this classification, it is high-

lighted that only a very small number of LA families have thus far been considered,

compared to the large number of possible families. The properties of the different

classes of LA families are considered, and algorithms which can be used to construct

such families for a wide range of input parameters are presented.

LS sequences

The construction of LS sequences is shown to be similar to the construction of

complete complementary codes. An expression for the cross-correlation terms of an

LS family in terms of the parameters used to construct the family is given. Using

this expression and numerical results, the effect each of the input parameters has

on the resulting correlation performance is considered.

LAS sequences

It is demonstrated that the IFW of an LA sequence is only preserved under the

method of absolute encoding when another IFW family is used as the encoding

family. Two methods of employing an LAS family in a given system are presented

and compared. A method to greatly reduce the interference exhibited just outside

the IFW of an LAS family by assigning multiple users to each sequence is given.

This claim of reduced interference is supported by numerical results.

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1.3. Publications

1.2.3 Bit error rate results

The performance of the modified UCHT, combination Oppermann and LAS famil-

ies is examined through simulation of their BERs in different systems. It is shown

that the advantages the combination Oppermann sequences have over Oppermann

sequences are not at the cost of an increased BER. When the number of users in

an LAS system is relatively low, it is demonstrated that for delays close to, but

outside, the IFW there is significant advantage in employing multiple full LAS se-

quences per user rather than base LAS. Finally, under some circumstances a cellular

system shows significant improvement in BER when permutation LAS sequences are

employed, rather than reusing the same LAS sequences in different cells.

1.3 Publications

This thesis is supported by the following publications:

(P.1) G. Cresp, H.-J. Zepernick, and H. H. Dam, Periodic Oppermann sequences

for spread spectrum systems, in Proc. IEEE Inform. Theory Workshop, Ro-

torua, New Zealand, Sept. 2005, pp. 3135.

(P.2) G. Cresp, H.-J. Zepernick, and H. H. Dam, Combination Oppermann se-

quences for spread spectrum systems, in Proc. IEEE Int. Symp. Inform. The-

ory, Adelaide, Australia, Sept. 2005, pp. 20452049.

(P.3) G. Cresp, H. H. Dam, and H.-J. Zepernick, Subset family design using a

branch and bound technique, in Proc. Australian Commun. Theory Work-

shop, Perth, Australia, Feb. 2006, pp. 5357.

(P.4) G. Cresp, H. H. Dam, and H.-J. Zepernick, Design of modified UCHT se-

quences, in Proc. Symp. on Trends in Commun., Bratislava, Slovakia, June

2006, pp. 4043.

(P.5) G. Cresp, H.-J. Zepernick, and H. H. Dam, On the classification of Large

Area sequences, in Proc. IEEE Inform. Theory Workshop, Bergen, Norway,

July 2007, pp. 153157.

(P.6) G. Cresp, H.-J. Zepernick, and H. H. Dam, Bit error rates of Large Area

Synchronous systems in the presence of adjacent cell interference to appear

in Proc. General Assembly Int. Union of Radio Science, Chicago, USA, July

2008.

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Chapter 2

Background

Spreading sequences have a number of areas of application, including spread spec-

trum communications, covert radar and range finding [1]. Throughout the following,

the most common application considered will be spread spectrum communications.

Regardless of the application, a number of preliminaries are required to proceed

with any in-depth examination of spreading sequences. First, spreading sequences

must be defined, and their use outlined.

All applications require some means of categorising how suited a given family

of sequences is to the task at hand. A number of quantitative measures suitablefor such categorisation are given here, most based on the concept of correlation.

Of particular interest are the mean squared correlation measures, the maximum

aperiodic correlation measures and the energy efficiency. Theoretical bounds relating

these measures are also given.

The concepts of orthogonal sequences and Interference Free Window (IFW) fam-

ilies are presented. A number of spreading sequence families from the literature are

introduced, including Gold codes, Walsh-Hadamard sequences, UCHT sequences,

Frank-Zadoff-Chu (FZC) sequences and Oppermann sequences. These sequences

serve as useful examples for the classes of spreading sequences and their properties,

and will be used in the construction of new sequence families in later chapters.

The design of sequence families can often be presented as an optimisation prob-

lem. To this end, two different methods of solving a given optimisation problem are

introduced, a genetic algorithm and a branch and bound algorithm. A genetic al-

gorithm mimics the operation of survival of the fittest, aiming to select the element

of the search space with the lowest cost. A branch and bound algorithm uses the

natural structure of the problem to map the search space onto a tree structure, then

removes sections of this tree to reduce the search space.

