This document summarizes a PhD thesis on signal processing techniques for modern multi-constellation GNSS receivers. The thesis proposes a computationally efficient algorithm for mitigating multipath signals. It also discusses the problem of satellite selection and designs a method based on convex hull theory. Additionally, it performs a study on the approximation between maximum volume and minimum GDOP. The thesis was approved at the Instituto Superior Técnico of the University of Lisbon.

1. UNIVERSIDADE T ´ECNICA DE LISBOA
INSTITUTO SUPERIOR T ´ECNICO
Signal Processing Techniques in
Modern Multi-Constellation GNSS Receivers
Nuria Blanco Delgado
Supervisor: Doctor Fernando Duarte Nunes
Thesis approved in public session to obtain the PhD Degree in
Electrical and Computer Engineering
Jury final classification: Pass With Merit
Jury
Chairperson: Chairman of the IST Scientific Board
Members of the Committee:
Doctor Jos´e Manuel Nunes Leit˜ao
Doctor Gonzalo Seco-Granados
Doctor Jos´e Manuel Bioucas Dias
Doctor Paulo Jorge Coelho Ramalho Oliveira
Doctor S´ergio Reis Cunha
Doctor Fernando Duarte Nunes
June 2011

3. UNIVERSIDADE T ´ECNICA DE LISBOA
INSTITUTO SUPERIOR T ´ECNICO
Signal Processing Techniques in
Modern Multi-Constellation GNSS Receivers
Nuria Blanco Delgado
Supervisor: Doctor Fernando Duarte Nunes
Thesis approved in public session to obtain the PhD Degree in
Electrical and Computer Engineering
Jury final classification: Pass With Merit
Jury
Chairperson: Chairman of the IST Scientific Board
Members of the Committee:
Doctor Jos´e Manuel Nunes Leit˜ao, Professor Catedr´atico Aposentado do Instituto Superior T´ecnico,
Universidade T´ecnica de Lisboa
Doctor Gonzalo Seco-Granados, Professor Associado da Universidade Aut´onoma de Barcelona, Espanha
Doctor Jos´e Manuel Bioucas Dias, Professor Associado (com Agregac¸˜ao) do Instituto Superior T´ecnico,
Universidade T´ecnica de Lisboa
Doctor Paulo Jorge Coelho Ramalho Oliveira, Professor Associado do Instituto Superior T´ecnico,
Universidade T´ecnica de Lisboa
Doctor S´ergio Reis Cunha, Professor Auxiliar da Faculdade de Engenharia da Universidade do Porto
Doctor Fernando Duarte Nunes, Professor Auxiliar do Instituto Superior T´ecnico, Universidade T´ecnica
de Lisboa
FUNDING INSTITUITIONS
European Commission (Marie Curie Actions, Project SIGNAL)
June 2011

5. To my family,

6. Resumo
Os Sistemas de Navegac¸˜ao Global via Sat´elite (GNSS), como o GPS e o sistema europeu Galileo, est˜ao a encon-
trar novas e desafiadoras aplicac¸˜oes em ambientes onde o receptor ´e obrigado a operar com relac¸˜oes sinal-ru´ıdo
muito fracas e com m´ultiplos sinais refletidos (multipercurso). Novas t´ecnicas podem ser criadas incentivadas
pela evoluc¸˜ao da tecnologia, ou seja, pelo grande n´umero de correladores e pelas maiores larguras de banda de
processamento dispon´ıveis nos receptores.
Por outro lado, dada a tendˆencia dos receptores para diminuir de tamanho, a reduc¸˜ao do consumo de energia
no processamento ´e outro aspecto de grande preocupac¸˜ao hoje em dia. Com a disponibilizac¸˜ao de m´ultiplas
constelac¸˜oes de sat´elites, processos como a escolha do melhor subconjunto de sat´elites para o c´alculo de posi-
cionamento, ser˜ao tarefas ainda mais dif´ıceis. Esta tarefa j´a consome uma quantidade importante de tempo de
processamento do receptor. No entanto, na presenc¸a de mais sat´elites, tal pode aumentar consideravelmente o
consumo/tempo do receptor para o c´alculo de uma correc¸˜ao ´a soluc¸˜ao de navegac¸˜ao. Al´em disso, uma r´apida
selec¸˜ao do melhor subconjunto de sat´elites ajudaria a reduzir o tempo requerido pelo receptor para verificar a
consistˆencia das informac¸˜oes fornecidas pelo sistema quando s˜ao aplicados algoritmos de RAIM.
Estes aspectos s˜ao abordados na tese sendo proposto um algoritmo de mitigac¸˜ao do multipercurso computa-
cionalmente eficiente. O problema da selec¸˜ao de sat´elites ´e tamb´em discutido e ´e concebido um m´etodo de
selec¸˜ao baseado na teoria do Convex Hull. Por fim, ´e igualmente realizado um estudo sobre a aproximac¸˜ao entre
o volume m´aximo e o GDOP m´ınimo.
Palavras-chave: Sistemas de navegac¸˜ao global por sat´elite, mitigac¸˜ao de multipercurso, receptor multicorrelac¸˜ao,
inv´olucro convexo, volume m´aximo, selecc¸˜ao do sat´elites, diluic¸˜ao da precis˜ao geom´etrica, erros n˜ao gaussianos,
algoritmos de navegac¸˜ao, malha de seguimento vetorial
i

7. Abstract
The Global Navigation Satellite Systems (GNSSs), such as GPS and the European system Galileo, are finding new
and challenging applications in demanding environments where the receiver is required to operate with very weak
signal-to-noise ratios and multiple reflected signals (multipath). New techniques can be devised fostered by the
technological evolution, i.e., the large number of correlators and the wider processing bandwidths implemented
in the receivers.
On the other hand, given the tendency of the receivers to decrease in size, processing power reduction is another
aspect of high concern nowadays. With the availability of multiple constellations of satellites, processes such as
the selection of the best set of satellites for positioning computation, will become an even more difficult task. This
task already takes an important amount of processing at the receiver. However, in the presence of more satellites,
this can increase notably the power/time consumption of the receiver for the computation of a fix. Furthermore,
a fast selection of the best satellite subset would help to reduce the time it takes to the receiver to check the
consistency of the information provided by the system when RAIM algorithms are applied.
These aspects are dealt with in the thesis where a computationally efficient multipath mitigation method is pro-
posed. The satellite selection problem is also tackled and a satellite selection method based on the Convex Hull
theory is designed. Finally, a study of the approximation between maximum volume and minimum GDOP is also
performed.
Key-words: Global positioning systems, multipath mitigation, multicorrelator receiver, convex hull, maximum
volume, satellite selection, geometric dilution of precision, non-gaussian errors, navigation algorithm, vector-
tracking loops.
”If we knew what we were doing, it wouldn’t be called research, would it?”
Albert Einstein
ii

8. Acknowledgements
Writing a thesis can be like a rollercoaster, with ups and downs filling the way. The support that I have received
during the difficult times has been a key factor for the successful completion of my doctoral thesis. Now that the
journey has finished I would like to express my gratitude to all of them.
First of all, I would like to thank my supervisor, Prof. Fernando Duarte Nunes for his guidance and support during
all this period. Then, I also thank my colleagues at the Department of Engenharia Electrot´ecnica e de Computa-
dores at Instituto Superior T´ecnico of Lisbon. The combined expertise of its members has been of great help
during my stay in Portugal. After that, I would like to thank Prof. Jo˜ao M. Xavier for our technical discussions
which have enlightened my work during my research and the members of my Comiss˜ao de Acompanhamento de
Tese for their useful comments on this work. Finally, I thank the reviewers of this thesis, Prof. Gonzalo Seco
Granados, Prof. Paulo Jorge Oliveira, and specially Prof. Elena Simona Lohan, for their careful reviews and
suggestions.
This work was carried out at the Instituto de Telecomunicac¸˜oes, Instituto Superior T´ecnico de Lisboa. I thank
them for providing the office space and facilities I needed to carry out this work. I gratefully acknowledge the
funding for my research from the European Union Marie Curie Actions, through the SIGNAL project (contract
MEST-CT-2005-021175).
Over the last years I have had the fortune to collaborate with some of the most knowledgeable persons in GNSS
signal processing. This has enriched not only the research that I am presenting in this document, but also my
professional career in general. In particular I would like to thank Dr. Gonzalo Seco Granados for all the time that
he has devoted to my research. His contribution has been crucial during my stay at the Universidad Aut´onoma de
Barcelona.
My colleagues in Lisboa and Barcelona have been of extreme help over these years. Working in a new coun-
try represents many challenges. However, my adaptation to the cities of Barcelona and Lisbon has been very
easy thanks to them. It was a pleasure working with you. I thank also the members of Centre de Tecnologia
Aeroespacial (CTAE) in Barcelona, for facilitating me finishing this work and the encouragement received by my
colleagues.
Finally I would like to thank my family and friends their immense patience and unconditional support. I simply
cannot conceive the completion of this work without them.
iii

9. iv

10. Contents
Notation xvii
Acronyms xxi
1 Introduction 1
1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Research Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Journal Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 International Conferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 National Conferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4 Patents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.5 Technical Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Principles of GNSS 9
2.1 Global Navigation Satellite Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 GPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Galileo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.3 Other GNSSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Traditional Navigation Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 Tracking Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.2 Satellite Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.3 Carrier-to-Noise Density Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.4 Position Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.5 Discrete-Time Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Vector Code/Phase Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.1 Vector code tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.2 Vector phase tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.4 GNSS Error Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4.1 Broadcast Ephemeris Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4.2 Satellite Clock Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4.3 Receiver Thermal and Tracking Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4.4 Satellite Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4.5 Atmospheric Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4.6 Multipath Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4.7 GNSS Enhancements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
v

11. 2.5 Satellite Selection Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.5.1 Satellite Health . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.5.2 Geometric Dilution of Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.5.3 User Range Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.5.4 Carrier-to-Noise Ratio, C{N0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.5.5 Satellite Elevation Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.5.6 External Aids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3 State of the Art Techniques 65
3.1 SoA Techniques in Multipath Mitigation for Single-Antenna GNSS Receivers . . . . . . . . . . 65
3.1.1 Effect in the Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.1.2 DLL-based Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.1.3 Maximum Likelihood Estimation Techniques . . . . . . . . . . . . . . . . . . . . . . . 75
3.1.4 CRLB of Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2 SoA Techniques in Satellite Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2.1 Brute Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2.2 Highest Elevation Satellite Selection Algorithm . . . . . . . . . . . . . . . . . . . . . . 85
3.2.3 Maximum Volume Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2.4 Four-Step Satellite Selection Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.2.5 Recursive Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.2.6 Quasi-Optimal Satellite Selection Algorithm . . . . . . . . . . . . . . . . . . . . . . . 87
3.2.7 Fast Satellite Selection Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.2.8 Multi-Constellations Satellite Selection Algorithm . . . . . . . . . . . . . . . . . . . . 88
3.3 Simplified VDLL based on Cell Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.3.1 Proposed tracking algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4 Multipath Mitigation Method 101
4.1 Multicorrelator GNSS Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1.1 Noise Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2 Estimation Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.3 Interpolation for Increased Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3.1 Piecewise Linear Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3.2 Polynomial Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4 Theoretical Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.5 Cram´er-Rao Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.6 Computational Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.8 Comparison with existing Multipath Mitigation Methods . . . . . . . . . . . . . . . . . . . . . 123
4.8.1 MEDLL Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.8.2 Narrow Correlator Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.8.3 HRC Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.9 Conclusions and Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5 Satellite Selection Method 131
5.1 Convex Hull Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.3 Convex Hull Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
vi

12. 5.3.1 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.4.1 2-Dimensional Convex Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.4.2 3-Dimensional Convex Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.5 Unequal UREs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.5.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.6 2-D Complementary Reduction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.7 Conclusions and Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6 Minimum DOP vs Maximum Volume 153
6.1 Determinant Maximization versus GDOP Minimization . . . . . . . . . . . . . . . . . . . . . . 153
6.2 Volume vs Determinant for a h m matrix with h ¥ m . . . . . . . . . . . . . . . . . . . . . . 160
6.3 Conclusions and Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7 Conclusions 167
7.1 Suggestions for Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
A Autocorrelation Functions 171
B Power Spectral Densities 173
C Derivation of the Cram´er-Rao Lower Bound 177
D Positioning accuracy 183
E Noise Samples Generation 187
F Impact of elevation angle on GDOP 189
G Simulation of Satellite Orbits 193
H Majorization Theory 195
I Volume and Determinants 197
J Eigenvalues and Eigenvectors 199
K Minimum GDOP in 2-D 201
Bibliography 212
vii

13. viii

14. List of Figures
2.1 GNSS system segments. Source: The Aerospace Corporation. . . . . . . . . . . . . . . . . . . 10
2.2 Autocorrelation function of a BPSK(1) modulated signal. . . . . . . . . . . . . . . . . . . . . . 14
2.3 Normalized power spectral density of the BPSK(1) modulation. . . . . . . . . . . . . . . . . . 14
2.4 Block diagram of the L2C signal generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Block diagram of the L5 signal generation process. . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Galileo frequency plan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.7 Autocorrelation function of a sBOC(1,1) modulated signal. . . . . . . . . . . . . . . . . . . . . 20
2.8 Normalized power spectral densities of sBOC(1,1) and cBOC(1,1) modulations. . . . . . . . . . 21
2.9 Autocorrelation function of a MBOC(6,1,1/11) modulated signal. . . . . . . . . . . . . . . . . . 21
2.10 Normalized power spectral densities of MBOC(6,1,1/11) and sBOC(1,1) modulations. . . . . . 22
2.11 Shapes of the data and pilot AltBOC(15,10) subcarriers. . . . . . . . . . . . . . . . . . . . . . 23
2.12 Normalized power spectral density of AltBOC(15,10) modulation. . . . . . . . . . . . . . . . . 23
2.13 Generic GNSS receiver block diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.14 Generic digital GNSS receiver channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.15 Code and carrier tracking loops. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.16 S-curve for different DLL discriminators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.17 Integrated random walk model for a PV dynamics process. . . . . . . . . . . . . . . . . . . . . 37
2.18 Dynamics model for receiver’s clock. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.19 Scalar and vector-tracking architectures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.20 Simplified structure of the VDLL receiver to simultaneously track N satellites with ∆Tc denoting
the code discriminator’s early late spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.21 Noise variance of the early-late power discriminator versus the signal-to-noise ratio for GPS C/A
signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.22 x-direction estimation errors using h 4 satellites in view. . . . . . . . . . . . . . . . . . . . . 45
2.23 Clock-offset estimation errors using h 4 satellites in view. . . . . . . . . . . . . . . . . . . . 45
2.24 Simplified block diagram of the receiver with co-op tracking to simultaneously track N satellites. 47
2.25 Phase errors of the PLLs without (conventional receiver) and with PLL coupling (co-op receiver)
for N 6 tracked satellites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.26 Effect of different geometries between two satellites and a receiver. . . . . . . . . . . . . . . . . 51
2.27 Effect that the number of satellites used has in the geometric error. . . . . . . . . . . . . . . . . 52
2.28 Example of a 1σ ionospheric delay error versus the elevation angle. . . . . . . . . . . . . . . . 53
2.29 Example of a 1σ tropospheric delay error versus the elevation angle. . . . . . . . . . . . . . . . 54
2.30 Multipath effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.31 Impact of in-phase and out-of-phase multipath on a sBOC(1,1) correlation signal. . . . . . . . . 55
2.32 S-curve affected with multipath. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.33 1σ multipath error as a function of the elevation angle. . . . . . . . . . . . . . . . . . . . . . . 56
ix

15. 2.34 GNSS/Inertial navigation systems architectures. . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.35 1σth error as a function of the C{N0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.36 1σ pseudorange error as a function of the elevation angle for different C{N0 T values. . . . . . 63
3.1 Multipath error envelopes for the C/A code signal as a function of the pre-correlation bandwidth
for a noncoherent early-minus-late discriminator with ∆ 1 chip and SMR = 6 dB. . . . . . . . 66
3.2 Multipath error envelopes for the C/A code signal as a function of the pre-correlation bandwidth
for a noncoherent early-minus-late power discriminator with ∆ 0.1 chip and SMR = 6 dB. . . 67
3.3 Multipath error envelope for a narrow correlator discriminator with ∆ 0.1 chips and B
12 MHz and different modulated signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4 S-curve of a HRC discriminator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5 Multipath error envelopes for a Double Delta correlator discriminator with ∆ 0.1 chips and
B 12 MHz and different modulated signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6 C/A code signal and corresponding gated C/A signal. . . . . . . . . . . . . . . . . . . . . . . . 71
3.7 Multipath Error Envelopes of a narrow correlator with EL spacing of 0.1 chips, SMR = 6 dB and
MEDLL algorithm considering 61 correlators. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.8 Multipath error envelopes of MEDLL algorithm as a function of the number of correlators and
different modulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.9 Multipath error envelopes of MEDLL algorithm as a function of the number of correlators con-
sidering a lowpass precorrelation filter with bandwidth B 6 MHz and different modulations. . 79
3.10 Multipath error envelopes of MEDLL algorithm as a function of the shape of the autocorrelation
signal considering a lowpass precorrelation filter with bandwidth B 6 MHz and different
number of correlators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.11 RMSEs for different post-integration carrier-to-noise ratios, M 30, and a lowpass precorrela-
tion filter with bandwidth B 6 MHz. (a) pC{N0qT = 46 dB. (b) pC{N0qT = 30 dB. . . . . . 80
3.12 Mean delay error of MEDLL algorithm for different post-integration carrier-to-noise ratios and a
lowpass precorrelation filter with bandwidth B 6 MHz. . . . . . . . . . . . . . . . . . . . . . 81
3.13 Pre-processing part of the proposed architecture. . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.14 Steps that constitute an iteration of the tracking algorithm. . . . . . . . . . . . . . . . . . . . . 93
3.15 Cell selection operation assuming that observations from N satellites are available. . . . . . . . 94
3.16 Cell selection criteria using the MW and the MV schemes. . . . . . . . . . . . . . . . . . . . . 95
3.17 Transitions between cells (incomplete) in the prediction step for a 2-D spatial trajectory. . . . . . 96
3.18 Illustration of a typical 2-D indoor trajectory. 3000 points are considered with ∆T 1 s, i.e., a
50-minutes simulation is performed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.19 RMSE in the estimation of the receiver position as a function of the cell size for a typical 2-
D indoor trajectory with BPSK(1) modulation for different cell selection criteria and pC{N0qT
values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.20 RMSE in the estimation of the receiver position as a function of the cell size for a typical 2-D
indoor trajectory with sBOC(1,1) modulation for different cell selection criteria and pC{N0qT
values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.21 Estimates of the x-local coordinate for BPSK(1) and sBOC(1,1) modulations using MV and a
pC{N0qT of 10 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.22 Estimates of the receiver’s clock bias error xφ in meters using MV and a pC{N0qT of 10 dB for
BPSK(1) and sBOC(1,1) modulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.23 RMSE in the estimation of the receiver position as a function of the post-integration SNR using
MV for BPSK(1) and sBOC(1,1) modulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
x

16. 3.24 Estimation of a linear trajectory using the proposed tracking algorithm with and without motion
constraints using MW with a pC{N0qT of 12 dB and BPSK(1) modulation. . . . . . . . . . . . 100
3.25 Estimates of the x local coordinate using MV with a pC{N0qT of 15 dB and sBOC(1,1) modu-
lation. In the interval between the arrows the number of satellites was reduced to only two. . . . 100
4.1 Power spectral density of the bandpass noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2 Power spectral density of the baseband noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.3 Illustration of the relationship between the delays of the P received rays τp and the delays of the
bank of correlators δk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.4 Structure of a bank of correlators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.5 Grid employed when P 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.6 Illustration of piecewise linear interpolation and polynomial interpolation. . . . . . . . . . . . . 111
4.7 Condition Number vs. Multipath Delay for several modulation types with ∆ 0.1 chips pM
10q, ∆ 0.033 chips, pM 30q, and pC{N0qT 40 dB. . . . . . . . . . . . . . . . . . . . . 112
4.8 Autocorrelation functions of BPSK(1), sBOC(1,1) and MBOC(6,1,1/11) modulated signals. . . 112
4.9 Effect of the delay separation between the line-of-sight and the multipath signal on the minimum
variance of pθp, with p 0, 1, pC{N0qT = 46 dB, and normalized amplitude. . . . . . . . . . . 115
4.10 Distribution of delay estimates for different SNRs, ∆ 0.05 Tc (M 20) and φ1 φ0 random. 118
4.11 Distribution of delay estimates for different SNRs, ∆ 0.025 Tc (M 40) and φ1 φ0 random. 118
4.12 Distribution of delay estimates for different SNRs, ∆ 0.05 Tc (M 20) and φ1 φ0. . . . . 119
4.13 Distribution of delay estimates for different SNRs, ∆ 0.05 Tc (M 20) and φ1 φ0 π. . 119
4.14 Distribution of delay estimates for different SNRs, ∆ 0.025 Tc (M 40) and φ1 φ0. . . . 120
4.15 Distribution of delay estimates for different SNRs, ∆ 0.025 Tc (M 40) and φ1 φ0 π. . 120
4.16 RMSE in the estimation of the line-of-sight code delay as a function of the interpolation method
for MBOC(6,1,1/11) with pC{N0qT 46 dB and M 20 (∆ = 15 m). . . . . . . . . . . . . . 121
4.17 RMSE in the estimation of the line-of-sight code delay as a function of the number of correlators
for MBOC(6,1,1/11) with pC{N0qT 46 dB and linear interpolation. . . . . . . . . . . . . . . 122
4.18 RMSE in the estimation of the real part of pθp, for p 0, 1 with pC{N0qT 46 dB and
MBOC(6,1,1/11) modulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.19 RMSE in the estimation of the line-of-sight code delay as a function of the pC{N0qT with inter-
polation for MBOC(6,1,1/11) and M 20 (∆ 15 m). . . . . . . . . . . . . . . . . . . . . . 123
4.20 RMSE in the estimation of the real part of pθp, with p 0, 1, M 30, p∆ 0.03Tcq and
MBOC(6,1,1/11) modulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.21 RMSE in the estimation of the line-of-sight code delay as a function of the modulation type with
pC{N0qT 20 dB, M 20, and linear interpolation. . . . . . . . . . . . . . . . . . . . . . . 124
4.22 RMSE in the estimation of the line-of-sight code delay for MBOC(6,1,1/11). . . . . . . . . . . 125
4.23 Estimation errors for MBOC(6,1,1/11) and M 30. . . . . . . . . . . . . . . . . . . . . . . . 125
4.24 Estimation errors for different correlators spacing, pC{N0qT 30 dB and MBOC(6,1,1/11). . . 126
4.25 Multipath error envelope for a narrow correlator discriminator with δ 1{10 chip, a pC{N0qT
of 30 dB, M 20, and B 12 MHz and different modulated signals. . . . . . . . . . . . . . . 127
4.26 Multipath error envelope for a Double Delta correlator discriminator with δ 1{20 chip, a
pC{N0qT of 30 dB, M 20, and B 12 MHz and different modulated signals. . . . . . . . . 128
5.1 A point set and its convex hull in R2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.2 Graham’s Scan algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.3 Chan’s algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
xi