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BACKGROUND

Structure

This chapter is structured as follows. Section 2.1 introduces the main notational

conventions used in this work. Section 2.2 introduces spread spectrum systems, and

hence spreading sequences. The performance measures of particular interest here

are given in Section 2.3. Several examples of spreading sequences from literature are

discussed in Section 2.4. Bounds relating the performance measures are discussed

in Section 2.5. Theoretical calculation of Bit Error Rates (BERs) is considered in

Section 2.6. Finally, two optimisation algorithms are given in Section 2.7.

2.1 Notation

The following notation is used in this thesis:

A sequence family is a finite set containing distinct sequences. Sequence fam-ilies are always denoted by script capitals, for example

U= {uk| k S}, S Z+. (2.1)

Here S is referred to as the index set of U. Index sets are also denoted byscript capitals and are finite subsets of the positive integers. Unless otherwise

noted the index set is be structured as

S= {1, 2, , S}, (2.2)

where S denotes the cardinality ofS, usually denoted by S.

A sequence is a row vector whose elements are taken from some alphabet.Sequences are denoted by bold lower case letters, sequences from a family use

the same letter as the family. Different sequences from a family are denoted

by unique subscripts. Unless otherwise noted, distinct elements of the index

set of a family denote distinct sequences, that is

uk = ur, k, r S, k = r. (2.3)

The elements of a spreading sequence are referred to as chips. Chips are

indexed starting at zero, and the notation

u = [u(0), , u(L 1)] (2.4)

is used. Sequence length is generally denoted by L, as in (2.4), or alternatively

N.

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2.2. Spread Spectrum Systems

In addition to index sets, script capitals are also used to denote subsets ofintegers, for example A Z. For such a set the following operations aredefined:

A + s = {a + s | a A}, (2.5)A s = {a s | a A}, (2.6)

and

|A| = {|a| | a A}, (2.7)

where |a| is the magnitude of a.

Bold capitals such as B denote matrices. The term in row r and column cof B is br,c, the first row and first column correspond to r = 1 and c = 1,

respectively. A matrix can be presented as B = [br,c].

A matrix is said to be orthogonal if the dot product of any two distinct rows

is zero.

Binary values are, except where stated otherwise, assumed to be bipolar, thatis valued

1.

2.2 Spread Spectrum Systems

Spread spectrum techniques arise from radar systems and missile anti-jamming

measures developed before and during the Second World War, but in the time since

their application has expanded. A spread spectrum system has the following char-

acteristics [1]

The carrier [signal] is an unpredictable, or pseudorandom, wideband signal. The bandwidth of the carrier is much wider than the bandwidth of the data

modulation.

Reception is accomplished by cross correlation of the received wide-band signalwith a synchronously generated replica of the wide-band carrier.

The third characteristic is not shared by all spread spectrum systems, for example

radar systems may alternatively employ an instrumental variables based receiver

[27].

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BACKGROUND

Data

Channel User kReceiver

Received

Data

Data

User k

User r

.

.

.

Modulator

SpreadingSequence

Modulator

Spreading

Sequence

Figure 2.1: Block diagram of a DS-CDMA communications system.

A common method used in spread spectrum communications is Direct Sequence

Code Division Multiple Access (DS-CDMA). DS-CDMA systems, generally em-

ployed in wireless applications, provide a useful framework for the discussion of

spreading sequence design. As such, most of the following work is performed with

reference to such systems.In a DS-CDMA system a number of users share the same frequency range and

transmit simultaneously. Each user is assigned a unique spreading sequence, also

referred to as a signature sequence or code, which is used to identify their signal.

The collection of these sequences is referred to as a sequence family. The elements

of the spreading sequences are referred to as chips. The rate at which chips are

transmitted is referred to as the chip rate, and is much higher than the data rate of

the system.

In short sequence DS-CDMA a user employs their entire spreading sequence foreach symbol of data transmitted. This scheme is shown in Figure 2.1. User ks

data symbols, dk(i), are first multiplied by the users spreading sequence, uk to

produce the spread symbols dk(i)uk. These spread symbols are them passed to the

modulator, for example pairs of symbols may be combined via Quadrature Phase

Shift Key (QPSK) modulation. The effect of the terms of the spreading sequence is

to alter the modulated signals phase at the chip rate. In such a system the ratio of

the chip rate to the symbol rate is equal to the length of the spreading sequence.

There are three different types of spreading sequence which will be considered

here. The types are distinguished by the range of values the sequence takes, referred

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2.2. Spread Spectrum Systems

to as the alphabet of the sequence family.

Binary Sequences

Binary sequences are valued +1 and 1. For chip values of +1, the users signal istransmitted as normal. For values of1 the signal is transmitted with a 180o phaseshift. For example, when QPSK modulation is employed the spread signal is still

QPSK, but its phase inverts at a rate much higher than the data rate.

Ternary Sequences

Ternary sequences take on values 0, +1, 1. They operate similarly to binary se-quences, except that where the sequence is 0 valued no signal is transmitted. The

result is a signal whose phase inverts at a high rate, which also cuts out for small

periods of time.

Polyphase Sequences

Polyphase sequences have chip values on the unit circle in the complex plane. Binary

sequences are thus a special case of polyphase sequences. The phase of the complex

value corresponds to the phase shift which is performed on the transmitted signal

during that chip. A polyphase sequence family may either have a regular alphabet,

where the sequence values are evenly spaced around the unit circle, generally com-

plex roots of unity, or the values can be irregularly spaced. Quadriphase sequences

are a special case of polyphase sequences whose alphabet is the 4th roots of unity,

+1, +j, 1, j, and have the advantage of easy implementation in practical systems.Polyphase sequences are of particular interest for two reasons. Firstly, polyphase

sequences can potentially out-perform binary sequences [11]. Secondly, polyphase

families generally have a larger number of parameters to be tuned, and hence can

be more specifically designed for a particular application.

2.2.1 Interference

The key factor in DS-CDMA systems is the interference between users. Every

user on the channel is a source of interference to every other user. Furthermore,

due to multipath and frequency selective fading effects, each user also experiences

self-interference. Unlike other multiple access schemes, which have hard limits on

channel capacity in terms of data rates and number of users, CDMA systems are

interference limited. A system can always support more users, or a higher data rate,

at the cost of larger interference, and hence higher error rates.

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BACKGROUND

Interference can be considered in terms of the anti-jamming capabilities of spread

spectrum systems. A non-spread spectrum system can be jammed by transmitting a

large amplitude but narrow bandwidth signal at the same frequency being employed

by the transmitter. Spread spectrum systems are relatively unaffected by suchjamming [1]. A jammer has two choices. Firstly, the jammer may transmit over the

entire frequency range occupied by the spread spectrum system. Given the wide

bandwidth of spread spectrum systems, this requires either a significantly reduced

power spectral density, or greatly increased transmission power. Alternatively, they

may transmit their own spread spectrum signal which matches the signal to be

jammed as closely as possible. Such an approach has two main problems. Firstly,

whilst spreading sequences are not generally cryptographically secure [28], their

pseudo-random nature and high chip rate can make acquiring an unknown sequencedifficult. Secondly, in order to be effective, the jamming signal must arrive at the

receiver in phase with the original signal. Particularly with a moving receiver or one

at an unknown location, this can be difficult to achieve.

Users on a DS-CDMA system appear to each other as unintentional jammers.

As is the case with intentional jamming, the interference relates to how similar the

users spreading sequences are. The interference between users is thus minimal as

long as different users spreading sequences are sufficiently different. The concept of

difference between sequences will be considered in the form of correlation measures

in Section 2.3. The interference limitation on the number of users in a system thus

directly relates to the existence of sequence families of a given size whose sequences

satisfy the required measure of being sufficiently different from each other, and

from delayed versions of themselves [29].

In addition to being resistent to narrow-band jamming, the interference caused

by a spread spectrum system on nearby narrow-band systems is generally minimal.

Interference is only caused by the proportion of the spread spectrum systems power

which lies inside the narrow-band systems frequency range [30], which, noting the

wide bandwidth of a spread spectrum system, will be small.

2.3 Performance Measures

As discussed in Section 2.2, the interference observed in a DS-CDMA system directly

affects the error rate experienced by users, and hence the usefulness of the system.

Rather than rely on poorly defined statements such as sequences being sufficiently

different, some quantitative means of measuring a sequence familys performance is

required. Here correlation measures which can be used to judge performance in this

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2.3. Performance Measures

manner are presented.

Throughout this section, Uwill be a sequence family containing S sequences oflength N, as described in (2.1).

2.3.1 Energy efficiency

It is of particular interest to measure interference as a fraction of the energy of the

relevant sequences. As such, it is necessary to have some measure of sequence energy.