17. 5.4 Mean computational complexity for different satellite selection algorithms as a function of the
number of satellites in view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.5 Mean number of satellites in the CH solution as a function of the number of satellites in view. . 137
5.6 GNSS satellite visibility at 38 441N latitude, 9 91W longitude and zero altitude, for a 5 eleva-
tion mask angle averaged over a 24-h time span. . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.7 Minimum and maximum GDOP values for 2-D positioning when computing the 2D-CH. . . . . 139
5.8 GDOP values in 2-D positioning for different number of satellites considering several combina-
tion of satellite constellations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.9 Reduction in the number of satellites by computation of the 2D-CH for several combinations of
satellite constellations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.10 Minimum and maximum GDOP values for 3-D positioning from 2D-CH . . . . . . . . . . . . 141
5.11 Minimum and maximum GDOP values for 3-D positioning from 3D-CH. . . . . . . . . . . . . 142
5.12 Reduction in the number of satellites by computation of the 3D-CH for several combinations of
satellite constellations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.13 GDOP values in 3-D positioning from 3D-CH for different number of satellites considering sev-
eral combination of satellite constellations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.14 Maximum discrepancy in the GDOP value obtained from selecting the best set of h satellites using
the maximum volume determined by the user-satellite unit vectors and the maximum volume
expanded by the user-satellite non-unit vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.15 Probability of selecting a satellite from each constellation as a function of the number of satellites
in the 2D-CH solution, h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.16 Distribution of the GDOP difference between the CH and the all-in-view solutions. . . . . . . . 145
5.17 Mean number of satellites in the CH solution as a function of the total number of satellites in view.147
5.18 Minimum and maximum WGDOP values for 2-D positioning when computing the 2D-CH. . . . 147
5.19 Minimum and maximum WGDOP values for 3-D positioning from 3D-CH. . . . . . . . . . . . 148
5.20 Minimum and maximum WGDOP values for 3-D positioning from 3D-CHp1q. . . . . . . . . . . 148
5.21 Distribution of the WGDOP difference between the CH and the all-in-view solutions. . . . . . . 149
5.22 Reduction method based on distance computation. . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.23 Minimum and maximum GDOP values for 2-D positioning obtained with the set of 4 satellites
selected from the 2D-CH using the minimum-distance criterion. . . . . . . . . . . . . . . . . . 150
5.24 Computational complexity in the computation of the 2D-CH when decreasing the set of satellites
in the 2D-CH to a total of three using the minimum-distance criterion as a function of the number
of satellites in view, N. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.1 (a) Cumulative sum of two points belonging to the same solution set. (b) Zoom over the region
where the two lines cross each other. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.2 (a) Level curves of the inverse of trpL 1
q. (b) Level curves of detpLq. Curves computed in R2
,
i.e., h = 3 for a generic matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.3 Contour plots for detpLq and the inverse of the trace of the GDOP matrix in R2
. Only the level
curves corresponding to the two largest values of both functions are plotted. . . . . . . . . . . . 159
6.4 Contour plots for the determinant and the inverse of the trace of the GDOP matrix in R2
with
h 3. Each blue dotted region corresponds to the points for which the determinant and the trace
would provide a different result if any of those points is feasible at the same time as the circled
one for each region. The grey shadow region corresponds to the feasible values of the eigenvalues
considering the restrictions in Table 6.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
xii

18. 6.5 Discrepancy plots of determinant versus the inverse of GDOP for all possible solutions in R2
. (a)
h 3. (b) h 4. (c) h 5. (d) h 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.6 Area of a parallelogram as a cross-product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.7 Discrepancy plots of determinant versus the area for all possible solutions in R2
. (a) h 3. (b)
h 4. (c) h 5. (d) h 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.8 Areas generated at several points of Fig. 6.7(c), i.e., with h 5 in R2
. . . . . . . . . . . . . . . 164
6.9 Areas generated at several points of Fig. 6.7(d), i.e., with h 6 in R2
. . . . . . . . . . . . . . . 165
6.10 Areas generated at several points of Fig. 6.7(c), i.e., with h 5 in R2
. . . . . . . . . . . . . . . 166
F.1 DOP parameters as a function of the elevation angle above the horizon. . . . . . . . . . . . . . 192
xiii

19. xiv

20. List of Tables
2.1 GPS SPS performance parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Performance specification of the OS, CS, and SoL Galileo services. . . . . . . . . . . . . . . . 18
2.3 Main characteristics of the Galileo signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Galileo message allocation and general data content. . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 DLL discriminator types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Typical Allan variance parameters for various timing standards. . . . . . . . . . . . . . . . . . . 39
2.7 1σ Error Budget for stand-alone and differential GPS. . . . . . . . . . . . . . . . . . . . . . . . 49
2.8 Satellite elevation dependent errors for a receiver located at 38 441N latitude for different eleva-
tion angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.1 Code delay RMSE considering a lowpass precorrelation filter with bandwidth B 6 MHz. . . . 81
3.2 Code delay RMSE considering a lowpass precorrelation filter with bandwidth B 12 MHz and
M 30. (Experimental / Theoretical values) . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3 GDOP of optimal geometries. The middle column is the number of satellites that lie at the zenith.
Other satellites are selected as explained in steps 3 and 4. . . . . . . . . . . . . . . . . . . . . . 88
4.1 RMSEs in the estimation of the line-of-sight code delay for several correlators spacings, SNRs
and modulation types, in meters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.1 Parameters of the different constellations considered [1]. . . . . . . . . . . . . . . . . . . . . . 137
6.1 Eigenvalues boundaries in Rn
when h satellites are selected. . . . . . . . . . . . . . . . . . . . 158
6.2 Limits on the values that every eigenvalue can take in R2
assuming that h satellites are selected. 158
G.1 RAAN and Argument of Latitude parameters for the optimized 24 GPS constellation. . . . . . . 194
xv

21. xvi

22. Notation
Boldface upper-case letters denote matrices and boldface lower-case letters denote column vectors. Scalar vari-
ables are in italics and estimated or predicted values have a “hat” on top.
Functions and derivatives
x © y x majorizes y or, equivalently, y is majorized by x.
arg max
x
fpxq Value of x that maximizes fpxq.
arg min
x
fpxq Value of x that minimizes fpxq.
Bfpxq
Bx Gradient of function fpxq with respect to vector x.
Bfpxq
Bxi
Partial derivative of function fpxq with respect to the variable xi.
rxsr The rth
vector element.
ln p q Natural logarithm (base e).
t u, t u Real and imaginary parts, respectively.
Xpfq Fourier transform of xptq, Xpfq Ftxptqu.
Rrxx Correlation function of the signals rxptq and xptq,
Rrxxpτq 1
T
» T
0
rxptqx pt τqBt.
Rxx Normalized autocorrelation function of the signal xptq as,
Rxxpτq 1
T
» T
0
xptqx pt τqdt.
Constants
Tc Chip time duration.
c Speed of light in vacuum.
Probability
N pµ, Σq Multivariate Gaussian distribution with mean µ and covariance matrix Σ.
xvii

23. U pa, bq Uniform distribution in the interval ra, bs.
i.i.d. independent identically distributed
r.v. random variable
Variables
rn Satellite n position at transmit time, pxn, yn, znq.
ru Receiver position at receive time, pxu, yu, zuq.
∆T Sampling interval.
∆t Receiver clock bias in seconds.
∆t Receiver clock bias with respect to the satellite clock.
δt Satellite-receiver traveling time of a radiofrequency signal.
∆x Residual error in the estimation of parameter x.
∆ Spacing between correlators.
δk kth
correlator delay in chips with δ0 0 by convention.
ρ Composite of errors produced by ionospheric/tropospheric uncompensated delays, satellite
ephemerides mismodeling, receiver noise, etc.
γ Satellite elevation angle.
ρn Pseudorange relative to satellite n
σ2
τ Tracking error variance.
px Estimation and true value of parameter x.
B Pre-correlation filter bandwidth.
BL Code tracking loop bandwidth.
bu Receiver clock bias in meters.
Bθ Carrier tracking loop bandwidth.
f Frequency value.
f0 Nominal carrier frequency.
fc Chip rate of the BPSK-R waveform, fc 1{Tc.
fs Square-wave subcarrier frequency, fs 1{Ts.
h Number of selected satellites.
N Number of satellites in view.
P Number of rays reaching the receiver antenna.
xviii

24. rptq RF radionavigation signal.
sptq Direct Sequence Spread Spectrum signal without data modulation.
T Integration time.
xptq Direct Sequence Spread Spectrum signal with data modulation.
xf Receiver clock drift error.
xφ Receiver clock bias error.
Some Specific Sets
R, C The set of real and complex numbers, respectively.
Rn
The set of real n-vectors (n 1) matrices.
Rn m
, Cn m
The set of n m matrices with real- and complex-valued entries, respectively.
R The set of nonnegative real numbers.
Topology and convex analysis
CH pSq Convex hull of set S.
Vectors and Matrices
G Direction cosine matrix of line-of-sight vectors to the satellites.
I Identity matrix. A subscript can be used to indicate the dimension.
R Measurement noise covariance matrix.
X Complex conjugate of matrix X (also applied to scalars).
XH
Complex conjugate and transpose (Hermitian) of matrix X.
XT
Transpose of matrix X.
det pXq Determinant of matrix X.
rXsr,c The X matrix element located in row r and column c.
diagpxq A diagonal matrix whose diagonal entries are given by x.
trpXq Trace of matrix X, tr pXq
N¸
n1
rXsnn.
xix

25. xx

26. Acronyms
C{N0 Carrier-to-Noise Density Ratio.
2-D 2-Dimensional.
3-D 3-Dimensional.
A-GNSS Assisted GNSS.
AltBOC Alternative BOC.
APME A-Posteriori Multipath Estimation.
BLUE Best Linear Unbiased Estimator.
BOC Binary Offset Carrier.
bps bits per second.
BPSK Binary Phase Shift Keying.
C/A Coarse/Acquisition.
CBOC Composite BOC.
cBOC cosine-phase modulated BOC.
CDMA Code Division Multiple Access.
CRLB Cram´er-Rao Lower Bound.
CS Commercial Service.
CWGN Complex White Gaussian Noise.
DGNSS Differential GNSS.
Diff2 second-order Differentiation.
DLL Delay Locked Loop.
DOP Dilution Of Precision.
DSSS Direct Sequence Spread Spectrum.
ECEF Earth Centered Earth Fixed.
ECI Earth-Centered Inertial.
EGNOS European Geostationary Navigation Overlay Sevice.
EKF Extended Kalman Filter.
ELS Early Late Slope.
ENU East-North-Up.
FAA Federal Aviation Administration.
xxi

27. FDE Fault Detection and Exclusion.
FDMA Frequency Division Multiple Access.
FEC Forward Error Correction.
FIMLA Fast Iterative Maximum-Likelihood Algorithm.
FLL Frequency Locked Loop.
GAGAN GPS Aided Geo Augmented Navigation.
GDOP Geometrical Dilution Of Precision.
GLONASS GLObal´naya NAvigatsionnaya Sputnikovaya Sistema.
GNSS Global Navigation Satellite System.
GPS Global Positioning System.
HDOP Horizontal Dilution Of Precision.
HRC High Resolution Correlator.
IF Intermediate Frequency.
LADGPS Local-Area Differential GPS.
LNA Low Noise Amplifier.
LO Local Oscillator.
MBOC Multiplexed Binary Offset Carrier.
MCRW Modified Correlator Reference Waveforms.
MCS Master Control Station.
MCSSA multi-constellations satellite selection algorithm.
MEDLL Multipath Estimating Delay Lock Loop.
MEO Medium Earth Orbit.
MET Multipath Elimination Technology.
MF Matched Filter.
MGD Multiple Gate Delay.
ML Maximum Likelihood.
MLE Maximum Likelihood Estimation.
MMT Multipath Mitigation Technology.
MSAS Multifunctional Transport Satellite.
MTLL Mean Time to Lose Lock.
MV Majority Voting.
MVU Minimum Variance Unbiased.
MW Maximum Weighting.
N/A Not Applicable.
NCO Numerically Controlled Oscillator.
xxii

28. NH Newmann-Hoffman.
NRZ Non-Return to Zero.
NSC Non-Standard C/A.
NSY Non-Standard Y.
OS Open Service.
PAC Pulse Aperture Correlator.
pdf probability density function.
PDOP Position Dilution Of Precision.
PLL Phase Lock Loop.
POCS Projections Onto Convex Sets.
PPS Precise Positioning Service.
PRN Pseudo-Random Noise.
PRS Public Regulated Service.
PT Peak Tracking.
PV Position and Velocity.
PVA Position, Velocity and Acceleration.
PVT Position, Velocity and Time.
RAAN Right Ascension of the Ascending Node.
RAIM Receiver Autonomous Integrity Monitoring.
RC Replacement Code.
RF Radio Frequency.
RHCP Right Handed Circularly Polarized.
rms root mean square.
RMSE Root Mean Square Error.
SA Selective Availability.
SBAS Satellite-Based Augmentation System.
sBOC sine-phase modulated BOC.
SIS Signal In Space.
SMR Signal-to-Multipath Ratio.
SNR Signal-to-Noise Ratio.
SoL Safety of Life Service.
SPS Standard Positioning Service.
TDOP Time Dilution Of Precision.
TK Teager Kaiser.
TMBOC Time-Multiplexed BOC.
TTFF Time-To-First-Fix.
xxiii

29. UEE User-Equivalent Error.
UERE User-Equivalent Range Error.
URE User Range Error.
UTC Coordinated Universal Time.
VC Vision Correlator.
VDLL Vector Delay Lock Loop.
VDOP Vertical Dilution Of Precision.
VFLL Vector Frequency Lock Loop.
VTEC Vertical TEC.
WAAS Wide Area Augmentation System.
WCDMA Wireless Code Division Multiple Access.
WDOP Weighted Dilution Of Precision.
WGDOP Weighted Geometrical Dilution Of Precision.
WHDOP Weighted Horizontal Dilution Of Precision.
WPDOP Weighted Position Dilution Of Precision.
WTDOP Weighted Time Dilution Of Precision.
WVDOP Weighted Vertical Dilution Of Precision.
xxiv

30. Chapter 1
Introduction
Radionavigation systems exploit the basic principles of propagation of radio waves. Radio waves correspond
to a range of electromagnetic wave frequencies from 10 kHz to 300 GHz. The radio signals travel in free
space in straight lines at the speed of light. Free space is an idealization with no electric or magnetic fields, nor
any obstructions. Space meets these requirements, but the earth’s atmosphere does not. Signal propagation in the
earth’s atmosphere can be very complicated depending upon the signal frequency and the environment. Radio
signals propagating in the earth’s environment suffer from reflection, refraction and diffraction by the ground,
buildings, and surface of water. They also interfere with each other and are attenuated by the earth’s atmosphere.
If the transmitting station position is known and the transmit time can be measured, the distance between the
transmitter and receiver can be determined. In a Global Navigation Satellite System (GNSS), like Global Po-
sitioning System (GPS) or GLObal´naya NAvigatsionnaya Sputnikovaya Sistema (GLONASS), the transmitting
stations are the satellites. To determine its position, a navigation receiver computes its distance to several satel-
lites. These measured distances are called pseudoranges as receiver and satellites clocks are not synchronized.
The bias in the receiver clock affects all measured pseudoranges equally at a given time instant. Thus, the clock
synchronization problem can be overcome by introducing a further unknown, the receiver clock bias. Therefore,
a minimum of three satellites are required to compute a 2-Dimensional (2-D) horizontal receiver position. An ad-
ditional satellite would be required to provide the 3-Dimensional (3-D) position. When operating with Receiver
Autonomous Integrity Monitoring (RAIM) redundant measurements are required [2, ch. 5] (at least 5 satellites
for 3-D positioning).
The computed receiver position is affected by range measurement errors and user-to-satellite geometry errors.
Altogether they sum up to an error of around 10 m. Most of the errors can be eliminated or mitigated by techniques
external or internal to the receiver. Only receiver thermal and tracking noise errors and multipath errors remain.
The effect of those errors on the estimated receiver’s position depends on the user-to-satellite geometry. Thus,
finding the best user-to-satellite geometry is important to minimize the positioning errors. Minimization of the
multipath effect and determination of the best user-to-satellite geometry are the subject of the study and different
techniques are developed to optimize them.
The objective of this work is to develop new architectures at receiver level able to alleviate problems encountered
when using GNSS signals for positioning purposes. The current chapter aims to provide the motivation for the
study and the research contributions. Section 1.2 contains a list of publications resulting from the work developed.
Finally, an outline of the document is presented in Section 1.3.
1

31. Chapter 1. Introduction
1.1 Background and Motivation
In GNSSs, the position of a receiver is computed by measuring its pseudorange to four or more satellites. The
common way to do it is by maximizing the correlation between the incoming signal from each satellite and a
locally-generated reference code. When code synchronization is attained, the receiver’s position is determined
by solving a system of nonlinear equations. However, if the signal reaches the receiver through several paths,
due to obstacles that refract and reflect the rays, the contribution of these rays distorts the cross-correlation
function resulting in a code synchronization error. Once errors caused by the ionosphere and troposphere are
conveniently eliminated, multipath constitutes the major source of errors in GNSS positioning as its influence
cannot be estimated a priori and has to be accounted for by the receiver.
In urban canyon environments, this effect is significant due to the great amount of obstacles such as buildings that
make closely-delayed replicas of the signal interfere at the receiver. Moreover, the orientation of mass-market
applications to urban scenarios makes it necessary to develop multipath mitigation techniques that prevent the
harmful effects of these undesired replicas, while keeping receiver hardware complexity, power consumption,
and cost, to a minimum.
Most of the existing techniques were born from a tradeoff between receiver complexity and mitigation per-
formance. However, it should be noted that the burden of the receiver complexity decreases over time as the
technology evolves; for example, nowadays it is technically possible to integrate a large number of correlators in
the receiver and therefore other solutions that were left out due to receiver complexity can now be considered as
feasible.
Several techniques have been developed during the last decades to cope with the multipath problem. Some of
them are based on the typical Delay Locked Loop (DLL), like the Narrow Correlator [3], the Double-Delta Cor-
relator [4], the Modified Correlator Reference Waveforms [5], the Multiple Gate Delay (MGD) structures [6],
[7], [8], the A-Posteriori Multipath Estimation (APME) technique [9], and the Multipath Elimination Technology
(MET) [10]. These techniques require relatively simple hardware/software structures and are effective only in
mitigating errors caused by mid-distance multipath (corresponding to secondary path induced extra delays larger
than approximately 30 m for GPS Coarse/Acquisition (C/A) signals). More complex structures, based on a bank
of correlators (multicorrelator receivers), have been proposed to improve the performance of the traditional mul-
tipath mitigation techniques [11]. Some methods are quite effective, especially for close-in multipath (secondary
path induced extra delays smaller than typically 30 m). They attempt to estimate the parameters (delay, ampli-
tude and phase) of the direct ray and one or more reflected rays by applying Maximum Likelihood Estimation
(MLE) theory [12]. Among them we refer to the Multipath Estimating Delay Lock Loop (MEDLL) [13], the
Saarnisaari’s Maximum Likelihood (ML) method [14], and other similar techniques with reduced complexity
(data compression) like the Multipath Mitigation Technology (MMT) [15], the Selva’s method [16], [17], the
Fast Iterative Maximum-Likelihood Algorithm (FIMLA) [18], and other alternative iterative solutions whose
convergence cannot be assured [19], [20], Projections Onto Convex Sets (POCS) [21], [22], the Peak Tracking
(PT) algorithm [23]. Finally, we mention a different technique, the discrete-time Teager Kaiser (TK) operator
[24], [25].
A different approach to the problem of multipath mitigation consists in the sequential estimation of a hidden
Markov process: the unknown channel parameters are estimated based on an evolving sequence of received noisy
channel outputs [26]. The estimation problem is cast in the framework of the Sequential Nonlinear Bayesian
estimation. Since the propagation of the a posteriori probability density function is computationally intractable,
a solution based on particle filters was proposed e.g., in [26], [27]. However, the underlying computational
complexity is still very high, making these algorithms unsuitable for mass-market receivers.
Another important concept concerning accuracy is Dilution Of Precision (DOP). DOP refers to a set of parameters
2