Such a measure is also of interest in its own right, in the calculation of the energy

efficiency, or duty cycle, of the sequences. A family with low energy efficiency can

cause problems with synchronisation, as a receiver can only synchronise on those

proportions of the signal where power is being transmitted. Secondly, a low energyefficiency will often correspond to a high Peak to Average Power Ratio (PAPR),

which can cause problems with the amplification of the transmitted signal [31,32].

The energy of a sequence uk, E(uk), is the sum of the squared magnitudes of its

chips, that is

E(uk) =N1i=0

|uk(i)|2. (2.8)

For ternary valued sequences, where each uk(i) takes values 1, 0 or 1, the energy

of each sequence is equal to the weight, that is the number of non-zero entries.The average energy of the family Uis

E(U) = 1UkS

E(uk), (2.9)

and the energy efficiency of the family is

(U) = E(U)N

. (2.10)

For sequence families where |uk(i)| = 1 for all k and i, for example binaryfamilies, the energy efficiency is (U) = 1.

2.3.2 Auto and cross-correlation

Correlation functions are a quantitative measure of the difference between sequences

at various relative shifts. They can be related directly to the integrate and sample

operation of the matched filters employed in simple receivers [33]. As will be dis-

cussed in Section 2.6, correlation values can be related to the resulting BER of the

system.

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BACKGROUND

Autocorrelation values relate to the difference between a sequence and delayed

versions of itself and cross-correlation values relate to the difference between offset

versions of distinct sequences. There are two main classes of correlation functions

considered here, aperiodic correlation and periodic correlation.

Aperiodic correlation

Although it has been noted that periodic correlation functions are generally simpler

to determine analytically [34], a direct connection can be made between the aperi-

odic correlation values and the received signal in a DS-CDMA system [29]. The

work of [33] provided a relationship between BERs in a CDMA system employing

Binary Phase Shift Key (BPSK) modulation over an Additive White Gaussian Noise

(AWGN) channel and the aperiodic correlation values for the sequence family em-

ployed. This relationship has since been expanded to a wider range of systems [35]

and is discussed further in Section 2.6. In light of this, aperiodic correlations are

almost exclusively considered in this work.

The aperiodic cross-correlation between the sequences uk, ur U at discreteshift l is defined as

Cuk,r(l) =

1E(uk)E(ur)

N1l

i=0uk(i)ur(i + l)

0 l N 11

E(uk)E(ur)

N1+li=0

uk(i l)ur(i) 1 N l < 0

0 otherwise,

(2.11)

where z is the complex conjugate of z.

In most applications every sequence in the family will have the same energy,

hence correlations can be normalised by 1/E(U). When the sequences are takenfrom two different families, for example U= {uk| k S} and V= {vk| k T }, thenotation

Cuk,vr(l) (2.12)

is used to denote the aperiodic cross-correlation between uk and vr at shift l.

When k = r, the expression in (2.11) gives the aperiodic autocorrelation of uk,

denoted Cuk (l). Noting that each autocorrelation is normalised by 1/E(uk), the

autocorrelation at zero shift is 1, that is Cuk (0) = 1.

Two useful properties of aperiodic correlation from [29] are given here.

Property 2.1. For any sequences u, v, both with length N, and any shift l, [29]

Cu,v(l) = Cv,u(l) (2.13)

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and hence for any sequence u,

Cu(l) = Cu(l). (2.14)

Property 2.2. Take two sequences u and v, both of length N, and construct the

reversed sequences u and v, also of length N, via

u(i) = u(N 1 i), 0 i < N,v(i) = v(N 1 i), 0 i < N.

(2.15)

The cross-correlation between these reversed sequences at shift l is [29]

Cu,v(l) = Cv,u(l), (2.16)

and hence the autocorrelation of a reversed sequence u is

Cu(l) = Cu(l). (2.17)

Periodic correlation

In calculating aperiodic correlations the sequences are assumed not to repeat, that

is they are N isolated terms surrounded by zeros. Any sequence uk of length N may

be extended to a periodic sequence by evaluating indices modulo N. Using such

extensions, the periodic cross-correlation of uk and ur may be calculated for anydiscrete shift l via [2]

uk,r(l) =1

E(uk)E(ur)

N1i=0

uk(i mod N)ur(i + l mod N). (2.18)

Analogously to aperiodic autocorrelation, uk,k(l) is referred to as the periodic auto-

correlation of uk at shift l and is denoted uk(l).

As in the aperiodic case, uk,vr(l) is used to denote the periodic cross-correlation

between uk and vr at shift l.