32. 1.1. Background and Motivation
that characterize the accuracy with which receivers determine position and time from noisy pseudoranges in
GNSSs. Smaller DOP values are better than larger ones but selection of the set of satellites providing the smallest
value can be a daunting task depending on the number of visible satellites at a given place and time.
Nowadays a low-cost navigation receiver can simultaneously process signals coming from 4 to 12 satellites.
GPS alone provides between 6 and 11 satellites in view nearly at every moment in almost every point of the
Earth’s surface for positioning purposes [28]. Under this condition, there are two approaches for computation of
position: use all N satellites in view or perform a satellite selection process for a selected number h of satellites
(with h N). Satellite selection techniques are based on brute force approach or recursive methods [29]. In
both cases, all possible combinations of h satellites are tested aiming to minimize the Geometrical Dilution Of
Precision (GDOP) [30]; for a certain value of h the set providing the minimum GDOP is chosen. Calculating a
single GDOP requires one p4 hq ph 4q matrix multiplication followed by one p4 4q matrix inversion.
Therefore, the total number of floating point operations required to find the optimal solution grows rapidly as the
number of visible satellites increases [31]. The corresponding number of combinations to test is N!{rh!pN hq!s,
where h is chosen according to the operational requirements [29]: for conventional positioning h ¥ 4, while for
RAIM purposes, h ¥ 5.
With the advent of Galileo and Compass and the modernization of GLONASS there will be soon four GNSS
constellations providing more than 40 satellites in view [32]. The significant increase in the number of satellites
available will make the use of recursive selection methods intractable. It would imply searching for the best
subset of satellites, h, out of N satellites, where N represents the total number of satellites in view provided by
the different constellations. For example, doing N 40 and h 5 would amount to search among 6.58 105
subsets. Note that, in general, more than 4 satellites are needed when combining measurements from more
than one constellation as the time references are not synchronized [33]. Other algorithms that present a low
computational complexity do not guarantee optimality [31] or are restricted to the selection of only four satellites
[34], [35].
On the other hand, reducing the signal processing time of the receiver dedicated to satellite selection, implies
both, increasing the processing capabilities available for other purposes and battery saving. These are aspects of
high concern nowadays given the tendency of the receivers to decrease in size. Furthermore, it would help to
reduce the time it takes the receiver to check the consistency of the information provided by the system when
RAIM algorithms are applied.
When Selective Availability (SA) was on, the impact of error sources depending on the elevation angle on the
positioning accuracy in GNSS receivers was irrelevant compared with that of SA. Now that SA has been dis-
continued, errors that depend on the elevation angle make User-Equivalent Range Error (UERE) values for each
satellite significantly different. A more accurate navigation solution requires proper weighting of individual
satellites range measurements.
As a first attempt, receivers use a mask angle (5 10 ) to reduce these elevation dependent errors. The remaining
errors need to be dealt with by adding a parameter that weights the optimality of a given satellite to be used in the
computation of the position solution. Therefore, instead of minimizing the GDOP, the optimal subset of satellites
should minimize the so called Weighted Dilution Of Precision (WDOP) parameter which takes into account the
measurement errors. WDOP was introduced in [36] to simultaneously consider the effect of GDOP and the
ranging errors to the satellites. Uncorrelated measurements with different uncertainties will be considered here
so that the weighting matrix will be diagonal as in [36], [37].
In general, GDOP is approximately minimized by maximizing the determinant of pGT
Gq, with G denoting
the receiver-satellite geometry matrix as the adjoint matrix of GT
G varies less strongly with geometry than
does its determinant [28, ch. 5]. It can be shown [38] that the determinant of pGT
Gq is directly proportional
3

33. Chapter 1. Introduction
to the size of the polytope whose vertices are determined by the user-to-satellite unit vector endpoints. The
relationship between GDOP and the volume of the polytope formed by the endpoints of the user-satellite unit
vectors when three satellites are selected in 2-D positioning (or four satellites in 3-D) is straightforward [39]. But,
the relationship when the number of satellites is larger has not been researched yet. Therefore, the relationship
when the position is computed from an overdetermined system of equations needs to be analyzed in detail.
1.2 Research Contributions
• A low computational complexity approach to multipath estimation in a multicorrelator receiver based on
the maximization of the log-likelihood function using a grid search approach is proposed. A closed-form
solution of the nonlinear estimation problem is considered; therefore, lack of convergence, initialization,
and computational complexity problems, are avoided. The closed-form solution is obtained by restricting
the estimation of the code delay to a grid of values defined by the correlators spacing, thus limiting its
resolution to half that spacing. To improve it, a solution based on interpolation is devised. The use of
interpolation gives also the possibility of decreasing the necessary number of correlators to achieve a spe-
cific accuracy. The performance of the algorithm for the legacy and the new GPS and Galileo navigation
signals is evaluated with special attention to the effects of filtering and the number of correlators required.
Results are presented where the methodology is demonstrated to outperform classical solutions like narrow
correlator and double-delta techniques. Its resolution depends on the correlators spacing and the Signal-to-
Noise Ratio (SNR). It has been shown to provide a good performance from pre-detection SNRs of 10 dB.
The algorithm is computationally efficient and fast enough to be considered for real time computations as
a large amount of operations can be precomputed and stored in memory, and the remaining operations can
be performed in parallel. This issue is addressed in [40], [41], and [42].
• A methodology in the context of convex geometry theory aiming to reduce the number of satellites used
for the position computation by a navigation receiver among all visible satellites is also proposed. This
reduction is based on the approximate equivalence between GDOP minimization and maximization of the
size of the polytope (polyhedron in 3-D or polygon in 2-D) expanded by the user-to-satellite unit vector
endpoints. The polytope is contained in the convex hull of all vector’s endpoints for the set of satellites
in view. An optimal solution for convex hull computation with low computational burden is proposed for
both static and time-varying scenarios. The method has a nearly linear computational complexity with the
number of satellites. Thus, the improvement achieved by the algorithm increases notably with the number
of satellites in view being several orders of magnitude faster than the brute force approach. Algorithms
proposed elsewhere (e.g., [34] and [35]) exhibit a low computational complexity, but they are restricted to
the selection of four satellites, whereas the solution herein proposed allows the selection of a larger number
of satellites. This feature may be critical as the time references for different GNSS constellations are not
synchronized; therefore, more than four satellites are needed when the time offsets are computed at the
receiver level. In addition, even for a single constellation, the application of RAIM techniques requires at
least six satellites in view to detect and isolate a failure. Once the smallest convex polytope is obtained,
we may need to reduce further the set of satellites. An efficient way to select a given number of satellites
among those that constitute the 2-D convex hull was proposed. It was shown that it provides GDOP values
close to the optimal ones. The research related with this topic was published in [43], [44], and [32].
• Furthermore, a solution for non-i.i.d. Gaussian measurements is also considered. The solution is based on
the computation of a weighted least-squares solution of the CH. Thus, it provides Weighted Geometrical
Dilution Of Precision (WGDOP) values that differ from the optimal ones only due to the disagreement
between volume maximization and WGDOP minimization. Although, the measurement noise covariance
4

34. 1.2. Research Contributions
matrix is assumed to be diagonal, results are also applicable in the presence of correlated measurements.
The solution was proposed in [45].
• Finally, this thesis presents an analysis of the relationship between maximization of the volume expanded
by the user-satellite unit vectors and GDOP minimization for any number of satellites. The analysis is
independent of the number of selected satellites. A discrepancy analysis between the solution provided
by the maximum volume approximation and the brute force approach (i.e., trace of the GDOP matrix) is
performed. Results show that both functions have a very similar behavior specially for large volumes and
thus, the error can be considered negligible. Only when the optimal solution corresponding to a regular
polytope is among the set of combinations possible at a given instant, we can assure that the solution
provided by both methods is equivalent. Note that the larger is the number of satellites in view, the more
likely a large volume will be obtained. The analysis can be found in [46].
Several publications resulted from the work developed. The list of thesis-related publications is provided
hereafter.
1.2.1 Journal Papers
[40] Nuria Blanco-Delgado, Fernando D. Nunes, ”Multipath Estimation in Multicorrelator GNSS Receivers
Using the Maximum Likelihood Principle,” IEEE Transactions on Aerospace and Electronic Systems, De-
cember 2009, status: submitted.
[43] Nuria Blanco-Delgado, Fernando D. Nunes, ”A Satellite Selection Method for Multi-Constellation GNSS
Using Convex Geometry,” IEEE Transactions on Vehicular Technology, IEEE Transactions on Vehicular
Technology, vol. 59, no. 9, pp. 4289 - 4297, Nov. 2010.
[46] Nuria Blanco-Delgado, Gonzalo Seco-Granados, Fernando D. Nunes, ”Minimum DOP versus Maximum
Volume for Positioning Purposes,” IEEE Transactions on Aerospace and Electronic Systems, June 2011,
status: to be submitted.
1.2.2 International Conferences
[42] Nuria Blanco-Delgado, Fernando D. Nunes, Jo˜ao M.F. Xavier, ”A Geometrical Approach for Maximum
Likelihood Estimation of Multipath,” in Proceedings of ION GNSS 2008, Savannah, GA, Sept. 2008.
[32] Nuria Blanco-Delgado, Fernando D. Nunes, Jo˜ao M.F. Xavier, ”GNSS Satellite Selection for Multiple-
Constellations Using Convex Geometry,” in 4th
ESA Workshop on Satellite Navigation User Equipment
Technologies, 2008.
[41] Nuria Blanco-Delgado, Fernando D. Nunes, ”Maximum Likelihood Estimation of Multipath and Theo-
retical Accuracy Limits for the New Navigation Signals,” in Proceedings of European Navigation Confer-
ence - Global Navigation Satellite Systems, ENC-GNSS 2009, Naples, May 2009.
[44] Nuria Blanco-Delgado, Fernando D. Nunes, ”A Convex Geometry Approach to Dynamic GNSS Satellite
Selection for a Multi-Constellation System,” in Proceedings of ION - GNSS 2009, Savannah, GA, Sept.
2009.
[47] Fernando D. Nunes, Fernando M.G. Sousa, Nuria Blanco-Delgado, ”A VDLL Approach to GNSS Cell
Positioning for Indoor Scenarios,” in Proceedings of ION - GNSS 2009, Savannah, GA, Sept. 2009.
[45] Nuria Blanco-Delgado, Fernando D. Nunes, ”Satellite Selection based on WDOP Concept and Convex
Geometry,” in 5th
ESA Workshop on Satellite Navigation User Equipment Technologies, 2010.
5

35. Chapter 1. Introduction
[48] Nuria Blanco-Delgado, Fernando D. Nunes, Gonzalo Seco-Granados, ”Relation between GDOP and the
Geometry of the Satellite Constellation,” in 1st
International Conference on Localization and GNSS, 2011.
1.2.3 National Conferences
[49] Nuria Blanco-Delgado, Fernando D. Nunes, ”Theoretical Performance of the MEDLL Algorithm for the
New Navigation Signals,” in 7th Conference on Telecomunications, ConfTele 2009, May 2009.
1.2.4 Patents
[50] Nuria Blanco-Delgado, ”Satellite Selection Method for Positioning Systems,” PT Provisional Patent
Application No 104741.
1.2.5 Technical Reports
[51] Nuria Blanco-Delgado, ”Accuracy Pointing of Dynamic Platforms through GPS Code Phase Measure-
ments,” Internal report, Instituto Superior T´ecnico, 2008.
[52] Nuria Blanco-Delgado, ”Multipath Estimating Delay Lock Loop,” Internal report, Instituto Superior
T´ecnico, 2008.
[53] Nuria Blanco-Delgado, ”A Tutorial on Bayesian Filters Applied to GNSS,” Internal report, Instituto
Superior T´ecnico, 2008.
1.3 Thesis Outline
The thesis consists of seven chapters, where review material and novel contributions are presented. The moti-
vation for the topics that I will deal with along the thesis were previously stated. Thesis contributions resulting
from the work are also listed in Section 1.2. The structure of the document is presented in this chapter, serving
as a guide to the reader.
The technical background required to understand the problems and the proposed solutions are presented in Chap-
ter 2. Chapter 3 provides a review of the literature on multipath mitigation and satellite selection. Chapter 4 to
Chapter 6 contain original techniques and results obtained along the research period.
Chapter 4 proposes a multipath mitigation technique for multicorrelator GNSS receivers that estimates the line-
of-sight and the multipath signal parameters in a computationally efficient manner. The maximum likelihood
principle is applied leading to a non-linear optimization problem. A closed-form solution of the nonlinear es-
timation problem is proposed where the estimation of the delay parameter is performed using a grid search
approach. Lack of convergence, initialization, and computational complexity problems are avoided and a low
computational complexity is achieved. A low-complexity complementary method based on interpolation is pro-
posed to increase the resolution of the estimated delay with respect to the limit set by the correlators’ spacing.
Theoretical performance and limits are determined considering that the delay is estimated through a grid search
approach and the observations are obtained in the presence of non-white Gaussian noise due to the correlation
between samples.
A methodology in the context of convex geometry theory aiming to reduce the number of satellites used for the
position computation by a navigation receiver among all visible ones is proposed in Chapter 5. This reduction is
6

36. 1.3. Thesis Outline
based on the approximate equivalence between GDOP minimization and maximization of the size of the polytope
(polyhedron in 3-D or polygon in 2-D) expanded by the user-to-satellite unit vector endpoints. The polytope is
contained in the convex hull of all vector’s endpoints for the set of satellites in view. An optimal solution for
convex hull computation with lower bound computational burden is suggested for both static and time-varying
scenarios.
The relationship between GDOP and maximum volume of the polytope expanded by the user-satellite unit vectors
endpoints is studied in Chapter 6. The analysis is independent of the number of satellites to be selected. The
solution set is constructed from a series of inequalities extracted from matrix algebra. Demonstration that the
solution provided by both functions is the same if the optimal point is part of the solution set is performed
whereas a discrepancy analysis is conducted for the rest of the cases.
In Chapter 7, the main conclusions of the thesis are drawn and topics for further research work are proposed.
The appendices present complementary analysis, results and studies that help supporting and explaining the work
carried out along the main part of the text.
7

37. 8

38. Chapter 2
Principles of GNSS
Global Navigation Satellite Systems (GNSSs) constitute one of the most important timing and positioning
systems. They rely on measuring the propagation time of a signal from a transmitting station to a receiver.
In a GNSS the transmitting stations are the satellites, which are provided with a very accurate timing generation
unit. Therefore, if the position of the transmitting station is known and the transmission time can be measured,
the distance between the transmitter and the receiver can be determined. Unfortunately, the computed receiver
position is affected by time and range measurement errors and by user-to-satellite geometry, altogether summing
up to an error of about 10 m [30]. Most errors can be eliminated or mitigated by techniques external or internal
to the receiver, only errors due to multipath and receiver thermal and tracking noise errors remain.
This chapter provides an overview of the GNSSs already implemented and under development, the signal pro-
cessing steps performed at the receiver to obtain a position solution and the different errors associated with the
system. Section 2.1 summarizes the main characteristics of the existing/planned GNSSs (number of satellites, or-
bits, signals, etc.). The structure of a conventional GNSS receiver including the position computation is described
in Section 2.2. An alternative structure based on vector code/frequency tracking is presented in Section 2.3. The
main GNSS error sources are reviewed in Section 2.4. The satellite selection parameters affecting the positioning
precision are presented in Section 2.5.
2.1 Global Navigation Satellite Systems
The navigation system can be divided into three segments: space, control and user segment, as illustrated in
Fig. 2.1. The space segment is constituted by the satellites. Each satellite constellation comprises a different
number of satellites. This number and its disposal are carefully designed to ensure that, at least, the minimum
number of satellites required for positioning computation is available at any point of the Earth for the maximum
percentage of time possible. Satellites are in charge of transmitting the navigation signals that will be processed
by the receiver to estimate its position. The control segment consists of a set of one or more Master Control
Station (MCS), several monitor stations and ground control stations transmitting information to the satellites.
These stations monitor the satellites and update the information that the satellites will then broadcast to the
receivers. Finally, the user segment is constituted by the receivers which are specifically designed to receive,
decode, and process the GNSS satellite signals.
Most of the satellite navigation systems use a Code Division Multiple Access (CDMA) scheme. Only GLONASS,
the Russian GNSS, uses Frequency Division Multiple Access (FDMA) but is now updating to CDMA [54].
Under the CDMA principle, each satellite transmits signals modulated by different Pseudo-Random Noise (PRN)
9

39. Chapter 2. Principles of GNSS
Figure 2.1: GNSS system segments. Source: The Aerospace Corporation.
codes to distinguish between signals transmitted in the same frequency. By contrast, all GLONASS satellites
transmit the same code but on different frequency channels. The receiver generates a replica of the PRN code that
correlates with the incoming signal. The receiver synchronizes with the signal sent by the satellite to measure the
time delay and decode the transmitted navigation message. In short, the code signal has the following goals:
• To give a means to measure the signal traveling time between satellite and receiver.
• To distinguish different signals in the same band.
• To minimize interference and jamming.
• To deny the access to the transmitted signals (in military signals).
As of 2010, the United States NAVSTAR Global Positioning System (GPS) is the only fully operational GNSS.
The Russian GLONASS is in the process of being restored to full operation (23 out of 24 satellites are operational
at the moment of writing this thesis). The European Union’s Galileo positioning system is in its initial deployment
phase and is scheduled to be partially operational in 2014. The People’s Republic of China has indicated it will
expand its regional Beidou navigation system into a GNSS, named Compass, by 2015-2017.
Global coverage for each system is generally achieved by a constellation of 20 to 30 Medium Earth Orbit (MEO)
satellites distributed among several orbital planes. The actual systems use orbit inclinations of slightly above
50 and orbital periods of approximately twelve hours with a height of about 20.000 km. They provide different
services on the transmitted signals (free, regulated or military).
2.1.1 GPS
The United States’ GPS is fully operational since 1994. The space segment consists of a nominal constellation
of 24 operating satellites distributed in six orbital planes with orbital periods of 11 hours and 58 minutes. Its
disposition has been carefully designed to provide four or more satellites in view at any instant in almost every
point of the Earth surface.
Two different service levels are provided by the GPS, the Precise Positioning Service (PPS) and the Standard
Positioning Service (SPS)
10

40. 2.1. Global Navigation Satellite Systems
• Precise Positioning Service (PPS): is an accurate positioning velocity and timing service which is avail-
able only to authorized users. The PPS is primarily intended for military purposes. PPS receivers can
use either the P(Y)-code or C/A-code or both. Maximum GPS accuracy is obtained using the P(Y)-code
on both L1 and L2. P(Y)-code capable receivers commonly use the C/A-code to initially acquire GPS
satellites.
• Standard Positioning Service (SPS): is a less accurate positioning and timing service which is available to
all GPS users. The SPS is primarily intended for civilian purposes. Table 2.1 displays the SPS performance
parameters.
The legacy GPS navigation signals - that is, those navigation signals that are broadcast by the GPS satellites up
through the Block IIR class [30, ch. 3] - are transmitted in the L-band of the electromagnetic spectrum. They
are allocated in two sub-bands referred to as L1 and L2. Although GPS will provide three new modernized civil
signals in the future, L2C, L5 (in the so called L5 band), and L1C, the L1 (1575.42 MHz) Coarse/Acquisition
(C/A) signal is the only civil GPS signal that has reached full operational capability at this time. These bands are
centered at:
- L1 : f0,L1 1575.42 MHz
- L2 : f0,L2 1227.60 MHz
- L5 : f0,L5 1176.45 MHz
GPS Performance
Standard Metric
SPS Signal Specification
August 1998
(user performance)
SPS Performance Standard
September 2008
(signal in space)
Global Accuracy
All-in-View Horizontal 95%
All-in-View Vertical 95%
¤ 100 meters
¤ 156 meters
¤ 9 meters
¤ 15 meters
Worst Site Accuracy
All-in-View Horizontal 95%
All-in-View Vertical 95%
¤ 100 meters
¤ 156 meters
¤ 17 meters
¤ 37 meters
User Range Error
(URE)
NONE
¤ 7.8 meters 95%
(Worst Satellite URE)
equivalent to 4 m rms
Geometry
(PDOP = 6)
¥ 95.87% global
¥ 83.92% worst site
¥ 98% global
¥ 88% worst site
Constellation
Availability
NONE
¥ 98% Probability of
21 Healthy Satellites
in 24 primary slots;
¥ 99.999% Probability
of 20 Healthy Satellites
in 24 primary slots;
Table 2.1: GPS SPS performance parameters [55]. The specifications apply only to users of the L1-C/A code
signal.
2.1.1.1 GPS Signals
The signal carriers are Direct Sequence Spread Spectrum (DSSS) modulated by spread spectrum codes with
unique PRN sequences associated to each satellite signal and by a common navigation data message. This way,
all satellites can transmit at the same carrier frequencies without interfering with each other. The DSSS signal
11