Periodic correlation can be calculated in terms of aperiodic correlation [29].

Property 2.3. Given two sequences u and v of length N, and any shift l with

N < l < N, [29]

u,v(l) =

Cu,v(l + N) + Cu,v(l), N < l < 0Cu,v(0), l = 0

Cu,v(l N) + Cu,v(l), 0 < l < N.(2.19)

Furthermore, u,v(l) is periodic with period N.

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BACKGROUND

Correlation at non-integral shift

Both the aperiodic and periodic correlations may also be calculated at shifts that

are not an integer multiple of the chip time. The result of such a correlation depends

on the pulse shape used for each chip. Where rectangular pulses with width equal

to the chip duration are used, for any integer l and 0 < < 1,

Cuk,r(l + ) = (1 )Cuk,r(l) + Cuk,r(l + 1). (2.20)

Since this value will be between Cuk,r(l) and Cuk,r(l + 1), any worst case analysis need

only consider correlation values at integer shifts [33].

The same result holds for periodic correlations.

2.3.3 Ratios of correlations

Performance measures can be constructed by considering ratios of correlation values.

Two examples of these are merit factor, which is the ratio of the energy in the

sidelobes of the aperiodic autocorrelation function of a sequence to the value at

the peak [2] and the main-to-sidelobe factor, which is the ratio of the zero shift

autocorrelation value to the maximum out-of-phase aperiodic autocorrelation value.

2.3.4 Mean squared correlation measures

In early work [33, 36] mean squared correlation measures were only considered

between a single pair of sequences. For two sequences uk and ur from a sequence

family Uthe sum squared correlation measure between them at shift i is [33]

uk,r(i) =N1

l=1N

Cuk,r(l)Cuk,r(l + i)

. (2.21)

Whilst the measure has been demonstrated to be a useful measure of interference[36], it does not provide an overall indication of the performance of the entire family.

The mean squared correlation measures for the entire family were first introduced

by [6] and have since been frequently used by other work, for example [35,3741].

Mean squared measures examine the average correlation performance of the se-

quence family over all sequences at all possible shifts. A low mean squared cross-

correlation indicates there will be little interference between users utilising different

sequences. A low mean squared autocorrelation facilitates synchronisation in an un-

known delay environment and provides resistance to frequency selective fading [42].

A low mean squared autocorrelation also ensures that in a DS-CDMA system the

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energy of each user will be evenly spread across the entire frequency range being em-

ployed [6]. Whilst a low value of both measures is desired, it follows from the mean

squared Welch bound, discussed in Section 2.5.4, that they cannot be simultaneously

zero.

Mean squared cross-correlation

The mean squared aperiodic cross-correlation (MSCC) of the sequence family U={uk| k S} is [6]

Rcc(U) = 1S(S 1)

kS

rSr=k

N1l=1N

Cuk,r(l)2 . (2.22)Using the measure given in (2.21), the MSCC can be expressed as

Rcc(U) = 1S(S 1)

kS

rSr=k

uk,r(0). (2.23)

Noting from [33] that

N1l=1N

Cuk,r(l)2 = N1l=1N

Cuk (l) (Cur (l))

, (2.24)

the MSCC can be calculated more efficiently via

Rcc(U) = 1S(S 1)

kS

rSr=k

N1l=1N

Cuk (l) (Cur (l))

. (2.25)

Mean squared autocorrelation

Defined similarly to MSCC, the mean squared aperiodic out-of-phase autocorrelation

(MSAC) of a family U is given by [6]

Rac(U) = 1S

kS

N1

l=1Nl=0

|Cuk (l)|2 . (2.26)

The term l = 0 is not included in the calculation of the MSAC since Cuk (0) = 1

for all sequences.

2.3.5 Maximum value measures

The maximum value of each correlation measure is also often considered. Whilst

in a system with time varying delays mean squared measures give a more accurate

indication of overall performance [36], maximum values give a measure of worst case

performance and can be more accurate in a static delay environment [40].

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BACKGROUND

Maximum cross-correlation

The maximum aperiodic cross-correlation ofUis defined by

ccm(U) = maxk,rSk=r

0lN1

Cuk,r(l) . (2.27)

The maximum periodic cross-correlation, cm(U), is defined analogously.Negative l need not be considered in this maximisation since, by Property 2.1,

Cuk,r(l) = Cur,k(l) . (2.28)Maximum autocorrelation

In a similar manner, the maximum

on combination and interference free window spreading sequences

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