41. Chapter 2. Principles of GNSS
is a composite signal generated by the modulo-2 addition of a PRN sequence and the 50 bits per second (bps)
downlink system data (referred to as navigation data).
Three different PRN spread spectrum codes are transmitted:
• The precision (P), code which has a 10.23 MHz chipping rate and a one-week period.
• The Y code (denoted as P(Y)), resulting from the encryption of the P code whenever the anti-spoofing
(A-S) mode of operation is activated. It has the same chipping rate as the P code.
• The Coarse/Acquisition (C/A) code which is used for acquisition of the P (or Y) code and as a civil ranging
signal. It consists of a 1023-chip sequence with a period of 1 ms and a chipping rate of 1.023 MHz.
The satellites may transmit intentionally ”incorrect” versions of the C/A and the P(Y) codes where needed to
protect the users from receiving and utilizing anomalous navigation signals as a result of a malfunction in the
satellites’ reference frequency generation system. These two ”incorrect” codes are termed Non-Standard C/A
(NSC) and Non-Standard Y (NSY) codes [56]. For the generation of the GPS C/A and P codes see [30, 4.3.1.1].
The carrier of the L1 signal consists of an in-phase and a quadrature component. The in-phase component
is biphase modulated by the navigation data stream and the C/A-code cptq prior to modulation with the L1
carrier. The quadrature phase component is also biphase modulated by the same navigation data stream but with
a different PRN code, the P-code pptq. The mathematical model of the L1 waveform is
rptq
a
2 PI dptqcptq cos pω0,L1 t θq
a
2 PQ dptqpptq sin pω0,L1 t θq (2.1)
where PI and PQ are the in-phase and quadrature-phase carrier power components, respectively, dptq is the data
signal,
dptq
8¸
k 8
dk Π
t k Tb
Tb
(2.2)
where dk t 1, 1u is the kth
data symbol, Tb
1
50 Hz, Πpt{Tq is a rectangular pulse defined by
Π
t
T
$
%
1, |t| ¤ T{2
0, otherwise
, (2.3)
and cptq and pptq are DSSS modulating waveforms [30, 4.2], [57].
Let us consider the general DSSS modulating waveform that uses the arbitrary pulse gptq of duration Tc,
sptq
8¸
k 8
ak gpt k Tcq (2.4)
where the PRN code values taku are assumed to be generated as a random coin-flip sequence.
In the case of the BPSK-R signals like cptq and pptq, the pulse gptq is
gBPSK Rptq Π
t k Tc
Tc
(2.5)
where Tc
1
1.023 MHz for the C/A code and Tc
1
10.23 MHz for the P(Y) code.
In Equation (2.1) ω0,L1 is the L1 carrier frequency in radians per second, and θ is a common phase in radians.
The quadrature carrier power PQ is approximately 3 dB less than PI. In contrast to the L1 signal, the L2 signal
12

42. 2.1. Global Navigation Satellite Systems
is modulated only with the P(Y)-code. The mathematical model of the L2 waveform is [56]
rptq
a
2 PI dptqpptq cos pω0,L2 t θq (2.6)
where ω0,L2 is the L2 carrier frequency. Note that new civilian signals are being implemented on the L2 frequency
but we do not enter into detail here [56].
Only the GPS C/A code is subject of detailed study along the thesis.
C/A code: Autocorrelation Function and Power Spectral Density The multiplication of the carrier by a
binary sequence cptq corresponds to a Binary Phase Shift Keying (BPSK) modulation. Therefore, the GPS C/A
code is a Binary Phase Shift Keying (BPSK) modulated Radio Frequency (RF) signal. The properties of the
BPSK signal will be presented here.
The autocorrelation function of the PRN sequence cptq is given by
Rccpτq 1
T
» T
0
cptqc pt τqdt. (2.7)
The autocorrelation function plays an important role in a GPS receiver as it forms the basis for code tracking and
accurate user-to-satellite range measurement. In fact, the receiver uses the correlation between the code of the
incoming signal and the locally-generated replicas to keep the incoming and local codes aligned. Since the code
sequence has a period T 1 ms, the autocorrelation function of the GPS C/A code is also periodic being well
approximated by
Rccpτq
$
'
'%
1
|τ|
Tc
, |τ| ¤ Tc
0, |τ| ¡ Tc
(2.8)
with chip duration Tc 10 3
{1023 s and Rccpτ k Tq Rccpτq, k 1, 2, . . .. The exact autocorrelation
functions are more complicated than indicated in Equation (2.8) and change slightly from code to code (for details
see [58, ch. 9]). However, we assume that the triangular shape of Equation (2.8) is a sufficient approximation for
our purposes.
The C/A codes are nearly orthogonal, meaning that any two codes, ciptqand cjptq, from different satellites, i j,
have cross-correlation
1
T
» T
0
ciptqcjpt τqdt 0, @ τ and @ i j (2.9)
to minimize interference. The cross-correlation levels for the C/A codes approaches a worst-case of -21.1 dB [30,
4.3]. High cross-correlation levels may lead to the synchronization of the receiver with the incorrect code.
The existence of different codes allows all satellites to transmit on the same frequency without incurring in
significant mutual interference. Therefore, a receiver can extract the signal from each satellite, and processes
them individually even though all signals are transmitted at the same frequency. The signals that are not of
interest will look much like white noise for the receiver.
Figures 2.2(a) and (b) show the BPSK(1) unfiltered and filtered autocorrelation function (see Equation (2.8)),
respectively. A 6th
order Butterworth low-pass filter of cutoff frequency equal to 4 MHz was assumed. The filter
has been implemented considering a zero-phase response for simplicity.
The power spectrum of a purely random code sequence with chip duration Tc 10 3
{1023 s and unit power is
13

43. Chapter 2. Principles of GNSS
−1 −0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Chip Offset [chips]
R(τ)
(a)
−1 −0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Chip offset [chips]
R(τ)
(b)
Figure 2.2: Autocorrelation function of a BPSK(1) modulated signal. (a) Unfiltered. (b) Filtered considering a
6th
order Butterworth low-pass filter with a bandwidth of 4 MHz.
(Fig. 2.3)
Spfq Tc sinc2
pf Tcq (2.10)
with sincpxq sinpπxq{πx. The bandwidth of the code sequence is often assumed to be equal to the position of
the first null of Spfq, i.e., 1.023 MHz.
−20 −15 −10 −5 0 5 10 15 20
−100
−95
−90
−85
−80
−75
−70
−65
−60
Frequency [MHz]
PowerSpectralDensity[dBW/Hz]
Figure 2.3: Normalized power spectral density of the BPSK(1) modulation.
When the GPS codes are combined with the navigation data message, there is an essentially imperceptible effect
on the resulting autocorrelation functions and on the power spectrum. When these are modulated onto the L-band
carrier, there is a shift to the L-band of the power spectrum from the baseband frequencies. The resulting RF
power spectrum is [30, ch. 4]
SLpfq Pc
2
rSP N pf fcq SP N pf fcqs (2.11)
where Pc is the unmodulated carrier power, fc is the carrier frequency (Hz), and SP N is the power spectrum of
the PRN code(s) (plus data). The bandwidth of the RF signal between nulls is equal to 2.046 MHz.
14

44. 2.1. Global Navigation Satellite Systems
2.1.1.2 Navigation Message
The GPS navigation message is repeated every 12.5 minutes with a bit-rate of 50 bps. The 50 bps data stream
conveys the navigation message, which includes, but is not limited to, the following information:
• Satellite Almanac Data. Each satellite transmits orbital data called the almanac, which enables the user to
calculate the approximate location of every satellite in the GPS constellation at any given time. Almanac
data are not accurate enough for determining position but can be stored in a receiver where they remain
valid for many days. They are used primarily to determine which satellites are visible at a given location so
that the receiver can search for those satellites when it is first turned on. They can also be used to determine
the approximate expected signal Doppler frequency shift and, therefore, contribute to a quick acquisition
of the satellite signals.
• Satellite Ephemeris Data. Ephemeris data are similar to almanac data but enable a much more accurate
determination of the satellite position needed to convert signal propagation delay into an estimate of the
user’s position. In contrast to almanac data, ephemeris data for a particular satellite are broadcast only by
that satellite, and the data are valid only for several hours.
• Signal Timing Data. Time tagging establishes the transmission time of specific points on the GPS signal.
This information is needed to determine the satellite-to-user propagation delay required for ranging.
• Ionospheric Delay Data. Ranging errors due to ionospheric effects can be partially canceled by using
estimates of the ionospheric delay that are broadcast in the data stream.
• Satellite Health Message. The data stream also contains information regarding the current health of the
satellite, so that the receiver can ignore that satellite if it is not operating properly.
2.1.1.3 GPS Modernization
Besides the legacy signals previously analyzed there is a set of new signals that will be briefly described next.
Fig. 2.4 shows the block diagram of the L2C signal generation process [59], [60].
CL code generator
period = 767,250
CM code generator
period = 10,230
CNAV message
25 bps
FEC
rate 1/2
Chip by chip
multiplexer
511.5 kcps
511.5 kcps
1.023 Mcps
r(t)
cos (2π fL2t)
Figure 2.4: Block diagram of the L2C signal generation.
The new code is sometimes called the Replacement Code (RC), because it will be used instead of the C/A code.
The RC has the same 1.023 106
chipping rate as the C/A code and, therefore, the null-to-null bandwidth is
2.046 MHz as well. However, the RC code results from multiplexing two codes: CM and CL. The CM code has
a length of 10, 230 chips and the CL code is even longer with a period of 767, 250 chips. The two codes alternate,
15

45. Chapter 2. Principles of GNSS
with the CM code controlling every second chip that is broadcast. The CL code is not modulated by navigation
data. This data-free signal is very helpful for operation in low-signal-to-noise environments. The CM code, by
contrast, is modulated by navigation data, encoded for Forward Error Correction (FEC).
Figure 2.5 shows the block diagram of the L5 signal generation process [60], [61].
SV codes
10.23 Mcps
10230 period
NAV data
50 bps
FEC
encoding
g1(t)
g2(t)
NH10(t)
NH20(t)
cos (2πfL5t)
sin (2πfL5t)
r(t)
Figure 2.5: Block diagram of the L5 signal generation process.
On one hand, the in-phase signal is modulated with the g1ptq code and the navigation data. The rate of the
navigation data is 50 bps, but forward error correction is applied and, as a consequence, the final symbol rate is
100 symbols per second. On the other hand, the quadrature component is modulated with the g2ptq code, but no
navigation data is applied.
Like the L2C signal, the L5 signal also provides a data free signal component to improve operation at low signal-
to-noise ratios. However, the L5 signal design did not have to resort to time multiplexing of the codes, because
both the in-phase and quadrature channels were available. Both codes g1ptq and g2ptq are 10, 230 chips long and
are transmitted at rates of 10.23 Mcps; and so the null-to-null bandwidth is 20.46 MHz. Newmann-Hoffman
(NH) codes also modulate both, the in-phase and the quadrature, channels. These codes are short and extend the
length of the g1ptq code to 102, 300 chips and the length of the g2ptq code to 204, 600 chips. The 10-symbol NH
code period, NH10ptq, is 10 ms long whereas the 20-symbol NH code, NH20ptq, has a period of 20 ms.
As mentioned above, both the L2C and the L5 signals have data-free components. These components are very
useful in low signal-to-noise ratio environments. Since the navigation bits are not known a priori, the receiver
must compute the square of the received signal to strip off the navigation bits. However, the squaring action
introduces squaring losses. Use of the data-free signal component avoids the squaring losses [60].
The latest of the new GPS signals to be deployed is the L1C. It provides a number of advanced features, including
[62]: 75% of power in a pilot component for enhanced signal tracking, advanced Weil-based spreading codes,
an overlay code on the pilot that provides data message synchronization, advanced forward error control coding,
and data symbol interleaving to combat fading.
The L1C signal consists of two components [63]: L1CP (pilot) and L1CD (data). The bitstream of the L1CP
signal is constructed by modulo-2 addition of the L1CP-code and the L1CO-code. The L1CO-code is a SV unique
overlay code. The bitstream of the L1CP signal modulates the L1 carrier frequency using a Time-Multiplexed
BOC (TMBOC) modulation scheme. This technique uses a mixture of sine-phase modulated BOC (sBOC)
spreading symbols: sBOC(1,1) and sBOC(6,1) spreading symbols. The sBOC modulation is presented in 2.1.2.1.
The pattern of spreading symbols repeats every 10,230 spreading symbols corresponding to a new bit of the
16

46. 2.1. Global Navigation Satellite Systems
L1CO-code. All spreading symbols in the L1CP component are sBOC(1,1) except for those that are sBOC(6,1)
which occur for chips tk 0, 4, 6, 29, 33 and repeat for t tk 33 n, n 1, 2, . . .. The bitstream of the L1CD
component is constructed by modulo-2 addition of the L1CD-code and the L1C message symbol train. The
bitstream of the L1CD signal modulates the L1 carrier frequency using sBOC(1,1) modulation, with a subcarrier
of 1.023 MHz and a chipping rate of 1.023 Mbps.
2.1.2 Galileo
The European Union and the European Space Agency agreed in March 2002 to introduce their own GNSS, called
Galileo. Galileo is being designed to be compatible with the modernized GPS system, thus it is also based on
CDMA techniques. Therefore, receivers will be able to do tracking of signals from both GNSSs satellites to
greatly increase the accuracy.
The Galileo satellite constellation plan consists of 30 satellites distributed in three orbital planes (27 operational
plus 3 active spares) positioned in MEO at 23,222 km with an inclination of the orbital planes of 56 with respect
to the equator (http://www.esa.int). The satellites will be spread evenly around each plane and will have an orbital
period of 14-h. Galileo has three slots allocated into the L-band, called E1, E5 and E6:
- E1 band: 1559-1591 MHz;
- E5 band: 1164-1215 MHz;
- E6 band: 1260-1300 MHz;
Each Galileo satellite will broadcast 10 different navigation signals. Several services will be provided by the
system: Open Service (OS), Safety of Life Service (SoL), Commercial Service (CS) and Public Regulated Service
(PRS):
• The OS is a free service for general applications. It is free of charge but no integrity information is provided.
The OSs is provided by the E1, E5a and E5b signals. Combination of signals at different bands is also
possible, such as dual frequency combinations using E1 and E5a (for better ionospheric error cancellation)
as well as single frequency operation (using E1, E5a, E5b or E5a and E5b together) where the ionospheric
error would be removed using a model. Triple frequency services using all the signals (E1, E5a and E5b)
can also be exploited for very precise, centimetric applications.
• The SoL Service presents the same characteristics as the OS but integrity information is provided in this
case. This aspect makes this service appropriate for its use in the transportation sector where real time
information about the status of the system is required (i.e., given that an error puts in danger human lives).
The SoL services are based on the measurements obtained from the open signal and use the integrity data
carried in special messages designated for this purpose within the open signals. The safety-of-life service
is like a data channel within the open signals.
• The CS will allow the transmission of additional encrypted data facilitating the development of professional
applications with some added value with respect to the performances of the OS. The CS is provided by two
additional signals centered at 1278.75 MHz band as well as by the E1, E5a and E5b signals.
• The PRS will be implemented with encrypted signals particularly resistent to interference (jamming, spoof-
ing, etc.); its access will be monitored by civil authorities and reserved to the needs of public institutions
regarding civil protection and national security. The PRS is provided by two signals, one in the L1 band
and the other in the E6 band. These signals are encrypted, allowing the implementation of an access control
scheme.
The main characteristics of the different services are provided in Table 2.2.
17

47. Chapter 2. Principles of GNSS
Galileo
Global Services
OS
Commercial
Service
SoL service
Coverage Global Global Global
Position accuracy
15 m or 24 m H – 35 m V 4 m H – 8 m V
(single frequency) (dual frequency)
4 m H – 8 m V
(dual frequency)
Timing
accuracy p95%q 30 ns 30 ns 30 ns
Integrity None None
Time to alert 12 m H – 20 m V
Integrity risk 6 seconds
Alert limit 3.5 10 7
{150 s
Continuity risk
Service availability 99.5% 99.5%
1 10e 5
{15 s
99.5%
Access control Free open access
Control access of ranging
codes and navigation data
message
Authentication of
integrity information
in the navigation
data message
Certification and
service guarantees
None
Guarantee of service possi-
ble
Build for certification
and guarantee of
service
Table 2.2: Performance specification of the OS, CS, and SoL Galileo services [30, ch. 10].
2.1.2.1 Galileo Signals
Like in GPS, the carrier frequencies are DSSS modulated by spread spectrum codes with unique PRN sequences
associated to each satellite signal and by a common navigation data message. In this way, all satellites can transmit
at the same carrier frequencies without interfering with each other. There are two distinct signals, those containing
navigation data (also known as data channels) and those carrying no data (also known as pilot channels). This is
highlighted in Fig. 2.6 by plotting the data channels and the pilot channels in orthogonal planes, a way to indicate
that the signals of data and pilot channels are shifted by 90 , which allows their separation at the receiver. Fig. 2.6
shows the frequency plan of the Galileo system.
Figure 2.6: Galileo frequency plan.
18

48. 2.1. Global Navigation Satellite Systems
Different codes with different characteristics are used by the different Galileo signals. The main characteristics
of the different codes are provided in Table 2.3. The design of the signals is a complicated process where several
variables should be considered. A compromise was found where different codes with different characteristics are
available on the various Galileo signals to accomplish the performances of the different services defined. The
avoidance of interference with other systems acting in the same band (like GPS) has also driven the definition of
the signal characteristics. The selected modulations allow several systems to occupy the same frequency while
avoiding inter-system interference.
In addition to the BPSK modulation (also used in GPS), the Galileo signals use the following signaling schemes:
cosine-phase modulated BOC (cBOC), Multiplexed Binary Offset Carrier (MBOC) and Alternative BOC (Alt-
BOC). A brief description of each one is provided:
Frequency
Bands
Channel
Modulation
type
Chip Rate
(Mchips/s)
Symbol
Rate (sps)
E5
E5a data
E5a pilot
E5b data
E5b pilot
AltBOC(15,10) 10.23
50
N/A
250
N/A
E6 E6P cBOC(10,5) 5.115 To be decided
E6C data
E6C pilot
BPSK-R(5) 5.115
1,000
N/A
E2 L1 E1 L1P cBOC(15,2.5) 2.5575 To be decided
E1B data
E1C pilot
MBOC(6,1,1/11) 1.023
250
N/A
Table 2.3: Main characteristics of the Galileo signals.
BOC modulation A Binary Offset Carrier (BOC)ppn, nq modulation can be seen as the multiplication of two
components: a BPSK-R waveform with a chip rate fc n 1.023 Mchip/s and a square-wave sub-carrier
signpsinp2πfstqq with fs pn 1.023 MHz, p a positive integer, and n P R . Thus, the sine-phase mod-
ulated BOC (sBOC) signal is a DSSS modulating waveform where gptq gBPSK Rptqsignpsinp2πfstqq (see
Equations (2.4) and (2.5)). That is,
sptq
8¸
k 8
ak Π
t kTc
Tc
signpsin p2πfstqq (2.12)
with Tc 1{fc and the cosine-phase modulated BOC (cBOC) signal is described as
sptq
8¸
k 8
ak Π
t kTc
Tc
signpcos p2πfstqq. (2.13)
The sBOC modulation concentrates more power on the inner part of the spectrum, while the cBOC modulation
main lobes have more power on the outer part as shown in Fig. 2.8.
The ideal autocorrelation function of a sBOCppn, nq signal is [64]
Rss pτq
$
'
'%
p 1qk 1
1
p
k2
2kp k p p4p 2k 1q |τ|
Tc
, |τ| Tc
0, otherwise
(2.14)
where k ceilp2 p |τ|{Tcq, with the ceiling function, i ceilpxq, denoting the smallest integer such that i ¥ x.
19

49. Chapter 2. Principles of GNSS
Figures 2.7(a) and (b) show the sBOC(1,1) unfiltered and filtered autocorrelation function, respectively. A 6th
order Butterworth low-pass filter with cutoff frequency equal to 12 MHz was assumed. The filter has been
implemented considering a zero phase response for simplicity.
−1 −0.5 0 0.5 1
−0.5
0
0.5
1
Chip Offset [chips]
R(τ)
(a)
−1 −0.5 0 0.5 1
−0.5
0
0.5
1
Chip offset [chips]
R(τ) (b)
Figure 2.7: Autocorrelation function of a sBOC(1,1) modulated signal. (a) Unfiltered. (b) Filtered considering a
6th
order Butterworth low-pass filter with a bandwidth of 12 MHz.
Assuming that the binary values of a BOC spreading sequence are equally likely, independent, and identically
distributed ideal rectangular waveforms, the normalized power spectral density of a sine-phased BOCppn, nq
modulation is [65] (see also Appendix B)
SsBOC pfq Tc sinc2
pfTcq tan2 πfTc
2 p
, k 2p 2 fs
fc
Tc
Ts
even (2.15)
and the power spectral density of a cosine-phased BOCppn, nq modulation is (see Appendix B)
ScBOC pfq 4 Tc sinc2
pfTcq
sin2 πf Tc
4 p
cos
πf Tc
2 p
2
, k even. (2.16)
Figure 2.8 shows the power spectral densities of sBOC(1,1) and cBOC(1,1) modulations, in red and green, re-
spectively. Note that multiplying the PRN sequence by a subcarrier results in a wider spectrum, concentrating
it around fs. In general, the subcarrier splits the main lobe of the PRN sequence into two lobes centered at
pn 1.023 MHz, where 2pn 1.023 MHz indicates the distance between these two lobes.
MBOC modulation The OS signal considered at the L1 frequency is a variant of the sBOC modulation
called MBOC. A MBOC signal results from multiplexing two sBOC signals. The signal recommended is the
MBOC(6,1,1/11) where the numbers denote the power spectral density mixture of the two sBOC components
[66]. Thus, the power spectral density of the MBOC(6,1,1/11) signal is [67]
SMBOCp6,1,1{11qpfq 10
11
SsBOCp1,1qpfq 1
11
SsBOCp6,1qpfq (2.17)
where SsBOCppn,nqpfq is the normalized-power spectral density of a sine-phased BOCppn, nq modulation (see
Equation (2.15)). The autocorrelation function of MBOC(6,1,1/11) and its power spectral density are plotted in
20